**Part 1**

**Hydrodynamic Lubrication** 

**1** 

Xie You-Bai

*China* 

**Theory of Tribo-Systems** 

*Shanghai Jiaotong University and Xi'an Jiaotong University* 

Some people say that tribology is friction, wear or lubrication. Others say that it is friction plus wear and plus lubrication. However both of them are not accurate enough. Tribology takes all theoretical and applied results from friction, wear and lubrication obtained in the past, inputs into them with much more new senses and contents based on the development of science and technology. It is striving to constitute a theoretical and technical platform to meet the future requirement. Tribology cannot be looked simply as equal only to friction,

The early stage of applying knowledge of friction, wear and lubrication in human productive and living practice can be traced back to 3000 BC or earlier (Dowson, 1979). This multi-disciplinary branch of science and technology and its application in comprehensive areas were studied in many different sub-subjects independently from very different points of view over a long period. A suggestion from H Peter Jost gave this old field a powerful impact and poured into it youthful vigor (Her Majesty's Stationery Office, 1966). It developed quicker and quicker thereafter. Tribology is a both old and young discipline. In the first phase of development of tribology since 1965, due to its universal existence in nature according to the definition given by Jost on one side and the belief of tribologists in many countries that they could make huge benefit for industry on another side, the influence of tribology increased dramatically fast in the seventies and eighties of the last century. Promise of saving 5 billion pounds per year in UK in the Jost Report pushed forward tribologists working on applying existed knowledge of friction, wear and lubrication to solve engineering problems. New techniques related to friction, wear and lubrication developed then rapidly in the following phase even though some people they did not like the name "tribology". Many books published in this stage with the title "Tribology" but no one discussed on the questions that what was tribology and why they

The later situation has shown that to achieve the potential benefit is not so easy (Xie, 1986; Xie & Zhang, 2009). A name, a definition and simply putting all knowledge components together subjectively are not enough. A concept system, theory system and method system, which can match the name, definition and nature of tribology and then can promote an

It is valuable to mention that "Tribology" was defined as one of the four major disciplines of Mechanical Systems by a Committee of NSF of US in 1983 (The Panel Steering Committee for the Mechanical Engineering and Applied Mechanics Division of the NSF, 1984) and then

**1. Introduction** 

What is tribology? Why people need tribology?

wear, lubrication or any other technique related.

used the word "tribology" except Jost did in his famous Report.

independent development and application of tribology, are expected.

## **Theory of Tribo-Systems**

Xie You-Bai *Shanghai Jiaotong University and Xi'an Jiaotong University China* 

#### **1. Introduction**

What is tribology? Why people need tribology?

Some people say that tribology is friction, wear or lubrication. Others say that it is friction plus wear and plus lubrication. However both of them are not accurate enough. Tribology takes all theoretical and applied results from friction, wear and lubrication obtained in the past, inputs into them with much more new senses and contents based on the development of science and technology. It is striving to constitute a theoretical and technical platform to meet the future requirement. Tribology cannot be looked simply as equal only to friction, wear, lubrication or any other technique related.

The early stage of applying knowledge of friction, wear and lubrication in human productive and living practice can be traced back to 3000 BC or earlier (Dowson, 1979). This multi-disciplinary branch of science and technology and its application in comprehensive areas were studied in many different sub-subjects independently from very different points of view over a long period. A suggestion from H Peter Jost gave this old field a powerful impact and poured into it youthful vigor (Her Majesty's Stationery Office, 1966). It developed quicker and quicker thereafter. Tribology is a both old and young discipline.

In the first phase of development of tribology since 1965, due to its universal existence in nature according to the definition given by Jost on one side and the belief of tribologists in many countries that they could make huge benefit for industry on another side, the influence of tribology increased dramatically fast in the seventies and eighties of the last century. Promise of saving 5 billion pounds per year in UK in the Jost Report pushed forward tribologists working on applying existed knowledge of friction, wear and lubrication to solve engineering problems. New techniques related to friction, wear and lubrication developed then rapidly in the following phase even though some people they did not like the name "tribology". Many books published in this stage with the title "Tribology" but no one discussed on the questions that what was tribology and why they used the word "tribology" except Jost did in his famous Report.

The later situation has shown that to achieve the potential benefit is not so easy (Xie, 1986; Xie & Zhang, 2009). A name, a definition and simply putting all knowledge components together subjectively are not enough. A concept system, theory system and method system, which can match the name, definition and nature of tribology and then can promote an independent development and application of tribology, are expected.

It is valuable to mention that "Tribology" was defined as one of the four major disciplines of Mechanical Systems by a Committee of NSF of US in 1983 (The Panel Steering Committee for the Mechanical Engineering and Applied Mechanics Division of the NSF, 1984) and then

Theory of Tribo-Systems 5

In the very early stage of tribology people have begun to think about system problems (Fleischer, 1970; Czichos, 1974; Salomon, 1974). A comprehensive study on applying system concepts to friction, wear and lubrication was given by Czichos which described how to use general systems theory and engineering system analysis in treating tribological problems (Czichos, 1978). Without an effective way for mathematic computation limited its application. Dai and Xue (Dai & Xue, 2003) tried to evaluate tribological behaviors with an entropy calculation in tribo-systems while Ge and Zhu (Ge & Zhu, 2005) worked out through a fractal analysis for a similar attempt. Either entropy calculation or fractal analysis cannot describe explicitly and quantitatively the character of movement of interacting surfaces in relative motion. Since they deal only with entropy or fractal parameters, transforming all other physical and geometric behaviors into an entropy or fractal change in calculation is unavoidable. It involves the transformation of a large amount of knowledge concerning with tribology getting together in the past into the thermodynamic or fractal knowledge and

Considering that relative motion is the first important character of tribology and is also a basic behavior studied in mechanisms, some concepts in mechanisms should be discussed

When several components are joined together by kinematic pairs it constructs a kinematic chain. The necessary condition that a kinematic chain becomes a mechanism, in other words a mechanical system is all components in the chain having definitive relative motions. This condition can be rewritten as that the number of motion conditions given (input) outside of the system equals to the number of residual degrees of freedom of the chain. For example, as shown in Figure 1 there is a plane kinematic chain of four components (1 - 4) with one fixed component (4, chassis), three revolute pairs (A – C) and one prismatic pair (D). Each movable component has three degrees of freedom while each revolute pair or each prismatic pair cancels two degrees of freedom. Revolute pairs and prismatic pairs all formed with surface contact are known as lower pairs and pairs formed with point contact or line contact are known as higher pairs in mechanisms. Each higher pair cancels one degree of freedom in plane analysis. Then the residual degrees of freedom of the plane chain can be calculated as

In formula (1) *MC*, *L* and *H* are the number of movable components, the number of lower pairs and the number of higher pairs respectively. The result shows 1 motion condition input is in need of becoming the chain to a crank-slider mechanism. In this example when a rotating speed of the crank (1) is given the relative motions of all other movable components

*RDOF MC L H* = 3 2 34 1 24 1 − − = − −⋅= ( ) (1)

before going to construct a *function based* systems theory for tribology.

is almost impossible in practice.

Fig. 1. A crank-slider mechanism

the "Journal of Lubrication Technology" was renamed as "Journal of Tribology" of Transaction of ASME. Only ten years later, a gentleman from US indicated in an informal speech in Beijing that a change under way was the gradual disappearance of the term "tribology" from programs and projects of NSF in US. In this period fewer papers which dealt with the relation between tribology and mechanical systems could be found in the journal. It implies that no enough effort has been made to carry out the original intention of the committee. Many famous tribologists prophesied that tribology would become or be replaced by surface engineering.

It shows some undesired situation in the development of tribology. There are at least three problems with it. Firstly tribology was born on the foundation of known appearances of friction, wear and lubrication but the difference between tribology and friction, wear and lubrication has not been paid attention to investigate into. Naturally the traditional way of studying friction, wear and lubrication independently is still having its visible influence on tribology. Secondly as tribology is so universal and so important to engineering and industry, much attention has been paid to the tribology-based applied techniques and a very fast development of the techniques has been achieved. Due to the nature of tribology, which will be discussed later, most of the techniques can be applied only to a specific branch of field for a specific target. Many people they work in the field of tribology but they don't think they are tribologists. Some of them think they are chemists, material scientists, biologists or mechanical engineers. Therefore the theoretical study of tribology, especially the efforts on finding a systematic framework for tribology cannot benefit further from such a fast development of technique. Thirdly people don't know how to use the results obtained under one condition to another condition and how to compare the results from one kind of test machines with what of another kind of test machines, in other words, there is no general model for tribology and almost no modern mathematic tool can be used in tribology. Engineers have to make each decision individually in design depending on experience or experiment. They cannot construct a tribological design in true sense for their products because no model can be found in simulating the behaviors of tribology other than friction, wear or lubrication individual. Therefore there is no strong enough attraction to take tribology as an independent discipline in industry further.

Tribology has been defined in 1965 as "the science and technology of interacting surfaces in relative motion and of the practices related thereto" (Her Majesty's Stationery Office, 1966) and modified later as "the science of behaviors of interaction surfaces in relative motion together with the active medium concerned (each of them is a tribo-element) in natural systems, their results and the technology related thereto" (Xie, 1996). A question arises then that why people need *the interacting surfaces in relative motion*? Both definitions deal with appearance aspects rather than functional aspects of tribology and cannot answer the question. Obviously any surface cannot exist independently and must be a part of a component. The relative motion of surfaces is defined by the relative motion of components and where the surfaces reside on. The interactions transmitted between surfaces are from the components in contact on the surfaces as well. In most (not all ) cases two interacting surfaces in relative motion function as *a joint* which permits only some kinds of relative motion and prevents other kinds of relative motion between two components in contact. Such joints are named kinematic pairs in mechanisms. The interacting surfaces in relative motion must function with other elements in a system or function with other elements for a system.

Therefore the problems with tribology are problems of systems science and systems engineering. In a sense, without system there would be no tribology.

the "Journal of Lubrication Technology" was renamed as "Journal of Tribology" of Transaction of ASME. Only ten years later, a gentleman from US indicated in an informal speech in Beijing that a change under way was the gradual disappearance of the term "tribology" from programs and projects of NSF in US. In this period fewer papers which dealt with the relation between tribology and mechanical systems could be found in the journal. It implies that no enough effort has been made to carry out the original intention of the committee. Many famous tribologists prophesied that tribology would become or be

It shows some undesired situation in the development of tribology. There are at least three problems with it. Firstly tribology was born on the foundation of known appearances of friction, wear and lubrication but the difference between tribology and friction, wear and lubrication has not been paid attention to investigate into. Naturally the traditional way of studying friction, wear and lubrication independently is still having its visible influence on tribology. Secondly as tribology is so universal and so important to engineering and industry, much attention has been paid to the tribology-based applied techniques and a very fast development of the techniques has been achieved. Due to the nature of tribology, which will be discussed later, most of the techniques can be applied only to a specific branch of field for a specific target. Many people they work in the field of tribology but they don't think they are tribologists. Some of them think they are chemists, material scientists, biologists or mechanical engineers. Therefore the theoretical study of tribology, especially the efforts on finding a systematic framework for tribology cannot benefit further from such a fast development of technique. Thirdly people don't know how to use the results obtained under one condition to another condition and how to compare the results from one kind of test machines with what of another kind of test machines, in other words, there is no general model for tribology and almost no modern mathematic tool can be used in tribology. Engineers have to make each decision individually in design depending on experience or experiment. They cannot construct a tribological design in true sense for their products because no model can be found in simulating the behaviors of tribology other than friction, wear or lubrication individual. Therefore there is no strong enough attraction to take

Tribology has been defined in 1965 as "the science and technology of interacting surfaces in relative motion and of the practices related thereto" (Her Majesty's Stationery Office, 1966) and modified later as "the science of behaviors of interaction surfaces in relative motion together with the active medium concerned (each of them is a tribo-element) in natural systems, their results and the technology related thereto" (Xie, 1996). A question arises then that why people need *the interacting surfaces in relative motion*? Both definitions deal with appearance aspects rather than functional aspects of tribology and cannot answer the question. Obviously any surface cannot exist independently and must be a part of a component. The relative motion of surfaces is defined by the relative motion of components and where the surfaces reside on. The interactions transmitted between surfaces are from the components in contact on the surfaces as well. In most (not all ) cases two interacting surfaces in relative motion function as *a joint* which permits only some kinds of relative motion and prevents other kinds of relative motion between two components in contact. Such joints are named kinematic pairs in mechanisms. The interacting surfaces in relative motion must function

with other elements in a system or function with other elements for a system.

engineering. In a sense, without system there would be no tribology.

Therefore the problems with tribology are problems of systems science and systems

replaced by surface engineering.

tribology as an independent discipline in industry further.

In the very early stage of tribology people have begun to think about system problems (Fleischer, 1970; Czichos, 1974; Salomon, 1974). A comprehensive study on applying system concepts to friction, wear and lubrication was given by Czichos which described how to use general systems theory and engineering system analysis in treating tribological problems (Czichos, 1978). Without an effective way for mathematic computation limited its application. Dai and Xue (Dai & Xue, 2003) tried to evaluate tribological behaviors with an entropy calculation in tribo-systems while Ge and Zhu (Ge & Zhu, 2005) worked out through a fractal analysis for a similar attempt. Either entropy calculation or fractal analysis cannot describe explicitly and quantitatively the character of movement of interacting surfaces in relative motion. Since they deal only with entropy or fractal parameters, transforming all other physical and geometric behaviors into an entropy or fractal change in calculation is unavoidable. It involves the transformation of a large amount of knowledge concerning with tribology getting together in the past into the thermodynamic or fractal knowledge and is almost impossible in practice.

Considering that relative motion is the first important character of tribology and is also a basic behavior studied in mechanisms, some concepts in mechanisms should be discussed before going to construct a *function based* systems theory for tribology.

Fig. 1. A crank-slider mechanism

When several components are joined together by kinematic pairs it constructs a kinematic chain. The necessary condition that a kinematic chain becomes a mechanism, in other words a mechanical system is all components in the chain having definitive relative motions. This condition can be rewritten as that the number of motion conditions given (input) outside of the system equals to the number of residual degrees of freedom of the chain. For example, as shown in Figure 1 there is a plane kinematic chain of four components (1 - 4) with one fixed component (4, chassis), three revolute pairs (A – C) and one prismatic pair (D). Each movable component has three degrees of freedom while each revolute pair or each prismatic pair cancels two degrees of freedom. Revolute pairs and prismatic pairs all formed with surface contact are known as lower pairs and pairs formed with point contact or line contact are known as higher pairs in mechanisms. Each higher pair cancels one degree of freedom in plane analysis. Then the residual degrees of freedom of the plane chain can be calculated as

$$\text{RDOF} = \text{3MC} - \text{2L} - \text{H} = \text{3(4} - \text{1)} - \text{2} \cdot \text{4} = \text{1} \tag{1}$$

In formula (1) *MC*, *L* and *H* are the number of movable components, the number of lower pairs and the number of higher pairs respectively. The result shows 1 motion condition input is in need of becoming the chain to a crank-slider mechanism. In this example when a rotating speed of the crank (1) is given the relative motions of all other movable components

Theory of Tribo-Systems 7

Tribology science and technology is very important in obtaining *the best way* (theory and application) to complete the motion guarantee function of tribo-systems. Tribology exists universally. Where there is relative motion there is tribology. Tribo-systems play sometimes very critical roles in machine systems and work sometimes under extreme severe condition. It implies that the motion guarantee function must be implemented with high reliability, low energy consumption, low cost, low pollution, human and environment

The study of friction, wear, lubrication and other tribological techniques are part of the efforts in finding the best way. Ignoring the fundamental function of tribo-systems and appreciating an *appearance based* study of friction, wear, lubrication or techniques from surface engineering, nanotechnology or biology etc, even though some physical or geometric results can be obtained, the study cannot give a clear overview on the relation between the results. Putting the results together in practice usually throws engineers into confusion. It will also increase the difficulty in tribo-system modeling and in looking for mathematic tools for an overall system and life cycle behavior simulation. Lake of overall system model and mathematic tool for simulation makes that the results from one working condition cannot be used in another condition, from one period of the life cannot be used in another period and from the study of friction cannot be used in the study of wear or the study of lubrication etc. Furthermore such a situation makes almost impossible to implement a tribological design since the tribo-design is the design of tribo-systems. It is well known that tribological design is a main channel for embedding tribology knowledge

Fig. 2. The block diagram of a typical tribo-system

friendship etc.

into products.

are defined and can be derived from the crank rotating speed. In the derivation each pair (interacting surfaces in relative motion) functions to permit some kinds of relative motion and prevent the others between two components joined by the pair, and each component functions to keep the surfaces of pairs having fixed positions on the component. Then the mechanism can be looked as consisting of two sub-systems: a component system and a pair system. The component system and the pair system work together to guarantee the mechanism with a definite motion when the number of motion conditions input is enough. Such function is a motion guarantee function.

Back to tribology, a tribo-pair is the physical realization of a kinematic pair and the tribopairs in total in a machine system constitutes the main part of a tribo-system.

Tribo-systems can be understood in another way. A system of higher rank can be divided into several systems of lower rank or sub-systems and vice versa. The division can be implemented in different ways. For example, a machine as a system can be divided into assembles, such as a rotor assemble, a chassis assemble, or can be divided according to the function of the sub-systems as well, such as a coolant circulation system, a brake system etc. A machine system can also be divided according to the character of behaviors of the subsystems, for example, dividing the machine into a mechanical system, a thermodynamic system, an electric system and what will be discussed in detail in tribology, a tribo-system, etc.

Such a division is making an abstraction of machine systems. It does not limit the division to groups of elements or kinds of functions but keep the investigation into a given category of behaviors and their results. For example, in general there is an electric circuit diagram for a machine and the diagram is just a description of the electric system abstracted from the machine.

When a machine system or another natural system is abstracted into a system consisting of tribo-elements and some supporting auxiliary sub-systems for studying behaviors on or between the interacting surfaces in relative motion, results of the behaviors and technology related to, a tribo-system is then constructed.

In most cases there will be a liquid, a gas or a fat lubricant film kept between the interacting surfaces in relative motion to reduce friction and wear. Solid lubricant films will not be included in this discussion and looked as parts of the surfaces with a motion similar to the surfaces. The auxiliary sub-system for fluid lubrication including at first a cycling subsystem, a cooling sub-system and a filtering sub-system operates to keep the fluid film staying between the interacting surfaces in relative motion and to make the surfaces work efficiently, reliably and friendly to human and environment.

Due to there is a time variable character with the tribo-systems which will be discussed later, the change in structure, behavior and function or the change of working condition in short should be monitored (necessary) and be adaptively controlled (suggested) to avoid low efficiency work, abnormal wear, environment pollution or catastrophic damage. Therefore the condition monitoring sub-system or the condition controlling sub-system usually takes place in the auxiliary sub-system of tribo-systems. Figure 2 gives a general construction diagram for tribo-systems (Xie, 1996).

It can be concluded that a machine system is consisted of a component system and a tribosystem from the view point of motion. The tribo-system together with the component system plays a motion guarantee function which keeps each part of the machine system with a definite motion when the number of motion conditions input outside of the system is enough.

are defined and can be derived from the crank rotating speed. In the derivation each pair (interacting surfaces in relative motion) functions to permit some kinds of relative motion and prevent the others between two components joined by the pair, and each component functions to keep the surfaces of pairs having fixed positions on the component. Then the mechanism can be looked as consisting of two sub-systems: a component system and a pair system. The component system and the pair system work together to guarantee the mechanism with a definite motion when the number of motion conditions input is enough.

Back to tribology, a tribo-pair is the physical realization of a kinematic pair and the tribo-

Tribo-systems can be understood in another way. A system of higher rank can be divided into several systems of lower rank or sub-systems and vice versa. The division can be implemented in different ways. For example, a machine as a system can be divided into assembles, such as a rotor assemble, a chassis assemble, or can be divided according to the function of the sub-systems as well, such as a coolant circulation system, a brake system etc. A machine system can also be divided according to the character of behaviors of the subsystems, for example, dividing the machine into a mechanical system, a thermodynamic system, an electric system and what will be discussed in detail in tribology, a tribo-system,

Such a division is making an abstraction of machine systems. It does not limit the division to groups of elements or kinds of functions but keep the investigation into a given category of behaviors and their results. For example, in general there is an electric circuit diagram for a machine and the diagram is just a description of the electric system abstracted from the

When a machine system or another natural system is abstracted into a system consisting of tribo-elements and some supporting auxiliary sub-systems for studying behaviors on or between the interacting surfaces in relative motion, results of the behaviors and technology

In most cases there will be a liquid, a gas or a fat lubricant film kept between the interacting surfaces in relative motion to reduce friction and wear. Solid lubricant films will not be included in this discussion and looked as parts of the surfaces with a motion similar to the surfaces. The auxiliary sub-system for fluid lubrication including at first a cycling subsystem, a cooling sub-system and a filtering sub-system operates to keep the fluid film staying between the interacting surfaces in relative motion and to make the surfaces work

Due to there is a time variable character with the tribo-systems which will be discussed later, the change in structure, behavior and function or the change of working condition in short should be monitored (necessary) and be adaptively controlled (suggested) to avoid low efficiency work, abnormal wear, environment pollution or catastrophic damage. Therefore the condition monitoring sub-system or the condition controlling sub-system usually takes place in the auxiliary sub-system of tribo-systems. Figure 2 gives a general

It can be concluded that a machine system is consisted of a component system and a tribosystem from the view point of motion. The tribo-system together with the component system plays a motion guarantee function which keeps each part of the machine system with a definite motion when the number of motion conditions input outside of the system is

pairs in total in a machine system constitutes the main part of a tribo-system.

Such function is a motion guarantee function.

related to, a tribo-system is then constructed.

efficiently, reliably and friendly to human and environment.

construction diagram for tribo-systems (Xie, 1996).

etc.

machine.

enough.

Fig. 2. The block diagram of a typical tribo-system

Tribology science and technology is very important in obtaining *the best way* (theory and application) to complete the motion guarantee function of tribo-systems. Tribology exists universally. Where there is relative motion there is tribology. Tribo-systems play sometimes very critical roles in machine systems and work sometimes under extreme severe condition. It implies that the motion guarantee function must be implemented with high reliability, low energy consumption, low cost, low pollution, human and environment friendship etc.

The study of friction, wear, lubrication and other tribological techniques are part of the efforts in finding the best way. Ignoring the fundamental function of tribo-systems and appreciating an *appearance based* study of friction, wear, lubrication or techniques from surface engineering, nanotechnology or biology etc, even though some physical or geometric results can be obtained, the study cannot give a clear overview on the relation between the results. Putting the results together in practice usually throws engineers into confusion. It will also increase the difficulty in tribo-system modeling and in looking for mathematic tools for an overall system and life cycle behavior simulation. Lake of overall system model and mathematic tool for simulation makes that the results from one working condition cannot be used in another condition, from one period of the life cannot be used in another period and from the study of friction cannot be used in the study of wear or the study of lubrication etc. Furthermore such a situation makes almost impossible to implement a tribological design since the tribo-design is the design of tribo-systems. It is well known that tribological design is a main channel for embedding tribology knowledge into products.

Theory of Tribo-Systems 9

more complex system consisted of simplest systems and their supporting auxiliary sub-

systems (Xie, 2010). The first axiom focuses on the relationship of structures.

**2.2 The second axiom: The property of tribo-elements and then the systems** 

In comparison with the material in the body of a component, the material of any element in a tribo-pair bears much more intensive load and works under a much more severe condition. As shown in Figure 4 when the roughness of surfaces in contact is considered, the real contact area is much smaller than the nominative contact area. Through the much smaller contact area it transmits a load equal to what transmitted by the body of the component with an area of the body section. The load density is then very high at the real contact area. On the other hand the transmission is implemented between different materials in a pair while it is through the same material in the case of the body of a component. Additional physical or chemical reaction between different materials may occur under such a condition. Furthermore there is a relative motion. It accelerates the change of their physical property, chemical composition and geometric configuration, especially due to the relative motion the change is continuously repeated, sometimes with very high frequency. The relative motion produces heat and then high temperature and other kinds of active energy. They will no doubt promote the change in physical aspects and chemical aspects. All of them make the change of the property of each element in the tribo-pair much more fast in comparison with what in the body of a component. Due to the severity in most cases the change is irrecoverable. Therefore as in performance analysis or design people usually consider what they deal with is a time-invariable system they must consider it as a time-variable system in tribological analysis and tribological design. As shown in Fig. 5 the speed of change of performance is variable in a life cycle. In the earlier stage of work of a new system the speed of change is high and this is a running in stage. Afterwards the speed of change will be slow in a stable operation stage. At last the speed of change increases faster and it predicts the end of life. The variation of speed of change is very complex in

Fig. 3. A simplest tribo-system

many systems.

**containing tribo-elements are time dependent** 

Therefore different from the independently study of friction, wear, lubrication or any technique related, tribology should be studied with a system viewpoint and cannot overlook the basic function of tribo-systems, the motion guarantee function for a machine system. Tribology study is to find the best way (theory and application) to complete the motion guarantee function. It is no doubt that all results from the study of friction, wear, lubrication and other technique related are indispensable in reaching the goal.

#### **2. How tribo-systems behave?**

After a detailed discussion on the basic function of tribo-systems the question arises that how a tribo-system behaves to complete the function?

Some basic knowledge about the character and pattern of change of the systems is necessary in describing their behaviors. For example, one character of a mechanical system or a mechanism is that all components in the system have definitive relative motion and it can be checked with formula (1). The pattern of change of the mechanism is governed by principles in kinematics from which the behaviors can be derived. For the crank-slider mechanism shown in Figure 1 the relative motions of all movable components can be derived with a given rotating speed of the crank. Such a requirement will be similar for thermodynamic systems, electric systems and all other systems abstracted from a system of higher rank according to a given category of behaviors. Tribology is a multi-disciplinary area. Any principle in other disciplines cannot describe and govern the character and pattern of change of tribo-systems accurately. A task of top priority is to organize the basic knowledge which matches the name, the definition and the nature of tribology rather than matches what concerning with only friction, wear, lubrication or any individual technique related.

After the investigation in many years the author suggests that there are three axioms in tribology and they can be used as a base or a start point to study into the character and pattern of change of tribo-systems. So-called an axiom means what people cannot find any opposite example with the axiom even though they cannot prove it theoretically (Suh, 1990). The three axioms in tribology (Xie, 2001) are: (1) The first Axiom: Tribological behaviors are system dependent. (2) The second Axiom: The property of tribo-elements and then the systems containing tribo-elements are time dependent. (3) The third Axiom: The results of tribological behaviors are the results of mutual action and strong coupling of many behaviors of other disciplines under a tribological condition consisted of interacting surfaces in relating motion. The three axioms will be discussed in more detail in the following.

#### **2.1 The first axiom: Tribological behaviors are system dependent**

Changes taking place on or between the interacting surfaces in relative motion are what to be investigated into in tribology and called tribological behaviors. Interactions and relative motion are causes of the behaviors. Results of the behaviors include a recoverable and irrecoverable change of intrinsic property of elements, a change of the state of the system consisted of the elements and a material, energy and information exchange with the environment in the forms of input and output. The intrinsic property includes geometric, physical and historic aspects and will be discussed later. A single surface or medium substance cannot implement any tribological behavior. In Fig. 3 there is a simplest tribosystem including three elements. The system is enveloped with a system block and exchanges material, energy and information via input and output with environment. The system dependent character governs not only the behaviors of simplest systems but also any

Therefore different from the independently study of friction, wear, lubrication or any technique related, tribology should be studied with a system viewpoint and cannot overlook the basic function of tribo-systems, the motion guarantee function for a machine system. Tribology study is to find the best way (theory and application) to complete the motion guarantee function. It is no doubt that all results from the study of friction, wear, lubrication

After a detailed discussion on the basic function of tribo-systems the question arises that

Some basic knowledge about the character and pattern of change of the systems is necessary in describing their behaviors. For example, one character of a mechanical system or a mechanism is that all components in the system have definitive relative motion and it can be checked with formula (1). The pattern of change of the mechanism is governed by principles in kinematics from which the behaviors can be derived. For the crank-slider mechanism shown in Figure 1 the relative motions of all movable components can be derived with a given rotating speed of the crank. Such a requirement will be similar for thermodynamic systems, electric systems and all other systems abstracted from a system of higher rank according to a given category of behaviors. Tribology is a multi-disciplinary area. Any principle in other disciplines cannot describe and govern the character and pattern of change of tribo-systems accurately. A task of top priority is to organize the basic knowledge which matches the name, the definition and the nature of tribology rather than matches what concerning with only friction, wear, lubrication or any individual technique related. After the investigation in many years the author suggests that there are three axioms in tribology and they can be used as a base or a start point to study into the character and pattern of change of tribo-systems. So-called an axiom means what people cannot find any opposite example with the axiom even though they cannot prove it theoretically (Suh, 1990). The three axioms in tribology (Xie, 2001) are: (1) The first Axiom: Tribological behaviors are system dependent. (2) The second Axiom: The property of tribo-elements and then the systems containing tribo-elements are time dependent. (3) The third Axiom: The results of tribological behaviors are the results of mutual action and strong coupling of many behaviors of other disciplines under a tribological condition consisted of interacting surfaces in relating motion. The three axioms will be discussed in more detail in the following.

and other technique related are indispensable in reaching the goal.

**2.1 The first axiom: Tribological behaviors are system dependent** 

Changes taking place on or between the interacting surfaces in relative motion are what to be investigated into in tribology and called tribological behaviors. Interactions and relative motion are causes of the behaviors. Results of the behaviors include a recoverable and irrecoverable change of intrinsic property of elements, a change of the state of the system consisted of the elements and a material, energy and information exchange with the environment in the forms of input and output. The intrinsic property includes geometric, physical and historic aspects and will be discussed later. A single surface or medium substance cannot implement any tribological behavior. In Fig. 3 there is a simplest tribosystem including three elements. The system is enveloped with a system block and exchanges material, energy and information via input and output with environment. The system dependent character governs not only the behaviors of simplest systems but also any

how a tribo-system behaves to complete the function?

**2. How tribo-systems behave?** 

more complex system consisted of simplest systems and their supporting auxiliary subsystems (Xie, 2010). The first axiom focuses on the relationship of structures.

Fig. 3. A simplest tribo-system

#### **2.2 The second axiom: The property of tribo-elements and then the systems containing tribo-elements are time dependent**

In comparison with the material in the body of a component, the material of any element in a tribo-pair bears much more intensive load and works under a much more severe condition. As shown in Figure 4 when the roughness of surfaces in contact is considered, the real contact area is much smaller than the nominative contact area. Through the much smaller contact area it transmits a load equal to what transmitted by the body of the component with an area of the body section. The load density is then very high at the real contact area. On the other hand the transmission is implemented between different materials in a pair while it is through the same material in the case of the body of a component. Additional physical or chemical reaction between different materials may occur under such a condition. Furthermore there is a relative motion. It accelerates the change of their physical property, chemical composition and geometric configuration, especially due to the relative motion the change is continuously repeated, sometimes with very high frequency. The relative motion produces heat and then high temperature and other kinds of active energy. They will no doubt promote the change in physical aspects and chemical aspects. All of them make the change of the property of each element in the tribo-pair much more fast in comparison with what in the body of a component. Due to the severity in most cases the change is irrecoverable. Therefore as in performance analysis or design people usually consider what they deal with is a time-invariable system they must consider it as a time-variable system in tribological analysis and tribological design. As shown in Fig. 5 the speed of change of performance is variable in a life cycle. In the earlier stage of work of a new system the speed of change is high and this is a running in stage. Afterwards the speed of change will be slow in a stable operation stage. At last the speed of change increases faster and it predicts the end of life. The variation of speed of change is very complex in many systems.

Theory of Tribo-Systems 11

meet such a need other than tribology, even though for any individual behavior there are principles from the discipline related which can predict its results. The difficulty for tribologists is that they have to know all the relative disciplines together with tribology simultaneously. Such a character of inter-disciplines and multi-disciplines requires a new methodology for tribology different from what for friction, wear or lubrication. In distinguish with the first axiom the third axiom focuses on the relationship of behaviors.

*The structure* is a description of intrinsic facts of a tribo-system while *the behaviors* are a description of change of the tribo-system. The structure of a system exists regardless whether there is an input. The behaviors must follow an input and can be derived from the

Czichos (Czichos, 1978) described the structure of a tribo-system with a parameter set as

where *E ee e* = { 1 2 , ,...., *<sup>N</sup>* } is a sub-set showing that there are *N* tribo-elements in total in the system, *P pp p* = { *e e eN* 1 2 , ,....., } is a sub-set describing the property of each element in the system and *Rr r r r* = { *e e e e e eN e e* 12 13 1 23 , ,.. , ,...} is a sub-set collecting all relations between

Such description carries out a problem. Since the relative motion is the first important character of tribology and then the relative displacements between elements changing with the motion condition input, therefore the relative displacements between elements cannot be

where *H hh h h* = { *w e e eN* , , ,...... 1 2 } is a sub-set including the history of the system as a whole

Each element *ei* , *i* = 1….*N,* in the sub-set *E* represents a surface or a medium substance, for example a journal surface, a bearing surface, a cylinder bore surface, a piston skirt surface or

Each element *pei , i* = 1….*N,* in the sub-set *P* describes the property of element *ei* . In more detail the contents of property of each element can be divided into two groups, i.e. *pei* = {*pg*, *pp*}*i*, *i* = 1….*N,* where *pg* is the geometric parameter group of property of the element, for example the diameter and width of the bearing surface in macro scale and the roughness in micro scale, and *pp* is the physical parameter group of property of the element. *pp* should be understood in a generalized sense including all physical, chemical and biological features besides geometric. It usually can be described by a group of physical, chemical and biological parameters, such as hardness, viscosity, acidity, activity etc. but there are some exceptions. Such features are affected by material composition, manufacturing process,

To avoid the problem the author modified the description of a structure as (Xie, 2010)

*S EPR* = { , , } (2)

*S EPH* = { , , } (3)

**3. How to model a tribo-system and simulate its behaviors?** 

treated as an intrinsic fact. Several other examples can be listed as well.

input for a given structure in principle.

**3.1 The structure of a tribo-system** 

elements in the system.

and of each element.

the lubricant film between the surfaces.

service history, surrounding temperature, atmosphere, etc.

Fig. 4. The severity of work condition of a tribo-pair

Fig. 5. Change of performance of a tribo-system in a life cylce

#### **2.3 The third axiom in tribology: The results of tribological behaviors are the results of mutual action and strong coupling of many behaviors of other disciplines under a tribological condition consisted of interacting surfaces in relating motion**

Obviously from the simplest tribo-system shown in Fig. 1 the force interaction, relative motion of the surfaces and the medium substance between the surfaces are a mechanical behavior. Transformation of the mechanical energy consumed in motion into heat energy and the diffusion of heat in surrounding, which makes a stable or unstable temperature field, are thermodynamic behaviors and heat transfer behaviors. The molecular interaction (including transferring) between surfaces and surfaces with medium is a physical or physical-chemical behavior. The reactions in ion level and atomic level are chemical behaviors. If there is any electric or magnetic field, which produces attractive or repelling interaction, or changes the arrangement of molecules in materials, or induces eddy current and heat, all of them are electric behaviors or magnetic behaviors, and so on. Most of them behavior simultaneously and inevitably change the structure of the system recoverably and irrecoverably. Then in turn they bring about the final results different from the results when they behavior singly. The results are different also from a simple addition of the results of individual behaviors. There is strong coupling between such behaviors. Tribology is the science and technology, which provides theories and techniques to describe and control the pattern of coupling between behaviors under tribological condition. No other discipline can

Fig. 4. The severity of work condition of a tribo-pair

Fig. 5. Change of performance of a tribo-system in a life cylce

**2.3 The third axiom in tribology: The results of tribological behaviors are the results of mutual action and strong coupling of many behaviors of other disciplines under a** 

Obviously from the simplest tribo-system shown in Fig. 1 the force interaction, relative motion of the surfaces and the medium substance between the surfaces are a mechanical behavior. Transformation of the mechanical energy consumed in motion into heat energy and the diffusion of heat in surrounding, which makes a stable or unstable temperature field, are thermodynamic behaviors and heat transfer behaviors. The molecular interaction (including transferring) between surfaces and surfaces with medium is a physical or physical-chemical behavior. The reactions in ion level and atomic level are chemical behaviors. If there is any electric or magnetic field, which produces attractive or repelling interaction, or changes the arrangement of molecules in materials, or induces eddy current and heat, all of them are electric behaviors or magnetic behaviors, and so on. Most of them behavior simultaneously and inevitably change the structure of the system recoverably and irrecoverably. Then in turn they bring about the final results different from the results when they behavior singly. The results are different also from a simple addition of the results of individual behaviors. There is strong coupling between such behaviors. Tribology is the science and technology, which provides theories and techniques to describe and control the pattern of coupling between behaviors under tribological condition. No other discipline can

**tribological condition consisted of interacting surfaces in relating motion** 

meet such a need other than tribology, even though for any individual behavior there are principles from the discipline related which can predict its results. The difficulty for tribologists is that they have to know all the relative disciplines together with tribology simultaneously. Such a character of inter-disciplines and multi-disciplines requires a new methodology for tribology different from what for friction, wear or lubrication. In distinguish with the first axiom the third axiom focuses on the relationship of behaviors.

#### **3. How to model a tribo-system and simulate its behaviors?**

*The structure* is a description of intrinsic facts of a tribo-system while *the behaviors* are a description of change of the tribo-system. The structure of a system exists regardless whether there is an input. The behaviors must follow an input and can be derived from the input for a given structure in principle.

#### **3.1 The structure of a tribo-system**

Czichos (Czichos, 1978) described the structure of a tribo-system with a parameter set as

$$S = \{E\_\prime P\_\prime R\} \tag{2}$$

where *E ee e* = { 1 2 , ,...., *<sup>N</sup>* } is a sub-set showing that there are *N* tribo-elements in total in the system, *P pp p* = { *e e eN* 1 2 , ,....., } is a sub-set describing the property of each element in the system and *Rr r r r* = { *e e e e e eN e e* 12 13 1 23 , ,.. , ,...} is a sub-set collecting all relations between elements in the system.

Such description carries out a problem. Since the relative motion is the first important character of tribology and then the relative displacements between elements changing with the motion condition input, therefore the relative displacements between elements cannot be treated as an intrinsic fact. Several other examples can be listed as well.

To avoid the problem the author modified the description of a structure as (Xie, 2010)

$$S = \{E\_\prime P\_\prime H\} \tag{3}$$

where *H hh h h* = { *w e e eN* , , ,...... 1 2 } is a sub-set including the history of the system as a whole and of each element.

Each element *ei* , *i* = 1….*N,* in the sub-set *E* represents a surface or a medium substance, for example a journal surface, a bearing surface, a cylinder bore surface, a piston skirt surface or the lubricant film between the surfaces.

Each element *pei , i* = 1….*N,* in the sub-set *P* describes the property of element *ei* . In more detail the contents of property of each element can be divided into two groups, i.e. *pei* = {*pg*, *pp*}*i*, *i* = 1….*N,* where *pg* is the geometric parameter group of property of the element, for example the diameter and width of the bearing surface in macro scale and the roughness in micro scale, and *pp* is the physical parameter group of property of the element. *pp* should be understood in a generalized sense including all physical, chemical and biological features besides geometric. It usually can be described by a group of physical, chemical and biological parameters, such as hardness, viscosity, acidity, activity etc. but there are some exceptions. Such features are affected by material composition, manufacturing process, service history, surrounding temperature, atmosphere, etc.

Theory of Tribo-Systems 13

for example the electric current *i* in the coil of the electric magnet of an adaptive magnetic

For tribo-systems the situation will be complex. There are three possible ways to be selected. 1. If in behavior simulation the change of structure is not considered there will be a time-

2. If in behavior simulation the recoverable change of structure is considered only there

For any artifact system a requirement of behavior repeatability in an observation of short period is obviously necessary for reuse. Therefore the state *X* is repeatable. The recoverable change of structure implies that the structure is a function of the state and independent to time. Whenever a similar input applied on a system with a similar state the system will have a similar state change and similar output. In other words the system behaves similarly. In an observation of short period the irrecoverable change due to very small in value in

In an observation of short period, *pg* or *pp* changes with *X* due to many causes under the tribological condition, i.e. on or between the interacting surfaces in relative motion. Because *X* is repeatable and *pg* or *pp* is a function of *X* only, the patterns of change of *pg* or *pp* are relative simple. For each cause there will be some principles dealing with how the cause affects the change of parameters of *pg* or *pp*. These principles are in general relative to a discipline independent to tribology*.* Meanwhile a governing equation system, which may be a theoretical, experimental or statistical one, can be found in the discipline to describe the patterns of change of parameters of *pg* or *pp* under the tribological condition. As discussed before, for an elastic deformation the governing equation system can be found in the theory of elasticity and dynamics for a temperature distribution change the governing equation system can be found in the thermodynamics and heat transfer, for a change of viscosity of lubricants in terms of relative motion the governing equation system can be found in

3. Irrecoverable changes are performed in entire processes of manufacturing, assembling, packaging, storing and transporting and will accumulate with service time and reach a comparable extent at last. It is history depended. In behavior simulation a time-variable

() () () ()

<sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> (8)

*A AXH B BXH C CXH D DXH*

( ) ( )

, or more accurate that , and , , , , , , ,

*S SXt S SXH*

= =

*S const* = (6)

*A const B const C const D const* = ,,, = = = (6a)

*P pg pp P X pg X pp X* = = = { , , } ( ) { ( ) ( )} (7a)

*S SX A AX B BX C CX D DX* = = = = = ( ), ,, , ( ) ( ) ( ) ( ) (7)

bearing.

rheology, etc.

invariable linear system, i.e.

will be a time-invariable non-linear system, i.e.

comparison with the recoverable change is negligible.

non-linear system have to be treated, i.e.

Simultaneously there will be also

and simultaneously

According to the second axiom in tribology, the property of elements and systems is time depended. The structure is a description of intrinsic facts but it is not invariable for a tribosystem. There are recoverable changes and irrecoverable changes in the structure due to the interaction and relative motion of surfaces. As described in formula (3), *E* is obvious invariable, the only variable things in *S* are *P* and *H.* Each element *hw*, *hei , i* = 1….*N,* in the sub-set *H* are too complex to be described with parameters, usually they are a series of records in natural language. Using *H* rather than using a time parameter *t* here is because of that *t* notes only a time scale but what happened at *t* is more important for understanding the change of the structure. The elements of *H* do not act directly upon the structure but affect the values of parameters in *pg* and *pp.* For each effect some principles which govern the progress of effect can be found in related discipline. For example an elastic deformation of the surfaces is a recoverable change of *pg* which follows the change of interacting load on the surfaces governed by principles in the theory of elasticity, while a plastic deformation or wear of the surfaces is an irrecoverable change of *pg*, it is defined by what happened in the history and governed by principles in the theory of plasticity and tribology.

#### **3.2 The behavior simulation of a tribo-system**

Different from what used in references (Dai & Xue, 2003; Ge & Zhu, 2005), a state space method is applied here to simulate the behaviors. The state space method is a combination of general systems theory with engineering systems analysis and has wide application in dynamic system analysis, control engineering and many non-engineering analysis (Ogata, 1970, 1987). It takes a vector quantity called *state* as a scale to coordinate and evaluate the results of behaviors. When an input is applied upon a system, the system behaves from one state to another state and gives an output. For a time-invariable linear system a state equation (4) and an output equation (5) can be used to describe the results of behaviors:

$$
\dot{X} = AX + B\mathcal{U} \tag{4}
$$

$$Y = CX + D\mathcal{U} \tag{5}$$

where *X, U, Y* are the state vector, input vector and output vector of the system respectively. *A* and *B* are the system matrix, input matrix for equation (4) while *C* and *D* the output matrix for equation (5) respectively. All of them consist of the elements of structure of the system. *A*, *B, C* and *D* are constant for a time-invariable linear system.

In general the elements in a state vector are what concerned with the results of behaviors. As discussed previously, the first important behavior to be studied in tribo-systems is the relative motion. Any surface cannot exist independently and must be a part of a component of the machine system from which the tribo-system abstracted. The relative motion of surfaces is defined by the relative motion of components and where the surfaces reside on. Therefore for tribo-systems in the state vectors there are usually the parameters of displacements and time derivatives of displacements of components. For example the state of a single mass moving horizontally can be written as

$$\mathbf{X} = \begin{bmatrix} \mathbf{x}\_{\prime\prime} \dot{\mathbf{x}} \end{bmatrix}^T$$

in which, *x* is the coordinate of the mass in *x* direction. When there are behaviors besides mechanics to be studied, parameters of related disciplines may emerge in the state vector,

According to the second axiom in tribology, the property of elements and systems is time depended. The structure is a description of intrinsic facts but it is not invariable for a tribosystem. There are recoverable changes and irrecoverable changes in the structure due to the interaction and relative motion of surfaces. As described in formula (3), *E* is obvious invariable, the only variable things in *S* are *P* and *H.* Each element *hw*, *hei , i* = 1….*N,* in the sub-set *H* are too complex to be described with parameters, usually they are a series of records in natural language. Using *H* rather than using a time parameter *t* here is because of that *t* notes only a time scale but what happened at *t* is more important for understanding the change of the structure. The elements of *H* do not act directly upon the structure but affect the values of parameters in *pg* and *pp.* For each effect some principles which govern the progress of effect can be found in related discipline. For example an elastic deformation of the surfaces is a recoverable change of *pg* which follows the change of interacting load on the surfaces governed by principles in the theory of elasticity, while a plastic deformation or wear of the surfaces is an irrecoverable change of *pg*, it is defined by what happened in the

Different from what used in references (Dai & Xue, 2003; Ge & Zhu, 2005), a state space method is applied here to simulate the behaviors. The state space method is a combination of general systems theory with engineering systems analysis and has wide application in dynamic system analysis, control engineering and many non-engineering analysis (Ogata, 1970, 1987). It takes a vector quantity called *state* as a scale to coordinate and evaluate the results of behaviors. When an input is applied upon a system, the system behaves from one state to another state and gives an output. For a time-invariable linear system a state equation (4) and an output equation (5) can be used to describe the results of behaviors:

where *X, U, Y* are the state vector, input vector and output vector of the system respectively. *A* and *B* are the system matrix, input matrix for equation (4) while *C* and *D* the output matrix for equation (5) respectively. All of them consist of the elements of structure of the

In general the elements in a state vector are what concerned with the results of behaviors. As discussed previously, the first important behavior to be studied in tribo-systems is the relative motion. Any surface cannot exist independently and must be a part of a component of the machine system from which the tribo-system abstracted. The relative motion of surfaces is defined by the relative motion of components and where the surfaces reside on. Therefore for tribo-systems in the state vectors there are usually the parameters of displacements and time derivatives of displacements of components. For example the state

[ ] , *<sup>T</sup> X xx* =

in which, *x* is the coordinate of the mass in *x* direction. When there are behaviors besides mechanics to be studied, parameters of related disciplines may emerge in the state vector,

system. *A*, *B, C* and *D* are constant for a time-invariable linear system.

of a single mass moving horizontally can be written as

*X AX BU* = + (4)

*Y CX DU* = + (5)

history and governed by principles in the theory of plasticity and tribology.

**3.2 The behavior simulation of a tribo-system** 

for example the electric current *i* in the coil of the electric magnet of an adaptive magnetic bearing.

For tribo-systems the situation will be complex. There are three possible ways to be selected.

1. If in behavior simulation the change of structure is not considered there will be a timeinvariable linear system, i.e.

$$S = const$$

and simultaneously

$$A = const, B = const, C = const, D = const$$

2. If in behavior simulation the recoverable change of structure is considered only there will be a time-invariable non-linear system, i.e.

$$\mathcal{S} = \mathcal{S}(X),\\ A = A\begin{pmatrix} X \\ \end{pmatrix}, B = B\begin{pmatrix} X \\ \end{pmatrix}, \mathcal{C} = \mathcal{C}\begin{pmatrix} X \\ \end{pmatrix}, \\ D = D\begin{pmatrix} X \\ \end{pmatrix} \\ \tag{7}$$

Simultaneously there will be also

$$P = \{p\mathbb{g}, pp\} = P(X) = \{p\mathbb{g}(X), pp(X)\}\tag{7a}$$

For any artifact system a requirement of behavior repeatability in an observation of short period is obviously necessary for reuse. Therefore the state *X* is repeatable. The recoverable change of structure implies that the structure is a function of the state and independent to time. Whenever a similar input applied on a system with a similar state the system will have a similar state change and similar output. In other words the system behaves similarly. In an observation of short period the irrecoverable change due to very small in value in comparison with the recoverable change is negligible.

In an observation of short period, *pg* or *pp* changes with *X* due to many causes under the tribological condition, i.e. on or between the interacting surfaces in relative motion. Because *X* is repeatable and *pg* or *pp* is a function of *X* only, the patterns of change of *pg* or *pp* are relative simple. For each cause there will be some principles dealing with how the cause affects the change of parameters of *pg* or *pp*. These principles are in general relative to a discipline independent to tribology*.* Meanwhile a governing equation system, which may be a theoretical, experimental or statistical one, can be found in the discipline to describe the patterns of change of parameters of *pg* or *pp* under the tribological condition. As discussed before, for an elastic deformation the governing equation system can be found in the theory of elasticity and dynamics for a temperature distribution change the governing equation system can be found in the thermodynamics and heat transfer, for a change of viscosity of lubricants in terms of relative motion the governing equation system can be found in rheology, etc.

3. Irrecoverable changes are performed in entire processes of manufacturing, assembling, packaging, storing and transporting and will accumulate with service time and reach a comparable extent at last. It is history depended. In behavior simulation a time-variable non-linear system have to be treated, i.e.

$$\begin{aligned} S &= S(X, t) \text{ or more accurate that } S = S(X, H) \\ \text{and } A &= A(X, H), B = B(X, H), C = C(X, H), D = D(X, H) \end{aligned} \tag{8}$$

Theory of Tribo-Systems 15

For time-variable non-linear systems the situation will be a little complex. Since matrix *A, B, C* or *D* is a function of the state and time (history related), integrating formula (9) and (10) analytically is in general impossible. The problem is similar with time-invariable non-linear systems when the matrixes *A, B, C* and *D* are functions of state *X* as shown in formula (7)

Numerical method is used for solving formula (9) and (10) for a time-variable non-linear system. The equations are discretized and integrated in a small time increment ∆*t* step by step. When the ∆*t* is small enough one can suppose that matrix *A, B, C* or *D* is independent to *X* and *t* and is constant in the time interval ∆*t*, i.e. the system becomes a time-invariable linear system. In the integration, matrix *A, B, C* or *D* as a constant matrix and the values of their elements are calculated base on the results of last step with state *X*1 and time *t*1. After integration, there will be a change for both state and time, i.e. *X*2 = *X*1+∆*X* and *t*2 = *t*1 +∆*t*. Afterwards the elements in matrixes *A, B, C* and *D* should be recalculated according to *X*<sup>2</sup> and *t*2 for the next step of integration if any of them is state and time related. Similar to the time-invariable and linear assumption made in the integration, a decoupling assumption is made also that the effect of any behavior on the values of elements in matrixes *A, B, C* and *D* can be calculated independently with the governing equations of related discipline or obtained from an experiment under a condition considering only the change of *X* and *t* ignoring other coupling effects. For example, in the simulation of the lubrication behavior in a piston skirt – cylinder bore pair, the lubricant film between the skirt surface and the bore surface undergoes a viscosity change when the piston changes its position along the bore due to a non-uniform distribution of temperature. The viscosity is a parameter in *pp* and its change may affect some elements in matrix *A, B, C* or *D*. A viscosity *η*1 corresponding to temperature *T*1 at *y*1, the coordinate of shirt in the bore, is used for obtaining matrix *A, B, C* or *D*. After integrating over a ∆*t*, *y*1 becomes to *y*2, *T*1 becomes to *T*2, *η*1 becomes to *η*2 and the matrix *A, B, C* or *D* will be recalculated with *η*2 for the next integration. For recoverable change in an observation of short period the function *η*(*T*) can be obtained by fitting experiment data and accurate enough. For irrecoverable change in an observation of long period a function in the form of *η*(*T,H*) is necessary. In the history, many causes of very different kinds can affect the relation between *η* and *T* and make the lubricant aging. The causes before service include the kind of base oil, the technology and process of refining, the additive used etc. while the causes after service include the service temperature, service atmosphere, pollution condition and filtration efficiency in service etc. Knowledge of *η*(*T,H*) have to be acquired for each application. Aging is a long period change and progresses very slowly. In numerical integration one can use a relative long time interval for such kinds of irrecoverable change other than recoverable change while a small time interval has to be used to keep the

accuracy of simulation for recoverable change in time-invariable non-linear system.

while a larger time step is used in integration (Xu, 2007).

**4. Examples of modeling and simulation** 

**4.1 Example1** 

There are many mathematic tools which make such an application available, for example, the Runge-Kutta Procedure (Chen, 1982). The difficulty in solving the problem is to find a balance between time consuming while a smaller time step (∆*t* ) is used and low precision

The cylinder – piston – conrod – crank system of a single cylinder internal combustion engine is shown in Fig. 6. The system can be abstracted into a tribo-system with following

and (7a) and will not be discussed separately in the following.

Since formula (3) and that the elements of *H* do not act directly upon the structure but affect the values of parameters in *pg* and *pp,* the following formula can be established

$$P = \{pg\_{\prime}, pp\} = P(X\_{\prime}H) = \{pg(X\_{\prime}H), pp(X\_{\prime}H)\} \tag{8a}$$

It shows that the property of a tribo-system changes with the system state and the history of the system.

In an observation of long period, *pg* or *pp* changes not only with *X* but also with *H*. There are many issues concerning with irrecoverable changes of the structure of machine systems. Wear, fatigue, plastic flow, creep, aging and corrosion are the most important irrecoverable changes. It is no doubt that wear is one of the issues studied in tribology. Fatigue takes place on the surfaces bringing forth a kind of fatigue wear. Plastic flow or creep carries out a permanent deformation of surfaces in macro scale which harms the motion guarantee function. Plastic flow in micro scale makes a change of elastic contact to plastic contact and will generate origins of surface fatigue after a number of cycles of repeat. Aging changes parameters in *pp* for solid surface materials and makes them inclining to failure. Aging spoils the performance of lubricants, increases corrosiveness and decreases the capability of lubrication. Corrosion of interacting surfaces in relative motion is also a kind of wear due to the chemical reaction of some compositions in lubricant or atmosphere with the materials of surfaces or due to the mechanical effect of break of air bubbles in the lubricant film. Obviously most issues concerning with irrecoverable changes are taken place in tribosystems and studied in tribology.

According to the third axiom in tribology, the results of tribological behaviors are the results of mutual action and strong coupling of behaviors of many disciplines under a tribological condition constituted by interacting surfaces in relative motion. Because of that history or time is unrepeatable, the irrecoverable change is more complex in description than the recoverable change and almost no simple equation system can be found in any discipline. The different causes occurred singly or jointly at different moment in the history and their results were accumulated or coupled each other and result an irrecoverable change of the structure at a given time. In other words the structure is a carrier of mutual action and strong coupling of behaviors of many disciplines and gives a structure change in total at last as the results.

#### **3.3 How to solve the state equations and output equations**

In the behavior simulation of tribo-systems a time-variable non-linear system must be faced. The state equations and output equations will be as

$$
\dot{X} = A\begin{pmatrix} X, H \end{pmatrix} \cdot X + B\begin{pmatrix} X, H \end{pmatrix} \cdot \mathcal{U}(t) \tag{9}
$$

$$Y = \mathbb{C}(X, H) \cdot X + D(X, H) \cdot \mathcal{U}(t) \tag{10}$$

Solving state equations is an initial value problem.

For a time-invariable linear system formula (4) can be integrated analytically when in formula (6a) *A* and *B* are constant. At any instant *t*1 an input *U* (*t*) is applied to a system in an initial state *X*1, then the system behaves to a state *X*2 at an instant *t*2 = *t*1 +∆*t* and give an output *Y* based on formula (5). It implies that similar initial state and similar input result similar change of state and similar output after a similar time interval ∆*t*. After obtaining a new *X*2 the new output *Y*2 can be computed accordingly with formula (5) and constant matrixes *C* and *D*.

It shows that the property of a tribo-system changes with the system state and the history of

In an observation of long period, *pg* or *pp* changes not only with *X* but also with *H*. There are many issues concerning with irrecoverable changes of the structure of machine systems. Wear, fatigue, plastic flow, creep, aging and corrosion are the most important irrecoverable changes. It is no doubt that wear is one of the issues studied in tribology. Fatigue takes place on the surfaces bringing forth a kind of fatigue wear. Plastic flow or creep carries out a permanent deformation of surfaces in macro scale which harms the motion guarantee function. Plastic flow in micro scale makes a change of elastic contact to plastic contact and will generate origins of surface fatigue after a number of cycles of repeat. Aging changes parameters in *pp* for solid surface materials and makes them inclining to failure. Aging spoils the performance of lubricants, increases corrosiveness and decreases the capability of lubrication. Corrosion of interacting surfaces in relative motion is also a kind of wear due to the chemical reaction of some compositions in lubricant or atmosphere with the materials of surfaces or due to the mechanical effect of break of air bubbles in the lubricant film. Obviously most issues concerning with irrecoverable changes are taken place in tribo-

According to the third axiom in tribology, the results of tribological behaviors are the results of mutual action and strong coupling of behaviors of many disciplines under a tribological condition constituted by interacting surfaces in relative motion. Because of that history or time is unrepeatable, the irrecoverable change is more complex in description than the recoverable change and almost no simple equation system can be found in any discipline. The different causes occurred singly or jointly at different moment in the history and their results were accumulated or coupled each other and result an irrecoverable change of the structure at a given time. In other words the structure is a carrier of mutual action and strong coupling of behaviors of many disciplines and gives a structure change in total at last as the results.

In the behavior simulation of tribo-systems a time-variable non-linear system must be faced.

For a time-invariable linear system formula (4) can be integrated analytically when in formula (6a) *A* and *B* are constant. At any instant *t*1 an input *U* (*t*) is applied to a system in an initial state *X*1, then the system behaves to a state *X*2 at an instant *t*2 = *t*1 +∆*t* and give an output *Y* based on formula (5). It implies that similar initial state and similar input result similar change of state and similar output after a similar time interval ∆*t*. After obtaining a new *X*2 the new output *Y*2 can be computed accordingly with formula (5) and constant

*X AXH X BXH Ut* = ⋅+ ⋅ ( , , ) ( ) ( ) (9)

*Y CXH X DXH Ut* = ⋅+ ⋅ ( , , ) ( ) ( ) (10)

**3.3 How to solve the state equations and output equations** 

The state equations and output equations will be as

Solving state equations is an initial value problem.

matrixes *C* and *D*.

the system.

systems and studied in tribology.

Since formula (3) and that the elements of *H* do not act directly upon the structure but affect the values of parameters in *pg* and *pp,* the following formula can be established

*P pg pp P X H pg X H pp X H* = = = { , , ,, , } ( ) { ( ) ( )} (8a)

For time-variable non-linear systems the situation will be a little complex. Since matrix *A, B, C* or *D* is a function of the state and time (history related), integrating formula (9) and (10) analytically is in general impossible. The problem is similar with time-invariable non-linear systems when the matrixes *A, B, C* and *D* are functions of state *X* as shown in formula (7) and (7a) and will not be discussed separately in the following.

Numerical method is used for solving formula (9) and (10) for a time-variable non-linear system. The equations are discretized and integrated in a small time increment ∆*t* step by step. When the ∆*t* is small enough one can suppose that matrix *A, B, C* or *D* is independent to *X* and *t* and is constant in the time interval ∆*t*, i.e. the system becomes a time-invariable linear system. In the integration, matrix *A, B, C* or *D* as a constant matrix and the values of their elements are calculated base on the results of last step with state *X*1 and time *t*1. After integration, there will be a change for both state and time, i.e. *X*2 = *X*1+∆*X* and *t*2 = *t*1 +∆*t*. Afterwards the elements in matrixes *A, B, C* and *D* should be recalculated according to *X*<sup>2</sup> and *t*2 for the next step of integration if any of them is state and time related. Similar to the time-invariable and linear assumption made in the integration, a decoupling assumption is made also that the effect of any behavior on the values of elements in matrixes *A, B, C* and *D* can be calculated independently with the governing equations of related discipline or obtained from an experiment under a condition considering only the change of *X* and *t* ignoring other coupling effects. For example, in the simulation of the lubrication behavior in a piston skirt – cylinder bore pair, the lubricant film between the skirt surface and the bore surface undergoes a viscosity change when the piston changes its position along the bore due to a non-uniform distribution of temperature. The viscosity is a parameter in *pp* and its change may affect some elements in matrix *A, B, C* or *D*. A viscosity *η*1 corresponding to temperature *T*1 at *y*1, the coordinate of shirt in the bore, is used for obtaining matrix *A, B, C* or *D*. After integrating over a ∆*t*, *y*1 becomes to *y*2, *T*1 becomes to *T*2, *η*1 becomes to *η*2 and the matrix *A, B, C* or *D* will be recalculated with *η*2 for the next integration. For recoverable change in an observation of short period the function *η*(*T*) can be obtained by fitting experiment data and accurate enough. For irrecoverable change in an observation of long period a function in the form of *η*(*T,H*) is necessary. In the history, many causes of very different kinds can affect the relation between *η* and *T* and make the lubricant aging. The causes before service include the kind of base oil, the technology and process of refining, the additive used etc. while the causes after service include the service temperature, service atmosphere, pollution condition and filtration efficiency in service etc. Knowledge of *η*(*T,H*) have to be acquired for each application. Aging is a long period change and progresses very slowly. In numerical integration one can use a relative long time interval for such kinds of irrecoverable change other than recoverable change while a small time interval has to be used to keep the accuracy of simulation for recoverable change in time-invariable non-linear system.

There are many mathematic tools which make such an application available, for example, the Runge-Kutta Procedure (Chen, 1982). The difficulty in solving the problem is to find a balance between time consuming while a smaller time step (∆*t* ) is used and low precision while a larger time step is used in integration (Xu, 2007).

#### **4. Examples of modeling and simulation**

#### **4.1 Example1**

The cylinder – piston – conrod – crank system of a single cylinder internal combustion engine is shown in Fig. 6. The system can be abstracted into a tribo-system with following

Theory of Tribo-Systems 17

Fig. 7. The system block diagram of a cylinder bore-piston skirt

are balanced by a resistant torque moment (load) on the crankshaft.

*P P P P X X*

follows.

β

β

θ

θ

Piston ring package is considered separately also and the friction force between ring surfaces and cylinder bore is treated as an input (*FRN* in Fig. 6) applied on the piston. Other inputs are the gas pressure *Q*(*t*) on the top of the piston, the thrust force from the cylinder bore surface on the piston skirt surface *S*, the force on the wrist pin *FP*. All of them

The state matrix equation of the system and the output matrix equation can be written as

*X A X U*

′ ⎡⎤⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ <sup>=</sup> ⎥⎢ ⎥ ⎢ <sup>+</sup> ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ <sup>⎥</sup> ⎢⎥⎢ ⎣⎦⎣ ⎥⎢ ⎥ ⎢ ⎦⎣ ⎦ ⎣ ⎥⎦ ⎣

 β

 β

 θ

 θ

When the hydrodynamic behavior between the skirt surface and bore surface is looked as an input applied on the system (via skirt surface), the resultant force of the hydrodynamic film pressure *S* and the resultant force of the resistant shear stress *FSK* will be the elements in *U*2

01000 0 000000 0 00000 010000 00010 0 000000 0 00000 000100 00000 1 000000 0 00000 000001

26 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥

(11)

⎥ ⎢ ⎥ ⎢ ⎥⎦

46 4

*A U*

66 6

*A U*

The output can be selected according to what one wants to know in the simulation.

tribo-pairs: piston skirt – cylinder bore, wrist pin – small end bearing, journal of crank – big end bearing and journal of crankshaft and main bearing, i.e. one prismatic pair and three revolute pairs in the system totally.

Fig. 6. Cylinder-piston-conrod-crank mechanism

A system block diagram for the piston skirt – cylinder bore pair is shown in Fig. 7 which gives a survey on the relationship between the skirt-bore tribo-pair and environment.

In the simulation the secondary motion of the piston and the change of inertia of the conrod are considered. The influence of the offset of the wrist pin can be considered as well and is taken as zero, positive or negative for comparison. The behaviors of lubricant film between the skirt surface and bore surface are treated according to the theory of thermalhydrodynamic lubrication. The configuration of the skirt and bore can be given in simulation and a thermal distortion, force deformation and wear process can be calculated separately with the theory of heat transfer, theory of elasticity and regression of measured wear data. Their effects will couple between each other and with what of other behaviors in the iteration but a rigid skirt and a rigid bore without wear are supposed in the example. All of the eccentricities of journals in bearings and all of the elastic deformation of other components in the system are neglect. Their behaviors can be simulated separately and is decoupling with other behaviors in the global simulation.

The tribological behavior concerned in the system is a hydrodynamic lubrication behavior between the skirt surface and the bore surface. It results a thrust force *S* which balances the interacting load on the lubricant film and a shear resistant (friction) force *FSK* against the relative motion. In general there are two ways to treat the hydrodynamic behavior. One is looking the forces produced in the film like the inputs of the system. The other is taking the lubricant film as a structure element between surfaces. In this example the first way of treatment is applied.

tribo-pairs: piston skirt – cylinder bore, wrist pin – small end bearing, journal of crank – big end bearing and journal of crankshaft and main bearing, i.e. one prismatic pair and three

A system block diagram for the piston skirt – cylinder bore pair is shown in Fig. 7 which gives a survey on the relationship between the skirt-bore tribo-pair and environment. In the simulation the secondary motion of the piston and the change of inertia of the conrod are considered. The influence of the offset of the wrist pin can be considered as well and is taken as zero, positive or negative for comparison. The behaviors of lubricant film between the skirt surface and bore surface are treated according to the theory of thermalhydrodynamic lubrication. The configuration of the skirt and bore can be given in simulation and a thermal distortion, force deformation and wear process can be calculated separately with the theory of heat transfer, theory of elasticity and regression of measured wear data. Their effects will couple between each other and with what of other behaviors in the iteration but a rigid skirt and a rigid bore without wear are supposed in the example. All of the eccentricities of journals in bearings and all of the elastic deformation of other components in the system are neglect. Their behaviors can be simulated separately and is

The tribological behavior concerned in the system is a hydrodynamic lubrication behavior between the skirt surface and the bore surface. It results a thrust force *S* which balances the interacting load on the lubricant film and a shear resistant (friction) force *FSK* against the relative motion. In general there are two ways to treat the hydrodynamic behavior. One is looking the forces produced in the film like the inputs of the system. The other is taking the lubricant film as a structure element between surfaces. In this example the first way of

revolute pairs in the system totally.

Fig. 6. Cylinder-piston-conrod-crank mechanism

decoupling with other behaviors in the global simulation.

treatment is applied.

Fig. 7. The system block diagram of a cylinder bore-piston skirt

Piston ring package is considered separately also and the friction force between ring surfaces and cylinder bore is treated as an input (*FRN* in Fig. 6) applied on the piston. Other inputs are the gas pressure *Q*(*t*) on the top of the piston, the thrust force from the cylinder bore surface on the piston skirt surface *S*, the force on the wrist pin *FP*. All of them are balanced by a resistant torque moment (load) on the crankshaft.

The output can be selected according to what one wants to know in the simulation.

The state matrix equation of the system and the output matrix equation can be written as follows.

$$
\begin{bmatrix} X\_p \\ \dot{X}\_p \\ \dot{\theta} \\ \dot{\theta} \\ \theta \\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & A\_{26} \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & A\_{46} \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & A\_{66} \end{bmatrix} \begin{bmatrix} X\_p \\ \dot{X}\_p \\ \dot{\theta} \\ \dot{\theta} \\ \theta \\ \theta \\ \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \tag{11}
$$

When the hydrodynamic behavior between the skirt surface and bore surface is looked as an input applied on the system (via skirt surface), the resultant force of the hydrodynamic film pressure *S* and the resultant force of the resistant shear stress *FSK* will be the elements in *U*2

Theory of Tribo-Systems 19

hydrodynamic film pressure in values and distribution and changes the shear stress. It

Table 1 shows a comparison on the friction power loss between different values of wrist pin

**Computation number Wrist Pin Offset Friction Power Loss in 720o** 

CC=+4.E-5 m 2.32121 Nm

CC=0.m 2.31236 Nm

CC=-4.E-5 m 2.30477 Nm

CC=+4.E-5 m 1.97164 Nm

CC=0.m 1.97038 Nm

CC=-4.E-5 m 1.96907 Nm

Left Offset

Zero Offset

Right Offset

Left Offset

Zero Offset

Right Offset

Table 1. Effects of wrist pin offset and skirt profile on piston skirt friction power loss

If the forces transmitted in the pairs *P, A* and *O* are interesting there will be another output

shows that the barrel skirt has a smaller friction loss.

Fig. 9. Influence of skirt configuration on the friction power loss

Linear Skirt LS99-2-C-1

Linear Skirt LS99-2-C-0

Linear Skirt LS99-2-C-2

Barrel Skirt BS99-2-C-1

Barrel Skirt BS99-2-C-0

Barrel Skirt BS99-2-C-2

matrix equation as

offset. The linear skirt is more sensitive to the offset than the barrel skirt is.

and *U*4. The hydrodynamic behavior depends on the gap geometry, the relative motion of surfaces and the lubricant viscosity. The gap geometry is changed with the wrist pin center displacement *XP* and the piston tilting angle *β* in this case. The relative motion includes a tangential and normal component. The lubricant viscosity changes with temperature which has a distribution along the cylinder wall in *y* direction. The temperature distribution changes with the engine working condition but keeps unchanged in the example. All of them will be calculated in a separate program based on Reynolds Equation (Pinkus & Sternlicht, 1961).

$$
\begin{array}{c|cccc}
\dot{\theta} & \dot{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{16} \\
 P\_{\rm LOS} & \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & \mathbf{C}\_{16} \\ 0 & 0 & 0 & 0 & 0 & \mathbf{C}\_{26} \\ 0 & 0 & 0 & 0 & 0 & \mathbf{C}\_{36} \\ 0 & 0 & 0 & 0 & 0 & \mathbf{C}\_{46} \\ 0 & 0 & 0 & 0 & 0 & \mathbf{C}\_{56} \\ 0 & 0 & 0 & 0 & 0 & \mathbf{C}\_{66} \end{bmatrix} \end{array} \begin{array}{c|cccc} \dot{X}\_P \\ \dot{X}\_P \\ \dot{\theta} \\ \dot{\theta} \\ \dot{\theta} \\ \dot{\theta} \\ \dot{\theta} \end{array} \tag{12}
$$

Fig. 8 gives the change of output in 7200 crankshaft rotating angle by formula (12) , where (a), (b), (c), (d), (e) and (f) are the deviation of crankshaft speed θ , change of friction power loss *PLOSS* in the skirt-bore pair, displacement *XP* of the wrist pin center in *X* direction, tilting angle *β* around the wrist pin center, thrust force *FRHT* on the right side of the skirt and thrust force *FLFT* on the left side of the skirt from the hydrodynamic lubrication film respectively.

Fig. 8. Output of the system in 720° rotating angle of crankshaft

Fig. 9 gives a comparison on the friction power loss when different skirt configurations are used. The geometry of skirt influences the gap between surfaces and then changes the

and *U*4. The hydrodynamic behavior depends on the gap geometry, the relative motion of surfaces and the lubricant viscosity. The gap geometry is changed with the wrist pin center displacement *XP* and the piston tilting angle *β* in this case. The relative motion includes a tangential and normal component. The lubricant viscosity changes with temperature which has a distribution along the cylinder wall in *y* direction. The temperature distribution changes with the engine working condition but keeps unchanged in the example. All of them will be calculated in a separate program based on Reynolds Equation (Pinkus &

*P C X X C*

*F C F C*

*P*

θ

*RHT LFT*

(a), (b), (c), (d), (e) and (f) are the deviation of crankshaft speed

Fig. 8. Output of the system in 720° rotating angle of crankshaft

β

*LOSS P*

⎡ ⎤ <sup>⎡</sup> ⎤⎡ ⎤ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>=</sup> <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎢⎣ ⎥⎢ ⎥ ⎦⎣ ⎦

Fig. 8 gives the change of output in 7200 crankshaft rotating angle by formula (12) , where

loss *PLOSS* in the skirt-bore pair, displacement *XP* of the wrist pin center in *X* direction, tilting angle *β* around the wrist pin center, thrust force *FRHT* on the right side of the skirt and thrust force *FLFT* on the left side of the skirt from the hydrodynamic lubrication film respectively.

Fig. 9 gives a comparison on the friction power loss when different skirt configurations are used. The geometry of skirt influences the gap between surfaces and then changes the

*C*

*C X*

*P*

β

β

(12)

, change of friction power

θ

θ

θ

Sternlicht, 1961).

hydrodynamic film pressure in values and distribution and changes the shear stress. It shows that the barrel skirt has a smaller friction loss.

Fig. 9. Influence of skirt configuration on the friction power loss

Table 1 shows a comparison on the friction power loss between different values of wrist pin offset. The linear skirt is more sensitive to the offset than the barrel skirt is.


Table 1. Effects of wrist pin offset and skirt profile on piston skirt friction power loss

If the forces transmitted in the pairs *P, A* and *O* are interesting there will be another output matrix equation as

Theory of Tribo-Systems 21

The derivation of elements *A*<sup>16</sup> to *A*<sup>66</sup> , *C*<sup>16</sup> to*C*<sup>66</sup> , *U*<sup>2</sup> to*U*<sup>6</sup> and *C*16′ to*C*66′ in formulas (11),

As shown in Fig. 11 there is a rotor-bearing system of a 300MW turbo-generator set consisted of the rotor of a high pressure cylinder (HP), an intermediate pressure cylinder (IP), a low pressure cylinder (LP), a generator, an exciter and eight hydrodynamic bearings (1# - 8#) on pedestals. A simplification is made in the example that the eight bearings are all plane bearings to reduce the amount of computation. The rotor in total is an elastic component supported by the bearings and can vibrate laterally. Obviously it is a statically indeterminate problem. The load on each bearing is determined by the relationship between the elevations of journal centers which are controlled by a camber curve checked at last in installation. There are many reasons which can change the relationship, for example the journals may float with different eccentricity *e* (Fig. 17) on the hydrodynamic film and the pedestals may change their heights due to the changes of working temperatures during different turbine output and then change the bearing loads under a statically indeterminate

The tribological behaviors considered in the example are the hydrodynamic behaviors in

1. For a hydrodynamic bearing the rotating journal is floating on the hydrodynamic film and there is an eccentricity between the journal center and the bearing center. During installation the journal is dropped upon the bottom surface of the bearing bore. The

2. The change of the load or eccentricity changes the geometric property and physical property (*pg, pp –* see section 3.1) of the film when taking it as a structure element

3. If the change of *pp* approaching to some extent the film will excite a kind of severe vibration of the system called oil whirl or oil resonance (Hori, 2002) and may result a

In general it is recognized that the oil whirl begins at the threshold of instability of the rotorbearing system and usually has a frequency half the rotor speed. It is a tribological behavior

The treatment of the hydrodynamic behavior in the film looks like inserting a structure element between surfaces and is different from what has done in example 1 (see section 4.1). In this case the film is a linearized spring-damper in time interval ∆*t* and its *pp* can be represented by four constant stiffness coefficients *kxx, kxy, kyx, kyy* and four constant damping coefficients *dxx, dxy, dyx, dyy*. It implies an assumption of using *pp*=*const* instead of *pp*=*pp*(*X*)

induced vibration and indicates a decrease or loss of motion guarantee function.

Fig. 11. The rotor-bearing system of a 300MW turbo-generator set

bearings. There are three points to be considered.

between surfaces.

eccentricity changes with the load on the bearing.

catastrophic damage of the turbo-generator set.

(12) and (13) can be found in Appendix.

**4.2 Example 2** 

condition.

$$
\begin{bmatrix} F\_{PX} \\ F\_{PY} \\ F\_{AX} \\ F\_{AY} \\ F\_{AY} \\ F\_{OX} \\ F\_{OY} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & C\_{16}' \\ 0 & 0 & 0 & 0 & 0 & C\_{26}' \\ 0 & 0 & 0 & 0 & 0 & C\_{36}' \\ 0 & 0 & 0 & 0 & 0 & C\_{46}' \\ 0 & 0 & 0 & 0 & 0 & C\_{56}' \\ 0 & 0 & 0 & 0 & 0 & C\_{66}' \end{bmatrix} \begin{bmatrix} X\_P \\ \dot{X}\_P \\ \dot{\theta} \\ \dot{\theta} \\ \dot{\theta} \\ \dot{\theta} \end{bmatrix} \tag{13}
$$

Where *FPX*, *FPY*, *FAX*, *FAY*, *FOX* and *FOY* are the force components transmitted in the small end bearing of conrod, in the big end bearing of conrod and in the main bearings (in total) of crankshaft respectively of the IC engine in discussion. The change of such forces in 7200 crankshaft rotating angle is shown in Fig. 10.

Fig. 10. Forces transmitted in the bearing of an IC engine. (a) Small end bearing of conrod. (b) Big end bearing of conrod. (c) Main bearing of crankshaft

The derivation of elements *A*<sup>16</sup> to *A*<sup>66</sup> , *C*<sup>16</sup> to*C*<sup>66</sup> , *U*<sup>2</sup> to*U*<sup>6</sup> and *C*16′ to*C*66′ in formulas (11), (12) and (13) can be found in Appendix.

#### **4.2 Example 2**

20 Tribology - Lubricants and Lubrication

*PX P PY P*

<sup>⎡</sup> ⎤ ⎡ ′ ⎤⎡ ⎤ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ ′ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ <sup>⎢</sup> ⎥ ⎢ ′ ⎥⎢ ⎥ <sup>⎢</sup> ⎥ ⎢ <sup>=</sup> ⎥⎢ ⎥ ′ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ ′ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ ⎢⎣ ⎥ ⎢ ⎦ ⎣ ′ ⎥⎢ ⎥ ⎦⎣ ⎦

Where *FPX*, *FPY*, *FAX*, *FAY*, *FOX* and *FOY* are the force components transmitted in the small end bearing of conrod, in the big end bearing of conrod and in the main bearings (in total) of crankshaft respectively of the IC engine in discussion. The change of such forces in 7200

*F C X F C X F C F C F C F C*

*AX AY OX OY*

crankshaft rotating angle is shown in Fig. 10.

(a)

(b) (c)

(b) Big end bearing of conrod. (c) Main bearing of crankshaft

Fig. 10. Forces transmitted in the bearing of an IC engine. (a) Small end bearing of conrod.

β

β

(13)

θ

θ

As shown in Fig. 11 there is a rotor-bearing system of a 300MW turbo-generator set consisted of the rotor of a high pressure cylinder (HP), an intermediate pressure cylinder (IP), a low pressure cylinder (LP), a generator, an exciter and eight hydrodynamic bearings (1# - 8#) on pedestals. A simplification is made in the example that the eight bearings are all plane bearings to reduce the amount of computation. The rotor in total is an elastic component supported by the bearings and can vibrate laterally. Obviously it is a statically indeterminate problem. The load on each bearing is determined by the relationship between the elevations of journal centers which are controlled by a camber curve checked at last in installation. There are many reasons which can change the relationship, for example the journals may float with different eccentricity *e* (Fig. 17) on the hydrodynamic film and the pedestals may change their heights due to the changes of working temperatures during different turbine output and then change the bearing loads under a statically indeterminate condition.

Fig. 11. The rotor-bearing system of a 300MW turbo-generator set

The tribological behaviors considered in the example are the hydrodynamic behaviors in bearings. There are three points to be considered.


In general it is recognized that the oil whirl begins at the threshold of instability of the rotorbearing system and usually has a frequency half the rotor speed. It is a tribological behavior induced vibration and indicates a decrease or loss of motion guarantee function.

The treatment of the hydrodynamic behavior in the film looks like inserting a structure element between surfaces and is different from what has done in example 1 (see section 4.1). In this case the film is a linearized spring-damper in time interval ∆*t* and its *pp* can be represented by four constant stiffness coefficients *kxx, kxy, kyx, kyy* and four constant damping coefficients *dxx, dxy, dyx, dyy*. It implies an assumption of using *pp*=*const* instead of *pp*=*pp*(*X*)

Theory of Tribo-Systems 23

hydrodynamic film and isotropic solid material. The hydrodynamic film then plays the role of a component of the system. It should be emphsized that the height of the journal center is determined by the sum of the height of bearing center controlled by pedestal and the project of ecentricity *e* of the journal center on ordinate axis while the load on each bearing is

Fig. 16. Angular displacements and inertia moments of a station in X-Z and Y-Z plane

Fig. 17. A linearized model of the hydrodynamic film

determined by the journal height under a static inderminate condition.

Fig. 15. The lateral deformation of a field

during integration in time interval ∆*t*. The eight coefficients can be calculated before integration with a separate program for a given film configuration (bearing bore geometry, eccentricity and attitude angle) and relative motion (tangential and normal) between journal surface and bearing surface (Pinkus & Sternlicht, 1961). The eight spring-dampers together with the distributed mass-stiffness-damping of the rotor defines the threshold of instability. To constitute the state space equation the rotor is discretized into 194 sections (Fig. 12) according to a concentrated mass treatment which can be found in rotor-bearing system dynamics (Glienicke, 1972) and its detail is omitted in the example.

Fig. 12. A discretized model of the rotor

Each section (Fig. 13) consists of a field of length *l* with stiffness but without mass and a station with mass, inertia moment but without length.

Fig. 13. A section of the rotor with a field and a station

The forces and moments applied on both side of a field and the related deformations are shown in Fig. 14 and Fig. 15.

Fig. 14. The forces and moments on a field

The angular displacements and inertia monents of a station are described in Fig. 16. All of the inputs (forces and moments) apply only on the station. They make a balance between the forces and moments appling by the fields (right and left) and the inertia forces and moments. If there is a bearing attached to a section then the station is looked like supported by a linearized spring-damper with four direct stiffness and damping coefficients *kxx, kyy, dxx, dyy* and four cross stiffness and damping coefficients *kxy, kyx, dxy, dyx* as shown in Fig. 17. The cross stiffness and damping coefficients show an important difference between the

during integration in time interval ∆*t*. The eight coefficients can be calculated before integration with a separate program for a given film configuration (bearing bore geometry, eccentricity and attitude angle) and relative motion (tangential and normal) between journal surface and bearing surface (Pinkus & Sternlicht, 1961). The eight spring-dampers together with the distributed mass-stiffness-damping of the rotor defines the threshold of instability. To constitute the state space equation the rotor is discretized into 194 sections (Fig. 12) according to a concentrated mass treatment which can be found in rotor-bearing system

Each section (Fig. 13) consists of a field of length *l* with stiffness but without mass and a

The forces and moments applied on both side of a field and the related deformations are

The angular displacements and inertia monents of a station are described in Fig. 16. All of the inputs (forces and moments) apply only on the station. They make a balance between the forces and moments appling by the fields (right and left) and the inertia forces and moments. If there is a bearing attached to a section then the station is looked like supported by a linearized spring-damper with four direct stiffness and damping coefficients *kxx, kyy, dxx, dyy* and four cross stiffness and damping coefficients *kxy, kyx, dxy, dyx* as shown in Fig. 17. The cross stiffness and damping coefficients show an important difference between the

dynamics (Glienicke, 1972) and its detail is omitted in the example.

Fig. 12. A discretized model of the rotor

station with mass, inertia moment but without length.

Fig. 13. A section of the rotor with a field and a station

shown in Fig. 14 and Fig. 15.

Fig. 14. The forces and moments on a field

Fig. 15. The lateral deformation of a field

hydrodynamic film and isotropic solid material. The hydrodynamic film then plays the role of a component of the system. It should be emphsized that the height of the journal center is determined by the sum of the height of bearing center controlled by pedestal and the project of ecentricity *e* of the journal center on ordinate axis while the load on each bearing is determined by the journal height under a static inderminate condition.

Fig. 16. Angular displacements and inertia moments of a station in X-Z and Y-Z plane

Fig. 17. A linearized model of the hydrodynamic film

Theory of Tribo-Systems 25

solution. When *ai* takes a negative value the amplitude of vibration will increase with time and the solution is then instable. Only when it is positive the solution can be stable.

Back to formula (14), if the input vector [*px, py, Mx, Mk, Nk*]*T* is constant, most structure parameters are constant in a short period of observation except the eight stiffness and damping coefficients which are defined by the relative motion (the rotating speed of the rotor) and the load on the bearing. Under a given elevation distribution the change of

Therefore *ai*= 0 is a condition of threshold of instability of the system.

system damping can be expressed in another form, the logarithmic decrement

such phenomena. Many efforts have been given to understand it (Li, 2001).

Logarithmic Decrement

**5. Conclusion** 

 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6

Δ= 2π*ai*/*bi*

Figure 18 gives two logarithmic decrement curves versus rotor rotating speed. The intersection point of each curve and abscissa (Δ= 0) gives a margin of threshold of instability with related elevation distribution. The turbo-generator set in power plant must work under a speed of 3000 rpm. In Fig. 18 one can find that at a speed of 3000 rpm, before and after the change of elevation of 4# bearing (decreasing a value of 0.15 mm) and 7# bearing (increasing a value of 0.7 mm) the logarithmic decrement changes from 0.95 to - 0.05. It implies that the change makes the system becoming not stable. Some turbo-generator set works normally in full output but during low output in middle night a half frequency vibration component emerges. Elevation distribution change might be an important cause of

> After Elevation Change on 4# and 7# Bearing Before Elevation Change on 4# and 7# Bearing

2000 2500 3000 3500 4000 4500 5000

Fig. 18. Logarithmic decrement versus rotating speed for two different elevation distributions

The problems with tribology are problems of systems science and systems engineering. In a sense, without system there would be no tribology. A machine system is consisted of a

Rotating Speed (r/min)

Another form of formula (4) for one section, for example for section *j*, can be written as

3 2 3 2 2 0 0 0 0 0 0 0 0 000 0 0 00 0 0 0 0 0 00 <sup>12</sup> <sup>6</sup> 0 0 12 6 0 0 6 2 0 0 <sup>6</sup> <sup>0</sup> *xx xy xx xy yx yy yx yy x <sup>y</sup> <sup>j</sup> j j j j mx d d x x k k my d d y kk y J J J J EJ EJ l l EJ EJ l l EJ EJ l l* θ θ ϕ ω ϕ ϕ ψ ω ψ ψ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + + ⎢ ⎥ ⎢ ⎥ <sup>−</sup> ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ − − + − − 3 2 3 2 2 1 2 2 1 1 3 2 3 2 2 2 <sup>12</sup> <sup>6</sup> 0 0 12 6 0 0 6 4 0 0 2 6 <sup>4</sup> 0 00 <sup>12</sup> <sup>6</sup> 0 0 <sup>12</sup> <sup>6</sup> 0 0 6 4 0 0 6 4 0 0 *j j j j EJ EJ l l x x EJ EJ y y l l EJ EJ l l EJ EJ EJ EJ l l l l EJ EJ l l EJ EJ l l EJ EJ l l EJ EJ l* ϕ ϕ ψ ψ + + + ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ <sup>+</sup> ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ − − + − − 3 2 3 2 2 1 2 <sup>12</sup> <sup>6</sup> 0 0 <sup>12</sup> <sup>6</sup> 0 0 6 2 0 0 6 2 0 0 *x y k j j k j j j EJ EJ l l x x EJ EJ P y y l l P EJ EJ M l l N EJ EJ l l l* ϕ ϕ ψ ψ − ⎡ ⎤ ⎡ ⎤ − − ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ − − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (14)

where *E* is the Young's module of the rotor material and *J* is the area moment inertia, other parameters can be found in Fig. 12 to 17. The state space equation for the rotor bearing system can be obtained by assembling formula (14) for *j*=1 to *j*=n with free boundary condition at the two terminal ends. The assembled result formula will not be presented in the example.

A question arises that how the change of elevation distribution influences the threshold of instability of the system? It can be transformed into an eigenvalue problem. In general the solutions of equation are as follows

$$\begin{aligned} \mathbf{x}\_i &= \mathbf{x}\_{0i} e^{\nu\_i t} = \mathbf{x}\_{0i} e^{-a\_i t} e^{j b\_i t} \\ \mathbf{y}\_i &= \mathbf{y}\_{0i} e^{-a\_i t} e^{j b\_i t} \\ \mathbf{q}\_i &= \mathbf{q}\_{0i} e^{-a\_i t} e^{j b\_i t} \\ \mathbf{y}\_i &= \mathbf{y}\_{0i} e^{-a\_i t} e^{j b\_i t}, i = \mathbf{1} \sim \mathcal{N}. \end{aligned} \tag{15}$$

*N* is defined by the practical requirement and the computational facility. Only some interesting solutions should be paid attention to, for example the solution *i* in this discussion to explain the tribological behavior. In formula (15) the item *jb ti e* , the virtual part of the solution where *j* = −1 , gives *bi* which is the frequency of vibration (oil whirl). Meanwhile the item *<sup>i</sup> a t e* <sup>−</sup> , the real part of the solution, gives *ai* which is the system damping of the system and predicts a speed of changing the amplitude of vibration concerning with the

 ϕ

 ψ

3 2

3 2

*x x EJ EJ P y y l l P*

*j j k j*

*EJ EJ M l l N EJ EJ*

 ϕ

 ψ

1

(15)

*e* , the virtual part of the

 ϕ

 ψ− *x y* (14)

*k*

⎢ ⎥ ⎣ ⎦

2

2 6 <sup>4</sup> 0 00

2

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ − − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

1

*EJ EJ EJ EJ l l l l*

3 2

*EJ EJ l l x x EJ EJ y y l l EJ EJ l l*

<sup>12</sup> <sup>6</sup> 0 0

12 6 0 0

6 4 0 0

⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ <sup>+</sup> ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

+

<sup>12</sup> <sup>6</sup> 0 0

<sup>12</sup> <sup>6</sup> 0 0

6 2 0 0

6 2 0 0

,

, 1~ .

<sup>−</sup> , the real part of the solution, gives *ai* which is the system damping of the

*j j*

1 1

3 2

*EJ EJ l l*

2

*j j*

*l l l*

where *E* is the Young's module of the rotor material and *J* is the area moment inertia, other parameters can be found in Fig. 12 to 17. The state space equation for the rotor bearing system can be obtained by assembling formula (14) for *j*=1 to *j*=n with free boundary condition at the two terminal ends. The assembled result formula will not be presented in

A question arises that how the change of elevation distribution influences the threshold of instability of the system? It can be transformed into an eigenvalue problem. In general the

> , ,

*ee i N*

*i i i*

*t at jb t*

−

0 0

*e e*

*x xe xe e*

*ii i a t jb t*

= =

− − −

ν

*ij i i i i i*

*a t jb t*

*a t jb t*

*N* is defined by the practical requirement and the computational facility. Only some interesting solutions should be paid attention to, for example the solution *i* in this discussion

solution where *j* = −1 , gives *bi* which is the frequency of vibration (oil whirl). Meanwhile

system and predicts a speed of changing the amplitude of vibration concerning with the

= =

0 0 0

*y ye e*

*i i*

= =

*i i*

ϕ ϕ

ψ ψ

to explain the tribological behavior. In formula (15) the item *jb ti*

*i i*

*j j*

+ +

Another form of formula (4) for one section, for example for section *j*, can be written as

0 0 0 0 0 0 0 0

000 0 0 00 0 0 0 0 0 00

ϕ

ψ

⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥

2 2

ϕ

ψ

⎣ ⎦ ⎣ ⎦

⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ − − ⎢ ⎥ ⎢ ⎥

ϕ

ψ

*<sup>y</sup> <sup>j</sup> j j j j*

*xx xy xx xy yx yy yx yy*

θ

⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

ω

*mx d d x x k k my d d y kk y*

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + + ⎢ ⎥ ⎢ ⎥ <sup>−</sup> ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

3 2

*EJ EJ l l*

*J J*

*J J*

−

<sup>12</sup> <sup>6</sup> 0 0

12 6 0 0

6 2 0 0

*EJ EJ l l*

2

2

the example.

the item *<sup>i</sup> a t e*

*x*

ϕ

ψ

−

+ −

+ − <sup>6</sup> <sup>0</sup>

−

3 2

*EJ EJ l l*

<sup>12</sup> <sup>6</sup> 0 0

−

<sup>12</sup> <sup>6</sup> 0 0

6 4 0 0

6 4 0 0

solutions of equation are as follows

2

*l*

−

*EJ EJ l l*

3 2

3 2

*EJ EJ*

*EJ EJ l l*

−

θ

 

ω

*EJ EJ l l*

solution. When *ai* takes a negative value the amplitude of vibration will increase with time and the solution is then instable. Only when it is positive the solution can be stable. Therefore *ai*= 0 is a condition of threshold of instability of the system.

Back to formula (14), if the input vector [*px, py, Mx, Mk, Nk*]*T* is constant, most structure parameters are constant in a short period of observation except the eight stiffness and damping coefficients which are defined by the relative motion (the rotating speed of the rotor) and the load on the bearing. Under a given elevation distribution the change of system damping can be expressed in another form, the logarithmic decrement

#### Δ= 2π*ai*/*bi*

Figure 18 gives two logarithmic decrement curves versus rotor rotating speed. The intersection point of each curve and abscissa (Δ= 0) gives a margin of threshold of instability with related elevation distribution. The turbo-generator set in power plant must work under a speed of 3000 rpm. In Fig. 18 one can find that at a speed of 3000 rpm, before and after the change of elevation of 4# bearing (decreasing a value of 0.15 mm) and 7# bearing (increasing a value of 0.7 mm) the logarithmic decrement changes from 0.95 to - 0.05. It implies that the change makes the system becoming not stable. Some turbo-generator set works normally in full output but during low output in middle night a half frequency vibration component emerges. Elevation distribution change might be an important cause of such phenomena. Many efforts have been given to understand it (Li, 2001).

Fig. 18. Logarithmic decrement versus rotating speed for two different elevation distributions

#### **5. Conclusion**

The problems with tribology are problems of systems science and systems engineering. In a sense, without system there would be no tribology. A machine system is consisted of a

Theory of Tribo-Systems 27

*mm m P PIS PIN* = +

2 cos sin tan

2 cos sin tan

<sup>⎡</sup> <sup>⎤</sup> ′ = −− <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

2 cos sin 2 tan cos cos

*l l* <sup>⎡</sup> <sup>⎤</sup> ⎛ ⎞ ′′ <sup>=</sup> <sup>⎢</sup> <sup>−</sup> <sup>⎥</sup> ⎜ ⎟ ⎢⎝ ⎠ <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup> θ

ϕ

ϕ

<sup>⎡</sup> <sup>⎤</sup> = −− <sup>⎢</sup> <sup>⎥</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

θ

> θ

> > ϕ

( )<sup>2</sup> 3 cos

*W rr*

ϕ

θ

ϕ

*r r <sup>W</sup>*

ϕ

*Wr r* 3 cos tan sin = ( θ

*W jr* 3 cos tan sin ′ = θ

( ) <sup>2</sup> <sup>4</sup> 2 tan 1 sin 2 tan

ϕ

2 2 cos 2 2 <sup>2</sup> 2 2 3 1 cos 3

θ

θϕ

θ θ

 θϕ

*IW m m W W j r <sup>j</sup> <sup>j</sup> <sup>W</sup>*

⎛ ⎞ ′′ = + +− + ′ ⎜ ⎟ ⎝ ⎠ ϕ

( ) ( ) <sup>2</sup> cos <sup>4</sup> 3 tan 1 cos 3tan

*<sup>r</sup> I I mhr I mW m r j W <sup>l</sup>* ⎛ ⎞ <sup>⎡</sup> <sup>⎤</sup> =+ + + + − + ′ ⎜ ⎟ ⎢⎣ ⎥⎦ ⎝ ⎠

> θ

 ϕ

( )

*P R R*

*g gr m* ( ) = ⎡ *PR C* (cos tan sin cos tan sin sin −+ −+ ) *m j* ( ) *m h* ⎤ ⎣ ⎦

2 3 2 2 1 sin cos 2 3 2

( ) ( ) <sup>5</sup> 2 *I*

θ

*I* ′ = − θ

 ϕ

*mW W mr j mW W*

+ −− + ′ ′

2 2

⎛ ⎞ ′ ′ = + −− + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

*Ir m m W W <sup>j</sup> <sup>r</sup> <sup>j</sup> <sup>j</sup> <sup>W</sup>*

θ

cos *r*

( )<sup>2</sup> 3 cos

cos *j r W rjr*

*RR R PP P*

*R RR P PP*

( ) ( ) 2

ϕ

⎛ ⎞⎛ ⎞ ⎛ ⎞ ′ <sup>=</sup> ⎜ ⎟⎜ ⎟ ⎜ ⎟ <sup>−</sup> ⎝ ⎠⎝ ⎠ ⎝ ⎠

cos cos 2 tan tan cos cos

> θ

) = −− ( ( ) *SK RN* ) ( os

*W*

*Qt Qt F F r* ( , c

θ

*m mm l*

cos *CC R P R*

2

*l l*

ϕ

θϕ

θ

*ml m m*

*l*

cos

cos

*R*

*r r I I*

θ

θ

θ ( )

θ

ϕ

θ

ϕ

*l*

(( ) ) <sup>2</sup> <sup>2</sup> <sup>1</sup> *W I m CC C* =+ − + *PIS PIS B A P* (1A)

θ ϕ

> θ ϕ

θ

ϕ

 θ

> θ

*W m CC* 1′ = − *PIS B A* ( ) (2A)

(3A)

(4A)

(5A)

 ϕ

> ϕ

 θ

tan − sin ) (13A)

(14A)

(8A)

(9A)

(10A)

(11A)

(12A)

− ) (6A)

− *r* (7A)

 θ

θ

θ

 θ

> θ

Following parameters are used for short in further discussion

component system and a tribo-system from the view point of motion. The tribo-system is consisted of tribo-elements and some supporting auxiliary sub-systems abstracted from a machine system for studying behaviors on or between the interacting surfaces in relative motion, results of the behaviors and technology related to. The tribo-system together with the component system plays a motion guarantee function which keeps each part of the machine system with a definite motion. Tribology science and technology is very important in obtaining the best way (theory and application) to complete the motion guarantee function of tribo-systems.

Tribological behaviors are system dependent. The property of tribo-elements and then the systems containing tribo-elements are time dependent. The results of tribological behaviors are the results of mutual action and strong coupling of many behaviors of other disciplines under a tribological condition consisted of interacting surfaces in relating motion.

A state space method which is a combination of general systems theory with engineering systems analysis can be successfully applied to simulate the behaviors. Two examples are given to show how the system structure can be connected with the system behaviors via the state space method. With the state space method the structure is a carrier in realizing the mutual action and coupling. The structure can have a recoverable change and an irrecoverable change while the behaviors can be repeatable and unrepeatable in the simulation.

#### **6. Acknowledgment**

This study is supported by the National Science Foundation of China in a long period especially the key item 50935004/E05067. The author wishes to thank Professor H. Xiao for his kind help on proofreading the whole chapter, Dr. Z. S. Zhang on having the calculation results of the example 2, Dr. Z. N. Zhang on preparing the manuscript and Professor J. Mao, she read the first draft and pointed out some mistakes.

#### **Appendix: Derivation of elements in the state space and output equations in example 1**

In this example, the study will focus mainly on the skirt – bore tribo-pair of a cylinder – piston – conrod – crank system of an internal combustion engine.

As shown in Fig. 6 and in the following formulas, symbols *Q* – gas pressure on the top of piston, *F* – force or friction force, S –thrust force in total on piston skirt, *T* – torque moment load on the crankshaft, *t* – time, *m* – mass of a component, *I* – inertial moment of a component, *P* – center of small end pair of conrod, *A* – center of big end pair of conrod, *O* – center of crankshaft pair on casing, *C* – center of mass of piston assembly, *R* – center of mass of conrod, *CR* – center of mass of crankshaft, *X,Y* – coordinate directions, *PIS* – piston, *PIN* – wrist pin, *SK* – skirt, *RN* – piston ring package, *R* – conrod respectively and *l* — length of conrod, *r* — length of crank, *jl* – distance from *A* to *R*, *hr* – distance from *O* to *CR*.

Suppose that the influence of secondary motion of piston on the motion and equilibrium of conrod and crankshaft can be neglected. The following formulas yield the geometry and motion relationship between the conrod and crankshaft:

$$l\sin\phi = r\sin\theta,\ \sin\phi = \frac{r}{l}\sin\theta,\ \dot{\phi} = \frac{\dot{\theta}r\cos\theta}{l\cos\phi},\ \ddot{\phi} = \dot{\theta}^2 \left[ \left(\frac{r\cos\theta}{l\cos\phi}\right)^2 \tan\phi - \frac{r\sin\theta}{l\cos\phi} \right] + \ddot{\theta}\frac{r\cos\theta}{l\cos\phi}$$

component system and a tribo-system from the view point of motion. The tribo-system is consisted of tribo-elements and some supporting auxiliary sub-systems abstracted from a machine system for studying behaviors on or between the interacting surfaces in relative motion, results of the behaviors and technology related to. The tribo-system together with the component system plays a motion guarantee function which keeps each part of the machine system with a definite motion. Tribology science and technology is very important in obtaining the best way (theory and application) to complete the motion guarantee

Tribological behaviors are system dependent. The property of tribo-elements and then the systems containing tribo-elements are time dependent. The results of tribological behaviors are the results of mutual action and strong coupling of many behaviors of other disciplines

A state space method which is a combination of general systems theory with engineering systems analysis can be successfully applied to simulate the behaviors. Two examples are given to show how the system structure can be connected with the system behaviors via the state space method. With the state space method the structure is a carrier in realizing the mutual action and coupling. The structure can have a recoverable change and an irrecoverable

This study is supported by the National Science Foundation of China in a long period especially the key item 50935004/E05067. The author wishes to thank Professor H. Xiao for his kind help on proofreading the whole chapter, Dr. Z. S. Zhang on having the calculation results of the example 2, Dr. Z. N. Zhang on preparing the manuscript and Professor J. Mao,

**Appendix: Derivation of elements in the state space and output equations in** 

In this example, the study will focus mainly on the skirt – bore tribo-pair of a cylinder –

As shown in Fig. 6 and in the following formulas, symbols *Q* – gas pressure on the top of piston, *F* – force or friction force, S –thrust force in total on piston skirt, *T* – torque moment load on the crankshaft, *t* – time, *m* – mass of a component, *I* – inertial moment of a component, *P* – center of small end pair of conrod, *A* – center of big end pair of conrod, *O* – center of crankshaft pair on casing, *C* – center of mass of piston assembly, *R* – center of mass of conrod, *CR* – center of mass of crankshaft, *X,Y* – coordinate directions, *PIS* – piston, *PIN* – wrist pin, *SK* – skirt, *RN* – piston ring package, *R* – conrod respectively and *l* — length of

Suppose that the influence of secondary motion of piston on the motion and equilibrium of conrod and crankshaft can be neglected. The following formulas yield the geometry and

> θ

<sup>=</sup> <sup>2</sup>

φ θ

<sup>2</sup> cos sin cos tan

φ

⎡ ⎤ ⎛ ⎞ = −+ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦ 

θ

φ

cos cos cos *r rr l ll*

θ

φ

 θ  θ

 φ

φ

conrod, *r* — length of crank, *jl* – distance from *A* to *R*, *hr* – distance from *O* to *CR*.

 cos , cos *r l* θ

under a tribological condition consisted of interacting surfaces in relating motion.

change while the behaviors can be repeatable and unrepeatable in the simulation.

she read the first draft and pointed out some mistakes.

motion relationship between the conrod and crankshaft:

θ

φ

 sin sin , *<sup>r</sup> l* φ=

piston – conrod – crank system of an internal combustion engine.

function of tribo-systems.

**6. Acknowledgment** 

**example 1** 

*l r* sin sin , φ=

θ

Following parameters are used for short in further discussion

$$
\boldsymbol{m}\_P = \boldsymbol{m}\_{\rm PIS} + \boldsymbol{m}\_{\rm PIN}
$$

$$
\boldsymbol{W1} = \boldsymbol{I}\_{\rm PIS} + \boldsymbol{m}\_{\rm PIS} \left( \left( \mathbf{C}\_B - \mathbf{C}\_A \right)^2 + \mathbf{C}\_P \right.\tag{1A}
$$

$$\mathcal{W}\mathbf{1}' = m\_{\mathrm{PLS}} \left( \mathbb{C}\_{\mathcal{B}} - \mathbb{C}\_{A} \right) \tag{2A}$$

$$\mathcal{W}\mathbf{2} = \left[\frac{\left(r\cos\theta\right)^2}{l\cos^3\theta} - r\cos\theta - r\sin\theta\tan\varphi\right] \tag{3A}$$

$$\mathcal{W}\mathcal{Z}' = \left[\frac{j\left(r\cos\theta\right)^2}{l\cos^3\varphi} - r\cos\theta - jr\sin\theta\tan\varphi\right] \tag{4A}$$

$$\mathcal{W}\mathfrak{V}^{\bullet} = \left[ \left( \frac{r\cos\theta}{l\cos\varphi} \right)^{2} \tan\varphi - \frac{r\sin\theta}{l\cos\varphi} \right] \tag{5A}$$

$$\mathcal{W}\mathfrak{F} = \left(r\cos\theta\tan\varphi - r\sin\theta\right) \tag{6A}$$

$$\mathcal{W}\mathcal{Y} = \dot{\mathcal{y}}r\cos\theta\tan\varphi - r\sin\theta\tag{7A}$$

$$\mathcal{W}\mathbf{4} = \frac{I\_R}{m\_P} \left(\frac{\mathcal{W}\mathbf{2}''}{l\cos\varphi}\right) + \frac{m\_R}{m\_P}\mathcal{W}\mathbf{2}'j\tan\varphi + \frac{m\_R}{m\_P}r(\mathbf{1} - j)j\sin\theta + \mathcal{W}\mathbf{2}\tan\varphi\tag{8A}$$

$$\mathcal{W}\mathbf{4}' = \frac{I\_R}{m\_P} \left( \frac{r \cos \theta}{\left(l \cos \phi\right)^2} \right) + \frac{m\_R}{m\_P} \mathcal{W}\mathbf{3}' j \tan \phi - \frac{m\_R}{m\_P} r \left(1 - j\right) j \cos \theta + \mathcal{W}\mathbf{3} \tan \phi \tag{9A}$$

$$I(\theta) = I\_C + m\_C h^2 r^2 + I\_R \left(\frac{r\cos\theta}{l\cos\varphi}\right)^2 + m\_P \mathcal{W} \mathcal{S}^2 + m\_R \left[r^2 \left(1 - j\right)^2 \cos^2\theta + \mathcal{W} \mathcal{S}^2\right] \tag{10A}$$

$$\begin{split} I'(\theta) &= 2I\_R \left( \frac{r \cos \theta}{l \cos \varphi} \right)^2 \left( \left( \frac{r \cos \theta}{l \cos \varphi} \right) \tan \varphi - \tan \theta \right) \\ &+ 2m\_P \mathcal{W} 3 \mathcal{W} 2 - 2m\_R r^2 \left( 1 - j \right)^2 \sin \theta \cos \theta + 2m\_R \mathcal{W} 3 \mathcal{W} \mathcal{W} 2' \end{split} \tag{11A}$$

$$\log\left(\theta\right) = \operatorname{gr}\left[m\_{\mathbb{P}}\left(\cos\theta\tan\varphi - \sin\theta\right) + m\_{\mathbb{R}}\left(j\cos\theta\tan\varphi - \sin\theta\right) + m\_{\mathbb{C}}l\sin\theta\right] \tag{12A}$$

$$Q(t,\theta) = \left(Q(t) - F\_{\rm SK} - F\_{\rm RN}\right)r\left(\cos\theta\tan\varphi - \sin\theta\right) \tag{13A}$$

$$\mathcal{W}\mathfrak{F} = -\frac{\mathcal{I}'(\theta)}{2\mathcal{I}(\theta)}\tag{14A}$$

Theory of Tribo-Systems 29

The equilibrium equations for the piston assembly, conrod and crankshaft can be written as

0, ( ) <sup>0</sup> <sup>Σ</sup>*F F F F Q t gm Y m PY PY SK RN* = ++ − − − = *P PP* (23A)

0, <sup>0</sup> <sup>∑</sup>*F Y m gm F F RY R R R PY AY* =− − − + = (26A)

0, <sup>0</sup> <sup>∑</sup>*F F F Xm CX OX AX C C* = −− = (28A)

0, <sup>0</sup> <sup>∑</sup>*F F F gm Y m CY OY AY C C C* = −− − = (29A)

θθ*C AX* (1 cos 1 sin cos sin 0 ) *AY* ( ) *OX*

Considering that the study focuses mainly on the piston skirt – cylinder bore tribo-pair, parameters relative to the motion of the piston and the parameters concerning with motion

<sup>2</sup> 4 4 *X W W FY <sup>P</sup>* =− − +

<sup>2</sup> 41 2 4 1 3 1

( ) <sup>2</sup> 4 54 5 4 *X W W W W W FY <sup>P</sup>* =− + − +

θ

2

θ θ

Inputting formula (33A) into formulas (31A) and (32A) yield

θ

*W W mWC W W mWC PIS P PIS P FY W M W WW*

⋅− ⋅ ⋅− ⋅ ′ ′ ′ ⋅ +′ = − <sup>−</sup> <sup>+</sup> (32A)

1 11

 θ

∑ =− ++ + + + − − = *M I T F h r F h r F hr F hr <sup>C</sup>* 0,

∑ =− − − − − − − = *M I F j l F j l F jl F jl <sup>R</sup>* 0,

 ϕ

condition input will be selected in the state vector , , , , , *<sup>T</sup>*

(1A) - (21A) and equilibrium conditions (22A) - (30A) into formula (22A) yield

θ

*P P*

*m m*

0, 1 1 0 Σ*M M XW Ym C W P PP* =+ + − = ′ *PIS <sup>P</sup>*

ϕ

θ

Similarly yield

β θ

follows

*<sup>S</sup> Q t F F gm gm j FY*

−+ ++ = −

( ) ( ) tan *SK RN P R*

ϕ

*FY Q t F F* ′ = −+ ( ) ( *SK RN* ) (21A)

0, <sup>0</sup> <sup>Σ</sup>*F F S Xm PX PX P P* = +− = (22A)

(24A)

θ*OY*

 βθ<sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> , i.e.. Inputting

ϕϕ

θ

β

 ϕ

(30A)

*R BX* (1 cos 1 sin cos sin 0 ) *BY* ( ) *AX AY* (27A)

0, <sup>0</sup> <sup>∑</sup>*F Xm F F RX R R PX AX* =− − + = (25A)

*XX X* = ⎡ *P P* β θ

′ (31A)

=⋅ + *W W* 5 5′ (33A)

′ ′′ (34A)

(20A)

$$\mathcal{W}\mathfrak{F}' = -\frac{\mathcal{g}\left(\theta\right) + \mathcal{Q}\left(t, \theta\right) - T}{I\left(\theta\right)}\tag{15A}$$

$$\mathbf{W}\mathbf{\hat{G}} = -m\_{\mathbb{C}} \cdot \mathbf{h} \cdot \mathbf{r} \sin\theta + m\_{\mathbb{R}} \cdot \mathbf{r} \begin{pmatrix} 1 - j \end{pmatrix} \sin\theta - m\_{\mathbb{P}} \cdot \mathbf{W}\mathbf{4} \tag{16A}$$

$$\mathbf{V}\mathbf{\hat{G}}' = m\_{\mathbf{C}} \cdot \mathbf{h} \cdot r \cos \theta - m\_{\mathbf{R}} \cdot r \begin{pmatrix} 1 - j \end{pmatrix} \cos \theta - m\_{\mathbf{P}} \cdot \mathbf{V}\mathbf{4}' \tag{17A}$$

$$\text{V}\heartsuit = m\_{\text{C}} \cdot \text{h} \cdot r \cos \theta + m\_{\text{R}} \cdot \text{V}\heartsuit 2' + m\_{\text{P}} \cdot \text{V}\heartsuit\text{2} \tag{18A}$$

$$\mathbf{V}\mathbf{V}\mathbf{7}' = \mathbf{m}\_{\mathbb{C}} \cdot \mathbf{h} \cdot \mathbf{r} \sin\theta + \mathbf{m}\_{\mathbb{R}} \cdot \mathbf{V}\mathbf{3}' + \mathbf{m}\_{\mathbb{P}} \cdot \mathbf{V}\mathbf{3} \tag{19A}$$

The displacements, the first derivatives and second derivatives of displacements of points *P, R* and *O* are given as follows

$$\begin{aligned} Y\_P &= r \cos \theta - l \cos \phi \\ \dot{Y}\_P &= -\dot{\theta} \cdot r (\sin \theta - \cos \theta \tan \phi) \\ \ddot{Y}\_P &= \dot{\theta}^2 \cdot \mathcal{W} \mathbf{2} + \ddot{\theta} \cdot \mathcal{W} \mathbf{3} \end{aligned}$$

, *X X P P* and *XP* will be given later because they need the values of secondary motion of piston which have to be obtained from the equations of equilibrium.

$$\begin{aligned} X\_R &= -r \sin \theta \cdot (1 - j) \\ Y\_R &= r \cos \theta - l \cdot j \cos \phi \end{aligned}$$

$$\begin{aligned} \dot{X}\_R &= -\dot{\theta} \cdot r (1 - j) \cos \phi \\ \dot{Y}\_R &= -\dot{\theta} (r \sin \theta - j \cdot r \cos \theta \tan \phi) \end{aligned}$$

$$\begin{aligned} \ddot{X}\_R &= \dot{\theta}^2 \cdot r (1 - j) \sin \theta - \ddot{\theta} \cdot r (1 - j) \cos \theta \\ \ddot{Y}\_R &= \dot{\theta}^2 \cdot W \mathcal{D}' + \ddot{\theta} \cdot W \mathcal{S}' \end{aligned}$$

For

$$\begin{aligned} X\_C &= h \cdot r \sin \theta \\ Y\_C &= -h \cdot r \cos \theta \end{aligned}$$

*R*

Then

$$\begin{aligned} \dot{X}\_{\mathbb{C}} &= \dot{\theta} \cdot h \cdot r \cos \theta \\ \dot{Y}\_{\mathbb{C}} &= \dot{\theta} \cdot h \cdot r \sin \theta \end{aligned}$$

$$\begin{aligned} \ddot{X}\_{\mathbb{C}} &= -\dot{\theta}^2 \cdot h \cdot r \sin\theta + \ddot{\theta} \cdot h \cdot r \cos\theta \\ \ddot{Y}\_{\mathbb{C}} &= \dot{\theta}^2 \cdot h \cdot r \cos\theta + \ddot{\theta} \cdot h \cdot r \sin\theta \end{aligned}$$

In the equilibrium analysis of the piston, conrod and crankshaft two other parameters are used for short also

, <sup>5</sup> *g Qt T <sup>W</sup> I* <sup>+</sup> <sup>−</sup> ′ = − θ

*W m hr m r* 6 =− ⋅ ⋅ + ⋅ − − ⋅ *CR P* sin 1 sin 4

*W m hr m r j m W* 6 cos 1 cos 4 ′ = ⋅⋅ − ⋅ − − ⋅ *CR P*

7 cos 2 2 *W m hr m W m W C RP* = ⋅⋅ + ⋅ + ⋅ θ

7 sin 3 3 *W m hr m W m W C RP* ′ = ⋅⋅ + ⋅ + ⋅ θ

The displacements, the first derivatives and second derivatives of displacements of points *P,* 

θ

 θ

, *X X P P* and *XP* will be given later because they need the values of secondary motion of

*Xr j Y r lj* =− ⋅ − = −⋅

( )

θ

2 3

*X hr Y hr* = ⋅ =− ⋅

*X hr Y hr* =⋅⋅ =⋅⋅

θ

*X hr hr Y hr hr* =− ⋅ ⋅ + ⋅ ⋅ = ⋅⋅ + ⋅⋅

 θ

*C C*

*C C*

2 2

θ

*C C*

 θ

 θ

*X rj rj*

= ⋅ − −⋅ −

θ

1 cos

( ) ( ) <sup>2</sup>

cos cos

θ

( )

 θφ

sin cos tan 2 3

sin 1( ) cos cos

( )

 φ

1 sin 1 cos

θθ

sin cos

θ

cos sin

θ

θ

sin cos cos sin

 θ

 θ

θθ

θθ

In the equilibrium analysis of the piston, conrod and crankshaft two other parameters are

θ

sin cos tan

φ

 θφ

> θ

θ

 φ

θ

θ

2

θ

*R R*

*X rj Y r jr* =− ⋅ − =− − ⋅

θ

θ

*Y WW*

 θ

= ⋅ +⋅ ′ ′ 

*Y WW*

*Yr l Y r*

θ

= − =− ⋅ − = ⋅ +⋅

*P P P*

piston which have to be obtained from the equations of equilibrium.

*R R*

2

θ

*R R*

 

*R* and *O* are given as follows

For

Then

used for short also

( ) ( ) ( )

θ

 θ

> θ

 θ

(15A)

( *j*) *m W* (16A)

( ) ′ (17A)

′ (18A)

′ (19A)

$$FY = \frac{S}{m\_P} - \frac{Q\left(t\right) - \left(F\_{SK} + F\_{\mathbb{RN}}\right) + gm\_P + gm\_R j}{m\_P} \tan \phi \tag{20A}$$

$$FY' = Q(t) - \left(F\_{SK} + F\_{RN}\right) \tag{21A}$$

The equilibrium equations for the piston assembly, conrod and crankshaft can be written as follows

$$
\Delta F\_{PX} = \mathbf{0}\_{\prime} F\_{PX} + \mathbf{S} - \ddot{X}\_P m\_P = \mathbf{0} \tag{22A}
$$

$$
\Sigma F\_{PY} = 0,\\
F\_{PY} + F\_{SK} + F\_{RN} - Q(t) - gm\_P - \dddot{Y}\_P m\_P = 0 \tag{23A}
$$

$$
\Sigma M\_P = 0,\\
M + \ddot{X}\_P \mathcal{W} \mathbf{1}' + \dddot{Y}\_P m\_{\text{PS}} \mathbf{C}\_P - \ddot{\mathcal{J}} \mathcal{W} \mathbf{1} = 0 \tag{24A}
$$

$$
\Sigma F\_{RX} = 0,\\
$$

$$
\Sigma F\_{RY} = \mathbf{0}\_{\prime} - \dddot{Y}\_R m\_R - g m\_R - F\_{PY} + F\_{AY} = \mathbf{0} \tag{26A}
$$

$$
\Sigma M\_R = 0,\\
$$

$$
\Sigma F\_{\rm CX} = 0,\\
F\_{\rm OX} - F\_{\rm AX} - \ddot{X}\_{\rm C}m\_{\rm C} = 0 \tag{28A}
$$

$$
\Sigma F\_{\rm CY} = 0,\\
F\_{\rmOY} - F\_{\rm AY} - g m\_{\rm C} - \ddot{Y}\_{\rm C} m\_{\rm C} = 0 \tag{29A}
$$

$$\sum M\_{\mathbb{C}} = 0,\\ -\ddot{\theta}I\_{\mathbb{C}} + T + F\_{AX} \left(1 + h\right)r\cos\theta + F\_{AY} \left(1 + h\right)r\sin\theta - F\_{\mathbb{O}X}lr\cos\theta - F\_{\mathbb{O}Y}lr\sin\theta = 0 \quad \text{(30A)}$$

Considering that the study focuses mainly on the piston skirt – cylinder bore tribo-pair, parameters relative to the motion of the piston and the parameters concerning with motion condition input will be selected in the state vector , , , , , *<sup>T</sup> XX X* = ⎡ *P P* β θ βθ <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> , i.e.. Inputting (1A) - (21A) and equilibrium conditions (22A) - (30A) into formula (22A) yield

$$
\ddot{X}\_p = -\dot{\theta}^2 \mathcal{W} \mathbf{4} - \ddot{\theta} \mathcal{W} \mathbf{4}' + FY \tag{31A}
$$

Similarly yield

$$\ddot{\beta} = -\dot{\theta}^2 \frac{\mathcal{W}\mathbf{4} \cdot \mathcal{W}\mathbf{1}\prime - m\_{\text{PLS}}\mathcal{W}\mathbf{2} \cdot \mathbb{C}\_p}{\mathcal{W}\mathbf{1}} - \ddot{\theta} \frac{\mathcal{W}\mathbf{4}\prime \cdot \mathcal{W}\mathbf{1}\prime - m\_{\text{PLS}}\mathcal{W}\mathbf{3} \cdot \mathbb{C}\_p}{\mathcal{W}\mathbf{1}} + \frac{\mathcal{F}\mathcal{Y} \cdot \mathcal{W}\mathbf{1}\prime + M}{\mathcal{W}\mathbf{1}} \tag{32A}$$

$$
\ddot{\theta} = \dot{\theta}^2 \cdot \mathcal{W}\mathbf{\tilde{5}} + \mathcal{W}\mathbf{\tilde{5}}'\tag{33A}
$$

Inputting formula (33A) into formulas (31A) and (32A) yield

$$\ddot{X}\_P = -\dot{\theta}^2 \left( \mathcal{W}\mathbf{4} + \mathcal{W}\mathbf{5}\mathcal{W}\mathbf{4}' \right) - \mathcal{W}\mathbf{5}'\mathcal{W}\mathbf{4}' + F\mathbf{Y} \tag{34A}$$

Theory of Tribo-Systems 31

( )( )

4 45 45 2 35 35

( )( )

( ) ( )

β

β

θ

θ

 θ

θ

 θ

*PRC*

 θ

θ

4 5 1 5 cos 2 35 2 35

=− + ⋅ + − − <sup>⎡</sup> ′ <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup>

*m W W FY m r j W S*

+− ⋅ − − − − ⎡ ⎤ ′ ′ ′ ⎣ ⎦ =⎡ + ⋅ + + ⋅ ⎤ ′ ⎣ ⎦ + ⋅ ++ ⋅ ++ ′ ′′ ′

( ) ( )( )

4 4 5 1 sin 5cos

 θ

θ

*Y*

θ

θ

θ

θ

θ

( ) ( )

*F m W W W m W W FY S F m W W W m W W g FY F m W W W mr j W*

=− + ⋅ − ⋅ − − ′ ′′ = + ⋅ + ⋅ ++ ′ ′

( ) ( )

*F mW W W mW W W*

6 56 5 6 7 57 5 7 *OX P*

*F W W W W W m FY S*

= + ⋅ + ⋅ +⋅− ′ ′′

*P R*

2 2 2

θ

 

θ

θ

*PX P P PY P P AX P R*

*AY P R P R*

( )

*AX AY OX OY*

2

2 2

θ

 

θ

Then the output matrix equation becomes

16 26

36

46

56 66

**7. References** 

θ

θ

θ

θ

θ

θ where

θ

( )( )

*mW W g mW W g F*

⎡ ⎤ ⎣ ⎦

*OY PRC*

*F C X F C X F C F C F C F C*

*PX P PY P*

<sup>⎡</sup> ⎤ ⎡ ′ ⎤⎡ ⎤ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ ′ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ <sup>⎢</sup> ⎥ ⎢ ′ ⎥⎢ ⎥ <sup>⎢</sup> ⎥ ⎢ <sup>=</sup> ⎥⎢ ⎥ ′ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ ′ <sup>⎢</sup> ⎥ ⎢ ⎥⎢ ⎥ ⎢⎣ ⎥ ⎢ ⎦ ⎣ ′ ⎥⎢ ⎥ ⎦⎣ ⎦

( ) ( )

 

( ) ( )( )

4 4 5 1 sin 5 cos

4 45 45 / 2 35 35 /

( ) ( )

( )( )

6 56 5 6 /

′ ′ = + ⋅ + ⋅ + + ++ ⎡ ⎤ ′ ′ ′ ⎣ ⎦

( ) ( )

Chen, P. (1982). *State Space Methods and Application*. Publishing House of Electronics

Czichos H. (1978). *Tribology: A Systems Approach to the Science and Technology of Friction,* 

Czichos H. The Principle of System Analysis and their Application to Tribology. *ASLE* 

7 57 5 7 /

*P*

35 3 5 /

( ) [ ]

*C W W W W W FY g m m m*

′ ′ = ⎡ + ⋅ + + ⋅ ⎤+ ′ ⎣ ⎦

4 5 1 5 cos / 2 35 2 3 5

′ ′ <sup>=</sup> ⎡− + ⋅ + − −⋅ +⎤ <sup>⎣</sup> <sup>⎦</sup>

′ ′′ =− + ⋅ −⎡ ⋅ − − ⎤ ′ ⎣ ⎦ ′ ′ = + ⋅ + ⋅ ++ ⎡ ⎤′ ⎣ ⎦

*C m W W W m W W FY S C m W W W m W W g FY*

*P P P P*

*P R*

*C m W W W mr j W m W W FY m r j W S C mW W W mW W W m W W g m W W g FY*

⎡ ⎤ − ⋅ − − −⋅ ⋅ − ′ ′ ′ ⎣ ⎦

( ) ( )

*P R*

*P R*

Industry, Beijing, China (In Chinese)

( )( )

*Lubrication and Wear*, ISBN 978-0444416766, Elsevier

*Trans*, Vol. 17, No. 4, (1974), pp. 300-306, ISSN 0569-8197

⎡ ⎤ ⋅ ++ ⋅ ++ ′ ′′ ′ ⎣ ⎦

*C W W W W W m FY S*

′ ′ == + ⋅ + ⋅ + ⋅ − ′ ′

*P R*

*F W W W W W FY m m m g*

= + ⋅ + ⋅ + + ++ ⋅ ′ ′′′

35 3 5

$$\begin{aligned} \ddot{\beta} &= -\dot{\theta}^2 \left[ \frac{W 4 W 1' - W 2 m\_{\text{PLS}} \text{C}\_P + W 5 \left( W 4' W 1' - W 3 m\_{\text{PS}} \text{C}\_P \right)}{W 1} \right] \\ &- \frac{W 5' \left( W 4' W 1' - W 3 m\_{\text{PS}} \text{C}\_P \right)}{W 1} + \frac{F Y W 1' + M}{W 1} \end{aligned} \tag{35A}$$

After reorganizing the state equation for the cylinder – piston – conrod – crank system can be derived as follows

$$
\begin{bmatrix} X\_P \\ \dot{X}\_P \\ \dot{\theta} \\ \dot{\theta} \\ \theta \\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & A\_{26} \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & A\_{46} \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & A\_{66} \end{bmatrix} \begin{bmatrix} X\_P \\ \dot{X}\_P \\ \dot{\theta} \\ \dot{\theta} \\ \theta \\ \theta \\ \theta \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}
$$

where

$$\begin{aligned} A\_{26} &= -\dot{\theta} \left( \mathcal{W} 4 + \mathcal{W} 5 \cdot \mathcal{W} 4' \right) \\ A\_{46} &= -\dot{\theta} \left[ \frac{\mathcal{W} 4 \cdot \mathcal{W} 1' - m\_{\text{IPS}} \mathcal{W} 2 \cdot \mathcal{C}\_P + \mathcal{W} 5 \left( \mathcal{W} 4' \cdot \mathcal{W} 1' - m\_{\text{PS}} \mathcal{W} 3 \cdot \mathcal{C}\_P \right)}{\mathcal{W} 1} \right] \\ A\_{66} &= \mathcal{W} \mathbf{5} \\ U\_2 &= -\mathcal{W} \mathbf{5}' \cdot \mathcal{W} \mathbf{4}' + F \mathbf{Y} \\ U\_4 &= -\frac{\mathcal{W} \mathbf{5}' \left( \mathcal{W} 4' \cdot \mathcal{W} 1' - m\_{\text{PS}} \mathcal{W} \mathbf{3} \cdot \mathcal{C}\_P \right)}{\mathcal{W} 1} + \frac{F \mathbf{Y} \cdot \mathcal{W} \mathbf{1'} + M}{\mathcal{W} 1} \\ U\_6 &= \mathcal{W} \mathbf{5'} \end{aligned}$$

When the behaviors of piston are interesting in study, the output equations can be written as

$$
\begin{bmatrix}
\dot{\theta} \\
 P\_{LOSS} \\
X\_P \\
\dot{\mathcal{B}} \\
F\_{RHT} \\
F\_{LFT}
\end{bmatrix} = \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & C\_{16} \\
0 & 0 & 0 & 0 & 0 & C\_{26} \\
0 & 0 & 0 & 0 & 0 & C\_{36} \\
0 & 0 & 0 & 0 & 0 & C\_{46} \\
0 & 0 & 0 & 0 & 0 & C\_{56} \\
0 & 0 & 0 & 0 & 0 & C\_{66} \\
\end{bmatrix} \begin{bmatrix}
X\_P \\
\dot{X}\_P \\
\dot{\theta} \\
\dot{\theta} \\
\dot{\theta} \\
\dot{\theta}
\end{bmatrix}
$$

where, 16 *C* = 1 and 26 36 46 56 66 *CCCCC* ,,,, concern the solution of Reynolds equation which governs the hydrodynamic lubrication behaviors between skirt and bore surfaces and cannot be presented explicitly. They will be computed numerically with a separate procedure before every integrating step from the value of elements in state vector obtained in last integrating.

If the forces transmitting in the pairs *P, A* and *O* are interesting the forces can be obtained with an equilibrium condition analysis for the piston on *P*, for the conrod on *A* and for the crankshaft on *O*. Replacing the first and second derivatives of displacements in formulas (22A) to (30A) and reordering yields

<sup>⎡</sup> ′ ′ −+ −′ <sup>⎤</sup> = − <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

After reorganizing the state equation for the cylinder – piston – conrod – crank system can

*X A X U*

′ ⎡⎤⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ <sup>=</sup> ⎥⎢ ⎥ ⎢ <sup>+</sup> ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎥⎢ ⎥⎢ ⎥ ⎢ <sup>⎥</sup> ⎢⎥⎢ ⎣⎦⎣ ⎦⎣ ⎦ ⎣ ⎥⎢ ⎥ ⎢ ⎦ ⎣⎥

 β

 β

 θ

 θ

41 2 54 1 3 1

<sup>⎡</sup> ⋅ − ⋅+ ⋅ − ⋅ ′ ′ ′ <sup>⎤</sup> = − <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

*W W mWC WW W mWC*

*W*

1 1

*W W*

When the behaviors of piston are interesting in study, the output equations can be written as

*P C X X C*

*F C F C*

*LOSS P*

<sup>⎡</sup> <sup>⎤</sup> <sup>⎡</sup> ⎤⎡ ⎤ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>=</sup> <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥ <sup>⎢</sup> ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎢⎣ ⎥⎢ ⎥ ⎦⎣ ⎦

where, 16 *C* = 1 and 26 36 46 56 66 *CCCCC* ,,,, concern the solution of Reynolds equation which governs the hydrodynamic lubrication behaviors between skirt and bore surfaces and cannot be presented explicitly. They will be computed numerically with a separate procedure before every integrating step from the value of elements in state vector obtained

If the forces transmitting in the pairs *P, A* and *O* are interesting the forces can be obtained with an equilibrium condition analysis for the piston on *P*, for the conrod on *A* and for the crankshaft on *O*. Replacing the first and second derivatives of displacements in formulas

01000 0 000000 0 00000 010000 00010 0 000000 0 00000 000100 00000 1 000000 0 00000 000001

1

*WW Wm C W W W Wm C W*

<sup>2</sup> 41 2 5 4 1 3

1 1

*PIS P*

*W W*

( )

*P P P P X X*

( )

*W W W mWC FY W M <sup>U</sup>*

′′′ ⋅− ⋅ ⋅ +′ = − <sup>+</sup>

541 3 1

*PIS P*

( )

=− + ⋅ ′

4 54

*P*

θ

*RHT LFT*

β

− +

β θ

β

β

θ

θ

26

46

*A*

66 2

*A W*

=

5

θ

θ

5

= ′

5 4

*U W W FY*

=− ⋅ + ′ ′

*A W WW*

4 6

in last integrating.

(22A) to (30A) and reordering yields

*U W*

be derived as follows

where

5 41 3 1

*W W W Wm C FYW M*

′ ′′ − ′ +

( )

(35A)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥

⎥ ⎢ ⎥ ⎢ ⎥⎦

*PIS P PIS P*

26 2

46 4

*A U*

66 6

( )

*A U*

*PIS P PIS P*

*C*

*C X*

*P*

β

β

θ

θ

$$\begin{aligned} F\_{PX} &= -\dot{\theta}^2 m\_P \Big( \mathcal{V} \mathcal{V} \mathcal{A} + \mathcal{V} \mathcal{V} \mathcal{V} \mathcal{S} \Big) - m\_P \Big( \mathcal{V} \mathcal{V} \mathcal{A} \cdot \mathcal{V} \mathcal{V} \mathcal{S} - F \mathcal{Y} \Big) - S \\ F\_{PY} &= \dot{\theta}^2 m\_P \Big( \mathcal{V} \mathcal{V} \mathcal{Z} + \mathcal{V} \mathcal{V} \mathcal{S} \cdot \mathcal{V} \mathcal{S} \Big) + m\_P \Big( \mathcal{V} \mathcal{V} \mathcal{S} \mathcal{V} \mathcal{S} + \mathcal{g} \Big) + F \mathcal{Y} \\ F\_{AX} &= \dot{\theta}^2 \Big[ -m\_P \Big( \mathcal{V} \mathcal{A} + \mathcal{V} \mathcal{U} \mathcal{V} \mathcal{V} \mathcal{S} \Big) + m\_R r \Big( \mathcal{I} - j \Big) \Big( \sin \theta - \mathcal{V} \mathcal{V} \mathcal{S} \cos \theta \Big) \Big] \\ &+ \Big[ -m\_P \Big( \mathcal{V} \mathcal{U} \mathcal{A}' \cdot \mathcal{V} \mathcal{V} \mathcal{S}' - F \mathcal{Y} \Big) - m\_R r \Big( \mathcal{I} - j \Big) \mathcal{V} \mathcal{V} \mathcal{S}' \cos \theta - S \Big] \\ F\_{AY} &= \dot{\theta}^2 \Big[ m\_P \Big( \mathcal{V} \mathcal{V} \mathcal{Z} + \mathcal{V} \mathcal{V} \mathcal{S} \mathcal{V} \mathcal{S} \Big) + m\_R \Big( \mathcal{V} \mathcal{V} \mathcal{L}' + \mathcal{V} \mathcal{V} \mathcal{S} \cdot \mathcal{V} \mathcal{S} \Big) \Big] \\ &+ \Big[ m\_P \Big( \mathcal{V} \mathcal{V} \mathcal{S} \mathcal{V} \mathcal{S}'$$

Then the output matrix equation becomes


where

$$\begin{aligned} C\_{16}' &= -\dot{\theta}m\_P \Big( \mathcal{W}\mathbf{4} + \mathcal{W}\mathbf{4}' \cdot \mathcal{W}\mathbf{5} \Big) - \Big[ m\_P \Big( \mathcal{W}\mathbf{4}' \cdot \mathcal{W}\mathbf{5}' - F\mathcal{Y} \Big) - \mathcal{S} \Big] / \dot{\theta} \\ C\_{26}' &= \dot{\theta}m\_P \Big( \mathcal{W}\mathbf{2} + \mathcal{W}\mathbf{3} \cdot \mathcal{W}\mathbf{5} \Big) + \Big[ m\_P \Big( \mathcal{W}\mathbf{3} \cdot \mathcal{W}\mathbf{5}' + \mathcal{g} \Big) + F\mathcal{Y}' \Big] / \dot{\theta} \end{aligned}$$

$$\begin{aligned} C\_{36}^{'} &= \dot{\theta} \left[ -m\_{P} \left( \mathcal{VV4} + \mathcal{VV4} \cdot \mathcal{VV5} \right) + m\_{R} r \left( 1 - j \right) \left( \sin \theta - \mathcal{VV5} \cdot \cos \theta \right) \right] + \\ \left[ -m\_{P} \left( \mathcal{VV4'} \cdot \mathcal{VV5'} - F \mathcal{Y} \right) - m\_{R} r \left( 1 - j \right) \cdot \mathcal{VV5'} \cdot \cos \theta - \mathcal{S} \right] / \dot{\theta} \\ C\_{46}^{'} &= \dot{\theta} \left[ m\_{P} \left( \mathcal{VV2} + \mathcal{VV3} \cdot \mathcal{VV5} \right) + m\_{R} \left( \mathcal{VV2'} + \mathcal{VV3'} \cdot \mathcal{VV5} \right) \right] + \\ \left[ m\_{P} \left( \mathcal{VV3} \cdot \mathcal{VV5'} + \mathcal{g} \right) + m\_{R} \left( \mathcal{VV3'} \cdot \mathcal{VV5'} + \mathcal{g} \right) + F \mathcal{Y'} \right] / \dot{\theta} \\ C\_{56}^{'} &= \dot{\theta} \left( \mathcal{VV6} + \mathcal{VV5} \cdot \mathcal{VV6'} \right) + \left[ \mathcal{VV5'} \cdot \mathcal{VV6'} + m\_{P} \cdot F \mathcal{Y} - \mathcal{S} \right] / \dot{\theta} \\ C\_{66}^{'} &= \dot{\theta} \left( \mathcal{VV7} + \mathcal{VV5} \cdot \mathcal{VV7'} \right) + \left[ \mathcal{VV5'} \cdot \mathcal{VV7'} + F \mathcal{Y'} + \mathcal{g} \left( m\_{P} + m\_{R} + m\_{\mathcal{C}} \right) \right] / \dot{\theta} \end{aligned}$$

#### **7. References**


**2** 

Jürgen Gegner

*Germany* 

*SKF GmbH, Department of Material Physics Institute of Material Science, University of Siegen* 

**Tribological Aspects of Rolling Bearing Failures** 

*Dedicated to Dipl.-Phys. Wolfgang Nierlich on the occasion of his 70th birthday* 

Rolling (element) bearings are referred to as *anti-friction* bearings due to the low friction and hence only slight energy loss they cause in service, especially compared to sliding or *friction* bearings. The minor wear occurring in proper operation superficially seems to suggest the question how rolling contact tribology should be of relevance to bearing failures. Satisfactorily proven throughout the 20th century primarily on small highly loaded ball bearings, the life prediction is actually based on material fatigue theories. Nonetheless, resulting subsurface spalling is usually called fatigue wear and therefore included in the discussion below. The influence of friction on the damage of rolling bearings, at first, is strikingly reflected, for instance, in foreign particle abrasion and smearing adhesion wear under improper running or lubrication conditions. On far less affected, visually intact raceways, however, temporary frictional forces can also initiate failure for common overall friction coefficients below 0.1. Larger size roller bearings with extended line contacts operating typically at low to moderate Hertzian pressure, generally speaking, are most susceptible to this surface loading. As large roller bearings are increasingly applied in the 21st century, e.g. in industrial gears, an attempt is made in the following to incorporate the rolling-sliding nature of the tribological contact into an extended bearing life model. By holding the established assumption that the stage of crack initiation still dominates the total lifetime, the consideration of the proposed competing normal stress hypothesis is deemed

The present chapter opens with a general introduction of the subsurface and (near-) surface failure mode of rolling bearings. Due to its particular importance to the identification of the damage mechanisms, the measuring procedure and the evaluation method of the material response analysis, which is based on an X-ray diffraction residual stress determination, are described in detail. In section 4, a metal physics model of classical subsurface rolling contact fatigue is outlined. Recent experimental findings are reported that support this mechanistic approach. The accelerating effect of absorbed hydrogen on rolling contact fatigue is also in agreement with the new model and verified by applying tools of material response analysis. It uncovers a remarkable impact of serious high-frequency electric current passage through bearings in operation, previously unnoticed in the literature. Section 5 provides an overview of state-of-the-art research on mechanical and chemical damage mechanisms by tribological

**1. Introduction** 

appropriate.


### **Tribological Aspects of Rolling Bearing Failures**

#### Jürgen Gegner

*SKF GmbH, Department of Material Physics Institute of Material Science, University of Siegen Germany* 

*Dedicated to Dipl.-Phys. Wolfgang Nierlich on the occasion of his 70th birthday* 

#### **1. Introduction**

32 Tribology - Lubricants and Lubrication

Dai, Z.; Xue, Q. Exploration Systematical Analysis and Quantitative Modeling of Tribo-

*Astronautics*, Vol. 25, No.6, (2003), pp.585 ~ 589 ISSN 1005-2615 (In Chinese)

Fleischer G. *Systembetrachtungen zur Tribologie*. Wiss. Z. TH Magdeburg, Vol. 14, (1970),

Ge, S.; Zhu, H. (2005). *Fractal in Tribology*. China Machine Press, ISBN 7-111-16014-2, Beijing,

Glienicke, J. (1972). Theoretische und experimentelle Ermittlung der Systemdaempfung

Her Majesty's Stationery Office. (1966). Lubrication (Tribology) Education and Research: A

Li, J. Analysis and Calculation of Influence of Steam Turbo-Generator Bearing Elevation

Ogata, K. (1970). *Modern Control Engineering*. Prentice-Hall, ISBN 9780135902325, New

Ogata, K. (1987). *Discrete-time Control Systems*. Prentice-Hall, ISBN 9780132161022, New

Pinkus, O.; Sternlicht, B. (1961). *Theory of Hydrodynamic Lubrication*. McGraw-Hill, New York,

Salomon G. Application of Systems Thinking to Tribology. *ASLE Trans*, Vol.17, No.4, (1974),

Division of the NSF. Research Needs in Mechanical Systems-Report of the Select Panel on Research Goals and Priorities in Mechanical Systems. *Trans ASME, Journal* 

Suh, N. (1990). *The Principle of Design*, Oxford University Press, ISBN 978-0195043457, USA The Panel Steering Committee for the Mechanical Engineering and Applied Mechanics

Xie, Y. On the Systems Engineering of Tribo-Systems. *Chinese Journal of Mechanical Engineering* (English Edition), No 2, (1996), pp. 89-99, ISSN 1000-9345 Xie, Y. On the System Theory and Modeling of Tribo-Systems. *Tribology*, Vol.30, No.1,

Xie, Y. *On the Tribological Database*. Lubrication Engineering, Vol.5, (1986), pp. 1-7, ISSN

Xie, Y. Three Axioms in Tribology. *Tribology*, Vol.21, No.3, (2001), pp.161-166, ISSN 1004-

Xie, Y.; Zhang, S. (Eds.). (2009). *Status and Developing Strategy Investigation on Tribology* 

Xu, S. (2007). *Digital Analysis and Methods*. China Machine Press, ISBN 978-7-111-20668-2,

*Science and Engineering Application: A Consulting Report of the Chinese Academy of Engineering (CAE)*. Higher Education Press, ISBN 978-7-04-026378-7, Beijing, China

*of Tribology*, Vol. 1, (1984), pp. 2~25, ISSN 0022-2305

(2010), pp.1-8, ISSN 1004-0595 (In Chinese)

Dowson, D. (1979). *History of Tribology*. Wiley, ISBN 978-1860580703, London

Report on the Present Position and Industry's Needs. London Hori, Y. (2005). *Hydrodynamic Lubrication*. Springer, ISBN 978-4431278986

pp.415-420

Duesseldorf

2691 (In Chinese)

pp.295-299, ISSN 0569-8197

0254-0150 (In Chinese)

Beijing, China (In Chinese)

0595 (In Chinese)

(In Chinese)

Jersey, USA

Jersey, USA

USA

China (In Chinese)

System Based on Entropy Concept. *Journal of Nanjing University of Aeronautics &* 

gleitgelagerter Rotoren und ihre Erhoehung durch eine aeussere Lagerdaempfung. Fortschritt Berichte der VDI Zeitschriften, Reihe 11, Nr. 13, VDI-Verlag GmbH,

Variation on Load. *North China Electric Power*, No. 11, (2001), pp.5 ~ 7, ISSN 1007-

Rolling (element) bearings are referred to as *anti-friction* bearings due to the low friction and hence only slight energy loss they cause in service, especially compared to sliding or *friction* bearings. The minor wear occurring in proper operation superficially seems to suggest the question how rolling contact tribology should be of relevance to bearing failures. Satisfactorily proven throughout the 20th century primarily on small highly loaded ball bearings, the life prediction is actually based on material fatigue theories. Nonetheless, resulting subsurface spalling is usually called fatigue wear and therefore included in the discussion below. The influence of friction on the damage of rolling bearings, at first, is strikingly reflected, for instance, in foreign particle abrasion and smearing adhesion wear under improper running or lubrication conditions. On far less affected, visually intact raceways, however, temporary frictional forces can also initiate failure for common overall friction coefficients below 0.1. Larger size roller bearings with extended line contacts operating typically at low to moderate Hertzian pressure, generally speaking, are most susceptible to this surface loading. As large roller bearings are increasingly applied in the 21st century, e.g. in industrial gears, an attempt is made in the following to incorporate the rolling-sliding nature of the tribological contact into an extended bearing life model. By holding the established assumption that the stage of crack initiation still dominates the total lifetime, the consideration of the proposed competing normal stress hypothesis is deemed appropriate.

The present chapter opens with a general introduction of the subsurface and (near-) surface failure mode of rolling bearings. Due to its particular importance to the identification of the damage mechanisms, the measuring procedure and the evaluation method of the material response analysis, which is based on an X-ray diffraction residual stress determination, are described in detail. In section 4, a metal physics model of classical subsurface rolling contact fatigue is outlined. Recent experimental findings are reported that support this mechanistic approach. The accelerating effect of absorbed hydrogen on rolling contact fatigue is also in agreement with the new model and verified by applying tools of material response analysis. It uncovers a remarkable impact of serious high-frequency electric current passage through bearings in operation, previously unnoticed in the literature. Section 5 provides an overview of state-of-the-art research on mechanical and chemical damage mechanisms by tribological

Tribological Aspects of Rolling Bearing Failures 35

Fig. 1. Normalized plot of the depth distribution of the σ*x*, σ*y*, and σ*z* main normal and of the

Fig. 2. Subsurface material loading and damage characterized, respectively, by (a) the residual stress distribution below the inner ring (IR) raceway of a deep groove ball bearing (DGBB) tested in an automobile gearbox rig, where the part is made of martensitically through hardened bearing steel and (b) a SEM image (secondary electron mode, SE) of fatigue spalling on the IR raceway of a rig tested DGBB with overrolling direction from left to right

v. Mises equivalent stress below the center line of the Hertzian contact area

stressing in rolling-sliding contact. The combined action of mixed friction and corrosion in the complex loading regime is demonstrated. Mechanical vibrations in bearing service, e.g. from adjacent machines, increase sliding in the contact area. Typical depth distributions of residual stress and X-ray diffraction peak width, which indicate microplastic deformation and (low-cycle) fatigue, are reproduced on a special rolling bearing test rig. The effect of vibrationally increased sliding friction on near-surface mechanical loading is described by a tribological contact model. Temperature rise and chemical lubricant aging are observed as well. Gray staining is interpreted as corrosion rolling contact fatigue. Material weakening by operational surface embrittlement is proven. Three mechanisms of *tribocracking* on raceways are discussed: tribochemical dissolution of nonmetallic inclusions and crack initiation by either frictional tensile stresses or shear stresses. Deep branching crack growth is driven by another variant of corrosion fatigue in rolling contact.

#### **2. Failure modes of rolling bearings**

Bearings in operation, in simple terms, experience pure rolling in elastohydrodynamic lubrication (EHL) or superimposed surface loading. With respect to the differing initiation sites of fatigue damage, a distinction is made between the classical subsurface and the (near-) surface failure mode (Muro & Tsushima, 1970). In the following simplified analysis, the evaluation of material stressing due to rolling contact (RC) loading is based on an extended static yield criterion by means of the distribution of the equivalent stress. The more complex surface failure mode, which predominates in today's engineering practice also due to the improved steelmaking processes and the tendency to use energy saving lower viscosity lubricants, comprises several damage mechanisms. Raceway indentations or boundary lubrication, for instance, respectively add edge stresses on Hertzian micro contacts and frictional sliding loading to the ideal elastohydrodynamic operating conditions.

#### **2.1 Subsurface failure mode**

The Hertz theory of elastic contact deformation between two solid bodies, specifically a rolling element and a ring of a bearing, is used to analyze the spatial stress state (Johnson, 1985). Initial yielding and generation of compressive residual stresses (CRS) is governed by the distortion energy hypothesis. In a normalized representation, Figure 1 plots the distance distributions of the three principal normal stresses σ*x*, σ*y* and σ*z* and the resulting v. Mises equivalent stress v.Mises σe below the center line of a purely radially loaded frictionless elastic line contact, where the maximum normal stress, i.e. the Hertzian pressure *p*0, occurs. In the coordinate trihedral, *x*, *y* and *z* respectively indicate the axial (lateral), tangential (overrolling) and radial (depth) direction. The v. Mises equivalent stress reaches its maximum max e,a <sup>0</sup> σ= × 0.56 *<sup>p</sup>* in a distance v.Mises <sup>0</sup>*z a* = 0.71× from the surface, which is valid in good approximation for roller and ball bearings (Hooke, 2003). The load is expressed as *p*<sup>0</sup> and *a* stands for the semiminor axis of the contact ellipse.

As illustrated in Figure 1 for a through hardened grade (*R*p0.2=const.), the v. Mises equivalent stress can locally exceed the yield strength *R*p0.2 of the steel that ranges between 1400 and 1800 MPa, depending, e.g., on the heat treatment and the degree of deformation of the material (segregations) or the operating temperature. From Hertzian pressures *p*0 of about 2500 to 3000 MPa, therefore, compressive residual stresses are built up. An example of a measured distance profile is shown in Figure 2a. By identifying the maximum position of the v. Mises and compressive residual stress, the Hertzian pressure is estimated to be 3500 MPa.

stressing in rolling-sliding contact. The combined action of mixed friction and corrosion in the complex loading regime is demonstrated. Mechanical vibrations in bearing service, e.g. from adjacent machines, increase sliding in the contact area. Typical depth distributions of residual stress and X-ray diffraction peak width, which indicate microplastic deformation and (low-cycle) fatigue, are reproduced on a special rolling bearing test rig. The effect of vibrationally increased sliding friction on near-surface mechanical loading is described by a tribological contact model. Temperature rise and chemical lubricant aging are observed as well. Gray staining is interpreted as corrosion rolling contact fatigue. Material weakening by operational surface embrittlement is proven. Three mechanisms of *tribocracking* on raceways are discussed: tribochemical dissolution of nonmetallic inclusions and crack initiation by either frictional tensile stresses or shear stresses. Deep branching crack growth is driven by

Bearings in operation, in simple terms, experience pure rolling in elastohydrodynamic lubrication (EHL) or superimposed surface loading. With respect to the differing initiation sites of fatigue damage, a distinction is made between the classical subsurface and the (near-) surface failure mode (Muro & Tsushima, 1970). In the following simplified analysis, the evaluation of material stressing due to rolling contact (RC) loading is based on an extended static yield criterion by means of the distribution of the equivalent stress. The more complex surface failure mode, which predominates in today's engineering practice also due to the improved steelmaking processes and the tendency to use energy saving lower viscosity lubricants, comprises several damage mechanisms. Raceway indentations or boundary lubrication, for instance, respectively add edge stresses on Hertzian micro contacts and

The Hertz theory of elastic contact deformation between two solid bodies, specifically a rolling element and a ring of a bearing, is used to analyze the spatial stress state (Johnson, 1985). Initial yielding and generation of compressive residual stresses (CRS) is governed by the distortion energy hypothesis. In a normalized representation, Figure 1 plots the distance distributions of the three principal normal stresses σ*x*, σ*y* and σ*z* and the resulting v. Mises equivalent stress v.Mises σe below the center line of a purely radially loaded frictionless elastic line contact, where the maximum normal stress, i.e. the Hertzian pressure *p*0, occurs. In the coordinate trihedral, *x*, *y* and *z* respectively indicate the axial (lateral), tangential (overrolling) and radial (depth) direction. The v. Mises equivalent stress reaches its

good approximation for roller and ball bearings (Hooke, 2003). The load is expressed as *p*<sup>0</sup>

As illustrated in Figure 1 for a through hardened grade (*R*p0.2=const.), the v. Mises equivalent stress can locally exceed the yield strength *R*p0.2 of the steel that ranges between 1400 and 1800 MPa, depending, e.g., on the heat treatment and the degree of deformation of the material (segregations) or the operating temperature. From Hertzian pressures *p*0 of about 2500 to 3000 MPa, therefore, compressive residual stresses are built up. An example of a measured distance profile is shown in Figure 2a. By identifying the maximum position of the v. Mises and compressive residual stress, the Hertzian pressure is estimated to be 3500 MPa.

<sup>0</sup>*z a* = 0.71× from the surface, which is valid in

frictional sliding loading to the ideal elastohydrodynamic operating conditions.

another variant of corrosion fatigue in rolling contact.

e,a <sup>0</sup> σ= × 0.56 *<sup>p</sup>* in a distance v.Mises

and *a* stands for the semiminor axis of the contact ellipse.

**2. Failure modes of rolling bearings** 

**2.1 Subsurface failure mode** 

maximum max

Fig. 1. Normalized plot of the depth distribution of the σ*x*, σ*y*, and σ*z* main normal and of the v. Mises equivalent stress below the center line of the Hertzian contact area

Fig. 2. Subsurface material loading and damage characterized, respectively, by (a) the residual stress distribution below the inner ring (IR) raceway of a deep groove ball bearing (DGBB) tested in an automobile gearbox rig, where the part is made of martensitically through hardened bearing steel and (b) a SEM image (secondary electron mode, SE) of fatigue spalling on the IR raceway of a rig tested DGBB with overrolling direction from left to right

Tribological Aspects of Rolling Bearing Failures 37

Cyclic loading of the Hertzian micro contacts induces continuously increasing compressive residual stresses near the surface up to a depth that is connected with the regular (e.g., lognormal) size distribution of the indentations. In the case of Figure 4, the superimposed profile modification by the basic macro contact is marginal, which means that the maximum Hertzian pressure of 3300 MPa is only applied for a short time. Compressive residual stresses in the edge zone are generated up to 60 µm depth. The high surface value reflects

The stress analysis for evaluation of the v. Mises yield criterion in Figure 1 refers to the ideal undisturbed EHL rolling contact in a bearing with fully separating lubricating film, where (fluid) friction only occurs. In an extension of this scheme, the surface mode of rolling contact fatigue (RCF) is illustrated in Figure 5 on the example of indentations (size *a*micro) that cover the raceway densely in the form of a statistical waviness at an early stage of

Fig. 5. Scheme of the v. Mises stress as a function of the distance from the Hertzian contact

The resulting peak of the v. Mises equivalent stress, max σe,surf. , is influenced by the sharpedged indentations of hard foreign particles (cf. Figure 3a). However, lubricant contamination by hardened steel acts most effectively because of the larger size. The contact area of the rolling elements also exhibits a statistical waviness of indentations. The stress concentrations on the edges of the Hertzian micro contacts promote material fatigue and damage initiation on or near the surface. Consequently, bearing life is reduced (Takemura & Murakami, 1998). It is shown in section 5.1 that, by creating tangential forces, additional sliding in frictional rolling contact can cause equivalent and hence residual stress distributions similar to Figures 5 and 4, respectively, on indentation-free raceways. The occurrence or dominance of the competing (near-) surface and subsurface failure mode depends on the magnitude of max <sup>σ</sup>e,surf. and the relative position of the (actually not varying) yield strength *R*p0.2, as

The ground area of an indentation is unloaded. On the highly stressed edges, the lubricating film breaks down and metal-to-metal contact results in locally most pronounced smoothing of the honing marks. Figure 6a reveals the back end of a metal span indentation in overrolling direction. Strain hardening by severe plastic deformation leads to material

with and without raceway indentations (roller on a smaller scale)

polishing of the raceway, associated with plastic deformation.

operation:

indicated in Figure 5.

Up to a depth *z* of 20 µm, the indicated initial state after hardening and machining is not changed, which manifests good lubrication. The residual stress is denoted by σres.

Fatigue spalling is eventually caused by subsurface crack initiation and growth to the surface in overrolling direction (OD), as evident from Figure 2b (Voskamp, 1996). In the scanning electron microscope (SEM) image, the still intact honing structure of the raceway confirms the adjusted ideal EHL conditions.

#### **2.2 Surface failure mode**

Hard (ceramic) or metallic foreign particles contaminating the lubricating gap at the contact area, however, result in indentations on the raceway due to overrolling in bearing operation. The SEM images of Figures 3a and 3b, taken in the SE mode, show examples of both types:

Fig. 3. SEM images (SE mode) of (a) randomly distributed dense hard particle raceway indentations (also track-like indentation patterns can occur, e.g. so-called frosty bands) from contaminated lubricant and (b) indentations of metallic particles on the smoothed IR raceway of a cylindrical roller bearing (CRB) that clearly reveal earlier surface conditions of better preserved honing structure

Fig. 4. Residual stress depth distribution of the martensitically hardened IR of a taper roller bearing (TRB) indicating foreign particle (e.g., wear debris) contamination of the lubricant

Up to a depth *z* of 20 µm, the indicated initial state after hardening and machining is not

Fatigue spalling is eventually caused by subsurface crack initiation and growth to the surface in overrolling direction (OD), as evident from Figure 2b (Voskamp, 1996). In the scanning electron microscope (SEM) image, the still intact honing structure of the raceway

Hard (ceramic) or metallic foreign particles contaminating the lubricating gap at the contact area, however, result in indentations on the raceway due to overrolling in bearing operation. The SEM images of Figures 3a and 3b, taken in the SE mode, show examples of both types:

 Fig. 3. SEM images (SE mode) of (a) randomly distributed dense hard particle raceway indentations (also track-like indentation patterns can occur, e.g. so-called frosty bands) from contaminated lubricant and (b) indentations of metallic particles on the smoothed IR raceway of a cylindrical roller bearing (CRB) that clearly reveal earlier surface conditions of

Fig. 4. Residual stress depth distribution of the martensitically hardened IR of a taper roller bearing (TRB) indicating foreign particle (e.g., wear debris) contamination of the lubricant

changed, which manifests good lubrication. The residual stress is denoted by σres.

confirms the adjusted ideal EHL conditions.

**2.2 Surface failure mode** 

better preserved honing structure

Cyclic loading of the Hertzian micro contacts induces continuously increasing compressive residual stresses near the surface up to a depth that is connected with the regular (e.g., lognormal) size distribution of the indentations. In the case of Figure 4, the superimposed profile modification by the basic macro contact is marginal, which means that the maximum Hertzian pressure of 3300 MPa is only applied for a short time. Compressive residual stresses in the edge zone are generated up to 60 µm depth. The high surface value reflects polishing of the raceway, associated with plastic deformation.

The stress analysis for evaluation of the v. Mises yield criterion in Figure 1 refers to the ideal undisturbed EHL rolling contact in a bearing with fully separating lubricating film, where (fluid) friction only occurs. In an extension of this scheme, the surface mode of rolling contact fatigue (RCF) is illustrated in Figure 5 on the example of indentations (size *a*micro) that cover the raceway densely in the form of a statistical waviness at an early stage of operation:

Fig. 5. Scheme of the v. Mises stress as a function of the distance from the Hertzian contact with and without raceway indentations (roller on a smaller scale)

The resulting peak of the v. Mises equivalent stress, max σe,surf. , is influenced by the sharpedged indentations of hard foreign particles (cf. Figure 3a). However, lubricant contamination by hardened steel acts most effectively because of the larger size. The contact area of the rolling elements also exhibits a statistical waviness of indentations. The stress concentrations on the edges of the Hertzian micro contacts promote material fatigue and damage initiation on or near the surface. Consequently, bearing life is reduced (Takemura & Murakami, 1998). It is shown in section 5.1 that, by creating tangential forces, additional sliding in frictional rolling contact can cause equivalent and hence residual stress distributions similar to Figures 5 and 4, respectively, on indentation-free raceways. The occurrence or dominance of the competing (near-) surface and subsurface failure mode depends on the magnitude of max <sup>σ</sup>e,surf. and the relative position of the (actually not varying) yield strength *R*p0.2, as indicated in Figure 5.

The ground area of an indentation is unloaded. On the highly stressed edges, the lubricating film breaks down and metal-to-metal contact results in locally most pronounced smoothing of the honing marks. Figure 6a reveals the back end of a metal span indentation in overrolling direction. Strain hardening by severe plastic deformation leads to material

Tribological Aspects of Rolling Bearing Failures 39

The investigation aims at characterizing the response of the steel in the highly stressed edge zone to rolling contact loading. Plastification (local yielding) and material aging (defect accumulation) is estimated by the changes of the (macro) residual stresses and the XRD peak width, respectively. Failure is related to mechanical damage by fatigue and tribological loading, (tribo-) chemical and thermal exposure. Mixed friction or boundary lubrication in rolling-sliding contact is reflected, for instance, by polishing wear on the surface. The operating condition of cyclically Hertzian loaded machine parts shall be analyzed. The key focus is put on rolling bearings but also other components, like gears or camshafts, can be examined. XRD material response analysis permits the identification of the relevant failure mode. In the frequent case of surface rolling contact loading, the acting damage mechanism, such as vibrations, poor or contaminated lubrication, is also deducible. The quantitative remaining life estimation in rig test evaluation supports, for instance, product development or design optimization. This analysis option receives great interest especially in automotive engineering. Drawing a comparison with the calculated nominal life is of high significance. Also, not too heavily damaged (spalled) field returns can be investigated in the framework

The practicable evaluation tools provided and applied in the following sections are derived from the basic research work of *Aat Voskamp* (Voskamp, 1985, 1996, 1998), who concentrates on residual stress evolution and microstructural alterations during classical subsurface rolling contact fatigue, and *Wolfgang Nierlich* (Nierlich et al., 1992; Nierlich & Gegner, 2002, 2008), who studies the surface failure mode and aligns the X-ray diffractometry technique from the 1970's on to meet industry needs. The application of the XRD line broadening for the characterization of material damage and the introduction of the peak width ratio as a quantitative measure represent the essential milestone in method development (Nierlich et al., 1992). The bearing life calibration curves for classical and surface rolling contact fatigue, deduced from rig test series, also make the connection to mechanical engineering failure analysis and design (Nierlich et al., 1992; Voskamp, 1998). The three stage model of material response allows the attribution of the residual stress and microstructure changes (Voskamp, 1985). With substantial modification on the surface (Nierlich & Gegner, 2002), this today accepted scheme proves applicable to both failure modes (Gegner, 2006a). The interdependent joint evaluation of residual stress and peak width depth profiles in the subsurface region of classical rolling contact fatigue completes the *Schweinfurt* methodology (Gegner, 2006a). Further developments of the XRD material response analysis, such as the application to other cyclically Hertzian loaded machine elements, are reported in the

To discuss the principles of material based bearing performance analysis, first a synopsis of the XRD measurement technique is provided. Data interpretation is subsequently described in section 3.3. The evaluation of a high number of measurements on run field and test bearings is necessary to create the appropriate scientific, engineering, and methodological foundations of XRD material response analysis. For efficient performance, the applied XRD technique must thus take into account the required fast specimen throughput at sufficient data accuracy. The rapid industrial-suited XRD measurement of residual stresses outlined below incorporates suggestions from the literature (Faninger & Wolfstieg, 1976). Usually, around ten depth positions are adequate for a profile determination. Residual stress free

**3.1 Intention and history of XRD material response analysis** 

of failure analysis and research.

literature (Gegner et al., 2007; Nierlich & Gegner, 2006).

**3.2 Residual stress measurement** 

embrittlement and subsequent crack initiation on the surface. Further failure development produces a so-called V pit of originally only several µm depth behind the indentation, as documented in Figure 6b. It is instructive to compare this shallow pit and the clearly smoothed raceway with the subsurface fatigue spall of Figure 2b that evolves from a depth of about 100 µm below an intact honing structure.

Fig. 6. SEM image (SE mode) of (a) incipient cracking and (b) beginning V pitting behind an indentation on the IR raceway of a TRB. Note the overrolling direction from left to right

#### **3. Material based bearing performance analysis**

Stressing, damage and eventually failure of a component occur due to a response of the material to the applied loading that generally acts as a combination of mechanical, chemical and thermal portions. The reliability of Hertzian contact machine elements, such as rolling bearings, gears, followers, cams or tappets, is of particular engineering significance. Advanced techniques of physical diagnostics permit the evaluation of the prevailing material condition on a microscopic scale. According to the collective impact of fatigue, friction, wear and corrosion and thus, for instance, depending on the type of lubrication, the degree of contamination, the roughness profile and the applied Hertzian pressure, failures are initiated on or below the raceway surface (see section 2). An operating rolling bearing represents a cyclically loaded tribological system. Depth resolved X-ray diffraction (XRD) measurements of macro and micro residual stresses provide an accurate estimation of the stage of material aging. The XRD material response analysis of rolling bearings is experimentally and methodologically most highly evolved. A quantitative evaluation of the changes in the residual stress distribution is proposed in the literature, for instance by integrating the depth profile to compute a characteristic deformation number (Böhmer et al., 1999). In the research reported in this chapter, however, the alternative XRD peak width based conception is used. The established procedure described in the following may be, due to its development to a powerful evaluation tool for scientific and routine engineering purposes in the SKF Material Physics laboratory under the guidance of *Wolfgang Nierlich*, referred to as the *Schweinfurt* methodology of XRD material response bearing performance analysis.

embrittlement and subsequent crack initiation on the surface. Further failure development produces a so-called V pit of originally only several µm depth behind the indentation, as documented in Figure 6b. It is instructive to compare this shallow pit and the clearly smoothed raceway with the subsurface fatigue spall of Figure 2b that evolves from a depth

 Fig. 6. SEM image (SE mode) of (a) incipient cracking and (b) beginning V pitting behind an indentation on the IR raceway of a TRB. Note the overrolling direction from left to right

Stressing, damage and eventually failure of a component occur due to a response of the material to the applied loading that generally acts as a combination of mechanical, chemical and thermal portions. The reliability of Hertzian contact machine elements, such as rolling bearings, gears, followers, cams or tappets, is of particular engineering significance. Advanced techniques of physical diagnostics permit the evaluation of the prevailing material condition on a microscopic scale. According to the collective impact of fatigue, friction, wear and corrosion and thus, for instance, depending on the type of lubrication, the degree of contamination, the roughness profile and the applied Hertzian pressure, failures are initiated on or below the raceway surface (see section 2). An operating rolling bearing represents a cyclically loaded tribological system. Depth resolved X-ray diffraction (XRD) measurements of macro and micro residual stresses provide an accurate estimation of the stage of material aging. The XRD material response analysis of rolling bearings is experimentally and methodologically most highly evolved. A quantitative evaluation of the changes in the residual stress distribution is proposed in the literature, for instance by integrating the depth profile to compute a characteristic deformation number (Böhmer et al., 1999). In the research reported in this chapter, however, the alternative XRD peak width based conception is used. The established procedure described in the following may be, due to its development to a powerful evaluation tool for scientific and routine engineering purposes in the SKF Material Physics laboratory under the guidance of *Wolfgang Nierlich*, referred to as the *Schweinfurt* methodology of XRD material response bearing performance

of about 100 µm below an intact honing structure.

**3. Material based bearing performance analysis** 

analysis.

#### **3.1 Intention and history of XRD material response analysis**

The investigation aims at characterizing the response of the steel in the highly stressed edge zone to rolling contact loading. Plastification (local yielding) and material aging (defect accumulation) is estimated by the changes of the (macro) residual stresses and the XRD peak width, respectively. Failure is related to mechanical damage by fatigue and tribological loading, (tribo-) chemical and thermal exposure. Mixed friction or boundary lubrication in rolling-sliding contact is reflected, for instance, by polishing wear on the surface. The operating condition of cyclically Hertzian loaded machine parts shall be analyzed. The key focus is put on rolling bearings but also other components, like gears or camshafts, can be examined. XRD material response analysis permits the identification of the relevant failure mode. In the frequent case of surface rolling contact loading, the acting damage mechanism, such as vibrations, poor or contaminated lubrication, is also deducible. The quantitative remaining life estimation in rig test evaluation supports, for instance, product development or design optimization. This analysis option receives great interest especially in automotive engineering. Drawing a comparison with the calculated nominal life is of high significance. Also, not too heavily damaged (spalled) field returns can be investigated in the framework of failure analysis and research.

The practicable evaluation tools provided and applied in the following sections are derived from the basic research work of *Aat Voskamp* (Voskamp, 1985, 1996, 1998), who concentrates on residual stress evolution and microstructural alterations during classical subsurface rolling contact fatigue, and *Wolfgang Nierlich* (Nierlich et al., 1992; Nierlich & Gegner, 2002, 2008), who studies the surface failure mode and aligns the X-ray diffractometry technique from the 1970's on to meet industry needs. The application of the XRD line broadening for the characterization of material damage and the introduction of the peak width ratio as a quantitative measure represent the essential milestone in method development (Nierlich et al., 1992). The bearing life calibration curves for classical and surface rolling contact fatigue, deduced from rig test series, also make the connection to mechanical engineering failure analysis and design (Nierlich et al., 1992; Voskamp, 1998). The three stage model of material response allows the attribution of the residual stress and microstructure changes (Voskamp, 1985). With substantial modification on the surface (Nierlich & Gegner, 2002), this today accepted scheme proves applicable to both failure modes (Gegner, 2006a). The interdependent joint evaluation of residual stress and peak width depth profiles in the subsurface region of classical rolling contact fatigue completes the *Schweinfurt* methodology (Gegner, 2006a). Further developments of the XRD material response analysis, such as the application to other cyclically Hertzian loaded machine elements, are reported in the literature (Gegner et al., 2007; Nierlich & Gegner, 2006).

#### **3.2 Residual stress measurement**

To discuss the principles of material based bearing performance analysis, first a synopsis of the XRD measurement technique is provided. Data interpretation is subsequently described in section 3.3. The evaluation of a high number of measurements on run field and test bearings is necessary to create the appropriate scientific, engineering, and methodological foundations of XRD material response analysis. For efficient performance, the applied XRD technique must thus take into account the required fast specimen throughput at sufficient data accuracy. The rapid industrial-suited XRD measurement of residual stresses outlined below incorporates suggestions from the literature (Faninger & Wolfstieg, 1976). Usually, around ten depth positions are adequate for a profile determination. Residual stress free

Tribological Aspects of Rolling Bearing Failures 41

detection angle of 2° in position 4. The adapted diffractometer arrangement, optimized with the courage to problem-oriented simplification, eventually provides a 10 times higher recorded X-ray intensity without noticeable loss in accuracy for the broad interference lines of hardened bearing steels. The effect on determining peak position and width is negligible. The dispersion, defining the line shift relevant to residual stress evaluation, remains

Stress determinations are generally based on the measurement of strain (Dally & Riley, 2005). The conversion occurs by elasticity theory. Residual stresses of the first kind result in lattice strains of the order of 1‰. This elastic distortion of the unit cell causes an anisotropic peak shift of interference lines, which is determinable by the XRD technique. The macro or volume residual stresses of a polycrystalline material are derived by measuring and evaluating the relative interplanar spacing for multiple specimen orientations bringing differently oriented sets of lattice plane into reflection. Larger line shifts preferable for a more sensitive strain determination occur in the backscattering region for 2θ>130°, as obvious from the total differential of the Bragg condition for the monochromatic radiation:

<sup>d</sup> cot d *<sup>D</sup>*

According to Figure 7, θ and 2θ respectively denote the glancing Bragg and the diffraction angle. Such favorable lattice spacings *D* of the reflecting planes increase the measuring accuracy. For martensitic or bainitic microstructures, the recorded α-Fe (211) diffraction peak, excited by the long-wave Cr Kα radiation (wavelength λ=0.229 nm), best meets this requirement with 2θ0=156.1° and is analyzable even for incomplete line detection (see Figure 8) when the rotating detector would touch the X-ray tube at 2θ>162°. Since its introduction in 1961 (Macherauch & Müller, 1961), the applied sin2ψ method is further developed and, for accuracy reasons (Macherauch, 1966), predominantly used for XRD macro residual stress measurements (Hauk, 1997; Noyan & Cohen, 1987). Due to the small penetration depth of the X-ray radiation of a few µm, a biaxial stress state exists in the specimen surface in good approximation. The strain can be measured from the line shift by Eq. (1). Poisson's ratio and Young's modulus are denoted ν and *E*, respectively. Applying Hooke's law, elemental

> ( ) ( ) , 0 <sup>2</sup> 0, 1 2

The azimuth and inclination Euler angles, ϕ and ψ, characterize the direction of the interplanar spacing *D*ϕ,ψ and the lattice strain εϕ,ψ. Furthermore, σ1 and σ2 are the principle stresses parallel to the specimen surface (σ3=0). The values *D*0 and θ0 refer to the strain-free undeformed lattice. For the surface stress component (ψ=90°) corresponding to ϕ, a

cot sin

0 ϕ ψ φ <sup>−</sup> +ν ν − θ−θ θ = = = σ ψ− σ +σ ε

1

*D EE*

2 2 σϕ1 2 =σ ϕ+σ ϕ cos sin (3)

geometry provides the fundamental equation of X-ray residual stress analysis:

0

*D D*

ϕ ψ

*<sup>D</sup>* <sup>=</sup> − θθ (1)

(2)

uninfluenced. The intensity corrections are not further discussed.

**ψ method** 

**3.2.2 Implementation of the sin2**

trigonometric relationship holds:

Substituting the X-ray elastic constants (XEC):

material removal with high precision occurs by electrochemical polishing. The spatial resolution is given by the low penetration power of the incident X-ray radiation to about 5 µm that is appropriate for the application.

XRD residual stress analysis is widely used in bearing engineering since the 1970's (Muro et al., 1973). In the investigations of the present chapter, computer controlled Ω goniometers with scintillation type counter tube are applied, which work on the principle of the focusing Bragg-Brentano coupled θ–2θ diffraction geometry (Bragg & Bragg, 1913; Hauk & Macherauch, 1984). The X-ray source is fixed and the detector gradually rotates with twice the angular velocity θ of the specimen to preserve a constant angle of 2θ between the incident and reflected beam.

#### **3.2.1 High intensity diffractometer**

The positions of major modifications of the conventional goniometer design are numbered consecutively in Figure 7. The severe difficulties of XRD measurements of hardened steels in the past from the broad asymmetrical diffraction lines of martensite are well known (Macherauch, 1966; Marx, 1966). Exploiting the negligible instrumental broadening, however, these large peak widths of about 5° to 7.5° only permit the implementation of such fundamental interventions in the beam path to increase the intensity of the incident and emergent X-ray radiation by tailoring the required resolution. In position 1, the square instead of the line focal spot is used. Thus, the intensity loss by vertical masking at the beam defining slit is reduced. Position 2 is also labeled in Figure 7. The distance from the horizontally and vertically adjustable defining slit to the focal spot is extended to two-thirds of the diffractometer (or measuring) circle radius. Whereas the lower resolution is of no significance, the intensity of the primary beam is further enhanced. The aperture α is indicated. The depicted scattering and Soller slits limit peak width and divergence of the diffracted beam on the expense of intensity loss. Position 3 signifies that parallelization of the radiation is dispensed with. For the same purpose, the receiving slit is opened to a

Fig. 7. Schematic diffractometer beam path with indicated modifications (1 to 4)

material removal with high precision occurs by electrochemical polishing. The spatial resolution is given by the low penetration power of the incident X-ray radiation to about 5

XRD residual stress analysis is widely used in bearing engineering since the 1970's (Muro et al., 1973). In the investigations of the present chapter, computer controlled Ω goniometers with scintillation type counter tube are applied, which work on the principle of the focusing Bragg-Brentano coupled θ–2θ diffraction geometry (Bragg & Bragg, 1913; Hauk & Macherauch, 1984). The X-ray source is fixed and the detector gradually rotates with twice

The positions of major modifications of the conventional goniometer design are numbered consecutively in Figure 7. The severe difficulties of XRD measurements of hardened steels in the past from the broad asymmetrical diffraction lines of martensite are well known (Macherauch, 1966; Marx, 1966). Exploiting the negligible instrumental broadening, however, these large peak widths of about 5° to 7.5° only permit the implementation of such fundamental interventions in the beam path to increase the intensity of the incident and emergent X-ray radiation by tailoring the required resolution. In position 1, the square instead of the line focal spot is used. Thus, the intensity loss by vertical masking at the beam defining slit is reduced. Position 2 is also labeled in Figure 7. The distance from the horizontally and vertically adjustable defining slit to the focal spot is extended to two-thirds of the diffractometer (or measuring) circle radius. Whereas the lower resolution is of no significance, the intensity of the primary beam is further enhanced. The aperture α is indicated. The depicted scattering and Soller slits limit peak width and divergence of the diffracted beam on the expense of intensity loss. Position 3 signifies that parallelization of the radiation is dispensed with. For the same purpose, the receiving slit is opened to a

Fig. 7. Schematic diffractometer beam path with indicated modifications (1 to 4)

of the specimen to preserve a constant angle of 2θ between the

µm that is appropriate for the application.

the angular velocity θ

incident and reflected beam.

**3.2.1 High intensity diffractometer** 

detection angle of 2° in position 4. The adapted diffractometer arrangement, optimized with the courage to problem-oriented simplification, eventually provides a 10 times higher recorded X-ray intensity without noticeable loss in accuracy for the broad interference lines of hardened bearing steels. The effect on determining peak position and width is negligible. The dispersion, defining the line shift relevant to residual stress evaluation, remains uninfluenced. The intensity corrections are not further discussed.

#### **3.2.2 Implementation of the sin2 ψ method**

Stress determinations are generally based on the measurement of strain (Dally & Riley, 2005). The conversion occurs by elasticity theory. Residual stresses of the first kind result in lattice strains of the order of 1‰. This elastic distortion of the unit cell causes an anisotropic peak shift of interference lines, which is determinable by the XRD technique. The macro or volume residual stresses of a polycrystalline material are derived by measuring and evaluating the relative interplanar spacing for multiple specimen orientations bringing differently oriented sets of lattice plane into reflection. Larger line shifts preferable for a more sensitive strain determination occur in the backscattering region for 2θ>130°, as obvious from the total differential of the Bragg condition for the monochromatic radiation:

$$\frac{\text{d}D}{D} = -\cot \theta \,\text{d}\theta \tag{1}$$

According to Figure 7, θ and 2θ respectively denote the glancing Bragg and the diffraction angle. Such favorable lattice spacings *D* of the reflecting planes increase the measuring accuracy. For martensitic or bainitic microstructures, the recorded α-Fe (211) diffraction peak, excited by the long-wave Cr Kα radiation (wavelength λ=0.229 nm), best meets this requirement with 2θ0=156.1° and is analyzable even for incomplete line detection (see Figure 8) when the rotating detector would touch the X-ray tube at 2θ>162°. Since its introduction in 1961 (Macherauch & Müller, 1961), the applied sin2ψ method is further developed and, for accuracy reasons (Macherauch, 1966), predominantly used for XRD macro residual stress measurements (Hauk, 1997; Noyan & Cohen, 1987). Due to the small penetration depth of the X-ray radiation of a few µm, a biaxial stress state exists in the specimen surface in good approximation. The strain can be measured from the line shift by Eq. (1). Poisson's ratio and Young's modulus are denoted ν and *E*, respectively. Applying Hooke's law, elemental geometry provides the fundamental equation of X-ray residual stress analysis:

$$- \left(\theta - \theta\_0\right) \cot \theta\_0 = \frac{D\_{\phi, \mathbf{v}} - D\_0}{D\_0} = \varepsilon\_{\phi, \mathbf{v}} = \frac{1 + \mathbf{v}}{E} \sigma\_{\phi} \sin^2 \psi - \frac{\mathbf{v}}{E} (\sigma\_1 + \sigma\_2) \tag{2}$$

The azimuth and inclination Euler angles, ϕ and ψ, characterize the direction of the interplanar spacing *D*ϕ,ψ and the lattice strain εϕ,ψ. Furthermore, σ1 and σ2 are the principle stresses parallel to the specimen surface (σ3=0). The values *D*0 and θ0 refer to the strain-free undeformed lattice. For the surface stress component (ψ=90°) corresponding to ϕ, a trigonometric relationship holds:

$$
\sigma\_{\varphi} = \sigma\_1 \cos^2 \varphi + \sigma\_2 \sin^2 \varphi \tag{3}
$$

Substituting the X-ray elastic constants (XEC):

Tribological Aspects of Rolling Bearing Failures 43

strategy with the modified arrangement of Figure 7, which equals the fastest up-to-date equipment also applied for the analyses in the present paper. Each individual residual stress determination on an irradiated area of 2×3 mm2 takes approximately 5 min. The single measured value scatter, expressing the measurement uncertainty by the standard deviation, is found to be about ±50 MPa, as correspondingly reported elsewhere in the literature

Unlike, for instance, several production processes (e.g., milling), rolling contact loading usually leads to the formation of similar depth distributions of the circumferential and axial residual stresses (Voskamp, 1987). Aside from rare exceptions such as the additional impact of severe three-dimensional vibrations (Gegner & Nierlich, 2008), deviations of maximum 20% to 30% reflect experience. As also the course of the depth profile is more important for the XRD material response analysis than the actual values of the single measurements, in the following the residual stresses are only determined in the circumferential (i.e., overrolling)

Due to the geometrical restrictions of the goniometer in Figure 7, the XRD line is only collected up to a diffraction angle of 162°. The peak width, expressed as *FWHM*, is measured at a specimen tilt of ψ=0° and provides information on the third kind (micro) residual stresses. For the extrapolation shown in Figure 9a, the background function is determined by a linear fit on the left of the line center. In the automated measurement procedure, the scintillation counter then moves to the onset of the diffraction peak. For the sake of simplicity, the background subtracted data of the subsequent line recoding in Figure 9b are fitted by an interpolating polynomial of high degree. The acquisition time per *FWHM* value and the measuring accuracy (one-fold standard deviation) amount to 3 to 5 min and 0.06° to

 Fig. 9. Illustration of (a) the programmed XRD peak width analysis with intervals of data fitting and (b) the evaluation of the *FWHM* value for the diffraction line of Figure 9a

It becomes clear in the following that the reliable interpretation of the measured depth distributions of residual stresses and XRD peak width, aside from optional auxiliary retained austenite determinations to further characterize material aging (Gegner, 2006a;

**3.2.5 Completion of investigation methods for material response analysis** 

(Voskamp, 1996).

direction.

0.09°, respectively.

**3.2.4 Automated XRD peak width evaluation** 

$$\frac{1}{2}S\_2 = \frac{1+\mathbf{v}}{E}, \quad S\_1 = -\frac{\mathbf{v}}{E} \tag{4}$$

into Eq. (2) ends up with the following expression:

$$\mathcal{L}\_{\phi,\psi} = \frac{1}{2} S\_2 \sigma\_{\phi} \sin^2 \psi + S\_1 \left(\sigma\_1 + \sigma\_2\right) \tag{5}$$

For hardened steel, isotropic grain distribution is assumed. The measurement of seven specimen tilt angles ψ from −45° to +45° symmetric about ψ=0° in equidistant sin2ψ steps is sufficient to reliably derive the desired σϕ value from the slope of the straight line fitted to the data of a *D*ϕ,ψ or εϕ,ψ plot against sin2ψ for constant ϕ (Nierlich & Gegner, 2008). High accuracy is already achieved by replacing *D*0 with the experimental *D*ϕ,ψ at ψ=0° (Voskamp, 1996). Recommendations for the X-ray elastic constants of the relevant steel microstructure are given in the literature (Hauk & Wolfstieg, 1976; Macherauch, 1966). For the XRD analyses reported in the present chapter, ½*S*2=5.811×10<sup>−</sup>6 MPa<sup>−</sup>1 is applied.

#### **3.2.3 Two stage diffraction line analysis and peak maximum method**

Besides X-ray intensity gain in the beam path, the second major task of rapid macro residual stress measurement is thus an efficient routine for the involved line shift evaluations. Accelerated determination of the diffraction peak position 2θ is achieved by an automated self-adjusting analysis technique tailored to the α-Fe (211) interference. The method is explained by means of Figure 8:

Fig. 8. Illustration of the self-developed peak finding procedure with a martensite diffraction reflex of full width at half maximum (*FWHM*) line breadth of 7.28°

The pre-measurement at reduced counting statistics across the indicated fixed angular range of 5° provides the peak maximum with an error of ±0.2°. This rough localization suffices to define appropriate symmetric evaluation points in an interval of 3° around the identified center for the subsequent highly accurate pulse controlled main run. The peak position is deduced from a fitting polynomial regression. A significant additional saving in acquisition time of 60%, compared to the standard procedure, is achieved by this skillful analysis

, 2 11 2

For hardened steel, isotropic grain distribution is assumed. The measurement of seven specimen tilt angles ψ from −45° to +45° symmetric about ψ=0° in equidistant sin2ψ steps is sufficient to reliably derive the desired σϕ value from the slope of the straight line fitted to the data of a *D*ϕ,ψ or εϕ,ψ plot against sin2ψ for constant ϕ (Nierlich & Gegner, 2008). High accuracy is already achieved by replacing *D*0 with the experimental *D*ϕ,ψ at ψ=0° (Voskamp, 1996). Recommendations for the X-ray elastic constants of the relevant steel microstructure are given in the literature (Hauk & Wolfstieg, 1976; Macherauch, 1966). For the XRD

Besides X-ray intensity gain in the beam path, the second major task of rapid macro residual stress measurement is thus an efficient routine for the involved line shift evaluations. Accelerated determination of the diffraction peak position 2θ is achieved by an automated self-adjusting analysis technique tailored to the α-Fe (211) interference. The method is

Fig. 8. Illustration of the self-developed peak finding procedure with a martensite diffraction

The pre-measurement at reduced counting statistics across the indicated fixed angular range of 5° provides the peak maximum with an error of ±0.2°. This rough localization suffices to define appropriate symmetric evaluation points in an interval of 3° around the identified center for the subsequent highly accurate pulse controlled main run. The peak position is deduced from a fitting polynomial regression. A significant additional saving in acquisition time of 60%, compared to the standard procedure, is achieved by this skillful analysis

reflex of full width at half maximum (*FWHM*) line breadth of 7.28°

<sup>1</sup> sin

2

analyses reported in the present chapter, ½*S*2=5.811×10<sup>−</sup>6 MPa<sup>−</sup>1 is applied.

**3.2.3 Two stage diffraction line analysis and peak maximum method** 

ε

*E E*

( ) <sup>2</sup>

ϕψ ϕ = *S S* σ ψ+ σ +σ (5)

<sup>+</sup> ν ν <sup>=</sup> , =− (4)

2 1 1 1 2 *S S*

into Eq. (2) ends up with the following expression:

explained by means of Figure 8:

strategy with the modified arrangement of Figure 7, which equals the fastest up-to-date equipment also applied for the analyses in the present paper. Each individual residual stress determination on an irradiated area of 2×3 mm2 takes approximately 5 min. The single measured value scatter, expressing the measurement uncertainty by the standard deviation, is found to be about ±50 MPa, as correspondingly reported elsewhere in the literature (Voskamp, 1996).

Unlike, for instance, several production processes (e.g., milling), rolling contact loading usually leads to the formation of similar depth distributions of the circumferential and axial residual stresses (Voskamp, 1987). Aside from rare exceptions such as the additional impact of severe three-dimensional vibrations (Gegner & Nierlich, 2008), deviations of maximum 20% to 30% reflect experience. As also the course of the depth profile is more important for the XRD material response analysis than the actual values of the single measurements, in the following the residual stresses are only determined in the circumferential (i.e., overrolling) direction.

#### **3.2.4 Automated XRD peak width evaluation**

Due to the geometrical restrictions of the goniometer in Figure 7, the XRD line is only collected up to a diffraction angle of 162°. The peak width, expressed as *FWHM*, is measured at a specimen tilt of ψ=0° and provides information on the third kind (micro) residual stresses. For the extrapolation shown in Figure 9a, the background function is determined by a linear fit on the left of the line center. In the automated measurement procedure, the scintillation counter then moves to the onset of the diffraction peak. For the sake of simplicity, the background subtracted data of the subsequent line recoding in Figure 9b are fitted by an interpolating polynomial of high degree. The acquisition time per *FWHM* value and the measuring accuracy (one-fold standard deviation) amount to 3 to 5 min and 0.06° to 0.09°, respectively.

Fig. 9. Illustration of (a) the programmed XRD peak width analysis with intervals of data fitting and (b) the evaluation of the *FWHM* value for the diffraction line of Figure 9a

#### **3.2.5 Completion of investigation methods for material response analysis**

It becomes clear in the following that the reliable interpretation of the measured depth distributions of residual stresses and XRD peak width, aside from optional auxiliary retained austenite determinations to further characterize material aging (Gegner, 2006a;

Tribological Aspects of Rolling Bearing Failures 45

Fig. 10. Three stage model of subsurface RCF with XRD peak width ratio based indication of dark etching region (DER) formation in the microstructure and *L*10 life calibration (DGBB)

Fig. 11. Three stage model of surface RCF with XRD peak width ratio based DER indication and actual *L*10 life calibration (roller bearings) that refers to the higher loaded inner ring

the progress of material loading in rolling contact fatigue with running time, expressed by the number *N* of inner ring revolutions. The changes are best described by the development of the maximum compressive residual stress, min <sup>σ</sup>res , and the RCF damage parameter, *b*/*B*, measured respectively in the depth and on (or near) the surface. The underlying alterations of the σres(*z*) and *FWHM*(*z*) distributions are demonstrated in Figure 12 for competing failure modes. The characteristic values are indicated in the profiles that in the subsurface region of classical RCF reveal an asymmetry towards higher depths (cf. Figure 1). The response of the steel to rolling contact loading is divided into the three stages of mechanical conditioning shakedown (1), damage incubation steady state (2), and material softening instability (3). Figures 10 to 12 provide schematic illustrations. The prevalently observed re-reduction of the compressive residual stresses in the instability phase of the surface mode, particularly

Jatckak & Larson, 1980), requires supportive investigation techniques for the condition of the raceway surface, microstructure, and oil or grease. Visual inspection, failure metallography, imaging and analytical scanning electron microscopy (SEM) and infrared spectroscopy of used lubricants are employed. Concrete examples of the application of these additional examination methods in the framework of XRD based material response bearing performance analysis are also discussed extensively in the literature (Gegner, 2006a; Gegner et al., 2007; Gegner & Nierlich, 2008, 2011a, 2011b, 2011c; Nierlich et al., 1992; Nierlich & Gegner, 2002, 2006, 2008).

#### **3.3 Evaluation methodology of XRD material response analysis**

The XRD peak width based *Schweinfurt* material response analysis (MRA) provides a powerful investigation tool for run rolling bearings. An actual life calibrated estimation of the loading conditions in the (near-) surface and subsurface failure mode represents the key feature of the evaluation conception (Nierlich et al., 1992; Voskamp, 1998).

The random nature of the effect of the large number of unpredictably distributed defects in the steel indicates a statistical risk evaluation of the failure of rolling bearings (Ioannides & Harris, 1985; Lundberg & Palmgren, 1947, 1952). The Weibull lifetime distribution is suitable for machine elements. The established mechanical engineering approach to RCF deals with stress field analyses on the basis, for instance, of tensor invariants or mean values (Böhmer et al., 1999; Desimone et al., 2006). On the microscopic level, however, the material experiences strain development when exposed to cyclic loading, which suggests a quantitative evaluation of the changes in XRD peak width during operation (Nierlich et al, 1992). Disregarding the intrinsic instrumental fraction, the physical broadening of an X-ray diffraction line is connected with the microstructural condition of the analyzed material (region) by several size and strain influences (Balzar, 1999). The peak width thus represents a measuring quantity for changing properties and densities of crystal defects. Lattice distortion provides the dominating contribution to the high line broadening of hardened steels. The average dimension of the coherently diffracting domains in martensite amounts to about 100 to 200 nm. Therefore, the XRD peak width is not directly correlated with the prior austenite grain size of few µm. The observed reduction of the line broadening by plastic deformation signifies a decrease of the lattice distortion. The minimum XRD peak width ratio, *b*/*B*, is the calibrated damage parameter of rolling contact fatigue that links materials to mechanical engineering (Weibull) failure analysis. The derived XRD equivalent values of the actual (experimental) *L*10 life at 90% survival probability (rating reliability) of a bearing population equal about 0.64 for the subsurface as well as 0.83 and 0.86, respectively for ball and roller bearings, for the surface mode of RCF (Gegner, 2006a; Gegner et al., 2007; Nierlich et al., 1992; Voskamp, 1998). Figures 10 and 11 display *b*/*B* data from calibrating rig tests. Here, *b* and *B* respectively denote the minimum *FWHM* in the depth region relevant to the considered (subsurface or near-surface) failure mode and the initial *FWHM* value. *B* is taken approximately in the core of the material or can be measured separately, e.g. below the shoulder of an examined bearing ring. The correlation between the statistical parameters representing a population of bearings under certain operation conditions and the state of aging damage (fatigue) of the steel matrix by the XRD peak width ratio measured on an accidentally selected part also reflects the intrinsic determinateness behind the randomness.

Based on Voskamp's three stage model for the subsurface and its extension to the surface failure mode (Gegner, 2006a; Voskamp, 1985, 1996), Figures 10 and 11 schematically illustrate

Jatckak & Larson, 1980), requires supportive investigation techniques for the condition of the raceway surface, microstructure, and oil or grease. Visual inspection, failure metallography, imaging and analytical scanning electron microscopy (SEM) and infrared spectroscopy of used lubricants are employed. Concrete examples of the application of these additional examination methods in the framework of XRD based material response bearing performance analysis are also discussed extensively in the literature (Gegner, 2006a; Gegner et al., 2007; Gegner & Nierlich, 2008, 2011a, 2011b, 2011c; Nierlich et al., 1992; Nierlich &

The XRD peak width based *Schweinfurt* material response analysis (MRA) provides a powerful investigation tool for run rolling bearings. An actual life calibrated estimation of the loading conditions in the (near-) surface and subsurface failure mode represents the key

The random nature of the effect of the large number of unpredictably distributed defects in the steel indicates a statistical risk evaluation of the failure of rolling bearings (Ioannides & Harris, 1985; Lundberg & Palmgren, 1947, 1952). The Weibull lifetime distribution is suitable for machine elements. The established mechanical engineering approach to RCF deals with stress field analyses on the basis, for instance, of tensor invariants or mean values (Böhmer et al., 1999; Desimone et al., 2006). On the microscopic level, however, the material experiences strain development when exposed to cyclic loading, which suggests a quantitative evaluation of the changes in XRD peak width during operation (Nierlich et al, 1992). Disregarding the intrinsic instrumental fraction, the physical broadening of an X-ray diffraction line is connected with the microstructural condition of the analyzed material (region) by several size and strain influences (Balzar, 1999). The peak width thus represents a measuring quantity for changing properties and densities of crystal defects. Lattice distortion provides the dominating contribution to the high line broadening of hardened steels. The average dimension of the coherently diffracting domains in martensite amounts to about 100 to 200 nm. Therefore, the XRD peak width is not directly correlated with the prior austenite grain size of few µm. The observed reduction of the line broadening by plastic deformation signifies a decrease of the lattice distortion. The minimum XRD peak width ratio, *b*/*B*, is the calibrated damage parameter of rolling contact fatigue that links materials to mechanical engineering (Weibull) failure analysis. The derived XRD equivalent values of the actual (experimental) *L*10 life at 90% survival probability (rating reliability) of a bearing population equal about 0.64 for the subsurface as well as 0.83 and 0.86, respectively for ball and roller bearings, for the surface mode of RCF (Gegner, 2006a; Gegner et al., 2007; Nierlich et al., 1992; Voskamp, 1998). Figures 10 and 11 display *b*/*B* data from calibrating rig tests. Here, *b* and *B* respectively denote the minimum *FWHM* in the depth region relevant to the considered (subsurface or near-surface) failure mode and the initial *FWHM* value. *B* is taken approximately in the core of the material or can be measured separately, e.g. below the shoulder of an examined bearing ring. The correlation between the statistical parameters representing a population of bearings under certain operation conditions and the state of aging damage (fatigue) of the steel matrix by the XRD peak width ratio measured on an accidentally selected part also reflects the intrinsic

Based on Voskamp's three stage model for the subsurface and its extension to the surface failure mode (Gegner, 2006a; Voskamp, 1985, 1996), Figures 10 and 11 schematically illustrate

**3.3 Evaluation methodology of XRD material response analysis** 

feature of the evaluation conception (Nierlich et al., 1992; Voskamp, 1998).

Gegner, 2002, 2006, 2008).

determinateness behind the randomness.

Fig. 10. Three stage model of subsurface RCF with XRD peak width ratio based indication of dark etching region (DER) formation in the microstructure and *L*10 life calibration (DGBB)

Fig. 11. Three stage model of surface RCF with XRD peak width ratio based DER indication and actual *L*10 life calibration (roller bearings) that refers to the higher loaded inner ring

the progress of material loading in rolling contact fatigue with running time, expressed by the number *N* of inner ring revolutions. The changes are best described by the development of the maximum compressive residual stress, min <sup>σ</sup>res , and the RCF damage parameter, *b*/*B*, measured respectively in the depth and on (or near) the surface. The underlying alterations of the σres(*z*) and *FWHM*(*z*) distributions are demonstrated in Figure 12 for competing failure modes. The characteristic values are indicated in the profiles that in the subsurface region of classical RCF reveal an asymmetry towards higher depths (cf. Figure 1). The response of the steel to rolling contact loading is divided into the three stages of mechanical conditioning shakedown (1), damage incubation steady state (2), and material softening instability (3). Figures 10 to 12 provide schematic illustrations. The prevalently observed re-reduction of the compressive residual stresses in the instability phase of the surface mode, particularly

Tribological Aspects of Rolling Bearing Failures 47

maximum orthogonal shear stress (Lundberg & Palmgren, 1947, 1952). However, the v. Mises equivalent stress, by which residual stress formation is governed, as well as each principal normal stress (cf. Figure 1) are pulsating in time. In the region of classical RCF below the raceway, the minimum XRD peak width occurs significantly closer to the surface than the maximum compressive residual stress (Gegner & Nierlich, 2011b; Schlicht et al., 1987). It is discussed in the literature which material failure hypothesis is best suited for predicting RCF loading (Gohar, 2001; Harris, 2001): Lundberg and Palmgren use the orthogonal shear stress approach but other authors prefer the Huber-von Mises-Hencky distortion or deformation energy hypothesis (Broszeit et al., 1986). The well-founded conclusion from the XRD material response analyses interconnects both views in a kind of paradox (Gegner, 2006a): whereas residual stress formation and the beginning of plastification conform to the distortion energy hypothesis, RCF material aging and damage evolution in the steel matrix, manifested in the XRD peak width reduction, responds to the

The detected location of highest damage of the steel matrix agrees with the observation that under ideal EHL rolling contact loading most fatigue cracks are initiated near the orthog.

depth (Lundberg & Palmgren, 1947). It is recently reported that the frequency of fracturing of sulfide inclusions in bearing operation due to the influence of the subsurface compressive stress field also correlates well with the distance distribution of the orthogonal shear stress below the raceway (Brückner et al., 2011). The three stages of the associated mechanism of butterfly formation, which occurs from a Hertzian pressure of about 1400 MPa, are documented in Figure 13: fracturing of the MnS inclusion (1), microcrack extension into the bulk material (2), development of a white etching wing microstructure along the crack (3). The light optical micrograph (LOM) and SEM image of Figures 14a and 14b, respectively, reveal in a radial (i.e., circumferential) microsection how the white etching area (WEA) of the butterfly wing virtually *emanates* from the matrix zone in contact with the pore like material separation of the initially fractured MnS inclusion into the surrounding steel

Fig. 13. Butterfly formation on sulfide inclusions observed in etched axial microsections of the outer ring of a CRB of an industrial gearbox after a passed rig test at *p*0=1450 MPa

0 *z*

alternating orthogonal shear stress.

microstructure.

typical of mixed friction running conditions, suggests relaxation processes. From experience, a residual stress limit of about –200 MPa is usually not exceeded, as included in the diagrams of Figures 11 and 12. The conventional logarithmic plot overemphasizes the differences in the slopes between the constant and the decreasing curves in the steady state and the instability stage of Figures 10 and 11. The existence of a third phase, however, is indicated by the reversal of the residual stress on the surface (cf. Figures 11 and 12) and also found in RCF component rig tests (Rollmann, 2000).

The first stage of shakedown is characterized by microplastic deformation and the quick build-up of compressive residual stresses when the yield strength, *R*p0.2, of the hardened steel is locally exceeded by the v. Mises equivalent stress representing the triaxial stress field in rolling contact loading (cf. section 2). Short-cycle cold working processes of dislocation rearrangement with material alteration restricted to the higher fatigue endurance limit, in which carbon diffusion is not involved, cause rapid mechanical conditioning (Nierlich & Gegner, 2008). Further details are discussed in section 4.2. The second stage of steady state arises as long as the applied load falls below the shakedown limit so that ratcheting is avoided (Johnson & Jefferis, 1963; Voskamp, 1996; Yhland, 1983). In this period of fatigue damage incubation, no significant microstructure, residual stress and XRD peak width alterations are observed. Elastic behavior of the pre-conditioned microstrained material is assumed. In the extended final instability stage, gradual microstructure changes occur (Voskamp, 1996). The phase transformations require diffusive redistribution of carbon on a micro scale, which is assisted by plastification. From *FWHM*/*B* of about 0.83 to 0.85 downwards, a dark etching region (DER) occurs in the microstructure by martensite decay. Note that this is in the range of the XRD *L*10 value for the surface failure mode but well before this life equivalent is reached for subsurface RCF (cf. Figures 11 and 12).

Fig. 12. Schematic residual stress and XRD peak width change with rising *N* during subsurface and surface RCF and prediction of the respective depth ranges (gray) of DER formation

Fatigue is damage (defect) accumulation under cyclic loading. Microplastic deformation is reflected in the XRD line broadening. The observed reduction of the peak width signifies a decrease of the lattice distortion. For describing subsurface RCF failure, the established Lundberg-Palmgren bearing life theory defines the risk volume of damage initiation on microstructural defects by the effect of an alternating load, thus referring to the depth of

typical of mixed friction running conditions, suggests relaxation processes. From experience, a residual stress limit of about –200 MPa is usually not exceeded, as included in the diagrams of Figures 11 and 12. The conventional logarithmic plot overemphasizes the differences in the slopes between the constant and the decreasing curves in the steady state and the instability stage of Figures 10 and 11. The existence of a third phase, however, is indicated by the reversal of the residual stress on the surface (cf. Figures 11 and 12) and also

The first stage of shakedown is characterized by microplastic deformation and the quick build-up of compressive residual stresses when the yield strength, *R*p0.2, of the hardened steel is locally exceeded by the v. Mises equivalent stress representing the triaxial stress field in rolling contact loading (cf. section 2). Short-cycle cold working processes of dislocation rearrangement with material alteration restricted to the higher fatigue endurance limit, in which carbon diffusion is not involved, cause rapid mechanical conditioning (Nierlich & Gegner, 2008). Further details are discussed in section 4.2. The second stage of steady state arises as long as the applied load falls below the shakedown limit so that ratcheting is avoided (Johnson & Jefferis, 1963; Voskamp, 1996; Yhland, 1983). In this period of fatigue damage incubation, no significant microstructure, residual stress and XRD peak width alterations are observed. Elastic behavior of the pre-conditioned microstrained material is assumed. In the extended final instability stage, gradual microstructure changes occur (Voskamp, 1996). The phase transformations require diffusive redistribution of carbon on a micro scale, which is assisted by plastification. From *FWHM*/*B* of about 0.83 to 0.85 downwards, a dark etching region (DER) occurs in the microstructure by martensite decay. Note that this is in the range of the XRD *L*10 value for the surface failure mode but well

before this life equivalent is reached for subsurface RCF (cf. Figures 11 and 12).

Fig. 12. Schematic residual stress and XRD peak width change with rising *N* during subsurface and surface RCF and prediction of the respective depth ranges (gray) of DER formation

Fatigue is damage (defect) accumulation under cyclic loading. Microplastic deformation is reflected in the XRD line broadening. The observed reduction of the peak width signifies a decrease of the lattice distortion. For describing subsurface RCF failure, the established Lundberg-Palmgren bearing life theory defines the risk volume of damage initiation on microstructural defects by the effect of an alternating load, thus referring to the depth of

found in RCF component rig tests (Rollmann, 2000).

maximum orthogonal shear stress (Lundberg & Palmgren, 1947, 1952). However, the v. Mises equivalent stress, by which residual stress formation is governed, as well as each principal normal stress (cf. Figure 1) are pulsating in time. In the region of classical RCF below the raceway, the minimum XRD peak width occurs significantly closer to the surface than the maximum compressive residual stress (Gegner & Nierlich, 2011b; Schlicht et al., 1987). It is discussed in the literature which material failure hypothesis is best suited for predicting RCF loading (Gohar, 2001; Harris, 2001): Lundberg and Palmgren use the orthogonal shear stress approach but other authors prefer the Huber-von Mises-Hencky distortion or deformation energy hypothesis (Broszeit et al., 1986). The well-founded conclusion from the XRD material response analyses interconnects both views in a kind of paradox (Gegner, 2006a): whereas residual stress formation and the beginning of plastification conform to the distortion energy hypothesis, RCF material aging and damage evolution in the steel matrix, manifested in the XRD peak width reduction, responds to the alternating orthogonal shear stress.

The detected location of highest damage of the steel matrix agrees with the observation that under ideal EHL rolling contact loading most fatigue cracks are initiated near the orthog. 0 *z* depth (Lundberg & Palmgren, 1947). It is recently reported that the frequency of fracturing of sulfide inclusions in bearing operation due to the influence of the subsurface compressive stress field also correlates well with the distance distribution of the orthogonal shear stress below the raceway (Brückner et al., 2011). The three stages of the associated mechanism of butterfly formation, which occurs from a Hertzian pressure of about 1400 MPa, are documented in Figure 13: fracturing of the MnS inclusion (1), microcrack extension into the bulk material (2), development of a white etching wing microstructure along the crack (3). The light optical micrograph (LOM) and SEM image of Figures 14a and 14b, respectively, reveal in a radial (i.e., circumferential) microsection how the white etching area (WEA) of the butterfly wing virtually *emanates* from the matrix zone in contact with the pore like material separation of the initially fractured MnS inclusion into the surrounding steel microstructure.

Fig. 13. Butterfly formation on sulfide inclusions observed in etched axial microsections of the outer ring of a CRB of an industrial gearbox after a passed rig test at *p*0=1450 MPa

Tribological Aspects of Rolling Bearing Failures 49

Critical butterfly wing growth up to the surface (see Figure 15), which leads to bearing failure by raceway spalling eventually, occurs very rarely (Schreiber, 1992). The metallurgically unweakened steel matrix in some distance to the inclusion can cause crack arrest. Multiple damage initiation, however, is found in the final stage of rolling contact fatigue. Subsurface cracks may then reach the raceway (Voskamp, 1996). Butterfly RCF damage develops by the microstructural transformation of low-temperature dynamic recrystallization of the highly strained regions along cracks rapidly initiated on stress raising nonmetallic inclusions in the steel (Böhm et al., 1975; Brückner et al., 2011; Furumura et al., 1993; Österlund et al., 1982; Schlicht et al., 1987; Voskamp, 1996), If this localized fatigue process occurs at Hertzian pressures below 2500 MPa (Brückner et al., 2011; Vincent et al., 1998), it is not recognizable alone by an XRD analysis that is sensitive to integral

orthogonal shear stress and its double amplitude depend on the footprint ratio between the semiminor and the semimajor axis of the pressure ellipse (Harris, 2001; Palmgren, 1964): the values respectively amount to 0.5×*a* and 0.5×*p*0 in line contact and are slightly lower for ball

minimum *b* significantly closer to the surface than the residual stresses, as it is illustrated in Figure 12 and apparent from the practical example of Figure 16a. This finding is exploited for XRD material response analysis (Gegner, 2006a). The residual stress and XRD peak width distributions are evaluated jointly in the subsurface region of classical rolling contact fatigue by applying the v. Mises and orthogonal shear stress interdependently. Data analysis is demonstrated in Figures 16a and 16b. Adjusting to the best fit improves the accuracy of deducing the Hertzian pressure *p*0 from the measured profiles. Superposition with the load stresses results in a slight gradual shift of the residual stress and XRD peak width distribution to larger depths with run duration (Voskamp, 1996), which is neglected in the evaluation (see Figure 12). In the example of Figure 16a, material aging is within the scattering range of the *L*10 life equivalent value for both, thus in this case competing, failure

<sup>0</sup> <sup>0</sup> *z az* = ×< 0.5 follows that the *FWHM* distance curve reaches its

Fig. 16. Graphical representation of (a) the residual stress and XRD peak width depth distribution measured below the IR raceway of a DGBB tested in an automobile gearbox rig with indication of the initial as-delivered condition and (b) the joint subsurface profile

<sup>0</sup> *z* of the maximum of the alternating

material loading (see section 3.2).

bearings. From orthog. v.Mises

evaluation

According to the Hertz theory, the depth orthog.

Butterflies become relevant in the upper bearing life range above *L*10. Inclusions of different chemical composition, shape, size, mechanical properties and surrounding residual stresses are technically unavoidable in steels from the manufacturing process. The potential for their reduction is limited also from an economic viewpoint and virtually fully tapped in the today's high cleanliness bearing grades. Local peak stresses on nonmetallic inclusions, i.e. internal metallurgical notches, below the contact surface can cause the initiation of microcracks. Operational fracture of embedded MnS particles (see Figures 13, 14) is quite often observed and represents a potential butterfly formation mechanism besides, e.g., decohesion of the interphase (Brückner et al., 2011). Subsequent fatigue crack propagation is driven by the acting main shear stress (Schlicht et al., 1987, 1988; Takemura & Murakami, 1995). The growing butterfly wings thus follow the direction of ideally 45° to the raceway tangent. Figure 15 shows a textbook example from a weaving machine gearbox bearing at around the nominal *L*50 life. The overrolling direction in the micrograph is from right to left. The white etching constituents show an extreme hardness of about 75 HRC (1200 HV) and consist of carbide-free nearly amorphous to nano-granular ferrite with grain sizes up to 20 to 30 nm.

Fig. 14. LOM micrograph (a) and corresponding SEM-SE image (b) of butterfly development on a cracked MnS inclusion in the etched radial microsection of the stationary outer ring of a spherical roller bearing (SRB) after a passed rig test at a Hertzian pressure *p*0 of 2400 MPa

Fig. 15. Butterfly wing growth from the depth to the raceway surface in overrolling direction (right-to-left) in the etched radial microsection of the IR of a CRB loaded at *p*0=1800 MPa

Butterflies become relevant in the upper bearing life range above *L*10. Inclusions of different chemical composition, shape, size, mechanical properties and surrounding residual stresses are technically unavoidable in steels from the manufacturing process. The potential for their reduction is limited also from an economic viewpoint and virtually fully tapped in the today's high cleanliness bearing grades. Local peak stresses on nonmetallic inclusions, i.e. internal metallurgical notches, below the contact surface can cause the initiation of microcracks. Operational fracture of embedded MnS particles (see Figures 13, 14) is quite often observed and represents a potential butterfly formation mechanism besides, e.g., decohesion of the interphase (Brückner et al., 2011). Subsequent fatigue crack propagation is driven by the acting main shear stress (Schlicht et al., 1987, 1988; Takemura & Murakami, 1995). The growing butterfly wings thus follow the direction of ideally 45° to the raceway tangent. Figure 15 shows a textbook example from a weaving machine gearbox bearing at around the nominal *L*50 life. The overrolling direction in the micrograph is from right to left. The white etching constituents show an extreme hardness of about 75 HRC (1200 HV) and consist of carbide-free nearly amorphous to nano-granular ferrite with grain sizes up to 20 to

 Fig. 14. LOM micrograph (a) and corresponding SEM-SE image (b) of butterfly development on a cracked MnS inclusion in the etched radial microsection of the stationary outer ring of a spherical roller bearing (SRB) after a passed rig test at a Hertzian pressure *p*0 of 2400 MPa

Fig. 15. Butterfly wing growth from the depth to the raceway surface in overrolling direction (right-to-left) in the etched radial microsection of the IR of a CRB loaded at *p*0=1800 MPa

30 nm.

Critical butterfly wing growth up to the surface (see Figure 15), which leads to bearing failure by raceway spalling eventually, occurs very rarely (Schreiber, 1992). The metallurgically unweakened steel matrix in some distance to the inclusion can cause crack arrest. Multiple damage initiation, however, is found in the final stage of rolling contact fatigue. Subsurface cracks may then reach the raceway (Voskamp, 1996). Butterfly RCF damage develops by the microstructural transformation of low-temperature dynamic recrystallization of the highly strained regions along cracks rapidly initiated on stress raising nonmetallic inclusions in the steel (Böhm et al., 1975; Brückner et al., 2011; Furumura et al., 1993; Österlund et al., 1982; Schlicht et al., 1987; Voskamp, 1996), If this localized fatigue process occurs at Hertzian pressures below 2500 MPa (Brückner et al., 2011; Vincent et al., 1998), it is not recognizable alone by an XRD analysis that is sensitive to integral material loading (see section 3.2).

According to the Hertz theory, the depth orthog. <sup>0</sup> *z* of the maximum of the alternating orthogonal shear stress and its double amplitude depend on the footprint ratio between the semiminor and the semimajor axis of the pressure ellipse (Harris, 2001; Palmgren, 1964): the values respectively amount to 0.5×*a* and 0.5×*p*0 in line contact and are slightly lower for ball bearings. From orthog. v.Mises <sup>0</sup> <sup>0</sup> *z az* = ×< 0.5 follows that the *FWHM* distance curve reaches its minimum *b* significantly closer to the surface than the residual stresses, as it is illustrated in Figure 12 and apparent from the practical example of Figure 16a. This finding is exploited for XRD material response analysis (Gegner, 2006a). The residual stress and XRD peak width distributions are evaluated jointly in the subsurface region of classical rolling contact fatigue by applying the v. Mises and orthogonal shear stress interdependently. Data analysis is demonstrated in Figures 16a and 16b. Adjusting to the best fit improves the accuracy of deducing the Hertzian pressure *p*0 from the measured profiles. Superposition with the load stresses results in a slight gradual shift of the residual stress and XRD peak width distribution to larger depths with run duration (Voskamp, 1996), which is neglected in the evaluation (see Figure 12). In the example of Figure 16a, material aging is within the scattering range of the *L*10 life equivalent value for both, thus in this case competing, failure

Fig. 16. Graphical representation of (a) the residual stress and XRD peak width depth distribution measured below the IR raceway of a DGBB tested in an automobile gearbox rig with indication of the initial as-delivered condition and (b) the joint subsurface profile evaluation

Tribological Aspects of Rolling Bearing Failures 51

The influence of hydrogen on rolling contact fatigue is also quantifiable this way, as applied

The characteristic subsurface microstructural alterations in hardened bearing steels occur due to shear induced carbon diffusion mediated phase transformations (Voskamp, 1996), for which a mechanistic metal physics model is introduced in the following. The local material fatigue aging of butterfly formation is already discussed in section 3.3. In Figures 18a to 18c, the XRD material response analysis of a rig tested automobile gearbox ball bearing is evaluated in the region of subsurface RCF. A Hertzian pressure of 3400 MPa is deduced. The joint interdependent profile evaluation is shown in Figure 18b. At the found relative decrease of the X-ray diffraction peak width to *b*/*B*≈0.71, i.e. still above the XRD *L*10 life equivalent value of roughly 0.64, rolling contact fatigue produces a distinct DER in the microstructure in the depth range predicted by the *FWHM*/*B* reduction below 0.84 (cf., Figures 10, 12 and 17a). This exact agreement is emphasized by a comparison of Figures 18a

Fig. 18. Subsurface RCF analysis of the IR of a run DGBB including (a) the measured depth distribution of residual stress and XRD peak width (*b*/*B*≈0.71) with DER prediction, (b) the joint XRD profile evaluation and (c) an etched axial microsection with actual DER extension Spatial differences in the etching behavior of the bearing steel matrix in metallographic microsections caused by high shear stresses below the raceway surface after a certain stage of material aging by cyclic rolling contact loading are known since 1946 (Jones, 1946). The localized weakening structural changes result from stress induced gradual partial decay of martensite into heavily plasticized ferrite, the development of regular deformation slip bands and alterations in the carbide morphology (Schlicht et al., 1987; Voskamp, 1996). Due to the appearance of the damaged zones after metallographic preparation in an optical microscope, these areas are referred to as dark etching regions (Swahn et al., 1976a). The small decrease in specific volume of less than 1% by martensite decomposition results in a tensile contribution to operational residual stress formation but the effects of opposed austenite decay and local yield strength reduction by phase transformation prevail (Voskamp, 1996). Recent reheating experiments also point to diffusion reallocation of carbon atoms from (partially) dissolving temper as well as globular carbides for dislocational

**4.1 Microstructural changes during subsurface rolling contact fatigue** 

to classical RCF in section 4.3.

and 18c.

modes of surface (*b*/*B*≈0.83) and subsurface RCF (*b*/*B*≈0.64): a relative XRD peak width reduction of *b*/*B*≥0.82 and *b*/*B*=0.67 is respectively taken from the diagram. The greater-orequal sign for the estimation of the surface failure mode considers the unknown small *FWHM* decrease on the raceway due to grinding and honing of the hardened steel in the asfinished condition (see Figures 12 and 16a) so that the alternatively used reference *B* in the core of the material or another uninfluenced region (e.g., below the shoulder of a bearing ring) exceeds the actual initial value at *z*=0 typically by about 0.02°. The original residual stress and XRD peak width level below the edge zone results from the heat treatment. The inner ring of Figure 16a, for instance, is made out of martensitically through hardened bearing steel.

The predicted dark etching regions at the surface and in a depth between 40 and 400 µm are well confirmed by failure metallography, as evident from a comparison of Figure 16a with Figures 17a and 17b. The DER-free intermediate layer is clearly visible in the overview micrograph. The dark etching region near the surface ranges to about 10 to 12 µm depth.

Fig. 17. LOM images of (a) the etched axial microsection of the inner ring of Figure 16a with evaluation of the extended subsurface DER and (b) a detail revealing the near-surface DER

#### **4. Subsurface rolling contact fatigue**

Since the historical beginnings with *August Wöhler* in the middle of the 19th century, today's research on material fatigue can draw from extensive experiences. Cyclic stressing in rolling contact, however, even eludes a theoretical description based on advanced multiaxial damage criteria, such as the Dang Van critical plane approach (Ciavarella et al., 2006; Desimone et al., 2006). Although little noticed in the young research field of very high cycle fatigue (VHCF) so far, RCF is the most important type of VHCF in engineering practice. Complex VHCF conditions occur under rapid load changes. The inhomogeneous triaxial stress state exhibits a large fraction of hydrostatic pressure *p*h=−(σ*x*+σ*y*+σ*z*)/3 (see Figure 1, maximum on the surface) and, in the ideal case of pure radial force transfer, no critical tensile stresses, which is favorable to brittle materials and makes the hardened steel behave ductilely. The number of cycles to failure defining the rolling bearing life is thus by orders of magnitude larger than in comparable push-pull or rotating bending loading (Voskamp, 1996). The RCF performance of hardened steels is difficult to predict. Fatigue damage evolution by gradual accumulation of microplasticity is associated with increasing probability of crack initiation and failure. Microstructural changes during RCF are usually evaluated as a function of the number of ring revolutions (Voskamp, 1996). For the scaled comparison of differently loaded bearings, however, the material inherent RCF progress measure of the minimum XRD peak width ratio, *b*/*B*, is more appropriate as it correlates with the statistical parameters of the Weibull life distribution of a fictive lot (see section 3.3).

modes of surface (*b*/*B*≈0.83) and subsurface RCF (*b*/*B*≈0.64): a relative XRD peak width reduction of *b*/*B*≥0.82 and *b*/*B*=0.67 is respectively taken from the diagram. The greater-orequal sign for the estimation of the surface failure mode considers the unknown small *FWHM* decrease on the raceway due to grinding and honing of the hardened steel in the asfinished condition (see Figures 12 and 16a) so that the alternatively used reference *B* in the core of the material or another uninfluenced region (e.g., below the shoulder of a bearing ring) exceeds the actual initial value at *z*=0 typically by about 0.02°. The original residual stress and XRD peak width level below the edge zone results from the heat treatment. The inner ring of Figure 16a, for instance, is made out of martensitically through hardened

The predicted dark etching regions at the surface and in a depth between 40 and 400 µm are well confirmed by failure metallography, as evident from a comparison of Figure 16a with Figures 17a and 17b. The DER-free intermediate layer is clearly visible in the overview micrograph. The dark etching region near the surface ranges to about 10 to 12 µm depth.

 Fig. 17. LOM images of (a) the etched axial microsection of the inner ring of Figure 16a with evaluation of the extended subsurface DER and (b) a detail revealing the near-surface DER

Since the historical beginnings with *August Wöhler* in the middle of the 19th century, today's research on material fatigue can draw from extensive experiences. Cyclic stressing in rolling contact, however, even eludes a theoretical description based on advanced multiaxial damage criteria, such as the Dang Van critical plane approach (Ciavarella et al., 2006; Desimone et al., 2006). Although little noticed in the young research field of very high cycle fatigue (VHCF) so far, RCF is the most important type of VHCF in engineering practice. Complex VHCF conditions occur under rapid load changes. The inhomogeneous triaxial stress state exhibits a large fraction of hydrostatic pressure *p*h=−(σ*x*+σ*y*+σ*z*)/3 (see Figure 1, maximum on the surface) and, in the ideal case of pure radial force transfer, no critical tensile stresses, which is favorable to brittle materials and makes the hardened steel behave ductilely. The number of cycles to failure defining the rolling bearing life is thus by orders of magnitude larger than in comparable push-pull or rotating bending loading (Voskamp, 1996). The RCF performance of hardened steels is difficult to predict. Fatigue damage evolution by gradual accumulation of microplasticity is associated with increasing probability of crack initiation and failure. Microstructural changes during RCF are usually evaluated as a function of the number of ring revolutions (Voskamp, 1996). For the scaled comparison of differently loaded bearings, however, the material inherent RCF progress measure of the minimum XRD peak width ratio, *b*/*B*, is more appropriate as it correlates with the statistical parameters of the Weibull life distribution of a fictive lot (see section 3.3).

bearing steel.

**4. Subsurface rolling contact fatigue** 

The influence of hydrogen on rolling contact fatigue is also quantifiable this way, as applied to classical RCF in section 4.3.

#### **4.1 Microstructural changes during subsurface rolling contact fatigue**

The characteristic subsurface microstructural alterations in hardened bearing steels occur due to shear induced carbon diffusion mediated phase transformations (Voskamp, 1996), for which a mechanistic metal physics model is introduced in the following. The local material fatigue aging of butterfly formation is already discussed in section 3.3. In Figures 18a to 18c, the XRD material response analysis of a rig tested automobile gearbox ball bearing is evaluated in the region of subsurface RCF. A Hertzian pressure of 3400 MPa is deduced. The joint interdependent profile evaluation is shown in Figure 18b. At the found relative decrease of the X-ray diffraction peak width to *b*/*B*≈0.71, i.e. still above the XRD *L*10 life equivalent value of roughly 0.64, rolling contact fatigue produces a distinct DER in the microstructure in the depth range predicted by the *FWHM*/*B* reduction below 0.84 (cf., Figures 10, 12 and 17a). This exact agreement is emphasized by a comparison of Figures 18a and 18c.

Fig. 18. Subsurface RCF analysis of the IR of a run DGBB including (a) the measured depth distribution of residual stress and XRD peak width (*b*/*B*≈0.71) with DER prediction, (b) the joint XRD profile evaluation and (c) an etched axial microsection with actual DER extension

Spatial differences in the etching behavior of the bearing steel matrix in metallographic microsections caused by high shear stresses below the raceway surface after a certain stage of material aging by cyclic rolling contact loading are known since 1946 (Jones, 1946). The localized weakening structural changes result from stress induced gradual partial decay of martensite into heavily plasticized ferrite, the development of regular deformation slip bands and alterations in the carbide morphology (Schlicht et al., 1987; Voskamp, 1996). Due to the appearance of the damaged zones after metallographic preparation in an optical microscope, these areas are referred to as dark etching regions (Swahn et al., 1976a). The small decrease in specific volume of less than 1% by martensite decomposition results in a tensile contribution to operational residual stress formation but the effects of opposed austenite decay and local yield strength reduction by phase transformation prevail (Voskamp, 1996). Recent reheating experiments also point to diffusion reallocation of carbon atoms from (partially) dissolving temper as well as globular carbides for dislocational

Tribological Aspects of Rolling Bearing Failures 53

Fig. 20. Subsurface RCF analysis of the IR of two run DGBB (N° 1, N° 2) including (a) the evaluated depth distribution of residual stress and XRD peak width (N° 1: *b*/*B*≈0.61, N° 2: *b*/*B*≈0.57, the given *B* values reflect different tempering temperature of martensite hardening of bearing steel) with DER prediction, (b) an etched axial microsection of IR-N° 1 and (c) an etched radial microsection of IR-N° 2, respectively with DER indication and

Fig. 21. SEM-SE detail of (a) Figure 20b (preparatively initiated cracks expose the DER) and (b) Figure 20c (βf = 22°) and (c) an etched radial microsection of the IR of a DGBB rig tested at a Hertzian pressure of 3700 MPa with indicated depth of maximum orthogonal shear

the preparatively lacerated material from the chemical attack by the etching process, acts as precursor of WEB formation (dark appearing phase, SEM-SE). The angles βf are determined

visible FWB

stress

segregation in severely deformed regions (Gegner et al., 2009), which is assumed to be inducible by cyclic material loading in rolling contact (see section 4.2).

The overall quite uniformly appearing DER (see Figures 17a and 18c) is displayed at higher magnification in the LOM micrograph of Figure 19a. On the micrometer scale, affected dark etching material evidently occurs locally preferred in zones of dense secondary cementite. As well as the spatial and size distribution of the precipitation hardening carbides, microsegregations (e.g., C, Cr) influence the formation of the DER spots.

Subsurface fatigue cracks usually advance in circumferential, i.e. overrolling, direction parallel to the raceway tangent in the early stage of their propagation (Lundberg & Palmgren, 1947), as exemplified in Figure 19b (Voskamp, 1996). The aged matrix material of the dark etching region exhibits embrittlement (see also section 5.5) that is most pronounced around the depth of maximum orthogonal shear stress, where the indicative X-ray diffraction line width is minimal and the microstructure reveals intense response to the damage sensitive preparative chemical etching process.

Fig. 19. LOM micrographs of (a) a detail of the DER of Figure 17a and (b) typical subsurface fatigue crack propagation parallel to the raceway around the depth of maximum orthogonal shear stress in the etched radial microsection of the inner ring of a deep groove ball bearing

In the upper subsurface RCF life range of the instability stage above the XRD *L*10 equivalent value, i.e. *b*/*B*<0.64 according to Figure 10, shear localization and dynamic recrystallization (DRX) induce (100)[110] and (111)[211] rolling textures that reflect the balance of plastic deformation and DRX (Voskamp, 1996). Regular flat white etching bands (WEB) of elongated parallel carbide-free ferritic stripes of inclination angles βf of 20° to 32° to the raceway tangent in overrolling direction occur inside the DER (Lindahl & Österlund, 1982; Swahn et al., 1976a, 1976b; Voskamp, 1996). For the automobile alternator and gearbox ball bearing from rig tests, N° 1 and N° 2 in Figure 20a, respectively, *b*/*B* equals about 0.61 and 0.57. Metallography of the investigated inner rings in Figures 20b and 20c confirms the dark etching region predicted by the relative XRD peak width reduction and indicates the discoid flat white bands (FWB) in the axial (N° 1) and radial microsection (N° 2).

Ferrite of the FWB is surrounded by reprecipitated highly carbon-rich carbides and remaining martensite (Lindahl & Österlund, 1982; Swahn et al., 1976a, 1976b). Note that the carbides originally dispersed in the hardened steel are dissolved in the WEB under the influence of the RCF damage mechanism (see section 4.2). The SEM images of Figures 21a and 21b imply that the aged DER microstructure, the embrittlement of which is reflected in

segregation in severely deformed regions (Gegner et al., 2009), which is assumed to be

The overall quite uniformly appearing DER (see Figures 17a and 18c) is displayed at higher magnification in the LOM micrograph of Figure 19a. On the micrometer scale, affected dark etching material evidently occurs locally preferred in zones of dense secondary cementite. As well as the spatial and size distribution of the precipitation hardening carbides, micro-

Subsurface fatigue cracks usually advance in circumferential, i.e. overrolling, direction parallel to the raceway tangent in the early stage of their propagation (Lundberg & Palmgren, 1947), as exemplified in Figure 19b (Voskamp, 1996). The aged matrix material of the dark etching region exhibits embrittlement (see also section 5.5) that is most pronounced around the depth of maximum orthogonal shear stress, where the indicative X-ray diffraction line width is minimal and the microstructure reveals intense response to the

 Fig. 19. LOM micrographs of (a) a detail of the DER of Figure 17a and (b) typical subsurface fatigue crack propagation parallel to the raceway around the depth of maximum orthogonal shear stress in the etched radial microsection of the inner ring of a deep groove ball bearing In the upper subsurface RCF life range of the instability stage above the XRD *L*10 equivalent value, i.e. *b*/*B*<0.64 according to Figure 10, shear localization and dynamic recrystallization (DRX) induce (100)[110] and (111)[211] rolling textures that reflect the balance of plastic deformation and DRX (Voskamp, 1996). Regular flat white etching bands (WEB) of elongated parallel carbide-free ferritic stripes of inclination angles βf of 20° to 32° to the raceway tangent in overrolling direction occur inside the DER (Lindahl & Österlund, 1982; Swahn et al., 1976a, 1976b; Voskamp, 1996). For the automobile alternator and gearbox ball bearing from rig tests, N° 1 and N° 2 in Figure 20a, respectively, *b*/*B* equals about 0.61 and 0.57. Metallography of the investigated inner rings in Figures 20b and 20c confirms the dark etching region predicted by the relative XRD peak width reduction and indicates the discoid

flat white bands (FWB) in the axial (N° 1) and radial microsection (N° 2).

Ferrite of the FWB is surrounded by reprecipitated highly carbon-rich carbides and remaining martensite (Lindahl & Österlund, 1982; Swahn et al., 1976a, 1976b). Note that the carbides originally dispersed in the hardened steel are dissolved in the WEB under the influence of the RCF damage mechanism (see section 4.2). The SEM images of Figures 21a and 21b imply that the aged DER microstructure, the embrittlement of which is reflected in

inducible by cyclic material loading in rolling contact (see section 4.2).

segregations (e.g., C, Cr) influence the formation of the DER spots.

damage sensitive preparative chemical etching process.

Fig. 20. Subsurface RCF analysis of the IR of two run DGBB (N° 1, N° 2) including (a) the evaluated depth distribution of residual stress and XRD peak width (N° 1: *b*/*B*≈0.61, N° 2: *b*/*B*≈0.57, the given *B* values reflect different tempering temperature of martensite hardening of bearing steel) with DER prediction, (b) an etched axial microsection of IR-N° 1 and (c) an etched radial microsection of IR-N° 2, respectively with DER indication and visible FWB

Fig. 21. SEM-SE detail of (a) Figure 20b (preparatively initiated cracks expose the DER) and (b) Figure 20c (βf = 22°) and (c) an etched radial microsection of the IR of a DGBB rig tested at a Hertzian pressure of 3700 MPa with indicated depth of maximum orthogonal shear stress

the preparatively lacerated material from the chemical attack by the etching process, acts as precursor of WEB formation (dark appearing phase, SEM-SE). The angles βf are determined

Tribological Aspects of Rolling Bearing Failures 55

Fig. 22. In the dislocation glide stability loss (DGSL) model of rolling contact fatigue, according to which gradual dissolution of (temper) carbides (spheres) occurs by diffusion (dotted arrows) mediated continuous carbon segregation at pinned dislocations (lines) bowing out under the influence of the cyclic shear stress τ (solid arrows), the smallest particles tend to disappear first due to their higher curvature-dependent surface energy so that the obstacles are passed successively and the level of localized microplasticity is

Rolling contact fatigue life is governed by the microcrack nucleation phase. Gradual dissolution of Fe2.2C temper carbides (spheres in Figure 22) driven by carbon segregation at initially pinned dislocations (lines), which bow out under the acting cyclic shear stress τ (arrows), causes successive overcoming of the obstacles and local restarting of plastic flow until activation of Frank-Read sources. Fatigue damage incubation in the steady state of apparent elastic material behavior is followed in the instability stage by the microstructural changes of DER formation, decay of globular secondary cementite (in the DGSL model due to dislocation-carbide interaction) and regular ferritic white etching bands developing inside the DER. Strain hardening, which embrittles the aged steel matrix and thus promotes crack initiation, compensates for the diminishing precipitation strengthening in the progress of rolling contact fatigue. This process results in further compressive residual stress build-up from the shakedown level and newly decreasing XRD peak width (see Figure 10). Gradual concentration of local microplasticity and microscopic accumulation of lattice defects characterize proceeding RCF damage. According to the DGSL model, Cottrell segregation of carbon atoms released from dissolving carbides at uncovered cores of dislocations, which are regeneratively generated by the glide movements during yielding, provides an additional contribution to the XRD peak width reduction by cyclic rolling contact loading (Gegner et al., 2009). The experimental proof of this essential prediction is discussed in detail below by means of Figures 23 and 24. The gradually increasing amount of localized dislocation microplasticity represents the fatigue defect accumulation mechanism of the DGSL model of RCF. It is thus associated with a rising probability for bearing failure (cf. Figure 10) due to material aging. The DGSL criterion for local microcracking is based on a critical dislocation density. Orientation and speed of fatigue crack propagation can then also

The proposed dislocation-carbide interaction mechanism explains (partial) fragmentation of uncuttable globular carbides of µm size, which is occasionally observed in microsections, and the increased energy level in the affected region. Localized microplastic deformation is related to energy dissipation. Note that the DGSL fatigue model involves the basic internal friction mechanism of Snoek-Köster dislocation damping under cyclic rolling contact loading. The increasing dislocation density of the aged, highly strained material eventually causes local dynamic recrystallization into the nanoscale microstructure of white etching areas, where carbides are completely dissolved. This approach also adumbrates an

increased accordingly

be analyzed.

to be 29° and 22° (see Figures 20c, 21b) for the inner ring of bearing N° 1 and N° 2, respectively. Texture development as initiating step of WEA evolution is suggested. Steep white bands (SWB) as shown in Figure 21c occur at an advanced RCF state, once a critical FWB density is reached, not until the actual *L*50 life (Voskamp, 1996), which amounts to 5.54×*L*10 for ball bearings with a typical Weibull modulus of 1.1. The inclination βs of 75° to 85° to the raceway in overrolling direction again relates to the stress field. The included angle βs-f between the FWB (30°-WEB) and the SWB (80°-WEB) thus equals about 50°. Note that in Figures 20c, 21b and 21 c, the overrolling direction is respectively from left to right. FWB appear weaker in the etched microstructure. The hardness loss is due to the increasing ferrite content. SWB reveal larger thickness and mutual spacing. The ribbon-like shaped carbide-free ferrite is highly plastically deformed (Gentile et al., 1965; Swahn et al., 1976a, 1976b; Voskamp, 1996).

#### **4.2 Metal physics model of rolling contact fatigue and experimental verification**

The classical Lundberg-Palmgren bearing life theory is empirical in nature (Lundberg & Palmgren, 1947, 1952). The application of continuum mechanics to RCF is limited. Material response to cyclic loading in rolling contact involves complex localized microstructure decay and cannot be explained by few macroscopic parameters. Moreover, fracture mechanics does not provide an approach to realistic description of RCF. The stage of crack growth, representing only about 1% of the total running time to incipient spalling (Yoshioka, 1992; Yoshioka & Fujiwara, 1988), is short compared to the phase of damage initiation in the brittle hardened steels. Without a fundamental understanding of the microscopic mechanisms of lattice defect accumulation for the prediction of material aging under rolling contact loading, which is reflected in (visible) changes of the cyclically stressed microstructure that are decisive for the resulting fatigue life, therefore, measures to increase bearing durability, for instance, by tailored alloy design cannot be derived. Physically based RCF models, however, are hardly available in the literature (Fougères et al., 2002). The reason might be that hardened bearing steels reveal complex microstructures of high defect density far from equilibrium. Precipitation strengthening due to temper carbides of typically 10 to 20 nm in diameter governs the fatigue resistance of the material in tempered condition. The mechanism proposed in the following therefore focuses on the interaction between dislocations and carbides or carbon clusters in the steel matrix.

The stress-strain hysteresis from plastic deformation in cyclic loading reflects energy dissipation (Voskamp, 1996). The vast majority of about 99% is generated as heat (Wielke, 1974), which produces a limited temperature increase under the conditions of bearing operation. The remaining 1% is absorbed as internal strain energy. This amount is associated with continuous lattice defect accumulation during metal fatigue and, therefore, damaging changes to the affected microstructure eventually. Gradual decay of retained austenite, martensite and cementite occurs in the instability stage of RCF (see Figure 10), with the dislocation arrangement of a fine sub-grain (cell) structure in the emerging ferrite and white etching band as well as texture development inside the DER in the upper life range (Voskamp, 1996). The phase transformations require diffusive redistribution of carbon on a micro scale, which is assisted by plastification. Strain energy dissipation and microplastic damage accumulation in rolling contact fatigue is described by the mechanistic Dislocation Glide Stability Loss (DGSL) model introduced in Figure 22. The different stages of compressive residual stress formation, XRD peak width reduction and microstructural alteration during advancing RCF are discussed in the framework of this metal physics scheme in the following.

to be 29° and 22° (see Figures 20c, 21b) for the inner ring of bearing N° 1 and N° 2, respectively. Texture development as initiating step of WEA evolution is suggested. Steep white bands (SWB) as shown in Figure 21c occur at an advanced RCF state, once a critical FWB density is reached, not until the actual *L*50 life (Voskamp, 1996), which amounts to 5.54×*L*10 for ball bearings with a typical Weibull modulus of 1.1. The inclination βs of 75° to 85° to the raceway in overrolling direction again relates to the stress field. The included angle βs-f between the FWB (30°-WEB) and the SWB (80°-WEB) thus equals about 50°. Note that in Figures 20c, 21b and 21 c, the overrolling direction is respectively from left to right. FWB appear weaker in the etched microstructure. The hardness loss is due to the increasing ferrite content. SWB reveal larger thickness and mutual spacing. The ribbon-like shaped carbide-free ferrite is highly plastically deformed (Gentile et al., 1965; Swahn et al., 1976a, 1976b; Voskamp, 1996).

**4.2 Metal physics model of rolling contact fatigue and experimental verification**  The classical Lundberg-Palmgren bearing life theory is empirical in nature (Lundberg & Palmgren, 1947, 1952). The application of continuum mechanics to RCF is limited. Material response to cyclic loading in rolling contact involves complex localized microstructure decay and cannot be explained by few macroscopic parameters. Moreover, fracture mechanics does not provide an approach to realistic description of RCF. The stage of crack growth, representing only about 1% of the total running time to incipient spalling (Yoshioka, 1992; Yoshioka & Fujiwara, 1988), is short compared to the phase of damage initiation in the brittle hardened steels. Without a fundamental understanding of the microscopic mechanisms of lattice defect accumulation for the prediction of material aging under rolling contact loading, which is reflected in (visible) changes of the cyclically stressed microstructure that are decisive for the resulting fatigue life, therefore, measures to increase bearing durability, for instance, by tailored alloy design cannot be derived. Physically based RCF models, however, are hardly available in the literature (Fougères et al., 2002). The reason might be that hardened bearing steels reveal complex microstructures of high defect density far from equilibrium. Precipitation strengthening due to temper carbides of typically 10 to 20 nm in diameter governs the fatigue resistance of the material in tempered condition. The mechanism proposed in the following therefore focuses on the interaction between

The stress-strain hysteresis from plastic deformation in cyclic loading reflects energy dissipation (Voskamp, 1996). The vast majority of about 99% is generated as heat (Wielke, 1974), which produces a limited temperature increase under the conditions of bearing operation. The remaining 1% is absorbed as internal strain energy. This amount is associated with continuous lattice defect accumulation during metal fatigue and, therefore, damaging changes to the affected microstructure eventually. Gradual decay of retained austenite, martensite and cementite occurs in the instability stage of RCF (see Figure 10), with the dislocation arrangement of a fine sub-grain (cell) structure in the emerging ferrite and white etching band as well as texture development inside the DER in the upper life range (Voskamp, 1996). The phase transformations require diffusive redistribution of carbon on a micro scale, which is assisted by plastification. Strain energy dissipation and microplastic damage accumulation in rolling contact fatigue is described by the mechanistic Dislocation Glide Stability Loss (DGSL) model introduced in Figure 22. The different stages of compressive residual stress formation, XRD peak width reduction and microstructural alteration during advancing RCF are discussed in the framework of this metal physics

dislocations and carbides or carbon clusters in the steel matrix.

scheme in the following.

Fig. 22. In the dislocation glide stability loss (DGSL) model of rolling contact fatigue, according to which gradual dissolution of (temper) carbides (spheres) occurs by diffusion (dotted arrows) mediated continuous carbon segregation at pinned dislocations (lines) bowing out under the influence of the cyclic shear stress τ (solid arrows), the smallest particles tend to disappear first due to their higher curvature-dependent surface energy so that the obstacles are passed successively and the level of localized microplasticity is increased accordingly

Rolling contact fatigue life is governed by the microcrack nucleation phase. Gradual dissolution of Fe2.2C temper carbides (spheres in Figure 22) driven by carbon segregation at initially pinned dislocations (lines), which bow out under the acting cyclic shear stress τ (arrows), causes successive overcoming of the obstacles and local restarting of plastic flow until activation of Frank-Read sources. Fatigue damage incubation in the steady state of apparent elastic material behavior is followed in the instability stage by the microstructural changes of DER formation, decay of globular secondary cementite (in the DGSL model due to dislocation-carbide interaction) and regular ferritic white etching bands developing inside the DER. Strain hardening, which embrittles the aged steel matrix and thus promotes crack initiation, compensates for the diminishing precipitation strengthening in the progress of rolling contact fatigue. This process results in further compressive residual stress build-up from the shakedown level and newly decreasing XRD peak width (see Figure 10). Gradual concentration of local microplasticity and microscopic accumulation of lattice defects characterize proceeding RCF damage. According to the DGSL model, Cottrell segregation of carbon atoms released from dissolving carbides at uncovered cores of dislocations, which are regeneratively generated by the glide movements during yielding, provides an additional contribution to the XRD peak width reduction by cyclic rolling contact loading (Gegner et al., 2009). The experimental proof of this essential prediction is discussed in detail below by means of Figures 23 and 24. The gradually increasing amount of localized dislocation microplasticity represents the fatigue defect accumulation mechanism of the DGSL model of RCF. It is thus associated with a rising probability for bearing failure (cf. Figure 10) due to material aging. The DGSL criterion for local microcracking is based on a critical dislocation density. Orientation and speed of fatigue crack propagation can then also be analyzed.

The proposed dislocation-carbide interaction mechanism explains (partial) fragmentation of uncuttable globular carbides of µm size, which is occasionally observed in microsections, and the increased energy level in the affected region. Localized microplastic deformation is related to energy dissipation. Note that the DGSL fatigue model involves the basic internal friction mechanism of Snoek-Köster dislocation damping under cyclic rolling contact loading. The increasing dislocation density of the aged, highly strained material eventually causes local dynamic recrystallization into the nanoscale microstructure of white etching areas, where carbides are completely dissolved. This approach also adumbrates an

Tribological Aspects of Rolling Bearing Failures 57

density increase in the defect-rich material of hardened bearing steel, reflects microstructure stabilization. An example of intense shakedown cold working is high plasticity ball burnishing. Figure 23a presents the result of the XRD measurement on the treated outer ring (OR) raceway of a taper roller bearing. The residual stress profile obeys the distribution of the v. Mises equivalent stress below the Hertzian contact (cf. Figure 1). The minimum XRD peak width *b* occurs closer to the surface. The applied Hertzian pressure is in the range of 6000 MPa (6 mm ball diameter). At the same *b*/*B* level of about 0.71 as in Figure 18a, in contrast to rolling contact fatigue, deep ball burnishing does not produce visible changes in the microstructure. The difference is evident from a comparison of the corresponding etched microsections in Figures 18c and 23b. Material alteration owing to mechanical conditioning by the build-up of compressive residual stresses in the shakedown cold working process is restricted to the higher fatigue endurance limit and based on yielding induced stabilization of the dislocation configuration but does not involve carbon diffusion (Nierlich & Gegner, 2008). Therefore, no dark etching region from martensite decay develops in the microstructure of the burnished ring displayed in Figure 23b, even in the depth zone indicated in Figure 23a by the XRD peak width relationship *FWHM*/*B*≤0.84. Mechanical surface enhancement treatments, like deep burnishing, shot peening, drum deburring and rumbling, as well as finishing operations (e.g. grinding, honing) and manufacturing processes, such as hard turning or (high-speed) cutting, are not associated with

microstructural fatigue damage (Gegner et al., 2009; Nierlich & Gegner, 2008).

Figure 23a indicates that an additional stabilization of the plastically deformed steel matrix by dislocational carbon segregation can also be induced thermally by reheating after deep ball burnishing. The associated slight compressive residual stress reduction does not affect a bearing application. The positive effect of this thermal post-treatment on RCF life, in the literature reported for surface finishing (Gegner et al., 2009; Luyckx, 2011), suggests only subcritical partial carbide dissolution. According to the DGSL model, the corresponding amount of *FWHM* decrease should be included in the reduced *b* value in rolling contact fatigue (cf. Figure 22). Therefore, no additional effect by similar reheating below the applied tempering temperature is to be expected. This crucial prediction of the model is confirmed by the experiment. In Figure 24a, the small thermal reduction of the absolute value of the residual stresses is comparable with the alterations for burnishing shown in Figure 23a. However, reheating after RCF loading leaves the XRD peak width unchanged. In Figures 23a and 24a, the plotted σres and *FWHM* values are deduced at separate sites of the raceway (i.e., one individual specimen for each depth) with increased reliability from three and eight repeated measurements, respectively, before and after the thermal treatment. The results of Figure 24a agree well with the XRD data of Figure 16a, determined by successive electrochemical polishing at one position of the racetrack of the same DGBB inner ring. This concordance is also evident for the indicated dark etching regions from a comparison of Figures 24b and 17a. The DGSL model is strongly supported by the discussed findings on the different *FWHM* response to reheating after rolling contact fatigue and cold working.

**4.3 Current passage through bearings − The aspect of hydrogen absorption and** 

The passage of electric current through a bearing causes damage by arcing across the surfaces of the rings and rolling elements in the contact zone. Fused metal in the arc results in the formation of craters on the racetrack, the appearance of which depends on the frequency. In the literature, the origin of causative shaft voltages in rotating machinery and

**accelerated rolling contact fatigue** 

Fig. 23. Investigation of cold working of a martensite hardened OR revealing (a) the residual stress and XRD peak width distributions, respectively after deep ball burnishing (*b*/*B*≈0.71) and subsequent reheating below the tempering temperature (unchanged hardness: 61 HRC) and (b) an etched axial microsection after burnishing free of visible microstructural changes

Fig. 24. Experimental investigation of reheating below tempering temperature (unchanged hardness: 60.5 HRC) after RCF loading on the martensite hardened IR of the endurance life tested DGBB of Figures 16 and 17 revealing (a) the initial and final residual stress and XRD peak width distributions (*b*/*B*≈0.68) and (b) an etched axial microsection (DER indicated)

interpretation of the development of (steep) white bands (see Figure 21c) differently from adiabatic shearing (Schlicht, 2008). The DGSL model suggests strain induced reprecipitation of carbon in the form of carbides at a later stage of RCF damage (Lindahl & Österlund, 1982; Shibata et al., 1996). Former austenite or martensite grain boundaries represent sites for heterogeneous nucleation. Reprecipitated carbide films tend to embrittle the material.

Shakedown in Figure 10 can be considered to be a cold working process (Nierlich & Gegner, 2008). As discussed in section 3.3, the XRD line broadening is sensitive to changes of the lattice distortion. The rapid peak width reduction during shakedown occurs due to glide induced rearrangement of dislocations to lower energy configurations, such as multipoles. This dominating influence, which surpasses the opposing effect of the limited dislocation

Fig. 24. Experimental investigation of reheating below tempering temperature (unchanged hardness: 60.5 HRC) after RCF loading on the martensite hardened IR of the endurance life tested DGBB of Figures 16 and 17 revealing (a) the initial and final residual stress and XRD peak width distributions (*b*/*B*≈0.68) and (b) an etched axial microsection (DER indicated)

interpretation of the development of (steep) white bands (see Figure 21c) differently from adiabatic shearing (Schlicht, 2008). The DGSL model suggests strain induced reprecipitation of carbon in the form of carbides at a later stage of RCF damage (Lindahl & Österlund, 1982; Shibata et al., 1996). Former austenite or martensite grain boundaries represent sites for heterogeneous nucleation. Reprecipitated carbide films tend to embrittle the material. Shakedown in Figure 10 can be considered to be a cold working process (Nierlich & Gegner, 2008). As discussed in section 3.3, the XRD line broadening is sensitive to changes of the lattice distortion. The rapid peak width reduction during shakedown occurs due to glide induced rearrangement of dislocations to lower energy configurations, such as multipoles. This dominating influence, which surpasses the opposing effect of the limited dislocation

Fig. 23. Investigation of cold working of a martensite hardened OR revealing (a) the residual stress and XRD peak width distributions, respectively after deep ball burnishing (*b*/*B*≈0.71) and subsequent reheating below the tempering temperature (unchanged hardness: 61 HRC) and (b) an etched axial microsection after burnishing free of visible microstructural changes

density increase in the defect-rich material of hardened bearing steel, reflects microstructure stabilization. An example of intense shakedown cold working is high plasticity ball burnishing. Figure 23a presents the result of the XRD measurement on the treated outer ring (OR) raceway of a taper roller bearing. The residual stress profile obeys the distribution of the v. Mises equivalent stress below the Hertzian contact (cf. Figure 1). The minimum XRD peak width *b* occurs closer to the surface. The applied Hertzian pressure is in the range of 6000 MPa (6 mm ball diameter). At the same *b*/*B* level of about 0.71 as in Figure 18a, in contrast to rolling contact fatigue, deep ball burnishing does not produce visible changes in the microstructure. The difference is evident from a comparison of the corresponding etched microsections in Figures 18c and 23b. Material alteration owing to mechanical conditioning by the build-up of compressive residual stresses in the shakedown cold working process is restricted to the higher fatigue endurance limit and based on yielding induced stabilization of the dislocation configuration but does not involve carbon diffusion (Nierlich & Gegner, 2008). Therefore, no dark etching region from martensite decay develops in the microstructure of the burnished ring displayed in Figure 23b, even in the depth zone indicated in Figure 23a by the XRD peak width relationship *FWHM*/*B*≤0.84. Mechanical surface enhancement treatments, like deep burnishing, shot peening, drum deburring and rumbling, as well as finishing operations (e.g. grinding, honing) and manufacturing processes, such as hard turning or (high-speed) cutting, are not associated with microstructural fatigue damage (Gegner et al., 2009; Nierlich & Gegner, 2008).

Figure 23a indicates that an additional stabilization of the plastically deformed steel matrix by dislocational carbon segregation can also be induced thermally by reheating after deep ball burnishing. The associated slight compressive residual stress reduction does not affect a bearing application. The positive effect of this thermal post-treatment on RCF life, in the literature reported for surface finishing (Gegner et al., 2009; Luyckx, 2011), suggests only subcritical partial carbide dissolution. According to the DGSL model, the corresponding amount of *FWHM* decrease should be included in the reduced *b* value in rolling contact fatigue (cf. Figure 22). Therefore, no additional effect by similar reheating below the applied tempering temperature is to be expected. This crucial prediction of the model is confirmed by the experiment. In Figure 24a, the small thermal reduction of the absolute value of the residual stresses is comparable with the alterations for burnishing shown in Figure 23a. However, reheating after RCF loading leaves the XRD peak width unchanged. In Figures 23a and 24a, the plotted σres and *FWHM* values are deduced at separate sites of the raceway (i.e., one individual specimen for each depth) with increased reliability from three and eight repeated measurements, respectively, before and after the thermal treatment. The results of Figure 24a agree well with the XRD data of Figure 16a, determined by successive electrochemical polishing at one position of the racetrack of the same DGBB inner ring. This concordance is also evident for the indicated dark etching regions from a comparison of Figures 24b and 17a. The DGSL model is strongly supported by the discussed findings on the different *FWHM* response to reheating after rolling contact fatigue and cold working.

#### **4.3 Current passage through bearings − The aspect of hydrogen absorption and accelerated rolling contact fatigue**

The passage of electric current through a bearing causes damage by arcing across the surfaces of the rings and rolling elements in the contact zone. Fused metal in the arc results in the formation of craters on the racetrack, the appearance of which depends on the frequency. In the literature, the origin of causative shaft voltages in rotating machinery and

Tribological Aspects of Rolling Bearing Failures 59

surface. The weaker operational high-frequency electric current passage of another bearing from the same rig test series documented in Figure 26a results only in a slightly increased content of 1.3 ppm H. The original honing structure of the raceway is displayed in Figure

An XRD material response analysis is performed in the load zone of the raceway of the hydrogen loaded outer ring of the bearing of Figure 25. According to Figures 27a, a high

Fig. 26. SEM-SE image of the raceway (a) of the OR of an identical DGBB tested in the same alternator rig as the bearing of Figure 25 after moderate high-frequency electric current passage and (b) in as-delivered (non-overrolled) surface condition with original honing

The applied joint evaluation of the depth profiles of the residual stress and XRD peak width in the subsurface zone of classical rolling contact fatigue is shown in Figure 27b. The damage parameter equals *b*/*B*≈0.71. The XRD *L*10 life equivalent is thus not yet exceeded on the outer ring. The microsection in Figure 27c confirms a subsurface dark etching region, the

 Fig. 27. Material response analysis of the OR of the tested DGBB of Figure 25 including (a) the residual stress and XRD peak width distribution (*b*/*B*≈0.71, *B* measured below the shoulder), (b) the joint profile evaluation and (c) an axial microsection with pronounced DER

26b. For comparison, Figures 25a, 26a and 26b have similar magnification.

Hertzian pressure above 5000 MPa is deduced.

position of which reflects the contact angle.

marks

the sources of current flows, the electrical characteristics of a rolling bearing and the influence of the lubricant properties as well as the development of the typical surface patterns are discussed in detail (Jagenbrein et al., 2005; Prashad, 2006; Zika et al., 2007, 2009, 2010). Complex chemical reactions occur in the electrically stressed oil film (Prashad, 2006). However, the ability of hydrogen released from decomposition products to be absorbed by the steel under the prevailing specific circumstances and subsequently to affect rolling contact fatigue is not yet investigated so far (Gegner & Nierlich, 2011b, 2011c).

Depending on the design of the electric generator, e.g. in diesel engines, alternator bearings may operate under current passage. Possible damage mechanisms become more important today because of the increased use of frequency inverters. Grease lubricated deep groove ball bearings with stationary outer ring, stemming from an automobile alternator rig test, are investigated in the following. The running period is in accordance with the nominal *L*<sup>10</sup> life. Rings and balls are made out of martensitically hardened bearing steel. The racetrack in Figure 25a suffers from severe high-frequency electric current passage. Arc discharge in the lubricating gap causes a gray matted surface. The resulting shallow remelting craters of few µm in diameter and depth cover the racetrack densely. The indicated isolated indentation, magnified in Figure 25b, reveals the earlier condition of a less affected area. The tribological properties of the contact surface are still sufficient. The microsection of Figure 25c confirms the small influence zone by a thin white etching layer. However, continuous chemical decomposition of the lubricant and surface remelting promote hydrogen penetration. Thus, a highly increased content of more than 3 ppm by weight is measured for the DGBB outer ring of Figure 25 by carrier gas hot extraction (CGHE). Typical concentrations in the asdelivered state, after through hardening and machining, range from 0.5 to 1.0 ppm H.

Fig. 25. Characterization of severe high-frequency electric current passage through a DGBB by (a) a SEM-SE overview and (b) the indicated SEM-SE detail of the remelted OR raceway track and (c) a near-surface LOM micrograph of an etched axial microsection

The amount of hydrogen absorbed by the steel depends on the release from the decomposition products of the aging lubricant and the available catalytically active blank metal surface (Kohara et al., 2006). Both affecting factors are enhanced by current passage in service. Fresh blank metal from remelting on the raceway enables the process step from physi- to chemisorption with abstraction of hydrogen atoms, which is otherwise effectively inhibited by the regenerative formation of a passivating protective reaction layer on the

the sources of current flows, the electrical characteristics of a rolling bearing and the influence of the lubricant properties as well as the development of the typical surface patterns are discussed in detail (Jagenbrein et al., 2005; Prashad, 2006; Zika et al., 2007, 2009, 2010). Complex chemical reactions occur in the electrically stressed oil film (Prashad, 2006). However, the ability of hydrogen released from decomposition products to be absorbed by the steel under the prevailing specific circumstances and subsequently to affect rolling

Depending on the design of the electric generator, e.g. in diesel engines, alternator bearings may operate under current passage. Possible damage mechanisms become more important today because of the increased use of frequency inverters. Grease lubricated deep groove ball bearings with stationary outer ring, stemming from an automobile alternator rig test, are investigated in the following. The running period is in accordance with the nominal *L*<sup>10</sup> life. Rings and balls are made out of martensitically hardened bearing steel. The racetrack in Figure 25a suffers from severe high-frequency electric current passage. Arc discharge in the lubricating gap causes a gray matted surface. The resulting shallow remelting craters of few µm in diameter and depth cover the racetrack densely. The indicated isolated indentation, magnified in Figure 25b, reveals the earlier condition of a less affected area. The tribological properties of the contact surface are still sufficient. The microsection of Figure 25c confirms the small influence zone by a thin white etching layer. However, continuous chemical decomposition of the lubricant and surface remelting promote hydrogen penetration. Thus, a highly increased content of more than 3 ppm by weight is measured for the DGBB outer ring of Figure 25 by carrier gas hot extraction (CGHE). Typical concentrations in the asdelivered state, after through hardening and machining, range from 0.5 to 1.0 ppm H.

Fig. 25. Characterization of severe high-frequency electric current passage through a DGBB by (a) a SEM-SE overview and (b) the indicated SEM-SE detail of the remelted OR raceway

The amount of hydrogen absorbed by the steel depends on the release from the decomposition products of the aging lubricant and the available catalytically active blank metal surface (Kohara et al., 2006). Both affecting factors are enhanced by current passage in service. Fresh blank metal from remelting on the raceway enables the process step from physi- to chemisorption with abstraction of hydrogen atoms, which is otherwise effectively inhibited by the regenerative formation of a passivating protective reaction layer on the

track and (c) a near-surface LOM micrograph of an etched axial microsection

contact fatigue is not yet investigated so far (Gegner & Nierlich, 2011b, 2011c).

surface. The weaker operational high-frequency electric current passage of another bearing from the same rig test series documented in Figure 26a results only in a slightly increased content of 1.3 ppm H. The original honing structure of the raceway is displayed in Figure 26b. For comparison, Figures 25a, 26a and 26b have similar magnification.

An XRD material response analysis is performed in the load zone of the raceway of the hydrogen loaded outer ring of the bearing of Figure 25. According to Figures 27a, a high Hertzian pressure above 5000 MPa is deduced.

Fig. 26. SEM-SE image of the raceway (a) of the OR of an identical DGBB tested in the same alternator rig as the bearing of Figure 25 after moderate high-frequency electric current passage and (b) in as-delivered (non-overrolled) surface condition with original honing marks

The applied joint evaluation of the depth profiles of the residual stress and XRD peak width in the subsurface zone of classical rolling contact fatigue is shown in Figure 27b. The damage parameter equals *b*/*B*≈0.71. The XRD *L*10 life equivalent is thus not yet exceeded on the outer ring. The microsection in Figure 27c confirms a subsurface dark etching region, the position of which reflects the contact angle.

Fig. 27. Material response analysis of the OR of the tested DGBB of Figure 25 including (a) the residual stress and XRD peak width distribution (*b*/*B*≈0.71, *B* measured below the shoulder), (b) the joint profile evaluation and (c) an axial microsection with pronounced DER

Tribological Aspects of Rolling Bearing Failures 61

indication of microcrack initiation on white etching bands by interfacial delamination is confirmed by Figure 30c. It is not observed in pure mechanical rolling contact fatigue (Voskamp, 1996), where actually an influence of WEB (as well as of butterfly) formation on bearing life is not proven (Schlicht, 2008). Therefore, hydrogen induced cracking propensity on WEB suggests higher hardness of the white etching areas and hydrogen embrittlement.

 Fig. 30. Etched radial microsection of the OR of Figure 27c revealing (a) a LOM overview with indicated DER, (b) the SEM-SE detail b and (c) the SEM-SE detail c, where the corresponding LOM inset highlights the WEA precursor effect of the surrounding DER

As also emphasized in Figure 31a by grain boundary etching, flat and steep white bands evolve from the distinctive surrounding DER material. The SWB seem to develop in an earlier stage prior to the complete dense formation of FWB (cf. Figure 21c). Particularly the oriented slip bands of FWB exhibit more intense white etching microstructure (cf. Figure 21c). Figure 31b presents the corresponding SEM-SE image of this extended detail of Figure 30b in the center of Figure 30a. The gradual evolution of white etching bands from the DER precursor, as particularly evident from Figure 31a, indicates advancing fatigue processes, e.g. as outlined in section 4.2, presumably correlated with texture development and dynamic recrystallization during rolling contact loading (Voskamp, 1996). On the other hand, this microstructural finding speaks against causative adiabatic shearing (Schlicht, 2008). The preferred occurrence of white etching bands in ball bearings should rather be connected with the higher Hertzian pressure compared to a corresponding roller contact. Note that no WEA of premature rolling contact fatigue damage are formed in the case of Figure 26. This moderate high-frequency electric current passage in operation is connected

Despite the occurrence of white etching bands in the outer ring of the rig tested DGBB of Figure 25, as documented in Figures 28 to 31, the XRD material aging parameter deduced from Figure 27a amounts to just *b*/*B*≈0.71. The same value is derived from the peak width distribution in Figure 18a, where for pure mechanical subsurface RCF, however, no WEA are formed inside the DER (see Figure 18c). As for the bearing operating under severe highfrequency electric current passage, the XRD *L*10 equivalent of classical rolling contact fatigue without additional chemical loading is not yet exceeded but well developed white etching bands, particularly SWB, already occur, hydrogen charging noticeably accelerates the

with only slight hydrogen enrichment in the bearing steel.

Note again the pronounced DER microstructure around the WEA in Figure 30c.

Inside the wide DER of Figure 27c, extended white etching areas are located (cf. Figure 28a), which evolve from the steel matrix. In the used clean material, butterfly formation is irrelevant and only two early stages are found (see inset of Figure 28a). Etching accentuates the actual RCF damage: the DER identified as brittle by the observed preparative cracking is clearly distinguishable from the chemically less affected material above and below in the indicated SEM-SE detail of Figure 28b. The WEA inside the DER appear smooth black.

Fig. 28. Etched axial microsection of the DGBB outer ring of Figure 27c revealing (a) a LOM overview (the inset shows an embryo butterfly) and (b) the indicated SEM-SE detail

The LOM micrograph in Figure 29a reveals dense dark etching regions adjacent to the WEA zones. Although reported contrarily in the literature (Martin et al., 1966), the embrittled dark etching region evidently acts as precursor of further phase transformation. The SEM-SE detail of Figure 29b also points to interfacial delamination (see indication) as pre-stage of fatigue crack initiation.

Fig. 29. Etched axial microsection of the DGBB outer ring of Figure 27c revealing (a) a LOM image and (b) the indicated SEM-SE detail

The development of white etching bands is identified in the radial microsection of the investigated outer ring shown in Figure 30a. Dense FWB and distinct SWB of inclinations βf=25° and βs=76°, respectively, are visible inside the indicated DER. The central SEM-SE detail of Figure 30b reveals the included angle βs-f of 51° (see section 4.1, Figure 21c). The

Inside the wide DER of Figure 27c, extended white etching areas are located (cf. Figure 28a), which evolve from the steel matrix. In the used clean material, butterfly formation is irrelevant and only two early stages are found (see inset of Figure 28a). Etching accentuates the actual RCF damage: the DER identified as brittle by the observed preparative cracking is clearly distinguishable from the chemically less affected material above and below in the indicated SEM-SE detail of Figure 28b. The WEA inside the DER appear smooth black.

 Fig. 28. Etched axial microsection of the DGBB outer ring of Figure 27c revealing (a) a LOM overview (the inset shows an embryo butterfly) and (b) the indicated SEM-SE detail

The LOM micrograph in Figure 29a reveals dense dark etching regions adjacent to the WEA zones. Although reported contrarily in the literature (Martin et al., 1966), the embrittled dark etching region evidently acts as precursor of further phase transformation. The SEM-SE detail of Figure 29b also points to interfacial delamination (see indication) as pre-stage of

 Fig. 29. Etched axial microsection of the DGBB outer ring of Figure 27c revealing (a) a LOM

The development of white etching bands is identified in the radial microsection of the investigated outer ring shown in Figure 30a. Dense FWB and distinct SWB of inclinations βf=25° and βs=76°, respectively, are visible inside the indicated DER. The central SEM-SE detail of Figure 30b reveals the included angle βs-f of 51° (see section 4.1, Figure 21c). The

fatigue crack initiation.

image and (b) the indicated SEM-SE detail

indication of microcrack initiation on white etching bands by interfacial delamination is confirmed by Figure 30c. It is not observed in pure mechanical rolling contact fatigue (Voskamp, 1996), where actually an influence of WEB (as well as of butterfly) formation on bearing life is not proven (Schlicht, 2008). Therefore, hydrogen induced cracking propensity on WEB suggests higher hardness of the white etching areas and hydrogen embrittlement. Note again the pronounced DER microstructure around the WEA in Figure 30c.

Fig. 30. Etched radial microsection of the OR of Figure 27c revealing (a) a LOM overview with indicated DER, (b) the SEM-SE detail b and (c) the SEM-SE detail c, where the corresponding LOM inset highlights the WEA precursor effect of the surrounding DER

As also emphasized in Figure 31a by grain boundary etching, flat and steep white bands evolve from the distinctive surrounding DER material. The SWB seem to develop in an earlier stage prior to the complete dense formation of FWB (cf. Figure 21c). Particularly the oriented slip bands of FWB exhibit more intense white etching microstructure (cf. Figure 21c). Figure 31b presents the corresponding SEM-SE image of this extended detail of Figure 30b in the center of Figure 30a. The gradual evolution of white etching bands from the DER precursor, as particularly evident from Figure 31a, indicates advancing fatigue processes, e.g. as outlined in section 4.2, presumably correlated with texture development and dynamic recrystallization during rolling contact loading (Voskamp, 1996). On the other hand, this microstructural finding speaks against causative adiabatic shearing (Schlicht, 2008). The preferred occurrence of white etching bands in ball bearings should rather be connected with the higher Hertzian pressure compared to a corresponding roller contact. Note that no WEA of premature rolling contact fatigue damage are formed in the case of Figure 26. This moderate high-frequency electric current passage in operation is connected with only slight hydrogen enrichment in the bearing steel.

Despite the occurrence of white etching bands in the outer ring of the rig tested DGBB of Figure 25, as documented in Figures 28 to 31, the XRD material aging parameter deduced from Figure 27a amounts to just *b*/*B*≈0.71. The same value is derived from the peak width distribution in Figure 18a, where for pure mechanical subsurface RCF, however, no WEA are formed inside the DER (see Figure 18c). As for the bearing operating under severe highfrequency electric current passage, the XRD *L*10 equivalent of classical rolling contact fatigue without additional chemical loading is not yet exceeded but well developed white etching bands, particularly SWB, already occur, hydrogen charging noticeably accelerates the

Tribological Aspects of Rolling Bearing Failures 63

Near-surface loading is often superimposed by the impact of externally generated threedimensional mechanical vibrations that represents a common cause of disturbed EHL operating conditions, e.g., in paper making or weaving machines, coal pulverizers, wind turbines, cranes, trains, tractors and fans. Ball bearings in car alternators of four-cylinder

The SEM image of Figure 32a shows the completely smoothed raceway in the rotating main load zone of a CRB inner ring after a rig test time of about 40% of the calculated nominal *L*<sup>10</sup> life (Nierlich & Gegner, 2002). Only parts of the deepest original honing grooves are left over on the surface. Causative mixed friction results from inadequate lubrication conditions without sufficient film formation (fuel addition to the oil). Initial micropitting by isolated material delamination of less than 10 µm depth is observed. Figure 32b provides a comparison with the non-overrolled as-finished raceway condition. On the damaged inner ring, a residual stress material response analysis is performed. The result is shown in Figure 33a. No changes of the measured XRD parameters in the depth of the material are found, whereas the XRD peak width on the surface decreases to *b*/*B*≥0.79. The relation symbol accounts for the small *FWHM* reduction of about 0.2° due to the honing process (see section 3.3, Figure 16a). Material aging considerably exceeds the XRD *L*10 equivalent value of 0.86 for the relevant surface failure mode of RCF. The corresponding re-increase of the residual stress on the raceway, discussed in the context of Figures 11 and 12 in section 3.3, reaches

The residual stress distribution of Fig. 33a is identified as a type B profile of vibrational loading in rolling-sliding contact (Gegner & Nierlich, 2008). The characteristic compressive residual stress side maximum in a short distance from the surface (here 40 µm), clearly

reflected in the corresponding reduction of the XRD peak width. The monotonically increasing type A vibration residual stress profile occurs more frequently in practical applications. The result of a material response analysis on a CRB outer ring, the raceway of which does not reveal indentations, represents a prime example in Figure 33b. Bainitic through hardening of the bearing steel results in compressive residual stresses in the core of

<sup>0</sup>*z* of maximum v. Mises equivalent stress for pure radial load, is

Fig. 32. SEM-SE image of (a) the damaged raceway of the inner ring of a CRB after rig testing under engine vibrations and (b) an original honing structure at the same

**5.1 Vibrational contact loading and tribological model** 

diesel engines are another familiar example.

–230 MPa.

magnification

above the depth v.Mises

evolution of microstructural RCF damage (hydrogen accelerated rolling contact fatigue, H-RCF). The dark etching region extends to zones of *FWHM*/*B*>0.84 near the surface, as evident from a comparison of Figures 27a, 28a and 30a. The calibration relationship between the *L*10 life and the evidently reduced *b*/*B* equivalent is modified by the hydrogen embrittled DER.

Fig. 31. Detail (approx. b) of Figure 30 comparing (a) a LOM and (b) a SEM-SE micrograph

The metal physics dislocation glide stability loss model, introduced in section 4.2, provides an approach to the mechanistic description of rolling contact fatigue in bearing steels. Hydrogen interacts with lattice defects (Gegner et al., 1996). The response to cyclic loading reflects its high atom mobility even at low temperature. The effect of hydrogen can be illustrated by the DGSL model of Figure 22. The microscopic fatigue processes are considerably promoted by intensifying the increase of dislocation density and glide mobility. Mechanisms of hydrogen enhanced localized plasticity (HELP) are discussed in the literature (Birnbaum & Sofronis, 1994). A comparison of chemically assisted with pure mechanical rolling contact fatigue and shakedown cold working at constant reference level of *b*/*B*≈0.71 in Figures 18, 23 and 27 to 31, completed by Figures 20, 21 and 24, suggests that material aging is accelerated by enhancing the microplasticity. At the same stage of *b*/*B* reduction, microstructural RCF damage is much more advanced. Premature formation of ribbon like or irregularly oriented white etching areas, for instance, might yet occur at lower loads.

#### **5. Surface failure induced by mixed friction in rolling-sliding contact**

The practically predominating surface failure mode involves various damage mechanisms. Besides indentations, discussed in detail in section 2.2, mixed friction or boundary lubrication in the rolling contact area occurs frequently in bearing applications. Polishing wear on the raceway, resulting in differently pronounced smoothing of the machining marks, is a characteristic visual indication. The depth of highest material loading is shifted towards the surface by sliding friction in rolling contact. The effect on the distribution and the maximum of the equivalent stress is similar to the scheme shown in Figure 5. The mechanisms of crack initiation on the surface are of utmost technical importance (Olver, 2005). New aspects of rolling contact tribology in bearing failures are presented in the following.

evolution of microstructural RCF damage (hydrogen accelerated rolling contact fatigue, H-RCF). The dark etching region extends to zones of *FWHM*/*B*>0.84 near the surface, as evident from a comparison of Figures 27a, 28a and 30a. The calibration relationship between the *L*10 life and the evidently reduced *b*/*B* equivalent is modified by the hydrogen embrittled

 Fig. 31. Detail (approx. b) of Figure 30 comparing (a) a LOM and (b) a SEM-SE micrograph The metal physics dislocation glide stability loss model, introduced in section 4.2, provides an approach to the mechanistic description of rolling contact fatigue in bearing steels. Hydrogen interacts with lattice defects (Gegner et al., 1996). The response to cyclic loading reflects its high atom mobility even at low temperature. The effect of hydrogen can be illustrated by the DGSL model of Figure 22. The microscopic fatigue processes are considerably promoted by intensifying the increase of dislocation density and glide mobility. Mechanisms of hydrogen enhanced localized plasticity (HELP) are discussed in the literature (Birnbaum & Sofronis, 1994). A comparison of chemically assisted with pure mechanical rolling contact fatigue and shakedown cold working at constant reference level of *b*/*B*≈0.71 in Figures 18, 23 and 27 to 31, completed by Figures 20, 21 and 24, suggests that material aging is accelerated by enhancing the microplasticity. At the same stage of *b*/*B* reduction, microstructural RCF damage is much more advanced. Premature formation of ribbon like or irregularly oriented white etching areas, for instance, might yet occur at lower

**5. Surface failure induced by mixed friction in rolling-sliding contact** 

The practically predominating surface failure mode involves various damage mechanisms. Besides indentations, discussed in detail in section 2.2, mixed friction or boundary lubrication in the rolling contact area occurs frequently in bearing applications. Polishing wear on the raceway, resulting in differently pronounced smoothing of the machining marks, is a characteristic visual indication. The depth of highest material loading is shifted towards the surface by sliding friction in rolling contact. The effect on the distribution and the maximum of the equivalent stress is similar to the scheme shown in Figure 5. The mechanisms of crack initiation on the surface are of utmost technical importance (Olver, 2005). New aspects of rolling contact tribology in bearing failures are presented in the

DER.

loads.

following.

#### **5.1 Vibrational contact loading and tribological model**

Near-surface loading is often superimposed by the impact of externally generated threedimensional mechanical vibrations that represents a common cause of disturbed EHL operating conditions, e.g., in paper making or weaving machines, coal pulverizers, wind turbines, cranes, trains, tractors and fans. Ball bearings in car alternators of four-cylinder diesel engines are another familiar example.

The SEM image of Figure 32a shows the completely smoothed raceway in the rotating main load zone of a CRB inner ring after a rig test time of about 40% of the calculated nominal *L*<sup>10</sup> life (Nierlich & Gegner, 2002). Only parts of the deepest original honing grooves are left over on the surface. Causative mixed friction results from inadequate lubrication conditions without sufficient film formation (fuel addition to the oil). Initial micropitting by isolated material delamination of less than 10 µm depth is observed. Figure 32b provides a comparison with the non-overrolled as-finished raceway condition. On the damaged inner ring, a residual stress material response analysis is performed. The result is shown in Figure 33a. No changes of the measured XRD parameters in the depth of the material are found, whereas the XRD peak width on the surface decreases to *b*/*B*≥0.79. The relation symbol accounts for the small *FWHM* reduction of about 0.2° due to the honing process (see section 3.3, Figure 16a). Material aging considerably exceeds the XRD *L*10 equivalent value of 0.86 for the relevant surface failure mode of RCF. The corresponding re-increase of the residual stress on the raceway, discussed in the context of Figures 11 and 12 in section 3.3, reaches –230 MPa.

Fig. 32. SEM-SE image of (a) the damaged raceway of the inner ring of a CRB after rig testing under engine vibrations and (b) an original honing structure at the same magnification

The residual stress distribution of Fig. 33a is identified as a type B profile of vibrational loading in rolling-sliding contact (Gegner & Nierlich, 2008). The characteristic compressive residual stress side maximum in a short distance from the surface (here 40 µm), clearly above the depth v.Mises <sup>0</sup>*z* of maximum v. Mises equivalent stress for pure radial load, is reflected in the corresponding reduction of the XRD peak width. The monotonically increasing type A vibration residual stress profile occurs more frequently in practical applications. The result of a material response analysis on a CRB outer ring, the raceway of which does not reveal indentations, represents a prime example in Figure 33b. Bainitic through hardening of the bearing steel results in compressive residual stresses in the core of

Tribological Aspects of Rolling Bearing Failures 65

addition to the radial load, controlled uni- to triaxial vibrations can be applied in axial, tangential and radial direction. Figure 35 displays a photograph of the rig. It represents a view of the housing of the test bearing and the equipment for the transmission of axial and tangential vibrations (radial excitation from below) with thermocouples and displacement

A micro friction model of the rolling-sliding contact is introduced by means of Figure 36. It describes the effect of vibrational loading. As shown in Figure 36, tangential forces by sliding friction acting on a rolling contact increase the equivalent stress and shift its maximum toward the surface on indentation-free raceways (Broszeit et al., 1977). A transition, indicated by solid-line curves, occurs between friction coefficients μ of 0.2 and 0.3: above and below μ=0.25, the increasing maximum of the Tresca equivalent stress is located directly on or near the surface, respectively. If the yield strength of the material is exceeded (cf. Figure 5), therefore, type A or B residual stress depth profiles are generated.

Fig. 35. Housing of the test bearing with devices for vibration generation

Material response to vibrational loading, which causes increased mixed friction, is described in the tribological model by partitioning the nominal contact area *A* into microscopic sections of different friction coefficients (Gegner & Nierlich, 2008). The inset of Figure 36 illustrates the basic idea. In some subdomains, arranged e.g. in the form of dry spots or bands, peak values from μ>≈0.2 (type B) to μ>≥0.3 (type A) are supposed to be reached intermittently for short periods. The thixotropy effect supports this concept because shearing of the lubricant by vibrational loading reduces the viscosity, which increases the tendency to mixed friction. In the other subareas of the contact, μ< is much lower so that the average friction coefficient μ(eff), meeting a mixing rule, remains below 0.1 as typical of running rolling bearings. Besides the verified compressive residual stress buildup, nonuniform cyclic mechanical loading of the contact area by, in general, complex three-

sensors.

the material. The XRD life parameter *b*/*B*≥0.82 is taken from the diagram. The running time of 2×108 revolutions indicates low-cycle fatigue under the influence of intermittently acting severe vibrations (Nierlich & Gegner, 2008). The residual stress analysis of the inner ring of a taper roller bearing from a harvester in Figure 34a provides another instructive example. Mixed short-term deeper reaching type A vibrational and near-surface Hertzian micro contact loading of the material are superimposed. Figure34b reveals indentations on the partly smoothed raceway. The applied Hertzian pressure *p*0 amounts to 2000 MPa. For comparison, the depth of maximum v. Mises equivalent stress for incipient plastic deformation in pure radial contact loading, i.e. *p*0 above 2500 to 3000 MPa, equals about 180 µm.

Fig. 33. The two types of vibration residual stress-XRD line width profiles, i.e. (a) type B with near-surface side peaks measured on the IR raceway of a CRB from a motorcycle gearbox test rig and (b) type A with monotonically increasing curves from a field application

Fig. 34. Investigation of the IR of a vibration-loaded harvester TRB revealing (a) the obtained type A residual stress pattern and (b) a SEM-SE image of the raceway with indentations

Both types of residual stress distributions are simulated experimentally in a specially designed vibration test rig for rolling bearings (Gegner & Nierlich, 2008). A type N CRB is used. The stationary lipless outer ring of the test bearing is displaced and experiences high vibrational loading via the sliding contact to the rollers. It thus becomes the specimen. In

the material. The XRD life parameter *b*/*B*≥0.82 is taken from the diagram. The running time of 2×108 revolutions indicates low-cycle fatigue under the influence of intermittently acting severe vibrations (Nierlich & Gegner, 2008). The residual stress analysis of the inner ring of a taper roller bearing from a harvester in Figure 34a provides another instructive example. Mixed short-term deeper reaching type A vibrational and near-surface Hertzian micro contact loading of the material are superimposed. Figure34b reveals indentations on the partly smoothed raceway. The applied Hertzian pressure *p*0 amounts to 2000 MPa. For comparison, the depth of maximum v. Mises equivalent stress for incipient plastic deformation in pure radial contact loading, i.e. *p*0 above 2500 to 3000 MPa, equals about 180

 Fig. 33. The two types of vibration residual stress-XRD line width profiles, i.e. (a) type B with near-surface side peaks measured on the IR raceway of a CRB from a motorcycle gearbox test rig and (b) type A with monotonically increasing curves from a field

 Fig. 34. Investigation of the IR of a vibration-loaded harvester TRB revealing (a) the obtained type A residual stress pattern and (b) a SEM-SE image of the raceway with indentations

Both types of residual stress distributions are simulated experimentally in a specially designed vibration test rig for rolling bearings (Gegner & Nierlich, 2008). A type N CRB is used. The stationary lipless outer ring of the test bearing is displaced and experiences high vibrational loading via the sliding contact to the rollers. It thus becomes the specimen. In

µm.

application

addition to the radial load, controlled uni- to triaxial vibrations can be applied in axial, tangential and radial direction. Figure 35 displays a photograph of the rig. It represents a view of the housing of the test bearing and the equipment for the transmission of axial and tangential vibrations (radial excitation from below) with thermocouples and displacement sensors.

A micro friction model of the rolling-sliding contact is introduced by means of Figure 36. It describes the effect of vibrational loading. As shown in Figure 36, tangential forces by sliding friction acting on a rolling contact increase the equivalent stress and shift its maximum toward the surface on indentation-free raceways (Broszeit et al., 1977). A transition, indicated by solid-line curves, occurs between friction coefficients μ of 0.2 and 0.3: above and below μ=0.25, the increasing maximum of the Tresca equivalent stress is located directly on or near the surface, respectively. If the yield strength of the material is exceeded (cf. Figure 5), therefore, type A or B residual stress depth profiles are generated.

Fig. 35. Housing of the test bearing with devices for vibration generation

Material response to vibrational loading, which causes increased mixed friction, is described in the tribological model by partitioning the nominal contact area *A* into microscopic sections of different friction coefficients (Gegner & Nierlich, 2008). The inset of Figure 36 illustrates the basic idea. In some subdomains, arranged e.g. in the form of dry spots or bands, peak values from μ>≈0.2 (type B) to μ>≥0.3 (type A) are supposed to be reached intermittently for short periods. The thixotropy effect supports this concept because shearing of the lubricant by vibrational loading reduces the viscosity, which increases the tendency to mixed friction. In the other subareas of the contact, μ< is much lower so that the average friction coefficient μ(eff), meeting a mixing rule, remains below 0.1 as typical of running rolling bearings. Besides the verified compressive residual stress buildup, nonuniform cyclic mechanical loading of the contact area by, in general, complex three-

Tribological Aspects of Rolling Bearing Failures 67

Fig. 37. SEM-SE image of a chemical surface attack on the outer ring raceway of a CRB

Fig. 38. LOM micrograph of the etched metallographic section of a sulfide inclusion line

On the inner ring raceway of a cylindrical roller bearing of a weaving machine examined in Figure 39, mixed friction is indicated by the mechanically smoothed honing structure. Due to aging of the lubricating oil, as detected under vibration loading, the gradually acidifying fluid attacks the steel surface. Tribochemical dissolution of manufacturing related MnS inclusion lines leaves crack-like defects on the raceway. Sulfur is continuously removed as

 MnS + H2 → H2S<sup>↑</sup> + Mn (6) The remaining manganese is then preferentially corroded out. This new mechanism of crack formation on tribologically loaded raceway surfaces is verified by chemical characterization using energy dispersive X-ray (EDX) microanalysis on the SEM. The EDX spectra in Figure 39, recorded at an acceleration voltage of 20 kV, confirm residues of manganese and sulfur at four sites (S1 to S4) of an emerging crack, thus excluding accidental intersection. The ring is made of martensitically hardened bearing steel. Reaction layer formation on the raceway

Crack initiation by tribochemical reaction is also found on lateral surfaces of rollers. In Figure 40, remaining manganese and sulfur are detected by elemental mapping in the insets

intersecting the surface of the inner ring raceway of a cylindrical roller bearing

gaseous H2S by hydrogen from decomposition products of the lubricant:

is reflected in the signals of phosphorus from lubricant additives and oxygen.

high level of cleanliness of bearing grades.

on the right.

As exemplified by Figure 38, some manganese sulfide lines intersect the rolling contact surface. Such inclusions are manufacturing related from the steelmaking process, despite the

dimensional vibrations is also evident from occasionally observed dent-like plastic deformation on the surface, spots of dark etching regions in the microstructure of the outermost material and varying preferred orientation of yielding across the raceway width, reflected in differing tangential and axial components of the residual stresses in the affected edge zone (Gegner & Nierlich, 2008). Friction increase is confirmed by temperature rise in the lubricating gap that correlates with the power loss per contact area. This effect can be exploited to easily assess the vibration resistance of specific oils or greases on the adapted bearing test rig (Gegner & Nierlich, 2008).

Fig. 36. Distribution of the Tresca equivalent stress below a rolling-sliding contact (*z*0 depth indicated for pure radial load, i.e. μ=0) and illustration of the tribological model of localized friction coefficient in the inset (*F*n is the normal force)

Under the influence of vibrations, disturbance of proper contact operating conditions in a way that high shearing stresses are induced in the lubricating film can promote lubricant degradation (Kudish & Covitch, 2010). Reduced lubricity enhances the effect of sliding friction, e.g. described in the tribological model of Figure 36. Further to the discussed mechanical and thermal influence, vibration loading induces chemical aging of the lubricant and its additives (Gegner & Nierlich, 2008). Contaminations, like water or wear debris, increase the effect. The gradual decomposition process and associated acidification of the lubricant promote, for instance, the initiation of surface cracks on the raceway by tribochemical dissolution of nonmetallic MnS inclusion lines, which is discussed in the next section.

#### **5.2 Tribochemically initiated surface cracks**

First, Figure 37 gives a demonstrative example of a corrosive attack by a decomposed lubricant. The etching pattern on the raceway reveals chemical smoothing of the surface. Copper contamination by abrasion from the graphite-brass ground brush of the diesel electric locomotive gets into the grease of the train wheel bearing and accelerates lubricant aging. As evident from Figure 37, the foreign particles also cause indentations on the raceway. An electrical oil sensor system can be used for online condition monitoring of the lubricant (Gegner et al., 2010). The application in industrial gearboxes, for instance of wind turbines, is of special practical interest. Note that the (e.g., extreme pressure) additives markedly influence the electrical properties of the lubricant (Prashad, 2006).

dimensional vibrations is also evident from occasionally observed dent-like plastic deformation on the surface, spots of dark etching regions in the microstructure of the outermost material and varying preferred orientation of yielding across the raceway width, reflected in differing tangential and axial components of the residual stresses in the affected edge zone (Gegner & Nierlich, 2008). Friction increase is confirmed by temperature rise in the lubricating gap that correlates with the power loss per contact area. This effect can be exploited to easily assess the vibration resistance of specific oils or greases on the adapted

Fig. 36. Distribution of the Tresca equivalent stress below a rolling-sliding contact (*z*0 depth indicated for pure radial load, i.e. μ=0) and illustration of the tribological model of localized

Under the influence of vibrations, disturbance of proper contact operating conditions in a way that high shearing stresses are induced in the lubricating film can promote lubricant degradation (Kudish & Covitch, 2010). Reduced lubricity enhances the effect of sliding friction, e.g. described in the tribological model of Figure 36. Further to the discussed mechanical and thermal influence, vibration loading induces chemical aging of the lubricant and its additives (Gegner & Nierlich, 2008). Contaminations, like water or wear debris, increase the effect. The gradual decomposition process and associated acidification of the lubricant promote, for instance, the initiation of surface cracks on the raceway by tribochemical dissolution of nonmetallic MnS inclusion lines, which is discussed in the next

First, Figure 37 gives a demonstrative example of a corrosive attack by a decomposed lubricant. The etching pattern on the raceway reveals chemical smoothing of the surface. Copper contamination by abrasion from the graphite-brass ground brush of the diesel electric locomotive gets into the grease of the train wheel bearing and accelerates lubricant aging. As evident from Figure 37, the foreign particles also cause indentations on the raceway. An electrical oil sensor system can be used for online condition monitoring of the lubricant (Gegner et al., 2010). The application in industrial gearboxes, for instance of wind turbines, is of special practical interest. Note that the (e.g., extreme pressure) additives

markedly influence the electrical properties of the lubricant (Prashad, 2006).

bearing test rig (Gegner & Nierlich, 2008).

friction coefficient in the inset (*F*n is the normal force)

**5.2 Tribochemically initiated surface cracks** 

section.

Fig. 37. SEM-SE image of a chemical surface attack on the outer ring raceway of a CRB

As exemplified by Figure 38, some manganese sulfide lines intersect the rolling contact surface. Such inclusions are manufacturing related from the steelmaking process, despite the high level of cleanliness of bearing grades.

Fig. 38. LOM micrograph of the etched metallographic section of a sulfide inclusion line intersecting the surface of the inner ring raceway of a cylindrical roller bearing

On the inner ring raceway of a cylindrical roller bearing of a weaving machine examined in Figure 39, mixed friction is indicated by the mechanically smoothed honing structure. Due to aging of the lubricating oil, as detected under vibration loading, the gradually acidifying fluid attacks the steel surface. Tribochemical dissolution of manufacturing related MnS inclusion lines leaves crack-like defects on the raceway. Sulfur is continuously removed as gaseous H2S by hydrogen from decomposition products of the lubricant:

$$\text{MnS} + \text{H}\_2 \rightarrow \text{H}\_2\text{S}^\uparrow + \text{Mn} \tag{6}$$

The remaining manganese is then preferentially corroded out. This new mechanism of crack formation on tribologically loaded raceway surfaces is verified by chemical characterization using energy dispersive X-ray (EDX) microanalysis on the SEM. The EDX spectra in Figure 39, recorded at an acceleration voltage of 20 kV, confirm residues of manganese and sulfur at four sites (S1 to S4) of an emerging crack, thus excluding accidental intersection. The ring is made of martensitically hardened bearing steel. Reaction layer formation on the raceway is reflected in the signals of phosphorus from lubricant additives and oxygen.

Crack initiation by tribochemical reaction is also found on lateral surfaces of rollers. In Figure 40, remaining manganese and sulfur are detected by elemental mapping in the insets on the right.

Tribological Aspects of Rolling Bearing Failures 69

The tribochemical dissolution of MnS lines on raceway surfaces during the operation of rolling bearings also agrees with the general tendency that inclusions of all types reduce the corrosion resistance of the steel. The chemical attack occurs by the lubricant aged in service. The example of an early stage of defect evolution in Figure 41a points out that continuous dissolution but not fracturing of MnS inclusions gradually initiates a surface crack. Three analysis positions, where residues of manganese and sulfur are found, are indicated in the SEM image. An exemplary EDX spectrum is shown in Figure 41b. The inner ring raceway of the ball bearing from a car alternator reveals high-frequency electric current passage (cf.

 Fig. 41. Tribochemically induced crack evolution on the IR raceway of a DGBB revealing (a) a SEM-SE image with indicated sites where EDX analysis proves the presence of residues of MnS dissolution and (b) a recorded EDX spectrum exemplarily of the analysis results

After defect initiation on MnS inclusions, further damage development involves shallow micropitting (Gegner & Nierlich, 2008). Figure 42a also suggests crack propagation into the depth. Four sites of verified MnS residues are indicated, for which Figure 42b provides a representative detection example. The partly smoothed raceway reflects the effect of mixed

 Fig. 42. Documentation of damage evolution by (a) a SEM-SE image of shallow material removals along dissolved MnS inclusions on the IR raceway of a TRB from an industrial gearbox with indication of four positions where EDX analysis reveals MnS residues and (b) EDX spectrum exemplarily of the analysis results recorded at the sites given in Figure 42a

Figure 26a) that promotes lubricant aging (see section 4.3).

friction.

Fig. 39. SEM-SE images of cracks on the IR raceway of a CRB from the gearbox of a weaving machine and EDX spectra S1 to S4 taken at the indicated analysis positions

Fig. 40. SEM-SE image of a crack on a CRB roller and elemental mapping (area as indicated)

Fig. 39. SEM-SE images of cracks on the IR raceway of a CRB from the gearbox of a weaving

Fig. 40. SEM-SE image of a crack on a CRB roller and elemental mapping (area as indicated)

machine and EDX spectra S1 to S4 taken at the indicated analysis positions

The tribochemical dissolution of MnS lines on raceway surfaces during the operation of rolling bearings also agrees with the general tendency that inclusions of all types reduce the corrosion resistance of the steel. The chemical attack occurs by the lubricant aged in service. The example of an early stage of defect evolution in Figure 41a points out that continuous dissolution but not fracturing of MnS inclusions gradually initiates a surface crack. Three analysis positions, where residues of manganese and sulfur are found, are indicated in the SEM image. An exemplary EDX spectrum is shown in Figure 41b. The inner ring raceway of the ball bearing from a car alternator reveals high-frequency electric current passage (cf. Figure 26a) that promotes lubricant aging (see section 4.3).

Fig. 41. Tribochemically induced crack evolution on the IR raceway of a DGBB revealing (a) a SEM-SE image with indicated sites where EDX analysis proves the presence of residues of MnS dissolution and (b) a recorded EDX spectrum exemplarily of the analysis results

After defect initiation on MnS inclusions, further damage development involves shallow micropitting (Gegner & Nierlich, 2008). Figure 42a also suggests crack propagation into the depth. Four sites of verified MnS residues are indicated, for which Figure 42b provides a representative detection example. The partly smoothed raceway reflects the effect of mixed friction.

Fig. 42. Documentation of damage evolution by (a) a SEM-SE image of shallow material removals along dissolved MnS inclusions on the IR raceway of a TRB from an industrial gearbox with indication of four positions where EDX analysis reveals MnS residues and (b) EDX spectrum exemplarily of the analysis results recorded at the sites given in Figure 42a

Tribological Aspects of Rolling Bearing Failures 71

The appearance of the micropits on the raceway is similar to shallow material removals on tribochemically dissolved MnS inclusions, as evident from a comparison of Figures 44a and 42a. Micropitting can occur on small cracks initiated on the loaded surface. The SEM image of Figure 45a indicates such causative shallow cracking induced by shear stresses, slightly inclined to the axial direction. The metallographic microsection in Figure 45b documents crack growth into the material in a flat angle to the raceway up to a small depth of few µm

 Fig. 45. Investigation of gray staining on the IR raceway of a rig tested automobile gearbox DGBB revealing (a) a SEM-SE image and (b) LOM micrographs of the etched (top) and

The SEM overview in Figure 46a illustrates how dense covering of the raceway with micropits results in the characteristic dull matte appearance of the affected surface. On the bottom left hand side of the detail of Figure 46b, damage evolution on axially inclined microcracks results in incipient material delamination. Micropitting on a honing groove illustrates typical band formation. Note that the *b*/*B* parameter is reduced on the raceway

 Fig. 46. Investigation of the smoothed damaged inner ring raceway of the deep groove ball bearing of Figure 45a presenting (a) a SEM-SE overview and (b) the indicated detail that reveals near-surface crack propagation in overrolling direction from the bottom to the top

followed by surface return to form a micropit eventually.

unetched section of a developing micropit

surface to 0.69.

The EDX reference analysis of bearing steel is provided in Figure 43. It allows comparisons with the spectra of Figures 39, 41b and 42b.

Fig. 43. EDX reference spectrum of bearing steel for comparison of the signals

#### **5.3 Gray staining – Corrosion rolling contact fatigue**

Gray staining by dense micropitting, well known as a surface damage on tooth flanks of gears, is also caused by mixed friction in rolling-sliding contact. The flatly expanded shallow material fractures of only few µm depth, which cover at least parts of an affected raceway, are frequently initiated along honing marks. In Figure 44a, propagation of material delamination to the right occurs into sliding direction. Typical features of the influence of corrosion are visible on the open fracture surfaces. The corresponding XRD material response analysis in Figure 44b shows that vibrational loading of the tribological contact can cause gray staining. Note that the shallow micropits do not affect the residual stress state considerably. The smoothed raceway of Fig. 44a, which indicates mixed friction, is virtually free of indentations. A characteristic type A vibration residual stress profile, maybe with some type B contribution in 100 µm depth (cf. Figures 33 and 36, *z*0 much larger), is obtained. The XRD rolling contact fatigue damage parameter of *b*/*B*≥0.83 reaches or slightly exceeds the *L*10 equivalent value of 0.86 for the surface failure mode of roller bearings.

Fig. 44. Investigation of gray staining on the IR raceway of a CRB revealing (a) a SEM-SE image and (b) the measured type A vibration residual stress and XRD peak width distribution

The EDX reference analysis of bearing steel is provided in Figure 43. It allows comparisons

Fig. 43. EDX reference spectrum of bearing steel for comparison of the signals

Gray staining by dense micropitting, well known as a surface damage on tooth flanks of gears, is also caused by mixed friction in rolling-sliding contact. The flatly expanded shallow material fractures of only few µm depth, which cover at least parts of an affected raceway, are frequently initiated along honing marks. In Figure 44a, propagation of material delamination to the right occurs into sliding direction. Typical features of the influence of corrosion are visible on the open fracture surfaces. The corresponding XRD material response analysis in Figure 44b shows that vibrational loading of the tribological contact can cause gray staining. Note that the shallow micropits do not affect the residual stress state considerably. The smoothed raceway of Fig. 44a, which indicates mixed friction, is virtually free of indentations. A characteristic type A vibration residual stress profile, maybe with some type B contribution in 100 µm depth (cf. Figures 33 and 36, *z*0 much larger), is obtained. The XRD rolling contact fatigue damage parameter of *b*/*B*≥0.83 reaches or slightly exceeds the *L*10 equivalent value of 0.86 for the surface failure mode of roller bearings.

 Fig. 44. Investigation of gray staining on the IR raceway of a CRB revealing (a) a SEM-SE image and (b) the measured type A vibration residual stress and XRD peak width

**5.3 Gray staining – Corrosion rolling contact fatigue** 

distribution

with the spectra of Figures 39, 41b and 42b.

The appearance of the micropits on the raceway is similar to shallow material removals on tribochemically dissolved MnS inclusions, as evident from a comparison of Figures 44a and 42a. Micropitting can occur on small cracks initiated on the loaded surface. The SEM image of Figure 45a indicates such causative shallow cracking induced by shear stresses, slightly inclined to the axial direction. The metallographic microsection in Figure 45b documents crack growth into the material in a flat angle to the raceway up to a small depth of few µm followed by surface return to form a micropit eventually.

Fig. 45. Investigation of gray staining on the IR raceway of a rig tested automobile gearbox DGBB revealing (a) a SEM-SE image and (b) LOM micrographs of the etched (top) and unetched section of a developing micropit

The SEM overview in Figure 46a illustrates how dense covering of the raceway with micropits results in the characteristic dull matte appearance of the affected surface. On the bottom left hand side of the detail of Figure 46b, damage evolution on axially inclined microcracks results in incipient material delamination. Micropitting on a honing groove illustrates typical band formation. Note that the *b*/*B* parameter is reduced on the raceway surface to 0.69.

Fig. 46. Investigation of the smoothed damaged inner ring raceway of the deep groove ball bearing of Figure 45a presenting (a) a SEM-SE overview and (b) the indicated detail that reveals near-surface crack propagation in overrolling direction from the bottom to the top

Tribological Aspects of Rolling Bearing Failures 73

surface of a roller from a rig tested TRB and gray staining on the cam race tracks of a camshaft, respectively. The even appearance of the separated grain boundaries points to intercrystalline cleavage fracture of embrittled surface material by frictional tensile stresses. The micropit on a raceway suffering from gray staining in Figure 49 suggests partly intercrystalline corrosion assisted crack growth. Striation-like crack arrest marks are clearly visible on the fracture surface. Microvoids in the indicated region point to corrosion

Fig. 49. SEM-SE image of a micropit on the IR raceway of a CRB from a field application

Possible mechanisms of gradual near-surface embrittlement during overrolling are (temper) carbide dissolution by dislocational carbon segregation (see section 4.2, Figure 22), carbide reprecipitation at former austenite or martensite grain boundaries, hydrogen absorption and work hardening by raceway indentations or edge zone plastification in the metal-to-metal contact under mixed friction. The occurrence of plate carbides, for instance, in micropits of gray staining is reported (Nierlich & Gegner, 2006). Due to lower chromium content than the steel matrix, these precipitates are obviously formed during rolling contact operation.

Premature bearing failures, characterized by the formation of heavily branching systems of cracks with borders partly decorated by white etching microstructure, occur in specific susceptible applications typically within a considerably reduced running time of 1% to 20% of the nominal *L*10 life. Therefore, ordinary rolling contact fatigue can evidently be excluded as potential root cause, which agrees with the general finding that only limited material response is detected by XRD residual stress analyses. As shown in Figure 50, axial cracks of length ranging from below 1 to more than 20 mm, partly connected with pock-like spallings, are typically found on the raceway in such rare cases. For an affected application, for instance, it is reported in the literature that the actual *L*10 bearing life equals only six months,

Particularly axial microsections often suggest subsurface damage initiation. An illustrative

In the literature, abnormal development of butterflies, material weakening by gradual hydrogen absorption through the working contact and severe plastic deformation in connection with adiabatic shearing are considered the potential root cause of premature

resulting in 60% failures within 20 months of operation (Luyckx, 2011).

processes (see section 5.3, C-RCF).

**5.5 White etching cracks** 

example is shown in Figure 51.

Pronounced striations on the open fracture surfaces of micropits prove a significant contribution of mechanical fatigue to the crack propagation. The SEM details of Figures 47a and 47b confirm this finding. Therefore, it is concluded that a variant of corrosion fatigue is the driving force behind crack growth of micropitting in gray staining.

Fig. 47. SEM-SE details of the inner ring raceway of the deep groove ball bearing of Figure 46a revealing (a) distinct striations on a micropit fracture surface and (b) the same microfractographic feature on the open fracture face of another micropit

The additional chemical loading is not considered in fracture mechanics simulations of micropit formation by surface initiation and subsequent propagation of fatigue cracks (Fajdiga & Srami, 2009). The findings discussed above, however, suggest that gray staining can be interpreted as corrosion rolling contact fatigue (C-RCF).

#### **5.4 Surface embrittlement in operation**

Although quickly obscured by subsequent overrolling damage in further operation, shallow intercrystalline fractures are sporadically observed on raceway surfaces (Nierlich & Gegner, 2006). Illustrative examples are shown in the SEM images of Figures 48a and 48b.

Fig. 48. SEM-SE images of the rolling contact surfaces of (a) a TRB roller and (b) a cam

The microstructure breaks open along former austenite grain boundaries. The affected raceway is heavily smoothed by mixed friction. Figure 48a and 48b characterize the lateral

Pronounced striations on the open fracture surfaces of micropits prove a significant contribution of mechanical fatigue to the crack propagation. The SEM details of Figures 47a and 47b confirm this finding. Therefore, it is concluded that a variant of corrosion fatigue is

 Fig. 47. SEM-SE details of the inner ring raceway of the deep groove ball bearing of Figure

The additional chemical loading is not considered in fracture mechanics simulations of micropit formation by surface initiation and subsequent propagation of fatigue cracks (Fajdiga & Srami, 2009). The findings discussed above, however, suggest that gray staining

Although quickly obscured by subsequent overrolling damage in further operation, shallow intercrystalline fractures are sporadically observed on raceway surfaces (Nierlich & Gegner,

 Fig. 48. SEM-SE images of the rolling contact surfaces of (a) a TRB roller and (b) a cam

The microstructure breaks open along former austenite grain boundaries. The affected raceway is heavily smoothed by mixed friction. Figure 48a and 48b characterize the lateral

46a revealing (a) distinct striations on a micropit fracture surface and (b) the same

2006). Illustrative examples are shown in the SEM images of Figures 48a and 48b.

microfractographic feature on the open fracture face of another micropit

can be interpreted as corrosion rolling contact fatigue (C-RCF).

**5.4 Surface embrittlement in operation** 

the driving force behind crack growth of micropitting in gray staining.

surface of a roller from a rig tested TRB and gray staining on the cam race tracks of a camshaft, respectively. The even appearance of the separated grain boundaries points to intercrystalline cleavage fracture of embrittled surface material by frictional tensile stresses. The micropit on a raceway suffering from gray staining in Figure 49 suggests partly

intercrystalline corrosion assisted crack growth. Striation-like crack arrest marks are clearly visible on the fracture surface. Microvoids in the indicated region point to corrosion processes (see section 5.3, C-RCF).

Fig. 49. SEM-SE image of a micropit on the IR raceway of a CRB from a field application

Possible mechanisms of gradual near-surface embrittlement during overrolling are (temper) carbide dissolution by dislocational carbon segregation (see section 4.2, Figure 22), carbide reprecipitation at former austenite or martensite grain boundaries, hydrogen absorption and work hardening by raceway indentations or edge zone plastification in the metal-to-metal contact under mixed friction. The occurrence of plate carbides, for instance, in micropits of gray staining is reported (Nierlich & Gegner, 2006). Due to lower chromium content than the steel matrix, these precipitates are obviously formed during rolling contact operation.

#### **5.5 White etching cracks**

Premature bearing failures, characterized by the formation of heavily branching systems of cracks with borders partly decorated by white etching microstructure, occur in specific susceptible applications typically within a considerably reduced running time of 1% to 20% of the nominal *L*10 life. Therefore, ordinary rolling contact fatigue can evidently be excluded as potential root cause, which agrees with the general finding that only limited material response is detected by XRD residual stress analyses. As shown in Figure 50, axial cracks of length ranging from below 1 to more than 20 mm, partly connected with pock-like spallings, are typically found on the raceway in such rare cases. For an affected application, for instance, it is reported in the literature that the actual *L*10 bearing life equals only six months, resulting in 60% failures within 20 months of operation (Luyckx, 2011).

Particularly axial microsections often suggest subsurface damage initiation. An illustrative example is shown in Figure 51.

In the literature, abnormal development of butterflies, material weakening by gradual hydrogen absorption through the working contact and severe plastic deformation in connection with adiabatic shearing are considered the potential root cause of premature

Tribological Aspects of Rolling Bearing Failures 75

increased portion of intercrystalline fractures, is restricted to the surrounding area of the original cracks (Nierlich & Gegner, 2011). The undamaged rolling contact surface is protected by a regenerative passivating reaction layer. Adiabatic shear bands (ASB) develop by local flash heating to austenitising temperature due to very rapid large plastic deformation characteristic of, for instance, high speed machining or ballistic impact. Such extreme shock straining conditions obviously do not arise during bearing operation. WEC reveal strikingly branched crack paths, whereas ASB form essentially straight regular ribbons of length in the mm range. Adiabatic shearing represents a localized transformation into white etching microstructure possibly followed by cracking of the brittle new ASB phase. WEC evolve contrary by primary crack growth. Parts of the paths are subsequently

The spidery pattern of the white etching areas in Figure 51 indicates irregular crack propagation prior to the microstructural changes on the borders. Equivalent stresses reveal uniform distribution in the subsurface region. The reason for the appearance of Figure 51 is the spreading and branching growth of the cracks in circumferential orientation. Cracks originated subsurface usually do not create axial raceway cracks but emerge at the surface mostly as erratically shaped spalling (cf. Figure 2b). Targeted radial microsections actually reveal the connection to the raceway. Figure 52 points to surface WEC initiation due to the overall orientation and depth extension of the crack propagation in overrolling direction from left to right. One can easily imagine how damage pattern similar to Figure 51 occur in

Fig. 52. LOM micrograph of the etched radial microsection of the case hardened inner ring of a CARB bearing from a paper making machine. The overrolling direction is left-to-right Another example is shown in Figure 53a. The overrolling direction is from left to right so that crack initiation on the surface is evident. Figure 53b reveals the view of the edge of this microsection. No crack is visible at the initiation site on the raceway in the SEM (see section 5.5.1) so that also the detection probability question arises. The intensity of the white microstructure decoration of individual crack segments depends, for instance, on the depth (e.g., magnitude of the orthogonal shear stress) and the orientation to the raceway surface (friction and wear between the flanks). The pronounced tendency of the propagating cracks to branch indicates no pure mechanical fatigue but high additional chemical loading. Together with the regularly observed transcrystalline crack growth, this is typical of

decorated with white etching constituents.

accidentally located etched axial microsections.

corrosion fatigue.

bearing damage by white etching crack (WEC) formation (Harada et al., 2005; Hiraoka et al., 2006; Holweger & Loos, 2011; Iso et al., 2005; Kino & Otani, 2003; Kohara et al., 2006; Kotzalas & Doll, 2010; Luyckx, 2011; Shiga et al., 2006). These hypotheses, however, conflict with essential findings from failure analyses (further details are discussed in the following). White etching cracks are observed in affected bearings without and with butterflies (Hertzian pressure higher than about 1400 MPa required, see section 3.3) so that evidently both microstructural changes are mutually independent. Depth resolved concentration determinations on inner rings with differently advanced damage show that hydrogen enrichment occurs as a secondary effect abruptly only after the formation of raceway cracks by aging reactions of the penetrating lubricant, i.e. rapidly during the last weeks to few months of operation but not continuously over a long running time (Nierlich & Gegner, 2011). Hydrogen embrittlement on preparatively opened raceway cracks, reflected in an

Fig. 50. Macro image of the raceway of a martensitically hardened inner ring out of bearing steel of a taper roller bearing from an industrial gearbox

Fig. 51. LOM micrograph of the etched axial microsection of the bainitically hardened inner ring of a spherical roller bearing from a crane lifting unit

bearing damage by white etching crack (WEC) formation (Harada et al., 2005; Hiraoka et al., 2006; Holweger & Loos, 2011; Iso et al., 2005; Kino & Otani, 2003; Kohara et al., 2006; Kotzalas & Doll, 2010; Luyckx, 2011; Shiga et al., 2006). These hypotheses, however, conflict with essential findings from failure analyses (further details are discussed in the following). White etching cracks are observed in affected bearings without and with butterflies (Hertzian pressure higher than about 1400 MPa required, see section 3.3) so that evidently both microstructural changes are mutually independent. Depth resolved concentration determinations on inner rings with differently advanced damage show that hydrogen enrichment occurs as a secondary effect abruptly only after the formation of raceway cracks by aging reactions of the penetrating lubricant, i.e. rapidly during the last weeks to few months of operation but not continuously over a long running time (Nierlich & Gegner, 2011). Hydrogen embrittlement on preparatively opened raceway cracks, reflected in an

Fig. 50. Macro image of the raceway of a martensitically hardened inner ring out of bearing

Fig. 51. LOM micrograph of the etched axial microsection of the bainitically hardened inner

steel of a taper roller bearing from an industrial gearbox

ring of a spherical roller bearing from a crane lifting unit

increased portion of intercrystalline fractures, is restricted to the surrounding area of the original cracks (Nierlich & Gegner, 2011). The undamaged rolling contact surface is protected by a regenerative passivating reaction layer. Adiabatic shear bands (ASB) develop by local flash heating to austenitising temperature due to very rapid large plastic deformation characteristic of, for instance, high speed machining or ballistic impact. Such extreme shock straining conditions obviously do not arise during bearing operation. WEC reveal strikingly branched crack paths, whereas ASB form essentially straight regular ribbons of length in the mm range. Adiabatic shearing represents a localized transformation into white etching microstructure possibly followed by cracking of the brittle new ASB phase. WEC evolve contrary by primary crack growth. Parts of the paths are subsequently decorated with white etching constituents.

The spidery pattern of the white etching areas in Figure 51 indicates irregular crack propagation prior to the microstructural changes on the borders. Equivalent stresses reveal uniform distribution in the subsurface region. The reason for the appearance of Figure 51 is the spreading and branching growth of the cracks in circumferential orientation. Cracks originated subsurface usually do not create axial raceway cracks but emerge at the surface mostly as erratically shaped spalling (cf. Figure 2b). Targeted radial microsections actually reveal the connection to the raceway. Figure 52 points to surface WEC initiation due to the overall orientation and depth extension of the crack propagation in overrolling direction from left to right. One can easily imagine how damage pattern similar to Figure 51 occur in accidentally located etched axial microsections.

Fig. 52. LOM micrograph of the etched radial microsection of the case hardened inner ring of a CARB bearing from a paper making machine. The overrolling direction is left-to-right

Another example is shown in Figure 53a. The overrolling direction is from left to right so that crack initiation on the surface is evident. Figure 53b reveals the view of the edge of this microsection. No crack is visible at the initiation site on the raceway in the SEM (see section 5.5.1) so that also the detection probability question arises. The intensity of the white microstructure decoration of individual crack segments depends, for instance, on the depth (e.g., magnitude of the orthogonal shear stress) and the orientation to the raceway surface (friction and wear between the flanks). The pronounced tendency of the propagating cracks to branch indicates no pure mechanical fatigue but high additional chemical loading. Together with the regularly observed transcrystalline crack growth, this is typical of corrosion fatigue.

Tribological Aspects of Rolling Bearing Failures 77

depth of around 60 µm. The original loading conditions relevant to damage initiation are not obscured by overrolling of spalls at a later stage of failure and only isolated indentations are found on the raceway. The characteristic type A residual stress profile in Figures 54a and 54b thus identifies the impact of vibrations. On the surface, advanced material aging of

Incipient hairline cracks on the raceway are almost undetectable even in the SEM. The virtually perspective view of the edge of a microsection in Figure 55 provides an example (cf. Figure 53b). A corresponding micrograph of the etched microsection is shown in Figure

Fig. 55. SEM-SE image of a hairline crack initiation site on the smoothed raceway surface and incipient fatigue crack growth into the material in overrolling direction from bottom to top visible in the cut microsection on the right. The SRB failure of Figure 54 is investigated

Fig. 56. LOM micrograph of the etched metallographic section on the right of Figure 55. The raceway surface is at the top of the image. The overrolling direction is from left to right

Shear stress control of surface fatigue crack initiation, under varying load and frictiondefining slip in the contact area, and subsequent propagation is apparent from crack advance in overrolling direction in a small angle to the raceway tangent. The mechanism is particularly evident from the unbranched crack in Figure 57. The inset zooms in on the edge zone. Compressive residual stresses near the surface (cf. Figure 54) demonstrate the effect of

*b*/*B*≥0.69 is deduced.

56.

Fig. 53. Investigation of a white etching crack system in the martensitically hardened inner ring of a taper roller bearing from a coal pulverizer revealing (a) a LOM micrograph of the etched radial microsection (overrolling direction from left to right) and (b) a near-surface SEM detail (backscattered electron mode) of the view of the edge of the same microsection

#### **5.5.1 Shear stress induced surface cracking and corrosion fatigue crack growth**

Mixed friction in rolling-sliding contact can cause surface cracks on bearing raceways. The shear stress induced initiation mechanism is introduced first. The result of the XRD material response analysis performed on both raceways of a double row spherical roller bearing is depicted in Figures 54a and 54b.

Fig. 54. Material response analysis showing a type A vibration residual stress and XRD peak width distribution below (a) the first and (b) the second raceway surface of the inner ring of a prematurely failed double row spherical roller bearing from a paper making machine

No subsurface changes of the XRD parameters occur. Note that for a Hertzian pressure of *p*0=2500 MPa, i.e. incipient plastic deformation in pure radial contact loading, the *z*0 depths of maximum v. Mises and orthogonal shear stress equal about 1.15 and 0.85 mm, respectively. Load induced butterfly microstructure transformations on nonmetallic inclusions are not observed in metallographic microsections of this large size roller bearing. Therefore, the maximum applied Hertzian pressure actually does not exceed about 1400 MPa (see section 3.3). Compressive residual stresses are formed near the surface up to a

Fig. 53. Investigation of a white etching crack system in the martensitically hardened inner ring of a taper roller bearing from a coal pulverizer revealing (a) a LOM micrograph of the etched radial microsection (overrolling direction from left to right) and (b) a near-surface SEM detail (backscattered electron mode) of the view of the edge of the same microsection

**5.5.1 Shear stress induced surface cracking and corrosion fatigue crack growth**  Mixed friction in rolling-sliding contact can cause surface cracks on bearing raceways. The shear stress induced initiation mechanism is introduced first. The result of the XRD material response analysis performed on both raceways of a double row spherical roller bearing is

 Fig. 54. Material response analysis showing a type A vibration residual stress and XRD peak width distribution below (a) the first and (b) the second raceway surface of the inner ring of a prematurely failed double row spherical roller bearing from a paper making machine

No subsurface changes of the XRD parameters occur. Note that for a Hertzian pressure of *p*0=2500 MPa, i.e. incipient plastic deformation in pure radial contact loading, the *z*0 depths of maximum v. Mises and orthogonal shear stress equal about 1.15 and 0.85 mm, respectively. Load induced butterfly microstructure transformations on nonmetallic inclusions are not observed in metallographic microsections of this large size roller bearing. Therefore, the maximum applied Hertzian pressure actually does not exceed about 1400 MPa (see section 3.3). Compressive residual stresses are formed near the surface up to a

depicted in Figures 54a and 54b.

depth of around 60 µm. The original loading conditions relevant to damage initiation are not obscured by overrolling of spalls at a later stage of failure and only isolated indentations are found on the raceway. The characteristic type A residual stress profile in Figures 54a and 54b thus identifies the impact of vibrations. On the surface, advanced material aging of *b*/*B*≥0.69 is deduced.

Incipient hairline cracks on the raceway are almost undetectable even in the SEM. The virtually perspective view of the edge of a microsection in Figure 55 provides an example (cf. Figure 53b). A corresponding micrograph of the etched microsection is shown in Figure 56.

Fig. 55. SEM-SE image of a hairline crack initiation site on the smoothed raceway surface and incipient fatigue crack growth into the material in overrolling direction from bottom to top visible in the cut microsection on the right. The SRB failure of Figure 54 is investigated

Fig. 56. LOM micrograph of the etched metallographic section on the right of Figure 55. The raceway surface is at the top of the image. The overrolling direction is from left to right

Shear stress control of surface fatigue crack initiation, under varying load and frictiondefining slip in the contact area, and subsequent propagation is apparent from crack advance in overrolling direction in a small angle to the raceway tangent. The mechanism is particularly evident from the unbranched crack in Figure 57. The inset zooms in on the edge zone. Compressive residual stresses near the surface (cf. Figure 54) demonstrate the effect of

Tribological Aspects of Rolling Bearing Failures 79

influence the tribochemical release of hydrogen. Accelerated lubricant aging due to vibration loading further supports the chemical assistance of corrosion fatigue cracking (CFC) and microstructure transformation into white etching constituents. Local material aging and embrittlement is manifested in the frequently observed formation of a dark etching region around the cracks. An example is given in the micrograph of Figures 59a. Regular etching induced preparative cracking along the branching CFC path in the corresponding SEM image of Figure 59b reflects plastification in the slip bands of the

Fig. 59. DER around CFC crack paths indicate localized material aging in (a) a LOM and (b) a SEM micrograph of an etched microsection of the IR of a TRB from an industrial gearbox

 Fig. 60. Carbide dissolution and distinct localized plastification at the multi-branching tip of a CFC crack visible in (a) a LOM and (b) a corresponding SEM micrograph of an etched radial microsection of the inner ring of a cylindrical roller bearing from a weaving machine Localized fatigue damage is promoted by hydrogen released from decomposition products and possibly contaminations of the lubricant, penetrating through the advancing crack from the raceway surface to the depth. The most intense microstructural changes thus occur on multi-branching sites of CFC cracks (cf. Figure 59). Particularly at these most effective hydrogen sources, pronounced carbide dissolution (see DGSL model, section 4.2) in the

embrittled DER material.

shear stresses required for crack development. According to Figure 58, extended white etching crack systems up to a depth of more than 1 mm are formed, where crack returns to the raceway result in pitting by break-out of the surface eventually. Note that in Figures 56 to 58, the overrolling direction from left to right strikingly indicates top-down WEC propagation.

Fig. 57. Same as Figure 56, another crack. The overrolling direction is from left to right

Fig. 58. Same as Figure 56, another WEC system. The overrolling direction from left to right and the orientation of repeated branching proves top-down growth of the CFC crack

Pronounced branching and deep, widely spreading propagation of the transcrystalline cracks essentially under moderate mechanical load of typically *p*0≈1500 MPa reveals corrosion fatigue in rolling contact as the driving force of crack growth. A comparison of Figure 56 and 57 suggests that also fracture of the new brittle ferritic phase can lead to the initiation of side cracks. Local phase transformation into white etching microstructure along the crack paths is caused by hydrogen (HELP mechanism) released from the highly stressed penetrating lubricant to the adjacent steel matrix. Wear between the crack flanks promotes the degradation reactions on blank metal faces (Kohara et al., 2006). Oil additives can

shear stresses required for crack development. According to Figure 58, extended white etching crack systems up to a depth of more than 1 mm are formed, where crack returns to the raceway result in pitting by break-out of the surface eventually. Note that in Figures 56 to 58, the overrolling direction from left to right strikingly indicates top-down WEC

Fig. 57. Same as Figure 56, another crack. The overrolling direction is from left to right

Fig. 58. Same as Figure 56, another WEC system. The overrolling direction from left to right and the orientation of repeated branching proves top-down growth of the CFC crack

Pronounced branching and deep, widely spreading propagation of the transcrystalline cracks essentially under moderate mechanical load of typically *p*0≈1500 MPa reveals corrosion fatigue in rolling contact as the driving force of crack growth. A comparison of Figure 56 and 57 suggests that also fracture of the new brittle ferritic phase can lead to the initiation of side cracks. Local phase transformation into white etching microstructure along the crack paths is caused by hydrogen (HELP mechanism) released from the highly stressed penetrating lubricant to the adjacent steel matrix. Wear between the crack flanks promotes the degradation reactions on blank metal faces (Kohara et al., 2006). Oil additives can

propagation.

influence the tribochemical release of hydrogen. Accelerated lubricant aging due to vibration loading further supports the chemical assistance of corrosion fatigue cracking (CFC) and microstructure transformation into white etching constituents. Local material aging and embrittlement is manifested in the frequently observed formation of a dark etching region around the cracks. An example is given in the micrograph of Figures 59a. Regular etching induced preparative cracking along the branching CFC path in the corresponding SEM image of Figure 59b reflects plastification in the slip bands of the embrittled DER material.

Fig. 59. DER around CFC crack paths indicate localized material aging in (a) a LOM and (b) a SEM micrograph of an etched microsection of the IR of a TRB from an industrial gearbox

Fig. 60. Carbide dissolution and distinct localized plastification at the multi-branching tip of a CFC crack visible in (a) a LOM and (b) a corresponding SEM micrograph of an etched radial microsection of the inner ring of a cylindrical roller bearing from a weaving machine

Localized fatigue damage is promoted by hydrogen released from decomposition products and possibly contaminations of the lubricant, penetrating through the advancing crack from the raceway surface to the depth. The most intense microstructural changes thus occur on multi-branching sites of CFC cracks (cf. Figure 59). Particularly at these most effective hydrogen sources, pronounced carbide dissolution (see DGSL model, section 4.2) in the

Tribological Aspects of Rolling Bearing Failures 81

Figure 62 completes the investigation of the SRB failure of Figures 54 to 58. The fracture face of a preparatively opened crack at the initiation site on the surface is shown. The inner ring raceway is visible at the top. Following the brittle incipient crack of about 5 µm depth, dense

striations indicate the fatigue nature of crack propagation almost from the surface.

Fig. 62. SEM-SE fractograph of the original fracture surface of a subsequently opened raceway crack on the inner ring of the spherical roller bearing of Figures 54 to 58

Fig. 63. SEM-SE image of the IR raceway of a CARB bearing and indicated elemental mapping (on the right) revealing an oxide inclusion (Al and Mg detected) that breaks off from the surface under frictional rolling contact loading to cause a micropit eventually

In Figure 63, the demonstrative elemental distribution images of magnesium and aluminum are mapped over the damaged region. Sharp-edged axial surface cracks on tribochemically dissolved MnS inclusions (see section 5.2, e.g. Figure 42a), which advance vertically downwards into the material (Nierlich & Gegner, 2006), as well as grain boundary cleavage (cf. Figure 48) further indicate the action of frictional tensile stresses. Another type of failure

**5.5.2 Frictional tensile stress induced surface cracking and normal stress hypothesis**  Figure 63 reveals a micropit on the smoothed inner ring raceway of a CARB bearing from a paper making machine. Material removal is caused by a brittle Mg-Al-O spinel inclusion that breaks off from the surface under tribomechanical loading of the rolling-sliding contact.

proceeding phase transformation is visible in the microsection. The region of the heavily branching tip of a CFC crack in the LOM micrograph of Figure 60a provides an illustration. Localized plasticity in the area of carbide dissolution is evident from the corresponding SEM image of Figure 60b. Weaker material aging and incipient phase transformation (DER) also occurs along unbranched crack paths. The etching process emphasizes the actual microstructure damage. The secondary hydrogen embrittlement around CFC cracks, linked to DER formation, is reflected in the increased susceptibility of the locally aged steel matrix to preparative stress corrosion cracking, which from its first detection is referred to as *Zang* structure. The example of Figures 61a and 61b documents that the local dark etching region around corrosion fatigue cracks can be perceived as precursor of WEA (see also section 4.3). The developed banana-shaped WEA, surrounded by the preliminary DER structure, nestles to the CFC crack at a multi-branching site. Its harder material (more than 1000 HV) appears smoothed and darker in the SEM detail of Figure 61b, where texturing is indicated by reorientation of the included cracks.

The observation of enhanced, evidently hydrogen induced phase transformation at (multi-) branching sites agrees with regular finding of pronounced white etching area decoration at these positions of WEC systems. Note that in Figure 61, the match of the curved shape of the WEA with the crack path excludes primary WEA evolution.

Fig. 61. Curved white etching area along a multi-branching site of a WEC with surrounding embrittled DER material, identified as WEA precursor, in (a) a LOM and (b) a SEM micrograph of an etched microsection of the inner ring of a TRB from an industrial gearbox

In the outer zone of the overrolled material, the shear stresses for dislocation glide in the described dynamic (nano-) recrystallization process of white etching microstructure formation around CFC cracks, which offer the hydrogen source for accelerated local fatigue aging, increase with depth. This is one reason why the decorating constituents in a WEC are often found less intense near the raceway surface (see, e.g., Figures 52, 56 and 57). The overall hydrogen content of 0.9 ppm measured at the inner ring of Figures 54 to 58 is consistent with the typical delivery condition. This finding reflects the limited damage of the investigated bearing. Depending on the density of the raceway cracks, gradual secondary hydrogen absorption from the surface to the bore is verified at the final stage of service life (Nierlich & Gegner, 2011).

proceeding phase transformation is visible in the microsection. The region of the heavily branching tip of a CFC crack in the LOM micrograph of Figure 60a provides an illustration. Localized plasticity in the area of carbide dissolution is evident from the corresponding SEM image of Figure 60b. Weaker material aging and incipient phase transformation (DER) also occurs along unbranched crack paths. The etching process emphasizes the actual microstructure damage. The secondary hydrogen embrittlement around CFC cracks, linked to DER formation, is reflected in the increased susceptibility of the locally aged steel matrix to preparative stress corrosion cracking, which from its first detection is referred to as *Zang* structure. The example of Figures 61a and 61b documents that the local dark etching region around corrosion fatigue cracks can be perceived as precursor of WEA (see also section 4.3). The developed banana-shaped WEA, surrounded by the preliminary DER structure, nestles to the CFC crack at a multi-branching site. Its harder material (more than 1000 HV) appears smoothed and darker in the SEM detail of Figure 61b, where texturing is indicated by

The observation of enhanced, evidently hydrogen induced phase transformation at (multi-) branching sites agrees with regular finding of pronounced white etching area decoration at these positions of WEC systems. Note that in Figure 61, the match of the curved shape of the

embrittled DER material, identified as WEA precursor, in (a) a LOM and (b) a SEM

Fig. 61. Curved white etching area along a multi-branching site of a WEC with surrounding

micrograph of an etched microsection of the inner ring of a TRB from an industrial gearbox In the outer zone of the overrolled material, the shear stresses for dislocation glide in the described dynamic (nano-) recrystallization process of white etching microstructure formation around CFC cracks, which offer the hydrogen source for accelerated local fatigue aging, increase with depth. This is one reason why the decorating constituents in a WEC are often found less intense near the raceway surface (see, e.g., Figures 52, 56 and 57). The overall hydrogen content of 0.9 ppm measured at the inner ring of Figures 54 to 58 is consistent with the typical delivery condition. This finding reflects the limited damage of the investigated bearing. Depending on the density of the raceway cracks, gradual secondary hydrogen absorption from the surface to the bore is verified at the final stage of service life

reorientation of the included cracks.

(Nierlich & Gegner, 2011).

WEA with the crack path excludes primary WEA evolution.

Figure 62 completes the investigation of the SRB failure of Figures 54 to 58. The fracture face of a preparatively opened crack at the initiation site on the surface is shown. The inner ring raceway is visible at the top. Following the brittle incipient crack of about 5 µm depth, dense striations indicate the fatigue nature of crack propagation almost from the surface.

Fig. 62. SEM-SE fractograph of the original fracture surface of a subsequently opened raceway crack on the inner ring of the spherical roller bearing of Figures 54 to 58

#### **5.5.2 Frictional tensile stress induced surface cracking and normal stress hypothesis**

Figure 63 reveals a micropit on the smoothed inner ring raceway of a CARB bearing from a paper making machine. Material removal is caused by a brittle Mg-Al-O spinel inclusion that breaks off from the surface under tribomechanical loading of the rolling-sliding contact.

Fig. 63. SEM-SE image of the IR raceway of a CARB bearing and indicated elemental mapping (on the right) revealing an oxide inclusion (Al and Mg detected) that breaks off from the surface under frictional rolling contact loading to cause a micropit eventually

In Figure 63, the demonstrative elemental distribution images of magnesium and aluminum are mapped over the damaged region. Sharp-edged axial surface cracks on tribochemically dissolved MnS inclusions (see section 5.2, e.g. Figure 42a), which advance vertically downwards into the material (Nierlich & Gegner, 2006), as well as grain boundary cleavage (cf. Figure 48) further indicate the action of frictional tensile stresses. Another type of failure

Tribological Aspects of Rolling Bearing Failures 83

Preferential surface cracking occurring vertically in axial direction on raceways of larger roller bearings (providing high *a* values) that run under (intermittently) increased mixed friction points to the validity of a normal stress fracture criterion (Nierlich & Gegner, 2011):

> nsh e

Modification of the equivalent normal stress nsh σ<sup>e</sup> , for instance by residual stresses (e.g. from surface finishing or cold working, cf. Figures 16a and 23a) or stress raising nonmetallic inclusions, is neglected in Figure 64 for the sake of simplicity. In a rough approximation, the relevant critical fracture strength σf of brittle spontaneous crack initiation is, due to almost deformationless material separation (see Figures 66 and 67 later in the text), estimated as the elastic limit *R*e≈800 MPa, which falls significantly below the yield strength for hardened bearing steel. In cyclic tension-compression tests, for instance, the material changes its response from elastic to microplastic at a stress level around 500 MPa (Voskamp, 1996). The failure range of the introduced normal stress hypothesis can then be determined as follows:


As spontaneous incipient crack formation is considered, the illustration of Figure 64 realistically refers to short-term loading of high Hertzian pressure *p*0≥2000 MPa and friction coefficient μ≥0.2. Rough indication of the relative σf/*p*0 level occurs accordingly. Note that the exact magnitude of the fracture strength σf≤*R*p0.2 does not make an essential difference to the validity of the introduced normal stress failure hypothesis but only influences the frequency of the rare events of raceway cracking as critical peak load operating conditions can cause tensile stresses 2μ*p*0≈2000 MPa on the surface. The length of the brittle mode I propagation of a frictionally initiated cleavage-like raceway crack depends on the stress intensity factor *K*I and the fracture toughness *K*Ic according to *K*I>*K*Ic. The depth effect of operational material embrittlement (see section 5.4) on the critical fracture strength σf (also valid for *K*Ic), which increases with the number *N* of ring revolutions, is schematically included in Figure 64, where larger size bearings with *a* in the range of 0.5 mm are considered. A concrete calculation example is given in the literature (Nierlich & Gegner, 2011). The semiminor axis *a* of the contact ellipse influences the extension of the failure range according to Eq. (12) and Figure 64. The micro friction model of Figure 36 is regarded. As deduced in section 5.1 from the effect of the induced equivalent shear stresses on plastification and the resulting type A or B residual stress patterns, vibrational loading can intermittently cause locally increased mixed friction. Under peak load operating conditions, such short-term states generally coincide with the impact of high Hertzian pressures. As the detection of type A residual stress distributions (see Figure 54) indicates, friction coefficients

*y a yy*

f

*y a yy*

<sup>=</sup> σ =σ (11)

<sup>=</sup> σ ≥ σ (12)

Fig. 65. Schematic representation of the macro contact area with elliptical Hertzian

distribution of the pressure *p* (maximum *p*0 in the center is indicated)

causing loading by differently disturbed bearing kinetics is thus reflected in brittle spontaneous crack initiation on raceway surfaces.

Application of the tribological model introduced in section 5.1 in the inset of Figure 36 allows the estimation of the development of the frictional tangential normal stresses *<sup>y</sup> <sup>a</sup> yy*−= <sup>σ</sup> with depth *z*. The classical analytical solution of a uniform infinite rolling-sliding line contact (Karas, 1941), for the highest tension level evaluated at the runout *y*=−*a*, is used for the approximation (μ=μ>):

$$\frac{\sigma\_{yy}}{p\_0} = \sinh\alpha \sin\beta \left(1 - \frac{\sinh 2\alpha + \mu \sin 2\beta}{\cosh 2\alpha - \cos 2\beta}\right) - (\sin\beta - 2\mu \cos \beta) \exp(-\alpha) \tag{7}$$

$$\sinh a = \frac{1}{a} \sqrt{\frac{1}{2} \left[ y^2 + z^2 - a^2 + \sqrt{\left( y^2 + z^2 - a^2 \right)^2 + 4a^2 z^2} \right]}, \text{ for } -a \le y \le 0 \tag{8}$$

$$y = a \cosh a \cos \beta, \ a \ge 0 \tag{9}$$

$$z = a \sinh \alpha \sin \beta, \text{ } \beta \ge 0 \tag{10}$$

The relationships of Eqs. (9) and (10) hold for the elliptic coordinates. Figure 64 shows a graphical representation of calculated depth distributions for increased friction coefficients μ of 0.2, 0.3 and 0.4. On the raceway surface at *z*=0, maximum tension of 2μ*p*0 is reached.

Fig. 64. Normalized distribution of the equivalent normal stress below a rolling-sliding contact (rolling occurs in *y* direction at velocity *vy*, see inset) and indication of the level of the critical fracture strength σf≈*R*e for typical peak loading with illustration of the expanding failure range by gradual in-service surface embrittlement (cf. section 5.4) with running time

Note that Figure 1 represents the stress field in the center of the Hertzian contact area. At the runout (*y*=−*a*, see inset of Figure 64), where the maximum sliding friction induced circumferential tensile stresses of Eq. (7) occur in the surface zone of the material, the hydrostatic pressure reduces to zero. A graphical illustration is provided in Figure 65.

causing loading by differently disturbed bearing kinetics is thus reflected in brittle

Application of the tribological model introduced in section 5.1 in the inset of Figure 36 allows the estimation of the development of the frictional tangential normal stresses *<sup>y</sup> <sup>a</sup>*

with depth *z*. The classical analytical solution of a uniform infinite rolling-sliding line contact (Karas, 1941), for the highest tension level evaluated at the runout *y*=−*a*, is used for

sinh 2 sin 2 sinh sin 1 sin 2 cos exp cosh 2 cos2

( )<sup>2</sup> 1 1 2 2 2 2 2 2 22 sinh 4 , for 0 <sup>2</sup> *y z a y z a az a y*

The relationships of Eqs. (9) and (10) hold for the elliptic coordinates. Figure 64 shows a graphical representation of calculated depth distributions for increased friction coefficients μ of 0.2, 0.3 and 0.4. On the raceway surface at *z*=0, maximum tension of 2μ*p*0 is reached.

Fig. 64. Normalized distribution of the equivalent normal stress below a rolling-sliding contact (rolling occurs in *y* direction at velocity *vy*, see inset) and indication of the level of the critical fracture strength σf≈*R*e for typical peak loading with illustration of the expanding failure range by gradual in-service surface embrittlement (cf. section 5.4) with running time Note that Figure 1 represents the stress field in the center of the Hertzian contact area. At the runout (*y*=−*a*, see inset of Figure 64), where the maximum sliding friction induced circumferential tensile stresses of Eq. (7) occur in the surface zone of the material, the hydrostatic pressure reduces to zero. A graphical illustration is provided in Figure 65.

⎡ ⎤ α = + − + + − + −≤ ≤ ⎢ ⎥ ⎣ ⎦

<sup>σ</sup> ⎛ ⎞ α+μ β <sup>=</sup> α β− ⎜ ⎟ − β − μ β −α ⎝ ⎠ α− β (7)

( ) ()

*y a* = cosh cos , 0 α β α≥ (9)

*z a* = sinh sin , 0 α β β≥ (10)

*yy*−= <sup>σ</sup>

(8)

spontaneous crack initiation on raceway surfaces.

the approximation (μ=μ>):

0

*a*

*yy p*

Fig. 65. Schematic representation of the macro contact area with elliptical Hertzian distribution of the pressure *p* (maximum *p*0 in the center is indicated)

Preferential surface cracking occurring vertically in axial direction on raceways of larger roller bearings (providing high *a* values) that run under (intermittently) increased mixed friction points to the validity of a normal stress fracture criterion (Nierlich & Gegner, 2011):

$$
\sigma\_{\mathfrak{e}}^{\text{nsh}} = \sigma\_{yy}^{y=\ast a} \tag{11}
$$

Modification of the equivalent normal stress nsh σ<sup>e</sup> , for instance by residual stresses (e.g. from surface finishing or cold working, cf. Figures 16a and 23a) or stress raising nonmetallic inclusions, is neglected in Figure 64 for the sake of simplicity. In a rough approximation, the relevant critical fracture strength σf of brittle spontaneous crack initiation is, due to almost deformationless material separation (see Figures 66 and 67 later in the text), estimated as the elastic limit *R*e≈800 MPa, which falls significantly below the yield strength for hardened bearing steel. In cyclic tension-compression tests, for instance, the material changes its response from elastic to microplastic at a stress level around 500 MPa (Voskamp, 1996). The failure range of the introduced normal stress hypothesis can then be determined as follows:

$$
\sigma\_{yy}^{y=\*t} \ge \sigma\_{\mathfrak{f}} \tag{12}
$$

As spontaneous incipient crack formation is considered, the illustration of Figure 64 realistically refers to short-term loading of high Hertzian pressure *p*0≥2000 MPa and friction coefficient μ≥0.2. Rough indication of the relative σf/*p*0 level occurs accordingly. Note that the exact magnitude of the fracture strength σf≤*R*p0.2 does not make an essential difference to the validity of the introduced normal stress failure hypothesis but only influences the frequency of the rare events of raceway cracking as critical peak load operating conditions can cause tensile stresses 2μ*p*0≈2000 MPa on the surface. The length of the brittle mode I propagation of a frictionally initiated cleavage-like raceway crack depends on the stress intensity factor *K*I and the fracture toughness *K*Ic according to *K*I>*K*Ic. The depth effect of operational material embrittlement (see section 5.4) on the critical fracture strength σf (also valid for *K*Ic), which increases with the number *N* of ring revolutions, is schematically included in Figure 64, where larger size bearings with *a* in the range of 0.5 mm are considered. A concrete calculation example is given in the literature (Nierlich & Gegner, 2011). The semiminor axis *a* of the contact ellipse influences the extension of the failure range according to Eq. (12) and Figure 64. The micro friction model of Figure 36 is regarded. As deduced in section 5.1 from the effect of the induced equivalent shear stresses on plastification and the resulting type A or B residual stress patterns, vibrational loading can intermittently cause locally increased mixed friction. Under peak load operating conditions, such short-term states generally coincide with the impact of high Hertzian pressures. As the detection of type A residual stress distributions (see Figure 54) indicates, friction coefficients

Tribological Aspects of Rolling Bearing Failures 85

56 to 61.

structure.

moment of surface cracking is suggested.

Fig. 68. SEM-SE details of Figure 67 as indicated (a) in the lower middle and (b) on the right In the area of the spontaneous crack of Figure 66, a mixed TiCN-MnS nonmetallic inclusion near the raceway surface in a depth of about 25 µm acts as stress raising crack nuclei. The appearance of the microsections, revealing white etching crack systems, is similar to Figures

Hydrogen releasing aging reactions of the lubricant during corrosion fatigue crack growth are proven by EDX microanalysis on preparatively opened fracture surfaces. As an example, Figure 69a shows an overview of the deep CFC region below the brittle lenticular crack of Figure 67. The area of the performed EDX analysis is marked in the SEM fractograph. Sulfur, phosphorus and zinc in the recorded spectrum of Figure 69b indicate reacted residues of oil additives near the crack tip in a depth of about 1 mm in higher concentration than on the low-deformation spontaneous incipient crack visible at top left of Figure 69a, where chemical attack is restricted to subsequent surface corrosion. Furthermore, numerous side cracks characterize corrosion fatigue fracture faces (see also, for instance, the microsections of Figures 56, 58 and 59). The forced rupture from preparative crack opening stands out clearly at the bottom and bottom left of Figure 69a against the dark original CFC fracture

The bearing applications of Figures 66 and 67 operate under vibrations. The observed local crack initiation on the raceway agrees with the approach of the tribological model in Figure 36 that subdivides the contact area into regions of different loading levels. Brittle spontaneous cracking occurs in subdomains of increased friction coefficient. Compared with the competing fatigue crack initiation mechanism discussed in section 5.5.1, lower slip in the

It is worth noting that post-machining thermal treatment (PMTT) of ground and honed rings and rollers, previously proposed in the literature for material reinforcement in the mechanically influenced edge zone (Gegner, 2006b; Gegner et al., 2009), is recently reported to be an effective countermeasure against premature bearing failures by white etching crack formation (Luyckx, 2011). The short reheating process of, e.g., 0.5 to 1 h after the finishing operation occurs below the tempering or transformation temperature to avoid undesired hardness loss (cf. section 4.2, Figure 23). As only the plastically deformed material in the outermost layer up to a depth of about 10 µm is microstructurally stabilized, a success of this simple treatment would provide further indication of surface WEC failure initiation.

above 0.3 can occur temporarily in subareas of the rolling contact. Larger size roller bearings are most sensitive to brittle cracking.

Further fractographic verification of normal stress failures is provided in the following. The steep gradient of the causative frictional tensile stress in Figure 64 indicates limited advance and rapid stop of an initiated brittle spontaneous mode I surface crack. The fracture faces of two preparatively opened vertical axial raceway cracks in the SEM images of Figures 66 and 67 confirm this prediction. The development of the (semi-) circular shape of the spontaneous cracks may be described by the depth dependence of the stress intensity factor and an energy balance criterion to minimize the interface energy.

Fig. 66. SEM-SE fractograph of the original fracture surface of a preparatively opened axial crack on the inner ring raceway of a failed taper roller bearing from an industrial gearbox

Fig. 67. SEM-SE fractograph of the original fracture surface of a preparatively opened axial crack on the inner ring raceway of a failed taper roller bearing from an industrial gearbox

The low-deformation transcrystalline lenticular cracks of about 150 µm depth act as incipient cracks of subsequent corrosion fatigue cracking into the depth, to the sides and on the surface. The distinct change of the fracture pattern in Figures 66 and 67, respectively with a demarcating bulge or crack network, on the latter of which Figures 68a and 68b zoom in, is evident. The crack arrest indicating numerous side cracks in Figure 68 reflect local material embrittlement as observed in the affected DER microstructure around CFC cracks in the SEM micrographs of etched microsections in Figures 59b and 60b.

above 0.3 can occur temporarily in subareas of the rolling contact. Larger size roller bearings

Further fractographic verification of normal stress failures is provided in the following. The steep gradient of the causative frictional tensile stress in Figure 64 indicates limited advance and rapid stop of an initiated brittle spontaneous mode I surface crack. The fracture faces of two preparatively opened vertical axial raceway cracks in the SEM images of Figures 66 and 67 confirm this prediction. The development of the (semi-) circular shape of the spontaneous cracks may be described by the depth dependence of the stress intensity factor and an

Fig. 66. SEM-SE fractograph of the original fracture surface of a preparatively opened axial crack on the inner ring raceway of a failed taper roller bearing from an industrial gearbox

Fig. 67. SEM-SE fractograph of the original fracture surface of a preparatively opened axial crack on the inner ring raceway of a failed taper roller bearing from an industrial gearbox The low-deformation transcrystalline lenticular cracks of about 150 µm depth act as incipient cracks of subsequent corrosion fatigue cracking into the depth, to the sides and on the surface. The distinct change of the fracture pattern in Figures 66 and 67, respectively with a demarcating bulge or crack network, on the latter of which Figures 68a and 68b zoom in, is evident. The crack arrest indicating numerous side cracks in Figure 68 reflect local material embrittlement as observed in the affected DER microstructure around CFC cracks

in the SEM micrographs of etched microsections in Figures 59b and 60b.

are most sensitive to brittle cracking.

energy balance criterion to minimize the interface energy.

Fig. 68. SEM-SE details of Figure 67 as indicated (a) in the lower middle and (b) on the right

In the area of the spontaneous crack of Figure 66, a mixed TiCN-MnS nonmetallic inclusion near the raceway surface in a depth of about 25 µm acts as stress raising crack nuclei. The appearance of the microsections, revealing white etching crack systems, is similar to Figures 56 to 61.

Hydrogen releasing aging reactions of the lubricant during corrosion fatigue crack growth are proven by EDX microanalysis on preparatively opened fracture surfaces. As an example, Figure 69a shows an overview of the deep CFC region below the brittle lenticular crack of Figure 67. The area of the performed EDX analysis is marked in the SEM fractograph. Sulfur, phosphorus and zinc in the recorded spectrum of Figure 69b indicate reacted residues of oil additives near the crack tip in a depth of about 1 mm in higher concentration than on the low-deformation spontaneous incipient crack visible at top left of Figure 69a, where chemical attack is restricted to subsequent surface corrosion. Furthermore, numerous side cracks characterize corrosion fatigue fracture faces (see also, for instance, the microsections of Figures 56, 58 and 59). The forced rupture from preparative crack opening stands out clearly at the bottom and bottom left of Figure 69a against the dark original CFC fracture structure.

The bearing applications of Figures 66 and 67 operate under vibrations. The observed local crack initiation on the raceway agrees with the approach of the tribological model in Figure 36 that subdivides the contact area into regions of different loading levels. Brittle spontaneous cracking occurs in subdomains of increased friction coefficient. Compared with the competing fatigue crack initiation mechanism discussed in section 5.5.1, lower slip in the moment of surface cracking is suggested.

It is worth noting that post-machining thermal treatment (PMTT) of ground and honed rings and rollers, previously proposed in the literature for material reinforcement in the mechanically influenced edge zone (Gegner, 2006b; Gegner et al., 2009), is recently reported to be an effective countermeasure against premature bearing failures by white etching crack formation (Luyckx, 2011). The short reheating process of, e.g., 0.5 to 1 h after the finishing operation occurs below the tempering or transformation temperature to avoid undesired hardness loss (cf. section 4.2, Figure 23). As only the plastically deformed material in the outermost layer up to a depth of about 10 µm is microstructurally stabilized, a success of this simple treatment would provide further indication of surface WEC failure initiation.

Tribological Aspects of Rolling Bearing Failures 87

also occur on nonmetallic inclusions. The generation and growth of butterflies are briefly

Section 4 focuses on classical subsurface RCF, which may lead to fatigue wear. Raceway spalling is initiated by cracks from the depth of the material eventually. The microstructural changes that characterize the progression of subsurface rolling contact fatigue in the steel matrix are metallographically examined, including scanning electron microscopy. A distinction is made from the shakedown stage during the short running-in period, which is identified as a cold working process of local (micro-) plastic deformation. Rapid compressive residual stress formation in this phase, in response to the exceedance of the yield strength by the v. Mises equivalent stress at Hertzian pressures above 2500 to 3000 MPa, occurs without visible microstructural alterations, the development of which requires carbon diffusion. The mechanistic metal physics dislocation glide stability loss (DGSL) model of rolling contact fatigue is introduced and examined by reheating experiments. As a new aspect of material damage by severe in-service high-frequency electric current passage through bearings, continuous absorption of hydrogen is found to accelerate subsurface RCF. Steep white bands that occur not until the *L*50 life (50% failure probability) in pure mechanical loading appear earlier at much lower *b*/*B* reduction in chemically promoted rolling contact fatigue. The accelerating effect of dissolved hydrogen is demonstrated by a comparison of the microstructures at *b*/*B*≈0.71, also considering cold working. The chosen reference level is yet above the XRD equivalent value of *b*/*B*≈0.64 of the rating *L*10 bearing life in pure mechanical subsurface rolling contact fatigue. The additional chemical loading accelerates material aging by enhancing the dislocation mobility and microplasticity, as evident from the DGSL model. Hydrogen absorption also causes crack initiation in the preembrittled microstructure by interfacial delamination at white etching bands that is not

In section 5 of the present chapter, the effect of mixed friction in the rolling contact area, which occurs frequently in bearing applications, is discussed in detail. Smoothing of the machining marks by polishing wear on the raceway is a characteristic visual indication. Several mechanisms of mixed friction induced failure initiation are introduced. The impact of externally generated three-dimensional mechanical vibrations represents a common cause of disturbed elastohydrodynamic lubrication conditions. Larger size roller bearings operating typically at low to moderate Hertzian pressure are most susceptible to frictional surface loading. Tangential forces by sliding friction acting on a rolling contact increase the v. Mises equivalent stress and shift its maximum, i.e. the position of incipient plastic deformation, toward the surface. The resulting build-up of compressive residual stresses in the edge zone at Hertzian pressures below 2500 MPa is observed for indentation-free raceways under the action of, e.g. engine, vibrations in operation. Material response is described by a tribological model that partitions the contact area into microscopic subdomains of intermittently different friction coefficients up to peak values above 0.3. The distinguishable type A and B vibrational residual stress distributions are explained. Vibrations can reduce the shear-sensitive viscosity of the lubricant. The generated

temperature increase is associated with the contact area related power loss.

Also, mixed friction or lubricant contamination, e.g. by water or wear debris, promotes chemical aging of the oil and its additives. As a consequence, the gradually acidifying fluid attacks the steel surface. Tribochemical dissolution of manufacturing related manganese sulfide inclusion lines leaves crack-like defects on the raceway. Further damage evolution by shallow micropitting occurs similar to gray staining that is also caused by, e.g. vibration

discussed, based on recent findings.

observed in pure mechanical RCF.

Fig. 69. Investigation of the fracture surface of Figure 67 in the deep corrosion fatigue crack region revealing (a) a SEM-SE fractograph and (b) the EDX spectrum taken at the indicated position where the presence of the tracer elements S, P and Zn of the oil additives proves the assistance of fatigue crack growth by chemical reactions, i.e. CFC, in a depth of 1 mm

#### **6. Conclusion**

The present chapter deals with important aspects of rolling contact tribology in bearing failures. Following the introduction, the fundamentals are presented in sections 2 and 3. The subsurface and (near-) surface failure modes of rolling bearings are outlined. X-ray diffraction (XRD) based residual stress analysis identifies the depth of highest loading and provides information about the material response and the stage of damage. The measurement technique, evaluation methodology and application procedure are discussed in detail. The loading induced reduction of the XRD peak width ratio *b*/*B* of minimum to initial value is used as a life calibrated measure of material aging to correlate the successive microstructural changes during rolling contact fatigue (RCF) with the Weibull bearing failure distribution. Therefore, it also permits the prediction of gradual alterations of the hardened steel matrix, which are roughly assigned to the corresponding bearing life in the final phase of the three stage model of RCF (shakedown, steady state, instability). Strong indication is given that dark etching regions (DER) from martensite decay act as the precursor of subsequently occurring ferritic white etching areas (WEA). The WEA are formed in regular parallel flat (30°) and steep (80°) bands within the aged matrix or along propagating corrosion fatigue cracks. Rolling contact fatigue in the subsurface region can

**6. Conclusion** 

Fig. 69. Investigation of the fracture surface of Figure 67 in the deep corrosion fatigue crack region revealing (a) a SEM-SE fractograph and (b) the EDX spectrum taken at the indicated position where the presence of the tracer elements S, P and Zn of the oil additives proves the assistance of fatigue crack growth by chemical reactions, i.e. CFC, in a depth of 1 mm

The present chapter deals with important aspects of rolling contact tribology in bearing failures. Following the introduction, the fundamentals are presented in sections 2 and 3. The subsurface and (near-) surface failure modes of rolling bearings are outlined. X-ray diffraction (XRD) based residual stress analysis identifies the depth of highest loading and provides information about the material response and the stage of damage. The measurement technique, evaluation methodology and application procedure are discussed in detail. The loading induced reduction of the XRD peak width ratio *b*/*B* of minimum to initial value is used as a life calibrated measure of material aging to correlate the successive microstructural changes during rolling contact fatigue (RCF) with the Weibull bearing failure distribution. Therefore, it also permits the prediction of gradual alterations of the hardened steel matrix, which are roughly assigned to the corresponding bearing life in the final phase of the three stage model of RCF (shakedown, steady state, instability). Strong indication is given that dark etching regions (DER) from martensite decay act as the precursor of subsequently occurring ferritic white etching areas (WEA). The WEA are formed in regular parallel flat (30°) and steep (80°) bands within the aged matrix or along propagating corrosion fatigue cracks. Rolling contact fatigue in the subsurface region can also occur on nonmetallic inclusions. The generation and growth of butterflies are briefly discussed, based on recent findings.

Section 4 focuses on classical subsurface RCF, which may lead to fatigue wear. Raceway spalling is initiated by cracks from the depth of the material eventually. The microstructural changes that characterize the progression of subsurface rolling contact fatigue in the steel matrix are metallographically examined, including scanning electron microscopy. A distinction is made from the shakedown stage during the short running-in period, which is identified as a cold working process of local (micro-) plastic deformation. Rapid compressive residual stress formation in this phase, in response to the exceedance of the yield strength by the v. Mises equivalent stress at Hertzian pressures above 2500 to 3000 MPa, occurs without visible microstructural alterations, the development of which requires carbon diffusion. The mechanistic metal physics dislocation glide stability loss (DGSL) model of rolling contact fatigue is introduced and examined by reheating experiments. As a new aspect of material damage by severe in-service high-frequency electric current passage through bearings, continuous absorption of hydrogen is found to accelerate subsurface RCF. Steep white bands that occur not until the *L*50 life (50% failure probability) in pure mechanical loading appear earlier at much lower *b*/*B* reduction in chemically promoted rolling contact fatigue. The accelerating effect of dissolved hydrogen is demonstrated by a comparison of the microstructures at *b*/*B*≈0.71, also considering cold working. The chosen reference level is yet above the XRD equivalent value of *b*/*B*≈0.64 of the rating *L*10 bearing life in pure mechanical subsurface rolling contact fatigue. The additional chemical loading accelerates material aging by enhancing the dislocation mobility and microplasticity, as evident from the DGSL model. Hydrogen absorption also causes crack initiation in the preembrittled microstructure by interfacial delamination at white etching bands that is not observed in pure mechanical RCF.

In section 5 of the present chapter, the effect of mixed friction in the rolling contact area, which occurs frequently in bearing applications, is discussed in detail. Smoothing of the machining marks by polishing wear on the raceway is a characteristic visual indication. Several mechanisms of mixed friction induced failure initiation are introduced. The impact of externally generated three-dimensional mechanical vibrations represents a common cause of disturbed elastohydrodynamic lubrication conditions. Larger size roller bearings operating typically at low to moderate Hertzian pressure are most susceptible to frictional surface loading. Tangential forces by sliding friction acting on a rolling contact increase the v. Mises equivalent stress and shift its maximum, i.e. the position of incipient plastic deformation, toward the surface. The resulting build-up of compressive residual stresses in the edge zone at Hertzian pressures below 2500 MPa is observed for indentation-free raceways under the action of, e.g. engine, vibrations in operation. Material response is described by a tribological model that partitions the contact area into microscopic subdomains of intermittently different friction coefficients up to peak values above 0.3. The distinguishable type A and B vibrational residual stress distributions are explained. Vibrations can reduce the shear-sensitive viscosity of the lubricant. The generated temperature increase is associated with the contact area related power loss.

Also, mixed friction or lubricant contamination, e.g. by water or wear debris, promotes chemical aging of the oil and its additives. As a consequence, the gradually acidifying fluid attacks the steel surface. Tribochemical dissolution of manufacturing related manganese sulfide inclusion lines leaves crack-like defects on the raceway. Further damage evolution by shallow micropitting occurs similar to gray staining that is also caused by, e.g. vibration

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induced, mixed friction. Reasons are given for the hypothesis that the crack propagation mechanism is a variant of corrosion fatigue in rolling contact. The material shows indication of in-service (near-) surface embrittlement.

White etching cracks can cause premature bearing failures in specific susceptible applications. The development of heavily branching and widely spreading transcrystalline crack systems at essentially low to moderate mechanical load indicate chemically assisted crack growth by corrosion fatigue under the influence of the penetrating aging lubricant. Released hydrogen locally induces collateral microstructural changes (HELP, DGSL) resulting in the decorating white etching constituents around parts of the crack paths eventually. Surface failure initiation by mixed friction is detected. Shear and tensile stress controlled damage mechanisms are identified. The formation of fatigue microcracks on the surface, comparable with gray staining, and initial crack extension in overrolling direction at a small angle to the raceway tangent are caused by the variation of load and frictiondefining slip in the contact area. The characteristic orientation of crack propagation reveals failure promoting shear stresses. The established tribological model also explains competing frictional tensile stress induced failure initiation in rolling-sliding contact. Vertical brittle spontaneous hairline cracking of limited depth and surface length of respectively about 0.1 to 0.2 mm occurs mainly in axial direction on the raceway. The normal stress hypothesis is thus proposed. Illustrative case examples are discussed. Failure metallography, fractography and residual stress analysis are applied. Whereas the circumferential tensile stress in the affected subdomains, referring to the introduced tribological model, must be high (maximum on the surface, ∝μ*p*0) to initiate cleavage-like raceway cracks, the contact area related frictional power loss (∝μ*p*0*v*s) is limited so that no smearing (adhesive wear) occurs. This interrelation leads to the conclusion that the rare events of brittle spontaneous raceway cracking in premature bearing failures can be considered as a consequence of specific (three-dimensional) vibration conditions of high Hertzian pressure *p*0 and local friction coefficient μ at low sliding speed *v*s (*gluing* effect). The shear stress induced inclined flat fatigue-like incipient microcracks, in contrast, are characterized by lower frictional tensile stresses, i.e. smaller μ*p*0 value (*v*s less important). From both of these crack initiation mechanisms, smearing is clearly differentiated by the much higher contact area related power loss.

#### **7. References**


induced, mixed friction. Reasons are given for the hypothesis that the crack propagation mechanism is a variant of corrosion fatigue in rolling contact. The material shows indication

White etching cracks can cause premature bearing failures in specific susceptible applications. The development of heavily branching and widely spreading transcrystalline crack systems at essentially low to moderate mechanical load indicate chemically assisted crack growth by corrosion fatigue under the influence of the penetrating aging lubricant. Released hydrogen locally induces collateral microstructural changes (HELP, DGSL) resulting in the decorating white etching constituents around parts of the crack paths eventually. Surface failure initiation by mixed friction is detected. Shear and tensile stress controlled damage mechanisms are identified. The formation of fatigue microcracks on the surface, comparable with gray staining, and initial crack extension in overrolling direction at a small angle to the raceway tangent are caused by the variation of load and frictiondefining slip in the contact area. The characteristic orientation of crack propagation reveals failure promoting shear stresses. The established tribological model also explains competing frictional tensile stress induced failure initiation in rolling-sliding contact. Vertical brittle spontaneous hairline cracking of limited depth and surface length of respectively about 0.1 to 0.2 mm occurs mainly in axial direction on the raceway. The normal stress hypothesis is thus proposed. Illustrative case examples are discussed. Failure metallography, fractography and residual stress analysis are applied. Whereas the circumferential tensile stress in the affected subdomains, referring to the introduced tribological model, must be high (maximum on the surface, ∝μ*p*0) to initiate cleavage-like raceway cracks, the contact area related frictional power loss (∝μ*p*0*v*s) is limited so that no smearing (adhesive wear) occurs. This interrelation leads to the conclusion that the rare events of brittle spontaneous raceway cracking in premature bearing failures can be considered as a consequence of specific (three-dimensional) vibration conditions of high Hertzian pressure *p*0 and local friction coefficient μ at low sliding speed *v*s (*gluing* effect). The shear stress induced inclined flat fatigue-like incipient microcracks, in contrast, are characterized by lower frictional tensile stresses, i.e. smaller μ*p*0 value (*v*s less important). From both of these crack initiation mechanisms, smearing is clearly differentiated by the much higher contact area related

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**3** 

**Мethodology of** 

*South Ural State University* 

*Russia* 

**Calculation of Dynamics and** 

**Structurally-Non-Uniform and** 

**Non-Newtonian Fluids** 

**Hydromechanical Characteristics of** 

**Heavy-Loaded Tribounits, Lubricated with** 

Juri Rozhdestvenskiy, Elena Zadorozhnaya, Konstantin Gavrilov,

Friction units, in which the sliding surfaces are separated by a film of liquid lubricant, generally, consist of three elements: a journal, a lubricating film and a bearing. Such tribounits are often referred to as journal bearings. Tribounits with the hydrodynamic lubrication regime and the time-varying magnitude and direction of load character are hydrodynamic, heavy-loaded (unsteady loaded). Such tribounits include connecting-rod and main bearings of crankshafts, a "piston-cylinder" coupling of internal combustion engines (ICE); sliding supports of shafts of reciprocating compressors and pumps, bearings of rotors of turbo machines and generators; support rolls of rolling mills, etc. The presence of lubricant in the friction units must provide predominantly liquid friction, in which the

The behavior of the lubricant film, which is concluded between the friction surfaces, is described by the system of equations of the hydrodynamic theory of lubrication, a heat transfer and friction surfaces are the boundaries of the lubricant film, which really have elastoplastic properties. During the simulation and calculation of heavy-loaded bearings researchers tend to take into account as many geometric, force and regime parameters as possible and they provide adequacy of the working capacity forecast of the hydrodynamic

In the classical hydrodynamic lubrication theory of fluid the motion in a thin lubricating film of friction units is described by three fundamental laws: conservation of a momentum, mass and energy. The equations of motion of movable elements of tribounits are added to the equations which are made on the basis of conservation laws for heavy-loaded bearings.

**1. Introduction** 

losses are small enough, and the wear is minimal.

tribounits on the early stages of the design.

**2. The system of equations** 

Igor Levanov, Igor Mukhortov and Nadezhda Khozenyuk


## **Мethodology of Calculation of Dynamics and Hydromechanical Characteristics of Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids**

Juri Rozhdestvenskiy, Elena Zadorozhnaya, Konstantin Gavrilov, Igor Levanov, Igor Mukhortov and Nadezhda Khozenyuk *South Ural State University Russia* 

#### **1. Introduction**

94 Tribology - Lubricants and Lubrication

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211, pp. 1-8

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Friction units, in which the sliding surfaces are separated by a film of liquid lubricant, generally, consist of three elements: a journal, a lubricating film and a bearing. Such tribounits are often referred to as journal bearings. Tribounits with the hydrodynamic lubrication regime and the time-varying magnitude and direction of load character are hydrodynamic, heavy-loaded (unsteady loaded). Such tribounits include connecting-rod and main bearings of crankshafts, a "piston-cylinder" coupling of internal combustion engines (ICE); sliding supports of shafts of reciprocating compressors and pumps, bearings of rotors of turbo machines and generators; support rolls of rolling mills, etc. The presence of lubricant in the friction units must provide predominantly liquid friction, in which the losses are small enough, and the wear is minimal.

The behavior of the lubricant film, which is concluded between the friction surfaces, is described by the system of equations of the hydrodynamic theory of lubrication, a heat transfer and friction surfaces are the boundaries of the lubricant film, which really have elastoplastic properties. During the simulation and calculation of heavy-loaded bearings researchers tend to take into account as many geometric, force and regime parameters as possible and they provide adequacy of the working capacity forecast of the hydrodynamic tribounits on the early stages of the design.

#### **2. The system of equations**

In the classical hydrodynamic lubrication theory of fluid the motion in a thin lubricating film of friction units is described by three fundamental laws: conservation of a momentum, mass and energy. The equations of motion of movable elements of tribounits are added to the equations which are made on the basis of conservation laws for heavy-loaded bearings.

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

ϕ

μ

*<sup>g</sup> if*

Where *r* is the radius of the journal;

Fig. 1. Cross section bearing

movement of the shaft (radial bearing).

homogeneous lubricant density;

width; *<sup>a</sup> p* is atmospheric pressure.

θ

If 2 1 ( )0 ω − = ω

where

ϕ*g* , ϕ

The degree of filling

, , ) is film thickness;

is time; *g* is switching function, 1, 1;

(Fig. 1); *h zt* (ϕ

factor; 1 2 ω ,ω

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 97

inertial coordinate system; 1 2 *w w*, are forward speed of bearing and journal, accordingly; *t*

0, 1. *if*

<sup>⎧</sup> <sup>≥</sup> <sup>=</sup> <sup>⎨</sup> <sup>&</sup>lt; <sup>⎩</sup>

θ

θ

is lubricant viscosity;

, then we get an equation for the tribounit with the forward movement of the

θ = ρ ρ

. (2)

θ ϕ

*<sup>c</sup>* is the lubricant density if a pressure is equal to the

ρ = ρ*c* and θ

, *z*) and

=± = = + (3)

ϕ

*<sup>c</sup>* , where

( , *z*) can be written as

determines the

ρis

journal (piston unit). If 2 1 ( )0 *w w*− = , we get the equation for the bearing with a rotational

has the double meaning. In the load region

mass content of the liquid phase (oil) per a unit of space volume between a journal and a

The equation (1) allows us to implement the boundary conditions by Jacobson-Floberga-

( ,) ( ,) ( ,) ; ( , / 2) ; ( , ) ( 2 , ), *g g ra a*

=∂ ∂ = =

*p z p zp zp p z B pp z p z*

β lnθ

> ϕ

*<sup>r</sup>* are the corners of the gap and restore of the lubricating film; *B* is bearing

 ϕπ

ϕ

*pp g* = +⋅ *<sup>c</sup>*

Olsen (JFO), which reflect the conservation law of mass in the lubricating film

 ϕϕ

ρ

pressure of cavitation *<sup>c</sup> p* . In the area of cavitation *<sup>c</sup> p* = *p* ,

bearing. The relation between hydrodynamic pressure *p*(

ϕ

ϕ

are the angular velocity of rotation of the bearing and the journal in the

, *z* are the angular and axial coordinates, accordingly

is lubricant compressibility

β

The problem of theory of hydrodynamic tribounits is characterized by the totality of methods for solving the three interrelated tasks:


Complex solution of these problems is an important step in increasing the reliability of tribounits, development of friction units, which satisfy the modern requirements. However, this solution presents great difficulties, since it requires the development of accurate and highly efficient numerical methods and algorithms.

The simulation result of heavy-loaded tribounits is accepted to assess by the hydromechanical characteristics. These are extreme and average per cycle of loading values for the minimum lubricant film thickness and maximum hydrodynamic pressure, the meanflow rate through the ends of the bearing, the power losses due to friction in the conjugation, the temperature of the lubricating film. The criterions for a performance of tribounits are the smallest allowable film thickness and maximum allowable hydrodynamic pressure.

#### **2.1 Determination of pressure in a thin lubricating film**

The following assumptions are usually used to describe the flow of viscous fluid between bearing surfaces: bulk forces are excluded from the consideration; the density of the lubricant is taken constant, it is independent of the coordinates of the film, temperature and pressure; film thickness is smaller than its length; the pressure is constant across a film thickness; the speed of boundary lubrication films, which are adjacent to friction surfaces, is taken equal to the speed of these surfaces; a lubricant is considered as a Newtonian fluid, in which the shear stresses are proportional to the shear rate; the flow is laminar; the friction surfaces microgeometry is neglected.

The hydrodynamic pressure field is determined most accurately by employment of the universal equation by Elrod (Elrod, 1981) for the degree of filling of the clearance θ by lubricant:

$$\frac{1}{r^2} \frac{\partial}{\partial \rho} \left[ \frac{h^3}{12\mu} \beta g \frac{\partial \theta}{\partial \rho} \right] + \frac{\partial}{\partial z} \left[ \frac{h^3}{12\mu} \beta g \frac{\partial \theta}{\partial z} \right] = \frac{(o\_2 - o\_1)}{2} \frac{\partial}{\partial \rho} (h\theta) + \frac{(w\_2 - w\_1)}{2r} \frac{\partial}{\partial z} (h\theta) + \frac{\partial}{\partial t} (h\theta) \tag{1}$$

Where *r* is the radius of the journal; ϕ, *z* are the angular and axial coordinates, accordingly (Fig. 1); *h zt* (ϕ, , ) is film thickness; μ is lubricant viscosity; β is lubricant compressibility factor; 1 2 ω ,ω are the angular velocity of rotation of the bearing and the journal in the inertial coordinate system; 1 2 *w w*, are forward speed of bearing and journal, accordingly; *t* θ

is time; *g* is switching function, 1, 1; 0, 1. *if <sup>g</sup> if* θ <sup>⎧</sup> <sup>≥</sup> <sup>=</sup> <sup>⎨</sup> <sup>&</sup>lt; <sup>⎩</sup>

96 Tribology - Lubricants and Lubrication

The problem of theory of hydrodynamic tribounits is characterized by the totality of

1. The hydrodynamic pressures in a thin lubricating film, which separates the friction surfaces of a journal and a bearing with an arbitrary law of their relative motion, are

2. The parameters of nonlinear oscillations of a journal on a lubricating film are detected

• the temperature parameters of the tribounit lubricant film during the period of loading,

• the elastic deformation of friction surfaces under the influence of hydrodynamic

• the parameters of the nonlinear oscillation of a journal on the lubricating film with a

Complex solution of these problems is an important step in increasing the reliability of tribounits, development of friction units, which satisfy the modern requirements. However, this solution presents great difficulties, since it requires the development of accurate and

The simulation result of heavy-loaded tribounits is accepted to assess by the hydromechanical characteristics. These are extreme and average per cycle of loading values for the minimum lubricant film thickness and maximum hydrodynamic pressure, the meanflow rate through the ends of the bearing, the power losses due to friction in the conjugation, the temperature of the lubricating film. The criterions for a performance of tribounits are the smallest allowable film thickness and maximum allowable hydrodynamic

The following assumptions are usually used to describe the flow of viscous fluid between bearing surfaces: bulk forces are excluded from the consideration; the density of the lubricant is taken constant, it is independent of the coordinates of the film, temperature and pressure; film thickness is smaller than its length; the pressure is constant across a film thickness; the speed of boundary lubrication films, which are adjacent to friction surfaces, is taken equal to the speed of these surfaces; a lubricant is considered as a Newtonian fluid, in which the shear stresses are proportional to the shear rate; the flow is laminar; the friction

The hydrodynamic pressure field is determined most accurately by employment of the universal equation by Elrod (Elrod, 1981) for the degree of filling of the clearance

1 ( ) ( ) 12 12 2 2 *h h w w*

 θ

β

 μ

*r z z rz t*

ω

∂ ∂ ∂ ∂ −∂ −∂ ∂ ⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ += + + ∂ ∂∂ ∂ ∂ ∂ ∂ ⎣ ⎦⎣ ⎦

2 1 2 1

*g g h hh*

ϕ

ω θby

. (1)

θ

( ) ( ) ( )

θθ

methods for solving the three interrelated tasks:

• the relative motion of the friction surfaces;

and the trajectories of the journal center are calculated. 3. The temperature of the lubricating film is calculated.

The field of hydrodynamic pressures in a thin lubricating film depends on:

sources of lubricant on these surfaces are taken into account;

• the characteristics of a lubricant, including its rheological properties.

pressure in the lubricating film and the external forces;

nonstationary law of variation of influencing powers; • the supplies-drop performance of a lubrication system;

highly efficient numerical methods and algorithms.

**2.1 Determination of pressure in a thin lubricating film** 

surfaces microgeometry is neglected.

3 3

θ

 ϕ

β

ϕμ

calculated.

pressure.

lubricant:

2

Fig. 1. Cross section bearing

If 2 1 ( )0 ω − = ω , then we get an equation for the tribounit with the forward movement of the journal (piston unit). If 2 1 ( )0 *w w*− = , we get the equation for the bearing with a rotational movement of the shaft (radial bearing).

The degree of filling θ has the double meaning. In the load region θ = ρ ρ*<sup>c</sup>* , where ρ is homogeneous lubricant density; ρ*<sup>c</sup>* is the lubricant density if a pressure is equal to the pressure of cavitation *<sup>c</sup> p* . In the area of cavitation *<sup>c</sup> p* = *p* , ρ = ρ*c* and θ determines the mass content of the liquid phase (oil) per a unit of space volume between a journal and a bearing. The relation between hydrodynamic pressure *p*(ϕ, *z*) and θ ϕ( , *z*) can be written as

$$p = p\_c + \lg \cdot \beta \ln \theta \,. \tag{2}$$

The equation (1) allows us to implement the boundary conditions by Jacobson-Floberga-Olsen (JFO), which reflect the conservation law of mass in the lubricating film

$$\begin{aligned} p(\boldsymbol{p}\_{\mathcal{S}}, \boldsymbol{z}) &= \partial p \, \Big/ \partial \, \boldsymbol{\varrho}(\boldsymbol{p}\_{\mathcal{S}}, \boldsymbol{z}) = p(\boldsymbol{\varrho}\_{r}, \boldsymbol{z}) = p\_{a}; \\ p(\boldsymbol{\varrho}, \boldsymbol{z} = \pm \mathbf{B} \; / \ \mathbf{2}) &= p\_{a} \, \boldsymbol{p}(\boldsymbol{\varrho}, \boldsymbol{z}) = p(\boldsymbol{\varrho} + \mathbf{2}\boldsymbol{\pi}, \boldsymbol{z}), \end{aligned} \tag{3}$$

where ϕ*g* , ϕ*<sup>r</sup>* are the corners of the gap and restore of the lubricating film; *B* is bearing width; *<sup>a</sup> p* is atmospheric pressure.

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

constant for the central shaft position in the bearing ( <sup>1</sup> *h Z* ( , ) const

circumferential and axial coordinates.

a non- radial tribounit (Prokopiev et al., 2010).

arbitrary, the film thickness is defined as

( <sup>1</sup> *h Z* ( , ) const ϕ

*h* ( ) const ϕ

Where ( ) \* *h t* ϕ

It is given by

The function ( ) \* *h ,t*

ϕ

constructive profiling.

deviations of the surfaces of tribounits and their possible elastic displacements.

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 99

Film thickness in the tribounit depends on the position of the journal center, the angle between the direct axis of a journal and a bearing, as well as on the macrogeometrical

We term the tribounit with a circular cylindrical journal and a bearing as a tribounit with a perfect geometry. In such a tribounit the clearance (film thickness) in any section is equal

For a tribounit with non-ideal geometry the function of the clearance isn't equal constant

If the tribounit geometry is distorted only in the axial direction, that is 1 *h Z*( ) const <sup>∗</sup> <sup>≠</sup> , we term it as a tribounit with non-ideal geometry in the axial direction, or a non-cylindrical tribounit. If the tribounit geometry is distorted only in the radial direction, that is

<sup>∗</sup> ≠ , we term it as a tribounit with a non-ideal geometry in the radial direction or

For a non- radial tribounit the macro deviations of polar radiuses of the bearing and the

Values Δ*i* don't depend on the position *z* and are considered positive (negative) if radiuses *<sup>i</sup>*<sup>0</sup> *r* are increased (decreased). In this case, the geometry of the journal friction surfaces is

( ) ( ) ( ) \* *hth te*

displacement of mass centers of the journal in relation to the bearing equals zero ( *e t*( ) = 0 ).

 ϕ

 ϕδ

, is the film thickness for the central position of the journal, when the

, , cos = ϕ

journal from the radiuses *<sup>i</sup>*<sup>0</sup> *r* of base circles (shown dashed) are denoted by Δ<sup>1</sup> (

( ) ( ) ( ) \* 01 2 *h t*

, , =Δ +Δ −Δ ϕ

can be defined by a table of deviations Δ*<sup>i</sup>* (

ϕ

ϕ

Fig. 2. Scheme of a bearing with the central position of a journal

of the second order) or approximated by series.

 <sup>∗</sup> ≠ ). This function takes into account profiles deviations of the journal and the bearing from circular cylindrical forms as a result of wear, manufacturing errors or

ϕ

<sup>∗</sup> <sup>=</sup> ). Where 1

− − . (7)

*t* , Δ= − 0 10 20 (*r r* ) . (8)

ϕ

ϕ

ϕ ) , Δ<sup>2</sup> (ϕ,*t*) .

,*t*) , analytically (functions

,*Z* are

The conditions of JFO can quite accurately determine the position of the load region of the film. The algorithms of the solution of equation (1), which implement them, are called "a mass conserving cavitation algorithm".

On the other hand the field of hydrodynamic pressures in a thin lubricating film is determined from the generalized Reynolds equation (Prokopiev et al., 2010):

$$\frac{1}{r^2} \frac{\partial}{\partial \rho} \left[ \frac{h^3}{12\,\mu} \frac{\partial p}{\partial \rho} \right] + \frac{\partial}{\partial z} \left[ \frac{h^3}{12\,\mu} \frac{\partial p}{\partial z} \right] = \frac{\{\alpha\_2 - \alpha\_1\}}{2} \frac{\partial h}{\partial \rho} + \frac{\{w\_2 - w\_1\}}{2r} \frac{\partial h}{\partial z} + \frac{\partial h}{\partial t} \tag{4}$$

The equation (4) was sufficiently widespread in solving problems of dynamics and lubrication of different tribounits.

When integrating the equation (4) in the area Ω= ∈ ∈− (ϕ π 0,2 ; /2, /2 *zB B* ) mostly often Stieber-Swift boundary conditions are used, which are written as the following restrictions on the function *p*(ϕ, *z*) :

$$p\left(\boldsymbol{\wp}, \boldsymbol{z} = \pm \boldsymbol{B} \mid \mathcal{D}\right) = p\_{\boldsymbol{a}} ; \; p(\boldsymbol{\wp}, \boldsymbol{z}) = p(\boldsymbol{\wp} + \boldsymbol{2}\boldsymbol{\pi}, \boldsymbol{z}) ; p(\boldsymbol{\wp}, \boldsymbol{z}) \ge p\_{\boldsymbol{a}} ,\tag{5}$$

If the sources of the lubricant feeding for the film locate on the friction surfaces, then equations (3) and (5) must be supplemented by

$$p\left(\boldsymbol{\varphi},\boldsymbol{z}\right) = p\_{\rm S} \quad \text{ } \mathsf{u}\left(\boldsymbol{\varphi},\boldsymbol{z}\right) \in \Omega\_{\rm S}, \text{ } \mathsf{S} = \mathsf{1}, \mathsf{2}...\mathsf{S}^{\*}, \tag{6}$$

where Ω*S* is the region of lubricant source, where pressure is constant and equal to the supply pressure *<sup>S</sup> p* ; \* *S* is the number of sources.

To solve the equations (1) and (3) taking into account relations (3), (5), (6) we use numerical methods, among which variational-difference methods with finite element (FE) models and methods for approximating the finite differences (FDM) are most widely used. These methods are based on finite-difference approximation of differential operators of the boundary task with free boundaries. They can most easily and quickly obtain solutions with sufficient accuracy for bearings with non-ideal geometry. These methods also can take into account the presence of sources of lubricant on the friction surface.

One of the most effective methods of integrating the Reynolds equation are multi-level algorithms, which allows to reduce significantly the calculation time. Equations (1) and (4) are reduced to a system of algebraic equations, which are solved, for example, with the help of Seidel iterative method or by using a modification of the sweep method.

#### **2.2 Geometry of a heavy-loaded tribounit**

The geometry of the lubricant film influences on hydromechanical characteristics the greatest. Changing the cross-section of a journal and a bearing leads to a change in the lubrication of friction pairs. Thus technological deviations from the desired geometry of friction surfaces or strain can lead to loss of bearing capacity of a tribounit. At the same time in recent years, the interest to profiled tribounits had increased. Such designs can substantially improve the technical characteristics of journal bearings: to increase the carrying capacity while reducing the requirements for materials; to reduce friction losses; to increase the vibration resistance. Therefore, the description of the geometry of the lubricant film is a crucial step in the hydrodynamic calculation.

The conditions of JFO can quite accurately determine the position of the load region of the film. The algorithms of the solution of equation (1), which implement them, are called "a

On the other hand the field of hydrodynamic pressures in a thin lubricating film is

*r z z r zt* ω ω

The equation (4) was sufficiently widespread in solving problems of dynamics and lubrication

Stieber-Swift boundary conditions are used, which are written as the following restrictions

If the sources of the lubricant feeding for the film locate on the friction surfaces, then

( ) ( ) \* , , ,1,2... , *S S pzp*

where Ω*S* is the region of lubricant source, where pressure is constant and equal to the

To solve the equations (1) and (3) taking into account relations (3), (5), (6) we use numerical methods, among which variational-difference methods with finite element (FE) models and methods for approximating the finite differences (FDM) are most widely used. These methods are based on finite-difference approximation of differential operators of the boundary task with free boundaries. They can most easily and quickly obtain solutions with sufficient accuracy for bearings with non-ideal geometry. These methods also can take into

One of the most effective methods of integrating the Reynolds equation are multi-level algorithms, which allows to reduce significantly the calculation time. Equations (1) and (4) are reduced to a system of algebraic equations, which are solved, for example, with the help

The geometry of the lubricant film influences on hydromechanical characteristics the greatest. Changing the cross-section of a journal and a bearing leads to a change in the lubrication of friction pairs. Thus technological deviations from the desired geometry of friction surfaces or strain can lead to loss of bearing capacity of a tribounit. At the same time in recent years, the interest to profiled tribounits had increased. Such designs can substantially improve the technical characteristics of journal bearings: to increase the carrying capacity while reducing the requirements for materials; to reduce friction losses; to increase the vibration resistance. Therefore, the description of the geometry of the lubricant

 ϕ

 ϕ , / 2 ; ( , ) ( 2 , ); , *z B* = ± = =+ ≥ ) *p p a a z p* ϕ

∂∂ −− ⎡ ⎤⎡ ⎤ <sup>∂</sup> ∂ ∂ ∂∂ ⎢ ⎥⎢ ⎥ += + + ∂ ∂ ∂ ∂ ∂ ∂∂ ⎣ ⎦⎣ ⎦

*h h p p h hh w w*

21 2 1

 π

= *на zSS* ∈Ω = (6)

. (4)

0,2 ; /2, /2 *zB B* ) mostly often

*z*) *p* , (5)

ϕ

ϕ

π *z p*(ϕ

determined from the generalized Reynolds equation (Prokopiev et al., 2010):

 μ

1 ( ) ( ) 12 12 2 2

3 3

When integrating the equation (4) in the area Ω= ∈ ∈− (

mass conserving cavitation algorithm".

2

of different tribounits.

on the function *p*(

ϕ

ϕ, *z*) :

> *p*(ϕ

equations (3) and (5) must be supplemented by

supply pressure *<sup>S</sup> p* ; \* *S* is the number of sources.

**2.2 Geometry of a heavy-loaded tribounit** 

film is a crucial step in the hydrodynamic calculation.

ϕ

account the presence of sources of lubricant on the friction surface.

of Seidel iterative method or by using a modification of the sweep method.

 μϕ Film thickness in the tribounit depends on the position of the journal center, the angle between the direct axis of a journal and a bearing, as well as on the macrogeometrical deviations of the surfaces of tribounits and their possible elastic displacements.

We term the tribounit with a circular cylindrical journal and a bearing as a tribounit with a perfect geometry. In such a tribounit the clearance (film thickness) in any section is equal constant for the central shaft position in the bearing ( <sup>1</sup> *h Z* ( , ) const ϕ <sup>∗</sup> <sup>=</sup> ). Where 1 ϕ,*Z* are circumferential and axial coordinates.

For a tribounit with non-ideal geometry the function of the clearance isn't equal constant ( <sup>1</sup> *h Z* ( , ) const ϕ <sup>∗</sup> ≠ ). This function takes into account profiles deviations of the journal and the bearing from circular cylindrical forms as a result of wear, manufacturing errors or constructive profiling.

If the tribounit geometry is distorted only in the axial direction, that is 1 *h Z*( ) const <sup>∗</sup> <sup>≠</sup> , we term it as a tribounit with non-ideal geometry in the axial direction, or a non-cylindrical tribounit. If the tribounit geometry is distorted only in the radial direction, that is *h* ( ) const ϕ <sup>∗</sup> ≠ , we term it as a tribounit with a non-ideal geometry in the radial direction or a non- radial tribounit (Prokopiev et al., 2010).

For a non- radial tribounit the macro deviations of polar radiuses of the bearing and the journal from the radiuses *<sup>i</sup>*<sup>0</sup> *r* of base circles (shown dashed) are denoted by Δ<sup>1</sup> (ϕ ) , Δ<sup>2</sup> (ϕ,*t*) . Values Δ*i* don't depend on the position *z* and are considered positive (negative) if radiuses *<sup>i</sup>*<sup>0</sup> *r* are increased (decreased). In this case, the geometry of the journal friction surfaces is arbitrary, the film thickness is defined as

$$h\left(\boldsymbol{\varrho},t\right) = h\left(\boldsymbol{\varrho},t\right) - e\cos\left(\boldsymbol{\varrho}-\boldsymbol{\delta}\right).\tag{7}$$

Where ( ) \* *h t* ϕ, is the film thickness for the central position of the journal, when the displacement of mass centers of the journal in relation to the bearing equals zero ( *e t*( ) = 0 ). It is given by

$$
\Delta h^\* \left( \varphi, t \right) = \Delta\_0 + \Delta\_1 \left( \varphi \right) - \Delta\_2 \left( \varphi, t \right),
\
\Delta\_0 = \left( r\_{10} - r\_{20} \right). \tag{8}
$$

The function ( ) \* *h ,t* ϕ can be defined by a table of deviations Δ*<sup>i</sup>* (ϕ,*t*) , analytically (functions of the second order) or approximated by series.

Fig. 2. Scheme of a bearing with the central position of a journal

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

*h Zt* ( , ,) ϕ

ideal geometry as

from a cylindrical shape (Fig. 3).

Fig. 3. Types of non-cylindrical journals

δ

position of the journal (Prokopiev et al., 2010) is given by

 and 2 δ

by the maximum deviations 1

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 101

Where Δ*<sup>i</sup>* ( ) *Z*<sup>1</sup> , *i* = 1,2 are the deviations of generating lines of bearing surfaces and the journal surfaces from the line (positive deviation is in the direction of increasing radius). Then, taking into account the expressions (8) and (14) we can write the general formula for a lubricant film thickness with the central position of the journal in the bearings with non-

A barreling, a saddle and a taper are the typical macro deviations of a journal and a bearing

The non-cylindrical shapes of the bearing and the journal in the axial direction are defined

described by the corresponding approximating curve. Then the film thickness at the central

1 2 \*

of a profile from the ideal cylindrical profile and are

1 0 11 21 ( ) *l l h Z kZ kZ* =Δ + + , (16)

 ϕϕ

*hZ Z Z* ( ) 1 0 11 21 ( ) ( ) <sup>∗</sup> =Δ +Δ −Δ . (14)

1 0 1 2 11 21 ( ) ( ,*tZ Z* ) ( ) ( ) <sup>∗</sup> =Δ +Δ −Δ +Δ −Δ . (15)

If a journal and a bearing have the elementary species of non-roundness (oval), their geometry is conveniently described by ellipses. For example, the oval bearing surface is represented as an ellipse (Fig. 2) and the journal surface is represented as a one-sided oval – a half-ellipse.

Using the known formulas of analytic geometry, we represent the surfaces deflection Δ*i* of a bearing and a journal from the radiuses of base surfaces 0*i i r b* = in the following form

$$\Delta\_i = b\_i \left\{ \nu\_i \left[ \nu\_i^2 - \left( \nu\_i^2 - 1 \right) \cos^2 \left( \varphi - \mathfrak{G}\_i \right) \right]^{-0.5} - 1 \right\},\tag{9}$$

where the parameter ν *<sup>i</sup>* is the ratio of high *<sup>i</sup> a* to low *<sup>i</sup> b* axis of the ellipse, ϑ*<sup>i</sup>* are angles which determine the initial positions of the ovals.

Due to fixing of the polar axis *O X*1 1 on the bearing, the angle ϑ1 doesn't depend on the time, and the angle ϑ<sup>20</sup> , which determines the location of the major axis of the journal elliptic surface with 0 *t t* = , is associated with a relative angular velocity ω21 by the following relation

$$\mathcal{B}\_2(t) = \mathcal{B}\_{20} + \int\_{t\_0}^t \phi\_{21} dt \,. \tag{10}$$

In an one-sided oval of a journal equation (9) is applied in the field 2 2 ( 2 ) (3 2 ) π + ≤≤ + ϑ ϕ πϑ, but off it 2 Δ = 0 .

If the macro deviations Δ<sup>1</sup> (ϕ ) , Δ2 2 (γ ) of journal and bearing radiuses *ri* (ϕ ) from the base circles radiuses *<sup>i</sup>*<sup>0</sup> *r* are approximated by truncated Fourier series, then they can be represented as (Prokopiev et al., 2010):

$$
\Delta\_i(\varphi') = \tau\_{i0} + \tau\_i \sin\left(k\_i \varphi + a\_i\right),
\tag{11}
$$

where 1 *i* = for a bearing, 2 *i* = for a journal; ψ = ϕ if 1 *i* = , ψ 2 12 = γ ϕϑ ϑ =+ − if 2 *i* = ; 2 21 ( ) 0 *t* ϑ ω <sup>=</sup> *t dt* ∫ ; *<sup>i</sup> <sup>k</sup>* is a harmonic number; *<sup>i</sup>* τ , α*<sup>i</sup>* are the amplitude and phase of the *k* -th

harmonic; *<sup>i</sup>*<sup>0</sup> τis a permanent member of the Fourier series, which is defined by

$$\pi\_{i0} = \frac{1}{2\pi} \int\_0^{2\pi} \Delta\_i(\wp) d\wp \,. \tag{12}$$

For elementary types of non-roundness (oval ( 2 *k* = ); a cut with three ( *k* = 3) or four ( ) *k* = 4 vertices of the profile) 0 0 *<sup>i</sup>* τ= .

The thickness of the lubricant film, which is limited by a bearing and a journal having elementary types of non-roundness, after substituting (12) in (7), is given by

$$h\left(\boldsymbol{\varphi},t\right) = \boldsymbol{\Lambda}\_0 + \tau\_1 \sin\left(k\_1 \boldsymbol{\varphi} + a\_1\right) - \tau\_2 \sin\left(k\_2 \boldsymbol{\chi}\_2 + a\_2\right) - e \cos\left(\boldsymbol{\varphi} - \boldsymbol{\delta}\right). \tag{13}$$

For tribounits with geometry deviations from the basic cylindrical surfaces in the axial direction the film thickness at the central position of the journal in an arbitrary cross-section *Z*1 is written by the expression

$$
\Delta h^\*(Z\_1) = \Delta\_0 + \Delta\_1(Z\_1) - \Delta\_2(Z\_1) \,. \tag{14}
$$

Where Δ*<sup>i</sup>* ( ) *Z*<sup>1</sup> , *i* = 1,2 are the deviations of generating lines of bearing surfaces and the journal surfaces from the line (positive deviation is in the direction of increasing radius). Then, taking into account the expressions (8) and (14) we can write the general formula for a lubricant film thickness with the central position of the journal in the bearings with nonideal geometry as

$$\ln^\*(\boldsymbol{\wp}, \boldsymbol{Z}\_1, t) = \boldsymbol{\Lambda}\_0 + \boldsymbol{\Lambda}\_1(\boldsymbol{\wp}) - \boldsymbol{\Lambda}\_2(\boldsymbol{\wp}, t) + \boldsymbol{\Lambda}\_1(\boldsymbol{Z}\_1) - \boldsymbol{\Lambda}\_2(\boldsymbol{Z}\_1) \,. \tag{15}$$

A barreling, a saddle and a taper are the typical macro deviations of a journal and a bearing from a cylindrical shape (Fig. 3).

Fig. 3. Types of non-cylindrical journals

100 Tribology - Lubricants and Lubrication

If a journal and a bearing have the elementary species of non-roundness (oval), their geometry is conveniently described by ellipses. For example, the oval bearing surface is represented as an ellipse (Fig. 2) and the journal surface is represented as a one-sided oval –

Using the known formulas of analytic geometry, we represent the surfaces deflection Δ*i* of a bearing and a journal from the radiuses of base surfaces 0*i i r b* = in the following form

( ) ( ) 0,5 22 2 1 cos <sup>1</sup> *i iii i <sup>i</sup> <sup>b</sup>*

 ϕϑ<sup>−</sup> <sup>⎧</sup> <sup>⎫</sup> <sup>Δ</sup> = −− − − ⎡ ⎤ <sup>⎨</sup> <sup>⎬</sup> ⎣ ⎦ ⎩ ⎭ , (9)

<sup>20</sup> , which determines the location of the major axis of the journal

) of journal and bearing radiuses *ri* (

if 1 *i* = ,

 ψα

ψ = ϕ

( )

 γ α

ψ ϕ ϑ

1 doesn't depend on the

21 by the following

) from the base

 ϑ=+ − if 2 *i* = ;

ϑ

*t d* = + *<sup>t</sup>* ∫ . (10)

<sup>0</sup> sin( *k* ) , (11)

ψ

= Δ∫ . (12)

 ϕδ

( ) sin( *k e* ) cos( ) . (13)

ω

ϕ

 2 12 = γ ϕϑ

*<sup>i</sup>* are the amplitude and phase of the *k* -th

*<sup>i</sup>* are angles

*<sup>i</sup>* is the ratio of high *<sup>i</sup> a* to low *<sup>i</sup> b* axis of the ellipse,

0

*t*

2 20 21 ( ) *t*

In an one-sided oval of a journal equation (9) is applied in the field

circles radiuses *<sup>i</sup>*<sup>0</sup> *r* are approximated by truncated Fourier series, then they can be

Δ =+ + *i ii i i* (

τ , α

is a permanent member of the Fourier series, which is defined by

2

1 <sup>2</sup> *i i <sup>d</sup>* π

π

0

For elementary types of non-roundness (oval ( 2 *k* = ); a cut with three ( *k* = 3) or four ( ) *k* = 4

The thickness of the lubricant film, which is limited by a bearing and a journal having

, sin ) =Δ + + − + − − 0 1 1 1 2 22 2

For tribounits with geometry deviations from the basic cylindrical surfaces in the axial direction the film thickness at the central position of the journal in an arbitrary cross-section

 τ

0

elementary types of non-roundness, after substituting (12) in (7), is given by

 ϕα

τ

ϑϑω

 ν

νν

Due to fixing of the polar axis *O X*1 1 on the bearing, the angle

elliptic surface with 0 *t t* = , is associated with a relative angular velocity

, but off it 2 Δ = 0 .

ψ ) τ τ

ϕ ) , Δ2 2 (γ

a half-ellipse.

where the parameter

time, and the angle

2 2 ( 2 ) (3 2 )

If the macro deviations Δ<sup>1</sup> (

 πϑ

represented as (Prokopiev et al., 2010):

where 1 *i* = for a bearing, 2 *i* = for a journal;

<sup>=</sup> *t dt* ∫ ; *<sup>i</sup> <sup>k</sup>* is a harmonic number; *<sup>i</sup>*

τ= .

ϕ

*Z*1 is written by the expression

*ht k* (

τ

 + ≤≤ + ϑ ϕ

relation

π

2 21 ( ) 0

harmonic; *<sup>i</sup>*<sup>0</sup>

τ

vertices of the profile) 0 0 *<sup>i</sup>*

 ω

*t*

ϑ

ν

ϑ

which determine the initial positions of the ovals.

The non-cylindrical shapes of the bearing and the journal in the axial direction are defined by the maximum deviations 1 δ and 2 δ of a profile from the ideal cylindrical profile and are described by the corresponding approximating curve. Then the film thickness at the central position of the journal (Prokopiev et al., 2010) is given by

$$h^\*(Z\_1) = \Delta\_0 + k\_1 Z\_1^{l\_1} + k\_2 Z\_1^{l\_2},\tag{16}$$

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

**2.3 The calculation of thermal processes** 

ρ

is density; 0 0 *c* ,

Where

ρ

described by the equation (Prokopiev&Karavayev, 2003)

λ

non-Newtonian fluid by the approximate expression

temperature ( ) *T t av* ; zones of elevated temperatures.

high. It reduces the accuracy of the results.

bearing and a lubrication groove.

is determined by solving the heat balance equation

into the ends of the tribounit during the loading cycle.

 ρ

are used about the temperature distribution in a thin lubricating film.

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 103

The theory of thermal processes in the heavy-loaded tribounit of fluid friction is based on a generalized equation of energy (heat transmission) for a thin film of viscous incompressible fluid, which is between two moving surfaces *S*1 and *S*<sup>2</sup> . If we assume a low thermal conductivity in the direction of the coordinate axes *Oxz* (the axis *Oy* is normal to the surface *S*<sup>1</sup> ) (Fig. 1), the temperature distribution *Txyzt* ( , , ,) in the lubricating film will be

> 0 0 *xyz* 0 2 *T TTT T c cV V V <sup>Д</sup> t xyz <sup>y</sup>*

(usually taken as constant); *t* is the time; *Д* is the dissipation function, which is defined for

\* *Д* <sup>2</sup> ≈ μ

The three approaches to the integration of the equation (21) (thermohydrodynamic (nonisothermal), adiabatic, isothermal) can be used, depending on the assumptions which

When thermohydrodynamic approach is applied the temperature will change in all directions, including across the oil film. In this case, the boundary conditions are stated quite simply and are the most adequate to the real thermal processes. With this approach, we get information about the local properties of the temperature field of lubricating film: a temperature distribution *Txyzt* ( , , ,); maximum temperature *T*max , instantaneous average

If adiabatic approach is applied the change of the temperature across the oil film (along the axis *Oy* ) is ignored, the journal and the bearing are assumed ideal thermal insulators. We introduce a computational averaged over the width of the bearing temperature ( ) \* \* *T T xt* = , . We substitute it into the equation (21) and receive a differential equation for the temperature distribution along the coordinate *x* . Since in this case the heat transfer to the journal and the bearing is not taken into account, the calculated temperatures are too

The isothermal approach assumes that the calculated current temperature ( ) *T Tt c c* = is the same at all points of the lubricant film. This temperature is a highly inertial parameter and it

This equation reflects the equality of the average values of the heat \* *AN* , which is dissipated in the lubricating film, and the average values of the heat \* *AQ* , which is drained by lubricant

The accurate definition of the current temperature can be performed: at each time step of the calculation; once per a cycle of loading the tribounit, at each time step of the calculation taking into account the thermal interaction between the lubricant film with a journal, with a

 ∂ ∂∂∂ ∂ ⎛ ⎞ + ⎜ ⎟ ++ − =

2

*I* . (22)

( ) ( ) \* \* *AN Q t At* = . (23)

λ

are specific heat capacity and thermal conductivity of lubricant

∂ ∂∂∂ ⎝ ⎠ <sup>∂</sup> . (21)

where *<sup>i</sup> k* defines the deviation of the approximating curve per unit of the width of the bearing, the degree of the parabola is accepted: 1 *il* = for the conical journals; 2 *il* = for barrel and saddle journals.

For the circular cylindrical bearing for 0 Δ*<sup>i</sup>* = the film thickness is determined by the wellknown formula:

$$\overline{h}\left(\boldsymbol{\varphi}, \overline{\boldsymbol{t}}\right) = 1 - \chi \cos\left(\boldsymbol{\varphi} - \boldsymbol{\delta}\right). \tag{17}$$

For the circular cylindrical journal its rotation axis is parallel to the axis *O Z*1 1 . In practice, the axis of the journal may be not parallel to the axis of the bearing, so there is a so-called "skewness". These deviations may be as due to technological factors (the inaccuracy of manufacturing during the production and repair) as to working conditions (wear, bending of shafts, etc.).

Position of the journal, which is regarded as a rigid body, in this case you can specify by two coordinates *e*,δ of the journal center *O*2 and by three angles ( γ , ε , θ <sup>2</sup> ). Angle γ is skewness of journal axis; ε is the deviation angle of skewness plane from the base coordinate plane; θ2 is the rotation angle of the journal on its own axis *O Z*2 2 .

When journal axis is skewed the film thickness at a random cross-section *Z*1*<sup>i</sup>* of the bearing depends on the eccentricity *ie* and the angle *<sup>i</sup>* δfor this cross-section

$$h(\boldsymbol{\wp}, \boldsymbol{Z}\_{1i}, \mathbf{t}) = h^\*(\boldsymbol{\wp}, \mathbf{Z}\_{1i}) - e\_i \cos(\boldsymbol{\wp} - \boldsymbol{\delta}\_i) \tag{18}$$

where \* <sup>1</sup> (, )*<sup>i</sup> h Z* ϕis the film thickness with the central journal position in *i* -th cross -section.

We term the tgγ = 2 /*s B* , where *s* is the distance between the geometric centers of the journal and the bearing at the ends of the tribounit; *B* is the width of the tribounit. The expression for the lubricant film thickness, taking into account the skewness, is written in the form

$$h(\boldsymbol{\wp}, \boldsymbol{Z}\_1, \mathbf{t}) = h^\*(\boldsymbol{\wp}, \mathbf{Z}\_1) - e \cos(\boldsymbol{\wp} - \boldsymbol{\delta}) - \boldsymbol{Z}\_1 \cdot \frac{2s}{B} \cos(\boldsymbol{\wp} - \boldsymbol{\varepsilon}) \,. \tag{19}$$

It should be also taken into account that the bearing surfaces are deformed under the action of hydrodynamic pressures. The value Δ( ) *p* is the radial elastic displacement of the bearing sliding surface under the action of hydrodynamic pressure *p* in the lubricant film. Function Δ( ) *p* is defined in the process of calculating of the bearing strain (for a "hard" bearing Δ = ( ) 0) *p* and is written in the form of a component in the equation for the lubricant film thickness.

Thus, the film thickness, taking into account the arbitrary geometry of friction surfaces of a journal and a bearing, the skewness of the journal and elastic displacements of the bearing, is determined by the equation:

$$h\left(\boldsymbol{\uprho}, \boldsymbol{Z}\_1, \mathbf{t}\right) = h\left(\boldsymbol{\uprho}, \boldsymbol{Z}\_1\right) - e\cos(\boldsymbol{\uprho} - \boldsymbol{\upsigma}) - \boldsymbol{Z}\_1 \cdot \boldsymbol{\uprho}\boldsymbol{\uprho}\mathbf{\mathcal{S}} \cdot \cos(\boldsymbol{\uprho} - \boldsymbol{\upsigma}) + \boldsymbol{\Lambda}\left(\boldsymbol{\uprho}\right) \tag{20}$$

where \* <sup>1</sup> *h Z* (, ) ϕ is the film thickness with the central position of the journal in the bearing with non-ideal geometry; *e t*( ) is displacement of journal mass centers in relation to the bearing; ε ( )*t* - an angle that takes into account the skewness of axes of a bearing and a journal . The values *et t t* ( ), , δ ε( ) ( ) are determined by solving the equations of motion.

#### **2.3 The calculation of thermal processes**

102 Tribology - Lubricants and Lubrication

where *<sup>i</sup> k* defines the deviation of the approximating curve per unit of the width of the bearing, the degree of the parabola is accepted: 1 *il* = for the conical journals; 2 *il* = for

For the circular cylindrical bearing for 0 Δ*<sup>i</sup>* = the film thickness is determined by the well-

, 1 cos ) = − − χ

For the circular cylindrical journal its rotation axis is parallel to the axis *O Z*1 1 . In practice, the axis of the journal may be not parallel to the axis of the bearing, so there is a so-called "skewness". These deviations may be as due to technological factors (the inaccuracy of manufacturing during the production and repair) as to working conditions (wear, bending

Position of the journal, which is regarded as a rigid body, in this case you can specify by two

 2 is the rotation angle of the journal on its own axis *O Z*2 2 . When journal axis is skewed the film thickness at a random cross-section *Z*1*<sup>i</sup>* of the bearing

δ

 = −− ϕ

journal and the bearing at the ends of the tribounit; *B* is the width of the tribounit. The expression for the lubricant film thickness, taking into account the skewness, is written in

> ϕδ

It should be also taken into account that the bearing surfaces are deformed under the action of hydrodynamic pressures. The value Δ( ) *p* is the radial elastic displacement of the bearing sliding surface under the action of hydrodynamic pressure *p* in the lubricant film. Function Δ( ) *p* is defined in the process of calculating of the bearing strain (for a "hard" bearing Δ = ( ) 0) *p* and is written in the form of a component in the equation for the lubricant film

Thus, the film thickness, taking into account the arbitrary geometry of friction surfaces of a journal and a bearing, the skewness of the journal and elastic displacements of the bearing,

with non-ideal geometry; *e t*( ) is displacement of journal mass centers in relation to the

11 1 *h Z t h Z e Z sB* ( , , ) ( , ) cos( ) 2 cos( )

 ϕδ= − − − ⋅ ⋅ − +Δ

( ) \*

( )*t* - an angle that takes into account the skewness of axes of a bearing and a

is the film thickness with the central position of the journal in the bearing

( ) ( ) are determined by solving the equations of motion.

of the journal center *O*2 and by three angles (

\* 1 1 ( , , ) ( , ) cos( ) *i i i i h Zt h Z e*

11 1 <sup>2</sup> ( , , ) ( , ) cos( ) cos( ) *<sup>s</sup> h Zt h Z e Z*

 ϕδ

( ) . (17)

γ , ε , θ

is the deviation angle of skewness plane from the base

for this cross-section

 ϕδ

*B*

 ϕε− −−⋅ − . (19)

> ϕ ε

is the film thickness with the central journal position in *i* -th cross -section.

= 2 /*s B* , where *s* is the distance between the geometric centers of the

<sup>2</sup> ). Angle

, (18)

*p* (20)

γis

*h t* (ϕ

ε

ϕ

\*

 ϕ

δ ε ϕ

barrel and saddle journals.

known formula:

of shafts, etc.).

coordinates *e*,

where \*

the form

thickness.

where \*

bearing;

coordinate plane;

δ

θ

depends on the eccentricity *ie* and the angle *<sup>i</sup>*

ϕ=

skewness of journal axis;

<sup>1</sup> (, )*<sup>i</sup> h Z* ϕ

γ

is determined by the equation:

<sup>1</sup> *h Z* (, ) ϕ

ε

ϕ

journal . The values *et t t* ( ), ,

We term the tg

The theory of thermal processes in the heavy-loaded tribounit of fluid friction is based on a generalized equation of energy (heat transmission) for a thin film of viscous incompressible fluid, which is between two moving surfaces *S*1 and *S*<sup>2</sup> . If we assume a low thermal conductivity in the direction of the coordinate axes *Oxz* (the axis *Oy* is normal to the surface *S*<sup>1</sup> ) (Fig. 1), the temperature distribution *Txyzt* ( , , ,) in the lubricating film will be described by the equation (Prokopiev&Karavayev, 2003)

$$
\rho \mathbf{c}\_0 \frac{\partial T}{\partial t} + \rho \mathbf{c}\_0 \left( V\_x \frac{\partial T}{\partial \mathbf{x}} + V\_y \frac{\partial T}{\partial y} + V\_z \frac{\partial T}{\partial z} \right) - \lambda\_0 \frac{\partial^2 T}{\partial y^2} = \mathbf{J} \tag{21}
$$

Where ρ is density; 0 0 *c* , λ are specific heat capacity and thermal conductivity of lubricant (usually taken as constant); *t* is the time; *Д* is the dissipation function, which is defined for non-Newtonian fluid by the approximate expression

$$
\mathbb{Z} \approx \mu^\* \mathbb{I}\_2 \,. \tag{22}
$$

The three approaches to the integration of the equation (21) (thermohydrodynamic (nonisothermal), adiabatic, isothermal) can be used, depending on the assumptions which are used about the temperature distribution in a thin lubricating film.

When thermohydrodynamic approach is applied the temperature will change in all directions, including across the oil film. In this case, the boundary conditions are stated quite simply and are the most adequate to the real thermal processes. With this approach, we get information about the local properties of the temperature field of lubricating film: a temperature distribution *Txyzt* ( , , ,); maximum temperature *T*max , instantaneous average temperature ( ) *T t av* ; zones of elevated temperatures.

If adiabatic approach is applied the change of the temperature across the oil film (along the axis *Oy* ) is ignored, the journal and the bearing are assumed ideal thermal insulators. We introduce a computational averaged over the width of the bearing temperature ( ) \* \* *T T xt* = , . We substitute it into the equation (21) and receive a differential equation for the temperature distribution along the coordinate *x* . Since in this case the heat transfer to the journal and the bearing is not taken into account, the calculated temperatures are too high. It reduces the accuracy of the results.

The isothermal approach assumes that the calculated current temperature ( ) *T Tt c c* = is the same at all points of the lubricant film. This temperature is a highly inertial parameter and it is determined by solving the heat balance equation

$$A\_N^\*\left(t\right) = A\_Q^\*\left(t\right). \tag{23}$$

This equation reflects the equality of the average values of the heat \* *AN* , which is dissipated in the lubricating film, and the average values of the heat \* *AQ* , which is drained by lubricant into the ends of the tribounit during the loading cycle.

The accurate definition of the current temperature can be performed: at each time step of the calculation; once per a cycle of loading the tribounit, at each time step of the calculation taking into account the thermal interaction between the lubricant film with a journal, with a bearing and a lubrication groove.

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

*XYZ X Y Z* =

*OZ* are excluded from the employed vectors.

γ γ

journal and a bearing are neglected:

γ

*OX OY* , .

repeated.

characteristics of tribounits.

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 105

Projections of linear and angular positions and velocities and loads *FMR* , ,,Μ onto the axis

In the case of planar motion of a journal on the lubricant film the solution to the problem of the dynamics of the radial bearings can be obtained more easily. The skewness of axes of a

The vectors of coordinates, velocities and loads include only their projections on the axes

The solution of the systems of equations of motion (24) or (25) can be found only numerically, because the loads, which are caused by the hydrodynamic pressure, are determined by the repeated numerical solution of the differential equations by Elrod (1) or by Reynolds (4). If we discretize the system of equations of motion over time, then the decision when passing to the next time step can be obtained by using the explicit or implicit method of calculation. In an explicit scheme the unknowns are the pressure and the coordinates of the journal center, in an implicit scheme the unknowns are the pressure and the rate of position change of the journal. However, the implementation of explicit schemes of integrating the motion equations is sensitive to the accumulation of rounding errors. Therefore, implicit schemes for integrating the equations of motion over time are realized in

The most common methods for solving equations of motion of type (24) are: Newton's method, Runge-Kutta's method with Merson's modification, the modified method of linear acceleration (Wilson's method), the method of non-central third-order differences (method by Habolt). To solve the system of the form (25) it is expedient to use special techniques, which are adapted to the systems of "stiff" differential equations (method based on the use of differentiation backward formulas (DBF) of the first- and second-order, method by Fowler, Wharton and others). The standard procedure for solving differential equations (25), which are unsolved relatively to derivatives, consists in the formal integration of the

several studies, which are dedicated to the dynamics of heavy-loaded tribounits.

equations ( , ) *U* = *f U t* and determining derivatives with the help of Newton method.

When the character of applied loads is periodical the initial values of variables *U* and their derivatives *U* can be set arbitrarily. With that the integration continues until the time when the values *U* and *U* , which are separated by a period *ct* of load changes, will not be

Ability to use a particular method of integration depends on the type of a tribounit, the character of acting loads and the possibility to set an initial approximation for the successful solution of (24) or (25). Currently, universal methods for solving the dynamics of heavyloaded tribounits are not designed. The result of calculating the dynamics of heavy-loaded bearings is a trajectory of mass center of the journal, as well as hydro-mechanical

The construction of the heavy-loaded tribounit is evaluated by parameters of the calculated trajectory and interconnected hydro-mechanical characteristics (HMCh). There is the lowest

max *<sup>p</sup>* , *MPa*; the unit load max *<sup>f</sup>* , \*

max *p* greater than allowable values *доп p* , %; mean-value losses due to friction \* *N* , W, the

α

min *h* , μ*m*;

*<sup>h</sup>* , where the values of min *h* less than

*<sup>p</sup>* , where the values of

α

and average per cycle of loading values of: the lubricant film thickness min inf *h* , \*

the hydrodynamic pressure in the lubricant film max sup*p* , \*

allowable values *доп h* , %; the relative total length of the regions *доп*

max *f* , *MPa*; the relative total length of the regions *доп*

0

= = = = = *MMM* , 0 Μ *XYZ* =Μ =Μ = . (26)

#### **2.4 The equations of heavy-loaded bearing dynamics**

To study the dynamics of bearings of liquid friction the motion of the journal on the lubricant film in the bearing is usually considered (Fig. 4). In the coordinates space OXYZ the movement of the journal, which rotates with the relative angular velocity and the angular acceleration, taking into account the axle skewness of a journal and a bearing, is described by approximate differential equations

$$
\tilde{m}\ddot{\mathcal{U}}(t) = \tilde{F}(t) + \tilde{R}\left(\mathcal{U}, \dot{\mathcal{U}}\right). \tag{24}
$$

Where *m* is the matrix of inertia of the journal: {*m mmmJ J J ii*} = { ,,, , , *XYZ*} , ,,, *mJ J J XYZ* are mass and moments of inertia of the journal, 0 *m ij i* <sup>≠</sup> = , 1,...,6 *i* = , *j* = 1,...,6 ;

Fig. 4. Scheme of a heavy-loaded bearing with arbitrary geometry of the lubricant film *U t XYZ* ( ) = { ,,, , , γ *XYZ* γ γ } is the vector of generalized coordinates of the journal centre; *Ft F F F M M M* ( ) <sup>=</sup> { *XYZ X Y Z* ,,, , , } is the vector of known loads on the journal, presented by the power *F* with its projections , , *FFF XYZ* on the axes of the coordinate system *OXYZ* and a moment of forces *M* with its projections , , *MXYZ M M* ; *RUU R R R* ( , ,,, , , ) = ΜΜΜ { *XYZ X Y Z*} is the vector of loads due to the hydrodynamic pressure in the lubricant film. The time derivatives are denoted by points. The forces of friction and weight, as well as gyroscopic moments of the rotating journal are considerably less than other loads, so they aren't taken into account in the equations of motion.

For the dynamics of radial bearings of ICE the level of loads *F* acting on the journal is higher than its own inertial forces. The system of equations of motion (24) in this case is rewritten as

$$
\tilde{F}(t) + \tilde{R}(\mathcal{U}, \dot{\mathcal{U}}) = 0 \,. \tag{25}
$$

To study the dynamics of bearings of liquid friction the motion of the journal on the lubricant film in the bearing is usually considered (Fig. 4). In the coordinates space OXYZ the movement of the journal, which rotates with the relative angular velocity and the angular acceleration, taking into account the axle skewness of a journal and a bearing, is

Where *m* is the matrix of inertia of the journal: {*m mmmJ J J ii*} = { ,,, , , *XYZ*} , ,,, *mJ J J XYZ* are

Fig. 4. Scheme of a heavy-loaded bearing with arbitrary geometry of the lubricant film

*Ft F F F M M M* ( ) <sup>=</sup> { *XYZ X Y Z* ,,, , , } is the vector of known loads on the journal, presented by the power *F* with its projections , , *FFF XYZ* on the axes of the coordinate system *OXYZ* and a moment of forces *M* with its projections , , *MXYZ M M* ; *RUU R R R* ( , ,,, , , ) = ΜΜΜ { *XYZ X Y Z*} is the vector of loads due to the hydrodynamic pressure in the lubricant film. The time derivatives are denoted by points. The forces of friction and weight, as well as gyroscopic moments of the rotating journal are considerably less than other loads, so they aren't taken

For the dynamics of radial bearings of ICE the level of loads *F* acting on the journal is higher than its own inertial forces. The system of equations of motion (24) in this case is

} is the vector of generalized coordinates of the journal centre;

*F t RUU* ( ) + (,) 0 = . (25)

mass and moments of inertia of the journal, 0 *m ij i* <sup>≠</sup> = , 1,...,6 *i* = , *j* = 1,...,6 ;

*mU t F t R U U* ( ) = + ( ) ( , ) . (24)

**2.4 The equations of heavy-loaded bearing dynamics** 

described by approximate differential equations

*U t XYZ* ( ) = { ,,, , ,

rewritten as

γ *XYZ* γ γ

into account in the equations of motion.

Projections of linear and angular positions and velocities and loads *FMR* , ,,Μ onto the axis *OZ* are excluded from the employed vectors.

In the case of planar motion of a journal on the lubricant film the solution to the problem of the dynamics of the radial bearings can be obtained more easily. The skewness of axes of a journal and a bearing are neglected:

$$\gamma\_X = \gamma\_Y = \gamma\_Z = M\_X = M\_Y = M\_Z = 0 \quad \tilde{\mathbf{M}}\_X = \tilde{\mathbf{M}}\_Y = \tilde{\mathbf{M}}\_Z = \mathbf{0} \tag{26}$$

The vectors of coordinates, velocities and loads include only their projections on the axes *OX OY* , .

The solution of the systems of equations of motion (24) or (25) can be found only numerically, because the loads, which are caused by the hydrodynamic pressure, are determined by the repeated numerical solution of the differential equations by Elrod (1) or by Reynolds (4). If we discretize the system of equations of motion over time, then the decision when passing to the next time step can be obtained by using the explicit or implicit method of calculation. In an explicit scheme the unknowns are the pressure and the coordinates of the journal center, in an implicit scheme the unknowns are the pressure and the rate of position change of the journal. However, the implementation of explicit schemes of integrating the motion equations is sensitive to the accumulation of rounding errors. Therefore, implicit schemes for integrating the equations of motion over time are realized in several studies, which are dedicated to the dynamics of heavy-loaded tribounits.

The most common methods for solving equations of motion of type (24) are: Newton's method, Runge-Kutta's method with Merson's modification, the modified method of linear acceleration (Wilson's method), the method of non-central third-order differences (method by Habolt). To solve the system of the form (25) it is expedient to use special techniques, which are adapted to the systems of "stiff" differential equations (method based on the use of differentiation backward formulas (DBF) of the first- and second-order, method by Fowler, Wharton and others). The standard procedure for solving differential equations (25), which are unsolved relatively to derivatives, consists in the formal integration of the equations ( , ) *U* = *f U t* and determining derivatives with the help of Newton method.

When the character of applied loads is periodical the initial values of variables *U* and their derivatives *U* can be set arbitrarily. With that the integration continues until the time when the values *U* and *U* , which are separated by a period *ct* of load changes, will not be repeated.

Ability to use a particular method of integration depends on the type of a tribounit, the character of acting loads and the possibility to set an initial approximation for the successful solution of (24) or (25). Currently, universal methods for solving the dynamics of heavyloaded tribounits are not designed. The result of calculating the dynamics of heavy-loaded bearings is a trajectory of mass center of the journal, as well as hydro-mechanical characteristics of tribounits.

The construction of the heavy-loaded tribounit is evaluated by parameters of the calculated trajectory and interconnected hydro-mechanical characteristics (HMCh). There is the lowest and average per cycle of loading values of: the lubricant film thickness min inf *h* , \* min *h* , μ*m*; the hydrodynamic pressure in the lubricant film max sup*p* , \* max *<sup>p</sup>* , *MPa*; the unit load max *<sup>f</sup>* , \* max *f* , *MPa*; the relative total length of the regions *доп* α*<sup>h</sup>* , where the values of min *h* less than allowable values *доп h* , %; the relative total length of the regions *доп* α *<sup>p</sup>* , where the values of max *p* greater than allowable values *доп p* , %; mean-value losses due to friction \* *N* , W, the

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

crankshaft.

estimates by 2-5%).

Newtonian behavior.

lubrication flow rate.

based on the concept of the first

viscosity value corresponds to the

grade of SAE may have different shear stability.

shear rate is defined as (Whilkinson, 1964)

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 107

It is assumed that the viscoelastic properties of thickened oils have a positive impact on the operation of sliding bearings, help to increase the thickness of the lubricant film. Qualitative influence of viscoelastic properties (relaxation time) of the lubricant is reflected in Fig. 6. With the increase of the relaxation time of lubrication, the mean-value of the minimum lubricating film thickness and power loss due to friction increase. It is seen that the character of the dependence is the same, but the values are shifted back to the rotation angle of the

Fig. 6. The dependence of the characteristics from the angle of rotation of crankshaft

Structural-viscous oils have the ability to temporarily reduce the viscosity during the shear, so they are called "energy saving", because they help to reduce power losses due to friction in internal combustion engines and, consequently, fuel consumption (according to various

The most well-known mathematical model describing the behavior of the structural-viscous oils, is a power law of Ostwald-Weyl, according to which the dependence of viscosity versus

\* 1 *n*

 γ

Where *k* – measure the fluid consistency; *n* – index characterizing the degree of non-

Gecim suggested the dependence of viscosity on the second invariant of shear rate, which is

( ) \* <sup>2</sup> 1

The higher *Kc* , the higher is the stability of the liquid with respect to the shift. At low shear

(Fig. 7). Experimental studies have established that multigrade oils of the same viscosity

The application of structural-viscous oils, along with a reduction of power losses to friction leads to a decrease in the lubricating film thickness, temperature and to the increase of

 μ

<sup>1</sup> (*T*) and the second

*c c K K*

<sup>+</sup> <sup>⋅</sup> <sup>=</sup> <sup>+</sup> <sup>⋅</sup> 

1

μ γ

μ γ

*k* <sup>−</sup> = . (28)

<sup>2</sup> (*T*) Newtonian viscosity, the

μ2

. (29)

μ

<sup>1</sup> , with increasing shear rate the viscosity tends to

μ

μ

parameter *K T <sup>c</sup>* ( ) , characterizing the shear stability of lubricants (Gecim, 1990):

μγ

μ

leakage of lubrication in the bearing ends \* 3 *Q* ,*м s* and temperature of the lubricant film *T C*, <sup>D</sup> .

#### **3. Lubrication with non-Newtonian and multiphase fluids**

The development of technology is inextricably linked with the improvement of lubricants, which today remain an important factor that ensures the reliability of machines. Currently, for lubrication of tribounits of ICE multigrade oils are widely used, rheological behavior of which does not comply with the law of Newton-Stokes equations on a linear relationship between shear stress and shear rate (Whilkinson, 1964):

$$
\pi = \mu \cdot \dot{\boldsymbol{\gamma}} \quad , \tag{27}
$$

where τ – shear stress; μ – dynamic viscosity, which is a function of temperature *T* and pressure *p* (Newtonian viscosity); γ – shear rate, 2 γ = *I* ; 2*I* – second invariant of shear rate ( ) ( ) 2 2 <sup>2</sup> *x z I V* ≈∂ ∂ +∂ ∂ *y V y* , , , *VVV <sup>x</sup> <sup>y</sup> <sup>z</sup>* – velocity component of the elementary volume lubrication, which is located between the two surfaces.

Particularly, the viscosity depends not only on the temperature and pressure, but also on the shear rate in a thin lubricating film separating the surfaces of friction pairs. These oils are called non-Newtonian.

Theoretical studies of the dynamics of friction pairs, which take into account non-Newtonian behavior of lubricant, are based on the modification of the equations for determining the field of hydrodynamic pressures by using different rheological models. One classification of a rheological model is shown in Fig. 5.

In general, non-Newtonian behavior includes any anomalies observed in the flow of fluid. In particular, the presence of viscous polymer additives in oils leads to a change in their properties. Oils with additives can be characterized as structurally viscous and viscoelastic

Viscoelastic fluids are those exhibiting both elastic recovery of form and viscous flow. There are various models of viscoelastic fluids, among which the best known model is the

Maxwell \* *t* τ τ λ μγ ∂ + = <sup>∂</sup> . Here λ– relaxation time, characterizing the delay of shear stress

changes in respect to changes of shear rates; \* μ (,,) *Т p* γ – dynamic viscosity (non-Newtonian viscosity). In this case, the liquid is called the Maxwell (Maxwell viscoelastic liquid).

Fig. 5. Classification of rheological models of lubricating fluids

leakage of lubrication in the bearing ends \* 3 *Q* ,*м s* and temperature of the lubricant film

The development of technology is inextricably linked with the improvement of lubricants, which today remain an important factor that ensures the reliability of machines. Currently, for lubrication of tribounits of ICE multigrade oils are widely used, rheological behavior of which does not comply with the law of Newton-Stokes equations on a linear relationship

– shear rate, 2

Particularly, the viscosity depends not only on the temperature and pressure, but also on the shear rate in a thin lubricating film separating the surfaces of friction pairs. These oils are

Theoretical studies of the dynamics of friction pairs, which take into account non-Newtonian behavior of lubricant, are based on the modification of the equations for determining the field of hydrodynamic pressures by using different rheological models. One

In general, non-Newtonian behavior includes any anomalies observed in the flow of fluid. In particular, the presence of viscous polymer additives in oils leads to a change in their properties. Oils with additives can be characterized as structurally viscous and viscoelastic Viscoelastic fluids are those exhibiting both elastic recovery of form and viscous flow. There are various models of viscoelastic fluids, among which the best known model is the

> μ (,,) *Т p* γ

viscosity). In this case, the liquid is called the Maxwell (Maxwell viscoelastic liquid).

<sup>2</sup> *x z I V* ≈∂ ∂ +∂ ∂ *y V y* , , , *VVV <sup>x</sup> <sup>y</sup> <sup>z</sup>* – velocity component of the elementary volume

γ

⋅ , (27)

= *I* ; 2*I* – second invariant of shear

– dynamic viscosity (non-Newtonian

– dynamic viscosity, which is a function of temperature *T* and

– relaxation time, characterizing the delay of shear stress

τ = μ γ

γ

**3. Lubrication with non-Newtonian and multiphase fluids** 

between shear stress and shear rate (Whilkinson, 1964):

μ

lubrication, which is located between the two surfaces.

classification of a rheological model is shown in Fig. 5.

λ

Fig. 5. Classification of rheological models of lubricating fluids

– shear stress;

rate ( ) ( ) 2 2

called non-Newtonian.

Maxwell \*

τ λ

*t* τ

∂ + =

 μγ

<sup>∂</sup> . Here

changes in respect to changes of shear rates; \*

pressure *p* (Newtonian viscosity);

*T C*, <sup>D</sup> .

where τ It is assumed that the viscoelastic properties of thickened oils have a positive impact on the operation of sliding bearings, help to increase the thickness of the lubricant film. Qualitative influence of viscoelastic properties (relaxation time) of the lubricant is reflected in Fig. 6.

With the increase of the relaxation time of lubrication, the mean-value of the minimum lubricating film thickness and power loss due to friction increase. It is seen that the character of the dependence is the same, but the values are shifted back to the rotation angle of the crankshaft.

Fig. 6. The dependence of the characteristics from the angle of rotation of crankshaft

Structural-viscous oils have the ability to temporarily reduce the viscosity during the shear, so they are called "energy saving", because they help to reduce power losses due to friction in internal combustion engines and, consequently, fuel consumption (according to various estimates by 2-5%).

The most well-known mathematical model describing the behavior of the structural-viscous oils, is a power law of Ostwald-Weyl, according to which the dependence of viscosity versus shear rate is defined as (Whilkinson, 1964)

$$
\mu^\* = k \dot{\gamma}^{n-1}.\tag{28}
$$

Where *k* – measure the fluid consistency; *n* – index characterizing the degree of non-Newtonian behavior.

Gecim suggested the dependence of viscosity on the second invariant of shear rate, which is based on the concept of the first μ<sup>1</sup> (*T*) and the second μ<sup>2</sup> (*T*) Newtonian viscosity, the parameter *K T <sup>c</sup>* ( ) , characterizing the shear stability of lubricants (Gecim, 1990):

$$
\mu^\* \left( \dot{\boldsymbol{\gamma}} \right) = \mu\_1 \frac{K\_c + \mu\_2 \cdot \dot{\boldsymbol{\gamma}}}{K\_c + \mu\_1 \cdot \dot{\boldsymbol{\gamma}}}.\tag{29}
$$

The higher *Kc* , the higher is the stability of the liquid with respect to the shift. At low shear viscosity value corresponds to the μ<sup>1</sup> , with increasing shear rate the viscosity tends to μ2 (Fig. 7). Experimental studies have established that multigrade oils of the same viscosity grade of SAE may have different shear stability.

The application of structural-viscous oils, along with a reduction of power losses to friction leads to a decrease in the lubricating film thickness, temperature and to the increase of lubrication flow rate.

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 109

Fig. 8. The dependence of the hydromechanical characteristics from the rotation angle of

μ

resistance to micro-rotation of particles. Length parameter A characterizes the size of

The presence of micro-particles in the lubricant leads to an increase in the resultant shear stress in the lubricating film. The calculations of heavy-loaded bearings using micropolar fluid theory suggest that this phenomenon significantly affects the HMCh of a bearing, in particular, leads to an increase of lubricating film thickness. The results of the calculation of the connecting rod bearing, taking into account the structural heterogeneity of lubricants (based on the model of micropolar fluids with the parameters <sup>2</sup> *L N* = = 10, 0,5 ) are reflected

> \* *QB* , *l/s*

Newtonian fluid 610,5 105,9 0,02355 4,416 280,3 1,93 0

Table 2. The results of the calculation of hydro-mechanical characteristics of the connecting

It is obvious, that the results will prove valuable for practice, only in case of experimental determination of the value of the micropolarity parameters *N* and *L*. Further studies of the authors are focused on the experimental basis of these values for modern thickened oils.

\* min *h* , μ*m*

727,4 110,6 0,0215 5,84 237,9 2,9 0

1 <sup>1</sup> 2

μ

μ μ 1 2

additionally characterized by two physical

μ

1 and the parameter

<sup>1</sup> , called the coefficient of eddy viscosity, takes into account the

⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠ <sup>+</sup> , <sup>0</sup> *<sup>h</sup> <sup>L</sup>* <sup>=</sup> <sup>A</sup> , (30)

max sup*p* , *MPa* 

min inf *h* , μ*m*

\* α *, %* 

crankshaft: 1) Newtonian fluid, and 2) the structural-viscous liquid (28)

microparticles or molecular lubricant. With the help of the coefficient

*N*

*T* , *º С*

rod bearing, taking into account the structural heterogeneity of lubrication

Micropolar fluid along with the viscosity

μ

A you can calculate the so-called micropolar parameters

\* *N* , *W* 

,A . Parameter

where 0 *h* – characteristic film thickness.

constants 1

μ

in Fig. 9 and Table. 2.

Hydromechanical characteristics

Structurally heterogeneous fluid (30)

Fig. 7. Fundamental character of the non-Newtonian oils viscosity

Comparative results of the calculation of hydro-mechanical characteristics of the connecting rod bearing for the dependence of oil viscosity versus shear rate and without it are presented in Table. 1 and Fig. 8.

All results were founded for connecting-rod bearing of engine type ЧН 13/15 (Co ltd. "ChTZ-URALTRAC") with follow parameters: rotating speed 219.91 c-1; length 0.033 m; journal radius 0.0475 m; radial clearance 51.5 μm.

The results indicate that the application of structural-viscous oils leads to a reduction of power losses due to friction in the range 15-20%. Consumption of lubricant through the bearing increases, the mean-value of the temperature decreases by 2-3° C. However, there is a decrease in the minimum lubricating film thickness by an average of 14-20%.

This fact confirms the view that the use of low-viscosity oil at high temperature and shear rate is justified only if it is allowed by the engine design, in particular, of crankshaft bearings.


Table 1. The results of the calculation of HMCh of the connecting rod bearing

In recent years, the oil, which has in its composition the so-called friction modifiers, for example, particles of molybdenum, is widespread. These additives are introduced into the base oil to improve its antiwear and extreme pressure properties to reduce friction and wear under semifluid and boundary lubrication regimes.

Oils with such additives are called "micropolar". They represent a mixture of randomly oriented micro-particles (molecules), suspended in a viscous fluid and having its own rotary motion.

Fig. 8. The dependence of the hydromechanical characteristics from the rotation angle of crankshaft: 1) Newtonian fluid, and 2) the structural-viscous liquid (28)

Micropolar fluid along with the viscosity μ additionally characterized by two physical constants 1 μ ,A . Parameter μ<sup>1</sup> , called the coefficient of eddy viscosity, takes into account the resistance to micro-rotation of particles. Length parameter A characterizes the size of microparticles or molecular lubricant. With the help of the coefficient μ1 and the parameter A you can calculate the so-called micropolar parameters

$$N = \left(\frac{\mu\_1}{2\mu + \mu\_1}\right)^{1/2}, \ L = \frac{h\_0}{\ell},\tag{30}$$

where 0 *h* – characteristic film thickness.

108 Tribology - Lubricants and Lubrication

Comparative results of the calculation of hydro-mechanical characteristics of the connecting rod bearing for the dependence of oil viscosity versus shear rate and without it are

All results were founded for connecting-rod bearing of engine type ЧН 13/15 (Co ltd. "ChTZ-URALTRAC") with follow parameters: rotating speed 219.91 c-1; length 0.033 m;

The results indicate that the application of structural-viscous oils leads to a reduction of power losses due to friction in the range 15-20%. Consumption of lubricant through the bearing increases, the mean-value of the temperature decreases by 2-3° C. However, there is

This fact confirms the view that the use of low-viscosity oil at high temperature and shear rate is justified only if it is allowed by the engine design, in particular, of crankshaft

> \* *QB* , *l/s*

Newtonian fluid 610,5 105,9 0,02345 4,416 280,3 1,93 0

liquid (28) 518,4 102,6 0,02512 3,75 309,8 1,52 16,9

liquid (29) 539,0 103,4 0,0246 3,789 307,8 1,66 11,9

In recent years, the oil, which has in its composition the so-called friction modifiers, for example, particles of molybdenum, is widespread. These additives are introduced into the base oil to improve its antiwear and extreme pressure properties to reduce friction and wear

Oils with such additives are called "micropolar". They represent a mixture of randomly oriented micro-particles (molecules), suspended in a viscous fluid and having its own rotary

\* min *h* , μ*m* 

max sup*p* , *MPa*

min inf *h* , μ*m*

\* α *, %* 

a decrease in the minimum lubricating film thickness by an average of 14-20%.

*T* , *º С*

Table 1. The results of the calculation of HMCh of the connecting rod bearing

Fig. 7. Fundamental character of the non-Newtonian oils viscosity

presented in Table. 1 and Fig. 8.

bearings.

motion.

Hydromechanical characteristics

Structural-viscous

Structural-viscous

journal radius 0.0475 m; radial clearance 51.5 μm.

\* *N* , *W* 

under semifluid and boundary lubrication regimes.

The presence of micro-particles in the lubricant leads to an increase in the resultant shear stress in the lubricating film. The calculations of heavy-loaded bearings using micropolar fluid theory suggest that this phenomenon significantly affects the HMCh of a bearing, in particular, leads to an increase of lubricating film thickness. The results of the calculation of the connecting rod bearing, taking into account the structural heterogeneity of lubricants (based on the model of micropolar fluids with the parameters <sup>2</sup> *L N* = = 10, 0,5 ) are reflected in Fig. 9 and Table. 2.


Table 2. The results of the calculation of hydro-mechanical characteristics of the connecting rod bearing, taking into account the structural heterogeneity of lubrication

It is obvious, that the results will prove valuable for practice, only in case of experimental determination of the value of the micropolarity parameters *N* and *L*. Further studies of the authors are focused on the experimental basis of these values for modern thickened oils.

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

surfaces of the friction.

surface; μ0 - viscosity in entirety.

hydrodynamic pressure by 4-5%.

Hydromechanical characteristics

numerical value 610,5 1)

(Mukhortov et al., 2010) and has the following form:

the HMCh of the rod bearing is illustrated in Fig.10 and Table. 3.

\* *N , W* 

681,2 2)

1 - Newtonian fluid; 2 - taking into account the highly viscous boundary film.

rod bearing in the light of high-viscosity boundary film lubrication

crankshaft: 1 - Newtonian fluid, 2 - with the boundary layer (33)

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 111

Based on experimental and theoretical studies it can be argued that under changing conditions of friction a repeated change of mechanisms of friction and wear occurs, in which the key role is played by the change of rheological properties of lubricants, depending on the thickness of the film, the contact pressure, surface roughness and the individual properties of the lubricant. Thus, there is a need for the computational models depending on the rheological properties of lubricating oil on the factors related to the availability, quantity and structure of the antifriction and antiwear additives and lubricants interaction with the

One model describing the dependence of viscosity of lubricant on thickness is proved in

<sup>0</sup> exp *<sup>i</sup>*

=+ −⎜ ⎟ ⎝ ⎠

where *lh* – characteristic parameter having the dimension of length, which value is specific for each combination of lubricant and the solid surface; *µS* – parameter having the meaning of the conditional values of the viscosity at infinitely small distance from the bounding

The impact of the availability of a highly viscous boundary film on the friction surfaces on

In the hydrodynamic friction regime the presence of adsorption films leads to an increase in the minimum lubricating film thickness by 40-45%, the temperature at 6-7%, the maximum

> *T , º С*

105,9 113,3

Table 3. The results of the calculation of hydro-mechanical characteristics of the connecting

Fig. 10. The dependence of the hydromechanical characteristics from the rotation angle of

*h h l*

\* min *h ,*  μ

4,416 5,665

*m* max sup*<sup>p</sup> , MPa* 

> 280,3 294,9

, (33)

min inf *h ,*  μ*m* 

> 1,93 3,59

⎛ ⎞

*i S*

μμμ

The calculation of the structural heterogeneity of the lubricant is a very complicated mathematical problem, since it is necessary to take into account many factors: the speed and shape of particles, their distribution, elasticity, etc.

Fig. 9. Dependence of the hydromechanical characteristics from the rotation angle of the crankshaft: 1 - Newtonian fluid; 2 - structurally heterogeneous fluid (30)

Sometimes simplified dependence is used. For example it is assumed that the viscosity of suspensions depends on the concentration volume of solid particles, which may be the wear products, external contaminants or finely divided special additives. In this case, the viscosity of the lubricant is sufficiently well described by the Einstein formula:

$$
\mu = \mu^"\left(1 + \xi \cdot \varphi\right). \tag{31}
$$

Where ξ – shape factor of particles, for asymmetric particles 2,5 ξ≥ .

Separate scientific problem is the availability of records in the lubricant gas component. Experimental studies have shown that the engine lubrication system always contains air dissolved in the form of gas bubbles. The proportion of bubbles in the total amount of oil may reach 30%.

The viscosity of gassy oils can be calculated with a sufficient degree of accuracy with the help of the formula:

$$
\mu = \mu^\* \left( 1 - \delta \right). \tag{32}
$$

Where the coefficient δ = *V V Г М* is equal to the ratio of the volume fraction of gas *VГ* in the bubble mixture to the volume fraction of pure oil *VМ* at temperature *T* .

When you select computer models you must take into account not only the working conditions, regime and geometric characteristics of tribounits under consideration, but also features of rheological behavior of used lubricants.

At present, as a result of parallel and interdependent modifications of ICE and production technologies of motor oils, the most loaded sliding bearings of an engine work at the minimum design film thickness of about 1 micron in the steady state and less - at low frequencies of crankshaft rotation, that is with film thicknesses comparable to twice the height of surface roughness of tribounits. In this case the life of one and the same friction unit can vary in 3 ... 5 times when using different motor oils, and be by orders of magnitude greater than the resource when using other grease lubricants at the same bulk rheological properties.

The calculation of the structural heterogeneity of the lubricant is a very complicated mathematical problem, since it is necessary to take into account many factors: the speed and

Fig. 9. Dependence of the hydromechanical characteristics from the rotation angle of the

\*

 ξϕ

Separate scientific problem is the availability of records in the lubricant gas component. Experimental studies have shown that the engine lubrication system always contains air dissolved in the form of gas bubbles. The proportion of bubbles in the total amount of oil

The viscosity of gassy oils can be calculated with a sufficient degree of accuracy with the

When you select computer models you must take into account not only the working conditions, regime and geometric characteristics of tribounits under consideration, but also

At present, as a result of parallel and interdependent modifications of ICE and production technologies of motor oils, the most loaded sliding bearings of an engine work at the minimum design film thickness of about 1 micron in the steady state and less - at low frequencies of crankshaft rotation, that is with film thicknesses comparable to twice the height of surface roughness of tribounits. In this case the life of one and the same friction unit can vary in 3 ... 5 times when using different motor oils, and be by orders of magnitude greater than the resource when using other grease lubricants at the same bulk rheological

( ) \*

 δ

= *V V Г М* is equal to the ratio of the volume fraction of gas *VГ* in the

(1 ) + ⋅ . (31)

1 − . (32)

ξ≥ .

Sometimes simplified dependence is used. For example it is assumed that the viscosity of suspensions depends on the concentration volume of solid particles, which may be the wear products, external contaminants or finely divided special additives. In this case, the

crankshaft: 1 - Newtonian fluid; 2 - structurally heterogeneous fluid (30)

viscosity of the lubricant is sufficiently well described by the Einstein formula:

μ = μ

– shape factor of particles, for asymmetric particles 2,5

μ = μ

bubble mixture to the volume fraction of pure oil *VМ* at temperature *T* .

shape of particles, their distribution, elasticity, etc.

Where ξ

may reach 30%.

help of the formula:

Where the coefficient

properties.

δ

features of rheological behavior of used lubricants.

Based on experimental and theoretical studies it can be argued that under changing conditions of friction a repeated change of mechanisms of friction and wear occurs, in which the key role is played by the change of rheological properties of lubricants, depending on the thickness of the film, the contact pressure, surface roughness and the individual properties of the lubricant. Thus, there is a need for the computational models depending on the rheological properties of lubricating oil on the factors related to the availability, quantity and structure of the antifriction and antiwear additives and lubricants interaction with the surfaces of the friction.

One model describing the dependence of viscosity of lubricant on thickness is proved in (Mukhortov et al., 2010) and has the following form:

$$
\mu\_i = \mu\_0 + \mu\_S \exp\left(-\frac{h\_i}{l\_h}\right),
\tag{33}
$$

where *lh* – characteristic parameter having the dimension of length, which value is specific for each combination of lubricant and the solid surface; *µS* – parameter having the meaning of the conditional values of the viscosity at infinitely small distance from the bounding surface; μ0 - viscosity in entirety.

The impact of the availability of a highly viscous boundary film on the friction surfaces on the HMCh of the rod bearing is illustrated in Fig.10 and Table. 3.

In the hydrodynamic friction regime the presence of adsorption films leads to an increase in the minimum lubricating film thickness by 40-45%, the temperature at 6-7%, the maximum hydrodynamic pressure by 4-5%.


1 - Newtonian fluid; 2 - taking into account the highly viscous boundary film.

Table 3. The results of the calculation of hydro-mechanical characteristics of the connecting rod bearing in the light of high-viscosity boundary film lubrication

Fig. 10. The dependence of the hydromechanical characteristics from the rotation angle of crankshaft: 1 - Newtonian fluid, 2 - with the boundary layer (33)

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

maximum hydrodynamic pressure is increased by 50-52%.

dependence of viscosity on one of the parameters ( *p T*,,,

occurring in the lubricant film, and ultimately, will improve accuracy.

of tribounits and must be taken into account in the methods of its calculation.

**4. Effect of elastic properties of the construction** 

pressure

design engineer.

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 113

thickness by 20-25%, the temperature by 1-2%. It is important to note that the instantaneous

Fig. 11. The hydromechanical characteristics: 1 - Newtonian fluid, 2 - viscosity depends on

Thus, accounting for one of the properties of the lubricant does not reflect the real process occurring in a thin lubricating film. Each of these properties of the lubricant and the

worsens the hydro-mechanical characteristics of tribounits. Therefore, the choice of rheological models used to calculate heavy-loaded tribounits, depends on the type of working conditions of lubricant and tribounits, as well as on the objectives pursued by the

Further research should be focused on experimental substantiation of the parameters of rheological models, as well as the creation of calculation methods for assessing the simultaneous influence of various non-Newtonian properties of the lubricant on the dynamics of heavy loaded tribounits. This will provide simulation of real processes

Elastohydrodynamic (EHD) regime of lubrication of bearings is characterized by a significant effect of dynamically changing strain of a bearing and (or) a journal on the clearance in the tribounit. Under unsteady loading the dynamic change in the geometry of the elements of tribounit caused by the finite stiffness of the bearing and the journal, leads to a change in the nature of the lubricant, hydromechanical parameters and supporting forces

The effect of finite stiffness of a bearing and a journal on the change of the profile of the clearance depends on the geometry of the bearing, the ratio of properties which are in contact through the lubricating film surfaces and other factors. In massive bearings local contact deformation of the surface film of the bearing and the journal prevail over the general changes of form of the bearing and the latter are usually neglected. These tribounits are usually referred to as contact-hydrodynamic (elasto-hydrodynamic). Examples of such

γ ϕ

, etc.) either improves or

These models should be used in accordance with the terms of the friction pairs. In this case, the use of non-Newtonian models of lubricants does not exclude taking into account the dependence of oil viscosity on temperature and pressure in the lubricating film of friction pairs (Prokopiev V. et al., 2010):

$$
\mu(T) = \mathbb{C}\_1 \cdot \exp\left\{ \mathbb{C}\_2 / \left( T + \mathbb{C}\_3 \right) \right\},
\tag{34}
$$

where 123 *CCC* , , – constants, which are the empirical characteristics of the lubricant. The coefficients *Ci* are calculated using the formula following from the dependence (34):

$$\begin{aligned} \mathbf{C}\_{3} &= \frac{-\left[T\_{1}\left(T\_{3} - T\_{2}\right)\ln\left(\frac{\mu\_{1}}{\mu\_{2}}\right) - T\_{3}\left(T\_{2} - T\_{1}\right)\ln\left(\frac{\mu\_{2}}{\mu\_{3}}\right)\right]}{\left[\left(T\_{3} - T\_{2}\right)\ln\left(\frac{\mu\_{1}}{\mu\_{2}}\right) - \left(T\_{2} - T\_{1}\right)\ln\left(\frac{\mu\_{2}}{\mu\_{3}}\right)\right]}; \end{aligned} \tag{35}$$
 
$$\mathbf{C}\_{2} = \frac{\ln\left(\frac{\mu\_{1}}{\mu\_{2}}\right) \cdot \left(T\_{1} + \mathbf{C}\_{3}\right) \cdot \left(T\_{2} + \mathbf{C}\_{3}\right)}{\left(T\_{2} - T\_{1}\right)}; \quad \mathbf{C}\_{1} = \frac{\mu\_{1}}{\exp\left(\mathbf{C}\_{2}/T\_{1}\right)}$$

To account for the dependence of viscosity on the hydrodynamic pressure the Barus formula is acceptable:

$$
\mu\_p = \mu\_0 e^{a \cdot p} \,, \tag{36}
$$

where μ<sup>0</sup> – viscosity of the lubricant at atmospheric pressure; *p* – hydrodynamic pressure in the lubricating film; α – piezoelectric coefficient of viscosity, which depends on temperature and chemical composition of lubricants.

On the base of a combination of models (28), (34) and (36) the authors propose to use a combined dependence of viscosity versus shear rate, pressure and temperature:

$$\boldsymbol{\mu}^\* = \boldsymbol{k} \bullet \dot{\boldsymbol{\mathcal{V}}}^{n-1} \bullet \boldsymbol{e}^{\alpha(T) \cdot \boldsymbol{p}} \cdot \mathbf{C}\_1 \cdot \boldsymbol{e}^{\left(\mathbf{C}\_2 \left\{ (T + \mathbf{C}\_3) \right\} \right)}.\tag{37}$$

The effect of hydrodynamic pressure in the film of lubricant on the HMCh of the connecting rod bearing is reflected in the Table 4 and Fig. 11.


1) - oil viscosity is independent of pressure, 2) - viscosity depends on pressure.

Table 4. The results of the calculation of hydro-mechanical characteristics of the connecting rod bearing for the dependence of viscosity on pressure

As seen from Table 4 and Figure 11, in the case of taking into account the effect of hydrodynamic pressure on the viscosity of the lubricant, all the values of HMCh of the bearing increase. In particular, the mean-power losses increase by 8-9%, the minimum film

These models should be used in accordance with the terms of the friction pairs. In this case, the use of non-Newtonian models of lubricants does not exclude taking into account the dependence of oil viscosity on temperature and pressure in the lubricating film of friction

where 123 *CCC* , , – constants, which are the empirical characteristics of the lubricant. The coefficients *Ci* are calculated using the formula following from the dependence (34):

( ) ( )

⎡ ⎤ ⎛ ⎞ ⎛ ⎞ − − −− ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ <sup>=</sup> ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎢ ⎥ − −− ⎜ ⎟ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠

13 2 32 1

μ

μ

*TT T TT T*

( ) ( )

μ

μ

*TT TT*

3 2 2 1

( )( )

*TC TC*

⎝ ⎠ <sup>=</sup> <sup>=</sup> <sup>−</sup>

μ μ

combined dependence of viscosity versus shear rate, pressure and temperature:

*T* , *ºС*

105,9 106,9

γ

2 1

*C C*

⎜ ⎟⋅+ ⋅+

13 23

( ) ( )

*T T C T*

;

2 1 2 1

2 1

To account for the dependence of viscosity on the hydrodynamic pressure the Barus formula

0 *p*

On the base of a combination of models (28), (34) and (36) the authors propose to use a

<sup>1</sup> ( ) ( ) \* ( ) 2 3 1 *<sup>n</sup> T p C TC k e Ce* α

The effect of hydrodynamic pressure in the film of lubricant on the HMCh of the connecting

Table 4. The results of the calculation of hydro-mechanical characteristics of the connecting

As seen from Table 4 and Figure 11, in the case of taking into account the effect of hydrodynamic pressure on the viscosity of the lubricant, all the values of HMCh of the bearing increase. In particular, the mean-power losses increase by 8-9%, the minimum film

\* *QB* , *l/s* 

0,02345 0,02420

<sup>0</sup> – viscosity of the lubricant at atmospheric pressure; *p* – hydrodynamic pressure

*<sup>p</sup> e* α

ln ln

1 2

2 3

1 2

ln ln

2 3

(*T C C TC* ) =⋅ + 1 23 exp( ( )) , (34)

 μ

 μ

 μ

 μ

exp

– piezoelectric coefficient of viscosity, which depends on

<sup>−</sup> <sup>⋅</sup> <sup>+</sup> <sup>=</sup> ⋅ ⋅ <sup>⋅</sup> <sup>⋅</sup> . (37)

\* min *h* , μ*m*

4,416 5,712 max sup*p* , *MPa*

> 280,3 588,6

min inf *h* , μ*m*

> 1,930 2,560

μ

<sup>⋅</sup> = , (36)

;

; (35)

μ

1

μ

⎛ ⎞

μ

ln

α

μ

\* *N* , *W* 

670,4 2)

rod bearing for the dependence of viscosity on pressure

1) - oil viscosity is independent of pressure, 2) - viscosity depends on pressure.

temperature and chemical composition of lubricants.

rod bearing is reflected in the Table 4 and Fig. 11.

pairs (Prokopiev V. et al., 2010):

3

*C*

is acceptable:

μ

in the lubricating film;

Hydromechanical characteristics

numerical value 610,5 1)

where

thickness by 20-25%, the temperature by 1-2%. It is important to note that the instantaneous maximum hydrodynamic pressure is increased by 50-52%.

Fig. 11. The hydromechanical characteristics: 1 - Newtonian fluid, 2 - viscosity depends on pressure

Thus, accounting for one of the properties of the lubricant does not reflect the real process occurring in a thin lubricating film. Each of these properties of the lubricant and the dependence of viscosity on one of the parameters ( *p T*,,, γ ϕ , etc.) either improves or worsens the hydro-mechanical characteristics of tribounits. Therefore, the choice of rheological models used to calculate heavy-loaded tribounits, depends on the type of working conditions of lubricant and tribounits, as well as on the objectives pursued by the design engineer.

Further research should be focused on experimental substantiation of the parameters of rheological models, as well as the creation of calculation methods for assessing the simultaneous influence of various non-Newtonian properties of the lubricant on the dynamics of heavy loaded tribounits. This will provide simulation of real processes occurring in the lubricant film, and ultimately, will improve accuracy.

#### **4. Effect of elastic properties of the construction**

Elastohydrodynamic (EHD) regime of lubrication of bearings is characterized by a significant effect of dynamically changing strain of a bearing and (or) a journal on the clearance in the tribounit. Under unsteady loading the dynamic change in the geometry of the elements of tribounit caused by the finite stiffness of the bearing and the journal, leads to a change in the nature of the lubricant, hydromechanical parameters and supporting forces of tribounits and must be taken into account in the methods of its calculation.

The effect of finite stiffness of a bearing and a journal on the change of the profile of the clearance depends on the geometry of the bearing, the ratio of properties which are in contact through the lubricating film surfaces and other factors. In massive bearings local contact deformation of the surface film of the bearing and the journal prevail over the general changes of form of the bearing and the latter are usually neglected. These tribounits are usually referred to as contact-hydrodynamic (elasto-hydrodynamic). Examples of such

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

1985; Bonneau 1995).

EY TU reaches 30%.

loading of mating parts such as the engine crank.

complex geometric shapes the easiest.

elastohydrodynamic lubrication problems, are used.

with the contribution of the displacement of the force nature.

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 115

Among the systemic methods the Newton-Raphson method is considered one of the most sustainable and effective solutions for elastohydrodynamic problems. In the literature it is known as the Newton-Kantorovich method or Newton (MN). The algorithm for system solutions of elastohydrodynamic problem consists of three nested iteration loops: the inner loop of implementation by the Newton method of simultaneous solution of hydrodynamic and elastic subproblems; the average - the cycle of calculation of the cavitation zone and the boundary conditions; external - the cycle of calculation of the trajectory of the journal center. Algorithm for the numerical realization of MN is based on the finite-difference or finite element discretization of the linearized system of equations of EHD problem (Oh&Genka

The application of the theory of elastohydrodynamic lubrication allows to predict lower mean-value as of the minimum lubricating film thickness as of the maximum hydrodynamic pressure. Thus, for the rod bearing of an engine, these changes may reach 35 ... 40%. The values of the maximum hydrodynamic pressure generated in the lubricating film of a EY bearing, are also smaller than for the "absolutely rigid" one. Reduction of the maximum hydrodynamic pressure is accompanied by an increase in the size of the bearing area. This fact, together with some increase in the clearance caused by the elastic deformation of the bearing, increases the flow of lubricating fluid through the ends of the bearing. Although the pressure gradient, on which the end consumption directly depends, is reduced. The difference in the instantaneous values of the mechanical flow between "absolutely rigid" and

Calculation of the bearing, taking into account the elastohydrodynamic lubrication regime, not only improves the quality of design of friction units, but also clarifies the dynamic

Thermoelastichydrodynamic (TEHD) regime of lubrication of journal bearings – is the mode of journal bearings, which are characterized by the influence on the magnitude of the clearance in tribounit thermoelastic deformations of a bearing and a journal, commensurate

Accounting for changes in the shape of thermoelastic friction surfaces of the journal and the bearing is possible in the case of inclusion in the resolution system of equations for the EY TU of energy equations and the relations of elasticity theory with the effects of temperature to determine the temperature fields and thermoelastic displacements caused by them. The sources of thermal fields can be either external to the tribounit, for example, a combustion chamber of an internal combustion engine for a connecting rod bearing, and internal lubricating film, in which heat generating is essential for the calculation of TEHD lubrication regime. Thus, the most complete version to solve the problem of TEHD lubrication of tribounits requires the joint consideration of problems of heat distribution in the journal, the bearing and lubricating film. The task is complicated by the fact that journal and the bearing are some idealized concepts. In reality they are rather complex shape parts (crankshaft, connecting rod, crankcase, etc.). Therefore, the methods for solving problems of TEHD lubricants are usually based on the method of FE, allowing to solve problems for bodies of

Transient thermal fields are typical for a lubricating film of heavy-loaded bearings, which requires the simultaneous solution of equations of fluid dynamics, energy, and elasticity at each step of the calculation of trajectories. In this case, to solve all the subproblems the method of FE and schemes, similar to the systemic methods of solving the

units can be gears, frictionless (rolling) bearings, journal bearings with an elastic liner and rigid housing. For their calculations it is reasonable to use the methods of contact hydrodynamics (elasto-hydrodynamic lubrication theory).

However, there is also a large group of hydrodynamic friction pairs in which the general corps deformations make a significant contribution to changing the profile of the clearance. They are characterized by the presence of a continuous gradient of the deformation field, which is independent of the load location, the significant (compared with the contact and hydrodynamic tribounits) values of lubricating film thickness and values of the displacement of the friction surfaces, caused by the bending deformation of the housing, which are commensurate with them. These tribounits are called elasto-yielding (EY TU) or elastohydrodynamic. The most typical representative of the EY TU is a connecting rod bearing of a crank mechanism (crank) of engine vehicles. The desire of engine designers to maximally reduce the weight of movable elements of a crank reduces the stiffness of the bearing (crank crosshead), which makes the mode of EHD lubrication working for the connecting rod bearings. The above features - comparable with the clearance of dynamically changing elastic displacements and a continuous gradient of deformations - prevent the direct application of methods of contact-hydrodynamic lubrication theory to the calculation of EY TU.

A mathematical model of EY TU differs from the "absolutely rigid" units model by the dependence of the instantaneous value of the lubricating film thickness *h zt* (ϕ, ,, *p*) on the elastic displacements of the friction surface of a bearing *W zt* (ϕ, ,, *p*) , which, in their turn, are determined by structural rigidity of the bearing and by the hydrodynamic pressure in the lubricating film *p* : *h zt* (φ φφ , ,, ,, ,,, *p*) = + *h zt W zt rig* ( ) ( *p*) . Where *h zt rig* (ϕ, , ) - film thickness in the "absolutely rigid" bearing. To determine it the expression (6) is used. Thus, the determination of pressures in the lubricating film and HMCh of EY TU is the related objective of the hydrodynamic lubrication theory and the theory of elasticity.

Modeling of EY TU, compared with "absolutely rigid" bearings is supplemented by an elastic subproblem the purpose of which is to determine the strain state of the friction surface of a crank crosshead under the influence of complex loads. The method of solving the elastic subproblem is chosen according to the accepted approximating model of an EY bearing. In today's solutions for EY TU the compliance and stiffness matrix of the bearing is usually constructed using the FE method.

The other side of modeling the elastic subsystem is adequate description of the entire complex of loads, causing the elastic deformation of the bearing housing and the conditions of fixing of FE model. One must consider not only the hydrodynamic pressure, but also the volume forces of rod inertia.

The known methods of solving the elastohydrodynamic lubrication problem can be classified as follows: direct methods or methods of successive approximations, in which the solutions of the hydrodynamic and elastic subtasks are performed separately, with the subsequent jointing of the results in the direct iterative process; and system, oriented for the joint solution of equations of fluid flow and elastic deformation.

In solving the problem of elastohydrodynamic lubrication of a bearing with the help of a direct iterative method, the hydrodynamic and elastic subproblems at each step of time discretization are solved sequentially in an iterative cycle. The main disadvantage of direct methods for the calculation of EHD is their slow convergence and the associated timeconsumption. These difficulties are partially overcome by carefully selected prediction scheme and a number of techniques that accelerate the convergence of the iterative process in the form of restrictions on movement, load and move calculation.

units can be gears, frictionless (rolling) bearings, journal bearings with an elastic liner and rigid housing. For their calculations it is reasonable to use the methods of contact

However, there is also a large group of hydrodynamic friction pairs in which the general corps deformations make a significant contribution to changing the profile of the clearance. They are characterized by the presence of a continuous gradient of the deformation field, which is independent of the load location, the significant (compared with the contact and hydrodynamic tribounits) values of lubricating film thickness and values of the displacement of the friction surfaces, caused by the bending deformation of the housing, which are commensurate with them. These tribounits are called elasto-yielding (EY TU) or elastohydrodynamic. The most typical representative of the EY TU is a connecting rod bearing of a crank mechanism (crank) of engine vehicles. The desire of engine designers to maximally reduce the weight of movable elements of a crank reduces the stiffness of the bearing (crank crosshead), which makes the mode of EHD lubrication working for the connecting rod bearings. The above features - comparable with the clearance of dynamically changing elastic displacements and a continuous gradient of deformations - prevent the direct application of

methods of contact-hydrodynamic lubrication theory to the calculation of EY TU.

dependence of the instantaneous value of the lubricating film thickness *h zt* (

objective of the hydrodynamic lubrication theory and the theory of elasticity.

elastic displacements of the friction surface of a bearing *W zt* (

joint solution of equations of fluid flow and elastic deformation.

in the form of restrictions on movement, load and move calculation.

φ

the lubricating film *p* : *h zt* (

usually constructed using the FE method.

volume forces of rod inertia.

A mathematical model of EY TU differs from the "absolutely rigid" units model by the

are determined by structural rigidity of the bearing and by the hydrodynamic pressure in

thickness in the "absolutely rigid" bearing. To determine it the expression (6) is used. Thus, the determination of pressures in the lubricating film and HMCh of EY TU is the related

Modeling of EY TU, compared with "absolutely rigid" bearings is supplemented by an elastic subproblem the purpose of which is to determine the strain state of the friction surface of a crank crosshead under the influence of complex loads. The method of solving the elastic subproblem is chosen according to the accepted approximating model of an EY bearing. In today's solutions for EY TU the compliance and stiffness matrix of the bearing is

The other side of modeling the elastic subsystem is adequate description of the entire complex of loads, causing the elastic deformation of the bearing housing and the conditions of fixing of FE model. One must consider not only the hydrodynamic pressure, but also the

The known methods of solving the elastohydrodynamic lubrication problem can be classified as follows: direct methods or methods of successive approximations, in which the solutions of the hydrodynamic and elastic subtasks are performed separately, with the subsequent jointing of the results in the direct iterative process; and system, oriented for the

In solving the problem of elastohydrodynamic lubrication of a bearing with the help of a direct iterative method, the hydrodynamic and elastic subproblems at each step of time discretization are solved sequentially in an iterative cycle. The main disadvantage of direct methods for the calculation of EHD is their slow convergence and the associated timeconsumption. These difficulties are partially overcome by carefully selected prediction scheme and a number of techniques that accelerate the convergence of the iterative process

φφ

ϕ

, ,, *p*) , which, in their turn,

ϕ

ϕ

, ,, ,, ,,, *p*) = + *h zt W zt rig* ( ) ( *p*) . Where *h zt rig* (

, ,, *p*) on the

, , ) - film

hydrodynamics (elasto-hydrodynamic lubrication theory).

Among the systemic methods the Newton-Raphson method is considered one of the most sustainable and effective solutions for elastohydrodynamic problems. In the literature it is known as the Newton-Kantorovich method or Newton (MN). The algorithm for system solutions of elastohydrodynamic problem consists of three nested iteration loops: the inner loop of implementation by the Newton method of simultaneous solution of hydrodynamic and elastic subproblems; the average - the cycle of calculation of the cavitation zone and the boundary conditions; external - the cycle of calculation of the trajectory of the journal center. Algorithm for the numerical realization of MN is based on the finite-difference or finite element discretization of the linearized system of equations of EHD problem (Oh&Genka 1985; Bonneau 1995).

The application of the theory of elastohydrodynamic lubrication allows to predict lower mean-value as of the minimum lubricating film thickness as of the maximum hydrodynamic pressure. Thus, for the rod bearing of an engine, these changes may reach 35 ... 40%. The values of the maximum hydrodynamic pressure generated in the lubricating film of a EY bearing, are also smaller than for the "absolutely rigid" one. Reduction of the maximum hydrodynamic pressure is accompanied by an increase in the size of the bearing area. This fact, together with some increase in the clearance caused by the elastic deformation of the bearing, increases the flow of lubricating fluid through the ends of the bearing. Although the pressure gradient, on which the end consumption directly depends, is reduced. The difference in the instantaneous values of the mechanical flow between "absolutely rigid" and EY TU reaches 30%.

Calculation of the bearing, taking into account the elastohydrodynamic lubrication regime, not only improves the quality of design of friction units, but also clarifies the dynamic loading of mating parts such as the engine crank.

Thermoelastichydrodynamic (TEHD) regime of lubrication of journal bearings – is the mode of journal bearings, which are characterized by the influence on the magnitude of the clearance in tribounit thermoelastic deformations of a bearing and a journal, commensurate with the contribution of the displacement of the force nature.

Accounting for changes in the shape of thermoelastic friction surfaces of the journal and the bearing is possible in the case of inclusion in the resolution system of equations for the EY TU of energy equations and the relations of elasticity theory with the effects of temperature to determine the temperature fields and thermoelastic displacements caused by them. The sources of thermal fields can be either external to the tribounit, for example, a combustion chamber of an internal combustion engine for a connecting rod bearing, and internal lubricating film, in which heat generating is essential for the calculation of TEHD lubrication regime. Thus, the most complete version to solve the problem of TEHD lubrication of tribounits requires the joint consideration of problems of heat distribution in the journal, the bearing and lubricating film. The task is complicated by the fact that journal and the bearing are some idealized concepts. In reality they are rather complex shape parts (crankshaft, connecting rod, crankcase, etc.). Therefore, the methods for solving problems of TEHD lubricants are usually based on the method of FE, allowing to solve problems for bodies of complex geometric shapes the easiest.

Transient thermal fields are typical for a lubricating film of heavy-loaded bearings, which requires the simultaneous solution of equations of fluid dynamics, energy, and elasticity at each step of the calculation of trajectories. In this case, to solve all the subproblems the method of FE and schemes, similar to the systemic methods of solving the elastohydrodynamic lubrication problems, are used.

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of

loading parameters listed in the table 5 are obtained.

these parameters should not exceed 20% (Fig. 12).

Engine group

Highly

Medium

Low

crankshaft

automotive internal combustion engines

relative total lengths of areas per the cycle of loading *per*

Maximum specific load max *f* , MPa

Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 117

of comparison of calculated results with those obtained experimentally or during operation. On the basis of experimental studies and modern methods of calculation the criteria of performance of hydrodynamic tribounits are developed: the smallest allowable film thickness *per h* , maximum allowable hydrodynamic pressure *per p* , minimum film thickness reduced to the diameter of the journal, the maximum unit load max *f* . According to calculations of crankshaft engine bearings of several dimensions the maximum permissible

Assessment of performance of bearings is also done according to the calculated value of the

min inf *h* are less, and max sup*p* are bigger than acceptable values. Experience has shown that

α

Loading parameters

Bearing Type Crank Main Crank Main Crank Main Antifriction material: SB - stalebronzovye inserts coated with lead bronze, SA - staleallyuminievye inserts coated aluminum alloy AMO 1-20

SB SA SB SA SB SA SB SA SB SA SB SA

accelerated 55 49 41 37 2,0 1,2 448 397 336 305

accelerated 52 46 34 30,5 2,3 1,5 420 377 275 245

accelerated 45 39,5 31 27 2,5 1,9 367 326 255 225

Table 5. Maximum permissible loading parameters of sliding bearings of a crankshaft of

Fig. 12. The dependence of the hydromechanical characteristics on the rotation angle of

Reduced to the diameter of the journal minimum film thickness, mμ/100 mm

*<sup>h</sup>* and *per* α

*<sup>p</sup>* , where the values of

The largest hydrodynamic pressure in the lubricating film max sup*p* , MPa

However, practically important solution of TEHD lubrication of a tribounits is obtained for the steady thermal state, which is justified by the high inertia of the thermal fields in comparison with the rapidly changing power impacts. For calculations of tribounits with TEHD lubrication regime, this approach allows the use of a simple iterative scheme to implement solutions using the method of FE once on the preliminary stage of calculating the EY TU with regard to thermal deformations.

Nonautonomous journal bearings include bearings of multisupporting shafts of piston and rotary engines. Their distinctive feature consists in the interconnectedness of the processes occurring in various tribounits. A typical representative of nonautonomous heavy-loaded bearings is the root supports of the crankshaft of ICE. Main bearings of ICE are a part of a complex tribomechanical system, which also typically includes a crankshaft and a crankcase. Crankshaft journals are interconnected through a resilient connection - crankshaft ICE. Bushings of main bearings are installed in the holes of crankcase walls, and thus, main bearings are interconnected via a flexible design of the crankcase. Therefore, in the most general formulation to calculate the bearings it is necessary to solve a related EHD (or TEHD) problem for a system of "crankshaft - lubricating films - the crankcase". An additional feature of this system is the dependence of the loads acting on the indigenous support on the elastic properties of the crankshaft and crankcase.

In the first methods of calculation of non-autonomous main bearings of ICE a crankshaft model was used as the core spatial frame mounted on a linear-elastic mounts. The use of approximate methods for calculating the trajectories of main crankshaft journal bearings allowed to evaluate the influence of nonlinear properties of the lubricant film on the dynamics and the loading of the crankshaft. Simultaneously, we took into account the effect of necks (journals) deviation, as well as linear and bending stiffness of tribounits on the HMCh of main bearings (Zakharov, 1996a). However, to obtain the results significant estimates and approximate models of lubricating films of main bearings are used.

Currently, for the calculation of bearings the development of computer technology allows to use the exact solution of the Reynolds (4) and Elrod (1) equation for the bearing of finite length, to carry out the assessments of the impact of the crankcase and crankshaft construction supports, and of misalignments of bearings and shaft journals on HMCh. To solve such problems it is advisable to use an iterative algorithm that requires consistent calculation of loads acting on each of the bearings of the shaft and the calculation of the trajectories of its journals in the bearings.

Such calculations of the system of engine bearings allow to determine not only the optimal geometric parameters and position of sources for supplying lubricant to the main bearings, but also limiting in terms of supports performance tolerances concerning the position and shape of the friction surfaces of bearings and of the crankshaft journals.

The application of modern methods of calculation of main bearings allows to specify the values of loads acting on the crankcase and crankshaft, and their strength characteristics. But the task of elastohydrodynamic lubrication for a system "crankshaft - lubricating films the crankcase" in the most general setting is still not solved.

#### **5. Performance criteria**

The development of criteria for evaluating the performance of heavy-loaded hydrodynamic bearings of piston and rotary engines is an integral part of modern methods of calculation and design. At the same time, the reliability of methods themselves is largely dependent on the applied criteria (Zakharov, 1996b). Reliability of criteria is usually assessed on the base

However, practically important solution of TEHD lubrication of a tribounits is obtained for the steady thermal state, which is justified by the high inertia of the thermal fields in comparison with the rapidly changing power impacts. For calculations of tribounits with TEHD lubrication regime, this approach allows the use of a simple iterative scheme to implement solutions using the method of FE once on the preliminary stage of calculating the

Nonautonomous journal bearings include bearings of multisupporting shafts of piston and rotary engines. Their distinctive feature consists in the interconnectedness of the processes occurring in various tribounits. A typical representative of nonautonomous heavy-loaded bearings is the root supports of the crankshaft of ICE. Main bearings of ICE are a part of a complex tribomechanical system, which also typically includes a crankshaft and a crankcase. Crankshaft journals are interconnected through a resilient connection - crankshaft ICE. Bushings of main bearings are installed in the holes of crankcase walls, and thus, main bearings are interconnected via a flexible design of the crankcase. Therefore, in the most general formulation to calculate the bearings it is necessary to solve a related EHD (or TEHD) problem for a system of "crankshaft - lubricating films - the crankcase". An additional feature of this system is the dependence of the loads acting on the indigenous

In the first methods of calculation of non-autonomous main bearings of ICE a crankshaft model was used as the core spatial frame mounted on a linear-elastic mounts. The use of approximate methods for calculating the trajectories of main crankshaft journal bearings allowed to evaluate the influence of nonlinear properties of the lubricant film on the dynamics and the loading of the crankshaft. Simultaneously, we took into account the effect of necks (journals) deviation, as well as linear and bending stiffness of tribounits on the HMCh of main bearings (Zakharov, 1996a). However, to obtain the results significant

Currently, for the calculation of bearings the development of computer technology allows to use the exact solution of the Reynolds (4) and Elrod (1) equation for the bearing of finite length, to carry out the assessments of the impact of the crankcase and crankshaft construction supports, and of misalignments of bearings and shaft journals on HMCh. To solve such problems it is advisable to use an iterative algorithm that requires consistent calculation of loads acting on each of the bearings of the shaft and the calculation of the

Such calculations of the system of engine bearings allow to determine not only the optimal geometric parameters and position of sources for supplying lubricant to the main bearings, but also limiting in terms of supports performance tolerances concerning the position and

The application of modern methods of calculation of main bearings allows to specify the values of loads acting on the crankcase and crankshaft, and their strength characteristics. But the task of elastohydrodynamic lubrication for a system "crankshaft - lubricating films -

The development of criteria for evaluating the performance of heavy-loaded hydrodynamic bearings of piston and rotary engines is an integral part of modern methods of calculation and design. At the same time, the reliability of methods themselves is largely dependent on the applied criteria (Zakharov, 1996b). Reliability of criteria is usually assessed on the base

estimates and approximate models of lubricating films of main bearings are used.

shape of the friction surfaces of bearings and of the crankshaft journals.

the crankcase" in the most general setting is still not solved.

EY TU with regard to thermal deformations.

trajectories of its journals in the bearings.

**5. Performance criteria** 

support on the elastic properties of the crankshaft and crankcase.

of comparison of calculated results with those obtained experimentally or during operation. On the basis of experimental studies and modern methods of calculation the criteria of performance of hydrodynamic tribounits are developed: the smallest allowable film thickness *per h* , maximum allowable hydrodynamic pressure *per p* , minimum film thickness reduced to the diameter of the journal, the maximum unit load max *f* . According to calculations of crankshaft engine bearings of several dimensions the maximum permissible loading parameters listed in the table 5 are obtained.

Assessment of performance of bearings is also done according to the calculated value of the relative total lengths of areas per the cycle of loading *per* α*<sup>h</sup>* and *per* α *<sup>p</sup>* , where the values of min inf *h* are less, and max sup*p* are bigger than acceptable values. Experience has shown that these parameters should not exceed 20% (Fig. 12).


Table 5. Maximum permissible loading parameters of sliding bearings of a crankshaft of automotive internal combustion engines

Fig. 12. The dependence of the hydromechanical characteristics on the rotation angle of crankshaft

**4** 

Attila Csobán

*Hungary* 

**The Bearing Friction of Compound** 

**Planetary Gears in the Early Stage** 

*Budapest University of Technology and Economics* 

**Design for Cost Saving and Efficiency** 

The efficiency of planetary gearboxes mainly depends on the tooth- and bearing friction losses. This work shows the new mathematical model and the results of the calculations to compare the tooth and the bearing friction losses in order to determine the efficiency of different types of planetary gears and evaluate the influence of the construction on the bearing friction losses and through it on the efficiency of planetary gears. In order to economy of energy transportation it is very important to find the best gearbox construction

In transmission system of gas turbine powered ships, power stations, wind turbines or other large machines in industry heavy-duty gearboxes are used with high gear ratio, efficiency of which is one of the most important issues. During the design of such equipment the main goal is to find the best constructions fitting to the requirements of the given application and to reduce the friction losses generated in the gearboxes. These heavy-duty tooth gearboxes are often planetary gears being able to meet the following requirements declared against the

There are some types of planetary gears which ensure high gear ratio, while their power flow is unbeneficial, because a large part of the rolling power (the idle power) circulate inside the planetary gearbox decreasing the efficiency. In the simple planetary gears there is no idle power circulation. Therefore heavy-duty planetary drives are set together of simple planetary gears in order to transmit megawatts or even more power, while they must be

The two- and three-stage planetary gears consisting of simple planetary gears are able to

for a given application and to reach the highest efficiency.

• High specific load carrying capacity

• Small mass/power ratio in some application

meet the requirements mentioned above [Fig. 1(a)–1(d).].

**1. Introduction** 

drive systems:

• High gear ratio • Small size

• High efficiency.

compact and efficient.

**2. Planetary gearbox types** 

#### **6. Conclusion**

Thus, the methodology of calculating the dynamics and HMCh of heavy-loaded tribounits lubricated by structurally heterogeneous and non-Newtonian fluids, consists of three interrelated tasks: defining the field of hydrodynamic pressures in a thin lubricating film that separates the friction surfaces of a journal and a bearing with an arbitrary law of their relative motion; calculation of the trajectory of the center of the journal; the calculation of the temperature of the lubricating film.

Mathematical models used in the calculation must reflect the nature of the live load, lubricant properties, geometry and elastic properties of a construction. The choice of models is built on the working conditions of tribounits in general and the properties of the lubricant. This will allow on the early stages of the design of tribounits to evaluate their bearing capacity, thermal stress and longevity.

#### **7. Acknowledgment**

The presented work is executed with support of the Federal target program «Scientific and scientifically pedagogical the personnel of innovative Russia» for 2009-2013.

#### **8. References**


## **The Bearing Friction of Compound Planetary Gears in the Early Stage Design for Cost Saving and Efficiency**

Attila Csobán *Budapest University of Technology and Economics Hungary* 

#### **1. Introduction**

118 Tribology - Lubricants and Lubrication

Thus, the methodology of calculating the dynamics and HMCh of heavy-loaded tribounits lubricated by structurally heterogeneous and non-Newtonian fluids, consists of three interrelated tasks: defining the field of hydrodynamic pressures in a thin lubricating film that separates the friction surfaces of a journal and a bearing with an arbitrary law of their relative motion; calculation of the trajectory of the center of the journal; the calculation of the

Mathematical models used in the calculation must reflect the nature of the live load, lubricant properties, geometry and elastic properties of a construction. The choice of models is built on the working conditions of tribounits in general and the properties of the lubricant. This will allow on the early stages of the design of tribounits to evaluate their bearing

The presented work is executed with support of the Federal target program «Scientific and

Elrod, H. (1981). A Cavitation Algorithm. *Journal of lubrication Technology*, Vol.103, No.3,

Prokopiev, V., Rozhdestvensky, Y. et al. (2010). The Dynamics and Lubrication of Tribounits

Prokopiev, V. & Karavayev, V. (2003). The Thermohydrodynamic Lubrication Problem of

Gecim, B. (1990). Non-Newtonian Effect of Multigrade Oils on Journal Bearing Perfomance,

Mukhortov, I., Zadorozhnaya, E., Levanov, I. et al. (2010). Improved Model of the

Oh, K. & Genka, P. (1985). The Elastohydrodynamic Solution of Journal Bearings Under

Bonneau, D. (1995). EHD Analysis, Including Structural Inertia Effect and Mass-Conserving Cavitation Model, *Journal of Tribology*, Vol. 117, (July 1995), pp. 540-547 Zakharov, S. (1996). Calculation of unsteady-loaded bearings, taking into account the

Zakharov, S. (1996). Tribological Evaluation Criteria of Efficiency of Sliding Bearings of

of Piston and Rotary Machines: Ponograph the Part 1, *South Ural State University*,

Heavy-loaded Journal Bearings by Non-Newtonian Fluids, *Herald of the SUSU*. *A* 

Rheological Properties of the Boundary layer of lubricant, *Friction and lubrication of* 

deviation of the shaft and the regime of mixed lubrication, *Friction and Wear,* Vol.17,

Crankshafts of Internal Combustion Engines, *Friction and Wear*, Vol.17, No 5, pp.

scientifically pedagogical the personnel of innovative Russia» for 2009-2013.

(July 1981), pp. 354-359, ISSN 0201-8160

ISBN 978-5-696-04036-3, Chelyabinsk

*series of "Engineering"*, Vol.3, No 1(17), pp. 55-66 Whilkinson, U. (1964). Non-Newtonian fluids, *Moscow: Mir* 

*Tribology Transaction*, Vol. 3, No 3, pp. 384-394.

Dynamic Loading, *Journal of Tribology*, No 3, pp. 70-76.

*machines and mechanisms*, No 5, pp. 8-19

No 4, pp. 425-434, ISSN 0202-4977

606 – 615, ISSN 0202-4977

**6. Conclusion** 

temperature of the lubricating film.

capacity, thermal stress and longevity.

**7. Acknowledgment** 

**8. References** 

The efficiency of planetary gearboxes mainly depends on the tooth- and bearing friction losses. This work shows the new mathematical model and the results of the calculations to compare the tooth and the bearing friction losses in order to determine the efficiency of different types of planetary gears and evaluate the influence of the construction on the bearing friction losses and through it on the efficiency of planetary gears. In order to economy of energy transportation it is very important to find the best gearbox construction for a given application and to reach the highest efficiency.

In transmission system of gas turbine powered ships, power stations, wind turbines or other large machines in industry heavy-duty gearboxes are used with high gear ratio, efficiency of which is one of the most important issues. During the design of such equipment the main goal is to find the best constructions fitting to the requirements of the given application and to reduce the friction losses generated in the gearboxes. These heavy-duty tooth gearboxes are often planetary gears being able to meet the following requirements declared against the drive systems:


There are some types of planetary gears which ensure high gear ratio, while their power flow is unbeneficial, because a large part of the rolling power (the idle power) circulate inside the planetary gearbox decreasing the efficiency. In the simple planetary gears there is no idle power circulation. Therefore heavy-duty planetary drives are set together of simple planetary gears in order to transmit megawatts or even more power, while they must be compact and efficient.

#### **2. Planetary gearbox types**

The two- and three-stage planetary gears consisting of simple planetary gears are able to meet the requirements mentioned above [Fig. 1(a)–1(d).].

The Bearing Friction of Compound Planetary Gears

**3. Friction loss model of roller bearings** 

bearing friction.

catalog [SKF 2005].

equivalent stress

can be searched in the following form:

be calculated using the following formula:

stresses, then the bending stresses are considered.

σ

in the Early Stage Design for Cost Saving and Efficiency 121

It is important to find the parameters (such as inner gear ratio, optimal power flow) of a compound planetary gear drive which result its highest performance for a given application. The power flow and the power distribution between the stages of a compound gearbox is also a function of the power losses generated mainly by the friction of mashing teeth and the

This is why it is beneficial, when, during the design of a planetary gear beside the tooth friction loss also the friction of rolling bearings is taken into consideration even in the early stage of design. In this work a new method is suggested for calculate the rolling bearing

In this model first the torque and applied loads (loading forces and, if possible, bending moments) originated from the tooth forces between the mating teeth have to be determined. Thereafter the average diameter of the bearing *dm* can be calculated as a function of the applied load and the prescribed bearing lifetime. Knowing the average diameter *dm*, the friction loss of bearings can be counted using the methods suggested by the bearing

For determining the functions between the bearing average diameter and between the basic dynamic, static load, inner and outer diameter [Fig. 2-6.], the data were collected from SKF

The functions between the bearing parameters (inner diameter *db,* dynamic basic load *C*) and the average diameters *dm* being necessary for calculation of the friction moment and the load

*<sup>d</sup> Y cd* = ⋅ *<sup>m</sup>*

The equations of the diagrams [Fig. 2-6.] give the values of *c* and *d* for the inner diameter of

diameter of the bearing for central gears (sun gear, ring gear) necessary to carry the load can

2;4 <sup>3</sup> 16

⋅

*c* τ π

*<sup>m</sup>* of planet gear pins, the bearing inner diameter *db* necessary to carry the

max

*h*

*M*

⋅

*c* σ π

*M*

*<sup>d</sup> <sup>m</sup>*

<sup>⋅</sup> <sup>=</sup> 

Calculating the tangential components of the tooth forces the applied radial loads of the planet gear shafts *Fr* can be determined (which are the resultant forces of the two tangential components *Ft2* and *Ft4*). The shafts of the planet gears are sheared and bended by the heavy radial forces, this is why, in this analysis, at the calculation of shaft diameter, once the shear

Calculating the maximal bending moment *Mhmax* of the planet gear shafts, and the allowable

applied load of the planet gear shaft and the average bearing diameter *dm3* can be calculated:

3 32

*<sup>d</sup> <sup>m</sup>*

<sup>⋅</sup> <sup>=</sup> 

(1)

τ*m,* σ

(2)

(3)

*<sup>m</sup>*) the mean

friction losses without knowing the exact sizes and types of the bearings.

manufacturers based on the Palmgren model [SKF 1989].

the bearings *db* and for the basic dynamic loads *C* of the bearings.

Knowing the torque *M24* and the strength of the materials of the shafts (

2;4

3

*d d*

*m*

( )

*d d*

*m*

( )

Varying the inner gear ratio (the ratio of tooth number of the ring gear and of the sun gear) of each simple planetary gear stage KB the performance of the whole combined planetary gear can be changed and tailored to the requirements.

There are special types of combined planetary gears containing simple KB units (differential planetary gears), which can divide the applied power between the planetary stages thereby increasing the specific load carrying capacity and efficiency of the whole planetary drives [Fig. 1(b)-1.(d)]. Proper connections between the elements of the stages in these differential planetary gears do not result idle power circulation.

Fig. 1. (a) Gearbox KB+KB; (b). Planetary gear PKG; (c). Planetary gear PV; (d). Planetary gear GPV

The efficiency of planetary gears depends on the various sources of friction losses developed in the gearboxes. The main source of energy loss is the tooth friction of meshing gears, which mainly depends on the arrangements of the gears and the power flow inside the planetary gear drives. The tooth friction loss is influenced by the applied load, the entraining speed and the geometry of gears, the roughness of mating tooth surfaces and the viscosity of lubricant. Designers of planetary gear drives can modify the geometry of tooth profile in order to lower the tooth friction loss and to reach a higher efficiency [Csobán, 2009].

#### **3. Friction loss model of roller bearings**

120 Tribology - Lubricants and Lubrication

Varying the inner gear ratio (the ratio of tooth number of the ring gear and of the sun gear) of each simple planetary gear stage KB the performance of the whole combined planetary

There are special types of combined planetary gears containing simple KB units (differential planetary gears), which can divide the applied power between the planetary stages thereby increasing the specific load carrying capacity and efficiency of the whole planetary drives [Fig. 1(b)-1.(d)]. Proper connections between the elements of the stages in these differential

Fig. 1. (a) Gearbox KB+KB; (b). Planetary gear PKG; (c). Planetary gear PV; (d). Planetary

order to lower the tooth friction loss and to reach a higher efficiency [Csobán, 2009].

The efficiency of planetary gears depends on the various sources of friction losses developed in the gearboxes. The main source of energy loss is the tooth friction of meshing gears, which mainly depends on the arrangements of the gears and the power flow inside the planetary gear drives. The tooth friction loss is influenced by the applied load, the entraining speed and the geometry of gears, the roughness of mating tooth surfaces and the viscosity of lubricant. Designers of planetary gear drives can modify the geometry of tooth profile in

(a) (b)

 (c) (d)

gear GPV

gear can be changed and tailored to the requirements.

planetary gears do not result idle power circulation.

It is important to find the parameters (such as inner gear ratio, optimal power flow) of a compound planetary gear drive which result its highest performance for a given application. The power flow and the power distribution between the stages of a compound gearbox is also a function of the power losses generated mainly by the friction of mashing teeth and the bearing friction.

This is why it is beneficial, when, during the design of a planetary gear beside the tooth friction loss also the friction of rolling bearings is taken into consideration even in the early stage of design. In this work a new method is suggested for calculate the rolling bearing friction losses without knowing the exact sizes and types of the bearings.

In this model first the torque and applied loads (loading forces and, if possible, bending moments) originated from the tooth forces between the mating teeth have to be determined. Thereafter the average diameter of the bearing *dm* can be calculated as a function of the applied load and the prescribed bearing lifetime. Knowing the average diameter *dm*, the friction loss of bearings can be counted using the methods suggested by the bearing manufacturers based on the Palmgren model [SKF 1989].

For determining the functions between the bearing average diameter and between the basic dynamic, static load, inner and outer diameter [Fig. 2-6.], the data were collected from SKF catalog [SKF 2005].

The functions between the bearing parameters (inner diameter *db,* dynamic basic load *C*) and the average diameters *dm* being necessary for calculation of the friction moment and the load can be searched in the following form:

$$Y = \tilde{\mathfrak{c}} \cdot \tilde{d}\_m^{\vec{d}} \tag{1}$$

The equations of the diagrams [Fig. 2-6.] give the values of *c* and *d* for the inner diameter of the bearings *db* and for the basic dynamic loads *C* of the bearings.

Knowing the torque *M24* and the strength of the materials of the shafts (τ*m,* σ*<sup>m</sup>*) the mean diameter of the bearing for central gears (sun gear, ring gear) necessary to carry the load can be calculated using the following formula:

$$d\_{m\_{2,4}}(d) = \sqrt[d]{\frac{16 \cdot M\_{2,4}}{\tau\_m \cdot \pi}}\tag{2}$$

Calculating the tangential components of the tooth forces the applied radial loads of the planet gear shafts *Fr* can be determined (which are the resultant forces of the two tangential components *Ft2* and *Ft4*). The shafts of the planet gears are sheared and bended by the heavy radial forces, this is why, in this analysis, at the calculation of shaft diameter, once the shear stresses, then the bending stresses are considered.

Calculating the maximal bending moment *Mhmax* of the planet gear shafts, and the allowable equivalent stress σ*<sup>m</sup>* of planet gear pins, the bearing inner diameter *db* necessary to carry the applied load of the planet gear shaft and the average bearing diameter *dm3* can be calculated:

$$d\_{m\_3}(d) = \sqrt[3]{\frac{32 \cdot M\_{h\_{\text{max}}}}{\sigma\_m \cdot \pi}}\tag{3}$$

The Bearing Friction of Compound Planetary Gears

d - dm

0 200 400 600 800 1000

C - dm

0 200 400 600 800 1000

y = 69,121x1,6675 R2 = 0,9808

y = 0,4167x1,0966 R2 = 0,9993

bearings [Fig. 3(a)-3(d)].

0

0

2000000

4000000

**C** [N]

6000000

8000000

400

800

**d** [mm] 1200

in the Early Stage Design for Cost Saving and Efficiency 123

The functions between the bearing geometry and load carrying capacity for cylindrical roller

D - dm

0 200 400 600 800 1000

C0 - dm

y = 22,552x1,9273 R2 = 0,9863

0 200 400 600 800 1000

dm [mm]

dm [mm]

y = 1,7114x0,9481 R2 = 0,9997

dm [mm]

dm [mm]

Fig. 3. (a) The average inner diameter of the cylindrical roller bearing as a function of its average diameter. (b). The average outer diameter of cylindrical roller bearing as a function of the average diameter. (c). The average basic dynamic load of the cylindrical roller bearing

as a function of its average diameter. (d). The average static load of different types of

(c) (d)

cylindrical roller bearing as a function of the average diameter

0

4000000

8000000

**C0** [N] 12000000

16000000

(a) (b)

0

400

800

**D** [mm] 1200

The functions between the bearing geometry and load carrying capacity for deep groove ball bearings [Fig. 2(a)-2(d)]. The points are the average data of the bearings taken from SKF Catalog [SKF 2005] and the continuous lines are the developed functions between the parameters.

Fig. 2. (a) The average inner diameter of the deep groove ball bearing as a function of its average diameter. (b). The average outer diameter of deep groove ball bearing as a function of the average diameter. (c). The average basic dynamic load of deep groove ball bearing as a function of the average diameter. (d). The average static load of deep groove ball bearing as a function of the average diameter

The functions between the bearing geometry and load carrying capacity for deep groove ball bearings [Fig. 2(a)-2(d)]. The points are the average data of the bearings taken from SKF Catalog [SKF 2005] and the continuous lines are the developed functions between the

D - dm

0 200 400 600 800 1000 1200 1400 1600

C0 - dm

0 200 400 600 800 1000 1200 1400 1600

y = 12,603x1,7564 R2 = 0,9912

dm [mm]

dm [mm]

y = 1,7374x0,9364 R2 = 0,9998

dm [mm]

dm [mm]

(c) (d)

Fig. 2. (a) The average inner diameter of the deep groove ball bearing as a function of its average diameter. (b). The average outer diameter of deep groove ball bearing as a function of the average diameter. (c). The average basic dynamic load of deep groove ball bearing as a function of the average diameter. (d). The average static load of deep groove ball bearing

0

1000000

2000000

**C0** [N] 3000000

4000000

(a) (b)

0

400

800

**D** [mm] 1200

1600

parameters.

0

0

400000

800000

**C** [N]

1200000

400

800

**d** [m m ] 1200

1600

d - dm

0 200 400 600 800 1000 1200 1400 1600

C - dm

0 200 400 600 800 1000 1200 1400 1600

as a function of the average diameter

y = 109,09x1,3236 R2 = 0,9682

y = 0,3984x1,1179 R2 = 0,9996

The functions between the bearing geometry and load carrying capacity for cylindrical roller bearings [Fig. 3(a)-3(d)].

Fig. 3. (a) The average inner diameter of the cylindrical roller bearing as a function of its average diameter. (b). The average outer diameter of cylindrical roller bearing as a function of the average diameter. (c). The average basic dynamic load of the cylindrical roller bearing as a function of its average diameter. (d). The average static load of different types of cylindrical roller bearing as a function of the average diameter

The Bearing Friction of Compound Planetary Gears

d - dm

0 500 1000 1500 2000

C - dm

0 500 1000 1500 2000

y = 188,21x1,5827 R2 = 0,987

y = 0,4331x1,0936 R2 = 0,9995

bearings [Fig. 5(a)-5(d)].

0

0

function of its average diameter

7000000

14000000

**C** [N]

21000000

28000000

400

800

**d** [m m ] 1200

1600

2000

in the Early Stage Design for Cost Saving and Efficiency 125

The functions between the bearing geometry and load carrying capacity for spherical roller

D - dm

0 500 1000 1500 2000

C0 - dm

0 500 1000 1500 2000

y = 45,547x1,9064 R2 = 0,9965

y = 1,72x0,9448 R2 = 0,9997

dm [mm]

dm [mm]

dm [mm]

dm [mm]

Fig. 5. (a) The average inner diameter of the spherical roller bearing as a function of its average diameter. (b). The average outer diameter of spherical roller bearing as a function of the average diameter. (c). The average basic dynamic load of spherical roller bearing as a function of its average diameter. (d). The average static load of spherical roller bearing as a

(c) (d)

0

14000000

28000000

**C0** [N] 42000000

56000000

(a) (b)

0

400

800

1200

**D** [m m ] 1600

2000

The functions between the bearing geometry and load carrying capacity for full complement cylindrical roller bearings [Fig. 4(a)-4(d)].

Fig. 4. (a) The average inner diameter of the full complement cylindrical roller bearing as a function of its average diameter. (b). The average outer diameter of the full complement cylindrical roller bearing as a function of the average diameter. (c). The average basic dynamic load of the full complement cylindrical roller bearing as a function of its average diameter. (d). The average static load of different types of full complement cylindrical roller bearing as a function of the average diameter

The functions between the bearing geometry and load carrying capacity for full complement

D - dm

0 200 400 600 800 1000 1200

C0 - dm

0 200 400 600 800 1000 1200

y = 164,13x1,6125 R2 = 0,9845

y = 1,6961x0,9399 R2 = 0,9993

dm [mm]

dm [mm]

dm [mm]

dm [mm]

Fig. 4. (a) The average inner diameter of the full complement cylindrical roller bearing as a function of its average diameter. (b). The average outer diameter of the full complement cylindrical roller bearing as a function of the average diameter. (c). The average basic dynamic load of the full complement cylindrical roller bearing as a function of its average diameter. (d). The average static load of different types of full complement cylindrical roller

(c) (d)

(a) (b)

0

0

4000000

8000000

**C0** [N] 12000000

16000000

400

800

**D** [mm] 1200

cylindrical roller bearings [Fig. 4(a)-4(d)].

y = 0,4595x1,0951 R2 = 0,9987

0

0

2000000

4000000

**C** [N]

6000000

8000000

400

800

**d** [mm] 1200

d - dm

0 200 400 600 800 1000 1200

C - dm

0 200 400 600 800 1000 1200

bearing as a function of the average diameter

y = 444,04x1,3465 R2 = 0,9621

The functions between the bearing geometry and load carrying capacity for spherical roller bearings [Fig. 5(a)-5(d)].

Fig. 5. (a) The average inner diameter of the spherical roller bearing as a function of its average diameter. (b). The average outer diameter of spherical roller bearing as a function of the average diameter. (c). The average basic dynamic load of spherical roller bearing as a function of its average diameter. (d). The average static load of spherical roller bearing as a function of its average diameter

The Bearing Friction of Compound Planetary Gears

τ

roller) bearing diameter (*dm res*).

lubricant and the bearing sizes.

When

and when

equivalent stress

in the Early Stage Design for Cost Saving and Efficiency 127

The *V* shear load of the planet gear shaft is equal with the applied load *Fr* divided by the number of sheared areas *A* of the shaft. Knowing the *V* shear load and the allowable

applied load of the planet gear shaft and the average bearing diameter *dm3* can be calculated:

<sup>3</sup> ( ) *<sup>d</sup> <sup>m</sup>*

<sup>10</sup> ( )

*<sup>a</sup> d L*

*d dd dL dd dL dd m m res mh m mh m* <sup>⎡</sup> <sup>⎤</sup> = +⋅ − + − <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

The sun gears and the ring gears are well balanced by radial components of tooth forces; the friction losses of their bearings are not depending on the applied load. The energy losses of these bearings are determined by the entraining speed of the bearings, the viscosity of

The calculation of the component of friction torque *M0* being independent of the bearing

0 0

*n*

⋅ <

2000

( ) 7 3 2/3

7 3

10 *<sup>m</sup>*

160 10 *<sup>m</sup>*

−

ν<sup>−</sup>

*M f nd*

0 0

1 11

Using the average bearing diameters the friction torques of the bearings can be determined:

*M f d*

At bearings of planet gears the component of friction torques *M1* depending on the bearing

2000

= ⋅⋅⋅ ⋅

16

<sup>⋅</sup> <sup>⋅</sup> <sup>=</sup> 

The average diameters of bearings necessary to reach the prescribed lifetime *L1h* was determined using the SKF modified lifetime equation [SKF 2005] (*C* is the basic dynamic load, *Fr* is the radial bearing load and *a1* is the bearing life correction factor) as follows:

*c* τ π

6 60

⋅ ⋅ <sup>⋅</sup> <sup>⋅</sup> <sup>=</sup>

From the two calculated average diameters of bearings (*dm(d)* and *dm(Lh)*) the larger ones have to be chosen. This biggest average diameter can be called resultant average (ball or

*c*

( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ) <sup>1</sup>

*<sup>p</sup> <sup>h</sup> <sup>r</sup> <sup>d</sup>*

*L n <sup>F</sup>*

*V*

⋅

3

*d d*

*m*

*m h*

2

load can be performed using the following equations [SKF 1989].

ν

*n*

ν

loads was calculated using the following simple equation [SKF 1989]:

⋅ ≥

**3.1 Calculating the friction losses and efficiency of roller bearings** 

*<sup>m</sup>* of planet gear pins, the bearing inner diameter *db* necessary to carry the

(4)

(5)

(6)

(7)

<sup>=</sup> ⋅ ⋅⋅ (8)

*a b M f* = ⋅ ⋅ *P dm* (9)

( ) 0 1 ( ) ( ) ... *res res res Md Md Md vm m m* <sup>=</sup> + + (10)

The functions between the bearing geometry and load carrying capacity for of CARB toroidal roller bearings [Fig. 6(a)-6(d)].

Fig. 6. (a) The average inner diameter of CARB toroidal roller bearing as a function of its average diameter. (b). The average outer diameter of CARB toroidal roller bearing as a function of the average diameter. (c). The average basic dynamic load of CARB toroidal roller bearing as a function of its average diameter. (d). The static load of CARB toroidal roller bearing as a function of the average diameter

The functions between the bearing geometry and load carrying capacity for of CARB

D - dm

0 200 400 600 800 1000 1200 1400

C0 - dm

0 200 400 600 800 1000 1200 1400

y = 59,869x1,8637 R2 = 0,9942

y = 1,4795x0,9674 R2 = 0,9997

dm [mm]

dm [mm]

toroidal roller bearings [Fig. 6(a)-6(d)].

0

0

5000000

10000000

**C** [N]

15000000

20000000

400

**d** [m m ] 800

1200

d - dm

0 200 400 600 800 1000 1200 1400

C - dm

0 200 400 600 800 1000 1200 1400

roller bearing as a function of the average diameter

y = 135,35x1,6435 R2 = 0,9915

y = 0,5691x1,0529 R2 = 0,9993

(a) (b)

dm [mm]

dm [mm]

(c) (d)

Fig. 6. (a) The average inner diameter of CARB toroidal roller bearing as a function of its average diameter. (b). The average outer diameter of CARB toroidal roller bearing as a function of the average diameter. (c). The average basic dynamic load of CARB toroidal roller bearing as a function of its average diameter. (d). The static load of CARB toroidal

0

10000000

20000000

**C0** [N] 30000000

40000000

0

400

800

**D** [m m ] 1200

1600

The *V* shear load of the planet gear shaft is equal with the applied load *Fr* divided by the number of sheared areas *A* of the shaft. Knowing the *V* shear load and the allowable equivalent stress τ*<sup>m</sup>* of planet gear pins, the bearing inner diameter *db* necessary to carry the applied load of the planet gear shaft and the average bearing diameter *dm3* can be calculated:

$$d\_{m\_3}(d) = \sqrt[d]{\frac{16 \cdot V}{3 \cdot \tau\_m \cdot \pi}}\tag{4}$$

The average diameters of bearings necessary to reach the prescribed lifetime *L1h* was determined using the SKF modified lifetime equation [SKF 2005] (*C* is the basic dynamic load, *Fr* is the radial bearing load and *a1* is the bearing life correction factor) as follows:

$$d\_m(L\_h) = \sqrt[3]{\frac{\frac{L\_h \cdot \text{GO} \cdot m}{10^6 \cdot a} \cdot F\_r}{\tilde{c}}} \tag{5}$$

From the two calculated average diameters of bearings (*dm(d)* and *dm(Lh)*) the larger ones have to be chosen. This biggest average diameter can be called resultant average (ball or roller) bearing diameter (*dm res*).

$$d\_{m\_{ms}} = \left[ d\_m \left( d \right) + \frac{1}{2} \cdot \left( \left| \left( d\_m \left( L\_h \right) - d\_m \left( d \right) \right) \right| + \left( d\_m \left( L\_h \right) - d\_m \left( d \right) \right) \right) \right] \tag{6}$$

#### **3.1 Calculating the friction losses and efficiency of roller bearings**

The sun gears and the ring gears are well balanced by radial components of tooth forces; the friction losses of their bearings are not depending on the applied load. The energy losses of these bearings are determined by the entraining speed of the bearings, the viscosity of lubricant and the bearing sizes.

The calculation of the component of friction torque *M0* being independent of the bearing load can be performed using the following equations [SKF 1989]. When

$$\begin{aligned} \nu \cdot n &\ge 2000\\ M\_0 &= 10^{-7} \cdot f\_0 \cdot \left(\nu \cdot n\right)^{2/3} \cdot d\_m^3 \end{aligned} \tag{7}$$

and when

$$\begin{aligned} \nu \cdot n &< 2000\\ M\_0 &= 160 \cdot 10^{-7} \cdot f\_0 \cdot d\_m^3 \end{aligned} \tag{8}$$

At bearings of planet gears the component of friction torques *M1* depending on the bearing loads was calculated using the following simple equation [SKF 1989]:

$$M\_1 = f\_1 \cdot P\_1^a \cdot d\_m^b \tag{9}$$

Using the average bearing diameters the friction torques of the bearings can be determined:

$$M\_{\upsilon} \left( d\_{m\_{ms}} \right) = M\_0 \left( d\_{m\_{ms}} \right) + M\_1 \left( d\_{m\_{ms}} \right) + \dots \tag{10}$$

The Bearing Friction of Compound Planetary Gears

Table 1. Parameters for the bearing selection

Fig. 7. The bearing selecting and efficiency calculation algorithm

in the Early Stage Design for Cost Saving and Efficiency 129

*Deep groove ball bearing* c <sup>d</sup>

*Cylindrical roller bearings* c d

*Spherical roller bearings* c d

*CARB® toroidal roller bearings* c d

*Full complement cylindrical roller bearings* c d

d – dm 0,3984 1,1179 D – dm 1,7374 0,9364 C – dm 109,09 1,3236 C0 – dm 12,603 1,7564

d – dm 0,4167 1,0966 D – dm 1,7114 0,9481 C – dm 69,121 1,6675 C0 – dm 22,552 1,9273

d – dm 0,4595 1,0951 D – dm 1,6961 0,9399 C – dm 444,04 1,3465 C0 – dm 164,13 1,6125

d – dm 0,4331 1,0936 D – dm 1,72 0,9448 C – dm 188,21 1,5827 C0 – dm 45,547 1,9064

d – dm 0,5691 1,0529 D – dm 1,4795 0,9674 C – dm 135,35 1,6435 C0 – dm 59,869 1,8637

**Bearing Types/function** 

Knowing the friction torques of the sun gear its bearing efficiency can be calculated using the following equation:

$$\log\_{2\_{\text{Roving}}} = \frac{\left\lfloor M\_2 - M\_v(d\_{m\_m}) \right\rfloor \cdot o\_2}{M\_2 \cdot o\_2} = 1 - \frac{M\_v(d\_{m\_m})}{M\_2} \tag{11}$$

The bearing efficiency of planet gears can be determined with the following equation:

$$\eta\_{3\_{\text{Rwing}}} = \frac{\left\lfloor M\_3 - M\_v(d\_{m\_m}) \right\rfloor \cdot o\_{3\_{\text{\%}}}}{M\_3 \cdot o\_{3\_{\text{\%}}}} = 1 - \frac{M\_v(d\_{m\_m})}{M\_3} \tag{12}$$

The power loss generated only by the bearings in the gearbox can be calculated as (the rolling efficiency of a simple stage and the gearbox efficiency is a function of only the bearing efficiencies):

$$\begin{aligned} \boldsymbol{\eta}\_{\mathcal{S}} &= \boldsymbol{\eta}\_{\mathcal{Z}\_{\text{Bearing}}} \cdot \boldsymbol{\eta}\_{\mathcal{S}\_{\text{Ruring}}} \to \boldsymbol{\eta}\_{\text{Garkbox}\_{\text{Raring}}}\\ \boldsymbol{\upsilon}\_{\text{Bearing}} &= \boldsymbol{P}\_{\text{in}} \cdot \left(\mathbf{1} - \boldsymbol{\eta}\_{\text{Garkbox}\_{\text{Ruring}}}\right) \end{aligned} \tag{13}$$

The power loss generated by the tooth friction can be calculated with the following equations (the rolling efficiency of a simple stage and the gearbox efficiency is a function of only the tooth efficiencies):

$$\begin{aligned} \boldsymbol{\eta}\_{\mathcal{S}} &= \boldsymbol{\eta}\_{z\_{29}} \cdot \boldsymbol{\eta}\_{z\_{34}} \to \boldsymbol{\eta}\_{\text{Gearbox}\_{\text{Toeth}}} \\ \boldsymbol{\upsilon}\_{\text{Toorth}} &= \boldsymbol{P}\_{\text{in}} \cdot \left( \mathbf{1} - \boldsymbol{\eta}\_{\text{Gearbox}\_{\text{Toch}}} \right) \end{aligned} \tag{14}$$

The rolling efficiency of a simple planetary gear stage can be calculated as:

$$
\eta\_{\mathcal{S}} = \eta\_{z\_{23}} \cdot \eta\_{z\_{94}} \cdot \eta\_{2\_{\text{Bearing}}} \cdot \eta\_{3\_{\text{Raring}}} \to \eta\_{\text{Cartan}} \tag{15}
$$

The total power loss generated in the planetary gear drive as a function of the gearbox efficiency:

$$
\Sigma \upsilon = P\_{in} \cdot \left(1 - \eta\_{\text{Gearhux}}\right) \tag{16}
$$

The power loss ratios show the dominant power loss component. The tooth power loss ratio is the tooth power loss component divided by the total power loss:

$$\frac{\sigma\_{\text{Toolt}}}{\Sigma v} \tag{17}$$

The bearing loss ratio is the power loss generated by the bearing friction divided by the total power loss:

$$\frac{\sigma\_{\text{Bearing}}}{\Sigma v} \tag{18}$$

The bearing selecting and efficiency calculation algorithm can be seen in figure 10.

Knowing the friction torques of the sun gear its bearing efficiency can be calculated using

2 2

ω

The bearing efficiency of planet gears can be determined with the following equation:

3 3

ω

1

*Bearing in Gearbox v P*

 η

=⋅→

*g Gearbox*

2 3

ηηη

The rolling efficiency of a simple planetary gear stage can be calculated as:

ηηη

is the tooth power loss component divided by the total power loss:

η*<sup>g</sup>* =

ηη

*g*

The power loss generated only by the bearings in the gearbox can be calculated as (the rolling efficiency of a simple stage and the gearbox efficiency is a function of only the

2 2 2 ( ) ( ) <sup>1</sup> *res res*

3 3 3 ( ) ( ) <sup>1</sup> *res res*

*vm g v m*

⎡ ⎤ − ⋅ ⎣ ⎦ <sup>=</sup> = − <sup>⋅</sup> (11)

⎡ ⎤ − ⋅ ⎣ ⎦ <sup>=</sup> <sup>=</sup> <sup>−</sup> <sup>⋅</sup> (12)

= ⋅− (13)

= ⋅− . (14)

) (16)

(17)

(18)

*M Md v m M d v m M M* ω

*M Md M d M M* ω

( )

 η

*Bearing*

*Tooth Tooth*

 η*z z Gearbox* ⋅⋅ ⋅ → (15)

*Bearing Bearing Bearing*

( ) 23 34 1

η

 η

 η

η

The power loss generated by the tooth friction can be calculated with the following equations (the rolling efficiency of a simple stage and the gearbox efficiency is a function of

> *g z z Gearbox Tooth in Gearbox v P*

23 34 2 3 *Bearing Bearing*

The total power loss generated in the planetary gear drive as a function of the gearbox

Σ= ⋅ − *v Pin Gearbox* (1 η

The power loss ratios show the dominant power loss component. The tooth power loss ratio

*Tooth v* Σ*v*

The bearing loss ratio is the power loss generated by the bearing friction divided by the total

*Bearing v* Σ*v*

The bearing selecting and efficiency calculation algorithm can be seen in figure 10.

=⋅→

the following equation:

bearing efficiencies):

only the tooth efficiencies):

efficiency:

power loss:

2

3

η

*Bearing*

η

*Bearing*


Table 1. Parameters for the bearing selection

Fig. 7. The bearing selecting and efficiency calculation algorithm

The Bearing Friction of Compound Planetary Gears

*GPV*

η

The efficiency of planetary gear GPV:

**5. Results of calculations** 

σF [MPa]

ηM [mPas]

in the Early Stage Design for Cost Saving and Efficiency 131

( ) ( "" ' ' )

<sup>⎡</sup> −⋅ ⋅ − ⋅ +⋅⋅ ⋅ <sup>⎤</sup> <sup>⎣</sup> <sup>⎦</sup> <sup>=</sup> −− +⋅ +⋅

 η

<sup>1</sup> 1 1 *<sup>b</sup>*

*b*

Calculations were to compare the tooth and the bearing friction losses in order to determine the efficiency of different types of planetary gears and evaluate the influence of the construction on the bearing friction losses and the efficiency of planetary gears. Comparing the calculated power losses caused by only the friction of tooth wheels or only by the bearing friction with the total power losses of the gearboxes, it is obvious that the bearing friction loss is a significant part of the whole friction losses. Behavior of various types of two- and three-stage and differential planetary gears were investigated and compared using the derived equations, following a row of systematical procedures. If the input power, the input speed and lubricant viscosity are known, the calculation can be performed. The first step is to choose various inner gear ratios for every stage and to combine them creating as many planetary gear ratios as possible. Using the equations presented above (1-29) the efficiency and the bearing power loss of every gear can be calculated. Some results are presented in diagrams (Fig. 8-17). Comparing the calculated values of efficiency and power loss ratios the optimal gearbox construction can be selected. The beneficial inner gear ratio of each stage and the power ratios were determined for all the four types of planetary gears. When the optimal inner gear ratios are known, the tooth profile ensuring the lowest tooth friction can be calculated for every planetary gear stage by varying the addendum modification of tooth wheels [Csobán 2009]. The calculations were performed for all planetary gears presented above for transmitting a power of 2000 kW at a driving speed of 1500 rpm.

*i i ii*

Power distribution between the stages (the power of the driver element of the first stage *P2"*

2' '

*P i* η

*b g*

*b g b g bb g g*

*b b bb bb*

*i i ii ii*

1 1 1

divided by the power of the driver element of the second stage *P2'*):

2"

In the calculations the parameters of Table 2 and 3 were used.

*R*a34 [μm]

Pin [kW]

500 63 0,63 1,25 2000 1500 0 0 3 0,8

a b f0 f1 c(db) d(db) c(L1h) d(L1h) 1 1 7,5 0,00055 0,4595 1,0951 444,04 1,3465

nin [1/min]

β [°]

*x2* [-]

N [-] b/dw [-]

*R*a23 [μm]

Table 2. Other important parameters for the analyses

Table 3. Parameters for calculate the bearing friction losses

*P i*

η

""'

( )"

*i*

"" ' 1 *GPV b b b b b b i i i i i ii* = − −+ ⋅+⋅ (27)

 ηη

⎛ ⎞ ⎜ − ⎟⋅ − ⎜ ⎟ <sup>⋅</sup> ⎝ ⎠ <sup>=</sup> (29)

(28)

#### **4. Comparing the properties of planetary gears**

The performance of a planetary gear drive depends on its kinematics, its inner gear ratios and the connections between the planetary stages. Only detailed calculations can reveal the behavior of planetary gears and show their best solutions for given applications. To calculate the gear ratios and the gearbox efficiencies of various planetary gears (Fig. 1(a).- 1(d).) the following equations were developed:

The gear ratio of planetary gear KB+KB (Fig. 1(a).) (sun gears drive and carriers are driven):

$$i\_{KB+KB} = \left(1 - i\_{b^\*}\right) \cdot \left(1 - i\_{b^\*}\right) \tag{19}$$

The efficiency of planetary gear KB+KB:

$$\eta\_{KB+KB} = \frac{\left(1 - i\_{b^{+}} \cdot \eta\_{\mathcal{g}^{+}}\right) \cdot \left(1 - i\_{b^{+}} \cdot \eta\_{\mathcal{g}^{+}}\right)}{\left(1 - i\_{b^{+}}\right) \cdot \left(1 - i\_{b^{+}}\right)}\tag{20}$$

The gear ratio of planetary gear PKG (Fig. 1(b).):

$$\dot{\mathbf{u}}\_{PKG} = \left(\dot{\mathbf{i}}\_{b^\*} + \dot{\mathbf{i}}\_{b^\*} - \dot{\mathbf{i}}\_{b^\*} \cdot \dot{\mathbf{i}}\_{b^\*}\right) \tag{21}$$

The efficiency of planetary gear PKG:

$$\eta\_{P\text{KG}} = \frac{\dot{\mathbf{i}}\_{b^{\*\*}} \cdot \eta\_{g^{\*\*}} + \dot{\mathbf{i}}\_{b^{\*}} \cdot \eta\_{g^{\*}} - \dot{\mathbf{i}}\_{b^{\*}} \cdot \dot{\mathbf{i}}\_{b^{\*}} \cdot \eta\_{g^{\*}} \cdot \eta\_{g^{\*}}}{\dot{\mathbf{i}}\_{b^{\*\*}} + \dot{\mathbf{i}}\_{b^{\*}} - \dot{\mathbf{i}}\_{b^{\*}} \cdot \dot{\mathbf{i}}\_{b^{\*}}} \tag{22}$$

Power distribution between the stages (the power of the driven element of the first stage *P4"* divided by the output power *Pout*):

$$\frac{P\_{4^{\ast}}}{P\_{out}} = \frac{1}{\left[1 + \frac{\dot{\mathbf{i}}\_{b^{\ast}} \cdot \eta\_{\mathcal{S}^{\ast}}}{\dot{\mathbf{i}}\_{b^{\ast}} \cdot \eta\_{\mathcal{S}^{\ast}}} - \dot{\mathbf{i}}\_{b^{\ast}} \cdot \eta\_{\mathcal{S}^{\ast}}\right]} \tag{23}$$

The gear ratio of planetary gear PV (Fig. 1(c).):

$$i\_{PV} = \mathbf{1} + i\_{b^\*} \cdot i\_{b^\*} - i\_{b^\*} \tag{24}$$

The efficiency of planetary gear PV:

$$\eta\_{PV} = \frac{1 - i\_{b^\*} \cdot \eta\_{\mathcal{g}^\*} + i\_{b^\*} \cdot i\_{b^\*} \cdot \eta\_{\mathcal{g}^\*} \cdot \eta\_{\mathcal{g}^\*}}{1 - i\_{b^\*} + i\_{b^\*} \cdot i\_{b^\*}} \tag{25}$$

Power distribution between the stages (the power of the driven element of the first stage *Pk"* divided by the output power *Pout*):

$$\frac{P\_{k^{\*\*}}}{P\_{out}} = \frac{1}{\left[1 + \frac{i\_{b^{\*\*}} \cdot i\_{b^{\*\*}} \cdot \eta\_{\mathcal{S}^{\*\*}} \cdot \eta\_{\mathcal{S}^{\*\*}}}{\left(1 - i\_{b^{\*\*}} \cdot \eta\_{\mathcal{S}^{\*\*}}\right)}\right]}\tag{26}$$

The gear ratio of planetary gear GPV (Fig. 1(d).):

$$i\_{GPV} = 1 - i\_{b^\*} - i\_b + i\_{b^\*} \cdot i\_b + i\_b \cdot i\_{b^\*} \tag{27}$$

The efficiency of planetary gear GPV:

130 Tribology - Lubricants and Lubrication

The performance of a planetary gear drive depends on its kinematics, its inner gear ratios and the connections between the planetary stages. Only detailed calculations can reveal the behavior of planetary gears and show their best solutions for given applications. To calculate the gear ratios and the gearbox efficiencies of various planetary gears (Fig. 1(a).-

The gear ratio of planetary gear KB+KB (Fig. 1(a).) (sun gears drive and carriers are driven):

1 1 1 1

η

( ) ( ) ( )( ) "" '' " '

" " ' ' "' " ' " ' "' *b g b g b b g g*

*i i ii*

<sup>⋅</sup> +⋅ − ⋅⋅ ⋅ <sup>=</sup> +−⋅

η

Power distribution between the stages (the power of the driven element of the first stage *P4"*

' '

<sup>=</sup> <sup>⎡</sup> <sup>⋅</sup> <sup>⎤</sup> <sup>⎢</sup> + −⋅ <sup>⎥</sup> <sup>⋅</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

η

1 *out b g*

*i*

1

1

*P i i*

η

<sup>−</sup> ⋅ + ⋅⋅ ⋅ <sup>=</sup> −+⋅

*i ii*

Power distribution between the stages (the power of the driven element of the first stage *Pk"*

1

*out bb g g*

*P i*

1

"

*k*

*P*

*PV*

η

" "

η

*b g*

η

4"

*P*

*b b bb*

1

*i i ii*

*i i i i*

*b g b g*

*b b*

*i ii KB KB b b* <sup>+</sup> =− ⋅− (1 1 " ' ) ( ) (19)

−⋅ ⋅−⋅ <sup>=</sup> − ⋅− (20)

*i i i ii PKG b b b b* = ( " ' "' +−⋅ ) (21)

"' " 1 *PV b b b i ii i* = +⋅− (24)

(22)

(23)

(25)

(26)

η

 ηη

' '

η

*b g*

*i*

" " "' " ' " "'

η η

*b g b b g g*

*b bb*

( )

*i* η η

<sup>=</sup> <sup>⎡</sup> <sup>⎤</sup> ⋅⋅ ⋅ <sup>⎢</sup> <sup>+</sup> <sup>⎥</sup> <sup>⎢</sup> − ⋅ <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

*b g*

η

"' " ' " "

1

*i ii*

**4. Comparing the properties of planetary gears** 

*KB KB*

η+

*PKG*

η

1(d).) the following equations were developed:

The gear ratio of planetary gear PKG (Fig. 1(b).):

The efficiency of planetary gear KB+KB:

The efficiency of planetary gear PKG:

divided by the output power *Pout*):

The efficiency of planetary gear PV:

divided by the output power *Pout*):

The gear ratio of planetary gear PV (Fig. 1(c).):

The gear ratio of planetary gear GPV (Fig. 1(d).):

$$\eta\_{GPV} = \frac{\left[\left(\mathbf{1} - \mathbf{i}\_b \cdot \eta\_{\mathcal{S}}\right) \cdot \left(\mathbf{1} - \mathbf{i}\_{b^\*} \cdot \eta\_{\mathcal{S}^\*}\right) + \mathbf{i}\_b \cdot \mathbf{i}\_{b^\*} \cdot \eta\_{\mathcal{S}} \cdot \eta\_{\mathcal{S}^\*}\right]}{\mathbf{1} - \mathbf{i}\_b - \mathbf{i}\_{b^\*} + \mathbf{i}\_b \cdot \mathbf{i}\_{b^\*} + \mathbf{i}\_b \cdot \mathbf{i}\_{b^\*}}\tag{28}$$

Power distribution between the stages (the power of the driver element of the first stage *P2"* divided by the power of the driver element of the second stage *P2'*):

$$\frac{P\_{2^\*}}{P\_{2^\*}} = \frac{\left(\frac{1}{i\_{b^\*} \cdot \eta\_{\mathcal{S}}} - 1\right) \cdot \left(1 - i\_{b^\*}\right)}{i\_{b^\*}}\tag{29}$$

#### **5. Results of calculations**

Calculations were to compare the tooth and the bearing friction losses in order to determine the efficiency of different types of planetary gears and evaluate the influence of the construction on the bearing friction losses and the efficiency of planetary gears. Comparing the calculated power losses caused by only the friction of tooth wheels or only by the bearing friction with the total power losses of the gearboxes, it is obvious that the bearing friction loss is a significant part of the whole friction losses. Behavior of various types of two- and three-stage and differential planetary gears were investigated and compared using the derived equations, following a row of systematical procedures. If the input power, the input speed and lubricant viscosity are known, the calculation can be performed. The first step is to choose various inner gear ratios for every stage and to combine them creating as many planetary gear ratios as possible. Using the equations presented above (1-29) the efficiency and the bearing power loss of every gear can be calculated. Some results are presented in diagrams (Fig. 8-17). Comparing the calculated values of efficiency and power loss ratios the optimal gearbox construction can be selected. The beneficial inner gear ratio of each stage and the power ratios were determined for all the four types of planetary gears. When the optimal inner gear ratios are known, the tooth profile ensuring the lowest tooth friction can be calculated for every planetary gear stage by varying the addendum modification of tooth wheels [Csobán 2009]. The calculations were performed for all planetary gears presented above for transmitting a power of 2000 kW at a driving speed of 1500 rpm. In the calculations the parameters of Table 2 and 3 were used.




Table 3. Parameters for calculate the bearing friction losses

The Bearing Friction of Compound Planetary Gears

**vi/**Σ**v [%]**

gearbox lifetime=5000[h]

**vi/**Σ**v [%]**

gearbox lifetime=50000[h]

in the Early Stage Design for Cost Saving and Efficiency 133

v tooth (ib"=2) v tooth (ib"=4) v tooth (ib"=6) v tooth (ib"=8) v bearing (ib"=2) v bearing (ib"=4) v bearing (ib"=6) v bearing (ib"=8)

v tooth (ib"=2) v tooth (ib"=4) v tooth (ib"=6) v tooth (ib"=8) v bearing (ib"=2) v bearing (ib"=4) v bearing (ib"=6) v bearing (ib"=8)

0 20 40 60 80 100 **i PV**

0 20 40 60 80 100 **i PV**

Fig. 11. The power loss ratio of planetary gear PV as a function of gear ratio. Prescribed

Fig. 10. The power loss ratio of planetary gear PV as a function of gear ratio. Prescribed

The results of calculation are presented in (Fig. 8-17). On the diagrams only those results can be seen, where the gears have no undercut or too thin top land.

Fig. 8. The power loss ratio of planetary gear KB+KB as a function of gear ratio. Prescribed gearbox lifetime=5000[h]

Fig. 9. The power loss ratio of planetary gear KB+KB as a function of gear ratio. Prescribed gearbox lifetime=50000[h]

The results of calculation are presented in (Fig. 8-17). On the diagrams only those results can

0 20 40 60 80 100 **i KB+KB**

0 20 40 60 80 100 **i KB+KB**

Fig. 9. The power loss ratio of planetary gear KB+KB as a function of gear ratio. Prescribed

v tooth (ib"=2) v tooth (ib"=4) v tooth (ib"=6) v tooth (ib"=8) v bearing (ib"=2) v bearing (ib"=4) v bearing (ib"=6) v bearing (ib"=8)

Fig. 8. The power loss ratio of planetary gear KB+KB as a function of gear ratio. Prescribed

v tooth (ib"=2) v tooth (ib"=4) v tooth (ib"=6) v tooth (ib"=8) v bearing (ib"=2) v bearing (ib"=4) v bearing (ib"=6) v bearing (ib"=8)

be seen, where the gears have no undercut or too thin top land.

**v i**/Σ

gearbox lifetime=50000[h]

**v [%]**

**v i**/Σ

gearbox lifetime=5000[h]

**v [%]**

Fig. 10. The power loss ratio of planetary gear PV as a function of gear ratio. Prescribed gearbox lifetime=5000[h]

Fig. 11. The power loss ratio of planetary gear PV as a function of gear ratio. Prescribed gearbox lifetime=50000[h]

The Bearing Friction of Compound Planetary Gears

**vi/**Σ**v [%]**

gearbox lifetime=5000[h], ib"=2

**vi/**Σ**v [%]**

gearbox lifetime=50000[h], ib"=2

in the Early Stage Design for Cost Saving and Efficiency 135

The power loss ratios of the three-stage GPV planetary gearbox were investigated at the

v tooth (ib"=2, ib'=2) v tooth (ib'=4) v tooth (ib'=6) v tooth (ib'=8) v bearing (ib"=2, ib'=2) v bearing (ib'=4) v bearing (ib'=6) v bearing (ib'=8)

v tooth (ib"=2, ib'=2) v tooth (ib'=4) v tooth (ib'=6) v tooth (ib'=8) v bearing (ib"=2, ib'=2) v bearing (ib'=4) v bearing (ib'=6) v bearing (ib'=8)

0 50 1 **i GPV** 00

Fig. 14. The power loss ratio of planetary gear GPV as a function of gear ratio. Prescribed

0 50 1 **i GPV** 00

Fig. 15. The power loss ratio of planetary gear GPV as a function of gear ratio. Prescribed

same gear ratio range as the two-stage differential gears have (Figure 14-15).

Fig. 12. The power loss ratio of planetary gear PKG as a function of gear ratio. Prescribed gearbox lifetime=5000[h]

Fig. 13. The power loss ratio of planetary gear PKG as a function of gear ratio. Prescribed gearbox lifetime=50000[h]

0 20 40 60 80 100 **i PKG**

0 20 40 60 80 100 **i PKG**

Fig. 13. The power loss ratio of planetary gear PKG as a function of gear ratio. Prescribed

Fig. 12. The power loss ratio of planetary gear PKG as a function of gear ratio. Prescribed

v tooth (ib"=2) v tooth (ib"=4) v tooth (ib"=6) v tooth (ib"=8) v bearing (ib"=2) v bearing (ib"=4) v bearing (ib"=6) v bearing (ib"=8)

v tooth (ib"=2) v tooth (ib"=4) v tooth (ib"=6) v tooth (ib"=8) v bearing (ib"=2) v bearing (ib"=4) v bearing (ib"=6) v bearing (ib"=8)

**vi/**Σ**v [%]**

gearbox lifetime=5000[h]

**vi/**Σ**v [%]**

gearbox lifetime=50000[h]

The power loss ratios of the three-stage GPV planetary gearbox were investigated at the same gear ratio range as the two-stage differential gears have (Figure 14-15).

Fig. 14. The power loss ratio of planetary gear GPV as a function of gear ratio. Prescribed gearbox lifetime=5000[h], ib"=2

Fig. 15. The power loss ratio of planetary gear GPV as a function of gear ratio. Prescribed gearbox lifetime=50000[h], ib"=2

The Bearing Friction of Compound Planetary Gears

higher (at the same types of bearings).

like efficiency, size and even cost can be compared easily.

*a1* factor for bearing life correction (*a1=0,21…1*) [-],

f0 coefficient (which is a function of bearing type and size) [-], f1 coefficient (which is a function of bearing type and load) [-],

*<sup>g</sup>* is the rolling efficiency of a simple planetary gear stage *KB*,

*ib* is the ratio of the number of teeth of sun gear and ring gear at the third stage, *ib'* is the ratio of the number of teeth of sun gear and ring gear at the second stage,

*c(db;L1h)* developed constant for bearing calculation *d(db;L1h)* developed constant for bearing calculation,

*<sup>M</sup>* viscosity at operating temperature [Pas],

 constant and exponent (table 1.), *dm* average diameter of bearing [mm], *dm res* resultant average bearing diameter [mm],

*C* is the basic dynamic load [N],

*Fr* is the radial bearing load [N],

*I* is the gear ratio,

change between 20% to 60%.

**7. Acknowledgment** 

**8. Nomenclature**  *2* sun gear, *3* planet gear, *4* ring gear, a, b exponents [-],

c ; d

η

η

Comparing the results of the analysis the following can be stated:

**6. Conclusions** 

gear PV.

in the Early Stage Design for Cost Saving and Efficiency 137

• It can be stated that thanks to relatively low predicted lifetime, smaller bearings have to be build in the gearbox. Having smaller bearings, the tooth power loss ratio will be

• If longer bearing life is needed larger bearings have to build in the gearbox. Larger

• Varying the inner gear ratios of the investigated planetary gear drives the values of the power loss rates change significantly only in the range of the lower gear ratios. • Depending on the gear ratios and prescribed lifetime the values of the tooth power loss ratio change between 80% to 40% while the values of the bearing power loss ratio

Using the bearing power loss model presented above all types of bearings can be considered for a given planetary gearbox optimization and application and all the important parameters

This work is connected to the scientific program of the " Development of quality-oriented and harmonized R+D+I strategy and functional model at BME" project. This project is supported by the New Hungary Development Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002).

bearings lead to higher power losses (at the same types of bearings).

• It is obvious that the bearing friction loss is a significant part of the friction losses. • Higher gear ratios can be realized with the planetary gear PKG than with planetary

The GPV gearbox can operate with higher gear ratios than the two-stage gears. The gear ratio range was changed with increasing the inner gear ratio of the first stage while the inner gear ratios of the second and third stage were changed and combined (Figure 16-17).

Fig. 16. The power loss ratio of planetary gear GPV as a function of gear ratio. Prescribed gearbox lifetime=5000[h], ib"=8

Fig. 17. The power loss ratio of planetary gear GPV as a function of gear ratio. Prescribed gearbox lifetime=50000[h], ib"=8

### **6. Conclusions**

136 Tribology - Lubricants and Lubrication

The GPV gearbox can operate with higher gear ratios than the two-stage gears. The gear ratio range was changed with increasing the inner gear ratio of the first stage while the inner

> v tooth (ib"=8, ib'=2) v tooth (ib'=4) v tooth (ib'=6) v tooth (ib'=8) v bearing (ib"=8, ib'=2) v bearing (ib'=4) v bearing (ib'=6) v bearing (ib'=8)

v tooth (ib"=8, ib'=2) v tooth (ib'=4) v tooth (ib'=6) v tooth (ib'=8) v bearing (ib"=8, ib'=2) v bearing (ib'=4) v bearing (ib'=6) v bearing (ib'=8)

gear ratios of the second and third stage were changed and combined (Figure 16-17).

0 50 **i GPV** 100

Fig. 16. The power loss ratio of planetary gear GPV as a function of gear ratio. Prescribed

0 50 1 **i GPV** 00

Fig. 17. The power loss ratio of planetary gear GPV as a function of gear ratio. Prescribed

**vi**/Σ

gearbox lifetime=5000[h], ib"=8

**vi**/Σ**v [%]**

gearbox lifetime=50000[h], ib"=8

**v [%]**

Comparing the results of the analysis the following can be stated:


Using the bearing power loss model presented above all types of bearings can be considered for a given planetary gearbox optimization and application and all the important parameters like efficiency, size and even cost can be compared easily.

### **7. Acknowledgment**

This work is connected to the scientific program of the " Development of quality-oriented and harmonized R+D+I strategy and functional model at BME" project. This project is supported by the New Hungary Development Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002).

### **8. Nomenclature**


*ib'* is the ratio of the number of teeth of sun gear and ring gear at the second stage,

**5** 

S. Sherbakov

σ *ij* , ( ) *<sup>p</sup> ij* ε)

*Belarus* 

**Three-Dimensional Stress-Strain** 

**Under Complex Loading** 

**State of a Pipe with Corrosion Damage** 

*Department of Theoretical and Applied Mechanics, Belarusian State University* 

In studies of the stress-stain state of models of pipeline sections without corrosion defects of a pipe, in the two-dimensional statement (cross section), pipes are usually modeled by a ring, whereas in the three-dimensional statement – by a thick-wall cylindrical shell [Ponomarev et al., 1958; Seleznev et al. 2005]. Usually, internal pressure or temperature is considered as a load applied to the pipe. The solution of the problems stated in this manner yields not bad results when a relatively not complicated procedure of calculation, both

The presence of corrosion damage at the inner surface of the pipe (Figures 1, 2), being a particular three-dimensional concentrator of stresses, requires a special approach to defening the stress-strain state. In addition, account should be taken of a simultaneous compound action of such loading factors as internal pressure and friction of the mineral oil

The analysis of the known references to articles shows that the problem of investigating the spatial stress-strain states of the pipe with regard to its corrosion damage with the account of various types of loading has not been stated up to now. In essence, the problems of

1982; Dertsakyan et al., 1977; Mirkin et al., 1991; O'Grady et al., 1992] are under consideration. The problem of determining stress-strain state caused by wall friction due to

problems of stress-strain state determination are usually being solved for shell models of a

Therefore the statement and solutions of the problem of determining three-dimensional stress-strain state of the models of pipes with corrosion defects under the action of internal pressure, friction caused by oil flow and temperature discussed in the present chapter are

) [Ainbinder et al., 1982; Borodavkin et al., 1984; Grachev et al.,

), as well as the most general problems of determining

σ

+ + has not been stated. In addition, the

*ij* is described for the three-

determining individual stress-strain states under the action of internal pressure ( ( ) *<sup>p</sup>*

 <sup>+</sup> <sup>+</sup> ( ) *<sup>p</sup> <sup>T</sup> ij* τ ε

**1. Introduction** 

or temperature ( ( ) *<sup>T</sup>*

( ) *p ij* τ σ

 <sup>+</sup> , ( ) *<sup>p</sup> ij* τ ε

viscous fluid motion ( ( )

 <sup>+</sup> ; ( ) *<sup>p</sup> <sup>T</sup>* σ*ij*

analytical and numerical, is adopted.

σ *ij* , ( ) *<sup>T</sup> ij* ε

flow over the inner surface of the pipe, as well as of soil.

*ij* τ σ , ( ) *ij* τ ε

<sup>+</sup> , ( ) *<sup>p</sup> <sup>T</sup> ij* ε

 <sup>+</sup> ; ( ) *<sup>p</sup> <sup>T</sup> ij* τ σ

dimensional model of the section of the pipe with corrosion damage.

pipe. Although, for example, in [Seleznev et al. 2005] ( ) *<sup>p</sup>*


#### **9. References**

Bartz, W.J.: Getriebeschmierung. Expert Verlag. Ehningen bei Böblingen, 1989


Niemann G.- Winter, H.: Maschinenelemente. Band II. Springer-Verlag, Berlin, 1989

Shell Co: The Lubrication of Industrial Gears. John Wright & Sons Ltd. London 1964.

SKF Főkatalógus, Reg. 47.5000 1989-12, Hungary

SKF General catalogue, 6000 EN, November 2005

S. Sherbakov

*Department of Theoretical and Applied Mechanics, Belarusian State University Belarus* 

#### **1. Introduction**

138 Tribology - Lubricants and Lubrication

*ib"* is the ratio of the number of teeth of sun gear and ring gear at the first stage,

*k* planetary carrier, *L1h* prescribed lifetime [h],

ν

σ

σ*m,* τ

Σ

ω

**9. References** 

M0 load independent friction torque [Nmm], M1 load dependent friction torque [Nmm],

kinematical viscosity at operating temperature [mm2/s],

*<sup>m</sup>* allowable equivalent and shear stress components [MPa],

Bartz, W.J.: Getriebeschmierung. Expert Verlag. Ehningen bei Böblingen, 1989

Tribology, Technische Akademie Esslingen, 2008

Erney György: Fogaskerekek, Műszaki Könyvkiadó, Budapest, 1983 Klein, H.: Bolygókerék hajtóművek. Műszaki Könyvkiadó, Budapest, 1968.

Kyoto, Japan, September 6 – 11, 2009

15-17 September, 2005. Sopron 134-141 Müller, H.W.: Die Umlaufgetriebe. Springer-Verlag, Berlin. 1971

SKF Főkatalógus, Reg. 47.5000 1989-12, Hungary SKF General catalogue, 6000 EN, November 2005

Csobán Attila., Kozma Mihály: Comparing the performance of heavy-duty planetary gears.

Csobán Attila, Kozma Mihály: Influence of the Power Flow and the Inner Gear Ratios on the

Csobán Attila, Kozma Mihály: A model for calculating the Oil Churning, the Bearing and

Duda, M.: Der geometrische Verlustbeiwert und die Verlustunsymmetrie bei geradverzahnten Stirnradgetrieben. Forschung im Ingenieurwesen 37 (1971) H. 1, VDI-Verlag

Kozma, M.: Effect of lubricants on the performance of gears. Proceedings of Interfaces'05.

Niemann G.- Winter, H.: Maschinenelemente. Band II. Springer-Verlag, Berlin, 1989 Shell Co: The Lubrication of Industrial Gears. John Wright & Sons Ltd. London 1964.

Proceedings of fifth conference on mechanical engineering Gépészet 2006 ISBN

Efficiency of Heavy-Duty Differential Planetary Gears, 16th International Colloquium

the Tooth Friction Generated in Planetary Gears, World Tribology Congress 2009,

*M2;4* sun or ring gear torque [Nm], *M3* planet gear torque [Nm], n bearing velocity [rpm], nin driving speed [rpm],

*Ra* average surface roughness (CLA),

*<sup>F</sup>* bending strength of teeth [MPa],

*V* shear load of planet gear pin [N], *vBearing* bearing friction loss component [W], *vtooth* tooth power loss component [W],

Y calculated parameter (*di, C*).

9635934653

*3g* angle velocity of planet gear [rad/s],

P1 load of the bearing [N], *Pin* driving power [W],

*v* total power loss [W], *v* entraining speed [m/s],

In studies of the stress-stain state of models of pipeline sections without corrosion defects of a pipe, in the two-dimensional statement (cross section), pipes are usually modeled by a ring, whereas in the three-dimensional statement – by a thick-wall cylindrical shell [Ponomarev et al., 1958; Seleznev et al. 2005]. Usually, internal pressure or temperature is considered as a load applied to the pipe. The solution of the problems stated in this manner yields not bad results when a relatively not complicated procedure of calculation, both analytical and numerical, is adopted.

The presence of corrosion damage at the inner surface of the pipe (Figures 1, 2), being a particular three-dimensional concentrator of stresses, requires a special approach to defening the stress-strain state. In addition, account should be taken of a simultaneous compound action of such loading factors as internal pressure and friction of the mineral oil flow over the inner surface of the pipe, as well as of soil.

The analysis of the known references to articles shows that the problem of investigating the spatial stress-strain states of the pipe with regard to its corrosion damage with the account of various types of loading has not been stated up to now. In essence, the problems of determining individual stress-strain states under the action of internal pressure ( ( ) *<sup>p</sup>* σ *ij* , ( ) *<sup>p</sup> ij* ε)

or temperature ( ( ) *<sup>T</sup>* σ *ij* , ( ) *<sup>T</sup> ij* ε ) [Ainbinder et al., 1982; Borodavkin et al., 1984; Grachev et al., 1982; Dertsakyan et al., 1977; Mirkin et al., 1991; O'Grady et al., 1992] are under consideration. The problem of determining stress-strain state caused by wall friction due to viscous fluid motion ( ( ) *ij* τ σ , ( ) *ij* τ ε ), as well as the most general problems of determining ( ) *p ij* τ σ <sup>+</sup> , ( ) *<sup>p</sup> ij* τ ε <sup>+</sup> ; ( ) *<sup>p</sup> <sup>T</sup>* σ *ij* <sup>+</sup> , ( ) *<sup>p</sup> <sup>T</sup> ij* ε <sup>+</sup> ; ( ) *<sup>p</sup> <sup>T</sup> ij* τ σ <sup>+</sup> <sup>+</sup> ( ) *<sup>p</sup> <sup>T</sup> ij* τ ε + + has not been stated. In addition, the problems of stress-strain state determination are usually being solved for shell models of a pipe. Although, for example, in [Seleznev et al. 2005] ( ) *<sup>p</sup>* σ *ij* is described for the threedimensional model of the section of the pipe with corrosion damage.

Therefore the statement and solutions of the problem of determining three-dimensional stress-strain state of the models of pipes with corrosion defects under the action of internal pressure, friction caused by oil flow and temperature discussed in the present chapter are

Fig. 2. Solid model of the neighborhood of the pipe with corrosion damage

on displacements of the outer surface of the pipe:

a. no pipeline fixing

place

direction on the right end

The distinctive feature of computer finite-element modeling of pipes of trunk pipelines with corrosion damage lies in the opportunity to assign different boundary conditions for the outer surface of the pipe. So, for loads (1)–(3), consideration is made of different restrictions

> 2 0, *<sup>r</sup> r r*

b. no displacements of the outer surface of the pipe in the *x* and *y* directions and in the *z*

2 2 2

d. contact between the pipe and soil where no displacements of the soil outer surface take

222

0

where the superscript 1 means the pipe, whereas 2 – soil, στ is the tangential component of

*<sup>n</sup> rr rr r r*

= = =

,

*f*

2 2

 σ

= =

 τ

σσσ

= =

(1) (2) (1)

=− =

3 3

*x y r r r r*

= =

*u u*

the stress vector, *f* is the friction coefficient, and *r*3 is the soil outer radius.

τ

σ

(1) (2)

*r r rr rr*

= −

<sup>=</sup> = (5)

0, 0, *xy z r r r r z L uu u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = = (6)

0, *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = = (7)

,

(8)

σ

2 2

c. no displacements of the outer surface of the pipe in all directions

important for pipeline systems and such related disciplines as solid mechanics, fluid mechanics and tribology.

#### **2. Statement of the problem**

The present Chapter deals with some of the results of investigation of the three-dimensional stress-strain state of the model of a pipe with corrosion damage (Figure 2).

Fig. 1. Simplified scheme of elliptical corrosion damage at the inner surface of the pipe displaying reduced wall thickness inside the damage

In calculations, the following basic loads applied to the pipe were taken into consideration:

• internal pressure

$$\left. \sigma\_r \right|\_{r=r\_1} = p\_r \tag{1}$$

where *r*1 is the inner radius of the pipe;

• mineral oil friction over the inner surface of the pipe, thus exciting wall tangential stresses

$$\left.\tau\_{rz}\right|\_{r=r\_1} = \tau\_{0\prime} \tag{2}$$

τ0 – tangential forces modeling the viscous fluid friction force over the inner surface of the pipe;

• change of the thermodynamic state (temperature) of the pipe

$$\left| T\_{r\_1} - T\_{r\_2} \right| = \Delta T\_{\prime} \tag{3}$$

where *r*2 is the outer radius of the pipe.

It should be emphasized that in the presence of corrosion damage

$$r\_1 = r\_1(\,\varphi, z\right),\tag{4}$$

where ϕ and *z* are the components of the cylindrical coordinate system (*r*, ϕ, *z*).

Fig. 2. Solid model of the neighborhood of the pipe with corrosion damage

The distinctive feature of computer finite-element modeling of pipes of trunk pipelines with corrosion damage lies in the opportunity to assign different boundary conditions for the outer surface of the pipe. So, for loads (1)–(3), consideration is made of different restrictions on displacements of the outer surface of the pipe:

a. no pipeline fixing

140 Tribology - Lubricants and Lubrication

important for pipeline systems and such related disciplines as solid mechanics, fluid

The present Chapter deals with some of the results of investigation of the three-dimensional

Fig. 1. Simplified scheme of elliptical corrosion damage at the inner surface of the pipe

σ

τ

• change of the thermodynamic state (temperature) of the pipe

It should be emphasized that in the presence of corrosion damage

In calculations, the following basic loads applied to the pipe were taken into consideration:

1 , *<sup>r</sup> r r*

• mineral oil friction over the inner surface of the pipe, thus exciting wall tangential

<sup>1</sup> <sup>0</sup> , *rz r r*

τ0 – tangential forces modeling the viscous fluid friction force over the inner surface of the

*rr z* 1 1 = (ϕ

where ϕ and *z* are the components of the cylindrical coordinate system (*r*, ϕ, *z*).

τ

*p* <sup>=</sup> = (1)

<sup>=</sup> = (2)

1 2 , *TT T r r* − = Δ (3)

, , ) (4)

displaying reduced wall thickness inside the damage

where *r*1 is the inner radius of the pipe;

where *r*2 is the outer radius of the pipe.

stress-strain state of the model of a pipe with corrosion damage (Figure 2).

mechanics and tribology.

• internal pressure

stresses

pipe;

**2. Statement of the problem** 

$$\left. \sigma\_r \right|\_{r=r\_2} = 0,\tag{5}$$

b. no displacements of the outer surface of the pipe in the *x* and *y* directions and in the *z* direction on the right end

$$\left.u\_x\right|\_{r=r\_2} = \left.u\_y\right|\_{r=r\_2} = 0, \left.u\_z\right|\_{z=L} = 0,\tag{6}$$

c. no displacements of the outer surface of the pipe in all directions

$$\left.u\_x\right|\_{r=r\_2} = \left.u\_y\right|\_{r=r\_2} = \left.u\_z\right|\_{r=r\_2} = 0,\tag{7}$$

d. contact between the pipe and soil where no displacements of the soil outer surface take place

$$\begin{aligned} \left. \sigma\_r^{(1)} \right|\_{r=r\_2} &= -\sigma\_r^{(2)} \Big|\_{r=r\_2} ' \\ \left. \sigma\_r^{(1)} \right|\_{r=r\_2} &= -\sigma\_r^{(2)} \Big|\_{r=r\_2} = f \left. \sigma\_n^{(1)} \right|\_{r=r\_2} \\ \left. \mu\_x \right|\_{r=r\_3} &= \mu\_y \Big|\_{r=r\_3} = 0 \end{aligned} \tag{8}$$

where the superscript 1 means the pipe, whereas 2 – soil, στ is the tangential component of the stress vector, *f* is the friction coefficient, and *r*3 is the soil outer radius.

10m /sec\*0.612m Re 43714.3. *<sup>K</sup>* 1.4\*10 m /sec

The critical Reynolds number (a transition from a laminar to a turbulent flow) for a viscous fluid moving in a round pipe is Re*cr* ≈ 2300. Thus, the turbulent flow motion should be considered in our problem. The software Fluent calculations used the turbulence *k* – ε model

As boundary conditions the following parameters were used: at the incoming flow surface the initial turbulence level equal to 7% was assigned; at the pipe walls the fixing conditions and the logarithmic velocity profile were predetermined; in the pipe the fluid pressure equal

Calculations of the steady regime of the fluid flow (quasi-parabolic turbulent velocity profile of the incoming flow) and of the unsteady regime (rectangular velocity profile of the

<sup>0</sup> <sup>1</sup> , *x r <sup>x</sup>*

In the problems with a quasi-parabolic turbulent velocity profile, at the entrance surface of the pipe the empirically found profile of the initial velocity was assigned, which is

> 1 <sup>7</sup> <sup>0</sup>

0 2

=

*r*

⎝ ⎠

1

 υ

0

=

*r*

⎝ ⎠

*r*

*r r*

0 max max 0 1

<sup>7</sup> 2 2 0 max max 0 1

 υ

<sup>1</sup> , 1.1428 ,0 2 , <sup>2</sup> *<sup>x</sup> <sup>x</sup>*

1 , 1.2244 , ,0 . *<sup>x</sup> <sup>x</sup>*

The calculation results have shown that the motion becomes steady (as the flow moves in the pipe, the quasi-parabolic turbulent profile of the longitudinal velocity *Vx* develops) at some distance from the entrance (left) surface of the pipe (Figure 3). So, from Figure 4 it is seen that for the quasi-parabolic velocity profile of the incoming flow the zone of the steady

Further, we will consider the results obtained for the velocity profiles of the incoming flow

Consider the flow turbulence intensity being the ratio of the root-mean-square fluctuation

' , *avg <sup>u</sup> <sup>I</sup> u*

At the surface of the incoming flow, the turbulence intensity is calculated by the formula

⎛ ⎞ <sup>=</sup> ⎜ ⎟ − = = + ≤≤

υ

⎛ ⎞ <sup>−</sup> = − ⎜ ⎟ = ≤≤

υ

υ

<sup>−</sup> = = <sup>=</sup> (12)

<sup>=</sup> = (13)

*r r*

*r y z rr*

= (16)

(14)

(15)

 υ

0 10 m/sec 4 2

for modeling turbulent flow viscosity [Launder et al., 1972; Rodi, 1976].

In the problems with a rectangular velocity profile of the incoming flow

The unsteady regime of the fluid flow was considered.

 υ

motion begins earlier than for the rectangular profile.

velocity *u*′ to the average flow velocity *uavg* (Figure 5).

to 4 МPа was set.

incoming flow) were made.

determined by the formula: - for the two-dimensional case

υ

υυ

calculated in accordance to (14) and (15).


υ *D* ν

With the use of the pipeline fixing (type a), tests of the pipe dug out of soil (in the air) are modeled. The stress-strain state of a pipe lying in hard soil without friction in the axial direction is modeled by means of pipe fixing (type b) while that of a pipe lying in hard soil and rigidly connected with it – by means of pipe fixing (type c). Subject to boundary conditions (8) (type d), a pipe lying in soil having particular mechanical characteristics is modeled.

Thus, the problem has been stated to make a comparative analysis of the stress-strain states of the pipe with corrosion damage for different combinations of boundary conditions (1)– (3), (6)–(8):

$$
\sigma\_{\vec{\imath}}^{(p)}, \varepsilon\_{\vec{\imath}}^{(p)}; \sigma\_{\vec{\imath}}^{(\tau)}, \varepsilon\_{\vec{\imath}}^{(\tau)}; \sigma\_{\vec{\imath}}^{(T)}, \varepsilon\_{\vec{\imath}}^{(T)}; \quad \tag{9}
$$

$$
\sigma\_{\vec{\imath}}^{(p+\tau)}, \varepsilon\_{\vec{\imath}}^{(p+\tau)}; \sigma\_{\vec{\imath}}^{(p+T)}, \varepsilon\_{\vec{\imath}}^{(p+T)}; \sigma\_{\vec{\imath}}^{(p+\tau+T)}, \varepsilon\_{\vec{\imath}}^{(p+\tau+T)}. \tag{9}
$$

where the superscripts *p*, τ, and *T* correspond to the stress states caused by internal pressure, friction force over the inner surface of the pipe, and temperature.

In the case of the elastic relationship between stresses and strains, the stress states in (9) are connected by the following relations

$$\begin{aligned} \sigma\_{ij}^{(p+\tau)} &= \sigma\_{ij}^{(p)} + \sigma\_{ij}^{(\tau)}, \\ \sigma\_{ij}^{(p+T)} &= \sigma\_{ij}^{(p)} + \sigma\_{ij}^{(T)}, \\ \sigma\_{ij}^{(p+\tau+T)} &= \sigma\_{ij}^{(p)} + \sigma\_{ij}^{(\tau)} + \sigma\_{ij}^{(T)}. \end{aligned} \tag{10}$$

Further, some of the solutions to more than 70 problems of studying the stress-strain state of the pipe cross section in the damage area (dot-and-dash line in Figure 1) [Kostyuchenko et al., 2007a; 2007b; Sherbakov et al., 2007b; 2008a; 2008b; Sherbakov, 2007b; Sosnovskiy et al., 2008] are analyzed. These two-dimensional problems mainly describe the stress-strain states of straight pipes with different-profile damage along the axis. Also, with the use of the finite-element method implemented in the software ANSYS, the essentially threedimensional stress-strain state of the pipe in the three-dimensional damage area (Figure 1) was investigated.

#### **3. Wall friction in the turbulent mineral oil flow in the pipe with corrosion damage**

Within the framework of the present work, hydrodynamic calculation was made of the motion characteristics of a viscous, incompressible, steady, isothermal fluid in a cylindrical channel that models a pipe and in a cylindrical channel with geometric characteristics with regard to the peculiarities of a pipe with corrosion damage (see, Sect. 2). Calculations were performed for the initial incoming flow velocities υ0: 1 m/sec and 10 m/sec.

The kinematic viscosity of fluid was taken equal to *vK* = 1.4 10-4 m2/sec, the viscous fluid density – 865 kg/m3. The calculated Reynolds numbers will be, respectively,

$$\mathrm{Re}\_{1\,\mathrm{m}/\mathrm{sec}} = \frac{\nu\_0 D}{\nu\_\mathrm{K}} = \frac{1\,\mathrm{m} \,/\,\mathrm{sec}^\* 0.612\,\mathrm{m}}{1.4^\* 10^{-4} \,\mathrm{m}^2 \,/\,\mathrm{sec}} = 4371.43\,\mathrm{s}.\tag{11}$$

$$\mathrm{Re}\_{10\text{ m/sec}} = \frac{\nu\_0 D}{\nu\_K} = \frac{10\,\text{m} \,/\,\text{sec}^\* 0.612\,\text{m}}{1.4^\* 10^{-4} \,\text{m}^2 \,/\,\text{sec}} = 43714.3. \tag{12}$$

The critical Reynolds number (a transition from a laminar to a turbulent flow) for a viscous fluid moving in a round pipe is Re*cr* ≈ 2300. Thus, the turbulent flow motion should be considered in our problem. The software Fluent calculations used the turbulence *k* – ε model for modeling turbulent flow viscosity [Launder et al., 1972; Rodi, 1976].

As boundary conditions the following parameters were used: at the incoming flow surface the initial turbulence level equal to 7% was assigned; at the pipe walls the fixing conditions and the logarithmic velocity profile were predetermined; in the pipe the fluid pressure equal to 4 МPа was set.

Calculations of the steady regime of the fluid flow (quasi-parabolic turbulent velocity profile of the incoming flow) and of the unsteady regime (rectangular velocity profile of the incoming flow) were made.

In the problems with a rectangular velocity profile of the incoming flow

$$\left. \boldsymbol{\nu}\_{\boldsymbol{x}} \right|\_{\mathbf{x}=\mathbf{0}} = \boldsymbol{\nu}\_{\boldsymbol{r}\mathbf{1}'} \tag{13}$$

The unsteady regime of the fluid flow was considered.

In the problems with a quasi-parabolic turbulent velocity profile, at the entrance surface of the pipe the empirically found profile of the initial velocity was assigned, which is determined by the formula:


142 Tribology - Lubricants and Lubrication

With the use of the pipeline fixing (type a), tests of the pipe dug out of soil (in the air) are modeled. The stress-strain state of a pipe lying in hard soil without friction in the axial direction is modeled by means of pipe fixing (type b) while that of a pipe lying in hard soil and rigidly connected with it – by means of pipe fixing (type c). Subject to boundary conditions (8) (type d), a pipe lying in soil having particular mechanical characteristics is

Thus, the problem has been stated to make a comparative analysis of the stress-strain states of the pipe with corrosion damage for different combinations of boundary conditions (1)–

> () () () () ( ) ( ) ( )( ) ( )( ) ( )( ) ,; ,; ,; ,; ,; , .

τ

 ε

> τ

 ε+ + + + ++ ++ (9)

.

<sup>−</sup> = = <sup>=</sup> (11)

(10)

*p p T T ij ij ij ij ij ij pp p T p T p T p T ij ij ij ij ij ij*

where the superscripts *p*, τ, and *T* correspond to the stress states caused by internal

In the case of the elastic relationship between stresses and strains, the stress states in (9) are

( ) () () ( )

=++

*pT p T ij ij ij ij*

Further, some of the solutions to more than 70 problems of studying the stress-strain state of the pipe cross section in the damage area (dot-and-dash line in Figure 1) [Kostyuchenko et al., 2007a; 2007b; Sherbakov et al., 2007b; 2008a; 2008b; Sherbakov, 2007b; Sosnovskiy et al., 2008] are analyzed. These two-dimensional problems mainly describe the stress-strain states of straight pipes with different-profile damage along the axis. Also, with the use of the finite-element method implemented in the software ANSYS, the essentially threedimensional stress-strain state of the pipe in the three-dimensional damage area (Figure 1)

, ,

τ

σσσ

τ

τ τ

> εσ

σεσεσ

( ) () ( ) ( ) () ( )

= + = +

**3. Wall friction in the turbulent mineral oil flow in the pipe with corrosion** 

performed for the initial incoming flow velocities υ0: 1 m/sec and 10 m/sec.

density – 865 kg/m3. The calculated Reynolds numbers will be, respectively,

0 1m/sec 4 2

υ *D* ν

Within the framework of the present work, hydrodynamic calculation was made of the motion characteristics of a viscous, incompressible, steady, isothermal fluid in a cylindrical channel that models a pipe and in a cylindrical channel with geometric characteristics with regard to the peculiarities of a pipe with corrosion damage (see, Sect. 2). Calculations were

The kinematic viscosity of fluid was taken equal to *vK* = 1.4 10-4 m2/sec, the viscous fluid

1m /sec\*0.612m Re 4371.43, *<sup>K</sup>* 1.4\*10 m /sec

*p p ij ij ij pT p T ij ij ij*

 σσ

 σσ

τ

+ + + +

σ

σ

σ

τ

τ

connected by the following relations

σεσ

 τ

pressure, friction force over the inner surface of the pipe, and temperature.

modeled.

(3), (6)–(8):

was investigated.

**damage** 

$$\left. \nu\_{\rm x} \right|\_{\rm x=0} = \nu\_{\rm max} \left( 1 - \frac{\left| r - 2r\_0 \right|}{2r\_0} \right)^{\frac{1}{7}}, \nu\_{\rm max} = 1.1428 \,\nu\_0, 0 \le r \le 2 \, r\_1. \tag{14}$$


$$\left. \nu\_{\rm x} \right|\_{\rm x=0} = \nu\_{\rm max} \left( 1 - \frac{r}{r\_0} \right)^{\frac{1}{7}}, \nu\_{\rm max} = 1.2244 \,\nu\_0 \,\,\, r = \sqrt{y^2 + z^2}, 0 \le r \le r\_1. \tag{15}$$

The calculation results have shown that the motion becomes steady (as the flow moves in the pipe, the quasi-parabolic turbulent profile of the longitudinal velocity *Vx* develops) at some distance from the entrance (left) surface of the pipe (Figure 3). So, from Figure 4 it is seen that for the quasi-parabolic velocity profile of the incoming flow the zone of the steady motion begins earlier than for the rectangular profile.

Further, we will consider the results obtained for the velocity profiles of the incoming flow calculated in accordance to (14) and (15).

Consider the flow turbulence intensity being the ratio of the root-mean-square fluctuation velocity *u*′ to the average flow velocity *uavg* (Figure 5).

$$I = \frac{\mu'}{\mu\_{avg}},$$
 
$$\text{(16)}$$

At the surface of the incoming flow, the turbulence intensity is calculated by the formula

Fig. 5. Turbulence intensity (two-dimensional pipe flow, quasi-parabolic turbulent velocity

Fig. 6. Transverse velocity *Vy* for the two-dimensional flow in the pipe with corrosion

At high initial flow velocity values the vortex formation rate is higher.

The zone of the unsteady turbulent motion is characterized by the higher turbulence intensity (vortex formation) in comparison with the remaining region of the pipe (Figure 5). The highest intensity is observed in the steady motion zone, which is especially noticeable in the calculations with the initial velocity of 1 m/sec in the pipe wall region, whereas the

profile, υ0 = 1 m/sec)

damage at υ0 = 10 m/sec

lowest one – at the flow symmetry axis.

$$I = 0.16 \left(\text{Re}\_{D\_H}\right)^{-1} \frac{1}{\nu}, \text{ Re}\_{D\_H} = \frac{\nu\_0 D\_H}{\nu},\tag{17}$$

where *DH* is the hydraulic diameter (for the round cross section: *DH* = 2*r*1 = 0.612 m), υ0 is the incoming flow velocity, and *v* is the kinematic viscosity of oil (*v* = 1.4⋅10–4 m2/sec).

Fig. 3. Longitudinal velocity *Vx* (two-dimensional flow) for the quasi-parabolic turbulent velocity profile of the incoming flow at υ0 = 1 m/sec

Fig. 4. Profiles of the longitudinal velocity *Vx.* over the pipe cross sections (three-dimensional flow) for the quasi-parabolic turbulent velocity profile of the incoming flow at υ0 = 1 m/sec

*H*

= = (17)

ν

υ

1 <sup>8</sup> <sup>0</sup> 0.16 Re , Re , *H H*

−

Fig. 3. Longitudinal velocity *Vx* (two-dimensional flow) for the quasi-parabolic turbulent

Fig. 4. Profiles of the longitudinal velocity *Vx.* over the pipe cross sections (three-dimensional flow) for the quasi-parabolic turbulent velocity profile of the incoming flow at υ0 = 1 m/sec

velocity profile of the incoming flow at υ0 = 1 m/sec

*D D <sup>D</sup> <sup>I</sup>*

where *DH* is the hydraulic diameter (for the round cross section: *DH* = 2*r*1 = 0.612 m), υ0 is the

( )

incoming flow velocity, and *v* is the kinematic viscosity of oil (*v* = 1.4⋅10–4 m2/sec).

Fig. 5. Turbulence intensity (two-dimensional pipe flow, quasi-parabolic turbulent velocity profile, υ0 = 1 m/sec)

Fig. 6. Transverse velocity *Vy* for the two-dimensional flow in the pipe with corrosion damage at υ0 = 10 m/sec

The zone of the unsteady turbulent motion is characterized by the higher turbulence intensity (vortex formation) in comparison with the remaining region of the pipe (Figure 5). The highest intensity is observed in the steady motion zone, which is especially noticeable in the calculations with the initial velocity of 1 m/sec in the pipe wall region, whereas the lowest one – at the flow symmetry axis.

At high initial flow velocity values the vortex formation rate is higher.

Fig. 8. Wall tangential stresses at the pipe wall: stresses at *y* = *f*(*x*), stresses at *y* = 2*r*1 for the

Fig. 9. Wall tangential stresses at the pipe wall: stresses at *y* = *f*(*x*), stresses at *y* = 2*r*1 for the

0.306 0.306

υ

4 0.1211 10 4 0.1211 1 15.83 Pa, 1.58 Pa.

⋅ ⋅ ⋅ ⋅ <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> (19)

 μ μ

⎛⎞ ⎛⎞ ∂ ∂ ∂ ∂

 υ  υ

(20)

 τ

The expression for the tangential stresses with regard to the turbulence is of the form

<sup>0</sup> ' ' ' ( ) . *y y x x xy xy x y <sup>t</sup> <sup>y</sup> x y <sup>x</sup>*

The last formula and the analysis of the calculations enable evaluating the turbulence influence on the value of tangential stresses at the pipe wall. As indicated above, at different profiles and initial velocity values the tangential stresses were obtained: at υ0 = 1 m/sec:

ρυ υ

=+ = + − = + + ⎜⎟ ⎜⎟ ∂ ∂ ∂ ∂ ⎝⎠ ⎝⎠

two-dimensional flow in the pipe with corrosion damage at υ0 = 10 m/sec

10 1 0 0

υ

 μ

Then τ0 for the velocities υ0 = 10 m/sec and υ0= 1 m/sec will be

τ

τ ττ

[Sedov, 2004]:

two-dimensional flow in the pipe with corrosion damage at υ0 = 1 m/sec

It should be emphasized that at a higher value of the initial flow velocity, the instability region is longer: at υ0 *=* 1 m/sec its length is about 2 m, while at υ0 *=* 10 m /sec its length is about 5 m.

The behavior of the motion (steady or unsteady) exerts an influence on the value of wall stresses. In the unsteady motion zone, they are essentially higher as against the appropriate stresses in the identical steady motion zone.

These figures illustrate that at that place of the pipe, where the fluid motion becomes steady, the value of tangential stress at υ0 *=* 1 m/sec is approximately equal to 8 Pa, whereas at υ0 *=*  10 m/sec it is about 240 Pa.

The results as presented above are peculiar for a pipe with corrosion damage and without it. At the same time, the presence of corrosion damage affects the kinematics of the moving flow in calculations with both the rectangular profile of the initial flow velocity and the quasi-parabolic turbulent one. In this domain of geometry, there appear transverse displacements that form a recirculation zone (Figure 7).

Fig. 7. Transverse velocity *Vz* for the three-dimensional flow in the pipe with corrosion damage at υ0 = 10 m/sec

The corrosion spot exerts a profound effect on changes in wall tangential stresses in the area of the pipe corrosion damage.

Figures 8 and 9 demonstrate that in the corrosion damage area, the values of wall tangential stresses undergo jumping.

For the laminar fluid motion, the value of tangential stresses at the pipe wall is calculated by the following formula [Sedov, 2004]:

$$
\pi\_0 = \mu \left( \frac{\partial \nu\_x}{\partial y} + \frac{\partial \nu\_y}{\partial x} \right) = \mu \frac{d\nu}{dr} = \frac{4\,\mu\nu\_0}{r\_0},
\tag{18}
$$

where μ = υ⋅ρ = 1.4⋅10–4⋅865 = 0.1211 kg/(m*\**sec) is the molecular viscosity, *r*0 = 0.306 m is the pipe radius.

It should be emphasized that at a higher value of the initial flow velocity, the instability region is longer: at υ0 *=* 1 m/sec its length is about 2 m, while at υ0 *=* 10 m /sec its length is

The behavior of the motion (steady or unsteady) exerts an influence on the value of wall stresses. In the unsteady motion zone, they are essentially higher as against the appropriate

These figures illustrate that at that place of the pipe, where the fluid motion becomes steady, the value of tangential stress at υ0 *=* 1 m/sec is approximately equal to 8 Pa, whereas at υ0 *=* 

The results as presented above are peculiar for a pipe with corrosion damage and without it. At the same time, the presence of corrosion damage affects the kinematics of the moving flow in calculations with both the rectangular profile of the initial flow velocity and the quasi-parabolic turbulent one. In this domain of geometry, there appear transverse

Fig. 7. Transverse velocity *Vz* for the three-dimensional flow in the pipe with corrosion

The corrosion spot exerts a profound effect on changes in wall tangential stresses in the area

Figures 8 and 9 demonstrate that in the corrosion damage area, the values of wall tangential

For the laminar fluid motion, the value of tangential stresses at the pipe wall is calculated by

υ

where μ = υ⋅ρ = 1.4⋅10–4⋅865 = 0.1211 kg/(m*\**sec) is the molecular viscosity, *r*0 = 0.306 m is the

<sup>4</sup> , *<sup>y</sup> <sup>x</sup> <sup>d</sup> y x dr r*

 μ υ

0

(18)

0

μυ

about 5 m.

stresses in the identical steady motion zone.

displacements that form a recirculation zone (Figure 7).

10 m/sec it is about 240 Pa.

damage at υ0 = 10 m/sec

of the pipe corrosion damage.

the following formula [Sedov, 2004]:

0

τμ

υ

⎛ ⎞ ∂ ∂ = + == ⎜ ⎟ ∂ ∂ ⎝ ⎠

stresses undergo jumping.

pipe radius.

Fig. 8. Wall tangential stresses at the pipe wall: stresses at *y* = *f*(*x*), stresses at *y* = 2*r*1 for the two-dimensional flow in the pipe with corrosion damage at υ0 = 1 m/sec

Fig. 9. Wall tangential stresses at the pipe wall: stresses at *y* = *f*(*x*), stresses at *y* = 2*r*1 for the two-dimensional flow in the pipe with corrosion damage at υ0 = 10 m/sec

Then τ0 for the velocities υ0 = 10 m/sec and υ0= 1 m/sec will be

$$
\tau\_0^{10} = \frac{4 \cdot 0.1211 \cdot 10}{0.306} = 15.83 \text{ Pa}, \ \tau\_0^1 = \frac{4 \cdot 0.1211 \cdot 1}{0.306} = 1.58 \text{ Pa}.\tag{19}
$$

The expression for the tangential stresses with regard to the turbulence is of the form [Sedov, 2004]:

$$
\sigma\_{xy} = \tau\_0 + \tau\_{xy}' = \mu \left( \frac{\partial \overline{\nu}\_x}{\partial y} + \frac{\partial \overline{\nu}\_y}{\partial x} \right) - \rho \, \overline{\nu \, \overline{\nu}\_x \, \nu \prime}\_x = (\mu + \mu\_t) \left( \frac{\partial \overline{\nu}\_x}{\partial y} + \frac{\partial \overline{\nu}\_y}{\partial x} \right). \tag{20}
$$

The last formula and the analysis of the calculations enable evaluating the turbulence influence on the value of tangential stresses at the pipe wall. As indicated above, at different profiles and initial velocity values the tangential stresses were obtained: at υ0 = 1 m/sec:

With the use of the relationship between stresses and strains, and also of Hook's law, it is possible to determine integration constants *С*1and *С2* under the boundary conditions of the

1

*r r r*

=

σ

σ

where *р*1 is the internal pressure; *р*2 is the external pressure.

Fig. 10. Loading diagram of the circular cavity of the pipe

*r*

σ

ϕ

σ

σ

where *kr*2 = *r* /*r*2, *kr*12 = *r*1 / *r*<sup>2</sup>

*r r r*

=

1

*p*

= −

*p*

,

2

.

= − (27)

(28)

(29)

2

In such a case, the general formulas for stresses at any pipe point have the following form:

*pr pr p p rr rr rr r pr pr p p rr rr rr r*

Assuming that the cylinder is loaded only with the internal pressure (*р*1 = *p*, *р*2 = 0), the

*r r r r k k*

To analyze the rigid fixing of the outer surface of the pipeline, as one of the equations of the boundary conditions we choose expression (26) for displacements, the value of which tends to zero at the outer surface of the model. As the secondary boundary condition we use an

1 2 <sup>1</sup> , 0. *r r r r r r*

*p u* <sup>=</sup> <sup>=</sup> =− = (30)

σϕ ⎛⎞ ⎛⎞ = −= + ⎜⎟ ⎜⎟ − − ⎝⎠ ⎝⎠

12 12 2 2 2 2 2 12 2 12 1 1 1 , 1 , 1 1

− − <sup>=</sup> <sup>−</sup> − −

− − <sup>=</sup> <sup>+</sup> − −

following expressions are obtained for the stresses based on the internal pressure:

*p p*

expression for stresses at the inner surface of the cylinder from (27):

Then, the expressions for the stresses will assume the form:

σ

( ) 2 2 ( )

*r r r*

*p p kk kk*

2 2 2 2 11 22 1 2 12 22 22 2 21 21 2 2 2 2 11 22 1 2 12 22 22 2 21 21

( ) <sup>1</sup> ,

( ) <sup>1</sup> .

form:

τ*xy* = τ*<sup>w</sup>* ≈ 8 Pa, at υ0 = 10 m/sec: τ*xy* = τ*<sup>w</sup>* ≈ 240 Pa. The value of the turbulent stress (Reynolds stress):

at υ0 = 1 m/sec :

$$
\boldsymbol{\sigma}'\_{xy} = -\rho \,\overline{\boldsymbol{\nu}'\_{x} \boldsymbol{\nu}'\_{y}} = \mu\_t \left( \frac{\partial \overline{\boldsymbol{\nu}}\_{x}}{\partial y} + \frac{\partial \overline{\boldsymbol{\nu}}\_{y}}{\partial \mathbf{x}} \right) = \boldsymbol{\tau}\_{xy} - \boldsymbol{\tau}\_{0} = \mathbf{8} - \mathbf{1.58} = \mathbf{6.42} \,\text{Pa},\tag{21}
$$

at υ0= 10 m/sec :

$$\begin{split} \pi\_{xy}^{\prime} &= -\rho \overline{\nu\_{x}^{\prime} \nu\_{y}^{\prime}} = \mu\_{t} \bigg( \frac{\partial \overline{\nu\_{x}}}{\partial y} + \frac{\partial \overline{\nu}\_{y}}{\partial x} \bigg) = \pi\_{xy} - \pi\_{0} = \\ &= 240 - 15.83 = 224.17 \,\text{Pa}\_{t} \end{split} \tag{22}$$

The results obtained are evident of the fact that the turbulence much contributes to the formation of wall tangential stresses. At the higher turbulence intensity (it is especially high in the pipe wall region), Reynolds stresses increase, too. I.e., the turbulence stresses are:

$$\text{at } \mathbf{u}\_0 = 1 \text{ m/sec}: \frac{\tau\_{xy} - \tau\_0}{\tau\_{xy}}\\100\% = \frac{8 - 1.58}{8} \\ 100\% = 80.25\%; \tag{23}$$

$$\text{at } u\_0 = 10 \text{ m/sec}: \frac{\tau\_{xy} - \tau\_0}{\tau\_{xy}}\\100\% = \frac{240 - 15.83}{240}\\100\% = 93.4\%. \tag{24}$$

The analysis as made above shows that the calculation of the motion of a viscous fluid in the pipe as laminar can result in a highly distorted distribution pattern of the tangential stresses at the inner surface of the pipe. It can be concluded that the analysis of viscous fluid friction, when the flow interacts with the pipe wall, must be performed on the basis of the calculation of flow motion as essentially turbulent one.

#### **4. Analytical solutions for the stress-strain state of the pipeline model under the action of internal pressure and temperature difference**

In the simplified analytical statement, the problem of calculating the stress-strain state of a long cylindrical pipe reduces to the problem of the strain of a thin ring loaded with a pressure *p*1 uniformly distributed over its inner wall and also with a pressure *p*2 uniformly distributed over the outer surface of the ring (Figure 10). Operating conditions of the ring do not vary depending on whether it is considered either as isolated or as a part of the long cylinder.

Work [Ponomarev et al., 1958] and many other publications contain the classical solution to this problem based on solving the following differential equation for radial displacements:

$$\frac{d^2 u\_r}{dr^2} + \frac{1}{r} \frac{du\_r}{dr} - \frac{1}{r^2} u\_r = 0. \tag{25}$$

The general solution of this equation is of the form:

$$
\mu\_r = \mathbf{C}\_1 r + \mathbf{C}\_2 \frac{1}{r}. \tag{26}
$$

With the use of the relationship between stresses and strains, and also of Hook's law, it is possible to determine integration constants *С*1and *С2* under the boundary conditions of the form:

$$\begin{aligned} \left. \sigma\_r \right|\_{r=r\_1} &= -p\_1, \\ \left. \sigma\_r \right|\_{r=r\_2} &= -p\_2. \end{aligned} \tag{27}$$

where *р*1 is the internal pressure; *р*2 is the external pressure.

148 Tribology - Lubricants and Lubrication

τ*xy* = τ*<sup>w</sup>* ≈ 8 Pa, at υ0 = 10 m/sec: τ*xy* = τ*<sup>w</sup>* ≈ 240 Pa. The value of the turbulent stress (Reynolds

<sup>0</sup> ' '' 8 1.58 6.42Pa, *<sup>y</sup> <sup>x</sup> xy x y t xy y x*

τ τ

υ

240

τ τ

<sup>−</sup> <sup>−</sup> <sup>=</sup> <sup>=</sup> (23)

<sup>−</sup> <sup>−</sup> <sup>=</sup> <sup>=</sup> (24)

+ − = (25)

= + (26)

(21)

(22)

υ

<sup>0</sup> ' ''

The results obtained are evident of the fact that the turbulence much contributes to the formation of wall tangential stresses. At the higher turbulence intensity (it is especially high in the pipe wall region), Reynolds stresses increase, too. I.e., the turbulence stresses are:

The analysis as made above shows that the calculation of the motion of a viscous fluid in the pipe as laminar can result in a highly distorted distribution pattern of the tangential stresses at the inner surface of the pipe. It can be concluded that the analysis of viscous fluid friction, when the flow interacts with the pipe wall, must be performed on the basis of the

**4. Analytical solutions for the stress-strain state of the pipeline model under** 

In the simplified analytical statement, the problem of calculating the stress-strain state of a long cylindrical pipe reduces to the problem of the strain of a thin ring loaded with a pressure *p*1 uniformly distributed over its inner wall and also with a pressure *p*2 uniformly distributed over the outer surface of the ring (Figure 10). Operating conditions of the ring do not vary depending on whether it is considered either as isolated or as a part of the long

Work [Ponomarev et al., 1958] and many other publications contain the classical solution to this problem based on solving the following differential equation for radial displacements:

*u*

1

2 2 1 1 0. *r r <sup>r</sup>*

1 2

. *u Cr C <sup>r</sup> <sup>r</sup>*

*dr r r dr*

⎛ ⎞ <sup>∂</sup> <sup>∂</sup> <sup>=</sup> − = + = −= ⎜ ⎟ ∂ ∂ ⎝ ⎠

*<sup>y</sup> <sup>x</sup> xy x y t xy y x* υ

υ

240 15.83 224.17 Pa,

 at υ0 = 1 m/sec : <sup>0</sup> 8 1.58 100% 100% 80.25%; <sup>8</sup> *xy xy*

τ

at υ0 = 10 m/sec : <sup>0</sup> 240 15.83 100% 100% 93.4%.

τ

*xy xy*

τ

**the action of internal pressure and temperature difference** 

2

*d u du*

calculation of flow motion as essentially turbulent one.

The general solution of this equation is of the form:

τ

τ

τ

=− =

⎛ ⎞ ∂ ∂ =− = + = − = − = ⎜ ⎟ ∂ ∂ ⎝ ⎠

stress):

at υ0 = 1 m/sec :

at υ0= 10 m/sec :

cylinder.

τ

 ρυ υ μ

τ

 ρυ υ μ

Fig. 10. Loading diagram of the circular cavity of the pipe

In such a case, the general formulas for stresses at any pipe point have the following form:

$$\begin{aligned} \sigma\_r &= \frac{p\_1 \, r\_1^2 - p\_2 \, r\_2^2}{r\_2^2 - r\_1^2} - \frac{(p\_1 - p\_2) \, r\_1^2 \, r\_2^2}{r\_2^2 - r\_1^2} \frac{1}{r^2}, \\ \sigma\_\phi &= \frac{p\_1 \, r\_1^2 - p\_2 \, r\_2^2}{r\_2^2 - r\_1^2} + \frac{(p\_1 - p\_2) \, r\_1^2 \, r\_2^2}{r\_2^2 - r\_1^2} \frac{1}{r^2}. \end{aligned} \tag{28}$$

Assuming that the cylinder is loaded only with the internal pressure (*р*1 = *p*, *р*2 = 0), the following expressions are obtained for the stresses based on the internal pressure:

$$\frac{\sigma\_r^{(p)}}{p} = \frac{1}{k\_{r2}^2} \left( \frac{k\_{r12}^2}{1 - k\_{r12}^2} - 1 \right) \prime \ \frac{\sigma\_\phi^{(p)}}{p} = \frac{1}{k\_{r2}^2} \left( \frac{k\_{r12}^2}{1 - k\_{r12}^2} + 1 \right) \prime \tag{29}$$

where *kr*2 = *r* /*r*2, *kr*12 = *r*1 / *r*<sup>2</sup>

To analyze the rigid fixing of the outer surface of the pipeline, as one of the equations of the boundary conditions we choose expression (26) for displacements, the value of which tends to zero at the outer surface of the model. As the secondary boundary condition we use an expression for stresses at the inner surface of the cylinder from (27):

$$\left. \sigma\_r \right|\_{r=r\_1} = -p\_1, \left. u\_r \right|\_{r=r\_2} = 0. \tag{30}$$

Then, the expressions for the stresses will assume the form:

( ) () ( ) , , ,, *pT p <sup>T</sup>*

These figures well illustrate the essential influence of the temperature and the procedure of

 ϕ

*ir z* <sup>+</sup> =+ = (36)

5°С-1, Δ*T* = 20 °C).

*i ii*

 σσ

σ

fixing the pipe on its stress-strain state.

*ν* = 0.3.

for *kr*12 = 0.8, *v* = 0.3, *E*1α*T* / *p* = 10 (for example, at *E*1 = 2⋅1011 Pa, α = 10-

Fig. 11. Radial stresses for problems (25), (27) and (33), (34) at *r*1 ≤ *r* ≤ *r*2

Fig. 12. Circumferential stresses for problems (25), (27) and (33), (34) at *r*1 ≤ *r* ≤ *r*2

Compare the distribution of the stresses calculated analytically with the use of (31) for a non-damaged pipe with the finite-element calculation results by plotting the graphs of the pipe thickness stress distribution (Figures 1.15–1.16). To make calculation, take the following initial data: inner and outer radii *r*1= 0.306 m and *r*2= 0.315 m, *p*1= 4М Pa, *p*2= 0, *Е* = 2⋅1011 Pa,

$$\begin{split} \frac{\sigma\_r^{(p)}}{p} &= -\frac{k\_{r12}^2}{k\_{r2}^2} \frac{k\_{r2}^2 (1+\nu\_1) - (\nu\_1 - 1)}{k\_{r12}^2 (1+\nu\_1) - (\nu\_1 - 1)},\\ \frac{\sigma\_\wp^{(p)}}{p} &= -\frac{k\_{r12}^2}{k\_{r2}^2} \frac{k\_{r2}^2 (1+\nu\_1) + (\nu\_1 - 1)}{k\_{r12}^2 (1+\nu\_1) - (\nu\_1 - 1)}. \end{split} \tag{31}$$

Consider a long thick-wall pipe, whose wall temperature *t* varies across the wall, but is constant along the pipe, i. e., *t = t*(*r*) [Ponomarev et al., 1958]*.*

If the heat flux is steady and if the temperature of the outer surface of the pipe is equal to zero and that of the inner surface is designated as *Т,* then from the theory of heat transfer it follows that the dependence of the temperature *t* on the radius *r* is given by the formula

$$t = \frac{T}{\ln k\_{r12}} \ln k\_{r2'} \tag{32}$$

Any other boundary conditions can be obtained by making uniform heating or cooling, which does not cause any stresses. Thus, the quantity *Т* in essence represents the temperature difference Δ*T* of the inner and outer surfaces of the pipe.

As the temperature is constant along the pipe, it can be considered that cross sections at a sufficient distance from the pipe ends remain plane, and the strain ε*z* is a constant quantity.

The temperature influence can be taken into account if the strains due to stresses are added with the uniform temperature expansion Δε = αΔ*T* where α is the linear expansion coefficient of material.

The stress-strain state in the presence of the temperature difference between the pipe walls can be determined by solving the differential equation [Ponomarev et al., 1958]:

$$\frac{d^2u}{dr^2} + \frac{1}{r}\frac{du}{dr} - \frac{u}{r^2} = \frac{1+\nu\_1}{1-\nu\_1}a\frac{dt}{dr}.\tag{33}$$

Subject to the boundary conditions

$$
\left.\sigma\_r\right|\_{r=r\_1} = 0, \left.\sigma\_r\right|\_{r=r\_2} = 0. \tag{34}
$$

Having solved boundary-value problem (33), (34), the expressions for stresses are of the form:

$$\begin{aligned} \sigma\_r^{(T)} &= \frac{E\_1 a \Delta T}{2(1 - \nu)} \frac{1}{\ln k\_{r12}} \left[ -\ln k\_{r2} - \frac{k\_{r12}^2}{1 - k\_{r12}^2} \left( 1 - \frac{1}{k\_{r2}^2} \right) \ln k\_{r12} \right], \\ \sigma\_\phi^{(T)} &= \frac{E\_1 a \Delta T}{2(1 - \nu)} \frac{1}{\ln k\_{r12}} \left[ -1 - \ln k\_{r2} - \frac{k\_{r12}^2}{1 - k\_{r12}^2} \left( 1 + \frac{1}{k\_{r2}^2} \right) \ln k\_{r12} \right], \\ \sigma\_z^{(T)} &= \frac{E\_1 a \Delta T}{2(1 - \nu)} \frac{1}{\ln k\_{r12}} \left[ -1 - 2 \ln k\_{r2} - \frac{2k\_{r12}^2}{1 - k\_{r12}^2} \ln k\_{r12} \right] \end{aligned} \tag{35}$$

Figures 11–14 show the distribution of dimensionless stresses (29), (31), (35) along *r* and their sums

12 2 1 1

+−− = − +−−

++− = − +−−

Consider a long thick-wall pipe, whose wall temperature *t* varies across the wall, but is

If the heat flux is steady and if the temperature of the outer surface of the pipe is equal to zero and that of the inner surface is designated as *Т,* then from the theory of heat transfer it follows that the dependence of the temperature *t* on the radius *r* is given by the formula

> 12 ln , ln *<sup>r</sup> r <sup>T</sup> t k*

Any other boundary conditions can be obtained by making uniform heating or cooling, which does not cause any stresses. Thus, the quantity *Т* in essence represents the

As the temperature is constant along the pipe, it can be considered that cross sections at a sufficient distance from the pipe ends remain plane, and the strain ε*z* is a constant quantity. The temperature influence can be taken into account if the strains due to stresses are added with the uniform temperature expansion Δε = αΔ*T* where α is the linear expansion

The stress-strain state in the presence of the temperature difference between the pipe walls

1 2 0, 0. *r r rr rr*

Having solved boundary-value problem (33), (34), the expressions for stresses are of the

2

12 12 2

*r r r*

*E T <sup>k</sup> k k k k k*

<sup>Δ</sup> <sup>⎡</sup> ⎛ ⎞ <sup>⎤</sup> <sup>=</sup> ⎢− −− + ⎜ ⎟ <sup>⎥</sup> <sup>−</sup> ⎢⎣ <sup>−</sup> ⎝ ⎠ ⎥⎦

1 1 1 ln 1 ln , 2 1 ln <sup>1</sup>

1 1 ln 1 ln , 2 1 ln <sup>1</sup>

*E T <sup>k</sup> k k k kk*

<sup>Δ</sup> ⎡ ⎤ ⎛ ⎞ = −− − ⎢ ⎥ ⎜ ⎟ <sup>−</sup> <sup>−</sup> ⎣ ⎦ ⎝ ⎠

12 12 2

*r r r*

12 12

<sup>1</sup> <sup>2</sup> 1 2ln ln , 2 1 ln <sup>1</sup>

*r r*

Figures 11–14 show the distribution of dimensionless stresses (29), (31), (35) along *r* and

*<sup>T</sup> <sup>r</sup> r r*

 σ

1 1 . <sup>1</sup> *d u du u dt dr r dr r dr*

1

α

2 1 2 2 2

2

2 1 2 2

2 1 2 2 2

2

+ −= <sup>−</sup> (33)

<sup>=</sup> <sup>=</sup> = = (34)

(35)

ν

+

ν

1

can be determined by solving the differential equation [Ponomarev et al., 1958]:

2 2

σ

1 12

1 12

*<sup>T</sup> <sup>r</sup> r rr*

1 12

*E T <sup>k</sup> k k k k*

<sup>Δ</sup> ⎡ ⎤ <sup>=</sup> ⎢ ⎥ −− − <sup>−</sup> <sup>−</sup> ⎣ ⎦

*<sup>T</sup> <sup>r</sup> z r <sup>r</sup>*

 ν

ν

ν

ν

2

*<sup>k</sup>* <sup>=</sup> (32)

2 12 1 1

(1 ) ( 1) , (1 ) ( 1)

ν

 ν

(31)

 ν

> ν

(1 ) ( 1) . (1 ) ( 1)

12 2 1 1

2 12 1 1

2 2

*k k*

*r rr*

*p k k*

*p k k*

temperature difference Δ*T* of the inner and outer surfaces of the pipe.

2

coefficient of material.

form:

their sums

Subject to the boundary conditions

( )

σ

( )

ϕ

σ

σ

( )

( )

ν

α

α

α

( )

ν

( )

ν

2 2

*r r*

2 2

*r r r r*

*k k*

2 2

( )

σ

*p*

( )

ϕ

σ

constant along the pipe, i. e., *t = t*(*r*) [Ponomarev et al., 1958]*.*

*p*

$$
\sigma\_i^{(p+T)} = \sigma\_i^{(p)} + \sigma\_i^{(T)}, \; i = r, \; \rho, z,\tag{36}
$$

for *kr*12 = 0.8, *v* = 0.3, *E*1α*T* / *p* = 10 (for example, at *E*1 = 2⋅1011 Pa, α = 10- 5°С-1, Δ*T* = 20 °C). These figures well illustrate the essential influence of the temperature and the procedure of fixing the pipe on its stress-strain state.

Fig. 11. Radial stresses for problems (25), (27) and (33), (34) at *r*1 ≤ *r* ≤ *r*2

Compare the distribution of the stresses calculated analytically with the use of (31) for a non-damaged pipe with the finite-element calculation results by plotting the graphs of the pipe thickness stress distribution (Figures 1.15–1.16). To make calculation, take the following initial data: inner and outer radii *r*1= 0.306 m and *r*2= 0.315 m, *p*1= 4М Pa, *p*2= 0, *Е* = 2⋅1011 Pa, *ν* = 0.3.

Fig. 12. Circumferential stresses for problems (25), (27) and (33), (34) at *r*1 ≤ *r* ≤ *r*2

σ

*<sup>r</sup>* ), for the three-dimensional computer model ( ( ) <sup>3</sup>*<sup>D</sup>*

), for the three-dimensional computer model ( ( ) <sup>3</sup>*<sup>D</sup>*

*p* <sup>=</sup> = = (37)

*<sup>r</sup>* ), for the two-

σ ϕ

), for the two-

σ ϕ)

σ*<sup>r</sup>* )

Fig. 15. Radial stress distribution for the analytical calculation ( ( ) *<sup>p</sup>*

Fig. 16. Circumferential stress distribution for the analytical calculation ( ( ) *<sup>p</sup>*

respectively, the length of the calculated pipe section *L*=3 m, sizes of elliptical corrosion

The pipe mateial had the following characteristics: elasticity modulus *E*1 = 2⋅1011 Pa, Poisson's coefficient *v*1 = 0.3, temperature expansion coefficient α = 10-5 °С-1, thermal conductivity *k* = 43 W/(m°С), and the soil parameters were: *E*2 = 1.5⋅109 Pa, Poisson's

4 MPa. *<sup>r</sup> r r*

coefficient *v*2 = 0.5. The coefficient of friction between the pipe and soil was μ = 0.5.

1

σ

σ ϕ

damage length × width × depth – 0.8 m × 0.4 m × 0.0034 m.

σ

dimensional computer model ( (2*D*)

dimensional computer model ( (2*D*)

The internal pressure in the pipe (1) is:

Fig. 13. Longitudinal stresses for problems (25), (30) and (33), (34) at *r*1 ≤ *r* ≤ *r*2

Fig. 14. Circumferential stresses for problems (25), (30) and (33), (34) at *r*1 ≤ *r* ≤ *r*2

As seen from Figures 15–16, the σ*r* and σϕ distributions obtained from the analytical calculation practically fully coincide with those obtained from the finite-element calculation, which points to a very small error of the latter.

#### **5. Stress-strain state of the three-dimenisonal model of a pipe with corrosion damage under complex loading**

Consider the problem of determining the stress-strain state of a two-dimenaional model of a pipe in the area of three-dimensional elliptical damage.

In calculations we used a model of a pipe with the following geometric characteristics (Figure 2): inner (without damage) and outer radii *r*1 = 0.306 m and *r*2= 0.315 m,

Fig. 13. Longitudinal stresses for problems (25), (30) and (33), (34) at *r*1 ≤ *r* ≤ *r*2

Fig. 14. Circumferential stresses for problems (25), (30) and (33), (34) at *r*1 ≤ *r* ≤ *r*2

which points to a very small error of the latter.

pipe in the area of three-dimensional elliptical damage.

**damage under complex loading** 

As seen from Figures 15–16, the σ*r* and σϕ distributions obtained from the analytical calculation practically fully coincide with those obtained from the finite-element calculation,

**5. Stress-strain state of the three-dimenisonal model of a pipe with corrosion** 

Consider the problem of determining the stress-strain state of a two-dimenaional model of a

In calculations we used a model of a pipe with the following geometric characteristics (Figure 2): inner (without damage) and outer radii *r*1 = 0.306 m and *r*2= 0.315 m,

Fig. 15. Radial stress distribution for the analytical calculation ( ( ) *<sup>p</sup>* σ *<sup>r</sup>* ), for the twodimensional computer model ( (2*D*) σ *<sup>r</sup>* ), for the three-dimensional computer model ( ( ) <sup>3</sup>*<sup>D</sup>* σ*<sup>r</sup>* )

Fig. 16. Circumferential stress distribution for the analytical calculation ( ( ) *<sup>p</sup>* σ ϕ ), for the twodimensional computer model ( (2*D*) σ ϕ ), for the three-dimensional computer model ( ( ) <sup>3</sup>*<sup>D</sup>* σ ϕ)

respectively, the length of the calculated pipe section *L*=3 m, sizes of elliptical corrosion damage length × width × depth – 0.8 m × 0.4 m × 0.0034 m.

The pipe mateial had the following characteristics: elasticity modulus *E*1 = 2⋅1011 Pa, Poisson's coefficient *v*1 = 0.3, temperature expansion coefficient α = 10-5 °С-1, thermal conductivity *k* = 43 W/(m°С), and the soil parameters were: *E*2 = 1.5⋅109 Pa, Poisson's coefficient *v*2 = 0.5. The coefficient of friction between the pipe and soil was μ = 0.5. The internal pressure in the pipe (1) is:

$$
\left.\sigma\_r\right|\_{r=r\_1} = p = 4 \text{ MPa}.\tag{37}
$$

The analysis of the calculation results will be mainly made for the normal (principal) stresses σ*x*, σ*y*, σ*z* in the Cartesian system of coordinates. It should be noted that for axissymmetrical models, among which is a pipe, the cylindrical system of coordinates is natural, in which the normal stresses in the radial σ*r*, circumferential σ*t*, and axial σ*z* directions are principal. Since the software ANSYS does not envisage stresses in the polar system of coordinates, the analysis of the stress state will be made on the basis of σ*x*, σ*y*, σ*z* in those domains where they coincide with σ*r*, σ*t*, σ*z* corresponding to the last principal stresses σ1,

Make a comparative analysis of the results of numerical calculation for boundary conditions (1), (6) and (1), (7) with those of analytical calculation as described in Sect. 1.4. Consider pipe

damage exerts an essential influence on the σ*t* distribution over the inner surface of the pipe. At the damage edge, the absolute value of circumferential σ*t* is, on average, by 15% higher than the one at the inner surface of the pipe with damage and, on average, by 30 % higher

Figures 18 and Figure 19 show that in the case of fixing 2 <sup>2</sup>

than the one inside damage. In the case of fixing 2 2 <sup>2</sup>

distributions are localized just in the damage area. The additional key condition 2

(coupling along the *z*-axis) is expressed in increasing |σ*t*| at the inner surface without damage in the calculation for (1), (7) approximately by 60% in comparison with the calculation for (1), (6). However in the calculation for (1), (7), the |σ*t*| differences between the damage edge, the inner surface without damage, and the inner surface with damage are, on average, only 6 and 3% , respectively. Maximum and minimum values of σ*t* in the

*<sup>t</sup>* =− ⋅ Pa and max <sup>5</sup> 7.96 10

The analysis of the stress distribution reveals a good coincidence of the results of the analytical and finite-element calculations for σ*t*. At *r*<sup>1</sup> ≤ *y* ≤ *r*2, *x=z=*0 in the vicinity of the

1.093 1.082 100% 1.03%, 1.093

1.175 1.165 100% 0.94%.

Thus, at the upper inner surface of the pipe the damage influence on the σ*t* variation is inconsiderable. A comparatively small error as obtained above is attributed to the fact that the three-dimensional calculation subject to (1), (6) was made at the same key conditions as the analytical calculation of the two-dimensional model. At the same time, owing to the

σ

*<sup>e</sup>* <sup>−</sup> = ⋅= (41)

*<sup>e</sup>* <sup>−</sup> = ⋅= (42)

0 *<sup>z</sup> r r u* <sup>=</sup> = the difference between the results of the analytical

*<sup>t</sup>* = − ⋅ Pa.

*a* − −

<sup>=</sup> = =⋅ = ⋅ (40)

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> , corrosion

<sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = = , the σ*<sup>t</sup>*

*<sup>t</sup>* =− ⋅ Pa; in the calculation

0 *<sup>z</sup> r r u* <sup>=</sup> =

( ) 2 42 <sup>0</sup> sin 260 10 3 / 2 2.25 10 Pa. *node*

1

stresses in the circumferential σ*t* and radial σ*r* directions.

τ

σ2, σ3 and also to the tangential stresses σ*yz*.

calculation for (1), (6) are: min <sup>6</sup> 1.27 10

pipe without damage, the error is at *r* = *r*<sup>1</sup>

for (1), (7) are: min <sup>6</sup> 1.72 10 σ

additonal condition 2

at *r* = *r*<sup>2</sup>

σ

*<sup>t</sup>* =− ⋅ Pa and max <sup>6</sup> 1.61 10

σ

1.175

calculation and the calculation for (1), (7) is much greater – about 45 %.

*rz FE FE r r*

τβ

The temperature diffference between the pipe walls is (3)

$$\left| T\_{r\_1} - T\_{r\_2} \right| = \Delta T = \mathfrak{D}0^o \mathbb{C}.\tag{38}$$

The value of internal tangential stresses (wall friction) (2) is determined from the hydrodynamic calculation of the turbulent motion of a viscous fluid in the pipe.

Calculations in the absence of fixing of the outer surface of the pipe and in the presence of the friction force over the inner surface (2) were made for 1/2 of the main model (Figure 2), since in this case (in the presence of friction) the calculation model has only one symmetry plane. In the absence of outer surface fixing, calculations were made for 1/4 of the model of the pipeline section since the boundary conditions of form (2) are also absent and, hence, the model has two symmetry planes.

The investigation of the stress state of the pipe in soil is peformed for 1/4 of the main model of the pipe placed inside a hollow elastic cylinder modeling soil (Figure 17).

In calculations without temperature load, a finite-element grid is composed of 20-node elements SOLID95 (Figure 17) meant for three-dimensional solid calculations. In the presence of temperature difference, a grid is composed of a layer of 10-node finite elements SOLID98 intended for three-dimensional solid and temperature calculations. The size of a finite element (fin length) *aFE* =10-2 m.

Fig. 17. General view and the finite-element partition of ¼ of the pipe model in soil

Thus, the pipe wall is composed of one layer of elements since its thickness is less than centermeter. During a compartively small computer time such partition allows obtaining the results that are in good agreement with the analytical ones (see, below).

Calculations for boundary conditions (8) with a description of the contact between the pipe and soil use elements CONTA175 and TARGE170.

As seen from Figure 17, the finite elements are mainly shaped as a prism, the base of which is an equivalateral triangle. The value of the tangential stresses 1 *rz r r* τ <sup>=</sup> applied to each node of the inner surface will then be calculated as follows:

$$\left. \pi\_{rz}^{(n\alpha dc)} \right|\_{r=r\_1} = \pi\_0 S\_\prime \tag{39}$$

where *S* is the area of the romb with the side *aFE* and with the acute angle β*FE* = π/3. Thus, the value of the tangential stress applied at one node will be

$$
\left.\tau\_{rz}^{(node)}\right|\_{r=r\_1} = \tau\_0 a\_{FE}^2 \sin \beta\_{FE} = 260 \cdot 10^{-4} \sqrt{3} \text{ J/2} = 2.25 \cdot 10^{-2} \text{Pa}.\tag{40}
$$

The analysis of the calculation results will be mainly made for the normal (principal) stresses σ*x*, σ*y*, σ*z* in the Cartesian system of coordinates. It should be noted that for axissymmetrical models, among which is a pipe, the cylindrical system of coordinates is natural, in which the normal stresses in the radial σ*r*, circumferential σ*t*, and axial σ*z* directions are principal. Since the software ANSYS does not envisage stresses in the polar system of coordinates, the analysis of the stress state will be made on the basis of σ*x*, σ*y*, σ*z* in those domains where they coincide with σ*r*, σ*t*, σ*z* corresponding to the last principal stresses σ1, σ2, σ3 and also to the tangential stresses σ*yz*.

Make a comparative analysis of the results of numerical calculation for boundary conditions (1), (6) and (1), (7) with those of analytical calculation as described in Sect. 1.4. Consider pipe stresses in the circumferential σ*t* and radial σ*r* directions.

$$\text{Figures 18 and Figure 19 show that in the case of fixing } \left. u\_x \right|\_{r=r\_2} = \left. u\_y \right|\_{r=r\_2} = 0 \text{ , cross-sectional area of } \left. u\_x \right|\_{r=r\_2} = \left. \frac{\partial u\_x}{\partial x} \right|\_{r=r\_2} = \frac{\partial u\_y}{\partial y} = 0$$

damage exerts an essential influence on the σ*t* distribution over the inner surface of the pipe. At the damage edge, the absolute value of circumferential σ*t* is, on average, by 15% higher than the one at the inner surface of the pipe with damage and, on average, by 30 % higher than the one inside damage. In the case of fixing 2 2 <sup>2</sup> <sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = = , the σ*<sup>t</sup>*

distributions are localized just in the damage area. The additional key condition 2 0 *<sup>z</sup> r r u* <sup>=</sup> =

(coupling along the *z*-axis) is expressed in increasing |σ*t*| at the inner surface without damage in the calculation for (1), (7) approximately by 60% in comparison with the calculation for (1), (6). However in the calculation for (1), (7), the |σ*t*| differences between the damage edge, the inner surface without damage, and the inner surface with damage are, on average, only 6 and 3% , respectively. Maximum and minimum values of σ*t* in the calculation for (1), (6) are: min <sup>6</sup> 1.27 10 σ *<sup>t</sup>* =− ⋅ Pa and max <sup>5</sup> 7.96 10 σ *<sup>t</sup>* =− ⋅ Pa; in the calculation for (1), (7) are: min <sup>6</sup> 1.72 10 σ *<sup>t</sup>* =− ⋅ Pa and max <sup>6</sup> 1.61 10 σ*<sup>t</sup>* = − ⋅ Pa.

The analysis of the stress distribution reveals a good coincidence of the results of the analytical and finite-element calculations for σ*t*. At *r*<sup>1</sup> ≤ *y* ≤ *r*2, *x=z=*0 in the vicinity of the pipe without damage, the error is at *r* = *r*<sup>1</sup>

$$e = \frac{1.093 - 1.082}{1.093} \cdot 100\% = 1.03\% \tag{41}$$

at *r* = *r*<sup>2</sup>

154 Tribology - Lubricants and Lubrication

The value of internal tangential stresses (wall friction) (2) is determined from the

Calculations in the absence of fixing of the outer surface of the pipe and in the presence of the friction force over the inner surface (2) were made for 1/2 of the main model (Figure 2), since in this case (in the presence of friction) the calculation model has only one symmetry plane. In the absence of outer surface fixing, calculations were made for 1/4 of the model of the pipeline section since the boundary conditions of form (2) are also absent and, hence, the

The investigation of the stress state of the pipe in soil is peformed for 1/4 of the main model

In calculations without temperature load, a finite-element grid is composed of 20-node elements SOLID95 (Figure 17) meant for three-dimensional solid calculations. In the presence of temperature difference, a grid is composed of a layer of 10-node finite elements SOLID98 intended for three-dimensional solid and temperature calculations. The size of a

Thus, the pipe wall is composed of one layer of elements since its thickness is less than centermeter. During a compartively small computer time such partition allows obtaining the

Calculations for boundary conditions (8) with a description of the contact between the pipe

As seen from Figure 17, the finite elements are mainly shaped as a prism, the base of which

1

where *S* is the area of the romb with the side *aFE* and with the acute angle β*FE* = π/3. Thus,

<sup>0</sup> , *node rz r r* τ

τ

( )

τ

*<sup>S</sup>* <sup>=</sup> <sup>=</sup> (39)

<sup>=</sup> applied to each node

Fig. 17. General view and the finite-element partition of ¼ of the pipe model in soil

results that are in good agreement with the analytical ones (see, below).

is an equivalateral triangle. The value of the tangential stresses 1 *rz r r*

and soil use elements CONTA175 and TARGE170.

of the inner surface will then be calculated as follows:

the value of the tangential stress applied at one node will be

hydrodynamic calculation of the turbulent motion of a viscous fluid in the pipe.

of the pipe placed inside a hollow elastic cylinder modeling soil (Figure 17).

1 2 20 . *<sup>о</sup> TT T r r* − =Δ = *<sup>С</sup>* (38)

The temperature diffference between the pipe walls is (3)

model has two symmetry planes.

finite element (fin length) *aFE* =10-2 m.

$$e = \frac{1.175 - 1.165}{1.175} \cdot 100\% = 0.94\%. \tag{42}$$

Thus, at the upper inner surface of the pipe the damage influence on the σ*t* variation is inconsiderable. A comparatively small error as obtained above is attributed to the fact that the three-dimensional calculation subject to (1), (6) was made at the same key conditions as the analytical calculation of the two-dimensional model. At the same time, owing to the additonal condition 2 0 *<sup>z</sup> r r u* <sup>=</sup> = the difference between the results of the analytical calculation and the calculation for (1), (7) is much greater – about 45 %.

Path 5. Along the straight line of the upper inner surface of the pipe

Path 6. Along the curve of the lower inner surface of the pipe – 0.8*L*/2 ≤ *z* ≤ 0.8*L*/2 at *x=*0,

*P*64(0, – *r*1, – 0.8*L*/2), *P*63(0, – *r*1, – *d*/2), *P*62(0, – *r*1, – 0.0025, –0.2), *P*61(0, – *r*1, – *h*, 0), *P*62(0, – *r*1, –

For 1/4 of the pipe model, paths 1–4 are the same as those for 1/2, whereas paths 5 and 6

Path 5. Along the strainght upper inner surface of the pipe 0 ≤ *z* ≤ 0.8*L*/2 at *x=*0, *y=r*1: from

Path 6. Along the curve of the lower inner surface of the pipe 0 ≤ *z* ≤ 0.8*L*/2 at *x=*0,

In the above descriptions of the paths, *d*=0.8 m is the length of corrosion damage along the *z* axis of the pipe. The function *f*(*z*) describes the inhomogeneity of the geometry of the inner

The analysis of the distributions shows that |σ*t*| increases up to 10% from the inner to the outer surface along paths 1, 2, 4 and decreases up to 2% along path 3. Thus, it is seen that at the corrosion damage edge over the cross section (path 3), the |σ*t*| distribution has a specific pattern. It should also be mentioned that if in the calculation for (1), (6), |σ*t*| inside the damage is approximately by 20% less than the one at the inner surface without damage,

Figure 20 shows the σ*r* distribution that is very similar to those in the calculations for (1), (6) and for (1), (7). I.e., the procedure of fixing the outer surface of the pipe practically does not influencesthe σ*r* distribution. At the corrorion damage edge of the inner surface of the pipe, the σ*r* distribution undergoes small variation (up to 1%). Maximum and

*<sup>r</sup>* =− ⋅ Pa and max <sup>6</sup> 3.91 10

*<sup>r</sup>* =− ⋅ Pa and max <sup>6</sup> 3.92 10

The numerical analysis of the resuts reveals a good agreement between the results of analytical and finite-element calculations for σ*r* ((1), (6)). For *r*<sup>1</sup> ≤ *y* ≤ *r*2, *x=z=*0 in the region

Make a comparative analysis of the results of these numerical calculations for (1), and (1), (8) with those of the analytical calculation described in Sect. 1.4 for the boundary conditions of

σ*r* directions under the action of internal pressure (1) for fixing absent at the outer surface

σ

σ

<sup>=</sup> = . Consider pipe stresses in the circumfrenetial σ*t* and radial

– 0.8*L*/2 ≤ *z* ≤ 0.8*L*/2 at *x =* 0, *y = r*1: from *P*51(0, *r*1, – 0.8*L*/2) to *P*52(0, *r*1, 0.8*L*/2).

through the points:

through the points:

*P*61(0, – *r*1, – *h*, 0), *P*62(0, – *r*1, – 0.0025, 0.2), *P*63(0, – *r*1, *d*/2), *P*64(0, – *r*1, 0.8*L*/2).

then in the calculation for (1), (7) this stress is approximately by 2% higher.

minimum values of σ*r* in the calculation for (1), (6) are: min <sup>6</sup> 4.02 10

*<sup>r</sup>* =− ⋅ Pa; in the calculation for (1), (7): min <sup>6</sup> 4.02 10

of the pipe without damage at *r* = *r*1*e is* >>1%, whereas at *r* = *r*2*e is* ≈1% for (1), (6).

Path 3. Cavity boundary over the cross section *z*=0: from *P* 31(0.186, – 0.243, 0) to *P* 32(0.192, – 0.25, 0). Path 4. Cavity boundary over the cross section *x*=0:

from *P* 41(0, *–r*1, *d/*2) to *P* 42(0, *–r*2, *d/*2).

, / 2 0.8 / 2 *r fz z d <sup>y</sup> rd z L* ⎧⎪−= ≤ ≤ <sup>=</sup> <sup>⎨</sup> ⎪− ≤≤ ⎩

*P*51(0, *r*1, 0) to *P*52(0, *r*1, 0.8*L*/2).

, /2 0.8 / 2

<sup>1</sup> ( )

*r fz z d <sup>y</sup> rd z L* ⎧− = ≤ ≤ <sup>⎪</sup> <sup>=</sup> <sup>⎨</sup> ⎪− ≤≤ ⎩

,0 /2

0.0025, 0.2), *P*63(0, – *r*1, *d*/2), *P*64(0, – *r*1, 0.8*L*/2).

,0 /2

surface of the pipe with corrosion damage.

<sup>1</sup> ( )

1

are of the form:

1

σ

σ

*<sup>r</sup>* =− ⋅ Pa.

the form 1 *<sup>r</sup> r r* σ

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2

σ

0 *<sup>r</sup> r r*

and at the contact between the the pipe and soil (1), (8).

Fig. 18. Distribution of the stress σ2(σ*t*) at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

Fig. 19. Distribution of the stress σ1 (σ*t*) at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup> <sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

A more detailed analysis of the stress-strain state can be made for distributions along the below paths.

For 1/2 of the pipe model: Path 1. Along the straight line *r*<sup>1</sup> ≤ *y* ≤ *r*2 at *x=z=*0: from *P*11(0, *r*1, 0) to *P*12(0, *r*2, 0). Path 2. Corrosion damage center (– *r*1 – *h* ≤ *y* ≤ – *r*2 at *x=z=*0): from *P* 21(0, *– r* 1*– h*, 0) to *P* 22(0, *– r*2, 0).

Path 3. Cavity boundary over the cross section *z*=0: from *P* 31(0.186, – 0.243, 0) to *P* 32(0.192, – 0.25, 0).

Path 4. Cavity boundary over the cross section *x*=0:

from *P* 41(0, *–r*1, *d/*2) to *P* 42(0, *–r*2, *d/*2).

156 Tribology - Lubricants and Lubrication

σ

σ

A more detailed analysis of the stress-strain state can be made for distributions along the

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup>

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup>

<sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

Fig. 18. Distribution of the stress σ2(σ*t*) at 1 *<sup>r</sup> r r*

Fig. 19. Distribution of the stress σ1 (σ*t*) at 1 *<sup>r</sup> r r*

Path 1. Along the straight line *r*<sup>1</sup> ≤ *y* ≤ *r*2 at *x=z=*0:

Path 2. Corrosion damage center (– *r*1 – *h* ≤ *y* ≤ – *r*2 at *x=z=*0):

below paths.

For 1/2 of the pipe model:

from *P*11(0, *r*1, 0) to *P*12(0, *r*2, 0).

from *P* 21(0, *– r* 1*– h*, 0) to *P* 22(0, *– r*2, 0).

Path 5. Along the straight line of the upper inner surface of the pipe – 0.8*L*/2 ≤ *z* ≤ 0.8*L*/2 at *x =* 0, *y = r*1: from *P*51(0, *r*1, – 0.8*L*/2) to *P*52(0, *r*1, 0.8*L*/2).

Path 6. Along the curve of the lower inner surface of the pipe – 0.8*L*/2 ≤ *z* ≤ 0.8*L*/2 at *x=*0,

$$\begin{array}{l} \text{If } y = \begin{cases} -r\_1 = f\left(z\right), 0 \le |z| \le d/2\\ -r\_1, d/2 \le |z| \le 0.8L / 2 \end{cases} \text{through the points: } \mathbf{z} \end{array}$$

*P*64(0, – *r*1, – 0.8*L*/2), *P*63(0, – *r*1, – *d*/2), *P*62(0, – *r*1, – 0.0025, –0.2), *P*61(0, – *r*1, – *h*, 0), *P*62(0, – *r*1, – 0.0025, 0.2), *P*63(0, – *r*1, *d*/2), *P*64(0, – *r*1, 0.8*L*/2).

For 1/4 of the pipe model, paths 1–4 are the same as those for 1/2, whereas paths 5 and 6 are of the form:

Path 5. Along the strainght upper inner surface of the pipe 0 ≤ *z* ≤ 0.8*L*/2 at *x=*0, *y=r*1: from *P*51(0, *r*1, 0) to *P*52(0, *r*1, 0.8*L*/2).

Path 6. Along the curve of the lower inner surface of the pipe 0 ≤ *z* ≤ 0.8*L*/2 at *x=*0,

$$y = \begin{cases} -r\_1 = f\left(z\right), 0 \le z \le d \ne 2\\ -r\_1, d \ne 2 \le z \le 0.8L \ne 2 \end{cases} \text{ through the points: } 1$$

*P*61(0, – *r*1, – *h*, 0), *P*62(0, – *r*1, – 0.0025, 0.2), *P*63(0, – *r*1, *d*/2), *P*64(0, – *r*1, 0.8*L*/2).

In the above descriptions of the paths, *d*=0.8 m is the length of corrosion damage along the *z* axis of the pipe. The function *f*(*z*) describes the inhomogeneity of the geometry of the inner surface of the pipe with corrosion damage.

The analysis of the distributions shows that |σ*t*| increases up to 10% from the inner to the outer surface along paths 1, 2, 4 and decreases up to 2% along path 3. Thus, it is seen that at the corrosion damage edge over the cross section (path 3), the |σ*t*| distribution has a specific pattern. It should also be mentioned that if in the calculation for (1), (6), |σ*t*| inside the damage is approximately by 20% less than the one at the inner surface without damage, then in the calculation for (1), (7) this stress is approximately by 2% higher.

Figure 20 shows the σ*r* distribution that is very similar to those in the calculations for (1), (6) and for (1), (7). I.e., the procedure of fixing the outer surface of the pipe practically does not influencesthe σ*r* distribution. At the corrorion damage edge of the inner surface of the pipe, the σ*r* distribution undergoes small variation (up to 1%). Maximum and minimum values of σ*r* in the calculation for (1), (6) are: min <sup>6</sup> 4.02 10 σ *<sup>r</sup>* =− ⋅ Pa and max <sup>6</sup> 3.91 10 σ *<sup>r</sup>* =− ⋅ Pa; in the calculation for (1), (7): min <sup>6</sup> 4.02 10 σ *<sup>r</sup>* =− ⋅ Pa and max <sup>6</sup> 3.92 10 σ*<sup>r</sup>* =− ⋅ Pa.

The numerical analysis of the resuts reveals a good agreement between the results of analytical and finite-element calculations for σ*r* ((1), (6)). For *r*<sup>1</sup> ≤ *y* ≤ *r*2, *x=z=*0 in the region of the pipe without damage at *r* = *r*1*e is* >>1%, whereas at *r* = *r*2*e is* ≈1% for (1), (6).

Make a comparative analysis of the results of these numerical calculations for (1), and (1), (8) with those of the analytical calculation described in Sect. 1.4 for the boundary conditions of the form 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 0 *<sup>r</sup> r r* σ <sup>=</sup> = . Consider pipe stresses in the circumfrenetial σ*t* and radial σ*r* directions under the action of internal pressure (1) for fixing absent at the outer surface and at the contact between the the pipe and soil (1), (8).

σ*p* <sup>=</sup> =

σ

Thus, at the upper inner surface of the pipe, the damage influence on the σ*t* variation is inconsiderable. A comparatively small error obtained says about the fit of the key condition

*p* <sup>=</sup> = in the three-dimensional calculation with the key condition for the two-

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

0 *<sup>r</sup> r r*

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2

σ

(1) (2) *r r rr rr*

<sup>=</sup> = in the analytical calculation. For (1), (8), because

 σ <sup>=</sup> <sup>=</sup> = − ,

Fig. 21. Distribution of the stress σ1 (σ*t*) at 1 *<sup>r</sup> r r*

Fig. 22. Distribution of the stress σ2 (σ*t*) at 1 *<sup>r</sup> r r*

σ

<sup>=</sup> <sup>=</sup> <sup>=</sup> =− = , 3 <sup>3</sup>

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2

σ

222

(1) (2) (1) *<sup>n</sup> rr rr rr <sup>f</sup>* σσσ

 τ

dimensional model 1 *<sup>r</sup> r r*

τ

<sup>1</sup> *<sup>r</sup> r r* σ

Fig. 20. Distribution of the stress σ3 (σ*r*) at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

From Figures 21 and 22 it is seen that in the case of pipe fixing 2 <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> the

corrosion damage exerts an essential influence on the σ*t* distribution over the inner surface of the pipe. The minimum of the tensile stress σ*t* is at the damage edge over the cross section, whereas the maximum – inside the damage. The σ*t* value at the damage edge is, on average, by 30% less than the one at the inner surface of the pipe without damage and by 60% less than the one inside the damage. The stress σ*t* is approximately by 50% less at the surface without damage as against the one inside the damage. At the contact between the pipe and soil, the σ*t* disturbances are localized just in the damage area. In the calculation for (1), (8), the σ*t* differences between the damage edge, the inner surface without damage, and the damage interior are, on average, 60 and 70%, respectively. The stress σ*t* is approximately by 30% less at the surface without damage as against the one inside the damage. In this calculation there appear essential end disturbances of σ*t*. Such a disturbance is the drawback of the calculation involvingh the modeling of the contact between the pipe and soil. Additional investigations are needed to eliminate this disturbance. On the whole, σ*t* at the inner surface of the pipe in the calculation for (1) is, on average, by 70% larger than the one in the calculation for (1), (8). Maximum and minimum values of σ*r* in the calculation for (1) are: min <sup>7</sup> 8.39 10 σ *<sup>t</sup>* = ⋅ Pa and max <sup>8</sup> 6.65 10 σ *<sup>t</sup>* = ⋅ Pa; in the calculation for (1), (8): min <sup>6</sup> 7.66 10 σ *<sup>t</sup>* = ⋅ Pa and max <sup>7</sup> 6.17 10 σ*<sup>t</sup>* = ⋅ Pa.

The numerical analysis of the results shows not bad coincidence of the results of the analytical and finite-element calculations for σ*t*, (1). At *r*<sup>1</sup> ≤ *y* ≤ *r*2, *x = z =* 0 in the region of the pipe without damage the error at *r* = *r*1 is approximately equal to

$$e = \frac{1.38 - 1.45}{1.38} \cdot 100\% = -6.71\% \,\tag{43}$$

at *r* = *r*<sup>2</sup>

$$e = \frac{1.34 - 1.305}{1.34} \cdot 100\% = 2.61\%. \tag{44}$$

Fig. 21. Distribution of the stress σ1 (σ*t*) at 1 *<sup>r</sup> r r* σ*p* <sup>=</sup> =

σ

From Figures 21 and 22 it is seen that in the case of pipe fixing 2 <sup>2</sup>

corrosion damage exerts an essential influence on the σ*t* distribution over the inner surface of the pipe. The minimum of the tensile stress σ*t* is at the damage edge over the cross section, whereas the maximum – inside the damage. The σ*t* value at the damage edge is, on average, by 30% less than the one at the inner surface of the pipe without damage and by 60% less than the one inside the damage. The stress σ*t* is approximately by 50% less at the surface without damage as against the one inside the damage. At the contact between the pipe and soil, the σ*t* disturbances are localized just in the damage area. In the calculation for (1), (8), the σ*t* differences between the damage edge, the inner surface without damage, and the damage interior are, on average, 60 and 70%, respectively. The stress σ*t* is approximately by 30% less at the surface without damage as against the one inside the damage. In this calculation there appear essential end disturbances of σ*t*. Such a disturbance is the drawback of the calculation involvingh the modeling of the contact between the pipe and soil. Additional investigations are needed to eliminate this disturbance. On the whole, σ*t* at the inner surface of the pipe in the calculation for (1) is, on average, by 70% larger than the one in the calculation for (1), (8). Maximum and minimum values of σ*r* in the calculation for (1)

The numerical analysis of the results shows not bad coincidence of the results of the analytical and finite-element calculations for σ*t*, (1). At *r*<sup>1</sup> ≤ *y* ≤ *r*2, *x = z =* 0 in the region of

1.38 1.45 100% 6.71%, 1.38

1.34 1.305 100% 2.61%.

1.34

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup>

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

*<sup>t</sup>* = ⋅ Pa; in the calculation for (1), (8): min <sup>6</sup> 7.66 10

*<sup>e</sup>* <sup>−</sup> = ⋅ =− (43)

*<sup>e</sup>* <sup>−</sup> = ⋅= (44)

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> the

σ

*<sup>t</sup>* = ⋅

Fig. 20. Distribution of the stress σ3 (σ*r*) at 1 *<sup>r</sup> r r*

*<sup>t</sup>* = ⋅ Pa and max <sup>8</sup> 6.65 10

σ

the pipe without damage the error at *r* = *r*1 is approximately equal to

are: min <sup>7</sup> 8.39 10 σ

at *r* = *r*<sup>2</sup>

Pa and max <sup>7</sup> 6.17 10 σ

*<sup>t</sup>* = ⋅ Pa.

Fig. 22. Distribution of the stress σ2 (σ*t*) at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 (1) (2) *r r rr rr* σ σ <sup>=</sup> <sup>=</sup> = − , 222 (1) (2) (1) *<sup>n</sup> rr rr rr <sup>f</sup>* σσσ τ τ <sup>=</sup> <sup>=</sup> <sup>=</sup> =− = , 3 <sup>3</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

Thus, at the upper inner surface of the pipe, the damage influence on the σ*t* variation is inconsiderable. A comparatively small error obtained says about the fit of the key condition <sup>1</sup> *<sup>r</sup> r r* σ *p* <sup>=</sup> = in the three-dimensional calculation with the key condition for the twodimensional model 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 0 *<sup>r</sup> r r* σ<sup>=</sup> = in the analytical calculation. For (1), (8), because

σ

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

σ

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 <sup>2</sup>

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2

σ

(1) (2) *r r rr rr*

 σ <sup>=</sup> <sup>=</sup> = − ,

Fig. 24. Distribution of the stress σ3 (σ*r*) at 1 *<sup>r</sup> r r*

 σ <sup>=</sup> <sup>=</sup> <sup>=</sup> =− = , 3 <sup>3</sup>

Fig. 25. Distribution of the stress σ*z* at 1 *<sup>r</sup> r r*

222

(1) (2) (1) *<sup>n</sup> rr rr rr <sup>f</sup>*

 τ σ

στ

of the presence of elastic soil the difference between the results of the analytical and finiteelement calculations and the calculation for (1), (7) is much larger – about 70 %.

The analysis shows that from the inner to the outer surface along paths 1, 2, 4, the stress σ*<sup>t</sup>* decreases approximately by 7, 36 and 43%, respectively, and increases approximately by 120% along path 3. Thus, it is seen that at the corrosion damage edge over cross section (path 3) the σ*t* distribution has an essentially peculiar pattern. The σ*t* variations in the calculation for (1), (8) along paths 1, 2, 3 are identical to those in the calculation for (1) and are approximately 3, 1.5 and 15 %, respectively. However unlike the calculation for (1), in the calculation for (1), (8) σ*t* increases a little (up to 1%) along path 4.

The stress σ*r* distributions shown in Figures 23 and 24 illustrate a qualitative agreement of the results of the analytical and finite-element calculations for (1). In the calculation for (1) |σ*r*| is approximately by 70% higher at the damage edge than the one at the inner surface without damage.

Fig. 23. Distribution of the stress σ3 (σ*r*) at 1 *<sup>r</sup> r r* σ*p* <sup>=</sup> =

In the calculation for (1), (8), because of the soil pressure, |σ*r*| practically does not vary in the damage vicinity.

Maximum and minimum values of σ*r* in the calculation for (1) are: min <sup>7</sup> 2.49 10 σ *<sup>r</sup>* =− ⋅ Pa and max <sup>5</sup> 4.64 10 σ *<sup>r</sup>* = ⋅ Pa; in the calculation for (1), (8): min <sup>7</sup> 1.62 10 σ *<sup>r</sup>* =− ⋅ Pa and max <sup>6</sup> 1.09 10 σ *<sup>r</sup>* = ⋅ Pa.

Figures 1.18– 1.28 plot the distributions of the principal stresses corresponding to the sresses σ*t*, σ*r*, σ*z* for different fixing types. From the comparison of theses distributions it is seen that four forms of boundary conditions form two qualitatively different types of the stress σ*<sup>t</sup>* distributions. So, in the case of rigid fixing of the outer surface of the pipe (at 2 2 <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> = = or 2 2 <sup>2</sup> <sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = = ) σ*t*<0. In the case, fixing is absent and

contact is present, σ*t*>0. At the contact interaction between the pipe and soil, the level due to the pressure soil in σ*t* is approximately three times less than in the absence of fixing. The

160 Tribology - Lubricants and Lubrication

of the presence of elastic soil the difference between the results of the analytical and finite-

The analysis shows that from the inner to the outer surface along paths 1, 2, 4, the stress σ*<sup>t</sup>* decreases approximately by 7, 36 and 43%, respectively, and increases approximately by 120% along path 3. Thus, it is seen that at the corrosion damage edge over cross section (path 3) the σ*t* distribution has an essentially peculiar pattern. The σ*t* variations in the calculation for (1), (8) along paths 1, 2, 3 are identical to those in the calculation for (1) and are approximately 3, 1.5 and 15 %, respectively. However unlike the calculation for (1), in

The stress σ*r* distributions shown in Figures 23 and 24 illustrate a qualitative agreement of the results of the analytical and finite-element calculations for (1). In the calculation for (1) |σ*r*| is approximately by 70% higher at the damage edge than the one at the inner surface

> σ*p* <sup>=</sup> =

Maximum and minimum values of σ*r* in the calculation for (1) are: min <sup>7</sup> 2.49 10

*<sup>r</sup>* = ⋅ Pa; in the calculation for (1), (8): min <sup>7</sup> 1.62 10

In the calculation for (1), (8), because of the soil pressure, |σ*r*| practically does not vary in

*<sup>r</sup>* =− ⋅ Pa and max <sup>5</sup> 4.64 10

Figures 1.18– 1.28 plot the distributions of the principal stresses corresponding to the sresses σ*t*, σ*r*, σ*z* for different fixing types. From the comparison of theses distributions it is seen that four forms of boundary conditions form two qualitatively different types of the stress σ*<sup>t</sup>* distributions. So, in the case of rigid fixing of the outer surface of the pipe (at

contact is present, σ*t*>0. At the contact interaction between the pipe and soil, the level due to the pressure soil in σ*t* is approximately three times less than in the absence of fixing. The

σ

<sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = = ) σ*t*<0. In the case, fixing is absent and

σ

*<sup>r</sup>* =− ⋅ Pa and max <sup>6</sup> 1.09 10

σ

*<sup>r</sup>* = ⋅

element calculations and the calculation for (1), (7) is much larger – about 70 %.

the calculation for (1), (8) σ*t* increases a little (up to 1%) along path 4.

Fig. 23. Distribution of the stress σ3 (σ*r*) at 1 *<sup>r</sup> r r*

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> = = or 2 2 <sup>2</sup>

without damage.

the damage vicinity.

2 2

σ

Pa.

Fig. 24. Distribution of the stress σ3 (σ*r*) at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 (1) (2) *r r rr rr* σ σ <sup>=</sup> <sup>=</sup> = − , 222 (1) (2) (1) *<sup>n</sup> rr rr rr <sup>f</sup>* στ τ σ σ <sup>=</sup> <sup>=</sup> <sup>=</sup> =− = , 3 <sup>3</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

Fig. 25. Distribution of the stress σ*z* at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

σ*t*<0 distributions over the inner surface of the pipe are qualitatively and quantitatively indentical in all calculations. The σ*z* distributions are essensially different for the considered

exist regions of both tensile and compressive stresses σ*z*. In the calculation for

<sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> = = = , the peculiarities of the σ*z*<0 distributions manefest themselves just in the damage region (fixing influence in all directions). At the contact interaction between the pipe and soil, the σ*z*>0 distribution in the damage region is similar to the

The bulk analysis of the stress distributions has shown that the results of calculation of the contact interaction of the pipe and soil are intermediate between the calculation results for the extreme cases of fixing. So, the σ*r*<0 distribution has a similar pattern in all calculations. By the σ*t* distribution, the case of the contact between the pipe and soil is close to that of absent fixing since in these calculations the boundary conditions allow the pipe to be expanded in the radial direction. By the σ*z* distributions, the case of the contact between the

of the pipe, displacements along the *z* axis of the pipe are possible and at the same time

σ

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2

σ

(1) (2) *r r rr rr*

 σ <sup>=</sup> <sup>=</sup> = − ,

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> and in the absence of fixing, there

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> , since in these calculations for the outer surface

calculations. In the calculations for 2 <sup>2</sup>

2 2 <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> .

pipe and soil is close for 2 <sup>2</sup>

displacements in the radial direction are limited.

Fig. 28. Distribution of the stress σ1 (σ*z*) at 1 *<sup>r</sup> r r*

 σ <sup>=</sup> <sup>=</sup> <sup>=</sup> =− = , 3 <sup>3</sup>

222

(1) (2) (1) *<sup>n</sup> rr rr rr <sup>f</sup>*

 τ σ

στ 2 2 2

distribution for

Fig. 26. Distribution of the stress σ*z* at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup> <sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

Fig. 27. Distirbution of the stress σ2 (σ*z*) at 1 *<sup>r</sup> r r* σ*p* <sup>=</sup> =

Fig. 26. Distribution of the stress σ*z* at 1 *<sup>r</sup> r r*

Fig. 27. Distirbution of the stress σ2 (σ*z*) at 1 *<sup>r</sup> r r*

σ

σ*p* <sup>=</sup> =

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup>

<sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

σ*t*<0 distributions over the inner surface of the pipe are qualitatively and quantitatively indentical in all calculations. The σ*z* distributions are essensially different for the considered calculations. In the calculations for 2 <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> and in the absence of fixing, there exist regions of both tensile and compressive stresses σ*z*. In the calculation for 2 2 2 <sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> = = = , the peculiarities of the σ*z*<0 distributions manefest themselves just in the damage region (fixing influence in all directions). At the contact interaction between the pipe and soil, the σ*z*>0 distribution in the damage region is similar to the distribution for 2 2 <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> .

The bulk analysis of the stress distributions has shown that the results of calculation of the contact interaction of the pipe and soil are intermediate between the calculation results for the extreme cases of fixing. So, the σ*r*<0 distribution has a similar pattern in all calculations. By the σ*t* distribution, the case of the contact between the pipe and soil is close to that of absent fixing since in these calculations the boundary conditions allow the pipe to be expanded in the radial direction. By the σ*z* distributions, the case of the contact between the pipe and soil is close for 2 <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> , since in these calculations for the outer surface

of the pipe, displacements along the *z* axis of the pipe are possible and at the same time displacements in the radial direction are limited.

Fig. 28. Distribution of the stress σ1 (σ*z*) at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 (1) (2) *r r rr rr* σ σ <sup>=</sup> <sup>=</sup> = − , (1) (2) (1) *<sup>n</sup> rr rr rr <sup>f</sup>* σττσ σ <sup>=</sup> <sup>=</sup> <sup>=</sup> =− = , 3 <sup>3</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

222

Fig. 31. Strains ε*r* at 1 *<sup>r</sup> r r*

Fig. 32. Strains ε*r* at 1 *<sup>r</sup> r r*

σ

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup>

<sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

σ

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup>

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

The corrosion damage disturbance of the strain state of the pipe as a whole corresponds to the disturbance of the stress state (Figures 29–34). The exception is only ε*t* (Figures 29, 30) that is tensile at the entire inner surface of the pipe, except for the damage edge where it becomes essentially compressive. This effect in principle corresponds to the effect of developing compressive strains inside the damage in a total compressive strain field. This effect was reaveled during full-scale pressure tests of pipes.

Fig. 29. Strains ε*t* at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

Fig. 30. Strains ε*t* at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup> <sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

Fig. 31. Strains ε*r* at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

The corrosion damage disturbance of the strain state of the pipe as a whole corresponds to the disturbance of the stress state (Figures 29–34). The exception is only ε*t* (Figures 29, 30) that is tensile at the entire inner surface of the pipe, except for the damage edge where it becomes essentially compressive. This effect in principle corresponds to the effect of developing compressive strains inside the damage in a total compressive strain field. This

effect was reaveled during full-scale pressure tests of pipes.

Fig. 29. Strains ε*t* at 1 *<sup>r</sup> r r*

Fig. 30. Strains ε*t* at 1 *<sup>r</sup> r r*

σ

σ

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup>

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup>

<sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

Fig. 32. Strains ε*r* at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup> <sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

Fig. 35. Distribution of the stress σ1 (σ*t*) in the absence of the outer surface fixing for

Fig. 36. Distribution of the stress σ1 (σ*t*) in the absence of the outer surface fixing for

σ *ij* , ( ) *<sup>T</sup>* σ

 *ij* , ( ) *<sup>p</sup> <sup>T</sup>* σ*ij*

<sup>+</sup> such that according to (10)

characteristic distribution types of the stresses ( ) *<sup>p</sup>*

( ) () *pT p* ( ) *T*

 σ *ij ij* <sup>+</sup> = + .

σ

σ*ij*

<sup>1</sup> *<sup>r</sup> r r* σ*p* <sup>=</sup> =

1 2 *TT T r r* − =Δ

Fig. 33. Strains ε*z* at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

Fig. 34. Strains ε*z* at 1 *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup> <sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

#### **6. Influnce of different loading types on the stress-strain state of threedimensional pipe models**

Figures 35, 36 present the distributions of the principal stresses corresponding to the stresses σ*t* for different loading types in the absence of fixing of the outer surface of the pipe. From the comparison of these distributions it is seen that three loading types form three

Fig. 33. Strains ε*z* at 1 *<sup>r</sup> r r*

Fig. 34. Strains ε*z* at 1 *<sup>r</sup> r r*

**dimensional pipe models** 

σ

σ

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , <sup>2</sup> <sup>2</sup>

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup>

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 2 2 <sup>2</sup>

<sup>0</sup> *xyz r r r r r r uuu* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> = =

Figures 35, 36 present the distributions of the principal stresses corresponding to the stresses σ*t* for different loading types in the absence of fixing of the outer surface of the pipe. From the comparison of these distributions it is seen that three loading types form three

**6. Influnce of different loading types on the stress-strain state of three-**

characteristic distribution types of the stresses ( ) *<sup>p</sup>* σ *ij* , ( ) *<sup>T</sup>* σ *ij* , ( ) *<sup>p</sup> <sup>T</sup>* σ *ij* <sup>+</sup> such that according to (10) ( ) () *pT p* ( ) *T* σ *ij* σ σ *ij ij* <sup>+</sup> = + .

Fig. 35. Distribution of the stress σ1 (σ*t*) in the absence of the outer surface fixing for <sup>1</sup> *<sup>r</sup> r r* σ*p* <sup>=</sup> =

Fig. 36. Distribution of the stress σ1 (σ*t*) in the absence of the outer surface fixing for 1 2 *TT T r r* − =Δ

Fig. 38. Distribution of the stress σ1 ( ( ) *<sup>p</sup>*

Fig. 39. Distribution of the stress σ1 ( ( ) *<sup>p</sup> <sup>T</sup>*

<sup>=</sup> = , 1 2 *TT T r r* − = Δ

τ

<sup>1</sup> *rz r r* <sup>0</sup>

<sup>1</sup> *<sup>r</sup> r r* σ

 *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 1 *rz r r* <sup>0</sup> τ

τ<sup>=</sup> =

τ

*ij* τ σ

*ij* τ σ

<sup>+</sup> <sup>+</sup> ) at 2 <sup>2</sup>

<sup>+</sup> ) at 2 <sup>2</sup>

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> , 0 *<sup>z</sup> z L <sup>u</sup>* <sup>=</sup> <sup>=</sup> for 1 *<sup>r</sup> r r*

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> , 0 *<sup>z</sup> z L <sup>u</sup>* <sup>=</sup> <sup>=</sup> for

σ

*p* <sup>=</sup> = ,

A comparative analysis of the stress distributions along the assigned paths shows that at the corrosion damage center (path 2) there is an almost two-fold increase of the stresses (σ*t*), as compared to the surface of the pipe without damage (path 1). The disturbing effect of corrosion damage (path 6) on the stress state is clearly seen.

Figures 37–39 plot the distributions of the principal stresses corresponding to the stresses σ*<sup>t</sup>* for different loading types when displacements are absent along the *x* and *y* axes of the outer surface of the pipe 2 <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> and along the *z* axis at the right end 0 *<sup>z</sup> z L <sup>u</sup>* <sup>=</sup> <sup>=</sup> when friction is present at the inner surface 1 0 *rz r r* τ <sup>=</sup> ≠ . From the comparison of these figures it is possible to single out several characteristic distribution types of the stresses ( ) *p* σ *ij* , ( ) *ij* τ σ , ( ) *<sup>T</sup>* σ *ij* , ( ) *<sup>p</sup> ij* τ σ <sup>+</sup> , ( ) *<sup>p</sup> <sup>T</sup>* σ *ij* <sup>+</sup> , ( ) *<sup>p</sup> <sup>T</sup> ij* τ σ<sup>+</sup> <sup>+</sup> related by (10).

Figures 1.37–1.38 illustrate a noticeable influence of the viscous fluid (oil) pipe wall friction ( ( ) *ij* τ σ ) on the ( ) *<sup>p</sup> ij* τ σ <sup>+</sup> formation. From Figure 39 it is seen that temeprature stresses are dominant, exceeding by no less than 2-3 times the stresses developed by the action of <sup>1</sup> *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> =4 MPa, 1 *rz r r* <sup>0</sup> τ τ <sup>=</sup> = =260 Pa. In view of the fact that the temperature difference 1 2 *TT T r r* − =Δ =20°C exerts a dramatic influence on the formation of the stress state of the pipe, the distributions of ( ) *<sup>p</sup> <sup>T</sup>* σ *ij* <sup>+</sup> and ( ) *<sup>p</sup> <sup>T</sup> ij* τ σ <sup>+</sup> <sup>+</sup> are qualitatively similar to the ( ) *<sup>p</sup> <sup>T</sup> ij* τ σ + + distribution, slightly differing in numerical values.

Fig. 37. Distribition pf the stress σ1 ( ( ) *<sup>p</sup>* σ *ij* ) at 2 <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> for 1 *<sup>r</sup> r r* σ*p* <sup>=</sup> =

A comparative analysis of the stress distributions along the assigned paths shows that at the corrosion damage center (path 2) there is an almost two-fold increase of the stresses (σ*t*), as compared to the surface of the pipe without damage (path 1). The disturbing effect of

Figures 37–39 plot the distributions of the principal stresses corresponding to the stresses σ*<sup>t</sup>* for different loading types when displacements are absent along the *x* and *y* axes of the

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> and along the *z* axis at the right end 0 *<sup>z</sup> z L <sup>u</sup>* <sup>=</sup> <sup>=</sup>

<sup>=</sup> ≠ . From the comparison of these

0 *rz r r*

<sup>+</sup> formation. From Figure 39 it is seen that temeprature stresses are

<sup>=</sup> = =260 Pa. In view of the fact that the temperature difference

<sup>+</sup> <sup>+</sup> are qualitatively similar to the ( ) *<sup>p</sup> <sup>T</sup>*

*ij* τ σ+ +

τ

figures it is possible to single out several characteristic distribution types of the stresses

dominant, exceeding by no less than 2-3 times the stresses developed by the action of

1 2 *TT T r r* − =Δ =20°C exerts a dramatic influence on the formation of the stress state of the

<sup>+</sup> and ( ) *<sup>p</sup> <sup>T</sup> ij* τ σ

 <sup>+</sup> <sup>+</sup> related by (10). Figures 1.37–1.38 illustrate a noticeable influence of the viscous fluid (oil) pipe wall friction

corrosion damage (path 6) on the stress state is clearly seen.

when friction is present at the inner surface 1

τ

σ*ij*

distribution, slightly differing in numerical values.

<sup>+</sup> , ( ) *<sup>p</sup> <sup>T</sup> ij* τ σ

 <sup>+</sup> , ( ) *<sup>p</sup> <sup>T</sup>* σ*ij*

outer surface of the pipe 2 <sup>2</sup>

 *ij* , ( ) *<sup>p</sup> ij* τ σ

> *ij* τ σ

> > τ

Fig. 37. Distribition pf the stress σ1 ( ( ) *<sup>p</sup>*

σ

*ij* ) at 2 <sup>2</sup>

<sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> for 1 *<sup>r</sup> r r*

σ*p* <sup>=</sup> =

*<sup>p</sup>* <sup>=</sup> <sup>=</sup> =4 MPa, 1 *rz r r* <sup>0</sup>

pipe, the distributions of ( ) *<sup>p</sup> <sup>T</sup>*

) on the ( ) *<sup>p</sup>*

( ) *p* σ *ij* , ( ) *ij* τ σ , ( ) *<sup>T</sup>* σ

( ( ) *ij* τ σ

<sup>1</sup> *<sup>r</sup> r r* σ

Fig. 38. Distribution of the stress σ1 ( ( ) *<sup>p</sup> ij* τ σ <sup>+</sup> ) at 2 <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> , 0 *<sup>z</sup> z L <sup>u</sup>* <sup>=</sup> <sup>=</sup> for 1 *<sup>r</sup> r r* σ *p* <sup>=</sup> = , <sup>1</sup> *rz r r* <sup>0</sup> τ τ<sup>=</sup> =

Fig. 39. Distribution of the stress σ1 ( ( ) *<sup>p</sup> <sup>T</sup> ij* τ σ <sup>+</sup> <sup>+</sup> ) at 2 <sup>2</sup> <sup>0</sup> *x y r r r r u u* <sup>=</sup> <sup>=</sup> <sup>=</sup> <sup>=</sup> , 0 *<sup>z</sup> z L <sup>u</sup>* <sup>=</sup> <sup>=</sup> for <sup>1</sup> *<sup>r</sup> r r* σ *<sup>p</sup>* <sup>=</sup> <sup>=</sup> , 1 *rz r r* <sup>0</sup> τ τ<sup>=</sup> = , 1 2 *TT T r r* − = Δ

[7] Launder B.E., Spalding D.B. Mathematical Models of Turbulence. London: Academic

[8] Mirkin А.Z., Usinysh V.V. Pipeline systems: Handbook Edition. М: Khimiya, 1991. – 286

[9] O'Grady T.J., Hisey D.Т., Kiefner J. F. Pressure calculation for corroded pipe developed

[10] Ponomarev S.D. Strength calculations in engineering industry / S.D. Ponomarev,

[11] Rodi W. A new algebraic relation for calculating the Reynolds stresses //ZAMM 56.

[12] Sedov L.I. Continuum mechanics: in 2 volumes. 6th edition, Saint-Petersburg: Lan', 2004.

[13] Seleznev V.Е., Aleshin V.V., Pryalov S.N. Fundamentals of numerical modeling of trunk

[14] Sherbakov S.S. Influence of fixing of a pipe with a corrosion defect on its stress-strain

[15] Sherbakov S.S. Modeling of the three-dimensional stress-strain state of a pipe with

[16] Sherbakov S.S. Modeling of the stress-strain state of a pipe with a corrosion defect

[17] Sherbakov S.S. Influence of wall friction in the turbulent oil flow motion in the pipe

[18] Sosnovskiy L.А. Modeling of the stress-strain state of pipes of trunk pipelines with

state / S.S. Sherbakov, N.А. Zalessky, P.A. Ivankin, V.V. Vorobiev // Reliability and safety of the trunk pipeline transportation: Proc. VI International Scientific-Technical Conference, Novopolotsk, 11–14 December 2007 / PSU; eds: V.K. Lipsky

a corrosion defect under complex loading / S.S. Sherbakov, N.А. Zalessky, P.S. Ivankin, L.А. Sosnovskiy// Reliability and safety of the trunk pipeline transportation: Proc. VI International Scientific-Technical Conference, Novopolotsk, 11–14 December 2007 / PSU; eds: V.K. Lipsky et al. – Novopolotsk, 2007 b. – P.

under complex loading / S.S. Shcherbakov, N.А. Zalessky, P.S. Ivankin // Х Belarusian Mathematical Conference: Abstract of the paper submitted to the International Scientific Conference, Minsk, 3–7 Novermber 2008 – Part 4. – Minsk: Press of the Institute of Mathematics of NAS of Belarus, 2008. – P. 53-

with a corrosion defect on the stress-strain state of the pipe / S.S. Sherbakov // Strength and reliability of trunk pipelines (Abstracts of the papers submitted to the International Scientific-Technical Conference "МТ-2008", Kiev, 5–7 June 2008). –

corrosion defects with regard to pressure, temperature, and interaction between the oil flow and the inner surface / L.А. Sosnovskiy, S.S. Sherbakov // Strength and safety of trunk pipelines (Abstracts of the papers submitted to the International

pipelines / Ed. by V.Е. Seleznev. – М: KomKniga, 2005. – 496 p.

V.D. Biderman, К.К. Likharev, V.M. Makushin, N.N. Malinin, V.I. Fedosiev. М: State Scientific-Technical Publishing House of Engineering Literature, 1958. Vol.

2007 b. – P. 78-80.

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et al. – Novopolotsk, 2007 a. – P. 52-55.

Kiev: IPS NAS Ukraine, 2008. – P.120-121.

Press, 1972.

2. – 974 p.

1976.

2nd vol.

55-58.

54.

p.

Novopolotsk, 11–14 December 2007 / PSU; eds: V.K. Lipsky et al. –Novopolotsk,

A comparative analysis of the stress distributions shows that at the corrosion damage center the stresses grow (almost two-fold increase for σ*t*) in comparison with the surface of the pipe without damage.

#### **7. Conclusion**

Within the framework of the investigations made, the method for evaluation of the influence of the process of friction of moving oil on the damage of the inner surface of the pipe has been developed. The method involves analytical and numerical calculations of the motion of the two-and three-dimensional flow of viscous fluid (oil) in the pipe within laminar and turbulent regimes, with different average flow velocities at some internal pipe pressure, in the presence or the absence of corrosion damage at the inner surface of the pipe.

The method allows defining a broad spectrum of flow motion characteristics, including: velocity, energy and turbulence intensity, a value of tangential stresses (friction force) caused by the flow motion at the inner surface of the pipe.

The method for evaluation of the stress-strain state of two-and three-dimensional pipe models as acted upon by internal pressure, uniformly distributed tangential stresses over the inner surface of the pipe (pipe flow friction forces), and temperature with regard to corrosion-erosion damages of the inner surface of the pipe has been developed, too. For finite-element pipe models with boundary conditions of type (1)–(7) mainly the circumferential stresses, being the largest, were considered.

The methof allows defining the variation in the values of the tensor components of stresses and strains in the pipe with corrosion damage for assigned pipe fixing under individual loading (temperature, pressure, fluid flow friction over the inner surface of the pipe) and their different combinations.

#### **8. References**


A comparative analysis of the stress distributions shows that at the corrosion damage center the stresses grow (almost two-fold increase for σ*t*) in comparison with the surface of the pipe

Within the framework of the investigations made, the method for evaluation of the influence of the process of friction of moving oil on the damage of the inner surface of the pipe has been developed. The method involves analytical and numerical calculations of the motion of the two-and three-dimensional flow of viscous fluid (oil) in the pipe within laminar and turbulent regimes, with different average flow velocities at some internal pipe pressure, in the presence or the absence of corrosion damage at the inner surface of

The method allows defining a broad spectrum of flow motion characteristics, including: velocity, energy and turbulence intensity, a value of tangential stresses (friction force)

The method for evaluation of the stress-strain state of two-and three-dimensional pipe models as acted upon by internal pressure, uniformly distributed tangential stresses over the inner surface of the pipe (pipe flow friction forces), and temperature with regard to corrosion-erosion damages of the inner surface of the pipe has been developed, too. For finite-element pipe models with boundary conditions of type (1)–(7) mainly the

The methof allows defining the variation in the values of the tensor components of stresses and strains in the pipe with corrosion damage for assigned pipe fixing under individual loading (temperature, pressure, fluid flow friction over the inner surface of the pipe) and

[1] Ainbinder А.B., Kamershtein А.G. Strength and stability calculation of trunk pipelines.

[4] Handbook on the designing of trunk pipelines / Ed, by А.К. Dertsakyan. L: Nedra, 1977.

[5] Kostyuchenko А.А. Influence of friction due to the oil flow on the pipe loading / А.А.

[6] Kostyuchenko А.А. Wall friction in the turbulent oil flow motion in the pipe with

V.K. Lipsky et al. – Novopolotsk, 2007 a. – P. 76-78.

Kostyuchenko, S.S. Sherbakov, N.А. Zalessky, P.A. Ivankin, L.А. Sosnovskiy // Reliability and safety of the trunk pipeline transportation: Proc. VI International Scientific-Technical Conference, Novopolotsk, 11–14 December 2007 / PSU; eds:

corrosion defect / А.А. Kostyuchenko, S.S. Sherbakov, N.А. Zalessky, P.S. Ivankin, L.А. Sosnovskiy // Reliability and safety of the trunk pipeline transportation: Proc. VI International Scientific-Technical Conference,

[2] Borodavkin P.P., Sinyukov А.М. Strength of trunk pipelines. М: Nedra, 1984. – 286 p. [3] Grachev V.V., Guseinzade М.А., Yakovlev Е.I. et al. Complex pipeline systems. М:

caused by the flow motion at the inner surface of the pipe.

circumferential stresses, being the largest, were considered.

without damage.

**7. Conclusion** 

the pipe.

their different combinations.

– 519 p.

М: Nedra, 1982. – 344 p.

Nedra, 1982. – 410 p.

**8. References** 

Novopolotsk, 11–14 December 2007 / PSU; eds: V.K. Lipsky et al. –Novopolotsk, 2007 b. – P. 78-80.


**Lubrication Tests and** 

**Biodegradable Lubricants** 

Scientific-Technical Conference "МТ-2008", Kiev, 5–7 June 2008). – Kiev: IPS NAS Ukraine 2008. – Pp. 107-108. **Part 2** 
