**Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical**

Jairo A. Martins and István Kövesdy

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46165

## **1. Introduction**

276 Finite Element Analysis – Applications in Mechanical Engineering

efficiency under an arbitrary operating condition.

elements, loss between rings, and their proportions.

*Honda R&D Co., Ltd. Automobile R&D Center, Japan* 

actuation slip control," 2004 CVT Congress

vol.221 Part J: J. Engineering Tribology

Pushing V-belt of CVT," SAE Paper, 2008-01-0415

Pushing V-belt with Feedback Control," SAE Paper, 2004-01-1326

element contact areas along the entire length of the V-belt.

transmission efficiency using flexible pulley components.

**7. Conclusion** 

**Author details** 

Toshihiro Saito

**8. References** 

Automobile Engineering

12. These are two reasons for the vibration. One is the impact forces are generated when the element gets into and out the pulleys. Another is the each element behavior itself. Figure 28 shows the element reaction force on V-surface and the friction coefficient on V-surface. The friction coefficient in these analyses shows the vibration in the pulleys. This presents that each

1. Feedback control to maintain speed ratios in an actual CVT has been implemented in a metal V-belt behavior simulation, making it possible to predict the CVT transmission

2. This simulation has successfully quantified sliding velocities and friction forces of

3. The simulation was used to calculate friction losses from sliding velocities and friction forces at different ratios—loss on the V-surface, loss on the saddle surface, loss between

4. The simulation technique was also available for actual CVT configuration to predict the

[1] Yamamoto, K., Sakaguchi, S., Kishida, M., Kimura, E. and Abe, H. "Development of Integrated Engine-CVT Control System," Honda R&D Technical review, Vol. 11, No. 1 [2] Akehurst, S., Vaughan, N. D., Parker, D. A. and Simner, D. "Modelling of loss mechanisms in a pushing metal V-belt continuously variable transmission. Part 1: torque losses due to band friction," Proc. Instn Mech. Engrs vol. 218 Part D: J.

[3] Veenhuizen, P. A., Bonsen, B., Klaassen, T.W.G.L., Albers, P.H.W.M., Changenet, C. and Poncy, S. "Pushbelt CVT efficiency improvement potential of servo-electromechanical

[4] Van der Noll, E., Van der Sluis, F., Van Dongen, T. and Van der Velde. A. "Innovative Self-optimising Clamping Force Strategy for the Pushbelt CVT," SAE Paper, 2009-01-1537 [5] Narita, K. and Priest, M. "Metal-metal friction characteristics and the transmission efficiency of a metal V-belt-type continuously variable transmission," Proc. IMechE

[6] Saito, T. "Application of Stress Simulation Under Transient Condition for Metal

[7] Saito, T. and Lewis, D. A. "Development of a Simulation Technique for CVT Metal

element could move individually to keep the contact and friction for adjacent parts.

The determination of stresses, deformations and the proper evaluation of calculations outputs are of extreme importance on the mechanical components and to assembly's effectiveness in the Mining Environmental. For instance, components of heavy duty plant machinery like Car Dumpers, Apron Feeders, Stacker-Reclaimers, etc…, and important components of Stacker-Reclaimers like Pulleys; which are under high responsibility must be calculated and designed properly and carefully. The unknown or ignorance of the complete environmental or data inputs (loads and constraints) where the component is applied, can bring tremendous damages to society and jeopardize entire businesses, mainly whether lives are involved. However, new technological tools, like Finite Element Method (FEM), have brought an even higher level to a better understanding of the complex products, those which have several parts in its conception. Like Klauss [2] describes, the Finite element methods are now widely used to solve structural, fluid, and multiphysics problems numerically. The methods are used extensively because engineers and scientists can mathematically model and numerically solve very complex problems. FEM is considered though a step further on the path on designing products, saving weight, consequently costs of design and manufacturing by the better understanding the pieces behaviors and performance prediction.

To evaluate the stresses in mechanical parts and/or components there are basically two manners; by the analytical approach and/or finite element method. This last, considered the most recent and complete tool to evaluate stresses and strains [2].

An example of the FEM simulation is shown briefly in this chapter when designing pulleys to Stacker-Reclaimers. We selected a standard pulley and generated it by analytical model (Redundant Structure Model) as well as by Finite Element Method (FEM) under linear

analysis. The analytical formulas presented in the text are those belonged to the classical mechanical engineering background. In fact, the analytical calculation has presented success along the time once most of the products in the field have performed properly. The model is considered robust enough to deliver products under high quality of project and which considers the material and manufacturing data in order to determine the allowable stress by safety factors.

Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical 279

c. having lower weights and consequently more effectiveness in energy savings to

d. providing refined visual presentation (3D) of stresses and displacements suffered by the

parts. Therefore bringing excellent power of analysis for engineers;

rotary/dynamical parts;

**Figure 1.** Car Dumper – overview, modeled by Inventor

**Figure 2.** Apron Feeder – overview, modeled by Inventor

This paper describes the limited resources when calculating pulleys (for Stacker-Reclaimers or belt conveyors) by analytical methods in comparison with the advantages of the Finite Element Method and its comparatively minor imprecision. The analytical calculation, particularly, presents an issue considered a constraint to overcome, which is related to the energy contribution by linear elastic deformation, of each component to the final sum of the stress x strain in the assembly.

To FEM simulation, the software Inventor 2010 [3] was used to develop the model, meanwhile the calculation by FEM made by Autodesk Simulation [4].

## **2. Overview of products modeled in 3D and simulated by FEM**

In the mining business the usage of software's in modeling components and machines 3D and afterwards simulated by FEM has increased potentially within the last decades. Machines like, Car Dumpers (Figure 1), Apron Feeders (Figure 2) and Stackers-Reclaimers (Figure 3) and their main components are modeled 3D and simulated by FEM software's. Several software's, specialized in modeling are available in the market, for example, Solidworks, Ideas, Inventor etc…, and in terms of FEM simulation, Nastran, Ansys and Autodesk Simulator are at the edge of this technology.

## **2.1. Advantages and drawbacks of modeling 3D and simulating by FEM in mining**

Due to the upgrade in the way of designing products along the last decades, the technologists and engineer´s had to change their minds when studying products in their initial phase of product design. Till the last decade the drawings were done basically in 2D environment, manually by clip boards and later on by CAD in computers. These required high imagination and capacity to evaluate technically and precisely the components in spatial views and in free space. However, the possibility of error by interferences between parts was relatively high and when happened, high costs were involved due to the required interven in the manufacturing or in the field. With the event of modeling by software, the need of another way of thinking about components and/or machines was strictly required. Despite of apparently complex, at first sight, due to the change of the way of designing, the FEM brings some advantages which worth, as follow;


**Figure 1.** Car Dumper – overview, modeled by Inventor

safety factors.

**mining** 

stress x strain in the assembly.

analysis. The analytical formulas presented in the text are those belonged to the classical mechanical engineering background. In fact, the analytical calculation has presented success along the time once most of the products in the field have performed properly. The model is considered robust enough to deliver products under high quality of project and which considers the material and manufacturing data in order to determine the allowable stress by

This paper describes the limited resources when calculating pulleys (for Stacker-Reclaimers or belt conveyors) by analytical methods in comparison with the advantages of the Finite Element Method and its comparatively minor imprecision. The analytical calculation, particularly, presents an issue considered a constraint to overcome, which is related to the energy contribution by linear elastic deformation, of each component to the final sum of the

To FEM simulation, the software Inventor 2010 [3] was used to develop the model,

In the mining business the usage of software's in modeling components and machines 3D and afterwards simulated by FEM has increased potentially within the last decades. Machines like, Car Dumpers (Figure 1), Apron Feeders (Figure 2) and Stackers-Reclaimers (Figure 3) and their main components are modeled 3D and simulated by FEM software's. Several software's, specialized in modeling are available in the market, for example, Solidworks, Ideas, Inventor etc…, and in terms of FEM simulation, Nastran, Ansys and

**2.1. Advantages and drawbacks of modeling 3D and simulating by FEM in** 

Due to the upgrade in the way of designing products along the last decades, the technologists and engineer´s had to change their minds when studying products in their initial phase of product design. Till the last decade the drawings were done basically in 2D environment, manually by clip boards and later on by CAD in computers. These required high imagination and capacity to evaluate technically and precisely the components in spatial views and in free space. However, the possibility of error by interferences between parts was relatively high and when happened, high costs were involved due to the required interven in the manufacturing or in the field. With the event of modeling by software, the need of another way of thinking about components and/or machines was strictly required. Despite of apparently complex, at first sight, due to the change of the way of designing, the

a. Mitigate or eliminate interferences, by visual analysis and software testing, decreasing

b. the possibility of designing lean shapes to particular application and loads;

meanwhile the calculation by FEM made by Autodesk Simulation [4].

Autodesk Simulator are at the edge of this technology.

FEM brings some advantages which worth, as follow;

substantially the re-work in manufacturing or in field;

**2. Overview of products modeled in 3D and simulated by FEM** 

**Figure 2.** Apron Feeder – overview, modeled by Inventor

Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical 281

combined and extended for the solution of problems to which they do not immediately apply, requires knowledge of certain principles and methods that are stated briefly in the pulleys calculations ahead. In determining stress by mathematical analysis, being analytical or FEM, for example, it is customary to assume the material as elastic, isotropic, homogeneous, and infinitely divisible without change in properties and in conforming to Hooke´s law, which states that strain, is proportional to stress. On the other hand, these assumptions despite of imposing certain limitations upon the conventional methods of stress analysis must be used in the form of safety factors. This precaution has given satisfactory results for nearly all problems

The pulley is basically composed by; expansion ring (when applicable), hub, shaft, disc and cylinder, as seen in the Figure 5 below. The calculation of individual components is still not an issue nowadays and classical formulas may be applied without main difficulties. But when there is an increase of components quantity and the interaction among them takes place, the analytical method cannot predict the real and accurate interaction, energy shared by each component in the assemble, due the imposed deformations. In other words, the proportion of deformation of each individual into the ensemble is a very complex to

The concepts of force´s flow are useful in the visualization of paths taken by the forces lines when crossing machines or structures from the load points till the support points. Whether the structure is simple and statically determined, the equations of equilibrium are enough to determine the reactions. On the other hand if redundant supports exist, it means additional supports to those required to satisfy the static equilibrium conditions, those simple equations are not enough anymore to explain the intensities (magnitude) in any one of the reactions. It happen due to the support works as a separated "spring", deflecting under load, proportionally to its stiffness, in a manner that all reactions are shared by all supports under an unknown way. Whether a stiffer rigid or under a rigid fixed deflection are in parallel with a less stiff spring or under a flexible deflection, the rigid deflection will absorb a higher portion of the loading. But whether a stiffer rigid or under a rigid fixed deflection are in series with a less stiff spring or under a flexible deflection, the loads absorbed are similar. The importance of such simple concept is applicable to all machines and real

**3.1. Division of forces in assemblies and redundant structures, pulley** 

structures where exist the combinations of parts ("springs"), in series or parallel (7).

divided between the spring 1 and spring 2, as follow;

As seen in the Figure 4 the springs can be arranged in parallel arrangements as well as in series. If the springs are arranged parallel the deflections are the same but the total force F is

(1) ଶܨ ଵܨൌܨ

in engineering, being in analytical or FEM models.

determine accurately and manually.

**application** 

once, ݕଵ ൌ ݕଵ ݕଶ

**Figure 3.** Stacker-reclaimer machine, overview – modeled by Inventor

Note – the 3D model in FEM however requires more deep knowledge of technologists and engineers in Stress x Strain analysis, stress tensors/matrixes, material properties, isotropy and anisotropy, stress states, residual stresses, von-Mises, Mohr circle and basic mechanics evaluations criteria. Even with the software advantages, the output in the FEM models still remains as the engineering duty;


There also drawbacks in the FEM simulations, like;


In order to overcome such deficiencies in the FEM calculations, safety stresses are applied and fatigue coefficients used within fatigue models like; Goodman, modified Goodman, Gerber and/or Soderberg (15-16) .

## **3. Description of an analytical method**

Most of the formulas of strength of materials express the relations among the form and dimensions of a member, the loads applied thereto, and the resulting stress or deformation. Any such formula is valid only within certain limitations and is applicable only to certain problems. An understanding of these limitations and of the way in which formulas may be combined and extended for the solution of problems to which they do not immediately apply, requires knowledge of certain principles and methods that are stated briefly in the pulleys calculations ahead. In determining stress by mathematical analysis, being analytical or FEM, for example, it is customary to assume the material as elastic, isotropic, homogeneous, and infinitely divisible without change in properties and in conforming to Hooke´s law, which states that strain, is proportional to stress. On the other hand, these assumptions despite of imposing certain limitations upon the conventional methods of stress analysis must be used in the form of safety factors. This precaution has given satisfactory results for nearly all problems in engineering, being in analytical or FEM models.

The pulley is basically composed by; expansion ring (when applicable), hub, shaft, disc and cylinder, as seen in the Figure 5 below. The calculation of individual components is still not an issue nowadays and classical formulas may be applied without main difficulties. But when there is an increase of components quantity and the interaction among them takes place, the analytical method cannot predict the real and accurate interaction, energy shared by each component in the assemble, due the imposed deformations. In other words, the proportion of deformation of each individual into the ensemble is a very complex to determine accurately and manually.

## **3.1. Division of forces in assemblies and redundant structures, pulley application**

The concepts of force´s flow are useful in the visualization of paths taken by the forces lines when crossing machines or structures from the load points till the support points. Whether the structure is simple and statically determined, the equations of equilibrium are enough to determine the reactions. On the other hand if redundant supports exist, it means additional supports to those required to satisfy the static equilibrium conditions, those simple equations are not enough anymore to explain the intensities (magnitude) in any one of the reactions. It happen due to the support works as a separated "spring", deflecting under load, proportionally to its stiffness, in a manner that all reactions are shared by all supports under an unknown way. Whether a stiffer rigid or under a rigid fixed deflection are in parallel with a less stiff spring or under a flexible deflection, the rigid deflection will absorb a higher portion of the loading. But whether a stiffer rigid or under a rigid fixed deflection are in series with a less stiff spring or under a flexible deflection, the loads absorbed are similar. The importance of such simple concept is applicable to all machines and real structures where exist the combinations of parts ("springs"), in series or parallel (7).

As seen in the Figure 4 the springs can be arranged in parallel arrangements as well as in series. If the springs are arranged parallel the deflections are the same but the total force F is divided between the spring 1 and spring 2, as follow;

$$F = F\_1 + F\_2 \tag{1}$$

once, ݕଵ ൌ ݕଵ ݕଶ

280 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 3.** Stacker-reclaimer machine, overview – modeled by Inventor

b. in order to produce manuals to operation and/or maintenance.

There also drawbacks in the FEM simulations, like;

remains as the engineering duty;

sensation of reality);

linear analysis;

machines are used.

Gerber and/or Soderberg (15-16) .

**3. Description of an analytical method** 

model;

Note – the 3D model in FEM however requires more deep knowledge of technologists and engineers in Stress x Strain analysis, stress tensors/matrixes, material properties, isotropy and anisotropy, stress states, residual stresses, von-Mises, Mohr circle and basic mechanics evaluations criteria. Even with the software advantages, the output in the FEM models still

a. beauty pictures to present products in commercial and marketing scenarios (high

a. limits when interferences between parts are present in the model, which require non-

b. usually residual stresses are present in the real component but are neglected in the

c. small details in the big picture sometimes need to be handled or suppressed in order to have allow enough capacity to run the model, even when powerful computational

In order to overcome such deficiencies in the FEM calculations, safety stresses are applied and fatigue coefficients used within fatigue models like; Goodman, modified Goodman,

Most of the formulas of strength of materials express the relations among the form and dimensions of a member, the loads applied thereto, and the resulting stress or deformation. Any such formula is valid only within certain limitations and is applicable only to certain problems. An understanding of these limitations and of the way in which formulas may be

$$\frac{F}{\nu} = \frac{F\_1}{\nu\_1} + \frac{F\_2}{\nu\_2} \tag{2}$$

Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical 283

the thickness of the disc. The final

 (8)

where LE it the distance between the block bearings in the shaft, LC it the length of cylinder, d is the cylinder internal diameter, b the disc thickness, Rd the radii of disc, r the internal radii

> �� <sup>=</sup> � ��

being yT the total deflection in mm, P the resultant load applied in N and kT the ensemble

It is quite easy to identify the contribution of each element by a simple comparison between the pulleys components in the formula above and in the Figure 5. The load is transmitted by the pulley cylinder toward the discs, which suffer the high deformation due to its low inertia, then to the shaft which is bent due to the reactions at the bearing blocks. This is a normal condition found in driven pulleys, the drive pulleys contain an additional load, torque, transmitted from the shaft to the discs and lately to the cylinder. The drive pulley

Like described previously, the finite element method (FEM) is a very powerful technique for determining stresses and deflections in complex structures when compared with analytical

of disc, E the Young Modulus, RE the shaft radii and

deflection of the ensemble is determined by;

constant (N.mm).

**Figure 5.** Diagram of pulleys loaded.

won´t be covered at this chapter.

**4.1. Method** 

**4. Description of Finite Element Method (FEM)** 

from where is obtained;

$$k\_p = k\_1 + k\_2 \tag{3}$$

to kp as being the spring constant to each spring in parallel

$$\begin{array}{c} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{1cm}} \mathsf{\hspace{$$

**Figure 4.** Different springs (stiffness – in parallel and series)

When the springs are arranged in series, the force F is the same to both springs, but the deflection of the spring 1 and spring 2 are associated to compose the total deflection, it means;

$$\frac{\mathcal{Y}}{F} = \frac{\mathcal{Y}\_1}{F\_1} + \frac{\mathcal{Y}\_2}{F\_2} \tag{4}$$

from where is obtained;

$$k\_S = \frac{1}{\frac{1}{k\_1} + \frac{1}{k\_2}}\tag{5}$$

to ks as the combined constant to springs in series

The diagram in the Figure 5 shows the pulley main components with imposed load. This load is transferred to all components and the total energy required to absorb such energy is composed by the sum of individual deformation. This deformation is directly related to the bending imposed in the Cylinder, the Disc and the Shaft. The sum of those deformations can be described as follow;

$$
\delta\_T = \delta\_{\mathcal{C} +} \mathcal{Z}. \delta\_D + \delta\_{\mathcal{S}} \tag{6}
$$

T is the total deformation, C the deformation of cylinder, S the deformation of shaft and the D deformation of discs. The deformation of hub is considered zero due to its superior stiffness.

In the case of pulleys the system can be considered the same as explained with springs, what means, the cylinder and the shaft are in series and the discs are parallel each other but in series with the other components. The assembling equation then can be arranged as follow;

$$k\_T = \frac{1}{\frac{4.L^3 E\_E}{6.E.\pi R\_E} + \frac{L^3 \mathcal{E}}{6.E.0, 4.d^3 \mathcal{A}} + b.\pi.(R\_d + r)}\tag{7}$$

where LE it the distance between the block bearings in the shaft, LC it the length of cylinder, d is the cylinder internal diameter, b the disc thickness, Rd the radii of disc, r the internal radii of disc, E the Young Modulus, RE the shaft radii and the thickness of the disc. The final deflection of the ensemble is determined by;

$$\mathcal{Y}\_T = \frac{P}{k\_T} \tag{8}$$

being yT the total deflection in mm, P the resultant load applied in N and kT the ensemble constant (N.mm).

**Figure 5.** Diagram of pulleys loaded.

It is quite easy to identify the contribution of each element by a simple comparison between the pulleys components in the formula above and in the Figure 5. The load is transmitted by the pulley cylinder toward the discs, which suffer the high deformation due to its low inertia, then to the shaft which is bent due to the reactions at the bearing blocks. This is a normal condition found in driven pulleys, the drive pulleys contain an additional load, torque, transmitted from the shaft to the discs and lately to the cylinder. The drive pulley won´t be covered at this chapter.

## **4. Description of Finite Element Method (FEM)**

#### **4.1. Method**

282 Finite Element Analysis – Applications in Mechanical Engineering

to kp as being the spring constant to each spring in parallel

**Figure 4.** Different springs (stiffness – in parallel and series)

to ks as the combined constant to springs in series

from where is obtained;

from where is obtained;

be described as follow;

stiffness.

follow;

� � <sup>=</sup> �� �� <sup>+</sup> �� ��

When the springs are arranged in series, the force F is the same to both springs, but the deflection of the spring 1 and spring 2 are associated to compose the total deflection, it means;

> �� <sup>=</sup> � � �� � � ��

The diagram in the Figure 5 shows the pulley main components with imposed load. This load is transferred to all components and the total energy required to absorb such energy is composed by the sum of individual deformation. This deformation is directly related to the bending imposed in the Cylinder, the Disc and the Shaft. The sum of those deformations can

T is the total deformation, C the deformation of cylinder, S the deformation of shaft and the D deformation of discs. The deformation of hub is considered zero due to its superior

In the case of pulleys the system can be considered the same as explained with springs, what means, the cylinder and the shaft are in series and the discs are parallel each other but in series with the other components. The assembling equation then can be arranged as

> ��� �.�.���.��.

��.�.������

�� <sup>=</sup> � �.��� �.�.�.���� 

� � <sup>=</sup> �� �� <sup>+</sup> �� ��

(2)

(4)

 (5)

 (7)

� � � � 2. � � � (6)

�� = �� + �� (3)

Like described previously, the finite element method (FEM) is a very powerful technique for determining stresses and deflections in complex structures when compared with analytical

methods. With this method the structure is divided into a network of small elements connected to each other at node points. Finite element method grew out of matrix methods for the analysis of structures when the widespread availability of the digital computer made it possible to solve system of hundred of simultaneous equations (8). The FEM is then a computerized method for predicting how a real-world object will react to forces, heat, vibrations, etc… in terms of whether it will break, wear out or function according to design. It is called "analysis", but in the product design cycle it is used to predict what will happen when the product is used (5).

Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical 285

The elements, on the other hand, can only communicate to one another via common nodes. Elements therefore must have common nodes to transfer loads from one to the next, such as

Computer programs usually have many options for types of elements to choose, below the

Since the applied load vector and element stiffnesses are known from the user input, the equation can be solved using matrix algebra by rearranging the equation as follow for the

where; {f} is the vector that represents all of the applied loads. [K] is the assemblage of all the individuals' element stiffness (AE/L) and {x} is the vector that represents the

x� � ������ ��� (9)

in the Figure 7 below.

most usual elements (9):

**Figure 8.** Most usual 3D elements

displacement vector:

**Figure 7.** Communication through Common Nodes

## **4.2. Nodes and elements**

A node is a coordinate location in space where the Degrees Of Freedom (DOFs) are defined. The DOFs of a node represent the possible movements of this point due to the loading of the structure. The DOFs also represent which forces and moments are transferred from one element to the next one. Also, deflection and stress results are usually given at the nodes. An element is a mathematical relation that defines how the DOFs of one node relate to the next. Elements can be lines (beams or trusses), 2-D areas, 3-D areas (plates) or solids (bricks and tetrahedra). The mathematical relation also defines how the deflections create strains and stresses. The degrees of freedom at a node characterize the response and represent the relative possible motion of a node. The type of element being used will characterize which DOFs a node will require. Some analysis types have only one DOF at a node. An example of this is temperature in a thermal analysis. A structural beam element, on the other hand, would have all the DOFs shown in Figure 6. "T" represents translational movement and "R" represents rotational movement about X, Y and Z axis direction, resulting in a maximum of six degrees of freedom.

**Figure 6.** Degrees of freedom of a node (DOFs)

The elements, on the other hand, can only communicate to one another via common nodes. Elements therefore must have common nodes to transfer loads from one to the next, such as in the Figure 7 below.

**Figure 7.** Communication through Common Nodes

284 Finite Element Analysis – Applications in Mechanical Engineering

when the product is used (5).

**4.2. Nodes and elements** 

six degrees of freedom.

**Figure 6.** Degrees of freedom of a node (DOFs)

methods. With this method the structure is divided into a network of small elements connected to each other at node points. Finite element method grew out of matrix methods for the analysis of structures when the widespread availability of the digital computer made it possible to solve system of hundred of simultaneous equations (8). The FEM is then a computerized method for predicting how a real-world object will react to forces, heat, vibrations, etc… in terms of whether it will break, wear out or function according to design. It is called "analysis", but in the product design cycle it is used to predict what will happen

A node is a coordinate location in space where the Degrees Of Freedom (DOFs) are defined. The DOFs of a node represent the possible movements of this point due to the loading of the structure. The DOFs also represent which forces and moments are transferred from one element to the next one. Also, deflection and stress results are usually given at the nodes. An element is a mathematical relation that defines how the DOFs of one node relate to the next. Elements can be lines (beams or trusses), 2-D areas, 3-D areas (plates) or solids (bricks and tetrahedra). The mathematical relation also defines how the deflections create strains and stresses. The degrees of freedom at a node characterize the response and represent the relative possible motion of a node. The type of element being used will characterize which DOFs a node will require. Some analysis types have only one DOF at a node. An example of this is temperature in a thermal analysis. A structural beam element, on the other hand, would have all the DOFs shown in Figure 6. "T" represents translational movement and "R" represents rotational movement about X, Y and Z axis direction, resulting in a maximum of

Computer programs usually have many options for types of elements to choose, below the most usual elements (9):

**Figure 8.** Most usual 3D elements

Since the applied load vector and element stiffnesses are known from the user input, the equation can be solved using matrix algebra by rearranging the equation as follow for the displacement vector:

$$\{\mathbf{x}\}\,\,\,=\![K]^{-1}.\{f\}\,\,\,\tag{9}$$

where; {f} is the vector that represents all of the applied loads. [K] is the assemblage of all the individuals' element stiffness (AE/L) and {x} is the vector that represents the

displacement. A is the area, E is the Modulus of Elasticity and L the length, and {f} = � � �� � � .

Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical 287

**Component Material Diameter (mm) Length (mm) Thickness (mm)**  Shaft SAE-1045 150 1600 diameter 150


When importing solid models that have thin parts, it is often better and simpler to analyze them using plate elements (5-10). Autodesk Simulation can be used to convert thin CAD solid models to plate elements. A plate element is drawn at the midplane of the part. Pulleys are commonly conditioned as described; it has solid elements, like shaft and hubs; and plate elements like discs and the cylinder. As shown ahead the difference is not too substantial but depending on the discrepancy of dimensions, comparatively between parts, the values (stresses outputs) can differ considerably. The DOFs associated with the plate elements are drawn in the Figure 10 and 11 that follows. Note that the out-of-plane rotation (Rz) is not

The components were calculated by Autodesk Simulation and each component received a particular 3D element type and meshing configuration as follow; the shaft received a brick condition with material AISI 1045 as-rolled, the discs received a midplane condition, material ASTM A36, isotropic, the cylinder simulated midplane condition, material ASTM A36, isotropic. The constraints were determined in the shaft region of block bearings (joint constraint). The command, which simulate block bearings with spherical roller bearing, is universal joint, which constraint the DOFs at Tx, Ty and Tz as well as Rz (longitudinal to the shaft length) in the simulation to the first side, and the DOFs Tx, Ty and Rz to the

taken into account because of plate theory, thus the plate elements have 5 DOFs.

Discs ASTM A-36 300 x 980 - 12 Cylinder ASTM A-36 1000 (outside) 1000 10 Hub ASTM A-36 150 x 300 - 70

Bearing block centre

**Table 1.** Pulley main characteristics

**Figure 10.** DOFs of midplane elements

opposite side.

distance

The strains are computed based on the classical differential equations. Stress can then be obtained from the strain using Hooke´s law. These basic equations do not require the use of a computer to solve. However, a computer is needed when complexity is added (4).

## **4.3. How to build the model**

Each individual piece is modeled 3D and then the final assembling built by each part gathering in the final component (product - pulley), see Figure 9 below. The boundary conditions have constraints between the shaft and the hub, the hub and the disc and the disc and the cylinder; all constraints are bonded surfaces. The shaft has at the extremes joint constraints due the presence of block bearings. The bearing blocks usually are composed by spherical roller bearings when pulleys for Stacker-Reclaimers are the case.

**Figure 9.** Pulley basic components (built in Inventor) (sectioned 90o for better visualization)

The pulley studied has its main characteristics shown in the Table 1 below;


**Table 1.** Pulley main characteristics

286 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 9.** Pulley basic components (built in Inventor)

The pulley studied has its main characteristics shown in the Table 1 below;

(sectioned 90o for better visualization)

**4.3. How to build the model** 

� � �� � � .

displacement. A is the area, E is the Modulus of Elasticity and L the length, and {f} =

The strains are computed based on the classical differential equations. Stress can then be obtained from the strain using Hooke´s law. These basic equations do not require the use of

Each individual piece is modeled 3D and then the final assembling built by each part gathering in the final component (product - pulley), see Figure 9 below. The boundary conditions have constraints between the shaft and the hub, the hub and the disc and the disc and the cylinder; all constraints are bonded surfaces. The shaft has at the extremes joint constraints due the presence of block bearings. The bearing blocks usually are composed by

a computer to solve. However, a computer is needed when complexity is added (4).

spherical roller bearings when pulleys for Stacker-Reclaimers are the case.

When importing solid models that have thin parts, it is often better and simpler to analyze them using plate elements (5-10). Autodesk Simulation can be used to convert thin CAD solid models to plate elements. A plate element is drawn at the midplane of the part. Pulleys are commonly conditioned as described; it has solid elements, like shaft and hubs; and plate elements like discs and the cylinder. As shown ahead the difference is not too substantial but depending on the discrepancy of dimensions, comparatively between parts, the values (stresses outputs) can differ considerably. The DOFs associated with the plate elements are drawn in the Figure 10 and 11 that follows. Note that the out-of-plane rotation (Rz) is not taken into account because of plate theory, thus the plate elements have 5 DOFs.

**Figure 10.** DOFs of midplane elements

The components were calculated by Autodesk Simulation and each component received a particular 3D element type and meshing configuration as follow; the shaft received a brick condition with material AISI 1045 as-rolled, the discs received a midplane condition, material ASTM A36, isotropic, the cylinder simulated midplane condition, material ASTM A36, isotropic. The constraints were determined in the shaft region of block bearings (joint constraint). The command, which simulate block bearings with spherical roller bearing, is universal joint, which constraint the DOFs at Tx, Ty and Tz as well as Rz (longitudinal to the shaft length) in the simulation to the first side, and the DOFs Tx, Ty and Rz to the opposite side.

**Figure 11.** Pulley meshing – Autodesk simulation

The elements in the joints (block bearings) were considered with a very high stiffness's, which guarantee not interference in the stresses results in the model (Figure 12).

In terms of loading, there was a force applied perpendicular to the surface, which resulted in a variable pressure (parabola) around 180o of cylinder, represented by the following equation;

$$P = 0.47.R^2 + 0.47\tag{10}$$

Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical 289

**Figure 12.** Constraint –Pin (universal joints)

**Table 2.** Stresses on the main components (MPa)

**Component Material von Mises** 

Mises stress (10-11).

Based on the fact the pulleys applications are dynamic (cyclic loading) the fatigue limit for each material was utilized in comparison with the stress range in reference to the equivalent stress (von Mises) by FEM (11-16). The Table 2 and Figure 19 reveal the main stresses on the pulley components. All the stresses are compared to the fatigue limit once this is the main phenomena the components is submitted. The stress range is calculated toward the von

**(MPa)** 

Shaft SAE-1045 90 180 230 Discs ASTM A-36 128 252 200 Cylinder ASTM A-36 45 90 200

**Stress range (MPa)** 

**Fatigue limit (MPa)** 

where P = pressure (MPa) R = pulley radius (mm), 0,47 = pressure (MPa).

The variable pressure is shown in the figure 13 below. The load applied on the cylinder outside and around 180o was 316kN. The analysis was done based on the previous description in the Autodesk simulation, being the von Mises stresses analyzed for each component, as follow by the Figures 13 to 18.

**Figure 12.** Constraint –Pin (universal joints)

**Figure 11.** Pulley meshing – Autodesk simulation

component, as follow by the Figures 13 to 18.

equation;

The elements in the joints (block bearings) were considered with a very high stiffness's,

In terms of loading, there was a force applied perpendicular to the surface, which resulted in a variable pressure (parabola) around 180o of cylinder, represented by the following

The variable pressure is shown in the figure 13 below. The load applied on the cylinder outside and around 180o was 316kN. The analysis was done based on the previous description in the Autodesk simulation, being the von Mises stresses analyzed for each

� � 0,47� �� + 0,47 (10)

which guarantee not interference in the stresses results in the model (Figure 12).

where P = pressure (MPa) R = pulley radius (mm), 0,47 = pressure (MPa).

Based on the fact the pulleys applications are dynamic (cyclic loading) the fatigue limit for each material was utilized in comparison with the stress range in reference to the equivalent stress (von Mises) by FEM (11-16). The Table 2 and Figure 19 reveal the main stresses on the pulley components. All the stresses are compared to the fatigue limit once this is the main phenomena the components is submitted. The stress range is calculated toward the von Mises stress (10-11).


**Table 2.** Stresses on the main components (MPa)

Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical 291

**Figure 14.** Stresses on the shaft (MPa)

**Figure 15.** Stresses on the discs (MPa)

**Figure 13.** Variable pressure

All the stresses are under the fatigue limit except the discs, which overtake the limit over 52MPa. At this case the re-analysis of the discs thickness should be done and the thickness most of times increased or another type or thickness of disc applied. After the re-calculation, as expected, all assemble components have a new and different stress level, being highly recommendable afterwards the revaluations due to the fatigue limit consideration.

The values found in the analytical model (Table 2) were also compared with the FEM and are described in the Table 3 below.

**Figure 13.** Variable pressure

are described in the Table 3 below.

All the stresses are under the fatigue limit except the discs, which overtake the limit over 52MPa. At this case the re-analysis of the discs thickness should be done and the thickness most of times increased or another type or thickness of disc applied. After the re-calculation, as expected, all assemble components have a new and different stress level, being highly

The values found in the analytical model (Table 2) were also compared with the FEM and

recommendable afterwards the revaluations due to the fatigue limit consideration.

**Figure 15.** Stresses on the discs (MPa)

Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical 293

**Figure 18.** Stresses at the interface hub and shaft (MPa)

**Figure 19.** Stresses on the components (MPa)

**Figure 16.** Stresses on the cylinder (MPa)

**Figure 17.** Stresses on the discs (MPa)

**Figure 18.** Stresses at the interface hub and shaft (MPa)

**Figure 16.** Stresses on the cylinder (MPa)

**Figure 17.** Stresses on the discs (MPa)

**Figure 19.** Stresses on the components (MPa)

Overview in the Application of FEM in Mining and the Study of Case: Stress Analysis in Pulleys of Stacker-Reclaimers: FEM vs. Analytical 295











environmental and cost savings due less weight and prior interference analysis.

*Machines and Components Product Engineering, Metso Minerals, Sorocaba, Brazil* 

data are not accurate like those presented by FEM either;

usage of models 3D and calculations by FEM;

drawings and awareness due interferences;

of stresses and strains are more eminent nowadays;

ones, being by analytical model or FEM are necessary;

and István Kövesdy

not demonstrated in this paper;

is wrong but they present certain differences in terms of stresses;

**5. Conclusions** 

technology;

**Author details** 

Corresponding Authors

Jairo A. Martins\*

 \*

machines suppliers;

**Figure 20.** Stresses on the components (MPa) Analytical versus FEM


**Table 3.** Difference between analytical and MEF methods

There are differences in the results between the Analytical and FEM models (11). The equivalent stresses on the shaft are closer, around 10% difference, showing the lower value found in FEM, the discs are those which have medium difference and around 26%, being the stresses on the FEM higher than the analytical model and the third is the cylinder which had its lower value found in the FEM and around minus 33%. The differences are not too high but in certain cases should be taken into account when safety factors are in the limit due lean projects purposes. Neither the analytical model nor FEM are described in details once the idea is to bring the basic concepts used in components designing.

## **5. Conclusions**

294 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 20.** Stresses on the components (MPa)

**Component Material Stress**

**Table 3.** Difference between analytical and MEF methods

idea is to bring the basic concepts used in components designing.

**(MPa)** 

There are differences in the results between the Analytical and FEM models (11). The equivalent stresses on the shaft are closer, around 10% difference, showing the lower value found in FEM, the discs are those which have medium difference and around 26%, being the stresses on the FEM higher than the analytical model and the third is the cylinder which had its lower value found in the FEM and around minus 33%. The differences are not too high but in certain cases should be taken into account when safety factors are in the limit due lean projects purposes. Neither the analytical model nor FEM are described in details once the

Shaft SAE-1050 102 90 -10% Disc ASTM A-36 100 128 +26% Cylinder ASTM A-36 54 45 -33%

**MEF (MPa)**  **Difference (%)** 

Analytical versus FEM


## **Author details**

Jairo A. Martins\* and István Kövesdy *Machines and Components Product Engineering, Metso Minerals, Sorocaba, Brazil* 

<sup>\*</sup> Corresponding Authors

## **Acknowledgement**

The acknowledgements are addressed to CNPq (National Council for Scientific and Technological Development – Brazil), Mr. Rubens Costa, Vice President, Mining Operations, South America at Metso Brazil, Mr. Misael Ramalho and the Bulk-Machines Team and Mr. Vanderson L. Zangerolamo from MAPDATA due their valuable technical support.

**Chapter 14** 

© 2012 Sakai and Nakayama, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Sakai and Nakayama, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Optimization and Improvement of Throwing** 

**Finite Element Analysis** 

Shinobu Sakai and Hitoshi Nakayama

http://dx.doi.org/10.5772/46159

**1. Introduction** 

Additional information is available at the end of the chapter

**Performance in Baseball Pitching Machine Using** 

Pitching machines for baseball are widely used in venues ranging from professional baseball stadiums to amusement facilities (Adair, 1994). The most important purpose of the pitching machine is to reproduce the throws of an adversary pitcher, which will be useful for the improvement batting technique. The most common commercial pitching machines for baseball are the "arm" type and the "two-roller" type. Some pitching machines can pitch a high-speed ball (fastball) and a breaking ball, but these machines have certain limitations. In particular, it is very difficult to change ball speed and direction simultaneously (Mish et al., 2001). Therefore, the throwing performance of conventional pitching machines used for batting practice is not very high. Balls pitched to change in instant at various speeds and with different pitch types (ex. fastball, curveball, screwball and forkball) are easily achieved by a new pitching machine equipped with three rollers which has been developed by the authors. It is called a "three-roller" type pitching machine (Sakai et al., 2007). With the structure of three rollers, comes the production of a new pitching machine that can pitch balls repeatedly in the way the batter desires, controlling both ball speed and pitch type. However, as observed during our study, the seam of a baseball coming in contact with the rollers, the spin rate and projection angle of the ball delicately change. From the results, it

In this chapter, the throw simulation of the three roller-type pitching machine is analyzed using a commercial dynamic finite element analysis code (ANSYS/ LS-DYNA). The moving behavior (such as velocity and spin rate) and contact stress state of the ball pitched by the pitching machine are clearly observed. The effect of the throw accuracy by the seam posture of the ball in the machine is examined. In addition, the shapes and materials of the rollers do not

became clear experimentally that from the throw accuracy deteriorates.

## **6. References**

