**Group-3: Models for Gear Dynamics**

A model that considers tooth compliance and the flexibility of the relevant components. Typically these models include torsional flexibility of shafts and lateral flexibility of bearings and shafts along the line of action.

#### **Group-4: Models for Geared Rotor Dynamics**

This group of models consider transverse vibrations of gear carrying shafts as well as the lateral component *(NOTE: Transverse: along the Plane of Action, Lateral: Normal to the Plane of Action).* Movement of the gears is considered in two mutually perpendicular directions to simulate, for example, whirling.

#### **Group-5: Models for Torsional Vibrations**

The models in the third and fourth groups consider the flexibility of the gear teeth by including a constant or time varying mesh stiffness. The models belonging to this group differentiate themselves from the third and fourth groups by having rigid gears mounted on flexible shafts. The flexibility at the gearmesh is neglected. These models are used in studying pure (low frequency) torsional vibration problems.

Table 2.4.1. Classification of Gear Dynamic Models. (Ozguven & Houser [6])

functionality. Traditionally lumped parameter modelling (LPM) has been a common technique that has been used to study the dynamics of gears. Wang [7] introduced a simple LPM to rationalize the dynamic factor calculation by the laws of mechanics. He proposed a model that relates the GTE and the resulting dynamic loading. A large number of gear dynamic models that are being used widely today are based on this work. The result of an additional literature survey on more recently published materials by Bartelmus [8], Lin & Parker [9, 10], Gao & Randall [11, 12], Amabili & Rivola [13], Howard et al [14], Velex & Maatar [15], Blankenship & Singh [17], Kahraman & Blankenship [18] show that the fundamentals of the modelling technique in gear simulations have not changed and the LMP method still serves as an efficient technique to model the wide range of gear dynamics behaviour. More advanced LPM models incorporate extra functions to simulate specialized phenomena. For example, the model presented by P. Velex and M. Maatar [15] uses the individual gear tooth profiles as input and calculates the GTE directly from the gear tooth profile. Using this method they simulated how the change in contact behaviour of meshing gears due to misalignment affects the resulting TE.

FEA has become one of the most powerful simulation techniques applied to broad range of modern Engineering practices today. There have been several groups of researchers who attempted to develop detailed FEA based gear models, but they were troubled by the

Most early models belong to this group. The model was used to study gear dynamic load and to determine the value of dynamic factor that can be used in gear root stress formulae. Empirical, semi-empirical models and dynamic models constructed specifically for

Models that consider tooth stiffness as the only potential energy storing element in the system. Flexibility of shafts, bearings etc is neglected. Typically, these models are single DOF spring-mass systems. Some of the models from this group are classified in group-1 if

A model that considers tooth compliance and the flexibility of the relevant components. Typically these models include torsional flexibility of shafts and lateral flexibility of

This group of models consider transverse vibrations of gear carrying shafts as well as the lateral component *(NOTE: Transverse: along the Plane of Action, Lateral: Normal to the Plane of Action).* Movement of the gears is considered in two mutually perpendicular directions to

The models in the third and fourth groups consider the flexibility of the gear teeth by including a constant or time varying mesh stiffness. The models belonging to this group differentiate themselves from the third and fourth groups by having rigid gears mounted on flexible shafts. The flexibility at the gearmesh is neglected. These models are used in

functionality. Traditionally lumped parameter modelling (LPM) has been a common technique that has been used to study the dynamics of gears. Wang [7] introduced a simple LPM to rationalize the dynamic factor calculation by the laws of mechanics. He proposed a model that relates the GTE and the resulting dynamic loading. A large number of gear dynamic models that are being used widely today are based on this work. The result of an additional literature survey on more recently published materials by Bartelmus [8], Lin & Parker [9, 10], Gao & Randall [11, 12], Amabili & Rivola [13], Howard et al [14], Velex & Maatar [15], Blankenship & Singh [17], Kahraman & Blankenship [18] show that the fundamentals of the modelling technique in gear simulations have not changed and the LMP method still serves as an efficient technique to model the wide range of gear dynamics behaviour. More advanced LPM models incorporate extra functions to simulate specialized phenomena. For example, the model presented by P. Velex and M. Maatar [15] uses the individual gear tooth profiles as input and calculates the GTE directly from the gear tooth profile. Using this method they simulated how the change in contact behaviour of meshing

FEA has become one of the most powerful simulation techniques applied to broad range of modern Engineering practices today. There have been several groups of researchers who attempted to develop detailed FEA based gear models, but they were troubled by the

**Group-1: Simple Dynamic Factor Models** 

**Group-2: Models with Tooth Compliance** 

**Group-3: Models for Gear Dynamics** 

simulate, for example, whirling.

bearings and shafts along the line of action. **Group-4: Models for Geared Rotor Dynamics** 

**Group-5: Models for Torsional Vibrations** 

determination of dynamic factor are included in this group.

they are designed solely for determining the dynamic factor.

studying pure (low frequency) torsional vibration problems.

gears due to misalignment affects the resulting TE.

Table 2.4.1. Classification of Gear Dynamic Models. (Ozguven & Houser [6])

challenges in efficiently modelling the rolling Hertzian contact on the meshing surfaces of gear teeth. Hertzian contact occurs between the meshing gear teeth which causes large concentrated forces to act in very small area. It requires very fine FE mesh to accurately model this load distribution over the contact area. In a conventional finite element method, a fully representative dynamic model of a gear requires this fine mesh over each gear tooth flank and this makes the size of the FE model prohibitively large.

Researchers from Ohio State University have developed an efficient method to overcome the Hertzian contact problem in the 1990s' [16]. They proposed an elegant solution by modelling the contact by an analytical technique and relating the resulting force distribution to a coarsely meshed FE model. This technique has proven so efficient that they were capable of simulating the dynamics of spur and planetary gears by [19, 20] (see Figure-2.4.1). For more details see the CALYX user's manuals [21, 22].

For the purpose of studies, which require a holistic understanding of gear dynamics, a lumped parameter type model appears to provide the most accessible and computationally economical means to conduct simulation studies.

A simple single stage gear model is used to explain the basic concept of gear dynamic simulation techniques used in this chapter. A symbolic representation of a single stage gear system is illustrated in Figure-2.4.2. A pair of meshing gears is modelled by rigid disks representing their mass/moment of inertia. The discs are linked by line elements that represent the stiffness and the damping (representing the combined effect of friction and fluid film damping) of the gear mesh. Each gear has three translational degrees of freedom (one in a direction parallel to the gear's line of action, defining all interaction between the gears) and three rotational degree of freedoms (DOFs). The stiffness elements attached to the centre of the disks represent the effect of gear shafts and supporting mounts. NOTE: Symbols for the torsional stiffnesses are not shown to avoid congestion.

Fig. 2.4.1. (a) Parker's planetary gear model and (b) FE mesh of gear tooth. Contacts at the meshing teeth are treated analytically. It does not require dense FE mesh. (Courtesy of Parker et al. [20])

where,

modelling.

intelligence.

**2.5.1 Modelling rolling element bearings** 

Angular position of pinion.

damping at the gearmesh.

*F* Static force vector.

refers to stiffness at the gearmesh.

Gearbox Simulation Models with Gears and Bearings Faults 27

*x* , *x* , *x* Vectors of translational and rotational displacement, velocity and acceleration.

*h* An 'on/off' switch governing the contact state of the meshing gear teeth.

*te* , *te* A vector representing the combined effect of tooth topography deviations.

**2.5 Modelling rolling element bearings and gearbox casing** 

*K, Kmb* Stiffness matrices (where *K* includes the contribution from *Kmb*). The subscript '*mb*'

*C, Cmb* Damping matrices (*C* including contribution from *Cmb*). The subscript '*mb*' refers to

For many practical purposes, simplified models of gear shaft supports (for example, the effect of rolling element bearings (REBs) and casing were modelled as simple springs with constant stiffnesses) can be effective tools. However, fuller representations of these components become essential in the pursuit of more complete and accurate simulation

For a complete and more realistic modelling of the gearbox system, detailed representations of the REBs and the gearbox casing are necessary to capture the interaction amongst the

Understanding the interaction between the supporting structure and the rotating components of a transmission system has been one of the most challenging areas of designing more detailed gearbox simulation models. The property of the structure supporting REBs and a shaft has significant influence on the dynamic response of the system. Fuller representation of the REBs and gearbox casing also improves the accuracy of the effect transmission path that contorts the diagnostic information originated from the faults in gears and REBs. It is desired in many applications of machine health monitoring that the method is minimally intrusive on the machine operation. This requirement often drives the sensors and/or the transducers to be placed in an easily accessible location on the machine, such as exposed surface of gearbox casing or on the machine skid or on an exposed

The capability to accurately model and simulate the effect of transmission path allows more realistic and effective means to train the diagnostic algorithms based on the artificial

A number of models of REBs exist in literatures [24, 25, 26, 27] and are widely employed to study the dynamics and the effect of faults in REBs. The authors have adopted the 2 DoF model originally developed by Fukata [27] in to the LPM of the gearbox. Figure-2.5.1 (a)

gears, the REBs and the effects of transfer path and dynamics response of the casing.

and readily accessible structural frame which the machine is mounted on.

*M t t <sup>s</sup> x Cx e Kx e F* (2.4.1)

*s mb mb M t t x Cx Kx F hC e hK e* (2.4.2)

*xi, yi, zi* Translation at *ith* Degrees of Freedom.

xi, yi, zi Rotation about a translational axis at ith Degrees of Freedom.


$$i \text{ } \huge\text{Index: } i \text{=1,2,3 \dots etc.}$$

Fig. 2.4.2. A Typical Lumped Parameter Model of Meshing Gears.

The linear spring elements representing the Rolling Element Bearings (REB) are a reasonable simplification of the system that is well documented in many papers on gear simulation. For the purpose of explaining the core elements of the gear simulation model, the detail of REB as well as the casing was omitted from this section; more comprehensive model of a gearbox, with REB and casing, will be presented later in section 2.5.

Vibration of the gears is simulated in the model as a system responding to the excitation caused by a varying TE, '*et*' and mesh stiffness '*Kmb*'. The dominant force exciting the gears is assumed to act in a direction along the plane of action (PoA). The angular position dependent variables '*et*' and '*Kmb*' are expressed as functions of the pinion pitch angle (*θy1)* and their values are estimated by using static simulation. Examples of similar techniques are given by Gao & Randall [11, 12], Du [23] and Endo and Randall [61].

Equations of motion derived from the LPM are written in matrix format as shown in Equation-2.4.1. The equation is rearranged to the form shown in the Euqation-2.4.2; the effect of TE is expressed as a time varying excitation in the equation source. The dynamic response of the system is simulated by numerically solving the second order term (accelerations) for each step of incremented time. The effect of the mesh stiffness variation is implemented in the model by updating its value for each time increment.

$$\mathbf{M}\ddot{\underline{\mathbf{x}}} + \mathbf{C}\left(\dot{\underline{\mathbf{x}}} - \dot{\underline{\mathbf{e}}}\_{t}\right) + \mathbf{K}\left(\underline{\mathbf{x}} - \underline{\mathbf{e}}\_{t}\left(\boldsymbol{\Theta}\right)\right) = \mathbf{F}\_{s} \tag{2.4.1}$$

$$\mathbf{M}\ddot{\mathbf{x}} + \mathbf{C}\dot{\underline{x}} + \mathbf{K}\underline{x} = F\_s + h\mathbf{C}\_{mb}\left(\boldsymbol{\theta}\right)\dot{\underline{e}}\_t + h\mathbf{K}\_{mb}\left(\boldsymbol{\theta}\right)\underline{e}\_t\tag{2.4.2}$$

where,

26 Mechanical Engineering

*C, Cmb* Damping matrices. The subscript '*mb*' refers to damping at the gearmesh. Typically for Cmb, ζ

*te* A vector representing the combined effect of tooth topography deviations and misalignment

The linear spring elements representing the Rolling Element Bearings (REB) are a reasonable simplification of the system that is well documented in many papers on gear simulation. For the purpose of explaining the core elements of the gear simulation model, the detail of REB as well as the casing was omitted from this section; more comprehensive model of a

Vibration of the gears is simulated in the model as a system responding to the excitation caused by a varying TE, '*et*' and mesh stiffness '*Kmb*'. The dominant force exciting the gears is assumed to act in a direction along the plane of action (PoA). The angular position dependent variables '*et*' and '*Kmb*' are expressed as functions of the pinion pitch angle (*θy1)* and their values are estimated by using static simulation. Examples of similar techniques are

Equations of motion derived from the LPM are written in matrix format as shown in Equation-2.4.1. The equation is rearranged to the form shown in the Euqation-2.4.2; the effect of TE is expressed as a time varying excitation in the equation source. The dynamic response of the system is simulated by numerically solving the second order term (accelerations) for each step of incremented time. The effect of the mesh stiffness variation is

*K, Kmb* Linear stiffness elements. The subscript '*mb*' refers to stiffness at the gearmesh. *h* An 'on/off' switch governing the contact state of the meshing gear teeth.

*xi, yi, zi* Translation at *ith* Degrees of Freedom.

= 3 ~ 7%.

 of the gear pair. *i* Index: *i*=1,2, 3 …etc.

xi, yi, zi Rotation about a translational axis at ith Degrees of Freedom.

Fig. 2.4.2. A Typical Lumped Parameter Model of Meshing Gears.

gearbox, with REB and casing, will be presented later in section 2.5.

given by Gao & Randall [11, 12], Du [23] and Endo and Randall [61].

implemented in the model by updating its value for each time increment.


*te* , *te* A vector representing the combined effect of tooth topography deviations.
