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

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 modelling.

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

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

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 intelligence.

### **2.5.1 Modelling rolling element bearings**

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)

Gearbox Simulation Models with Gears and Bearings Faults 29

The load deflection factor *<sup>b</sup> k* depends on the geometry of contacting bodies, the elasticity of the material, and exponent n. The value of n=1.5 for ball bearings and n=1.1 for roller bearings. Using Equation-2.5.4 and summing the contact forces in the *x* and *y* directions for a ball bearing with *nb* balls, the total force exerted by the bearings to the supporting

1.5

**2.5.2 Gearbox casing model – Component mode synthesis method** 

comparison between the simulated and measured vibration signals.

cos

 

*f k* and 1.5

The stiffness of the given REB model is non-linear, and is time varying as it depends on the positions of the rolling elements that determine the contact condition. The effect of slippage was introduced to the model by adding random jitters of 0.01-0.02 radians to the nominal

Lumped parameter modelling (LPM) is an efficient means to express the internal dynamics of transmission systems; masses and inertias of key components such as gears, shafts and bearings can be lumped at appropriate locations to construct a model. The advantage of the LPM is that it provides a method to construct an effective dynamic model with relatively small number of degrees-of-freedoms (DOF), which facilitates computationally economical method to study the behaviour of gears and bearings in the presence of nonlinearities and

One of the limitations of the LPM method is that it does not account for the interaction between the shaft and the supporting structure; i.e. casing flexibility, which can be an important consideration in light weight gearboxes, that are common aircraft applications. Not having to include the appropriate effect of transmission path also results in poor

Finite Element Analysis (FEA) is an efficient and well accepted technique to characterize a dynamic response of a structure such as gearbox casings. However, the use of FEA results in a large number of DOF, which could cause some challenges when attempt to solve a vibrodynamic model of a combined casing and the LMP of gearbox internal components. Solving a large number of DOFs is time consuming even with the powerful computers available today and it could cause a number of computational problems, especially when attempting to simulate a dynamic response of gear and bearing faults which involves nonlinearities.

To overcome this shortcoming, a number of reduction techniques [36, 37] have been proposed to reduce the size of mass and stiffness matrix of FEA models. The simplified gearbox casing model derived from the reduction technique is used to capture the key characteristics of dynamic response of the casing structure and can be combined with the

The Craig-Bampton method [37] is a dynamic reduction method for reducing the size of the finite element models. In this method, the motion of the whole structure is represented as a combination of boundary points (so called master degree of freedom) and the modes of the structure, assuming the master degrees of freedom are held fixed. Unlike the Guyan reduction [38], which only deals with the reduction of stiffness matrix, the Craig-Bumpton

1

 

*nb y b i j j i*

sin

 

*f k* (2.5.5)

1

 

*nb x b jj j j*

structure can be calculated as follows:

position of the cage at each step.

geometrical faults [32, 33, 34, 35].

LPM models of gears and REBs.

illustrates the main components of the rolling element bearing model and shows the load zone associated with the distribution of radial loads in the REB as it supports the shaft. Figure-2.5.1 (b) explains the essentials of the bearing model as presented by [28]. The two degree-of-freedom REB model captures the load-deflection relationships, while ignoring the effect of mass and the inertia of the rolling elements. The two degrees of freedom (*xs*, *ys*) are related to the inner race (shaft). Contact forces are summed over each of the rolling elements to give the overall forces on the shaft.

Fig. 2.5.1. (a) Rolling element bearing components and load distribution; (b) Two degree of freedom model. [28]

The overall contact deformation (under compression) for the j'th -rolling element *j* is a function of the inner race displacement relative to the outer race in the *x* and *y* directions (( ) *<sup>s</sup> <sup>p</sup> x x* ,( ) *<sup>s</sup> <sup>p</sup> y y* ), the element position *<sup>j</sup>* (time varying) and the clearance ( *c* ). This is given by:

$$\boldsymbol{\delta}\_{j} = (\mathbf{x}\_{s} - \mathbf{x}\_{p})\cos\phi\_{j} + (y\_{s} - y\_{p})\sin\phi\_{j} - \boldsymbol{c} - \boldsymbol{\beta}\_{j}\mathbf{C}\_{d} \tag{2.5.1} \text{ (}\text{ $i = 1, 2$ .)}$$

Accounting for the fact that compression occurs only for positive values of *<sup>j</sup>* , *<sup>j</sup>* (contact state of *<sup>j</sup>* the rolling elements) is introduced as:

$$\gamma\_{\dot{j}} = \begin{cases} 1, & \text{if } \mathcal{S}\_{\dot{j}} > 0 \\ 0, & \text{otherwise} \end{cases} \tag{2.5.2}$$

The angular positions of the rolling elements *<sup>j</sup>* are functions of time increment *dt*, the previous cage position *<sup>o</sup>* and the cage speed *c* (can be calculated from the REB geometry and the shaft speed *<sup>s</sup>* assuming no slippage) are given as:

$$\phi\_j = \frac{2\,\pi(j-1)}{n\_b} + \alpha\_c dt + \phi\_o \quad \text{with} \quad \alpha\_c = (1 - \frac{D\_b}{D\_p})\frac{\alpha\_s}{2} \tag{2.5.3}$$

The ball raceway contact force *f* is calculated by using traditional Hertzian theory (nonlinear stiffness) from:

$$f = k\_b \, \delta^v \tag{2.5.4}$$

illustrates the main components of the rolling element bearing model and shows the load zone associated with the distribution of radial loads in the REB as it supports the shaft. Figure-2.5.1 (b) explains the essentials of the bearing model as presented by [28]. The two degree-of-freedom REB model captures the load-deflection relationships, while ignoring the effect of mass and the inertia of the rolling elements. The two degrees of freedom (*xs*, *ys*) are related to the inner race (shaft). Contact forces are summed over each of the rolling elements

Fig. 2.5.1. (a) Rolling element bearing components and load distribution; (b) Two degree of

The overall contact deformation (under compression) for the j'th -rolling element *j* is a function of the inner race displacement relative to the outer race in the *x* and *y* directions

Accounting for the fact that compression occurs only for positive values of *<sup>j</sup>* , *<sup>j</sup>* (contact

<sup>0</sup>

*0, otherwise*

*<sup>o</sup>* and the cage speed *c* (can be calculated from the REB geometry

with (1 ) <sup>2</sup>

 *b s <sup>c</sup> p D*

*<sup>j</sup>*

 *c o*

The ball raceway contact force *f* is calculated by using traditional Hertzian theory (non-

*n*

*1, if*

 *<sup>s</sup> pj j <sup>d</sup> x x y y c C* (*j* = 1,2..) (2.5.1)

*<sup>j</sup>* (time varying) and the clearance ( *c* ). This is

(2.5.2)

*D* (2.5.3)

*<sup>j</sup>* are functions of time increment *dt*, the

*<sup>b</sup> f k* (2.5.4)

 *<sup>j</sup>* ( )cos ( )sin *<sup>s</sup> p j* 

*j*

to give the overall forces on the shaft.

freedom model. [28]

previous cage position

linear stiffness) from:

given by:

(( ) *<sup>s</sup> <sup>p</sup> x x* ,( ) *<sup>s</sup> <sup>p</sup> y y* ), the element position

state of *<sup>j</sup>* the rolling elements) is introduced as:

The angular positions of the rolling elements

*j*

and the shaft speed *<sup>s</sup>* assuming no slippage) are given as:

2 ( 1)

*<sup>j</sup> dt*

*b*

*n*

The load deflection factor *<sup>b</sup> k* depends on the geometry of contacting bodies, the elasticity of the material, and exponent n. The value of n=1.5 for ball bearings and n=1.1 for roller bearings. Using Equation-2.5.4 and summing the contact forces in the *x* and *y* directions for a ball bearing with *nb* balls, the total force exerted by the bearings to the supporting structure can be calculated as follows:

$$f\_x = k\_b \sum\_{j=1}^{n\_b} \gamma\_j \delta\_j^{1.5} \cos \phi\_j \quad \text{and} \quad f\_y = k\_b \sum\_{i=1}^{n\_b} \gamma\_i \delta\_j^{1.5} \sin \phi\_j \tag{2.5.5}$$

The stiffness of the given REB model is non-linear, and is time varying as it depends on the positions of the rolling elements that determine the contact condition. The effect of slippage was introduced to the model by adding random jitters of 0.01-0.02 radians to the nominal position of the cage at each step.
