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

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

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 comparison between the simulated and measured vibration signals.

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 LPM models of gears and REBs.

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

Gearbox Simulation Models with Gears and Bearings Faults 31

By applying this transformation, the number of DOFs of the component will be reduced. The new reduced mass and stiffness matrices can be extracted using Equations 2.5.11 &

*<sup>t</sup>*

0

0 0

Thus Equation-2.5.7 can be re-written in the new reduced form using the reduced mass and

Where *Mbb* is the boundary mass matrix i.e. total mass properties translated to the boundary points. *bb k* is the interface stiffness matrix, i.e. stiffness associated with displacing one boundary DOF while the others are held fixed. The *Mbq* is the component matrix ( *Mqb*

\ 0

0 \ *kqq i* 

\ 0

0 \ 

0 0 0

Finally the dynamic equation of motion (including damping) using the Craig-Bampton

 

*M M u u uF k*

*bb bq m m bb m m*

 = fraction of critical damping) For more detailed explanation of the techniques related to the modelling of rolling element bearings and the application of component mode synthesis (CMS) techniques, refer to the

*qb*

works by Sawalhi, Deshpande and Randall [41].

= modal damping (

*M M u uF k*

*reduced reduced reduced <sup>M</sup> <sup>k</sup> <sup>F</sup> bb bq m bb m m*

 

*qb qq qq*

If the mode shapes have been mass normalized (typically they are) then:

*ii i i k m*/

, and,

*i* is the eigenvalues; <sup>2</sup> 

stiffness matrices as well as the modal coordinates as follows:

*<sup>t</sup> Mreduced T MT* (2.5.11)

*reduced k T kT* (2.5.12)

*M I qq* (2.5.15)

2

0 2 0 0

*MIq q q* (2.5.16)

(2.5.14)

*MM q <sup>k</sup> <sup>q</sup>* (2.5.13)

2.5.12 respectively:

is the transpose of *Mbq* ).

where

where 2

transform can be written as:

and

method accounts for both the mass and the stiffness. Furthermore, it enables defining the frequency range of interest by identifying the modes of interest and including these as a part of the transformation matrix. The decomposition of the model into both physical DOFs (master DOFs) and modal coordinates allows the flexibility of connecting the finite elements to other substructures, while achieving a reasonably good result within a required frequency range. The Craig-Bumpton method is a very convenient method for modelling a geared transmission system as the input (excitation) to the system is not defined as forces, but as geometric mismatches at the connection points (i.e. gear transmission error and bearing geometric error). The following summary of the Craig-Bampton method is given based on the references [39-41].

In the Craig-Bampton reduction method, the equation of motion (dynamic equilibrium) of each superelement (substructure), without considering the effect of damping, can be expressed as in Equation-2.5.6:

$$[M]\{\ddot{u}\} + [k]\{u\} = \{F\} \tag{2.5.6}$$

Where [ ] *M* is the mass matrix, [ ] *k* is the stiffness matrix, *F* is the nodal forces, *u* and *u* are the nodal displacements and accelerations respectively. The key to reducing the substructure is to split the degrees of freedom into masters *um* (at the connecting nodes) and slaves *us* (at the internal nodes). The mass, the stiffness and the force matrices are rearranged accordingly as follows:

$$
\begin{aligned}
\overbrace{\begin{bmatrix}
\boldsymbol{M}\_{mm} & \boldsymbol{M}\_{ms} \\
\boldsymbol{M}\_{sm} & \boldsymbol{M}\_{ss}
\end{bmatrix}}
\end{aligned}
\begin{aligned}
\begin{bmatrix}
\ddot{\boldsymbol{u}}\_{m} \\
\ddot{\boldsymbol{u}}\_{s}
\end{bmatrix}
\end{aligned}
\begin{aligned}
\begin{aligned}
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{s}
\end{aligned}
\end{aligned}
\begin{aligned}
\begin{aligned}
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{s}
\end{aligned}
\end{aligned}
\begin{aligned}
\begin{aligned}
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{s}
\end{aligned}
\end{aligned}
\begin{aligned}
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{s}
\end{aligned}
\end{aligned}
\begin{aligned}
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{s}
\end{aligned}
\end{aligned}
\begin{aligned}
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{s}
\end{aligned}
\end{aligned}
\begin{aligned}
\boldsymbol{\underline{k}}\_{m} = \begin{bmatrix}
\boldsymbol{F}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\boldsymbol{\underline{k}}\_{m} \\
\end{aligned}}
\end{aligned}$$

The subscript *m* denotes master, *s* denotes slave. Furthermore, the slave degrees of freedom (internals) can be written by using generalized coordinates (modal coordinates by ( *q* ) using the fixed interface method, i.e. using the mode shapes of the superelement by fixing the master degrees of freedom nodes (connecting/ boundary nodes). The transformation matrix (*T* ) is the one that achieves the following:

$$
\begin{Bmatrix} \mu\_m\\ \mu\_s \end{Bmatrix} = T \begin{Bmatrix} \mu\_m\\ q \end{Bmatrix} \tag{2.5.8}
$$

For the fixed interface method, the transformation matrix (*T* ) can be expressed as shown in Equation-2.5.9:

$$T = \begin{bmatrix} I & \mathbf{0} \\ G\_{\text{sur}} & \phi\_s \end{bmatrix} \tag{2.5.9}$$

where,

$$\mathbf{G}\_{sm} = -k\_{ss}^{-1} k\_{sm} \tag{2.5.10}$$

and *<sup>s</sup>* is the modal matrix of the internal DOF with the interfaces fixed. By applying this transformation, the number of DOFs of the component will be reduced. The new reduced mass and stiffness matrices can be extracted using Equations 2.5.11 & 2.5.12 respectively:

$$M\_{reduced} = \mathbf{T}^t \mathbf{M} \mathbf{T} \tag{2.5.11}$$

and

30 Mechanical Engineering

method accounts for both the mass and the stiffness. Furthermore, it enables defining the frequency range of interest by identifying the modes of interest and including these as a part of the transformation matrix. The decomposition of the model into both physical DOFs (master DOFs) and modal coordinates allows the flexibility of connecting the finite elements to other substructures, while achieving a reasonably good result within a required frequency range. The Craig-Bumpton method is a very convenient method for modelling a geared transmission system as the input (excitation) to the system is not defined as forces, but as geometric mismatches at the connection points (i.e. gear transmission error and bearing geometric error). The following summary of the Craig-Bampton method is given

In the Craig-Bampton reduction method, the equation of motion (dynamic equilibrium) of each superelement (substructure), without considering the effect of damping, can be

Where [ ] *M* is the mass matrix, [ ] *k* is the stiffness matrix, *F* is the nodal forces, *u* and *u* are the nodal displacements and accelerations respectively. The key to reducing the substructure is to split the degrees of freedom into masters *um* (at the connecting nodes) and slaves *us* (at the internal nodes). The mass, the stiffness and the force matrices are re-

> 

The subscript *m* denotes master, *s* denotes slave. Furthermore, the slave degrees of freedom (internals) can be written by using generalized coordinates (modal coordinates by ( *q* ) using the fixed interface method, i.e. using the mode shapes of the superelement by fixing the master degrees of freedom nodes (connecting/ boundary nodes). The transformation matrix

> *m m*

For the fixed interface method, the transformation matrix (*T* ) can be expressed as shown in

 <sup>0</sup> *sm s* 

*I*

*u u T*

*s*

*T*

*<sup>s</sup>* is the modal matrix of the internal DOF with the interfaces fixed.

*M M u kk u F*

*sm ss s sm ss s*

*mm ms m mm ms m m*

 

*M k*

[ ] [] *M u ku F* (2.5.6)

0

*u q* (2.5.8)

*<sup>G</sup>* (2.5.9)

<sup>1</sup> *G kk sm ss sm* (2.5.10)

*MM u kk u* (2.5.7)

based on the references [39-41].

expressed as in Equation-2.5.6:

arranged accordingly as follows:

(*T* ) is the one that achieves the following:

Equation-2.5.9:

where,

and 

$$k\_{reduced} = T^{\ddagger} k T \tag{2.5.12}$$

Thus Equation-2.5.7 can be re-written in the new reduced form using the reduced mass and stiffness matrices as well as the modal coordinates as follows:

$$
\begin{aligned}
\overbrace{\begin{bmatrix}
\boldsymbol{M}\_{bb} & \boldsymbol{M}\_{hq} \\
\boldsymbol{M}\_{qb} & \boldsymbol{M}\_{qq}
\end{bmatrix}}
\begin{Bmatrix}
\ddot{\boldsymbol{u}}\_{m} \\
\ddot{\boldsymbol{q}}
\end{Bmatrix}+
\begin{Bmatrix}
\boldsymbol{k}\_{radw} \\
\boldsymbol{0}
\end{Bmatrix}
\begin{Bmatrix}
\boldsymbol{u}\_{m} \\
\boldsymbol{q}
\end{Bmatrix}
\end{aligned}
\begin{aligned}
\begin{aligned}
\boldsymbol{\tilde{\boldsymbol{u}}}\_{radw} \\
\boldsymbol{\tilde{\boldsymbol{\eta}}}
\end{aligned}
\end{aligned}
\begin{aligned}
\boldsymbol{\tilde{\boldsymbol{\eta}}}\_{radw} \\
\boldsymbol{\tilde{\boldsymbol{\eta}}}
\end{aligned}
\begin{aligned}
\boldsymbol{\tilde{\boldsymbol{\eta}}}\_{radw} \\
\boldsymbol{\tilde{\boldsymbol{\eta}}}
\end{aligned}
\tag{2.5.13}
\end{aligned}
\tag{2.5.13}
\end{aligned}$$

Where *Mbb* is the boundary mass matrix i.e. total mass properties translated to the boundary points. *bb k* is the interface stiffness matrix, i.e. stiffness associated with displacing one boundary DOF while the others are held fixed. The *Mbq* is the component matrix ( *Mqb* is the transpose of *Mbq* ).

If the mode shapes have been mass normalized (typically they are) then:

$$\mathcal{A}\_{qq} = \begin{bmatrix} \ & & 0 \\ & \mathcal{A}\_i & \\ 0 & & \ \end{bmatrix} \tag{2.5.14}$$

where *i* is the eigenvalues; <sup>2</sup> *ii i i k m*/ , and,

$$\mathcal{M}\_{q\eta} = \begin{bmatrix} \ & & \ & \\ & I & & \\ 0 & & \ & \end{bmatrix} \tag{2.5.15}$$

Finally the dynamic equation of motion (including damping) using the Craig-Bampton transform can be written as:

$$
\begin{bmatrix} M\_{bb} & M\_{bq} \\ M\_{qb} & I \end{bmatrix} \begin{bmatrix} \ddot{u}\_m \\ \ddot{q} \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 2\zeta a \end{bmatrix} \begin{bmatrix} \dot{u}\_m \\ \dot{q} \end{bmatrix} + \begin{bmatrix} k\_{bb} & 0 \\ 0 & a^2 \end{bmatrix} \begin{bmatrix} u\_m \\ q \end{bmatrix} = \begin{bmatrix} F\_m \\ 0 \end{bmatrix} \tag{2.5.16}
$$

where 2 = modal damping ( = fraction of critical damping)

For more detailed explanation of the techniques related to the modelling of rolling element bearings and the application of component mode synthesis (CMS) techniques, refer to the works by Sawalhi, Deshpande and Randall [41].

Gearbox Simulation Models with Gears and Bearings Faults 33

Discontinuities

Stiffness

Fig. 2.6.2. Non-linearity due to contact loss in meshing gears; a) Force vs. Displacement, b)

order terms above the 1st derivative are ignored for linearization, which leaves the following expression (Equation-2.6.3). The differential term '*J'* is called the Jacobian matrix (or Jacobian in short). The problems involving gear contact losses are "Stiff" problems because of the discontinuity in system derivatives (Jacobians). For a more detailed discussion on this

The study of gear faults has long been an important topic of research for the development of gear diagnostic techniques based on vibration signal analysis. Understanding how different types of gear tooth faults affect the dynamics of gears is useful to characterise and predict the symptoms of the damage appearing in vibration signals [44, 45]. The strong link between the TE and the vibration of the gears was explained earlier. The effect of different types of gear tooth faults on TE can be studied by using the static simulation models. The result of static simulation can be then used to determine how different types of gear faults can be

Gears can fail for a broad range of reasons. Finding a root cause of damage is an important part of developing a preventative measure to stop the fault from recurring. Analysis of gear failure involves a lot of detective works to link the failed gear and the cause of the damage. Comprehensive guidelines for gear failure analysis can be found in Alban [46], DeLange [47] and DANA [48]. AGMA (American Gear Manufacturers Association) recognizes four types of gear failure mode and a fifth category which includes everything else: Wear, Surface

0 0 , , *<sup>x</sup>*

0

*d f f xt f x t x x dx* (2.6.2)

*f xt f x t J x x* , , 0 0 (2.6.3)

, *dx f xt dt* (2.6.1)

Displacement

derivative of former, i.e. Stiffness vs. Displacement

Backlash tolerance

(a) (b)

Displacement

Force

topic refer to the work presented by Singh [42].

**3. Modelling gearbox faults** 

modelled into the dynamic simulation.

Fatigue, Plastic Flow, Breakage and associated gear failures [49].
