**2.6 Solving the gear dynamic simulation models**

A block diagram summarizing the time integral solution of a typical dynamic model with some time varying parameters is shown in Figure-2.6.1. There are a range of numerical algorithms available today to give solution of the dynamic models: direct time integration, harmonic balancing and shooting techniques, to name some commonly recognized methods.

(NOTE: '∫' stands for integration over a single step of incremented time)

Fig. 2.6.1. Dynamic Simulation Process

Sometimes the solution for gears requires simulation of highly non-linear events, for example, rattling and knocking in gears, which involve modelling of the contact loss. The works presented by R. Singh [42], Kahraman & Singh [43], Kahraman & Blankenship [18] and Parker & Lin [9, 10] show some examples of the "stiff" problems involving non-linearity due to contact loss and clearances.

The solution for these Vibro-Impact problems presents difficulties involving ill-conditioning and numerical "stiffness". In [42] Singh explains that ill-conditioning of a numerical solution occurs when there is a component with a large frequency ratio: ratio of gear mesh frequency to the natural frequency of the component.

The numerical stiffness in the gear dynamic simulation becomes a problem when gears lose contact. The relationship between the elastic force, relative deflection and gear mesh stiffness is illustrated in Figure-2.6.2. The gradient of the curve represent the gear mesh stiffness.

Contact loss between the gears occurs when the force between the gears becomes zero. The gears are then unconstrained and free to move within the backlash tolerance. The presence of a discontinuity becomes obvious when the derivative of the curve in Figure-2.6.2 (i.e. mesh stiffness) is plotted against the relative deflection. The discontinuity in the stiffness introduces instability in the numerical prediction.

In more formalized terms the "stiffness" of a problem is defined by local Eigen-values of the Jacobian matrix. Consider an equation of motion expressed in simple first order vector form '*f(x, t)*', (Equation-2.6.1). Typically, the solution of an equation of motion is obtained by linearizing it about an operating point, say '*x0*', (Equation-2.6.2). Usually, most of the higher

A block diagram summarizing the time integral solution of a typical dynamic model with some time varying parameters is shown in Figure-2.6.1. There are a range of numerical algorithms available today to give solution of the dynamic models: direct time integration, harmonic balancing and shooting techniques, to name some commonly recognized methods.

**- -** 

Sometimes the solution for gears requires simulation of highly non-linear events, for example, rattling and knocking in gears, which involve modelling of the contact loss. The works presented by R. Singh [42], Kahraman & Singh [43], Kahraman & Blankenship [18] and Parker & Lin [9, 10] show some examples of the "stiff" problems involving non-linearity

The solution for these Vibro-Impact problems presents difficulties involving ill-conditioning and numerical "stiffness". In [42] Singh explains that ill-conditioning of a numerical solution occurs when there is a component with a large frequency ratio: ratio of gear mesh frequency

The numerical stiffness in the gear dynamic simulation becomes a problem when gears lose contact. The relationship between the elastic force, relative deflection and gear mesh stiffness is illustrated in Figure-2.6.2. The gradient of the curve represent the gear mesh

Contact loss between the gears occurs when the force between the gears becomes zero. The gears are then unconstrained and free to move within the backlash tolerance. The presence of a discontinuity becomes obvious when the derivative of the curve in Figure-2.6.2 (i.e. mesh stiffness) is plotted against the relative deflection. The discontinuity in the stiffness

In more formalized terms the "stiffness" of a problem is defined by local Eigen-values of the Jacobian matrix. Consider an equation of motion expressed in simple first order vector form '*f(x, t)*', (Equation-2.6.1). Typically, the solution of an equation of motion is obtained by linearizing it about an operating point, say '*x0*', (Equation-2.6.2). Usually, most of the higher

Σ ∫ ∫ *<sup>x</sup> <sup>x</sup> <sup>x</sup> <sup>M</sup>*

*M C*

Time varying damping

Time varying stiffness

*M K*

**+**

**+**

1

*M* 1

(NOTE: '∫' stands for integration over a single step of incremented time)

**2.6 Solving the gear dynamic simulation models** 

Generate Input Parameters

*<sup>t</sup> e* , *<sup>t</sup> e*

*Fs*

*Kmb, Cmb*

Fig. 2.6.1. Dynamic Simulation Process

due to contact loss and clearances.

stiffness.

to the natural frequency of the component.

introduces instability in the numerical prediction.

Fig. 2.6.2. Non-linearity due to contact loss in meshing gears; a) Force vs. Displacement, b) derivative of former, i.e. Stiffness vs. Displacement

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

$$
\underline{f}(\mathbf{x},t) = \begin{cases} \underline{\mathbf{x}} \underline{\mathbf{v}} \\ \underline{\mathbf{v}} \end{cases} \tag{2.6.1}
$$

$$\underline{f}(\mathbf{x},t) \equiv \underline{f}(\mathbf{x}\_0, t) + \left(\bigwedge\_{\mathbf{x}\_0}^{df} \underline{\mathbf{x}}\right)\_{\mathbf{x}\_0} (\mathbf{x} - \mathbf{x}\_0) \tag{2.6.2}$$

$$
\underline{f}(\mathbf{x},t) \equiv \underline{f}\left(\mathbf{x}\_0, t\right) + \underline{f}\left(\mathbf{x} - \mathbf{x}\_0\right) \tag{2.6.3}
$$

### **3. Modelling gearbox faults**

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 modelled into the dynamic simulation.

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

Gearbox Simulation Models with Gears and Bearings Faults 35

Fig. 3.1.2. Effect of backup ratio (mb) and initial crack location on propagation path.

A set of spur gears later used in the validation of the simulation result have a "backup ratio" (rim thickness divided by tooth height) greater than mb=1.3. Therefore based on the Lewicki's prediction the cracks occurring in the tooth fillet region are most likely to propagate in the trajectory shown in Pattern Figure-3.1.2 (i); roughly 30~45° in to the tooth relative to the radial line path through the symmetric axis of the spur gear tooth profile.

Full 3D modelling of a propagating gear tooth crack is one of the actively researched areas. Some examples of simulation studies using the Boundary Element Method (BEM) are given in [52, 53, 54, 55]. The simulation studies using 3D models show complex behaviour crack growth from the small crack seeded at the middle of the gear tooth fillet. An example is shown in Figure-3.1.3 from "Modelling of 3D cracks in split spur gear", by Lewicki [52]. The crack front expands rapidly across the width of the gear tooth as it progresses into the thickness of the tooth. The tooth fillet crack (TFC) model used in this work assumes 2D

(Lewicki [51])

The effect of gear tooth fillet cracks (TFC) and spalls on gear transmission error was studied in detail by using a static simulation models (FEA and LTCA (HyGears [50])). A pair of meshing gears were modelled and analysed in step incremented non-linear static environment. Note: the transmission error obtained from the static simulation models are referred to as Motion Errors (ME) here forth by following the HyGears convention.

It was explained earlier that the interaction between two meshing gears can be expressed in the dynamic model as time-varying stiffness, damping and gear tooth topological error elements linking the two lumped mass moments of inertia. The effect of gear tooth faults can be implemented into the dynamic simulation model as changes to these parameters. The understanding gained from the detailed simulation model studies of TFCs and spalls on gear motion has lead to the method of modelling the effect of the faults in dynamic model. The relevance between the types of gear faults to the selected parameters will be explained through subsequent sections.

Further to the simulation of gear tooth faults, this chapter also briefly touches on the modelling of spalls in rolling element bearings (REB), which is also a common type of faults in geared transmission systems.
