**2.4 Modelling gear dynamics**

It is a standardized design procedure to perform STE analysis to ensure smoothly meshing gears in the loaded condition. It was explained in this section how the strong correlation between the TE and the gear vibration makes the TE a useful parameter to predict the quietness of the gear drives. However, a more realistic picture of the gear's dynamic properties can not be captured without modelling the dynamics of the assembled gear drive system. Solution of engineering problems often requires mathematical modelling of a physical system. A well validated model facilitates a better understanding of the problem and provides useful information for engineers to make intelligent and well informed decisions.

A comprehensive summary of the history of gear dynamic model development is given by Ozguven and Houser [6]. They have reviewed 188 items of literature related to gear dynamic simulation existing up to 1988. In Table-2.4.1, different types of gear dynamics models were classified into five groups according to their objectives and

Gearbox Simulation Models with Gears and Bearings Faults 25

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

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

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

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:

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])

flank and this makes the size of the FE model prohibitively large.

Symbols for the torsional stiffnesses are not shown to avoid congestion.

(a) (b)

details see the CALYX user's manuals [21, 22].

economical means to conduct simulation studies.
