The Importance of Numerical Simulation
Performing numerical simulations to extract noise and vibrations generated from gear transmissions is a challenging task due to the complex nonlinear phenomena involved, such as, for example, multiple movable contact points, impacts and influence friction. Dynamic analysis is a crucial tool to achieve the optimal design of transmissions, and the scientific literature reports several investigations on the topic.
The tools proposed to improve the design process of gear transmissions are classified into three main categories: 1) analytical methodologies; 2) Finite Element Method (FEM) modified models; 3) multibody models. Experiments are also widely used to assess model results, but they are not suitable when searching for optimization, due to the high effort required and the complexity of the testing apparatus, especially when high-performance gears are tested with large power throughput. Moreover, the scientific literature lacks enough experimental data available for comparison. The dynamic behaviour of gears has been historically first approached with analytical models, that represent the real system with simplified lumped parameters, in which the gear system is reduced to simple masses connected by discrete elastic and dissipative elements. A reduced single degree of freedom (SDOF) model is composed from two rigid gears linked through a single lumped spring element defined Mesh Stiffness (MS). With this approach, the vibrations induced by the shaft, the bearings, and the gearbox are neglected.
History of the Transmission Error
The first studies about gear dynamics data through analytical models go back to the second decades of 1900 [1, 2]. Ozguven and Houser in  and then Wang et al.  reviewed several analytical methodologies aiming to predict the noise produced by a transmission. A fundamental breakthrough is made by Harris , introducing the definition of the Transmission Error (TE). Effects of gear backlash, friction, and corner contact are investigated in [6, 7]. Parey et al. in  introduced the effects due to time-varying mesh stiffness. Kahraman et al. in [9, 10] analysed the back-side tooth impact induced by backlash and time-varying MS. Backlash produces a significant increase in the noise level which can be generated either in driving conditions (i.e. the “rattle” noise) or in neutral conditions (“idle” noise). Plenty of researchers focused on the analysis of profile modification for noise reduction . Multi degrees of freedom (MDOF) models  take into analysis shafts and bearings, considering such components as spring-damper elements. A widespread approach adopted in gear simulation is the FEM, which due to the high number of needed elements in the contact zone, is not suitable for dynamic analysis, whilst finding a great deal of room in quasi-static analysis such as for MS identification. Parker in  developed an innovative FEM model (FECO) to investigate the dynamic response of spur gears. In FECO the contact between teeth is computed at each time step through an analytical solution, giving the possibility of adopting a coarse mesh for the gear discretization with no need of requiring the identification of MS a priori. This approach, however, requires far more computational resources than the analytical models.
Multibody Dynamics Methodology
Multibody dynamics (MUBO) simulations are numerical methods developed to simulate the dynamic behaviour of mechanisms with many members, called bodies. In its native form, MUBO considers bodies as rigid elements, connected through constraints such as kinematic joints, dynamic constraints, and contact relations. The study of contact is a key point in MUBO, and the most adopted approach is called “penalty method” [14, 15, 16]. When applied to gear dynamics, the main limitation of MUBO is represented by the lack of bodies’ flexibility. Recently, a novel contact-based multibody model able to include the flexibility of the teeth through the pseudo-rigid body method has been developed . These studies demonstrate that concentrating tooth elasticity into an elastic element is a reliable approach, which allows evaluation of the MS, study of nonlinear dynamics phenomena, and moreover, profile optimization . Another method to consider flexibility in MUBO is based on the modal representation of elastic body displacements [19, 20]. The approach can be readily applied to flexible bodies subjected to small displacements, but by adopting a sub-structuring approach, can also be applied to large displacement . Compared to classical FEM, because of the small number of elastic free coordinates needed to represent the elastic behaviour of flexible bodies, it requires lower computational resources. With this approach, the constraint and load entities can be applied to bodies and individual nodes.