Assessment of Modelling Fidelity for Predicting Transmission Error in Dynamically Coupled Spur Gear Systems

In the present study, the transmission error (TE) behaviour of an identical involute spur gear pair dynamically coupled with an attached subsystem is investigated through numerical modelling. The surrounding mechanical system is represented by an equivalent spring–mass subsystem, and the TE response is examined over a range of operating conditions associated with different resonance regimes. Two finite element–based modelling approaches are considered to describe the local elastic deformations occurring at the gear mesh. In the fully coupled finite element formulation, the interaction between meshing gear teeth is described using nonlinear contact conditions implemented through an augmented Lagrangian approach, in which tooth contact behaviour, elastic deformation, and subsystem dynamics are treated simultaneously. In the reduced hybrid finite element formulation, elastic deformations arising from tooth bending and contact compliance are evaluated separately under quasi-static equilibrium and subsequently combined to estimate the overall mesh deformation and TE. Quasi-static analyses are conducted to assess the accuracy of both modelling approaches through comparison with published benchmark results at multiple load levels. Dynamic simulations based on the fully coupled formulation are then used to investigate TE behaviour in quasi-static, near-resonant, resonant, and post-resonant operating regimes of the attached subsystem. The effect of the subsystem resonance on the amplitude and harmonic content of TE is examined. The results indicate that subsystem dynamics have a pronounced influence on TE behaviour, with peak-to-peak values increasing by up to approximately 70% at the fundamental resonance of the attached subsystem and with associated changes in harmonic content that persist beyond the resonance region. The modelling approaches are compared in terms of predictive capability across resonance regimes, computational effort, and relevance to automotive gear transmission applications.