Nonlinear Time-Varying Dynamic Analysis of a Multi-Mesh Spur Gear Train

The nonlinear dynamics of a multi-mesh spur gear train is considered in this study. The gear train consists of three spur gears, with one of the gears in mesh with the other two. Dynamic model includes gear backlash in the form of clearance-type displacement functions and time variation of gear mesh stiffness. The system is reduced to a two-degree-of-freedom definite model by using the relative gear mesh displacements as the coordinates. The equations of motion are solved for periodic steady-state response by using Harmonic Balance Method (HBM). The accuracy of the HBM solutions is demonstrated by comparing them to direct numerical integration solutions. Floquet theory is applied to determine the stability of the steady-state solutions. Two different loading conditions, where the system is driven by the middle gear and driven by one of the end gears, are considered. Phase difference between the two gear meshes is determined under each loading condition and natural modes are predicted for each loading condition. The forced response due to the combination of parametric excitation and static transmission error excitation is obtained and effects of loading conditions and asymmetric positioning on the response are explored.