Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method

A non-linear time-varying dynamic model of a typical multi-mesh gear train is proposed in this study. The physical system includes three rigid shafts coupled by two gear pairs. The lumped parameter dynamic model includes the gear backlash in the form of clearance-type displacement functions and parametric variation of gear mesh stiffness values dictated by the gear contact ratios. The system is reduced to a two degree of freedom definite model by using the relative gear mesh displacements as the coordinates. Dimensionless equations of motion are solved for the steady state periodic response by using a multi-term Harmonic Balance Method (HBM) in conjunction with Discrete Fourier Transforms and a Parametric Continuation scheme. Super-harmonic, harmonic and sub-harmonic motions were predicted. 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 harmonic balance solutions. An example gear train is used to investigate the influence of key system parameters including alternating mesh stiffness amplitudes, gear mesh damping, static torque transmitted, and the gear mesh frequency ratio. Influence of the amount of backlash is also investigated as well as the modal coupling issues. The equations of motions and their solutions were expanded to a general multi-mesh parallel axis gear system with flexible connecting shafts.