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

In this study, a non-linear time-varying dynamic model is used to investigate sub-harmonic and chaotic motions exhibited by a typical multi-mesh gear train. The purely torsional system is formed by three rigid shafts connected to each other by two spur gear pairs. The lumped parameter dynamic model includes both gear backlash clearances and parametric gear mesh stiffness fluctuations. Steady state period-one motions of the same system were studied in another by using a multi-term harmonic balance method in conjunction with discrete Fourier transforms. This study expands the same solution technique for an investigation of sub-harmonic resonances of the forced response. The accuracy of the predictions is demonstrated by comparing them to the direct numerical integration results. Effect of several system parameters such as alternating mesh stiffness amplitudes, gear mesh damping and static torque transmitted on sub-harmonic motions are described. It is shown that stable sub-harmonic motions mostly in the form of softening type resonances dictate the frequency ranges in which the period-one motions are unstable due to parametric excitations. Other non-linear phenomena including long sub-harmonic motions and period-doubling bifurcations leading to chaotic behavior are also predicted.