Analysis of parametric stability for a spur gear pair system considering the effect of extended tooth contact

In spur gear dynamic analysis, rectangular waves are often used to approximate the mesh stiffness alternating between one and two pairs of teeth in contact. But in actual practice, extended tooth contact (ETC) Occurs due to gear tooth deflection under load. Considering the effect of ETC, the mesh stiffness in the pre-mature and post-mature contact regions is gradually rather than abruptly varying with time, which would influence the parametric stability of the geared system significantly. Therefore, research on parametric stability for a spur gear pair system considering the effect of ETC is carried out in this article. First, a torsional parametric vibration model for a spur gear pair system is established and the periodically time-varying mesh stiffness is approximated linearly by trapezoidal waveforms (with the effect of ETC) and rectangular waves (without the effect of ETC). Then, the Floquet theory for stability analysis (including two key elements: one is the derivation of state transition matrix (STM); the other is the stability criterion for parametric vibration system) is presented briefly. Based on these, the stabilities (stable and unstable regions) of a practically used high-speed and heavy-load Spur gear pair with and without taking into account ETC are determined utilizing Floquet theory, respectively, and the differences between the two cases in three ranges of operating speeds for the system (low speed range, middle speed range, and high speed range) are contrasted in detail. In addition, various values of operating torques and mesh damping of the gear pair are also simulated and discussed for their influences upon unstable regions.