The torsional vibration model of vibration lapping process for spiral bevel gear under displacement excitation was established

and the method of local maps between pre-impact and post-impact states was introduced.The Poincaré map relation was deduced in consideration of the oblique-impact vibration between teeth under the influence of sliding friction.The calculation method for Lyapunov exponent of vibration lapping system of spiral bevel gear was obtained.The dynamic response for a given gear lapping system with the change of excitation parameter was analyzed by employing phase plane plots

Poincaré map

Lyapunov exponent and bifurcation plot compositely.The results show that

with the increase of excitation amplitude

the gear lapping system exhibits a period from doubling route to chaos in single-side tooth impact state

and the period window can be found in chaotic region.With the continued increase of excitation amplitude

a double-side tooth impact phenomenon of the lapping system appears.