Coexisting solutions and bifurcations in mechanical oscillators with backlash

Lightly damped geared systems have been shown to exhibit unwanted noise and vibration problems. We present a nonlinear analysis of this behaviour, based on freeplay. We derive a simple model of a pair of meshing spur gears as a single degree of freedom oscillator with backlash. We consider the behaviour of such a system with low damping, and with both large finite and infinite stiffness values. We show that the solution where the gears remain permanently in contact can coexist with many other stable rattling solutions which we compute analytically. We calculate the regions of existence and stability of the families of rattling solutions on two-parameter bifurcation diagrams, and show that to leading order the large finite and infinite stiffness models give the same results. We provide numerical simulation to support our analysis, and we also draw practical conclusions for machine design