Coexisting solutions and bifurcations in mechanical oscillators with backlash

Lightly damped backlash systems have been shown to exhibit unwanted noise and vibration problems and we present a nonlinear analysis of this behaviour. As a representative example, we derive a simple model for 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 permanent contact solution 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.