Mathematical models of gear rattle in Roots blower vacuum pumps

This paper is concerned with the modelling of gear rattle in Roots blower vacuum pumps. Analysis of experimental data reveals that the source of the noise and vibration problem is the backlash nonlinearity due to gear teeth losing and re-establishing contact. We develop non-smooth ordinary differential equation models for the dynamics of the pump. The models include a time-dependent forcing term which arises from the imperfect, eccentric mounting of the gears. We use a combination of explicit construction, asymptotic methods and numerical techniques to classify complicated dynamic behaviour in realistic parametric regimes. We first present a linear analysis of motions where the gears do not lose contact, and develop upper bounds on eccentricity for quiet operation. We then develop a nonlinear analysis of `backlash oscillations', where the gears lose and re-establish contact, corresponding to noisy pump operation. It is found that noisy solutions can coexist with silent ones, explaining why geared systems can rattle intermittently. We then consider several possible design solutions, and show their implications for pump design in terms of the existence and stability of silent and noisy solutions. Finally, we present conclusions and possibilities for future work