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 derive simple non-smooth models for the lightly damped, lightly loaded dynamics of the pump. The models include a time-dependent forcing term which arises from the eccentric mounting of the gears acting at the gross rotation rate. 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 permanent-contact motions, and derive upper bounds on eccentricity for silent operation. We then develop a nonlinear analysis of 'backlash oscillations', where the gears lose and re-establish contact, corresponding to noisy pump operation. We show that noisy solutions can coexist with silent ones, explaining why geared systems can rattle intermittently. Finally, we consider possible design solutions, and show implications for pump design in terms of existence and stability of solutions.