Mathematical Models of Gear Rattle in Roots Blower Vacuum Pumps


Article Preview

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.



Edited by:

Patrick Sean Keogh




J. F. Mason et al., "Mathematical Models of Gear Rattle in Roots Blower Vacuum Pumps ", Applied Mechanics and Materials, Vols. 5-6, pp. 21-28, 2006

Online since:

October 2006




[1] M.H. Hablanian. Design and performance of oil-free pumps. Vacuum, 41: 1814-1818, (1990).


[2] C.K. Halse. Nonlinear Dynamics of the Automotive Driveline. PhD thesis, University of Bristol, (2004).

[3] C.K. Halse, R.E. Wilson, M.E. Homer, and M. di Bernardo. Coexisting solutions and bifurcations in mechanical oscillators with backlash, preprint, (2006).

[4] W.T. Holmes. Plane Geometry of Rotors in Pumps and Gears. The Scientific Publishing Company, (1978).

[5] J.F. Mason. Mathematical modelling of gear rattle in dual-shaft vacuum pumps. Master's thesis, University of Bristol, (2004).

[6] H.E. Merritt. Gear Engineering. Pitman Publishing, (1971).

[7] A.P. Troup and N.T.M. Dennis. Six years of dry pumping,: A review of experiences and issues. J. Vac. Sci. Technol. A, 9: 2048-2052, (1990).

[8] H. Wycliffe. Mechanical high-vacuum pumps with oil-free swept volume. J. Vac. Sci. Technol. A, 5: 2608-2611, 1987. JM gratefully acknowledges a CASE award from EPSRC and BOC Edwards Ltd.