Basins of attraction in nonsmooth models of gear rattle

JF Mason, PT Piiroinen, RE Wilson, ME Homer

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Abstract

This paper is concerned with the analysis of attractors found in the modelling of gear rattle in vacuum pumps. We use nonsmooth ordinary differential equation models, proposed by Halse et al, for the dynamics of the pump, where the nonlinearity arises from the backlash between the gear teeth. We find that rattling and quiet solutions can coexist, which may explain why geared systems can rattle intermittently. To develop an understanding of the pumps' dependence on parameters, we compute basins of attraction using cell-to-cell mapping techniques. These reveal rich and delicate dynamics, and we present an explanation of some of the transitions in the system's behaviour in terms of smooth and discontinuity-induced bifurcations. We discuss the important role that the discontinuity in our system plays in the intricate stretching and folding of the phase space. In addition, we use DsTool (Dynamical Systems Toolkit) to calculate stable manifolds, which form the basin boundaries.
Original languageEnglish
Pages24 p
Publication statusUnpublished - Jan 2008

Bibliographical note

Sponsorship: JM gratefully acknowledges the support of a CASE award from BOC Edwards Ltd. and the Engineering and Physical Sciences Research
Council. PP gratefully acknowledges the support from the European Union (FP5 Project
SICONOS, grant no. IST-201-37172)

Keywords

  • gear rattle
  • basins of attraction
  • nonsmooth

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