Dry friction can seriously affect the performance of a wide range of systems. More specifically, the stiction phenomenon in friction can induce the presence of equilibrium sets. The stability properties of such equilibrium sets is of major interest when analysing the global dynamic behaviour of these systems.

In this joint research project, we derive conditions under which the equilibrium set of multi-degree-of-freedom non-linear mechanical systems with an arbitrary number of frictional unilateral constraints is attractive. The theorems for attractivity are proved by using the framework of measure differential inclusions together with a Lyapunov-type stability analysis and a generalisation of LaSalle’s invariance principle for non-smooth systems. The special structure of mechanical multi-body systems allows for a natural Lyapunov function and an elegant derivation of the proof. Moreover, an instability theorem for assessing the instability of equilibrium sets of non-linear mechanical systems with frictional bilateral constraints is formulated. These results are illustrated by means of examples with both unilateral and bilateral frictional constraints.

Rod with two frictional constraints and phase-plane
 

Publications:

Leine, R.I. & van de Wouw, N. Stability and Convergence of Mechanical Systems with Unilateral Constraints, Lecture Notes in Applied and Computational Mechanics Vol. 36, Berlin Heidelberg New-York, Springer-Verlag, 2008. ISBN: 978-3-540-76974-3

van de Wouw, N. and Leine. R.I., "Attractivity of equilibrium sets of systems with dry friction", Nonlinear Dynamics , Vol. 35,1,  pp. 19–39, 2004. PDF(270kb)

Leine. R.I. and van de Wouw, N., "Stability properties of equilibrium sets of nonlinear mechanical systems with dry friction and impact", Nonlinear Dynamics, Vol. 51, No. 4, pp. 551-583, 2008 PDF (1077kb)

Leine, R.I. and Van de Wouw, N., "Dry Friction Induced Attractivity of Equilibrium Sets in Mechanical Multibody Systems", Proceedings of the Design Engineering Technical Conferences & Computers and Information in Engineering Conference (DETC 2005), ASME Symposium on Nonlinear Dynamics In Engineering Systems, Long Beach, USA, 2005. PDF (93kb)

Van de Wouw, N., Leine, R.I. and Nijmeijer, H., "Controlling attractivity of friction-induced equilibrium sets", Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference (CDC2005), Sevilla, Spain, pp. 2610-2615, 2005. PDF (229kb)

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09/12/11 | Remco Leine | ZfM | ETH