Attractivity of Equilibrium Sets of Systems with Dry Friction

Abstract

Abstract. The dynamics of mechanical systems with dry friction elements, modelled by set-valued force laws, can be described by differential inclusions. An equilibrium set of such a differential inclusion corresponds to a stationary mode for which the friction elements are sticking. The attractivity properties of the equilibrium set are of major importance for the overall dynamic behaviour of this type of systems. Conditions for the attractivity of the equilibrium set of MDOF mechanical systems with multiple friction elements are presented. These results are obtained by application of a generalisation of LaSalle's principle for differential inclusions of Filippov-type. Besides passive systems, also systems with negative viscous damping are considered. For such systems, only local attractivity of the equilibrium set can be assured under certain conditions. Moreover, an estimate for the region of attraction is given for these cases. The effectiveness of the results is illustrated by means of both 1DOF and MDOF examples.

Bifurcations of equilibrium sets in mechanical systems with dry friction

“…The assumption F s > 0 does reduce the number of topological distinct systems of (12). However, the case F s < 0 yields topologically equivalent systems when time is reversed.…”

“…In general, the sliding motion of the system (2) yields a nonzero sliding vector field, whereas the sliding motion of (1) contains a set of equilibria. Therefore, in this section bifurcations of (1) are studied, restricting ourselves to planar systems as given in (12). Such bifurcations occur in systems (2) as well, but will have a higher codimension.…”

“…Using Assumption 2, in Section 3 several types of topologically distinct planar systems (12) are identified, which are structurally stable in a neighbourhood of the equilibrium set. Hence, it seems a reasonable step to consider parameter-dependent systems, and study the parameters where the given conditions on the differential inclusion (12) no longer hold.…”

“…Although quite some results exist on the asymptotic stability and attractivity of equilibrium sets of mechanical systems with dry friction, see [9][10][11][12]6,13], few results exist that describe bifurcations of equilibrium sets, see e.g. [14,15].…”

“…[14,15]. In [12,6,11], Leine and van de Wouw derive sufficient conditions for attractivity and asymptotic stability of equilibrium sets using Lyapunov theory and invariance results. In the papers of Adly et al [10,9], conditions are presented under which trajectories converge to the equilibrium set in finite time.…”