Zeno chattering of rigid bodies with multiple point contacts

Abstract

Ideally rigid objects establish sustained contact with one another via complete chatter (a.k.a. Zeno behavior), i.e. an infinite sequence of collisions accumulating in finite time. Alternatively, such systems may also exhibit a finite sequence of collisions followed by separation (sometimes called incomplete chatter). Earlier works concerning the chattering of slender rods in two dimensions determined the exact range of model parameters, where complete chatter is possible. We revisit and slightly extend these results. Then the bulk of the paper examines the chattering of threedimensional objects with multiple points hitting an immobile plane almost simultaneously. In contrast to rods, the motion of these systems is complex, nonlinear, and sensitive to initial conditions and model parameters due to the possibility of various impact sequences. These difficulties explain why we model this phenomenon as a nondeterministic discrete dynamical system. We simplify the analysis by assuming linearized kinematics, frictionless interaction, by neglecting the effect of external forces, and by investigating objects with rotational symmetry. Application and extension of the theory of common invariant cones of multiple linear operators enable us to find sufficient conditions of the existence of initial conditions, which give rise to complete chatter. Additional analytical and numerical investigations predict that our sufficient conditions are indeed exact, moreover solving a simple eigenvalue problem appears to be enough to judge the possibility of complete chatter.

Imperfections, impacts, and the singularity of Euler's disk

“…Similarly to the case of c < 2 we can now conclude that if t is very close to t f , then the effect of continuous motion on p during the time interval (t, t f ) is negligible in comparison the effect of impacts. The bouncing motion of polygonal objects in the absence of gravity has been investigated by the paper [22]. Our results show that without the effect of gravity, a regular n-gon shaped object will always separate from the ground after a finite number of impacts provided that the radius of gyration is above a certain threshold ρ 0 (n, γ), which depends on n and γ.…”

“…The exact range of initial conditions corresponding to this scenario are determined in a previous paper by us (Sec. 3 of [22]).…”

“…We have seen that for low γ and ρ values, a rod either undergoes motion cumulating in a simultaneous impact, or complete chatter if gravity is absent. In a previous paper [22], we have demonstrated by using analytical and numerical tools that a flat polygonal disk with multiple contact points behaves in the same way in the absence of gravity. In that paper, we have found that the motion always leads to an infinite sequence of impacts at two vertices (terminating in finite time), followed by a simultaneous impact if it satisfies the following criterion: Definition 2.…”

On generalized Coulomb–Amontons’ law in the context of rigid body dynamics

“…Further, these planes will be referred to as the table and the cushion. 1 In the absence of contact between the ball and the cushion, we are in the setting of the previously discussed case. The impact of the ball on the cushion has also been repeatedly considered in the literature (see, e.g.…”

“…In article [1] a mechanical model of a liquid flowmeter based on the kinematics of a two-point contact of a ball placed in an axisymmetric vessel is investigated. The authors derive the equations of motion of the ball forced to move along the edge of the vessel by swirling fluid flow from Newton's second law in the coordinate system rotating together with the center of the ball around the axis of symmetry of the vessel.…”