# Zeno chattering of rigid bodies with multiple point contacts

## TL;DR

This paper analyzes the complex phenomenon of Zeno chattering in rigid bodies with multiple contact points, extending previous 2D results to 3D objects, and provides conditions for the occurrence of complete chatter using invariant cone theory.

## Contribution

It extends the analysis of chattering from 2D slender rods to 3D objects with multiple contact points, introducing a nondeterministic model and eigenvalue-based conditions for complete chatter.

## Key findings

- Derived sufficient conditions for complete chatter in 3D systems.
- Showed that eigenvalue problems can predict the possibility of chattering.
- Extended previous 2D results to more complex 3D scenarios.

## 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 three-dimensional 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. \keywords{contact dynamics \and chattering \and Zeno behavior \and common invariant cone