Modelling elastic structures with strong nonlinearities with application to stick-slip friction

TL;DR

This paper introduces an exact transformation method that simplifies complex elastic structures with strong nonlinearities, like friction, into low-dimensional models with memory, aiding analysis of discontinuous contact phenomena.

Contribution

The paper presents a novel, general transformation technique that reduces continuum structures with point nonlinearities to manageable low-dimensional equations with memory.

Findings

Contact forces are Lipschitz continuous at sticking onset for certain structures.

The method effectively models Coulomb friction in elastic bodies.

The approach simplifies analysis of impact and friction at contact points.

Abstract

An exact transformation method is introduced that reduces the governing equations of a continuum structure coupled to strong nonlinearities to a low dimensional equation with memory. The method is general and well suited to problems with point discontinuities such as friction and impact at point contact. It is assumed that the structure is composed of two parts: a continuum but linear structure and finitely many discrete but strong nonlinearites acting at various contact points of the elastic structure. The localised nonlinearities include discontinuities, e.g., the Coulomb friction law. Despite the discontinuities in the model, we demonstrate that contact forces are Lipschitz continuous in time at the onset of sticking for certain classes of structures. The general formalism is illustrated for a continuum elastic body coupled to a Coulomb-like friction model.