Extended invariant cones as Nonlinear Normal Modes of inhomogeneous piecewise linear systems

TL;DR

This paper extends the concept of invariant cones to inhomogeneous piecewise linear systems to compute nonlinear normal modes, providing a new approach for analyzing nonsmooth mechanical systems with external forcing.

Contribution

It introduces a novel extension of invariant cone theory to inhomogeneous systems and demonstrates its application to nonlinear normal modes in mechanical oscillators.

Findings

Invariant cones of homogeneous systems serve as singularities in the theory.

The methodology effectively computes nonlinear normal modes with external forcing.

Results align with traditional methods like shooting and time integration.

Abstract

The aim of this paper is to explore the relationship between invariant cones and nonlinear normal modes in piecewise linear mechanical systems. As a key result, we extend the invariant cone concept, originally established for homogeneous piecewise linear systems, to a class of inhomogeneous continuous piecewise linear systems. The inhomogeneous terms can be constant and/or time-dependent, modeling nonsmooth mechanical systems with a clearance gap and external harmonic forcing, respectively. Using an augmented state vector, a modified invariant cone problem is formulated and solved to compute the nonlinear normal modes, understood as periodic solutions of the underlying conservative dynamics. An important contribution is that invariant cones of the underlying homogeneous system can be regarded as a singularity in the theory of nonlinear normal modes of continuous piecewise linear systems. In addition, we use a similar methodology to take external harmonic forcing into account. We illustrate our approach using numerical examples of mechanical oscillators with a unilateral elastic contact. The resulting backbone curves and frequency response diagrams are compared to the results obtained using the shooting method and brute force time integration.