Model order reduction of piecewise linear mechanical systems using invariant cones

TL;DR

This paper develops a novel model order reduction method for piecewise linear mechanical systems with nonsmooth contact laws by leveraging invariant cone theory and positive homogeneity, enabling explicit reduced models.

Contribution

It extends invariant manifold theory to nonsmooth systems using invariant cones and introduces two techniques for their computation, including a graph-style and arc-length parametrization.

Findings

Effective reduction of mechanical oscillators with unilateral supports

Explicit reduced models derived in closed form

Applicable to systems with continuous and discontinuous unilateral forces

Abstract

We present a methodology that extends invariant manifold theory to a class of autonomous piecewise linear systems with nonsmoothness at the equilibrium, providing a framework for model order reduction in mechanical structures with compliant contact laws. The key idea is to make the absence of a local linearization around the equilibrium tractable by leveraging the positive homogeneity property. This property simplifies the invariance equations defining the geometry of the invariant cones, from a set of partial differential equations to a system of ordinary differential equations, enabling their effective solution. We introduce two techniques to compute these invariant cones. First, an intuitive graph-style parametrization is proposed that utilizes Fourier expansions and Chebyshev polynomials to derive explicit reduced-order models in closed form. Second, an arc-length parametrization is introduced to robustly compute invariant cones with complex folding geometries, which are intractable with a standard graph-style technique. The approach is demonstrated on mechanical oscillators with unilateral visco-elastic supports, showcasing its applicability for systems with both continuous (unilateral elastic) and discontinuous (unilateral visco-elastic) unilateral force laws.