Model order reduction of piecewise linear mechanical systems using invariant cones

TL;DR

This work develops a geometric framework to reduce piecewise linear mechanical systems with nonsmooth equilibria by exploiting invariant cones under positive homogeneity. It introduces two parametrization schemes: a graph-style method for conic geometry and a robust arc-length approach that handles folding and discontinuities, enabling explicit two-dimensional reduced descriptions of nonlinear normal modes. The methodology is validated on a 3-DOF chain with unilateral contact, including internally resonant and folding cones, for both conservative and dissipative cases, demonstrating high fidelity of the resulting reduced-order models. The approach provides a principled basis for nonlinear modal analysis of nonsmooth PWL systems and lays groundwork for extending to forced responses while maintaining computational efficiency.

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.

Figures (14)

• Figure 1: Characteristic of the unilateral elastic contact law ($c_n=0$).
• Figure 2: A pictorial view of an invariant cone foliated by periodic orbits with $\mu=1$.
• Figure 3: A $3-$D representative sketch of an invariant cone and its generating set obtained as its intersection with the unit-cylinder.
• Figure 4: Schematic of the 3-DOF chain of oscillators with a discontinuous unilateral support. This system generalizes the benchmark from Attar et al. Attar2017 by including internal linear damping (via coefficient $d$) and unilateral contact damping (via coefficient $c_n$).
• Figure 5: Frequency-stiffness plot for the first NNM of the undamped $(c=0)$ 3-DOF chain of oscillators with no initial gap $(\delta=0)$. The vertical axis shows the dimensionless frequency, while the horizontal axis shows the dimensionless contact stiffness. Reference data from Attar et al. Attar2017.

Theorems & Definitions (8)

• Remark 1: Underlying Conservative Dynamics
• Proposition 1: Absence of Sliding Modes
• proof
• Definition 1: Poincaré Half-Maps
• Definition 2: Elementary Invariant Cone
• Definition 3: Generating Cycle of Rays
• Proposition 2
• Definition 4: $k$-crossing Invariant Cone