dorsal/arxiv
View SchemaModel order reduction of piecewise linear mechanical systems using invariant cones
| Authors | A. Yassine Karoui, Remco I. Leine |
|---|---|
| Categories | |
| ArXiv ID | 2601.10241vv1 |
| URL | https://arxiv.org/abs/2601.10241 |
| License | http://creativecommons.org/licenses/by/4.0/ |
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.
{
"annotation_id": "cdd89af0-76f6-4133-ab28-3c78d9d4ab33",
"date_created": "2026-02-17T05:53:24.220000Z",
"date_modified": "2026-02-17T05:53:24.220000Z",
"file_hash": "54c6070b6be30934dc89fbc68a15ea327db2ba0987b51ecf176c8a0f3569b8a6",
"private": false,
"record": {
"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.",
"arxiv_id": "2601.10241",
"authors": [
"A. Yassine Karoui",
"Remco I. Leine"
],
"categories": [
"math.DS"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Model order reduction of piecewise linear mechanical systems using invariant cones",
"url": "https://arxiv.org/abs/2601.10241",
"version": "v1"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "b1dad728-c477-46de-8786-3f7f320769dc",
"id": "arXiv Dataset",
"type": "Model",
"variant": "snapshot-2026-01-17",
"version": "0.1.0"
},
"user_id": 1000002
}