Model order reduction of piecewise linear mechanical systems using invariant cones
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| Format: | Preprint |
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2026
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| _version_ | 1866918291060031488 |
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| author | Karoui, A. Yassine Leine, Remco I. |
| author_facet | Karoui, A. Yassine Leine, Remco I. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_10241 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Model order reduction of piecewise linear mechanical systems using invariant cones Karoui, A. Yassine Leine, Remco I. Dynamical Systems 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. |
| title | Model order reduction of piecewise linear mechanical systems using invariant cones |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2601.10241 |