Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

A. Yassine Karoui, Remco I. Leine

TL;DR

The paper develops an augmented invariant-cone framework to link invariant cones with nonlinear normal modes (NNMs) in inhomogeneous continuous piecewise-linear 2CPL$_n$ systems. By augmenting the state with the gap variable and, for forced cases, the forcing phase, the authors recast inhomogeneous dynamics as autonomous homogeneous systems and formulate modified invariant-cone problems that yield NNM backbone curves and forced-responses. The approach recovers NNMs as invariant cones foliated by periodic orbits, with homogeneous cones representing a singular limit and high-energy behavior approaching bilinear limits; stability is assessed via Floquet monodromy. Forced responses are computed by extending the state with forcing variables and solving a constrained invariant-cone problem, enabling accurate frequency-response curves that agree with time integration and shooting methods. Overall, the work provides a coherent cone-based means to compute NNMs and FRCs in nonsmooth, inhomogeneous PWL mechanical systems, with demonstrated numerical validation against established techniques.

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.

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

TL;DR

The paper develops an augmented invariant-cone framework to link invariant cones with nonlinear normal modes (NNMs) in inhomogeneous continuous piecewise-linear 2CPL systems. By augmenting the state with the gap variable and, for forced cases, the forcing phase, the authors recast inhomogeneous dynamics as autonomous homogeneous systems and formulate modified invariant-cone problems that yield NNM backbone curves and forced-responses. The approach recovers NNMs as invariant cones foliated by periodic orbits, with homogeneous cones representing a singular limit and high-energy behavior approaching bilinear limits; stability is assessed via Floquet monodromy. Forced responses are computed by extending the state with forcing variables and solving a constrained invariant-cone problem, enabling accurate frequency-response curves that agree with time integration and shooting methods. Overall, the work provides a coherent cone-based means to compute NNMs and FRCs in nonsmooth, inhomogeneous PWL mechanical systems, with demonstrated numerical validation against established techniques.

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.
Paper Structure (13 sections, 1 theorem, 73 equations, 14 figures)

This paper contains 13 sections, 1 theorem, 73 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Continuous PWL systems
  3. Invariant cones of homogeneous 2CPL$_n$ systems
  4. NNM backbones as extended invariant cones of inhomogeneous 2CPL$_n$ systems
  5. Augmented invariant cone
  6. Stability
  7. Numerical example
  8. Computing forced response curves using invariant cones
  9. Numerical example - Forced system with gap $\delta =1$
  10. Numerical example - Forced system without gap $\delta =0$
  11. Conclusions
  12. Conflict of interest
  13. Shooting method and pseudo-arclength continuation

Key Result

Proposition 1

(see Carmona2002) All 2CPL$_n$ systems can be transformed to the Lure-like form where $\mathbf A \in \mathbb{R}^{n \times n}, \mathbf x,\mathbf b,\mathbf c \in \mathbb{R}^n$, $\mathbf e_1$ is the first basis vector of $\mathbb{R}^n$ and the matrices $\mathbf A^\pm$ being defined as $\mathbf A^- = \mathbf A$ and $\mathbf A^+ = \mathbf A + \mathbf c \mathbf e_1^{\mathop{\mathrm{T

Figures (14)

  • Figure 1: Characteristic of the unilateral elastic contact law.
  • Figure 2: A pictorial view of an invariant cone foliated by periodic orbits with $\mu=1$.
  • Figure 3: A 2CPL$_2$ mechanical system with a unilateral elastic contact.
  • Figure 4: Backbone curves showing the validity of the MIC approach by comparison to classical shooting.
  • Figure 5: Invariant cone with frequency $\omega=0.8105$. The red periodic orbit for $v_\delta=\delta=1$ corresponds to the first NNM.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Definition 1
  • Proposition 1
  • Remark 1
  • Remark 2