# Nonlinear modes for a discrete mechanical system with rigid contact

Sokly Heng, Stéphane Junca, Mathias Legrand

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Sokly Heng, Stéphane Junca, Mathias Legrand. Nonlinear modes for a discrete mechanical system with rigid contact. 11th World Congress on Computational Mechanics (WCCM XI), Jul 2014, Barcelona, Spain. hal-03186031

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Submitted on 30 Mar 2021

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Distributed under a Creative Commons Attribution 4.0 International License
NONLINEAR MODES FOR A DISCRETE MECHANICAL
SYSTEM WITH RIGID CONTACT

Sokly HENG¹, Stéphane JUNCA², and Mathias LEGRAND³

¹ Université de Nice Sophia-Antipolis, Laboratoire JAD & Team COFFEE INRIA, Parc Valrose, 06108 Nice, France, i03e05@yahoo.com

² Université de Nice Sophia-Antipolis, Laboratoire JAD & Team COFFEE INRIA, Parc Valrose, 06108 Nice, France, **Stephane.junca@unice.fr**

³ McGill University, Department of Mechanical Engineering, 817 Sherbrooke Street West
Montreal, Quebec H3A 0C3, Canada, mathias.legrand@mcgill.ca

**Key words:** nonlinear vibrations, perfectly elastic rigid unilateral contact, non-smooth
modal analysis.

Nonlinear normal modes (NNM) [2, 3] are useful tools to study nonlinear vibrations for
mechanical systems. We are interested in finding NNM in the context of perfectly elastic
unilateral contact. To this end is investigated a discrete model representing a flexible thin
rod in contact against a frictionless rigid foundation [4]. This one dimensional model is
essentially given by coupled harmonic oscillators with an unilateral rigid contact against a
wall. We assume that the contact preserves energy. A natural mathematical formulation
is derived and important features for this *N* dof system are explored: grazing contact,
periodic solutions.

Let us give a flavor of our results for the simple 2 dof case: ($u_1, v_1, u_2, v_2$) belonging
to the phase space $\mathbb{R}^4$ and $v_i = \dot{u}_i$. Families of periodic trajectories are computed and
the associated non-smooth invariant manifolds are investigated. The problem is clearly

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Discrete model with *N* = 2 dof</figcaption></figure>

nonlinear but we use a linear procedure introduced in [5] and generalized for *N* dof,
*N* > 2. The vibratory energy is continuously increased in order to reach grazing as well as
transverse unilateral contact conditions. The corresponding autonomous periodic orbits
<figure><img src="image_3.png" /><figcaption>Figure 2: Invariant manifolds supporting autonomous periodic trajectories in variables (left) $(u_1, v_1, u_2)$ and (right) $(u_1, v_1, v_2)$</figcaption></figure>

are organized on non-smooth invariant manifolds of dimension 2 in the state space of the system of interest.

This applied mathematical subject has many applications and opportunities in Mechanical Engineering, in particular, in aeronautic where unilateral contact occurrences are becoming common due to the use of more flexible and lighter structures implying larger displacements.

## REFERENCES

[1] J. Bastien, F. Bernardin, and C.-H. Lamarque, *Non smooth deterministic or stochastic discrete dynamical systems. Applications to models with friction or impact*. Wiley, 2013.

[2] D. Jiang, C. Pierre, and S. Shaw. Large-amplitude non-linear normal mode of piecewise linear systems. *Journal of Sound and Vibration*, Vol. **272**, 869–891, 2004.

[3] G. Kerschen, M. Peeters, J.C. Golinval, and A.F. Vakakis. Nonlinear normal modes, Part I: A useful framework for the structural dynamics. *Mechanical Systems and Signal Processing*, Vol. **23**, 170–194, 2009.

[4] D. Laxalde and M. Legrand. Nonlinear modal analysis of mechanical systems with frictionless contact interfaces. *Mechanical Systems and Signal Processing* Vol. **47**(4), 469–478, 2011.

[5] M. Pascal. Dynamics and stability of a two degree of freedom oscillator with an elastic stop. *J. Comput. Nonlinear Dynam.* (1), 94-102, 2006.