Nonsmooth modal analysis of a N-degree-of-freedom system undergoing a purely elastic impact law

Mathias Legrand,Stéphane Junca,Sokly Heng

doi:10.1016/j.cnsns.2016.08.022

Abstract

The dynamics of a N-degree-of-freedom autonomous oscillator undergoing an energy-preserving impact law on one of its masses is investigated in the light of nonlinear modal analysis. The impacted rigid foundation provides a natural Poincaré section of the investigated system from which is formulated a smooth First Return Map well-defined away from the grazing trajectory. In order to focus on the impact-induced nonlinearity, the oscillator is assumed linear. Continuous one-parameter families of T-periodic orbits featuring one impact per period and lying on two-dimensional invariant manifolds in the state-space are shown to exist. The geometry of these piecewise-smooth manifolds is such that a linear “flat” portion (on which contact is not activated) is continuously attached to a purely nonlinear portion (on which contact is activated once per period) exhibiting a velocity discontinuity through a grazing orbit. These features explain the newly introduced terminology “Nonsmooth modal analysis”. The stability of the periodic orbits lying on the invariant manifolds is also explored by calculating the eigenvalues of the linearized First Return Map. Internal resonances and multiple impacts per period are not addressed in this work. However, the pre-stressed case is succinctly described and extensions to multiple oscillators as well as self-contact are discussed.

Highlights

The modal dynamics can be regarded as a single degree-of-freedom second-order nonlinear oscillator from which all position and velocity coordinates can be functionally parametrized
We prove later that near an admissible periodic solution with initial data in H C, the First Return Map is locally well-defined and smooth
The second result means there is no admissible solution near the harmonics and subharmonics of the linear differential system (2.6a), while the third result says there is no nonlinear modes of high frequency

Summary

Introduction

In the framework of linear vibration theory of conservative autonomous systems, natural modes of vibration, uniquely defined as a countable sequence of pairs of natural frequencies and corresponding normalized modeshapes, exhibit interrelated properties appealing to the engineer and mathematician: (1) they span the state-space through the principle of superposition, (2) they are invariant (that is linearly independent), orthogonal, and uncouple the equations of motion enabling the construction of reduced-order models, (3) they efficiently predict potential vibratory resonances of periodically forced systems [17]. U.M.R. 6621 du CNRS, Université de Nice Sophia Antipolis Inria, Sophia Antipolis Méditerranée Research Centre, Project COFFEE to the invariance property of linear modes, such manifolds are invariant under the flow: trajectories stemming from an initial condition in the manifold will remain in the manifold as time unfolds [26, 14] On such manifolds, the modal dynamics can be regarded as a single degree-of-freedom second-order nonlinear oscillator from which all position and velocity coordinates can be functionally parametrized. As it is known in linear and smooth nonlinear modal analysis, modes of vibration of conservative autonomous systems accurately approximate the vibratory resonances of their periodically forced and slightly damped counterparts. This feature of interest to the designer is assumed to persist in the nonsmooth framework of this work.

Assumptions and formulations

Conclusion

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