Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes

Rémi Arquier,Sergio Bellizzi,Robert Bouc,Bruno Cochelin

doi:10.1016/j.compstruc.2006.01.011

Abstract

The aim of this paper is to present two methods for the calculation of the nonlinear normal modes of vibration for undamped nonlinear mechanical systems: the time integration periodic orbit method and the modal representation method. In the periodic orbit method, the nonlinear normal mode is obtained by making the continuation of branches of periodic orbits of the equation of motion. The terms “periodic orbits” means a closed trajectory in the phase space, which is obtained by time integration. In the modal representation method, the nonlinear normal mode is constructed in terms of amplitude, phase, mode shape, and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The methods are compared on two DOF strongly nonlinear systems.

Highlights

Extending the concept of normal modes of vibration to the case where the restoring forces contain nonlinear terms, has been a challenge to many authors
Two methods have been described for the determination of the nonlinear normal modes of undamped nonlinear mechanical systems
In the periodic orbit method, the nonlinear normal mode is constructed by making the continuation of branches of periodic solutions of the equation of motion

Summary

Introduction

Extending the concept of normal modes of vibration to the case where the restoring forces contain nonlinear terms, has been a challenge to many authors This has led to the so-called nonlinear normal modes (NNMs) which have great potential for applications in nonlinear vibrating systems. It is established that the knowledge of the NNMs, together with their bifurcations, can be very helpful to understand the dynamics of a nonlinear system [1] Some important phenomenon such as the localisation of the motion [2], the interaction between modes [3], the pumping of energy of a linear system by a pure nonlinear one [4], can be nicely explained using the NNM concept. Eventhough the principle of superposition does not hold for nonlinear equations, the NNM can be useful to generate effective reduced-order models for multi-degree-of-freedom nonlinear systems [6,7]

Objectives

Methods

Conclusion