Non Linear Modes As A Tool To Analyse Nonlinear Dynamical Systems.

Abstract

In structural analysis, the concept of normal modes is classically related to the linear vibration
theory. Extending the concept of normal modes to the case where the restoring forces contain non-linear
terms has been a challenge to many authors mainly because the principle of linear superposition does not
hold for non-linear systems. The aim of this paper is to show how the concept of the Noninear Modes
(NNMs) can be used to better understand the response of the nonlinear mechanical systems. The concept
of NNMs is introduced here in the framework of invariant manifold theory for dynamical systems. A
NNM is defined in terms of amplitude, phase, frequency, damping coefficient and mode shape, where
the last three quantities are amplitude and phase dependent. An amplitude-phase transformation is performed
on the nonlinear dynamical system, giving the time evolution of the nonlinear mode motion via
the two first-order differential equations governing the amplitude and phase variables, as well as the geometry
of the invariant manifold. The formulation adopted here is suitable for use with a Galerkin-based
computational procedure. It will be shown how the NNMs give access to the existence and stability of
periodic orbits such as limit cycle.