Large-amplitude non-linear normal modes of piecewise linear systems

Abstract

A numerical method for constructing non-linear normal modes (NNMs) for piecewise linear autonomous systems is presented. These NNMs are based on the concept of invariant manifolds, and are obtained using a Galerkin-based solution of the invariant manifold's non-linear partial differential equations. The accuracy of the constructed non-linear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that thisconstruction approach can accurately capture the NNMs over a wide range of amplitudes, including those with strong non-linear effects. Several interesting dynamic characteristics of the non-linear modal motion are found and compared to those of linear modes. A two-degree-offreedom example is used to illustrate the technique. The existence, stability and bifurcations of the NNMs for this example are investigated.

Introduction

“…By oscillatory dynamics we mean the time periodic motions or the nonlinear normal modes -NNMs that are supported by the finite granular chains. The discussion of NNMs in the literature has focused mainly on the dynamical systems with smooth stiffness nonlinearities [18,[77][78][79][80][81][82][83][84], although some works on the systems with non-smooth nonlinearities have also appeared [85][86][87]. NNMs are defined as the time-periodic oscillations of discrete or continuous oscillators that can be either synchronous or asynchronous.…”

The non-linear dynamic response of the Euler–Bernoulli beam with an arbitrary number of switching cracks

“…An appealing approach for the modal analysis of nonlinear systems, even with strong nonlinearities, has been proposed in the literature by means of the definition of the so called 'non-linear normal modes' [29].…”

Nonlinear normal modes and their application in structural dynamics

“…nonlinearities [3] and nonlinear systems with external forcing [4]. Consequently, the Galerkin method seamlessly interfaces with finite element models of engineering structures, and it allows the user to specify the vibration amplitude range and the accuracy of the model over that range.…”