Nonlinear modes: amplitude-phase formulation and bifurcation analysis

Abstract

Nonlinear Modes (NMs) are efficient tools for analysing the behaviour of dynamical mechanical systems[1]. The objective of this contribution is to show how this concept can be used to characterize periodic orbits and limit cycles. Following Shaw and Pierre[2] the concept of NMs is introduced here in the framework of the invariant manifold theory for dynamical systems. A NM is defined in terms of amplitude, phase, frequency, damping coefficient and mode shape with the distinctive feature that the last three quantities are amplitude and phase dependent. An amplitude-phase transformation is performed to give as well the time evolution of the NM motion (through the two first order differential equations governing the amplitude and phase variables) as the geometry of the invariant manifold. The conservative case was considered in [4]. The formulation is extended here to autonomous mechanical systems including gyroscopic and/or nonlinear damping terms. Our approach differs of that in [3] where the amplitude-phase transformation is based on the frequency of the linearized system. Bifurcation analysis, existence and stability of periodic orbits on the associated invariant manifold can be studied from the differential equations governing the amplitude and phase variables.The procedure is illustrated on a 2 DOF van der Pol mechanical system.