An amplitude-phase formulation for nonlinear modes and limit cycles through invariant manifolds

The aim of this paper is to show how the concept of nonlinear modes can be used to characterize periodic orbits and limit cycles in multi-degree-of-freedom nonlinear mechanical systems. In line with previous studies by Shaw and Pierre, the concept of nonlinear modes is introduced here in the framework of invariant manifold theory for dynamical systems. A nonlinear mode 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 system of formulation adopted here is suitable for use with a Galerkin-based computational procedure. The existence and stability of periodic orbits such as limit cycles on the associated invariant manifolds can be studied from the differential equations governing the amplitude and phase variables. The examples given here involve adding gyroscopic and/or “negative” nonlinear damping terms of Van der Pol type, and nonlinear restoring force to the system equations.