Abstract
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.
Highlights
Nonlinear modes (NMs) are efficient tools for analyzing the behavior of vibrating mechanical systems [1,2,3,4,5,6]
The invariant manifold geometry is characterized by a set of partial differential equations (PDEs) with respect to the ‘‘master coordinates’’
It is observed that in some cases periodic solutions may exist for the amplitude equation, denoting the presence of stable or unstable ‘‘limit cycles’’ on the invariant manifold
Summary
Linear modal frequency in Ref. [8], takes computational advantage of the 2p-periodic character of the parametric expressions for the invariant manifold in terms of the phase variable. It was established that the frequency and mode shape functions solve a 2p-periodic (with respect to the phase variable) nonlinear differential eigenvalue problem This method gives a parametric description of the associated invariant manifold. Three cases are considered: a sdof oscillator without damping but with inertial nonlinearity and a nonlinear restoring force, a sdof oscillator with a linear damping term, unit mass and a nonlinear restoring force, a sdof oscillator with ‘‘negative nonlinear damping’’ of the van der Pol type yielding two stable and one unstable limit cycles In this last example, it is observed that a small parameter approach leads to predictive analytical results as to the behavior of the oscillator. A system of two coupled van der Pol equations is discussed, including the stability results of the corresponding limit cycles on the associated invariant manifolds
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