摘要：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.
其他摘要：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.
