Normal Modal Analysis For A Two Degrees-of-freedom Piecewise Linear Systems

According to the system inherent dynamic characteristics, which are derived from the system modal analysis, the practical problem can be solved. That is why the modal analysis of nonlinear systems is carried on. The nonlinear normal modes(NNMs) for a two-degree-of-freedom piecewise linear vibratory systems is constructed through the invariant manifolds, which could be obtained with a Galerkin-based solution on the non-linear partial differential equations. By means of the theoretical analysis and numerical simulation, the stability of the NNMs is investigated. The main content is organized in the following aspects:First of all, the study significance is clarified, and the development about the nonlinear modal analysis theory is reviewed. The characteristics of the NNMs are interpreted. The problems to be resolved and the development trends in the nonlinear modal analysis are also summarized.The theory on NNMs and the non-smooth dynamical systems are introduced. The three definitions of the NNMs and its construction are presented. Especially, the process of the approach mentioned above is emphasized. In the context of the NNMs concept which is proposed by Shaw and Pierre, this approach utilizes the Galerkin method to solve the constraint relations, and the equation of the invariant manifolds can be drawn, thus the NNMs is constructed. Meanwhile, as the research object, the elastic collision vibration systems are often modeled by equations of motion with piecewise linear (PWL) terms. Following its mathematics model is established, the Floquet characteristic multipliers and Poincare map method are used to analyze its dynamics characteristic. The procedure for building its NNMs is shown, and its advantages are reflected.The NNMs of the piecewise linear vibratory systems with a clearance is depicted, including the free vibration and the forced vibration with damping. The model is established, and the Galerkin-based approach is applied to the PWL systems, then each NNMs is described. Their general dynamic behavior is performed by a further analysis. Through numerical simulation, the phase portraits of the periodic motion on the invariant manifold is depicted, and the response frequencies are obtained from the simulation of the system dynamics and that of the dynamics restricted to the manifold. The stability and post-critical dynamics of the nonlinear normal modes are investigated using characteristic multiplier method and the Poincare map.