Normal Modes of Vibration for Non-Linear Continuous Systems

International audienceA definition and a constructive methodology for normal nodes of motion are developed for a class of vibratory systems the dynamics of which are governed by non-linear partial differential equations. The definition for normal modes is given in terms of the dynamics on two-dimensional invariant manifolds in the system phase space. These manifolds are a continuation of the planes which represent the well-known normal modes of the linearized system. A local asymptotic approximation of the geometric structure of these manifolds can be obtained by using an approach which follows that used for generating center manifolds. The procedure also provides the non-linear ordinary differential equations which govern individual modal dynamics and a physical description of the system configuration when it is undergoing a modal motion. In this paper, the general theory is described for the application of vibrations of continuous media, but it can easily be extended to other situations. In order to demonstrate the power of the approach and its unique procedural aspects, three examples involving beam vibrations are worked out in detail. The examples are conservative, simply supported beams. The first demonstrates the methodology for a linear beam model, the second is a beam on a non-linear elastic foundation, and the third example is a beam with non-linear torsional springs attached at each end. In these examples, the simplicity of the mode shapes of the linearized model yields relatively simple calculations that do not obscure the important features of the procedure