Geometrical Theory of Nonlinear Modal Analysis.

摘要

This thesis presents a geometrical theory for modal analysis of nonlinear structural systems. The first objective of this thesis is to develop a coherent and consistent framework, wherein, dynamic analysis of a class of nonlinear systems of large order can be performed efficiently. Specifically, methods are presented to compute the periodic responses or nonlinear modes of the system, to assess their stability, and to perform bifurcation analysis in order to characterize the branches of solutions that emerge.;In this study, the definition of nonlinear eigensolutions (modes) is based on the concept of an Instantaneous Center Manifold (ICM) which was introduced by the author as the periodic quotient of the invariant manifolds that can be defined for a class of nonlinear systems. Both analytical and numerical methods for calculation of such invariant manifolds have been developed . Also an efficient numerical method for calculation of nonlinear modes, namely Multi-harmonic Multiple-point Collocation (MMC), has been developed which can identify multiple nonlinear modes in each solution and which does not require integrating the equations of motion.;The second objective is to study extensions to superposition for nonlinear systems, where the response of the system can be expressed as a function of its nonlinear eigensolutions (modes). More specificity, it is of interest to find the general form of these functions, which are called connecting functions and can be used to generate new solutions to the system from combinations of the nonlinear modes. The form of the connecting function for a class of nonlinear systems is presented and methods are presented for computing them numerically. This work could eventually allow one to obtain an arbitrary solution of the system from a set of its eigensolutions, namely nonlinear modes. Second, they can be used to decompose any arbitrary solution of the system onto a set of its neighboring eigensolutions in order to better understand the system characteristics that cause the response. Three numerical approaches have been also developed to identify connecting functions which provide interesting insights into the relationship between a system's eigensolutions and its general solution.