Numerical analysis of invariant manifolds characterized by quasi-periodic oscillations of nonlinear systems

Quasi-periodic motions systematically occur, when a system is subjected to multiple unrelated excitation mechanisms. Since unrelated mechanisms generically exhibit in dependent frequencies, the resulting motion is characterized by multiple fundamental frequencies. Consequently, the resulting motion is not periodic, but, due to the inde pendent frequencies, sort of periodic, namely quasi-periodic. Not being able to identify one unique fundamental frequency, established methods for the identification of periodic motions are impracticable. In order to calculate and analyze a quasi-periodic motion, approaches to the invariant toroidal manifold, which is the hull of a quasi-periodic motion, are utilized. This thesis provides the theoretical frameworks for the developed analyzing program Quont, which is capable of calculating, analyzing and continuing quasi-periodic motions of arbitrary dynamical systems. Introductory, the theoretical fundamentals of quasi-periodic motions are discussed and the connections between a quasi-periodic motion and an invariant toroidal manifold are explained. Furthermore, possible transition scenarios between periodic and quasi-periodic motions are examined, since these are often encountered in engineering applications. Two conceptually different equations to describe invariant toroidal manifolds are derived and analyzed, from which the hyper-time invariance equation is the superior approach to engineering systems. Having identified the governing equation, three numerical approaches to solve the hyper-time invariance equation are introduced, the multi-dimensional Fourier-Galerkin method, the finite difference method and a shooting method for quasi-periodic motions. From these, the shooting method represents a new approach developed within this thesis. Once the governing equation is solved, an invariant quasi-periodic motion is established. The sole knowledge of stationary motions is often insufficient in applications and the stability property is sought. Two approaches to the stability identification of quasi-periodic motions are developed in this thesis, a spatial and a temporal one. From both, the temporal approach is generally applicable and highly efficient, by which it is implemented in Quont. In order to verify and validate Quont, different nonlinear dynamical systems with varying complexity are analyzed. Regarded systems range from academical examples to a structure abstracted from an engineering application. Multiple new results and phenomena are identified, from which especially the additionally determined stability properties of quasi-periodic motions provide new insides.