The phenomenon of quasi-periodicity in deterministic dynamical systems describes stationary solutions, which neither exhibit a finite period length nor are chaotic. Recently, an increasing demand for robust numerical methods is driven by applied dynamics and industrial applications. In this context, direct time integration proves to be impractical due to extensive integration intervals. Therefore, in a first step, this contribution aims on giving an application oriented survey of the basic theory as well as alternative concepts. In the following, the focus is set on the direct computation of invariant manifolds (surfaces) on which quasi-periodic solutions evolve. This approach offers a unique framework from which classical methods (e.g., the multi-harmonic-balance) can be systematically deduced and mutual similarities between different methods may be revealed. This contribution starts with a brief summary of related mathematical basics, which is followed by an overview of available methods. Subsequently, the computation of invariant manifolds by means of solving a partial differential equation is emphasized. These PDEs may be formulated using different parametrization strategies. Here, the concept of hyper-time parametrization is particularly interesting, since it is a promising starting point for the development of numerical schemes with general applicability in engineering problems. In order to solve the underlying PDE, various methods may be used. The implementation of a Fourier-Galerkin method as well as a finite difference method is presented and compared on the basis of computational results of the van-der-Pol equation (with and without forcing). Moreover, it is demonstrated that both methods apply to periodic as well as quasi-periodic solutions alike. In order to exemplify the practical use, these methods are applied to a generic rotordynamic model problem.

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This contribution discusses the influence of fluid forces, stemming from compliant, contact-free annular rotor seals, on the steady state stability and bifurcation behaviour of a rotor. The model used in this work consists of a Laval-Rotor where the disc runs in a turbulently streamed seal. The compliance of the seal is reduced to a visco-elastically supported outer seal ring. In order to account for the fluid seal forces the Childs-Hirs-model is used. An investigation of the eigenvalues shows that the compliance of the seal support may lead to a significant increase in the stable operating range. A stability-loss via Hopf -, Hopf -Hopf or secondary Hopf -Bifurcations can occur depending on the system parameters.

The determination of stationary solutions of dynamical systems as well as analyzing their stability is of high relevance in science and engineering. For static and periodic solutions a lot of methods are available to find stationary motions and analyze their stability. In contrast, there are only few approaches to find stationary solutions to the important class of quasi-periodic motions-which represent solutions of generalized periodicity-available so far. Furthermore, no generally applicable approach to determine their stability is readily available. This contribution presents a unified framework for the analysis of equilibria, periodic as well as quasi-periodic motions alike. To this end, the dynamical problem is changed from a formulation in terms of the trajectory to an alternative formulation based on the invariant manifold as geometrical object in the state space. Using a so-called hypertime parametrization offers a direct relation between the frequency base of the solution and the parametrization of the invariant manifold. Over the domain of hypertimes, the invariant manifold is given as solution to a PDE, which can be solved using standard methods as Finite Differences (FD), Fourier-Galerkin-methods (FGM) or quasi-periodic shooting (QPS). As a particular advantage, the invariant manifold represents the entire stationary dynamics on a finite domain even for quasi-periodic motions -whereas obtaining the same information from trajectories would require knowing them over an infinite time interval. Based on the invariant manifold, a method for stability assessment of quasi-periodic solutions by means of efficient calculation of Lyapunov-exponents is devised. Here, the basic idea is to introduce a Generalized Monodromy Mapping, which may be determined in a pre-processing step: using this mapping, the Lyapunov-exponents may efficiently be calculated by iterating this mapping.