An engineering perspective on the numerics of quasi-periodic oscillations

Abstract

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.

A combined FD-HB approximation method for steady-state vibrations in large dynamical systems with localised nonlinearities

“…Within this contribution it is assumed that T = {t 0 +i t} i∈Z is a set of equidistant grid points and t is the time step. Here, α (1) k m , α (2) k m are the weights of the finite difference with index 8 k m ∈ M ⊂ Z. Further on, the approximations for the time derivations in Eq.…”

“…Here, the Kronecker product allocates the equations for the coefficients for the corresponding kth Fourier order in a global manner. Introducing the matrices α (1) , α (2) containing the difference weight factors, a global notation is analogously found for the difference quotients…”

“…by Newton-like schemes. To name some representatives, the Harmonic Balance (HB), Finite Difference (FD) and Shooting (SH) method [5,15,18] are capable of approximating periodic motions and can also be extended to quasi-periodic solutions [1,6,16]. These approaches based on an AES are suitable for both stable and unstable solutions, but can differ significantly in their numerical convergence speed depending on the problem.…”

“…This is typically the case for weakly non-linear problems [19, p. 131]. The FD may be better suited for strongly non-linear problems [1]. The SH method on the other hand becomes problematic when dealing with highly repulsive solutions.…”

“…The SH method on the other hand becomes problematic when dealing with highly repulsive solutions. 1 For the case of systems with strong local non-linearities discussed here, the combination of two different computation methods into a hybrid approximation framework is a promising approach. Strengths can be combined and weaknesses mitigated thereby increasing computational efficiency.…”