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Avoidance of artifacts in harmonic balance solutions for nonlinear dynamical systems

Lentz Lukas, von Wagner Utz

Journal of Theoretical and Applied Mechanics

2020

Vol. 58 nr 2

307--316

Although nonlinear mechanical systems have been the topic of numerous investigations during the last decades, the research on suitable analysis methods is still ongoing. One method that is commonly known and still sees a lot of interest is the Harmonic Balance method. In the basic version of this method, only one harmonic is used to approximate a periodic solution, which allows for fairly easy application. A drawback is that this approach may lead to solutions that are inaccurate or even artifacts which are solutions that possess no physical relevance. In this article, it is demonstrated how an error criterion can be used to access the accuraccy of solutions and how artifacts can be indentified based on this assessment. Subsequently, stability analysis is performed for solutions that possess small errors. The method is applied to an asymmetric Duffing oscillator as well as to a system that consists of two linearly coupled Duffing oscillators. The authors gave a corresponding presentation of their work at PCM-CCM Kraków 2019.

Dynamics of an Array of Duffing Oscillators Suspended on Elastic Etructures

Czołczyński K., Stefański A., Perlikowski P., Kapitaniak T.

Machine Dynamics Problems

2007

Vol. 31, No. 2

57-65

We consider the dynamics of externally excited chaotic oscillators suspended on the elastic structure. We show that for the given conditions of oscillations of the structure, initially uncorrelated chaotic oscillators become periodic and synchronous. In the periodic regime we observed synchronized clusters and multistability as different attractors coexist.

Synchronization of coupled mechanical oscillators

Perlikowski P., Stefański A.

Mechanics and Mechanical Engineering

2006

Vol. 10, nr 1

110-116

Complete synchronization of coupled chaotic systems is usually a primary and crucial issue. Coupling in mechanical systems introduces mutual perturbation of their dynamics. In case of identical systems such perturbation can lead to the synchronization. We can predict the synchronization threshold of such systems using a concept called Master Stability Function (MSF). As a tool of MSF we use transverse Lyapunov exponents, which characterize the stability of synchronization state. We show areas of synchronization in coupling parameters space in typical nonlinear systems: Duffing and Duffing - Van der Pol oscillators.