2023/07/16 by Thomas Thurnher, George Haller, Thurnher, Thomas +3 · 2 citations
Engineering · Physics and Astronomy · #Bladed Disk Vibration Dynamics #Dynamical Systems (math.DS) #Dynamics and Control of Mechanical Systems #FOS: Mathematics #Force Microscopy Techniques and Applications
paper · pdf · doi:10.48550/arxiv.2307.10240
openalex publication_date 2023/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We use the recent theory of Spectral Submanifolds (SSM) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization of SSMs and their reduced dynamics. We provide these results both for general first-order as well as second-order mechanical systems under periodic and quasiperiodic excitation using a multi-index based approach, thereby optimizing memory requirements and the computational procedure. We further provide theoretical results that simplify the SSM parametrization for general second-order dynamical systems. More practically, we show how the reduced dynamics on the SSM can be used to extract the resonance tongues and the forced response around the principal resonances in parametrically excited systems. In the case of two-dimensional SSMs, we formulate explicit expressions for computing the steady-state response as the zero-level set of a two-dimensional function for systems that are subject to external as well as parametric excitation. This allows us to parallelize the computation of the forced response over the range of excitation frequencies. We demonstrate our results on several examples of varying complexity, including finite-element type examples of mechanical systems. Furthermore, we provide an open-source implementation of all these results in the software package SSMTool.