Spectral collocation methods for the periodic solution of flexible multibody dynamics

Many flexible multibody systems of practical interest exhibit a periodic response. This paper focuses on the implementation of the collocation version of the Fourier spectral method to determine periodic solutions of flexible multibody systems modeled via the finite element method. To facilitate the analysis and obtain governing equations presenting low-order nonlinearities, the motion formalism is adopted. Application of Fourier spectral methods requires global interpolation schemes that approximate the unknown fields over the entire period of response with exponential convergence characteristics. The classical spectral interpolation schemes were developed for linear fields and hence do not apply to the nonlinear configuration manifolds, such as SO(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SO}(3)$$\end{document} or SE(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SE}(3)$$\end{document}, that are used to describe the kinematics of multibody systems. Furthermore, the configuration and velocity fields are related through nonlinear kinematic compatibility equations. Clearly, special procedures must be developed to adapt the Fourier spectral approach to flexible multibody systems. The spectral interpolation of motion is investigated; interpolation schemes based on the polar decomposition are proposed. Assembly of the linearized governing equations at all the grid points leads to the governing equations of the spectral method. Numerical examples illustrate the performance of the proposed approach.