Abstract
Abstract. Complex multibody system models that contain bodies with small mass or nearly singular inertia tensor may suffer from high frequency solution components that deteriorate the solver efficiency in time integration. Singular perturbation theory suggests to neglect these small mass and inertia terms to allow a more efficient computation of the smooth solution components. In the present paper, a recursive multibody formalism is developed to evaluate the equations of motion for a tree structured N body system with O(N) complexity even if isolated bodies have a rank-deficient body mass matrix. The approach is illustrated by some academic test problems in 2-D.
Highlights
Classical time integration methods in technical simulation are tailored to problems with smooth solution
Singular perturbation theory gives much insight in the analytical background of these phenomena and allows an efficient approximation of smooth solutions neglecting all terms that contain small parameters, see, e.g., Hairer and Wanner (1996). The application of these classical results to multibody dynamics is non-trivial since the numerical algorithms for evaluating the equations of motion efficiently are based on regularity assumptions that may be violated if small mass and inertia terms are neglected
A recursive multibody formalism is developed for tree structured systems with bodies that suffer from a rank-deficient body mass matrix and may be considered as limit case of systems with bodies of small mass or nearly singular inertia tensor
Summary
Classical time integration methods in technical simulation are tailored to problems with smooth solution. Singular perturbation theory gives much insight in the analytical background of these phenomena and allows an efficient approximation of smooth solutions neglecting all terms that contain small parameters, see, e.g., Hairer and Wanner (1996) The application of these classical results to multibody dynamics is non-trivial since the numerical algorithms for evaluating the equations of motion efficiently (multibody formalisms) are based on regularity assumptions that may be violated if small mass and inertia terms are neglected. The classical set of second order equations of motion in the joint coordinates q(t) is substituted by a suitable combination of second and first order ODEs describing the system dynamics in the limit case of a zero mass body These results extend the previous analysis for chain structured systems in Arnold et al (2010) and may be considered as a step to extend advanced multibody formalisms for flexible multibody systems to models with bodies of (very) small mass. Some basic information on Moore-Penrose pseudo-inverses is provided in Appendix A
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