Stabilization methods for simulations of constrained multibody dynamics

The descriptor form of constrained multibody systems, and any general formulation of such systems with closed loops, yield index-3 di erential-algebraic equations (DAEs). Generally, some index reduction techniques have to be used before numerical discretization can be safely applied. However, a direct di erentiation of the constraints introduces mild instability. Hence one must consider index reduction with stabilization. One popular method for index reduction with stabilization is Baumgarte's technique. However the di culty of choosing the parameters in practice makes this method's robustness unclear. Moreover our numerical experiments show that there are still large constraint drifts even with Baumgarte's stabilization. In the thesis, we employ concepts and techniques of dynamical systems in order to improve the situation. We rst study the relationship between a DAE and its underlying vector elds. A general form of vector elds with stabilized invariant manifolds is given. We propose a new numerical stabilization method for semi-explicit index-2 and index-3 DAEs of Hessenberg form. Our stabilization method improves on Baumgarte's stabilization which is widely used in engineering as well as in simulations of multibody systems. We then develop a numerical code based on the new stabilization method with an adaptive step-size control for the descriptor form of the Euler-Lagrange equations in multibody systems. Numerical simulations have been conducted with our code on a variety of multibody systems including a spatial ve-link-suspension model in a vehicle, Andrews squeezing mechanism, a two-arm manipulator with a prescribed motion and a mechanism with kinematic singularity. Our code is e cient, fast and therefore is more attractive for real-time simulations. ii When high speed and light-weight substructures are involved in the multibody system, the rigid body model is usually no longer valid. In such a case we compute the elastic deformations and oscillations of the substructures using the nite element method. Satisfactory numerical results using our method are presented for deformable multibody systems as well. iii Table of

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