Coupling of Ode- and Dae-Integration With the Time-Stepping Integration Method for Multibody Systems With Impacts and Friction

Abstract eng:
Multibody systems with unilateral constraints feature difficulties in the numerical integration because of their non-smooth characteristics. Depending on the number of unilateral contacts and their state transitions either time-stepping or event-driven schemes are commonly applied [1]. The hybrid method presented in [2] couples Moreau’s half-explicit time-stepping method on velocity level with a Livermore Solver for ordinary differential equations (LSODAR). It profits from the respective advantages and avoids many drawbacks. As an extension, the hybrid integration scheme [2] is complemented by an integration method for differential algebraic equations (DASKR). This optionally allows solving the non-impulsive phases as an index-2 DAE by replacing LSODAR with DASKR. Therefore, the non-impulsive contact conditions are formulated on velocity level. Hence, they are fulfilled on the same kinematic level as in the time-stepping mode reserved for impulsive transitions. In contrast to multibody systems with only bilateral constraints, the algebraic equations become more complicated due to possible state transitions. Complementarity conditions for forces and kinematic quantities are expressed via a proximal point formulation which transforms the inequality conditions into a system of non-smooth equations. This can be solved by specially designed root solving method like non-smooth variants of Newton’s method [3].For making the Newton iteration more robust, the corresponding iteration matrix can be provided partly analytically. Particularly, this includes analytic representations of the discontinuous Jacobians in the proximal point formulation. The behavior of the semi-analytical approach is compared to the fully numerical approximation of the iteration matrix provided automatically by DASKR. ODE- and DAE-type hybrid integration schemes are implemented in the multibody simulation software MBSim [4]. Test examples are used for a comparison. Therefore, the constraint conditions are regarded on different kinematic levels, as well as necessities to switch between time-stepping and non-impulsive integration modes, re-initialization procedures and numerical differences in the solutions are discussed.

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