Time Integration Methods

In flexible multibody dynamics, the time integration of the semi-discretized equations of motion represents a challenging problem due to the simultaneous presence of constraints and different time scales. This combination leads, as analyzed in the foregoing chapter, to a stiff differential-algebraic system. We investigate here the behavior of numerical methods for such problems, with particular focus on the well-established implicit integrators that are either based on the BDF (Backward Differentiation Formulas) methods or on implicit Runge–Kutta methods of collocation type. The chapter starts, however, with an overview on time integration methods for constrained mechanical systems. Using a model equation with smooth and highly oscillatory solution parts, we then show that stiff methods suffer from order reductions which are directly related to the limiting DAE of index 3 to which the stiff mechanical system converges. At the end of this chapter, we also introduce extensions of the generalized-α method and of the implicit midpoint rule to the differential-algebraic case and investigate their potential for mechanical multibody systems.

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