Application of Runge-Kutta-Rosenbrock methods to the analysis of flexible multibody systems

Numerical integration methods are discussed for general equations of motion for multibody systems with flexible parts, which are fairly stiff, time-dependent and non-linear. A family of semi-implicit methods, which belong to the class of Runge-Kutta-Rosenbrock methods, with rather weak non-linear stability properties, are developed. These comprise methods of first, second and third order of accuracy that are A-stable and L-stable and hence introduce numerical damping and the filtering of high frequency components. It is shown, both from theory and examples, that it is generally preferable to use deformation mode coordinates to global nodal coordinates as independent variables in the formulation of the equations of motion. The methods are applied to a series of examples consisting of an elastic pendulum, a beam supported by springs, a four-bar mechanism, and a robotic manipulator with collocated control.