Modeling and velocity stabilization of constrained mechanical systems

Abstract

An application of some one- and multistep methods for the numerical integration ofconstrained mechanical systems with and without stabilization is demonstrated to compare their efficiency. In particular, the use of (high order) multistep methods is considered. Choosingsupernumerary coordinates, constrained mechanical systems are formulated in descriptor form (DAE). For 2D systems at least, an assembling technique methodically different from, but equivalent to, the Lagrange formalism is presented for the computer generation of the equations of motion with constant mass matrix and quadratic constraints. This modeling, which can be generalized to 3D multibody systems, is applied on two non-stiff test problems. An example of the automatic modeling and simulation is given by a seven body mechanism. Computer programs are presented. By index reduction, the DAE model is transferred to an equivalent ODE representation whose unstable numerical solution is stabilized byprojection onto the constraint manifold. A perturbation analysis shows thatvelocity stabilization is the most efficient projection with regard to improvement of the numerical integration. How frequently the numerical solution of the transferred ODE should be stabilized is discussed. For a class of multistep methods, a strategy of stabilizing at certain time steps and performing a much less demandingquasi-stabilization at the others is suggested, especially for high order methods. With these high order methods, stabilization is not even necessary for the second test problem.