On the Numerical Solution of Constrained Multibody Dynamic Systems

A new formulation of the equations of motion of constrained multibody systems is presented using a coordinate-splitting (CS) technique. In contrast to the conventional index reduction techniques which apply diierentiation to the Lagrange formulation, this approach applies the coordinate-split operator to the variational form of constrained dynamics, yielding consistent dynamic equations of multibody systems. A Newton-type iteration for solving the discretized diierential-algebraic system is derived based on eecient computations for the CS operators and the iteration matrix. The CS formulation is advantageous because the numerical solution of the resulting DAE does not suuer from lack of smoothness of the CS matrix. It is demonstrated that for some classes of mechanical systems including highly oscillatory systems, convergence of the Newton iteration can be a problem. This is the case for both the Lagrangian formulation and, to a lesser extent , the CS formulation. An improved Newton-type iteration for these types of problems is proposed. Numerical results illustrate the eeectiveness of the new techniques.