A Rosenbrock–Nystrom state space implicit approach for the dynamic analysis of mechanical systems: I—theoretical formulation

When performing dynamic analysis of a constrained mechanical system, a set of index three differential-algebraic equations (DAE) describes the time evolution of the system. The paper presents a state space based method for the numerical solution of the resulting DAE. A subset of so-called independent generalized coordinates, equal in number to the number of degrees of freedom of the mechanical system, is used to express the time evolution of the mechanical system. The second-order state space ordinary differential equations (SSODE) that describe the time variation of independent coordinates are numerically integrated using a Rosenbrock-type formula specialized to second-order systems of differential equations. Rosenbrock methods are known to be efŽ cient for medium accuracy integration of stiff systems; they do not require the solution of non-linear systems for the stage values, and possess optimal linear stability properties for stiff integration. The computation of exact Jacobians needed by Rosenbrock formulas is discussed in the context of multibody systems. The companion paper [19] discusses a choice of method coefŽ cients based on a four-stage L-stable Rosenbrock formula and presents numerical results.