Solving Differential Algebraic Equation Systems: Alternative Index 2 and Index 1 Approaches for Constrained Mechanical Systems

Abstract

Regarding constrained mechanical systems, we are faced with index3 differentialalgebraic equation (DAE) systems. Direct discretization of the index3 DAE systems only enforces the position constraints to be fulfilled at the integrationtime points, but not the hidden constraints. In addition, order reduction effects are observed in the velocity variables and the Lagrange multipliers. In literature, different numerical techniques have been suggested to reduce the index of the system and to handle the numerical integration of constrained mechanical systems. This paper deals with an alternative concept, called collocated constraints approach. We present index2 and index1 formulations in combination with implicit Runge–Kutta methods. Compared with the direct discretization of the index3 DAE system, the proposed method enforces also the constraints on velocity and—in case of the index1 formulation—the constraints on acceleration level. The proposed method may very easily be implemented in standard Runge–Kutta solvers. Here, we only discuss mechanical systems. The presented approach can, however, also be applied for solving nonmechanical higherindex DAE systems.