Modified phase space formulation for constrained mechanical systems. Differential approach

Efficient treatment of holonomic and nonholonomic constraints in multibody analysis is widely recognized as a key feature of every reliable and efficient computational procedure for dynamics. In this paper a particular formulation of the problem of motion of constrained mechanical systems is discussed both from a theoretical and a numerical point of view. The method, originated in the context of finite elements in time, is here extended to a differential equation formulation. The state vector of the system is described in a modified phase space where the generalized moments are allowed a component in the unfeasible directions of the constraints, thus becoming free quantities not subject to constraint conditions. Moreover, the use of the classical Lagrange multipliers is avoided in favor of a different form of the multipliers which may be interpreted as their integrals with respect to time. violations compared with those obtained with other stabilization procedures. Significant examples cast light on the more relevant numerical properties of the present approach. Moreover, a penalty formulation is developed assuming the reaction forces as quantities proportional to the constraint violations. In this way a solution procedure is obtained which avoids all the potential difficulties and precision losses which may be experienced when singular configurations are met during the motion of the system. All the basic theoretical and numerical characteristics of the method discussed here are maintained in its penalty formulation.