Numerical Integration of a New Set of Equations of Motion for Mechanical Systems With Scleronomic Constraints

Abstract

Some new theoretical and numerical results are presented on the dynamic response of a class of mechanical systems with equality motion constraints. At the beginning, the equations of motion of the corresponding unconstrained system are presented, first in strong and then in a weak form. Next, the formulation is extended to systems with holonomic and/or nonholonomic constraints. The formulation is based on a new set of equations of motion, represented by a system of second order ordinary differential equations (ODEs) in both the coordinates and the Lagrange multipliers associated to the motion constraints. Moreover, the position, velocity and momentum type quantities are assumed to be independent, forming a three field set of equations. The weak formulation developed was first used to cast the equations of motion as a set of first order ODEs in the coordinates and the corresponding momenta. Then, the same formulation was also employed as a basis for producing a suitable time integration scheme for the systems examined. The validity and efficiency of this scheme was tested and illustrated by applying it to a number of characteristic example systems.