Variational Approaches for Dynamics: Numerical Studies

Abstract

Weak formulations in Analytical Dynamics are developed, paralleling the variational methods in elastostatics, and including a fundamental yet novel approach for treating constraints (both holonomic and nonholonomic). A general three field approach is presented, in which the momentum balance conditions, the compatibility conditions between displacement and velocity, the constitutive relations and the displacement and momentum boundary conditions are all enforced in weak form. A primal or kinematic formulation is developed from the general form by enforcing the compatibility conditions and displacement boundary conditions a priori. The conditonal stability of the kinematic formulation is the counterpart of locking phenomenon in elastostatics and may be avoided either by reduced order integration or by a mixed formulation. Toward this end, a two field mixed formulation is presented, which follows from the general form when the constitutive relations are satisfied a priori. A general set of the constraint equations are introduced into the kinematic and mixed formulations using a specific choice of multipliers which results in a modified variational principles. Two simple examples concerning rigid body dynamics are presented.