On the existence of limit cycles for a class of penalty formulations of dissipative multibody systems
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A convergence and stability result is presented for a class of penalty formulations of multibody dynamics. Despite the fact that the governing equations are nonlinear, the theoretical convergence of the formulation is guaranteed if the multibody system is natural and conservative. Unfortunately, the convergence theorem-theorem is not directly applicable to dissipative multibody systems, such as those encountered in control applications. However, it is shown that the approximate solutions of the dissipative system converge to a nearby collection of trajectories that can be characterized precisely using a Liapunov/Invariance Principle analysis. Specifically, the trajectories of the constraint violation time histories approach limit cycles that represent 'residual, uncontrollable' energy in the penalized dynamical system.