Nonholonomically constrained dynamics of rolling isolation systems
Rolling Isolation Systems provide a simple and effective means for protecting components from horizontal floor vibrations. In these systems a platform rolls on four steel balls which, in turn, rest within shallow bowls. The trajectories of the balls is uniquely determined by the horizontal and rotational components of the rolling platform, and thus provides nonholonomic constraints. In general, the bowls are not parabolic, so the potential energy function of this system is not quadratic. This paper presents the application of Gauss’s Principle of Least Constraint to the modeling of rolling isolation platforms. The equations of motion are described in terms of a redundant set of constrained coordinates. Coordinate accelerations are uniquely determined at any point in time via Gauss’s Principle by solving a linearly-constrained quadratic minimization. In the absence of any modeled damping, the equations of motion conserve energy. Simulations and experiments show that responses are highly sensitive to small changes in the initial conditions; peak responses can be predicted only statistically.
