The nonholonomic and chaotic nature of a rolling isolation system

This paper presents the modeling of a rolling isolation platform used to protect objects from the hazards of horizontal shaking. The platform is built from four pairs of recessed steel bowls: four concave-up, attached to a shaking floor via a steel frame, and four concave-down, attached to the isolated object via a steel frame. Four steel ball-bearings, located between these bowls, allow the bottom- and top-frames to roll with respect to one another. In order for the four ball-bearings to remain in contact with the bowls during times of large angular rotation, the top-frame is assumed to conform to a saddle shape. The kinematics of the rolling contact impose a set of eight nonholonomic constraints to the translational and rotational dynamics of the platform center. The nonholonomic equations of state are found from the fundamental form of Lagrange's equation, which incorporates velocity constraints via Lagrange multipliers. A viscous damping model calibrated to experimental free-responses depends upon the mass of the isolated object. Precise photogrammetric measurements of a sample of these steel bowls provide the data required to calibrate the potential energy function for this system. It is shown that for the non-quadratic potential energy function of this particular system, free response trajectories are strongly sensitive to slight perturbations in initial conditions, and that the system is, in fact, chaotic. Even in the absence of any mass or stiffness eccentricity, translational and rotational motions are nonlinearly coupled through the constraint of slip-free rolling. Uni-axial models cannot be used to predict responses of these systems.