Limit Trajectories in a Non-holonomic System of a Ball Moving Inside a Spherical Cavity

Abstract

The authors analyze the regular and distinctive patterns of the free motion of a ball type tuned mass damper. The governing differential system modeling movement of a heavy ball rolling inside a spherical cavity is formulated and investigated; six degrees of freedom with three non-holonomic constraints and no slipping are assumed. Predominance of the Appell-Gibbs approach over the conventional Lagrangian procedure is pointed out when complicated non-holonomic systems are in question. General properties of the differential system in the normal form are discussed and possibilities of further investigation using semi-analytical methods are outlined. Simultaneously, a wide program of numerical simulation is presented concerning the homogeneous system with a number of initial condition settings and other parameter variants. Seven types of limit solutions (or limit trajectories) have been found and physically interpreted together with their neighborhood. The set of limit trajectories represents boundaries separating solution groups of a certain character. The shape and general character of regular solutions within particular domains delimited by these limit solutions were analyzed to facilitate a practical application of this theoretical background. The paper represents a contribution to the theoretical background of the ball type tuned mass damper used in Civil Engineering. The analysis provides an insight to the possible character of a free motion of a ball type tuned mass damper under various configurations; this way it helps to analyze possible critical parameters of the system or interaction with the structure. Assumptions of further investigation are outlined.