Singular solutions of a non-holonomic system as limits separating solution groups of particular type

The non-holonomic system represents a model of the ball tuned mass damper (TMD), as used to absorb vibration of selected engineering structures, where conventional absorber types are inapplicable. The device consists of a ball moving within a spherical cavity fixed with the structure. To deduce a governing differential system the Appel-Gibbs formulation has been employed. The non-linear mathematical model includes six degrees of freedom and three non-holonomic constraints. The system has an auto-parametric character. The homogeneous differential system in the normal form is formulated. Its general properties are investigated for various settings of non-homogeneous initial conditions. Several singular solutions are extracted and physically interpreted. In principal, they represent limits separating solution groups of a certain character. The shape and general character of regular solutions within individual domains staked out by these limits are analyzed in order to facilitate a practical application of this theoretical background.