Abstract
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.
Highlights
The study is motivated as a theoretical background of a passive vibration absorber of a ball type
It reveals that the governing differential system in the homogeneous case exhibits singular solutions which can be considered as limits separating regular solutions into certain groups according to their individual properties
These singular solutions are characterized by nonzero initial conditions and correspond to the case when the ball starts its movement from unbalanced initial condition in the fixed cavity
Summary
The study is motivated as a theoretical background of a passive vibration absorber of a ball type. Various generalizations allow to extend the class of problems concerned slightly beyond conventional limits of Hamiltonian system This strategy is very effective, concerning 2D and simpler 3D systems. In this article we concentrate on one important effect stemming from a strongly nonlinear character of the system It reveals that the governing differential system in the homogeneous case (without external excitation) exhibits singular solutions which can be considered as limits separating regular solutions into certain groups according to their individual properties. These singular solutions are characterized by nonzero initial conditions and correspond to the case when the ball starts its movement from unbalanced initial condition in the fixed cavity. These singular solutions do not seem to be important from the viewpoint of separation of regular solutions
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