A Loop-Based Approach for Topological Processing of Mobility and Subsystem Rigidity in Complex Spatial Multibody Systems

For a given system of bodies whose relative motions are described by a set of constraints, the dimension of these allowed motions can provide valuable information about the physical properties of this system. The special case of rigidity is of particular interest for multidisciplinary research, e.g., for the analysis of crystal lattices, granular packings, the definition of placement conditions in CAD systems, or the synthesis of multibody systems. This thesis is dedicated to the recognition of rigid substructures in the context of multibody systems.

The number of possible motions is characterized by the concept of the degree of freedom, for which several established classifications are reviewed, and three additional categories, namely fully, transmitted, and structurally isolated degrees of freedom are introduced. Based on this classification, it is now possible to recognize not only absolutely rigid systems, but also those that are rigid in character, yet have isolated degrees of freedom, e.g., in the form of bodies that can rotate freely between two spherical joints.

For this purpose, a multibody system consisting of rigid bodies and joints is topologically understood as a network of coupled modular multibody loops, which simplifies the recognition of efficient recursive solutions and enables a uniform consideration of planar, spatial and mixed systems. The isolated degrees of freedom are first independently identified in the individual multibody loops based on their displacement groups. Afterward, they are tracked across the couplings of the network and are classified with respect to newly introduced categories. In a graph, rigid subsystems, both with and without isolated degrees of freedom, can be identified by means of simple graph cuts. Through subsequent replacement of the determined rigid subsystems by simple bodies, the complexity of a multibody system can be reduced. Overdetermined parts that lead to false assumptions counting the degree of freedom are thus identified, enabling a better determination of the actual degree of freedom of a system.
Finally, the procedure is successfully applied to well-known counter-examples in which conventional formulas for determining the degree of freedom lead to incorrect results.


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