The unification of finite, infinitesimal and hybrid kinematic synthesis

This thesis deals with the kinematic size-synthesis of mechanism design. Synthesis theories for which several finitely-separated positions of a rigid body are specified are called finite kinematic synthesis theories. Design theories for which a single position of a body and several derivatives are given are called infinitesimal kinematic synthesis theories. Hybrid finite-infinitesimal theories combine elements of both finite and infinitesimal kinematic synthesis. Since one can consider infinitesimal and hybrid synthesis theories as depending on infinitesimally-separated positions, there are conceptual limiting relationships among the theories. Although both the theories and the conceptual limiting relationships have been known for some time, no unifying methods have heretofore been developed.; In this dissertation, analytical limiting relationships among kinematic synthesis theories are presented. The infinitesimal theory is shown to be a limiting case of the finite theory. Each hybrid theory is also shown to be a limiting case of the finite theory. In addition, every hybrid theory is a limiting case of another hybrid theory. These analytical limiting relationships are arranged into an hierarchy of theories and limits. The existence of these limiting relationships means that finite, infinitesimal, and hybrid kinematic synthesis can be performed with one numerical algorithm, and new numerical methods are presented which accomplish finite, infinitesimal, and hybrid synthesis in one computer program. Furthermore, it is shown that, even though infinitesimal and hybrid kinematic synthesis are limiting cases of finite kinematic synthesis, the results of each theory are not always completely analogous. These apparent discrepancies in the results of corresponding theories are explained from the viewpoint of degeneracy. These new analytical limiting relationships serve to analytically unify finite, infinitesimal, and hybrid kinematic synthesis. Numerical unification is achieved through the implementation of new numerical methods.