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Thesis: Analogy between Equilibrium of Structures and Compatibility of Mechanisms by András Lengyel, University of Oxford
Abstract: Planar bar-and-joint mechanisms with one degree-of-freedom are widely used in deployable structures and machines. Such mechanisms are designed to undergo a specific motion, which can be described mathematically by plotting out the compatibility conditions, resulting in a curve called the compatibility path. It has been observed that compatibility paths can develop singularities similar to that of equilibrium paths of elastic structures.

This dissertation studies singularities occurring in compatibility paths with the aid of knowledge in the theory of structural stability. An analogy is set up between the equilibrium path of elastic structures and the compatibility path of mechanisms with a single degree-of-freedom incorporating different types of bifurcation, effects of imperfections and detection of singularities. It is shown that the fundamentally distinct critical points such as limit points and bifurcation points can also appear in the compatibility path. Methods used to study singularities for compatibility conditions of mechanisms and equilibrium of structures are unified so that they can be used for both cases. A formulation of potential energy for mechanisms is also proposed in analogy with the potential energy function used in structural analysis.
Thesis: Analogy between Equilibrium of Structures and Compatibility of Mechanisms by András Lengyel, University of Oxford

Further analysis of the mechanisms is carried out to demonstrate that singularities of compatibility paths can also be dealt with by elementary catastrophe theory similar to stability theory. A relationship is established between the mathematical formulation of different compatibility bifurcations and the canonical forms of catastrophe types. Examples of mechanisms demonstrating the existence of cuspoids of the compatibility conditions are given. An overall classification of the compatibility paths is also proposed.

Keywords: bifurcation, catastrophe types, compatibility condition, compatibility path, elementary catastrophe theory, equilibrium path, imperfection, Jacobian matrix, mechanism with one degree-of-freedom, potential energy











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