Nonlinear Strain Energy Formulation of a Generalized Bisymmetric Spatial Beam for Flexure Mechanism Analysis

Analytical load-displacement relations for flexure mechanisms, formulated by integrating the individual analytical models of their building-blocks (i.e., flexure elements), help in understanding the constraint characteristics of the whole mechanism. In deriving such analytical relations for flexure mechanisms, energy based approaches generally offer lower mathematical complexity, compared to Newtonian methods, by reducing the number of unknowns-specifically, the internal loads. To facilitate such energy based approaches, a closed-form nonlinear strain energy expression for a generalized bisymmetric spatial beam flexure is presented in this paper. The strain energy, expressed in terms of the end-displacement of the beam, considers geometric nonlinearities for intermediate deformations, enabling the analysis of flexure mechanisms over a finite range of motion. The generalizations include changes in the initial orientation and shape of the beam flexure due to potential misalignment or manufacturing. The effectiveness of this approach is illustrated via the analysis of a multilegged table flexure mechanism. The resulting analytical model is shown to be accurate using nonlinear finite elements analysis, within a load and displacement range of interest.