Nonlinear strain energy formulation of spatially deflected strip flexures

Modeling spatial deflections of strip flexures in compliant mechanisms has been one of the most challenging problems due to the nonlinear coupling effects between bending, torsion and stretching. Energy-based approaches can mitigate the modeling complexity by treating a compliant mechanism in its entirety and eliminating the analysis of load equilibriums between the adjacent elements. To leverage energy-based approaches, this work formulates the nonlinear strain energy stored in spatially deflected strip flexures. The strain energy, expressed as the integration of the product of stress and strain over the entire volume, is simplified using Taylor series expansion followed by truncation of higher order terms, leading to a closed-form expression with relevant nonlinearities retained. The coefficient matrices in the strain expression are validated by the load–deflection relations obtained in our previous work. The effectiveness of the energy expression combined with Castigliano's first theorem is demonstrated by two compliant mechanisms containing multiple strip flexures, and the results are validated by those of nonlinear finite element models. The energy expression offers a convenient and mathematically simple way for modeling compliant mechanisms containing spatially deflected strip flexures.