Closed-form solution for nonlinear spatial deflections of strip flexures of large aspect ratio considering second order load-stiffening

Modeling the nonlinear spatial deflections of strip flexures has been a challenging problem in compliant mechanism research. Although there have been several practically useful methods for modeling spatial deflections, few have derived the load-displacement directly from the governing equations which means they lack physical insights. This work achieves the load-displacement relations for strip flexures of large aspect ratio by solving the nonlinear governing differential equations of strip flexures. Power series method are employed to solve the governing equations because other analytical methods are not effective given the complexity of the equations. Then by simplifying the solution using Taylor series expansion and truncation, closed-form load-displacement relations are obtained. Two examples are employed to validate the accuracy of the solution. By comparing with previously reported models, our model is demonstrated to be capable of capturing relevant geometric nonlinearities in the intermediate deflection range defined as less than 10% of the length. The resulting load-displacement relation presented in this work offers a useful and parameterized tool for understanding the constraint behavior of strip flexures of large aspect ratio and synthesizing compliant mechanism designs with nonlinear kinetostatic behaviors.