椭圆型柔性铰链闭式扭转柔度研究
Research on closed-form torsional compliance of elliptical flexure hinges

摘要 针对变矩形截面梁扭转柔度难以解析求解问题，基于矩形截面梁扭转柔度公式及利用微分和积分思想，采用非线性曲线拟合方法得到了可用于推导变矩形截面梁闭式扭转柔度的设计公式，解决了变矩形截面梁扭转柔度不可积或可积情况下精度较低的问题，并用于解析推导椭圆型柔性铰链闭式扭转柔度公式.通过数值算例和有限元对闭式解析结果进行验证，并分析了结构参数对柔性铰链扭转柔度的影响，结果表明：闭式解与数值解结果一致，与有限元结果相比相对误差小于7.5%，且当铰链宽度和最小槽口厚度较小时对扭转柔度的影响较大，从而为变矩形截面柔性铰链受扭转力矩时的尺寸优化及结构设计奠定理论基础.
Abstract：Aiming at the problem that the torsional compliance of variable rectangle cross-section (VRC) beams is difficult to formulate, based on the existing equations for torsional compliance of rectangular cross-section beams and the use of calculus, a design formula used for deriving the closed-form torsional compliance of VRC beams is obtained by using nonlinear curve fitting method, which solves the problem of non-integrability or low-precision under integrable condition for torsional compliance of VRC beams. It is applied in analytical derivation of the equation for closed-form torsional compliance of elliptical flexure hinges. The closed-form solutions are validated through the numerical examples and finite element simulations, and the effect of structural parameters on torsional compliance of flexure hinges is also analyzed. The results show that the analytical results are consistent with the numerical results, and the relative errors are smaller than 7.5% compared with the finite element results. When the width and minimum-thickness values of flexure hinges are relatively small, they have a larger impact on the torsional compliance, which lays a theoretical foundation for size optimization and structural design of VRC flexure hinges under torsion load.