拟弧长延拓法在静电激励MEMS吸合特性研究中的应用

Application of the Pseudo-Arclength Continuation Algorithm to Investigate the Size-Dependent Pull-in Instability of the Electrostatically Actuated MEMS

在静电激励微机电系统MEMS(micro-electro-mechanical systems)吸合特性研究中,基于应变梯度理论的微梁结构的控制方程是非线性高阶微分方程,给方程的求解带来了困难.由于该问题的数学模型本质上是分叉问题,方程的解支上出现奇异点,而运用局部延拓法无法通过奇异点.因此,通过运用广义微分求积法将控制方程降阶离散,结合拟弧长延拓法使迭代顺利通过奇异点,求出了整个解曲线.结果表明,拟弧长延拓法能有效并准确地求解具有分叉现象的高阶微分方程问题,为精确预测静电激励MEMS的吸合电压提供有力帮助.

In the study of the electrostatically actuated MEMS （micro-electro-mechanical sys- tems）, based on the strain gradient elasticity theory, the governing equations for the microbeam are nonlinear differential equations that are difficult to solve. The mathematical model for this problem is of essential bifurcation, and the solution branches of the equations have in- flection points. The iteration process can＇ t go through the inflection points with the local con- tinuation method. Therefore, the generalized differential quadrature method was applied to dis- cretize and reduce the order of the governing equations, and the pseudo-arclength continuation algorithm was used to enable the iteration process to go smoothly through the inflection points, with the complete solution curve calculated. The numerical results show that the pseudo-arc- length continuation algorithm makes an effective way precisely solving the nonlinear high-order differential equations with bifurcation phenomenon embedded, and helps to accurately predict the pull-in voltage of the electrostatically actuated MEMS.