考虑卡西米尔力的静电激励NEMS吸合特性及其尺寸效应研究

Study on the Size Dependence and Pull-in Instability of Electrostatically Actuated NEMS with Casimir Force

基于应变梯度理论和哈密顿原理,并考虑卡西米尔力的影响,建立了静电激励纳米机电系统(NEMS)的尺寸效应模型,并得到模型的控制方程和边界条件.然后,引入广义微分求积法和拟弧长算法,得到模型的数值解.结果表明,当考虑卡西米尔力的影响时,系统两极的吸合电压有所减小.并且,当系统尺寸达到一个临界值时(即两电极间距小于"最小间距",或可变形电极长度超过"拉起长度"),系统会在没有外加电压的作用下自动发生吸合,这将为NEMS的优化设计和定量分析提供理论基础.

The size dependence and pull-in instability of electrostatically actuated Nano-Electro-Mechanical Systems （NEMS） are considered incorporating the influence of Casimir force. The governing equa- tion and boundary conditions are derived based on the strain gradient elasticity theory and Hamilton princi- ple. A discrete and reduced-order governing equation and boundary conditions for electrically actuated mi- crobeam-based NEMS with Casimir force are presented with the help of generalized differential quadrature method （GDQ）. Because the pull-in instability of NEMS is a saddle-node bifurcation, the local continuation methods are not applicable for solving the equation. The complete solution is instead obtained by implemen- ting the pesudo-arclength algorithm,which enables the iteration process to smoothly go through the inflec- tion points. The size dependence and pull-in instability of electrostatically actuated NEMS with Casimir force included is then studied. The results reveal that Casimir force can reduce the pull-in voltage of sys- tem. Although the effect of Casimir force is so small that it can be ignored in the pull-in behavior of NEMS actuators, this effect becomes obviously significant on the size dependence and pull-in instability of system when the size of structure decreases to the nanometer scale. With the Casimir force incorporated,once the scale of system reaches the critical value （i. e. the gap between two electrodes is smaller than the minimum gap,or the length of movable beam is larger than the detachment length） ,the pull-in instability will occur without any voltage applied. For the present model, when the Casimir force is ignored, the pull-in voltage increases with decreasing size scale; when Casimir force is considered,however,the pull-in voltage first increases and then decreases sharply to zero with decreasing size scale, which is quite different from that predicted by the classical beam model. Therefore, the effect of Casimir force should be taken into account in the process of production and application. This study may be helpful for the design and theoretical modeling of NEMS.