This paper deals with the frequency–amplitude response of superharmonic resonance of second order (order two) of electrostatically actuated Micro-Electro-Mechanical System (MEMS) cantilever resonators. The structure of MEMS resonators consists of a cantilever resonator over a parallel ground plate, with a given gap in between, and under AC voltage. This resonance results from hard excitations and AC voltage frequency near one-fourth of the natural frequency of the resonator. The forces acting on the resonator are the nonlinear electrostatic force to include fringe effect, and a linear damping force. In order to solve the dimensionless partial differential equation of motion along with boundary and initial conditions, two types of models are developed, namely Reduced Order Models (ROMs), and Boundary Value Problem (BVP) model. The BVP model is essentially a finite difference model with a discretization in time only. ROMs are developed using one through five modes of vibration. The Method of Multiple Scales (MMS), numerical integrations using MATLAB, as well as a continuation and bifurcation analysis are used to solve the ROMs. The BVP model, resulting from using finite differences for time derivatives, is also numerically integrated. Five modes of vibration ROM is found to make accurate predictions in all amplitudes. A softening effect of the response is predicted. The response consists of a bifurcation with a bifurcation point of amplitude one fourth of the gap, and a stable branch in larger frequencies with a pull-in instability end point at three fourths of the gap. The bifurcation point shifts to lower frequencies as the voltage and/or fringe effect increase, and/or damping decreases. If damping increases, the branches coalesce, peak amplitude decreases, and a linear behavior is experienced.

本文研究静电致动微机电系统（MEMS）悬臂谐振器的二阶（二阶）超谐谐振的频率-幅度响应。MEMS谐振器的结构由位于平行接地板上的悬臂谐振器组成，在它们之间以及交流电压之下具有给定的间隙。这种共振是由硬激励和交流引起的电压频率接近谐振器固有频率的四分之一。作用在谐振器上的力是包括边缘效应的非线性静电力和线性阻尼力。为了解决运动的无量纲偏微分方程以及边界条件和初始条件，开发了两种类型的模型，即降阶模型（ROM）和边界值问题（BVP）模型。BVP模型本质上是一个仅在时间上离散的有限差分模型。ROM是使用一种到五种振动模式开发的。使用多尺度方法（MMS），使用MATLAB进行数值积分以及连续和分叉分析来求解ROM。由于对时间导数使用有限差分而产生的BVP模型也进行了数值积分。发现了五种振动ROM模式，可以对所有振幅进行准确的预测。预计响应的软化效果。响应包括一个分叉，其分叉点的幅度为间隙的四分之一，并且在较大频率下具有稳定的分支，在该间隙的四分之三处有一个引入不稳定性终点。随着电压和/或条纹效应的增加和/或阻尼的减小，分叉点移至较低的频率。如果阻尼增加，则分支合并，峰值幅度减小，并且会经历线性行为。在较大的频率中有一个稳定的分支，在该间隙的四分之三处有一个引入不稳定性终点。随着电压和/或条纹效应的增加和/或阻尼的减小，分叉点移至较低的频率。如果阻尼增加，则分支合并，峰值幅度减小，并且会经历线性行为。在较大的频率中有一个稳定的分支，在该间隙的四分之三处有一个引入不稳定性终点。随着电压和/或条纹效应的增加和/或阻尼的减小，分叉点移至较低的频率。如果阻尼增加，则分支合并，峰值幅度减小，并且会经历线性行为。