Abstract This paper investigates the nonlinear dynamic behavior of a repulsive force actuator from the bifurcation perspective in the cases of primary and principal parametric resonances. The studied model consists of a specific configuration that produces out-of-plane repulsive (levitation) force. The governed equation of motion around static equilibrium is described in the form of a generalized forced Mathieu equation holding quadratic and cubic nonlinearities. While the particular system of interest is one with generated levitation force, the results can be applied to a wide class of MEMS in which the governing equation of motion is classified as the forced Mathieu equation. Coordinate transformation of the systems’ analytical solution from polar to Cartesian is done to preclude the singularity in origin. The frequency response is studied, and analytical expressions for equilibrium positions as fixed points are presented. Assessing the Jacobian matrix, the stability of each branch is examined, and the critical detuning values are defined in which the bifurcations occur. This criterion is then used to border the regions and illustrate the phase portraits of picked points within the regions. The results graphically show that inducing system by the initial conditions corresponding to the fixed points, triggers constant amplitude as well as angle, which ends up with periodic motion. Moreover, the outcome proves that perturbation analysis merely captures the periodic motions and fails to detect quasi-periodic motions. Our analysis provides a platform for further understanding of the repulsive electrostatic force, which is highly nonlinear. The results can be deployed in designing sensors with a wider detection range in which the amplitude jump is measured near the pitchfork bifurcation of principal parametric and cyclic fold bifurcation of primary resonances.

中文翻译：

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悬浮力MEMS执行器的分岔分析

摘要 本文从分叉的角度研究了在主参数共振和主参数共振情况下排斥力致动器的非线性动态行为。研究的模型由产生平面外排斥（悬浮）力的特定配置组成。围绕静态平衡的受控运动方程以广义强制 Mathieu 方程的形式描述，该方程包含二次和三次非线性。虽然感兴趣的特定系统是产生悬浮力的系统，但结果可以应用于一大类 MEMS，其中控制运动方程被归类为受迫 Mathieu 方程。系统的解析解从极坐标到笛卡尔坐标变换是为了排除原点的奇异性。研究频率响应，并给出了平衡位置作为不动点的解析表达式。评估雅可比矩阵，检查每个分支的稳定性，并定义发生分叉的临界失谐值。然后使用该标准来界定区域并说明区域内拾取点的相位图。结果以图形方式表明，通过固定点对应的初始条件诱导系统，触发恒定的幅度和角度，最终以周期性运动结束。此外，结果证明微扰分析仅捕获了周期运动，而未能检测到准周期运动。我们的分析为进一步理解高度非线性的排斥静电力提供了一个平台。