因微結構與靜電場的耦合(coupling)效應、靜電力的非線性、雜散電場(fringe field)、及微結構因殘留應力及應力梯度所產生的預變形(pre-deformation)等，使得靜電式元件的分析相當複雜而不易求解。本研究目的即在探討具預變形之微結構在承受靜電力作用下的吸附現象，建立一高精確度的解析模型，求得臨界電壓的近似解析解(approximate analytical solutions)。
本研究之方法乃先推導出微型曲樑承受靜電負載之彎曲應變能與電位能，得到具非線性靜電項之總能量函式，基於微小變形之假設，將非線性靜電項以泰勒級數展開(Taylor series expansion)，並分別忽略四次及五次以上之高次項，得到三階(3-order)與四階(4-order)之低度非線性(weak nonlinear)能量式；繼之以能量的觀念利用雷利-立茲法與假設模態法，並採用樑的第一個固有模態(natural mode)做為曲樑的假設撓曲函數，來求得曲樑之臨界電壓的近似解析解，並比較全階模型(full-order)、三階模型、及四階模型所得結果的差異。經與文獻相比較，發現本研究所建立的解析模型與解析解相當接近實際。續比較五種常見的假設撓曲函數對分析結果的影響，包括：固有模態函數、均佈負載撓曲函數、集中負載撓曲函數、複合型負載撓曲函數、及二次函數等。數值結果顯示，不論是全階模型或降階模型(三階與四階)，以結構之固有模態為假設撓曲函數均可得到最接近實驗值的結果，其中全階模型最大誤差約在8%以內，四階模型最大誤差則約在9%以內，三階模型最大誤差約在18%以內，而文獻中所提出之曲樑模型與實驗量測值誤差甚大，其最大誤差約為37%。

The analytical modeling of the electrostatic devices is quite complicated and difficult in virtue of the electric-mechanical coupling effect, the nonlinearity of the electrostatic force, the fringe field, and the pre-deformation of the micro-structure caused by the residual stress and stress gradient. This thesis is to investigate the pull-in phenomenon of the pre-deformed microstructures subjected to electrostatic loads. High precision analytical modeling of the threshold voltage is established in this thesis.
First of all, we use energy method to drive out the bending strain energy and electrical potential energy of the micro curled beam subjected to electrostatic loads. Based on the assumption of small deflection and adopting the Taylor series expansion, the expression of the total potential energy can be simplified as third-order and fourth-order models by omitting the terms with higher order than the third-order and the fourth-order respectively. Continuously, by the use of Rayleigh-Ritz method and assumed mode method, the approximate analytical solutions of the threshold voltage of curled beams are obtained based on the full-order, the third-order, and the fourth-order models respectively. The results obtained by this work agree more well to the experimental results compared to the published works. Five common used assumed deflection shape functions, including the natural mode, the uniform load deflection function, the concentrated load deflection function, the combined loads deflection function, and the square function are also compared with each other. The natural mode is verified to be the best choice. The numerical results show that the greatest error of the full-order model is below 8%, the one of the fourth-order model is below 9%, the one of the third-order model is below 18%, and the one of the literature is below 37%.

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