承受移動靜電力之微結構的電彈性分析

Abstract

本文旨在探討微結構在承受移動靜電力作用下的動態特性理論與穩定性分析，其採用的微結構以微型樑作為主要分析之微結構外形。研究主題包括微型樑承受來回移動力之動態分析、微型樑承受來回移動靜電力之動態分析與穩定性分析。 理論推導方面首先求得系統之能量式，再利用漢米頓原理(Hamilton principle)推導出具靜電項之非線性偏微分方程式，移動靜電負荷的位置以單位脈衝函數(Dirac delta function)定義之，基於微小變形之假設，將非線性之靜電項以泰勒級數展開(Taylor series expansion)並忽略二次以上之高次項，得到一結構與靜電耦合之線性化運動方程式，最後利用假設模態展開法求得微系統之離散化運動方程式，探討靜電外力為靜電常力或靜電簡諧力作用下之動態響應。 動態分析中以倫基•庫達二氏法(Runge-Kutta Method)求解，獲得外力為靜電常力與靜電簡諧力作用下之動態響應。文中討論移動外力頻率與簡諧力頻率對系統造成之頻率分歧現象，還有移動力與移動靜電力對系統影響的差異。本文中以弗洛蓋定理(Floquet’s theory)觀察微結構承受移動靜電力動態穩定性。發現靜電外力越大，其不穩定區域越大。

The dynamics and stabilities of a micro-beam subjected to a to and fro traveling electrostatic force were investigated in this thesis. Research topics include the dynamics of a micro-beam subjected to a traveling load, the dynamics and stabilities of a micro-beam subjected to and fro traveling electrostatic loads. In this study, Hamilton principle is first used to derive the nonlinear partial differential equation with the nonlinear electrostatic term. The position of the electrostatic load defined as Dirac delta function. Expand the electrostatic term by the Taylor series expansion. Based on the small deflection assumption, the second and higher order terms of the electrostatic expansion can be neglected, get a linearized equations of motion for the structure and electrostatic coupling effect, and finally the use of the mode expansion method yields the discrete equation of motion, then the dynamics of a micro-beam subjected to a to and fro traveling electrostatic force were investigated In the numerical analysis, Runge-Kutta Method is used to find dynamic responses. According to the analytical results, the resonant frequencies of a structure will be bifurcated by the traveling harmonic loads. Apart from Floquet’s theory, the beam may be unstable at some unstable traveling speeds. Furthermore the unstable regions expand with increasing driving voltage.