本論文旨在模擬與研討微機電系統中靜電式曲形微致動器之非線性靜、動態特性。
在研究非線性吸致變形的分析模型中，除了不同邊界條件的靜電式致動結構，例如懸臂樑、兩端固定樑，而且將電場的邊緣效應以及殘留應力也一併考慮。為求解非線性方程式的封閉解，本文首先利用Adomian分解法評估矩形樑與平坦電極所組成的微致動器，因為在求解非線性變形的過程中不需要迭代，分解法是評估靜電式微致動器不穩定吸致行為的最有效方法之一。
本研究中，為探討在微致動器模擬過程中對於準確性的影響，小變形與大變形的假設同時被執行，同時，本文進一步利用微分值積法來評估及檢驗不同形狀的微型樑與曲形電極所組成的微致動器對於非線性吸致行為的影響。再其次，本論文亦研究曲形微夾式致動器的接觸力與吸致變形，並且採用微分值積法來求解曲形靜電場力與不同形狀之懸臂式致動器變形間的非線性交互作用，數值分析的結果顯示微分值積法可以精確、有效及系統化的處理此種非線性致動器問題。
為評估靜電式微致動器的動態特性，利用微分值積法計算一種兩端固定的形狀樑在其受到靜電力作用下於靜態變形處振動的自然頻率，所分析的模型包含曲形靜電場力與形狀微型樑恢復力之間的非線性交互作用，以及中性面的拉伸效應、軸向殘留應力與電場的邊緣效應，分析結果顯示本分析模型的模擬結果與實測值非常吻合，本文同時亦研究形狀微型樑與曲形電極對於致動器系統自然頻率的影響，解析的結果顯示各種微型樑或電極形狀不僅會影響靜電場的分怖，而且嚴重改變微致動器的動態特性，解析的結果進一步證明形狀微型樑與曲形電極微致動器可以將工作電壓範圍增加為矩形樑與平坦電極微致動器的六倍。
今日，愈來愈多的微機電機構整合了電路設計，例如微開關、光學微鏡片…等，這些微致動器在電壓作用下要到達永久靜止的位置之前會有殘振的產生，因此，本研究進一步探討懸臂樑式微開關且考慮微型樑與基底間之空氣壓縮薄膜阻尼時的殘振現象。
在分析模擬不同形狀的微致動器之後，對於其非線性之靜、動態特性可以得到適切的瞭解，並進一步提供設計者在設計階段得以適當且精確的控制微機構之操作範圍。

This dissertation performs a simulation investigation into the nonlinear static and dynamic characteristics of electrostatically driven shaped micro-actuators in micro-electro-mechanical systems (MEMS).
The model proposed in the current nonlinear pull-in deflection study considers various boundary conditions for the electrostatically actuated structures, e.g. the cantilever beam and the fixed-fixed beam, and takes account of the electrical field fringing effect and the axial residual stress. Initially, the Adomian decomposition method is employed to evaluate the response of a micro-actuator incorporating a rectangular micro-beam and a flat electrode by obtaining the closed-form solution of the corresponding nonlinear equation. Since no iteration is required in solving the nonlinear deformation, this decomposition method is one of the most efficient methods available for evaluating the unstable pull-in behavior of an electrostatically driven micro-actuator.
The present study implements both small and large deflection assumptions when simulating the response of the micro-actuator in order to explore the possible effects of the two models on the accuracy of the simulation results. The shaped micro-beam with a curved electrode micro-actuator is further assessed using the differential quadrature method (DQM) to examine the influence of the nonlinear pull-in effect. This dissertation also studies the contact force and the pull-in deflection of shaped micro-tweezers. The DQM is employed to solve the nonlinear interaction between the curved electrostatic field force and the corresponding deflection of the shaped cantilever actuators. The numerical results confirm the ability of the DQM to treat this form of nonlinear actuator problem accurately, efficiently and systematically.
To evaluate the dynamic characteristics of the electrostatic micro-actuator, the DQM is applied to solve the natural frequencies of a fixed-fixed shaped beam vibrating around its statically deflected position under electrostatic loading. The proposed model not only takes account of the nonlinear interaction between the curved electrostatic field force and the restoring force of the shaped micro-beam, but also considers mid-plane stretching, axial residual stress, and electrical field fringing effects. It is shown that an excellent agreement exists between the simulation results obtained using the proposed model and those measured experimentally. This study also investigates the micro-beam and electrode shape effect on the natural frequencies of the actuator system. The analytical results indicate that variations in the shape of the micro-beam or of the electrode not only influence the electrostatic field distribution, but also significantly alter the dynamic characteristics of the micro-actuator. Furthermore, the results demonstrate that the shaped micro-beam with a curved electrode micro-actuator increases the working voltage range of the micro-actuator by a factor of approximately six times compared to that of a micro-actuator incorporating a rectangular micro-beam and a flat electrode.
A continuing trend nowadays is the integration of micro-electro-mechanical devices with electronic circuitry to fabricate MEMS devices such as micro-switches, optical micro-mirrors, etc. It is known that when an electrical voltage is applied to these devices, the micro-actuators will undergo a residual vibration before reaching their permanent position. Hence, this dissertation investigates the residual vibration phenomenon of cantilever beam type micro-switches with air squeeze-film damping between the micro-beam and substrate.
The present simulations of various shaped micro-actuators provide an understanding of the nonlinear static and dynamic behaviors of these devices and as such provide designers with the information required to properly and accurately control the device operating range during the design stage.

CONTENTS

Acknowledgements……………………………………………………………………………… i
ABSTRACT ……………………………………………………………………………………. ii
List of Tables …………………………………………………………………………………… x
List of Figures …………………………………………………………………………………..xi
Nomenclature ………………………………………………………………………………….. xiv
Chapter 1 Introduction…………………………………………………………………………1
1.1 Electrostatic Micro-Electro-Mechanical Systems (MEMS)…………………………….2
1.1.1 Micro-switches, Micro-tweezers and 3-D MEMS Simulator…..…………………...2
1.1.2 Micro-sensors and Micro-actuators…………………..……………………………4
1.1.3 Dynamic Characteristics and Damping……………..…………………………….9
1.2 Numerical Methods…………………………………………………..…………………..12
1.3 Outline of Dissertation………………………………………………………………….14
Chapter 2 Nonlinear Pull-in Effect of Electrostatic Micro-actuators………………………18
2.1 Different Actuator Devices……………………………………………………………..18
2.1.1 Fixed-fixed Beam Micro-actuator……..…………….……………………………18
2.1.2 Cantilever Beam Micro-actuator……………………………..…………………..20
2.2 Adomian Decomposition Method………………………………………………………20
2.3 Analysis of Different Micro-actuators………..………………………………………….22
2.3.1 Simulation Models of Fixed-fixed Beam Micro-actuators…...………………….25
2.3.1.1 1-D Continuous Beam Model…………………………………………….25
2.3.1.2 Lumped Model……………………………………………………………...27
2.3.2 Simulation Model of Cantilever Beam Micro-actuator…....………………………30
2.4 Conclusions……………………..………………………………………………………..32
Chapter 3 The Nonlinear Electrostatic Behavior for Shaped Electrode Micro-actuators.41
3.1 Shaped Electrode Micro-actuators…………………………………………………….41
3.2 Analysis Model of Nonlinear Pull-in Deflection………….……………………………42
3.2.1 Small Deflection Model of a Cantilever Beam Micro-actuator……..……………43
3.2.2 Large Deflection Model of a Cantilever Beam Micro-actuator…..………………..43
3.2.3 Small Deflection Model of a Fixed-fixed Beam Micro-actuator…..………………44
3.3 Differential Quadrature Method (DQM)……………………………………………….45
3.3.1 DQM Model for Small Deflection Theory of a Cantilever Beam Micro-actuator...48
3.3.2 DQM Model for Large Deflection Theory of a Cantilever Beam Micro-actuator...50
3.3.3 DQM Model for a Fixed-fixed Beam Micro-actuator……………………………..52
3.4 Analysis Results…………………………………………….……………………………53
3.4.1 Cantilever Rectangular Beam Micro-actuators with Different Curved Electrodes..53
3.4.2 Difference Introduced by Using the Large or Small Deflection Models……..……55
3.4.3 Fixed-fixed Rectangular Beam Micro-actuators with Different Residual Stresses..55
3.4.4 Different Combinations of Shaped Beam and Curved Electrode………………….56
3.5 Conclusions…………………………………………………………………………….57
Chapter 4 Electrostatic Micro-tweezers………………………………………………………68
4.1 Models of Micro-tweezers……………………………………………………………68
4.1.1 Analysis Model…………………………………………………………………….68
4.1.2 DQM Model………………………..………………………………………………69
4.2 Numerical Analyses of Different Micro-tweezers……………………………………71
4.2.1 Cantilever Rectangular Beam Type Micro-tweezers……………..…………..71
4.2.1.1 Simplified Micro-tweezers ……………………………………………72
4.2.1.2 Micro-tweezers without Silicon Dioxide Coating……………………...73
4.2.1.3 Micro-tweezers with Silicon Dioxide Coating…………………………...74
4.2.2 Shaped Cantilever Beam Micro-tweezers……………………………………….75
4.3 Conclusions………………………………………………………………………………75
Chapter 5 Dynamic Characteristics of Shaped Micro-actuators……………………………85
5.1 Analysis Model for a Fixed-fixed Beam Type Micro-actuator………………..…………85
5.2 DQM Models…………………………………………………………………………….89
5.2.1 Static Deflection Model of a Fixed-fixed Beam Micro-actuator………………….90
5.2.2 Mode Shapes and Natural Frequencies of the Fixed-fixed Beam
Micro-actuator……………………………………………………………………92
5.3 Numerical Results for Shaped Micro-actuators.……………………………………….93
5.3.1 Fixed-fixed Rectangular Micro-beams with Flat Electrodes..……………………93
5.3.2 Fixed-fixed Rectangular Micro-beam Actuators with Different Curved
Electrodes...…………………………………………….....………………………95
5.3.3 Shaped Micro-beam Actuators with a Flat Electrode………..…………………….96
5.3.4 Shaped Micro-beams with Curved Electrodes………………..……………………97
5.4 Conclusions……………………………………………………………………………....98
Chapter 6 Residual Vibrations of Micro-actuators………………………………………..110
6.1 Model Description……………………………………………………………………..111
6.1.1 Governing Equation of a Cantilever Micro-beam Type Actuator………………...111
6.1.2 Squeeze-film Damping………...………………..………………………………..112
6.2 Analysis Method………………………………………………………………………114
6.3 Numerical Examples…………………………………………………………………116
6.3.1 Natural Frequencies of the System……………………………………………….116
6.3.2 Residual Vibration of a Uniform Rectangular Micro-beam with a Full
Electrode Design…………………………………………………………………117
6.3.3 Residual Vibration of a Uniform Rectangular Micro-beam with a Partial
Electrode Design…………………………………………………………………118
6.3.4 Residual Vibration of a Shaped Micro-beam with a Partial Electrode Design…...119
6.3.4.1 A Shaped Beam with a Shape Parameter of m=0.5………………………...119
6.3.4.2 A Shaped Beam with a Shape Parameter of m=1.0………………………120
6.3.4.3 A Shaped Beam with a Shape Parameter of m=2.0.………………………120
6.4 Conclusions……………………………………………………………………..………121
Chapter 7 Conclusions and Future Research…………………………………………..……..129
7.1 Conclusions……………………………………………………………………………..129
7.2 Future Research…………………………………………………………………..…….130
References……………………………………………………………………………………131
Appendix ……………………………………………………………………………………143

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