本文應用混合微分轉換/有限差分法探討在微奈米結構系統與靜電場之耦合效應、殘留應力及雜散電場效應的影響下，受靜電驅動之微奈米結構系統的動態特性研究，並用非局域性理論探討在奈米尺度下的微奈米系統之動態行為。首先應用微分轉換法將微奈米橋狀樑之統御方程式轉換成迭代方程，並在頻率方程式上做代數計算，得到微奈米橋狀樑的自然頻率。接著利用混合微分轉換/有限差分法探討微奈米橋狀樑在靜電驅動下，殘留應力及壓膜阻尼對於吸附電壓的影響。
研究結果顯示模擬非局域微奈米橋狀樑在不同邊界條件下之自然頻率與文獻結果一致，誤差皆在0.003%內。接著應用非局域性彈性理論探討單層石墨烯之微奈米橋狀樑在靜電力驅動下之動態分析，結果顯示傳統彈性理論及非局域彈性理論在探討壓膜阻尼及殘留應力有同樣結果；壓膜阻尼對於靜電力驅動之微奈米橋狀樑的吸附電壓影響甚微，相較之下殘留應力對於微奈米橋狀樑之吸附電壓的影響較多。透過非局域參數的調整可以更加的貼近現實實驗數據，因為奈米尺度下需要考慮其分子之相互作用。故混合微分轉換/有限差分法是一種比其他分析方法更簡單，更快速求解非線性偏微分問題，尤其是在複雜的非局域性方程式，更能顯示出其快速收斂的優點。

In this study, the hybrid differential transformation/finite difference method is used to analyze the dynamic characteristic of micro / nano beams, which are electrostatically actuated under the influence of the coupling effect, the residual stress and the fringing field effect between the micro / nano system and electrostatic field. Furthermore, the nonlocal continuum field theory is applied to analyze the dynamic behavior of micro / nano beams.
To obtain the natural frequencies of the micro / nano beam, the governing equation is transformed to the algebraic equation by differential transformation. The effect of pull-in voltage by the residual stress and the squeeze damping is discussed by using hybrid differential transformation/finite difference method.
The results of this study show that the natural frequency of micro / nano beam under different boundary conditions is consistent with the literatures by the errors within 0.003%. The nonlocal elasticity theory is employed to analyze the behavior of an electrostatically actuated graphene micro / nano beams. It indicates that the results by traditional elasticity theory and nonlocal elasticity theory are the same as the beams subjected to residual stress and squeeze-film damping. However, unlike the residual stress, the effect of squeeze-film damping on pull-in voltage is very small. By consideration of the interaction on nanoscale, the data can be more real by adjustment of the nonlocal parameter. Therefore, the hybrid differential transformation / finite difference method is simpler and faster on nonlinear partial differential equations than other methods, especially on complex equation of nonlocal continuum field.

Chen, C. K., and Ho, S. H., “Application of Differential Transformation to Eigenvalue Problems”, Applied Mathematics and Computation, 79(2), 173-188, 1996.
Chen, C. O. K., and Ho, S. H., “Free Vibration Analysis of Non-uniform Timoshenko Beams Using Differential Transform. Transactions of the Canadian Society for Mechanical Engineering”, 22(3), 231-250, 1998.
Chen, C. L., and Liu, Y. C., “Solution of Two-Point Boundary-Value-Problems Using the Differential Transform Method”, Journal of Optimization Theory and Application, 99(1), 23–35, 1998.
Cheng, J., Zhe, J., Wu, X., Farmer, K. R., Modi, V., and Frechette, L., “Analytical and FEM Simulation Pull-in Study on Deformable Electrostatic Micro Actuators”, Technical Proceedings of the International Conference on Modeling and Simulation of Microsystems, 298–301, 2002.
Chiou, J. S., and Tzeng, J. R., “Application of the Taylor Transform to Nonlinear Vibration Problems”, Journal of Vibration and Acoustics, 118(1), 83-87, 2006.
Chowdhury, S., Ahmadi, M., and Miller, W. C., “A Comparison of Pull-in Voltage Calculation Methods for MEMS-Based Electrostatic Actuator Design”, 1st International Conference on Sensing Technology, 112-117, 2005.
Chang W.J and Lee H.L., “Free Asymmetric Transverse Vibration of Circular Monolayer Graphene”, Nano, 9(7), 1450072, 2014.
Eltaher, M. A., Alshorbagy, A. E., and Mahmoud, F. F., “Vibration Analysis of Euler–Bernoulli Nanobeams by Using Finite Element Method”, Applied Mathematical Modelling, 37(7), 4787-4797, 2013.
Eringen, A. C., “Nonlocal Continuum Field Theories”, Springer Science and Business Media, 2002.
Gupta, R. K., “Electrostatic Pull-in Test Structure Design for In-situ Mechanical Property Measurements of Microelectromechanical Systems (MEMS)”, Doctoral Dissertation, Massachusetts Institute of Technology, 1998.
Gupta, R. K., Osterberg, P. M., and Senturia, S. D., “Material Property Measurements of Micromechanical Polysilicon Beams”, In Micromachining and Microfabrication'96. International Society for Optics and Photonics, 39-45, 1996.
Hirai, Y., Marushima, Y., Nishikawa, K., and Tanaka, Y., “Young’s Modulus Evaluation of Si Thin Film Fabricated by Compatible Process with Si MEMS’s”, In Microprocesses and Nanotechnology Conference, 82-83,2000.
Hu, Y. C., Chang, C. M., and Huang, S. C., “Some Design Considerations on the Electrostatically Actuated Microstructures”, Sensors and Actuators A: Physical, 112(1), 155-161, 2004.
Kuo, B. L., and Chen, C. K., “Application of a Hybrid Method to the Solution of the Nonlinear Burgers’ equation”, Journal of Applied Mechanics, 70(6), 926-929, 2003.
Kuang, J. H., and Chen, C. J., “Adomian Decomposition Method Used for Solving Nonlinear Pull-In Behavior in Electrostatic Micro-Actuators”, Mathematical and Computer Modelling, 41(13), 1479-1491, 2005.
Lishchynska, M., Cordero, N., Slattery, O., and O'Mahony, C., “Modelling Electrostatic Behavior of Microcantilevers Incorporating Residual Stress Gradient and Non-ideal Anchors”, Journal of Micromechanics and Microengineering, 15(7), S10, 2005.
Liu, C. C., “Application of Hybrid Differential Transformation and Finite Difference Method on the Dynamic Analysis of Nonlinear Electrostatic-Actuating Microsystems”, Doctoral Dissertation, Ph. D. Thesis, National Chen-Kung University, 2009.
Mousavi, T., Bornassi, S., and Haddadpour, H., “The Effect of Small Scale on the Pull-in Instability of Nano-switches using DQM”, International Journal of Solids and Structures, 50(9), 1193-1202, 2013.
Mohammadi, M., Ghayour, M., and Farajpour, A., “Free Transverse Vibration Analysis of Circular and Annular Graphene Sheets with Various Boundary Conditions Using the Nonlocal Continuum Plate Model”, Composites Part B: Engineering, 45(1), 32-42, 2013.
Najafi, K., and Suzuki, K., “A Novel Technique and Structure for the Measurement of Intrinsic Stress and Young's Modulus of Thin Films, In Micro Electro Mechanical Systems, Proceedings”, An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots. IEEE, 96-97, 1989.
Nayfeh, A. H., and Younis, M. I., “A New Approach to the Modeling and Simulation of Flexible Microstructures Under the Effect of Squeeze-Film Damping”, Journal of Micromechanics and Microengineering, 14(2), 170-181, 2004.
Najar, F., El-Borgi, S., Reddy, J. N., and Mrabet, K., “Nonlinear Nonlocal Analysis of Electrostatic Nanoactuators”, Composite Structures, 120, 117-128, 2015.
O'Mahony, C., Hill, M., Duane, R., and Mathewson, A., “Analysis of Electromechanical Boundary Effects on the Pull-in of Micromachined Fixed–fixed Beams”, Journal of Micromechanics and Microengineering, 13(4), S75, 2003.
Osterberg, P. M., “Electrostatically Actuated Microelectromechancial Test Structures for Material Property Measurement”, Doctoral Dissertation, Massachusetts Institute of Technology, 1995.
Osterberg, P. M., and Senturia, S. D., “M-test: a Test Chip for MEMS Material Property Measurement Using Electrostatically Actuated Test Structures”, Journal of Microelectromechanical Systems, 6(2), 107-118, 1997.
Osterberg, P. M., Yie, H., Cai, X., White, J., and Senturia, S. D., “Self-Consistent Simulation and Modeling of Electrostatically Deformed Diaphragms, In Micro Electro Mechanical Systems”, MEMS'94, Proceedings, IEEE Workshop on, 28-32, 1994.
Pamidighantam, S., Puers, R., Baert, K., and Tilmans, H. A., “Pull-in Voltage Analysis of Electrostatically Actuated Beam Structures with Fixed-Fixed and Fixed-Free End Conditions”, Journal of Micromechanics and Microengineering, 12(4), 458-464, 2002.
Peddieson, J., Buchanan, G. R., and McNitt, R. P., “Application of Nonlocal Continuum Models to Nanotechnology”, International Journal of Engineering Science, 41(3), 305-312, 2003.
Ramezani, A., Alasty, A., and Akbari, J., “Closed-form solutions of the Pull-in Instability in Nano-cantilevers Under Electrostatic and Intermolecular Surface Forces”, International Journal of Solids and Structures, 44(14), 4925-4941, 2007.
Reddy, J. N., “Nonlocal Theories for Bending, Buckling and Vibration of Beams”, International Journal of Engineering Science, 45(2), 288-307, 2007.
Reddy, J. N., “Nonlocal Nonlinear Formulations for Bending of Classical and Shear Deformation Theories of Beams and Plates”, International Journal of Engineering Science, 48(11), 1507-1518, 2010.
Reddy, J. N., and El-Borgi, S., “Eringen’s Nonlocal Theories of Beams Accounting for Moderate Rotations”, International Journal of Engineering Science, 82, 159-177, 2014.
Roque, C. M. C., Ferreira, A. J. M., and Reddy, J. N., “Analysis of Timoshenko Nanobeams with a Nonlocal Formulation and Meshless Method”, International Journal of Engineering Science, 49(9), 976-984, 2010.
Younis, M. I., “MEMS Linear and Nonlinear Statics and Dynamics”, Springer Science and Business Media, 2011.
Yang, J., Jia, X. L., and Kitipornchai, S., “Pull-inInstability of Nano-switches Using Nonlocal Elasticity Theory”, Journal of Physics D: Applied Physics, 41(3), 035103, 2008.
劉晉嘉，「混合微分轉換與有限差分法在非線性靜電驅動微結構系統動態特性之分析」，國立成功大學機械工程研究所碩士論文，2009。
連致淵，「應用微分轉換與有限差分法在微結構樑與微平板之動態特性之分析」，國立成功大學機械工程研究所碩士論文，2013。
趙家奎，「微分轉換及其在電路中的應用」，華中理工大學出版社，1986。
陳柏佑，「混合微分轉換/有限差分法在非線性暫態熱傳問題之研究」，2005。