Electrostatic-driven microelectromechanical systems devices, in most cases, consist of couplings of such energy domains as electromechanics, optical electricity, thermoelectricity, and electromagnetism. Their nonlinear working state makes their analysis complex and complicated. This article introduces the physical model of pull-in voltage, dynamic characteristic analysis, air damping effect, reliability, numerical modeling method, and application of electrostatic-driven MEMS devices.
References
[1]
Akar, O.; Akin, T.; Najafi, K. A wireless batch sealed absolute capacitive pressure sensor. Sensor Actuator A-Phys？2001, 95, 29–38, doi:10.1016/S0924-4247(01)00753-1.
[2]
Schonhardt, S.; Korvink, J.G.; Mohr, J.; Hollenbach, U.; Wallrabe, U. Optimization of an electromagnetic comb drive actuator. Sensor Actuator A-Phys？2009, 154, 212–217, doi:10.1016/j.sna.2008.08.007.
[3]
Jang, L.S.; Kan, W.H.; Chen, M.K.; Chou, Y.M. Parameter extraction from BVD electrical model of PZT actuator of micropumps using time-domain measurement technique. Microfluid. Nanofluid？2009, 7, 559–568, doi:10.1007/s10404-009-0416-7.
[4]
Kamisuki, S.; Fujii, M.; Takekoshi, T.; Tezuka, C.; Atobe, M. A high resolution, electrostatically-driven commercial inkjet head. Proceedings of the IEEE Thirteenth Annual International Conference on MEMS, Miyazaki, Japan, 23–27 January 2000; pp. 793–798.
[5]
Nguyen, C.T.C.; Katehi, L.P.B.; Rebeiz, G.M. Micromachined devices for wireless communications. Proc. IEEE？1998, 86, 1756–1768, doi:10.1109/5.704281.
Krylov, S. The pull-in behavior of electrostatically actuated bistable microstructures. J. Micromech. Microeng？2008, 18, 055026, doi:10.1088/0960-1317/18/5/055026.
[8]
Osterberg, P.M.; Senturia, S.D. M-TEST: A test chip for MEMS material property measurement using electrostatically actuated test structures. J. Microelectromech. Syst？1997, 6, 107–118, doi:10.1109/84.585788.
[9]
Castaner, L.M.; Senturia, S.D. Speed-energy optimization of electrostatic actuators based on pull-in. J. Microelectromech. Syst？1999, 8, 290–298, doi:10.1109/84.788633.
[10]
Castaner, L.; Rodriguez, A.; Pons, J.; Senturia, S.D. Pull-in time-energy product of electrostatic actuators: Comparison of experiments with simulation. Sensor Actuator A-Phys？2000, 83, 263–269, doi:10.1016/S0924-4247(99)00394-5.
[11]
Fargas-Marques, A. Resonant pull-in condition in parallel-plate electrostatic actuators. J. Microelectromech. Syst？2007, 16, 1044, doi:10.1109/JMEMS.2007.900893.
[12]
Krylov, S.; Seretensky, S.; Schreiber, D. Pull-in behavior and multistability of a curved microbeam actuated by a distributed electrostatic force. Proceedings of MEMS 2008: The 21st IEEE International Conference on Micro Electro Mechanical System, Tucson, AZ, USA, 13–17 January 2008; pp. 499–502.
[13]
Krylov, S. Higher order correction of electrostatic pressure and its influence on the pull-in behavior of microstructures. J. Micromech. Microeng？2006, 16, 1382, doi:10.1088/0960-1317/16/7/036.
[14]
Krylov, S.; Maimon, R. Pull-in dynamics of an elastic beam actuated by continuously distributed electrostatic force. J. Vibr. Acoust. Trans. ASME？2004, 126, 332–342, doi:10.1115/1.1760559.
[15]
Zou, Q.; Li, Z.; Liu, L. New methods for measuring mechanical properties of thin films in micromachining: beam pull-in voltage method and long beam deflection (LBD) method. Sensor Actuator A-Phys？1995, 48, 137–143, doi:10.1016/0924-4247(95)00987-6.
[16]
Degani, O.; Socher, E.; Lipson, A.; Leitner, T.; Setter, D.J.; Kaldor, S.; Nemirovsky, Y. Pull-in study of an electrostatic torsion microactuator. J. Microelectromech. Syst？1998, 7, 373–379, doi:10.1109/84.735344.
[17]
Degani, O.; Nemirovsky, Y. Design considerations of rectangular electrostatic torsion actuators based on new analytical pull-in expressions. J. Microelectromech. Syst？2002, 11, 20–26, doi:10.1109/84.982859.
[18]
Nadal-Guardia, R.; Dehe, A.; Aigner, R.; Castaner, L.M. Current drive methods to extend the range of travel of electrostatic microactuators beyond the voltage pull-in point. J. Microelectromech. Syst？2002, 11, 255–263, doi:10.1109/JMEMS.2002.1007404.
[19]
Pamidighantam, S.; Puers, R.; Baert, K.; Tilmans, H.A.C. Pull-in voltage analysis of electrostatically actuated beam structures with fixed-fixed and fixed-free end conditions. J. Micromech. Microeng？2002, 12, 458–464, doi:10.1088/0960-1317/12/4/319.
[20]
Wei, L.C.; Mohammad, A.B.; Kassim, N.M. Analytical modeling for determination of pull-in voltage for an electrostatic actuated MEMS cantilever beam. Proceedings of IEEE International Conference on Consumer Electronics, Sydney, Australia, June 2002; pp. 233–238.
[21]
Xiao, Z.; Peng, W.; Wu, X.; Farmer, K.R. Pull-in study for round double-gimbaled electrostatic torsion actuators. J. Micromech. Microeng？2002, 12, 77–81, doi:10.1088/0960-1317/12/1/312.
[22]
O’Mahony, C.; Hill, M.; Duane, R.; Mathewson, A. Analysis of electromechanical boundary effects on the pull-in of micromachined fixed-fixed beams. J. Micromech. Microeng？2003, 13, S75–S80, doi:10.1088/0960-1317/13/4/312.
[23]
Cheng, J.; Zhe, J.; Wu, X. Analytical and finite element model pull-in study of rigid and deformable electrostatic microactuators. J. Micromech. Microeng？2004, 14, 57–68, doi:10.1088/0960-1317/14/1/308.
[24]
Chowdhury, S.; Ahmadi, M.; Miller, W.C. A closed-form model for the pull-in voltage of electrostatically actuated cantilever beams. J. Micromech. Microeng？2005, 15, 756–763, doi:10.1088/0960-1317/15/4/012.
[25]
Chao, P.C.P.; Chiu, C.W.; Tsai, C.Y. A novel method to predict the pull-in voltage in a closed form for micro-plates actuated by a distributed electrostatic force. J. Micromech. Microeng？2006, 16, 986–998, doi:10.1088/0960-1317/16/5/016.
[26]
Chowdhury, S.; Ahmadi, M.; Miller, W.C. Pull-in voltage study of electrostatically actuated fixed-fixed beams using a VLSI on-chip interconnect capacitance model. J. Microelectromech. Syst？2006, 15, 639–651, doi:10.1109/JMEMS.2005.863784.
[27]
Elata, D.; Bamberger, H. On the dynamic pull-in of electrostatic actuators with multiple degrees of freedom and multiple voltage sources. J. Microelectromech. Syst？2006, 15, 131–140, doi:10.1109/JMEMS.2005.864148.
[28]
Gorthi, S.; Mohanty, A.; Chatterjee, A. Cantilever beam electrostatic MEMS actuators beyond pull-in. J. Micromech. Microeng？2006, 16, 1800–1810, doi:10.1088/0960-1317/16/9/007.
[29]
Hu, Y.C. Closed form solutions for the pull-in voltage of micro curled beams subjected to electrostatic loads. J. Micromech. Microeng？2006, 16, 648–655, doi:10.1088/0960-1317/16/3/021.
[30]
Hu, Y.C.; Lin, D.T.W.; Lee, G.D. In A closed form solution for the pull-in voltage of the micro bridge with initial stress subjected to electrostatic loads. Proceedings of 1st IEEE International Conference on Nano/Micro Engineered and Molecular Systems, Zhuhai, China, 18–21 January 2006; pp. 757–761.
[31]
Stefano, L.; Giuseppe, R. Control of pull-in dynamics in a nonlinear thermoelastic electrically actuated microbeam. J. Micromech. Microeng？2006, 16, 390, doi:10.1088/0960-1317/16/2/025.
[32]
Zhang, Y.; Zhao, Y.P. Numerical and analytical study on the pull-in instability of micro-structure under electrostatic loading. Sensor Actuator A-Phys？2006, 127, 366–380, doi:10.1016/j.sna.2005.12.045.
[33]
Gusso, A.; Delben, G.J. Influence of the Casimir force on the pull-in parameters of silicon based electrostatic torsional actuators. Sensor Actuator A-Phys？2007, 135, 792–800, doi:10.1016/j.sna.2006.09.008.
[34]
Hu, Y.C.; Chang, P.Z.; Chuang, W.C. An approximate analytical solution to the pull-in voltage of a micro bridge with an elastic boundary. J. Micromech. Microeng？2007, 17, 1870–1876, doi:10.1088/0960-1317/17/9/016.
[35]
Hu, Y.C.; Lee, G.D. A closed form solution for the pull-in voltage of the micro bridge. Tamkang. J. Sci. Eng？2007, 10, 147–150.
Alsaleem, F.A.; Younis, M.I.; Ouakad, H.M. On the nonlinear resonances and dynamic pull-in of electrostatically actuated resonators. J. Micromech. Microeng？2009, 19, 045013, doi:10.1088/0960-1317/19/4/045013.
[38]
Zhang, W.M.; Meng, G.; Li, H.G. Electrostatic micromotor and its reliability. Microelectron. Reliab？2005, 45, 1230–1242, doi:10.1016/j.microrel.2004.12.017.
[39]
Quevy, E.; Bigotte, P.; Collard, D.; Buchaillot, L. Large stroke actuation of continuous membrane for adaptive optics by 3D self-assembled microplates. Sensor Actuator A-Phys？2002, 95, 183–195, doi:10.1016/S0924-4247(01)00735-X.
[40]
Chan, E.K.; Dutton, R.W. Electrostatic micromechanical actuator with extended range of travel. J. Microelectromech. Syst？2000, 9, 321–328, doi:10.1109/84.870058.
[41]
Hung, E.S.; Senturia, S.D. Extending the travel range of analog-tuned electrostatic actuators. J. Microelectromech. Syst？1999, 8, 497–505, doi:10.1109/84.809065.
[42]
Legtenberg, R.; Gilbert, J.; Senturia, S.D.; Elwenspoek, M. Electrostatic curved electrode actuators. J. Microelectromech. Syst？1997, 6, 257–265, doi:10.1109/84.623115.
[43]
Ye, W.J.; Mukherjee, S. Optimal shape design of three-dimensional MEMS with applications to electrostatic comb drives. Int. J. Numer. Methods Eng？1999, 45, 175–194, doi:10.1002/(SICI)1097-0207(19990520)45:2<175::AID-NME585>3.0.CO;2-7.
[44]
Ye, W.J.; Mukherjee, S.; MacDonald, N.C. Optimal shape design of an electrostatic comb drive in microelectromechanical systems. J. Microelectromech. Syst？1998, 7, 16–26, doi:10.1109/84.661380.
[45]
Kuang, J.H.; Chen, C.J. The nonlinear electrostatic behavior for shaped electrode actuators. Int. J. Mech. Sci？2005, 47, 1172–1190, doi:10.1016/j.ijmecsci.2005.04.006.
[46]
Rhoads, J.; Shaw, S.W.; Turner, K.L. Nonlinear Dynamics and Its Applications in Micro- and Nano-Resonators. Proceedings of DSCC 2008: The 2008 ASME Dynamic Systems and Control Conference, Ann Arbor, Michigan, USA, 20–22 October 2008.
[47]
Busta, H.; Amantea, R.; Furst, D.; Chen, J.M.; Turowski, M.; Mueller, C. A MEMS shield structure for controlling pull-in forces and obtaining increased pull-in voltages. J. Micromech. Microeng？2001, 11, 720–725, doi:10.1088/0960-1317/11/6/315.
[48]
Bao, M.H.; Yang, H.; Yin, H.; Shen, S.Q. Effects of electrostatic forces generated by the driving signal on capacitive sensing devices. Sensor Actuator A-Phys？2000, 84, 213–219, doi:10.1016/S0924-4247(00)00312-5.
[49]
McCarthy, B.; Adams, G.G.; McGruer, N.E.; Potter, D. A dynamic model, including contact bounce, of an electrostatically actuated microswitch. J. Microelectromech. Syst？2002, 11, 276–283, doi:10.1109/JMEMS.2002.1007406.
[50]
Lin, R.M.; Wang, W.J. Structural dynamics of microsystems—Current state of research and future directions. Mech. Syst. Sig. Process？2006, 20, 1015–1043, doi:10.1016/j.ymssp.2005.08.013.
[51]
Mahmoodi, S.N. Modeling, nonlinear dynamics, and identification of a piezoelectrically actuated microcantilever sensor. IEEE/ASME Trans. Mechatron？2008, 13, 58, doi:10.1109/TMECH.2008.915823.
[52]
Zhang, W.M.; Meng, G. Nonlinear dynamic analysis of electrostatically actuated resonant MEMS sensors under parametric excitation. IEEE Sens. J？2007, 7, 370–380, doi:10.1109/JSEN.2006.890158.
[53]
Ai, S.B.; Pelesko, J.A. Dynamics of a canonical electrostatic MEMS/NEMS system. J. Dyn. Differ. Equ？2008, 20, 609–641, doi:10.1007/s10884-007-9094-x.
[54]
Leus, V.; Elata, D. On the dynamic responses of electrostatic MEMS switches. J. Microelectromech. Syst？2008, 17, 236–243, doi:10.1109/JMEMS.2007.908752.
[55]
Li, L.J.; Uttamchandani, D. Dynamic response modelling and characterization of a vertical electrothermal actuator. J. Micromech. Microeng？2009, 19, 075014, doi:10.1088/0960-1317/19/7/075014.
[56]
Fargas, M.A.; Costa, C.R.; Shkel, A.M. Modeling the electrostatic actuation of MEMS: State of the art 2005. IOC-DT-P-2005-18;; Institute of Industrial and Control Engineering: Barcelona, Spain, 2005.
[57]
Batra, R.C.; Porfiri, M.; Spinello, D. Review of modeling electrostatically actuated microelectromechanical systems. Smart Mater. Struct？2007, 16, R23–R31, doi:10.1088/0964-1726/16/6/R01.
[58]
Bao, M.; Yang, H. Squeeze film air damping in MEMS. Sensor Actuator A-Phys？2007, 136, 3–27, doi:10.1016/j.sna.2007.01.008.
[59]
Veijola, T. Equivalent-circuit model of the squeezed gas film in a silicon accelerometer. Sensor Actuator A-Phys？1995, 48, 239, doi:10.1016/0924-4247(95)00995-7.
[60]
Wang, X.; Liu, Y.; Wang, M.; Chen, X. Study on the squeezed air-damping on MEMS parallel plates. China Mech. Eng？2005, 16, 1276–1278.
[61]
Wang, X.; Wang, M.; Liu, Y. The Analysis of Squeezed Air-Damping on the Planar MEMS Structures; American Society of Mechanical Engineers: New York, NY, USA, 2005; pp. 811–812.
[62]
Feng, C.; Zhao, Y.; Liu, D. Air damping in micro-beam resonators. J. Univ. Sci. Technol. Beijing？2007, 29, 841–844.
[63]
Blech, J.J. On isothermal squeeze films. J. Lubr. Technol. (Trans. ASME)？1983, 105, 615–620, doi:10.1115/1.3254692.
[64]
Lee, J.W.; Tung, R.; Raman, A.; Sumali, H.; Sullivan, J.P. Squeeze-film damping of flexible microcantilevers at low ambient pressures: Theory and experiment. J. Micromech. Microeng？2009, 19, 105029, doi:10.1088/0960-1317/19/10/105029.
[65]
Mol, L.; Rocha, L.A.; Cretu, E.; Wolffenbuttel, R.F. Squeezed film damping measurements on a parallel-plate MEMS in the free molecule regime. J. Micromech. Microeng？2009, 19, 074021, doi:10.1088/0960-1317/19/7/074021.
[66]
Suijlen, M.A.J.; Koning, J.J.; van Gils, M.A.J.; Beijerinck, H.C.W. Squeeze film damping in the free molecular flow regime with full thermal accommodation. Sensor Actuator A-Phys？2009, 156, 171–179, doi:10.1016/j.sna.2009.03.025.
[67]
Luo, A.C.J.; Wang, F.-Y. Nonlinear dynamics of a micro-electro-mechanical system with time-varying capacitors. J. Vib. Acoust？2004, 126, 77–83, doi:10.1115/1.1597211.
[68]
Liu, S.; Davidson, A.; Lin, Q. Simulation studies on nonlinear dynamics and chaos in a MEMS cantilever control system. J. Micromech. Microeng？2004, 14, 1064–1073, doi:10.1088/0960-1317/14/7/029.
[69]
Li, G.; Aluru, N.R. Linear, nonlinear and mixed-regime analysis of electrostatic MEMS. Sensor Actuator A-Phys？2001, 91, 278–291, doi:10.1016/S0924-4247(01)00597-0.
[70]
Zhang, W.M.; Meng, G. Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS. Sensor Actuator A-Phys？2005, 119, 291–299, doi:10.1016/j.sna.2004.09.025.
[71]
Adams, S.G.; Bertsch, F.M.; Shaw, K.A.; MacDonald, N.C. Independent tuning of linear and nonlinear stiffness coefficients [actuators]. J. Microelectromech. Syst？1998, 7, 172–180, doi:10.1109/84.679344.
[72]
Zhang, W.H.; Baskaran, R.; Tumer, K.L. Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor. Sensor Actuator A-Phys？2002, 102, 139–150, doi:10.1016/S0924-4247(02)00299-6.
[73]
Lee, S.; Kim, J.; Moon, W.; Choi, J.; Park, I.; Bae, D. A multibody-based dynamic simulation method for electrostatic actuators. Nonlinear Dyn？2008, 54, 53–68, doi:10.1007/s11071-007-9268-4.
[74]
Aluru, N.R.; White, J. Efficient numerical technique for electromechanical simulation of complicated microelectromechanical structures. Sensor Actuator A-Phys？1997, 58, 1–11, doi:10.1016/S0924-4247(97)80218-X.
[75]
Younis, M.I.; Abdel-Rahman, E.M.; Nayfeh, A. A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst？2003, 12, 672–680, doi:10.1109/JMEMS.2003.818069.
[76]
Lee, Y.C.; McCarthy, B.; Diao, J.; Zhang, Z.; Harsh, K.F. Computer-aided design for microelectromechanical systems (MEMS). Int. J. Mater. Prod. Technol？2003, 18, 356–380.
[77]
Senturia, S.D.; Harris, R.M.; Johnson, B.P.; Kim, S.; Nabors, K.; Shulman, M.A.; White, J.K. A computer-aided design system for microelectromechanical systems (MEMCAD). J. Microelectromech. Syst？1992, 1, 3–13, doi:10.1109/84.128049.
[78]
De, S.K.; Aluru, N.R. Full-Lagrangian schemes for dynamic analysis of electrostatic MEMS. J. Microelectromech. Syst？2004, 13, 737–758, doi:10.1109/JMEMS.2004.835773.
[79]
Li, G.Y.; Wang, L. Influence of bonding parameters on electrostatic force in anodic wafer bonding. Thin Solid Films？2004, 462–463, 334–338.
[80]
Hu, Y.C.; Chang, C.M.; Huang, S.C. Some design considerations on the electrostatically actuated microstructures. Sensor Actuator A-Phys？2004, 112, 155–161, doi:10.1016/j.sna.2003.12.012.
[81]
Chowdhury, S.; Ahmadi, M.; Miller, W.C. Proceedings of 2003 International Conference on MEMS, NANO and Smart Systems, Banff, Alberta, Canada, 20–23 July 2003; pp. 297–302.
[82]
Younis, M.I. A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn？2003, 31, 91, doi:10.1023/A:1022103118330.
[83]
Spearing, S.M. Materials issues in microelectromechanical systems (MEMS). Acta. Mater？2000, 48, 179–196, doi:10.1016/S1359-6454(99)00294-3.
[84]
Zhang, W.M.; Meng, G. Contact dynamics between the rotor and bearing hub in an electrostatic micromotor. Microsyst. Technol？2005, 11, 438–443, doi:10.1007/s00542-004-0493-1.
Zhang, W.M.; Meng, G. Stability, bifurcation and chaos of a high-speed rub-impact rotor system in MEMS. Sensor Actuator A-Phys？2006, 127, 163–178, doi:10.1016/j.sna.2005.11.014.
[87]
Ghisi, A.; Fachin, F.; Mariani, S.; Zerbini, S. Multi-scale analysis of polysilicon MEMS sensors subject to accidental drops: Effect of packaging. Microelectron. Reliab？2009, 49, 340–349, doi:10.1016/j.microrel.2008.12.010.
[88]
Ghisi, A.; Kalicinski, S.; Mariani, S.; De Wolf, I.; Corigliano, A. Polysilicon MEMS accelerometers exposed to shocks: Numerical-experimental investigation. J. Micromech. Microeng？2009, 19, 035023, doi:10.1088/0960-1317/19/3/035023.
[89]
Mariani, S.; Ghisi, A.; Corigliano, A.; Zerbini, S. Multi-scale analysis of MEMS sensors subject to drop impacts. Sensers？2007, 7, 1817–1833, doi:10.3390/s7081817.
[90]
Mariani, S.; Ghisi, A.; Corigliano, A.; Zerbini, S. Modeling impact-induced failure of polysilicon MEMS: A multi-scale approach. Sensors？2009, 9, 556–567, doi:10.3390/s90100556. 22389617
[91]
Mariani, S.; Ghisi, A.; Fachin, F.; Cacchione, F.; Corigliano, A.; Zerbini, S. A three-scale FE approach to reliability analysis of MEMS sensors subject to impacts. Meccanica？2008, 43, 469–483, doi:10.1007/s11012-008-9111-0.
[92]
Mariani, S.; Perego, U. Extended finite element method for quasi-brittle fracture. Int. J. Numer. Meth. Eng？2003, 58, 103–126, doi:10.1002/nme.761.
Osterberg, P.; Yie, H.; Cai, X.; White, J.; Senturia, S. Self-consistent simulation and modeling of electrostatically deformed diaphragms. Proceedings of MEMS’94, Oiso, Japan, January 1994; pp. 28–32.
[95]
Lishchynska, M.; Cordero, N.; Slattery, O.; O’Mahony, C. Modelling electrostatic behaviour of microcantilevers incorporating residual stress gradient and non-ideal anchors. J. Micromech. Microeng？2005, 15, S10–S14, doi:10.1088/0960-1317/15/7/002.
[96]
Hu, Y.C.; Wei, C.S. An analytical model considering the fringing fields for calculating the pull-in voltage of micro curled cantilever beams. J. Micromech. Microeng？2007, 17, 61–67, doi:10.1088/0960-1317/17/1/008.
[97]
Chuang, W.C.; Hu, Y.C.; Lee, C.Y.; Shih, W.P.; Chang, P.Z. Electromechanical behavior of the curled cantilever beam. J. Micro/Nanolithogr. MEMS MOEMS？2009, 8, 033020–033028.
Clark, J.V.; Zhou, N.; Pister, K.S.J. Modified Nodal Analysis for MEMS with Multi-Energy Domains; Computational Publications: Cambridge, MA, USA, 2000; pp. 723–726.
[100]
Wen, F.; Li, W.; Rong, H. Creation of the macromodel of equivalent circuit and IP library for clamped-clamped beam microsensor. Res. Prog. Solid State Electr？2004, 24, 68–72.
[101]
Zhang, J.; Li, W.H. Method of system-level dynamic simulation for a MEMS cantilever based on the mechanical description. Chinese J. Sens. Actuators？2006, 19, 1376–1380.
[102]
Yao, J.J.; MacDonald, N.C. A micromachined, single-crystal silicon, tunable resonator. J. Micromech. Microeng？1995, 5, 257, doi:10.1088/0960-1317/5/3/009.
[103]
Veijola, T.; Mattila, T.; Jaakkola, O.; Kiihamaki, J.; Lamminmaki, T.; Oja, A.; Ruokonen, K.; Seppa, H.; Seppala, P.; Tittonen, I. Large-displacement modelling and simulation of micromechanical electrostatically driven resonators using the harmonic balance method. IEEE MTT-S Int. Microwave Symp. Dig？2000, 1, 99–102.
Bergqvist, J.; Gobet, J. Capacitive microphone with a surface micromachined backplate using electroplating technology. J. Microelectromech. Syst？1994, 3, 69–75, doi:10.1109/84.294323.
[106]
Andrews, M.; Harris, I.; Turner, G. A comparison of squeeze-film theory with measurements on a microstructure. Sensor Actuator A-Phys？1993, 36, 79–87, doi:10.1016/0924-4247(93)80144-6.
[107]
Darling, R.B.; Hivick, C.; Xu, J.Y. Compact analytical modeling of squeeze film damping with arbitrary venting conditions using a Green’s function approach. Sensor Actuator A-Phys？1998, 70, 32–41, doi:10.1016/S0924-4247(98)00109-5.
[108]
Schaaf, S.A.; Chambr Paul, L. Flow of Rarefied Gases; Princeton University Press: Princeton, NJ, USA, 1961.
[109]
Li, B.Q.; Wu, H.Y.; Zhu, C.C.; Liu, J.H. The theoretical analysis on damping characteristics of resonant microbeam in vacuum. Sensor Actuator A-Phys？1999, 77, 191–194, doi:10.1016/S0924-4247(99)00072-2.
[110]
Yang, Y.J.; Kamon, M.; Rabinovich, V.L.; Ghaddar, C.; Deshpande, M.; Greiner, K.; Gilbert, J.R. Modeling Gas Damping and Spring Phenomena in MEMS with Frequency Dependent Macro-Models; Institute of Electrical and Electronics Engineers Inc: Interlaken, Switzerland, 2001; pp. 365–368.
[111]
Gao, R.; Wang, X.; Wang, M.; Yu, M.; Xie, M. Nonlinear dynamics of lateral micro-resonator including viscous air damping. Chinese J. Mech. Eng？2007, 20, 75–78.
[112]
Zhu, Y.M.; Jia, J.Y.; Fan, K.Q. Study on the air damp of the plate movement in the MEMS devices. J. Univ. Electr. Sci. Technol. China？2007, 36, 965–968.
[113]
Hung, E.S.; Yang, Y.J.; Senturia, S.D. Low-order models for fast dynamical simulation of MEMS microstructures. Proceedings of the IEEE International Conference on Solid State Sensors and Actuators, Chicago, IL, USA, 16–19 June 1997; pp. 1101–1104.
[114]
Stewart, J.T. Finite element modeling of resonant microelectromechanical structures for sensing applications. Proceedings of Micromachined Devices and Components, Austin, TX, USA, 23–24 October 1995; pp. 643–646.
[115]
Swart, N.R.; Bart, S.F.; Zaman, M.H.; Mariappan, M.; Gilbert, J.R.; Murphy, D. AutoMM: Automatic generation of dynamic macromodels for MEMS devices. Proceedings IEEE MEMS Workshop, Heidelberg, Germany, 25–29 January 1998; pp. 178–183.
[116]
Mukherjee, T. Emerging simulation approaches for micromachined devices. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst？2000, 19, 1572, doi:10.1109/43.898833.
[117]
Zhang, W.M.; Meng, G.; Chen, D. Stability, nonlinearity and reliability of electrostatically actuated MEMS devices. Sensors？2007, 7, 760–796, doi:10.3390/s7050760.
[118]
van Spengen, W.M.; Puers, R.; De Wolf, I. A physical model to predict stiction in MEMS. J. Micromech. Microeng？2002, 12, 702–713, doi:10.1088/0960-1317/12/5/329.
Komvopoulos, K. Surface engineering and microtribology for microelectromechanical systems. Wear？1996, 200, 305–327, doi:10.1016/S0043-1648(96)07328-0.
[126]
Bart, S.F.; Mehregany, M.; Tavrow, L.S.; Lang, J.H.; Senturia, S.D. Electric micromotor dynamics. IEEE Trans. Electron Devices？1992, 39, 566–575, doi:10.1109/16.123479.
[127]
Tai, Y.C.; Muller, R.S. Frictional study of IC-processed micromotors. Sensor Actuator A-Phys？1990, 21, 180–183, doi:10.1016/0924-4247(90)85034-2.
[128]
Ren, S.L.; Yang, S.R.; Wang, J.Q.; Liu, W.M.; Zhao, Y.P. Preparation and tribological studies of stearic acid self-assembled monolayers on polymer-coated silicon surface. Chem. Mater？2004, 16, 428–434, doi:10.1021/cm0345669.
[129]
Hu, Y.C.; Huang, P.Y. Wafer-level microelectromechanical system structural material test by electrical signal before pull-in. Appl. Phys. Lett？2007, 90, 121908, doi:10.1063/1.2714998.
[130]
Hu, Y.C.; Shih, W.P.; Lee, G.D. A method for mechanical characterization of capacitive devices at wafer level via detecting the pull-in voltages of two test bridges with different lengths. J. Micromech. Microeng？2007, 17, 1099–1106, doi:10.1088/0960-1317/17/6/001.
[131]
Hu, Y.C.; Shih, W.P.; Lee, G.D. A method for mechanical characterization of capacitive devices at wafer level via detecting the pull-in voltages of two test bridges with different lengths. J. Micromech. Microeng？2007, 17, 1099–1106, doi:10.1088/0960-1317/17/6/001.
[132]
Gorman, D.J. Free Vibration Analysis of Beams and Shafts; John Wiley & Sons: Hoboken, NJ, USA, 1975.
