Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v6.i6.50
pages 563-584

Parametric Excitation and Stabilization of Electrostatically Actuated Microstructures

Slava Krylov
School of Mechanical Engineering, Tel Aviv University, Israel

ABSTRACT

The parametric instability of double-clamped microscale beams actuated by a time-varying distributed electrostatic force provided by two electrodes symmetrically located at two sides of the beam and subjected to nonlinear squeeze film damping is investigated. A reduced-order model is built using the Galerkin decomposition with undamped linear modes as base functions. The stability analysis is performed by evaluating the sign of the largest Lyapunov exponent, which defines the character of the response. It is shown that this approach provides an efficient quantitative criterion for the evaluation of parametric instability, especially when combined with compact reduced-order models. Based on the Lyapunov exponent criterion, the influence of various parameters on the beam dynamic stability is investigated. We show that application of a time-dependent (ac) voltage in addition to a steady (dc) voltage exceeding the static stability limit may have a stabilizing influence while the structure, in accordance with the Lyapunov exponent criterion, remains stable. Parametric stabilization considered in this work represents an example of the strong influence of the fast-scale excitation on the slow-scale behavior.

REFERENCES

  1. Senturia, S. D., Microsystems Design.

  2. Pelesko, J. A. and Bernstein, D. H., Modeling of MEMS and NEMS.

  3. Nathanson, H. C., Newell, W. E., Wickstrom, R. A., and Davis, J. R., The Resonant Gate Transistor. DOI: 10.1109/T-ED.1967.15912

  4. Abdel-Rahman, E. M., Younis, M. I., and Nayfeh, A. H., Characterization of the Mechanical Behavior of an Electrically Actuated Microbeam. DOI: 10.1088/0960-1317/12/6/306

  5. Abdalla, M. M., Reddy, C. K., Faris, W. F., and Gurdal, Z., Optimal Design of an Electrostatically Actuated Microbeam for Maximum Pull-In Voltage. DOI: 10.1016/j.compstruc.2004.07.010

  6. Batra, R. C., Porfiri, M., and Spinello, D., Analysis of Electrostatic MEMS Using Meshless Local Petrov-Galerkin (MLPG) Method. DOI: 10.1016/j.enganabound.2006.04.008

  7. Chowdhury, S., Ahmadi, M., and Miller, W. C., A Closed-Form Model for the Pull-In Voltage of Electrostatically Actuated Cantilever Beams. DOI: 10.1088/0960-1317/15/4/012

  8. Elata, D., Bochobza-Degani, O., and Nemirovsky, Y., Analytical Approach and Numerical α-Lines Method for Pull-In Hyper-Surface Extraction of Electrostatic Actuators with Multiple Uncoupled Voltage Sources. DOI: 10.1109/JMEMS.2003.818456

  9. Li, G. and Aluru, N. R., A Lagrangian Approach for Electrostatic Analysis of Deformable Conductors.

  10. Li, G. and Aluru, N. R., Efficient Mixed- Domain Analysis of Electrostatic MEMS. DOI: 10.1109/ICCAD.2002.1167574

  11. Krylov, S. and Seretensky, S., Higher Order Corrections of Electrostatic Pressure and its Influence on the Pull-In Behavior of Microstructures. DOI: 10.1088/0960-1317/16/7/036

  12. Krylov, S., Ilic, B. R., Schreiber, D., Seretensky, S., and Craighead, H., Pull-In Behavior of Electrostatically Actuated Bistable Microstructures. DOI: 10.1088/0960-1317/18/5/055026

  13. Pamidighantam, S., Puers, R., Baert, K., and Tilmans, H. A. C., Pull-In Voltage Analysis of Electrostatically Actuated Beam Structures with Fixed-Fixed and Fixed-Free end Conditions. DOI: 10.1088/0960-1317/12/4/319

  14. Rochus, V., Rixen, D. J., and Golinval, J.-C., Monolithic Modelling of Electro-Mechanical Coupling in Micro-Structures. DOI: 10.1002/nme.1450

  15. Osterberg, P. M. and Senturia, S. D., M-Test: A Test Chip for MEMS Material Property Measurement Using Electrostatically Actuated Test Structures. DOI: 10.1109/84.585788

  16. Yang, F., Electromechanical Instability of Microscale Structures. DOI: 10.1063/1.1496123

  17. Younis, M. I., Abdel-Rahman, E. M., and Nayfeh, A. H., A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS. DOI: 10.1109/JMEMS.2003.818069

  18. Zhang, Y. and Zhao, Y., Numerical and Analytical Study on the Pull-In Instability of Micro-Structure under Electrostatic Loading. DOI: 10.1016/j.sna.2005.12.045

  19. Batra, R. C., Porfiri, M., and Spinello, D., Review of Modeling Electrostatically Actuated Microelectromechanical Systems. DOI: 10.1088/0964-1726/16/6/R01

  20. Fargas-Marques, A., Costa, C. R., and Shkel, A. M., Modelling the Electrostatic Actuation of MEMS: State of the Art 2005.

  21. Oberhammer, J., Liu, A. Q., and Stemme, G., Time-Efficient Quasi-Static Algorithm for Simulation of Complex Single-Sided Clamped Electrostatic Actuators. DOI: 10.1109/JMEMS.2007.892917

  22. Rebeiz, G. M., RF MEMS: Theory, Design, and Technology.

  23. Castaner, L. M. and Senturia, S. D., Speed- Energy Optimization of Electrostatic Actuators Based on Pull-In. DOI: 10.1109/84.788633

  24. Gupta, K. G. and Senturia, S. D., Pull-In Dynamics as a Measure of the Ambient Pressure. DOI: 10.1109/MEMSYS.1997.581830

  25. Gabbay, L. D., Mehner, J. E., and Senturia, S. D., Computer-Aided Generation of Nonlinear Reduced-Order Dynamic Macromodels-II: Stress-Stiffened Case. DOI: 10.1109/84.846708

  26. Krylov, S., Lyapunov Exponents as a Criterion for the Dynamic Pull-In Instability of Electrostatically Actuated Microstructures. DOI: 10.1016/j.ijnonlinmec.2007.01.004

  27. Hung, E. S. and Senturia, S. D., Generating Efficient Dynamical Models for Microelectromechanical Systems from a Few Finite-Element Simulation Runs. DOI: 10.1109/84.788632

  28. De, S. K. and Aluru, N. R., Full-Lagrangian Schemes for Dynamic Analysis of Electrostatic MEMS. DOI: 10.1109/JMEMS.2004.835773

  29. Flores, G., Mercado, G. A., and Pelesko, J. A., Dynamics and Touchdown in Electrostatic MEMS. DOI: 10.1109/ICMENS.2003.1221990

  30. Ai, S. and Pelesko, J. A., Dynamics of a Canonical Electrostatic MEMS/NEMS System. DOI: 10.1007/s10884-007-9094-x

  31. Krylov, S. and Maimon, R., Pull-In Dynamics of an Elastic Beam Actuated by Continuously Distributed Electrostatic Force. DOI: 10.1115/1.1760559

  32. Rocha, L. A., Cretu, E., andWolffenbuttel, R. F., Behavioural Analysis of the Pull-In Dynamic Transition. DOI: 10.1088/0960-1317/14/9/006

  33. McCarthy, B., Adams, G. G., McGruer, N. E., and Potter, D., A Dynamic Model, Including Contact Bounce, of an Electrostatically Actuated Microswitch. DOI: 10.1109/JMEMS.2002.1007406

  34. Lenci, S. and Rega, G., Control of Pull-In Dynamics in a Nonlinear Thermoelastic Electrically Actuated Microbeam. DOI: 10.1088/0960-1317/16/2/025

  35. Luo, A. C. J. andWang, F. Y., Chaotic motion in a Micro-Electro-Mechanical System with Non- Linearity from Capacitors. DOI: 10.1016/S1007-5704(02)00005-9

  36. Rocha, L. A., Cretu, E., and F. Wolffenbuttel, R. F., Analysis and Analytical Modeling of Static Pull-In with Application to MEMSBased Voltage Reference and Process Monitoring. DOI: 10.1109/JMEMS.2004.824892

  37. Elata, D. and Bamberger, H., On the Dynamic Pull-In of Electrostatic Actuators with Multiple Degrees of Freedom and Multiple Voltage Sources. DOI: 10.1109/JMEMS.2005.864148

  38. Leus, V. and Elata, D., On the Dynamic Responses of Electrostatic MEMS Switches. DOI: 10.1109/JMEMS.2007.908752

  39. Nielson, G. N. and Barbastathis, G., Dynamic Pull-In of Parallel-Plate and Torsional Electrostatic MEMS Actuators. DOI: 10.1109/JMEMS.2006.879121

  40. De, S. K. and Aluru, N. R., Complex Oscillations and Chaos in Electrostatic Microelectromechanical Systems under Superharmonic Excitations.

  41. Nayfeh, A. H. and Younis, M. I., Dynamics of MEMS Resonators under Superharmonic and Subharmonic Excitations. DOI: 10.1088/0960-1317/15/10/008

  42. Shi, F., Ramesh, P., and Mukherjee, S., Dynamic Analysis of Micro-Electro-Mechanical Systems. DOI: 10.1002/(SICI)1097-0207(19961230)39:24<4119::AID-NME42>3.0.CO;2-4

  43. Younis, M. I. and Nayfeh, A. H., A Study of the Nonlinear Rresponse of a Resonant Microbeam to an Electric Actuation. DOI: 10.1023/A:1022103118330

  44. Wang, P. K. C., Feedback Control of Vibrations in a Micromachined Cantilever Beam with Electrostatic Actuators. DOI: 10.1006/jsvi.1998.1525

  45. Fargas-Marques, A., Casals-Terte, J., and Shkel, A. M., Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators. DOI: 10.1109/JMEMS.2007.900893

  46. Nayfeh, A. H., Younis, M. I., and Abdel- Rahman, E. M., Dynamic Pull-In Phenomenon in MEMS Resonators. DOI: 10.1007/s11071-006-9079-z

  47. Turner, K. L., Miller, S. A., Hartwell, P. G., MacDonald, N. C., Strogatz, S. H., and Adams, S. G., Five Parametric Resonances in a Micromechanical System.

  48. Baskaran, R. and Turner, K. L., Mechanical Domain Coupled Mode Parametric Resonance and Amplification in a Torsional Mode Micro Electro Mechanical Oscillator. DOI: 10.1088/0960-1317/13/5/323

  49. Buks, E. and Roukes, M. L., Electrically Tunable Collective Response in a Coupled Micromechanical Array. DOI: 10.1109/JMEMS.2002.805056

  50. Hu, Y. C., Chang, C. M., and Huang, S. C., Some Design Considerations on the Electrostatically Actuated Microstructures. DOI: 10.1016/j.sna.2003.12.012

  51. Krylov, S., Harari, I., and Cohen, Y., Stabilisation of Electrostatically Actuated Microstructures Using Parametric Excitation. DOI: 10.1088/0960-1317/15/6/009

  52. Lifshitz, R. and Cross, M. C., Response of Parametrically Driven Nonlinear Coupled Oscillators with Application to Micromechanical and Nanomechanical Resonator Arrays. DOI: 10.1103/PhysRevB.67.134302

  53. Mestrom, R. M. C., Fey, R. H. B., van Beek, J. T. M., Phan, K. L., and Nijmeijer, H., Modelling the Dynamics of a MEMS Resonator: Simulations and Experiments. DOI: 10.1016/j.sna.2007.04.025

  54. Napoli, M., Baskaran, R., Turner, K. L., and Bamieh, B., Understanding Mechanical Domain Parametric Resonance in Microcantilevers. DOI: 10.1109/MEMSYS.2003.1189713

  55. Rhoads, J. F., Shaw, S.W., and Turner, K. L., The Nonlinear Response of Resonant Microbeam Systems with Purely-Parametric Electrostatic Actuation. DOI: 10.1088/0960-1317/16/5/003

  56. Ruar, D. and Grutter, P., Mechanical Parametric Amplification and Thermomechanical Noise Squeezing. DOI: 10.1103/PhysRevLett.67.699

  57. Lifshitz, R. and Cross, M. C., Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators. DOI: 10.1002/9783527626359

  58. Zhang, W., Baskaran, R., and Turner, K. L., Effect of Cubic Nonlinearity on Auto-Parametrically, Amplified Resonant MEMS Mass Sensor. DOI: 10.1016/S0924-4247(02)00299-6

  59. Zhang, W. and Turner, K. L., Application of Parametric Resonance Amplification in a Single-Crystal Silicon Micro-Oscillator Based Mass Sensor. DOI: 10.1016/j.sna.2004.12.033

  60. Younis, M. I. and Al Saleem, F. M., Switch Triggered by Mass Threshold.

  61. Oropeza-Ramos, L. A., Burgner, C. B., Olroyd, C., and Turner, K. L., Characterization of a Novel MEM Gyroscope Actuated by Parametric Resonance. DOI: 10.1115/DETC2007-35559

  62. Elata, D. and Leus, V., How Slender Can Comb-Drive Fingers be?. DOI: 10.1088/0960-1317/15/5/023

  63. Elata, D. and Abu-Salih, S., Analysis of a Novel Method for Measuring Residual Stress in Micro-Systems. DOI: 10.1088/0960-1317/15/5/004

  64. Abu-Salih, S. and Elata, D., Experimental Validation of Electromechanical Buckling. DOI: 10.1109/JMEMS.2006.886015

  65. Goldhirsch, I., Sulem, P. L., and Orszag, S. A., Stability and Lyapunov Stability of Dynamical Systems: A Differential Approach and a Numerical Method. DOI: 10.1016/0167-2789(87)90034-0

  66. Gilat, R. and Aboudi, J., Parametric Stability of Nonlinearly-Elastic Composite Plates by Lyapunov Exponents. DOI: 10.1006/jsvi.2000.2936

  67. Gilat, R. and Aboudi, J., The Lyapunov Exponents as a Quantitative Criterion for the Dynamic Buckling of Composite Plates. DOI: 10.1016/S0020-7683(01)00108-1

  68. Villagio, P., Mathematical Models for Elastic Structures.

  69. Batra, R. C., Porfiri, M., and Spinello, D., Capacitance Estimate for Electrostatically Actuated Narrow Microbeams. DOI: 10.1049/mnl:20065046

  70. Blech, J. J., On Isothermal Squeeze Films. DOI: 10.1115/1.3254692

  71. Burgdorfer, A., The Influence of the Molecular Mean-Free Path on the Performance of Hydrodynamic Gas Lubricated Bearings.

  72. Nayfeh, A. H., Younis, M. I., and Abdel- Rahman, E. M., Reduced-Order Models for MEMS Applications. DOI: 10.1007/s11071-005-2809-9

  73. Shampine, L. and Reichelt, M., The Matlab ODE Suite. DOI: 10.1137/S1064827594276424

  74. Bochobza-Degani, O., Elata, D., and Nemirovsky, Y., An Efficient DIPIE Algorithm for CAD of Electrostatically Actuated MEMS Devices. DOI: 10.1109/JMEMS.2002.803280

  75. Kierzenka, J. and Shampine, L. F., A BVP Solver Based on Residual Control and the MATLAB PSE. DOI: 10.1145/502800.502801

  76. Rand, R. H., Lecture Notes on Nonlinear Vibrations, vers. 52.

  77. De, S. K. and Aluru, N. R., Complex Nonlinear Oscillations in Electrostatically Actuated Microstructures.

  78. DeMartini, B. E., Butterfield, H. E., Moehlis, J., and Turner, K. L., Chaos for a Microelectromechanical Oscillator Governed by the Nonlinear Mathieu Equation. DOI: 10.1109/JMEMS.2007.906757


Articles with similar content:

MAGNETO-THERMOELASTIC RAYLEIGH WAVES IN A PRESTRESSED SOLID HALF-SPACE UNDER HEAT SOURCE USING GREEN AND LINDSAY'S MODEL
Computational Thermal Sciences: An International Journal, Vol.7, 2015, issue 3
Shikha Kakar, Rajneesh Kakar
Nonlinear Instability of Two Superposed Electrified Bounded Fluids Streaming Through Porous Medium in (2 + 1) Dimensions
Journal of Porous Media, Vol.12, 2009, issue 12
G. M. Moatimid, Mohamed F. El-Sayed, T. M. N. Metwaly
COMPUTER SIMULATION OF THE PHASE TRANSITIONS IN CLUSTERS
Nanoscience and Technology: An International Journal, Vol.3, 2012, issue 2
R. S. Berry , Boris M. Smirnov
On Synthesis of Stabilizing Controllers Using Linear Matrix Inequalities
Journal of Automation and Information Sciences, Vol.32, 2000, issue 12
Vladimir B. Larin
FREQUENCY SHIFTS INDUCED BY LARGE DEFORMATIONS IN PLANAR PANTOGRAPHIC CONTINUA
Nanoscience and Technology: An International Journal, Vol.6, 2015, issue 2
Dionisio Del Vescovo, Antonio Battista, Nicola Luigi Rizzi, Christian Cardillo, Emilio Turco