Skip to main content
SpringerLink
Log in

Reduced-Order Models for MEMS Applications

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

We review the development of reduced-order models for MEMS devices. Based on their implementation procedures, we classify these reduced-order models into two broad categories: node and domain methods. Node methods use lower-order approximations of the system matrices found by evaluating the system equations at each node in the discretization mesh. Domain-based methods rely on modal analysis and the Galerkin method to rewrite the system equations in terms of domain-wide modes (eigenfunctions). We summarize the major contributions in the field and discuss the advantages and disadvantages of each implementation. We then present reduced-order models for microbeams and rectangular and circular microplates. Finally, we present reduced-order approaches to model squeeze-film and thermoelastic damping in MEMS and present analytical expressions for the damping coefficients. We validate these models by comparing their results with available theoretical and experimental results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Zavracky, P. M., Majumder, S., and McGruer, N. E., ‘Micromechanical switches fabricated using nickel surface micromachining’, Journal of Microelectromechanical Systems 6, 1997, 3–9.

    Article  Google Scholar 

  2. Choi, B. and Lovell, E. G., ‘Improved analysis of microbeams under mechanical and electrostatic loads’, Journal of Microelectromechanical Systems 7, 1997, 24–29.

    Google Scholar 

  3. Guyan, R. J., ‘Reduction of stiffness and mass matrices’, AIAA Journal 3, 1965, 380.

    Google Scholar 

  4. Bechtold, T., Rudnyi, E. B., and Korvink, J. G., ‘Automatic order reduction of thermo-electric models for MEMS: Arnoldi versus Guyan’, in Proceedings of the Fourth International Conference on Advanced Semiconductor Devices and Microsystems, Smolenice, Slovakia, 2002, pp. 333–336.

  5. Ostergaard, D. F. and Gyimesi, M., ‘Finite element based reduced order modeling of Micro Electro Mechanical Systems (MEMS)’, in Proceedings of the International Conference on Modeling and Simulation of Microsystems, San Diego, CA, 2000, pp. 684–687.

  6. Bechtold, T., Rudnyi, E. B., and Korvink, J. G., ‘Automatic order reduction of thermo-electric model for micro-ignition unit’, in Proceedings of the International Conference on Simulation of Semiconductor Processes and Devices, Kobe, Japan, 2002, pp. 131–134.

  7. Wang, F. and White, J., ‘Automatic model order reduction of a microdevice using the Arnoldi approach’, in Proceedings of the International Mechanical Engineering Congress and Exposition, Anaheim, CA, 1998, pp. 527–530.

  8. Hung, E. S., Yang, Y.-J., and Senturia, S. D., ‘Low-order models for fast dynamical simulation of MEMS microstructures’, in Proceedings of the International Conference on Solid-State Sensors and Actuators: Transducers 1997, Vol. 2, Chicago, IL, 1997, pp. 1101–1104.

  9. Bai, Z., Bindel, D., Clark, J. V., Demmel, J., Pister, K. S. J., and Zhou, N., ‘New numerical techniques and tools in Sugar for 3D MEMS simulation’, in Proceedings of the International Conference on Modeling and Simulation of Microsystems, Hilton Head Island, SC, 2001, pp. 31–34.

  10. Clark, J. V., Bindel, D., Zhou, N., Bhave, S., Bai, Z., Demmel, J., and Pister, K. S. J., ‘Sugar: Advancements in a 3D multi-domain simulation package for MEMS’, in Proceedings of the Microscale Systems: Mechanics and Measurements Symposium, Portland, OR, 2001.

  11. Clark, J. V., Bindel, D., Kao, W., Zhu, E., Kuo, A., Zhou, N., Nie, J., Demmel, J., Bai, Z., Govindjee, S., Pister, K. S. J., Gu, M., and Agogino, A., ‘Addressing the needs of complex MEMS design’, in Proceedings of the International Conference on Micro Electro Mechanical Systems, Las Vegas, NV, 2002, pp. 204–209.

  12. Srinivasan, V., Jog, A., and Fair, R. B., ‘Scalable macromodels for microelectromechanical systems’, in Proceedings of the International Conference on Modeling and Simulation of Microsystems, Hilton Head Island, SC, 2001, pp. 72–75.

  13. Chen, Y. and White, J., ‘A quadratic method for nonlinear model order reduction’, in Proceedings of the International Conference on Modeling and Simulation of Microsystems, San Diego, CA, 2000, pp. 477–480.

  14. Ramaswamy, D. and White, J., ‘Automatic generation of small-signal dynamic macromodels from 3-D simulation’, in Proceedings of the International Conference on Modeling and Simulation of Microsystems, Hilton Head Island, SC, 2001, pp. 27–30.

  15. Chen, J. and Kang, S.-M., ‘Techniques for coupled circuit and micromechanical simulation’, in Proceedings of the International Conference on Modeling and Simulation of Microsystems, San Diego, CA, 2000, pp. 213–216.

  16. Chen, J. and Kang, S.-M., ‘An algorithm for automatic model-order reduction of nonlinear MEMS devices’, in Proceedings of the International Symposium on Circuits and Systems, Vol. 2, Geneva, Switzerland, 2000, pp. 445–448.

  17. Chen, J. and Kang, S.-M., ‘Model-order reduction of weakly nonlinear MEMS devices with Taylor series expansion and Arnoldi process’, in Proceedings of the IEEE Midwest Symposium on Circuits and Systems, Lansing, MI, 2000, pp. 248–251.

  18. Chen, J. and Kang, S.-M., ‘Dynamic macromodeling of MEMS mirror devices’, in Technical Digest of the International Electron Devices Meeting, Washington, DC, 2001, pp. 925–928.

  19. Rewieński, M. and White, J., ‘A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices’, in Proceedings of the IEEE/ACM International Conference on Computer-Aided Design, Vol. 1, San Jose, CA, 2001, pp. 252–257.

  20. Rewieński, M. and White, J., ‘A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices’, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 22, 2003, 155–170.

    Article  Google Scholar 

  21. Rewieński, M. and White, J., ‘Improving trajectory piecewise-linear approach to nonlinear model order reduction for micromachined devices using an aggregated projection basis’, in Proceedings of the International Conference on Modeling and Simulation of Microsystems, Vol. 1, San Juan, PR, 2002, pp. 128–131.

  22. Vasilyev, D., Rewieński, M., and White, J., ‘A TBR-based trajectory piecewise-linear algorithm for generating accurate low-order models for nonlinear analog circuits and MEMS’, in Proceedings of the Conference on Design Automation, Anaheim, CA, 2003, pp. 490–495.

  23. Liang, Y. C., Lin, W. Z., Lee, H. P., Lim, S. P., Lee, K. H., and Sun, H., ‘Proper orthogonal decomposition and its applications. Part II. Model reduction for MEMS dynamical analysis’, Journal of Sound and Vibration 256(3), 2002, 515–532.

    Article  Google Scholar 

  24. Hung, E. S. and Senturia, S. D., ‘Generating efficient dynamical models for microelectromechanical systems from a few finite-element simulations runs’, Journal of Microelectromechanical Systems 8, 1999, 280–289.

    Article  Google Scholar 

  25. Chen, J. and Kang, S.-M., ‘Model-order reduction of nonlinear MEMS devices through arclength-based Karhunen–Loève decomposition’, in Proceedings of the International Symposium on Circuits and Systems, Vol. 3, Sydney, Australia, 2001, pp. 457–460.

  26. Liang, Y. C., Lin, W. Z., Lee, H. P., Lim, S. P., Lee, K. H., and Feng, D. P., ‘A neural-network-based method of model reduction for the dynamic simulation of MEMS’, Journal of Micromechanics and Microengineering 11, 2001, 226–233.

    Article  CAS  Google Scholar 

  27. Lin, W. Z., Lee, K. H., Lim, S. P., and Liang, Y. C., ‘Proper orthogonal decomposition and component mode synthesis in macromodel generation for the dynamic simulation of a complex MEMS device’, Journal of Micromechanics and Microengineering 13, 2003, 646–654.

    Article  Google Scholar 

  28. Qiao, R. and Aluru, N. R., ‘Mixed-domain and reduced-order modeling of electroosmotic transport in Bio-MEMS’, in Proceedings of the International Workshop on Behavioral Modeling and Simulation, Orlando, FL, 2000, pp. 51–56.

  29. De, S. K. and Aluru, N. R., ‘Physical and reduced-order dynamic analysis of MEMS’, in Proceedings of the International Conference on Computer Aided Design, San Jose, CA, 2003, pp. 270–273.

  30. Anathasuresh, G. K., Gupta, R. K., and Senturia, S. D., ‘An approach to macromodelling of MEMS for nonlinear dynamic simulation’, in Proceedings of the Symposium on Mechanics in Microelectromechanical Systems (MEMS), ASME-DSC, Atlanta, GA, 1996, pp. 401–407.

  31. Grtétillat, M. A., Yang, Y. J., Hung, E. S., Rabinovich, V., Ananthasuesh, G. K., Rooij, N. F., and Senturia, S. D., ‘Nonlinear electromechanical behavior of an electrostatic microrelay’, in Proceedings of the Iternational Conference on Solid State Sensors and Actuators: Transducers 1997, Vol. 2, Chicago, IL, 1997, pp. 1141–1144.

  32. Gabbay, L. D., Mehner, J. E., and Senturia, S. D., ‘Computer-aided generation of nonlinear reduced-order dynamic macromodels. Part I. Non-stress-stiffened case’, Journal of Microelectromechanical Systems 9, 2000, 262–269.

    Article  Google Scholar 

  33. Varghese, M., Rabinovich, V. L., and Senturia, S. D., ‘Reduced-order modeling of Lorentz force actuation with modal basis functions’, in Proceedings of the International Conference on Modeling and Simulation of Microsystems, San Juan, PR, 1999, pp. 155–158.

  34. Mehner, J. E., Gabbay, L. D., and Senturia, S. D., ‘Computer-aided generation of nonlinear reduced-order dynamic macromodels. Part II. Stress-stiffened case’, Journal of Microelectromechanical Systems 9, 2000, 270–278.

    Article  Google Scholar 

  35. Bennini, F., Mehner, J., and Dötzel, W., ‘Computational methods for reduced order modeling of coupled domain simulations’, in Proceedings of the International Conference on Solid-State Sensors and Actuators: Transducers 2001, Munich, Germany, 2001, pp. 260–263.

  36. Mehner, J. E., Doetzel, W., Schauwecker, B., and Ostergaard, D., ‘Reduced order modeling of fluid structural interactions in MEMS based on modal projection techniques’, in Proceedings of the International Conference on Solid-State Sensors, Actuators and Microsystems: Transducers 2003, Vol. 2, Boston, MA, 2003, pp. 1840–1843.

  37. Westby, E. R. and Fjeldly, T. A., ‘Nonlinear analytical reduced-order models for MEMS’, in Proceedings of the International Conference on Modeling and Simulation of Microsystems, Vol. 1, San Juan, PR, 2002, pp. 150–153.

  38. Xie, W. C., Lee, H. P., and Lim, S. P., ‘Nonlinear dynamic analysis of MEMS switches by nonlinear modal analysis’, Nonlinear Dynamics 31, 2003, 243–256.

    Article  Google Scholar 

  39. Abdel-Rahman, E. M., Younis, M. I., and Nayfeh, A. H., ‘Characterization of the mechanical behavior of an electrostatically actuated microbeam’, Journal of Micromechanics and Microengineering 12, 2002, 759–766.

    Article  Google Scholar 

  40. Younis, M. I., Abdel-Rahman, E. M., and Nayfeh, A. H., ‘Static and dynamic behavior of an electrically excited resonant microbeam,’ in Proceedings of the AIAA Structures, Structural Dynamics, and Materials Conference, AIAA Paper 2002-1305, Denver, CO, 2002.

  41. Abdel-Rahman, E. M., Younis, M. I., and Nayfeh, A. H., ‘A nonlinear reduced-order model for electrostatic MEMS,’ in Proceedings of the Biennial ASME Conference on Mechanical Vibration and Noise, DETC2003/VIB-48517, Chicago, IL, 2003.

  42. Younis, M. I., Abdel-Rahman, E. M., and Nayfeh, A. H., ‘A reduced-order model for electrically actuated microbeam-based MEMS,’ Journal of Microelectromechanical Systems 12, 2003, 672–680.

    Article  Google Scholar 

  43. Tilmans, H. A. and Legtenberg, R., ‘Electrostatically driven vacuum-encapsulated polysilicon resonators. Part II. Theory and performance’, Sensors and Actuators A 45, 1994, 67–84.

    Article  Google Scholar 

  44. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.

    Google Scholar 

  45. Nayfeh, A. H., Nonlinear Interactions, Wiley, New York, 2000.

    Google Scholar 

  46. Zhao, X., Abdel-Rahman, E. M., and Nayfeh, A. H., ‘Mechanical behavior of an electrically actuated microplate’, in Proceedings of the Biennial ASME Conference on Mechanical Vibration and Noise, DETC2003/VIB-48531, Chicago, IL, 2003.

  47. Francais, O. and Dufour, I., ‘Normalized abacus for the global behavior of diaphragms: Pneumatic, electrostatic, piezoelectric or electromagnetic actuation’, Journal of Modeling and Simulation of Microsystems 2, 1999, 149–160.

    Google Scholar 

  48. Vogl, G. W. and Nayfeh, A. H., ‘A reduced-order model for electrically actuated clamped circular plates’, in Proceedings of the Biennial ASME Conference on Mechanical Vibration and Noise, DETC2003/VIB-48530, Chicago, IL, 2003.

  49. Osterberg, P. M., ‘Electrostatically Actuated Microelectromechanical Test Structures for Material Property Measurement’, PhD dissertation, Massachusetts Institute Technology, Cambridge, MA, 1995.

  50. Tilmans, H. A., Elwespoek, M., and Fluitman, J. H., ‘Micro resonant force gauges’, Sensors and Actuators A 30, 1992, 35–53.

    Article  Google Scholar 

  51. Starr, J. B., ‘Squeeze-film damping in solid-state accelerometers’, in Proceedings of the IEEE Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, 1990, pp. 44–47.

  52. Blech, J. J., ‘On isothermal squeeze films’, Journal of Lubrication Technology A 105, 1983, 615–620.

    Google Scholar 

  53. Darling, R. B., Hivick, C., and Xu, J., ‘Compact analytical modeling of squeeze film damping with arbitrary venting conditions using a Green’s function approach’, Sensors and Actuators A 70, 1998, 32–41.

    Article  Google Scholar 

  54. Gupta, M. C., Statistical Thermodynamics, Wiley, New Delhi, 1990.

    Google Scholar 

  55. Kàdàr, Z., Kindt, W., Bossche, A., and Mollinger, J., ‘Quality factor of torsional resonators in the low-pressure region’, Sensors and Actuators A 53, 1996, 299–303.

    Article  Google Scholar 

  56. Li, B., Wu, H., Zhu, C., and Liu, J., ‘The theoretical analysis on damping characteristics of resonant microbeam in vacuum’, Sensors and Actuators A 77, 1999, 191–194.

    Article  Google Scholar 

  57. Christian, R. G., ‘The theory of oscillating-vane vacuum gauges’, Vacuum 16, 1966, 175–178.

    Article  Google Scholar 

  58. Zook, J. D., Burns, D. W., Guckel, H., Sniegowski, J. J., Engelstad, R. L., and Feng, Z., ‘Characteristics of polysilicon resonant microbeams’, Sensors and Actuators A 35, 1992, 290–294.

    Article  Google Scholar 

  59. Bao, M., Yang, H., Yin, H., and Sun, Y., ‘Energy transfer model for squeeze-film air damping in low vacuum’, Journal of Micromechanics and Microengineering 12, 2002, 341–346.

    Article  Google Scholar 

  60. Roszhart, T. V., ‘The effect of thermoelastic internal friction on the Q of micromachined silicon resonators’, in Technical Digest of Solid-State Sensors and Actuators Workshop, Hilton Head Island, SC, 1990, pp. 13–16.

  61. Zener, C., ‘Internal friction in solids. Part I. Theory of internal friction in reeds’, Physical Review 52, 1937, 230–235.

    Article  Google Scholar 

  62. Lifshitz, R. and Roukes, M. L., ‘Thermoelastic damping in micro- and nanomechanical systems’, Physical Review B 61, 2000, 5600–5609.

    Article  CAS  Google Scholar 

  63. Nayfeh, A. H. and Younis, M. I., ‘Modeling and simulations of thermoelastic damping in microplates’, Journal of Micromechanics and Microengineering 14, 2004, 1711–1717.

    Article  Google Scholar 

  64. 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, 2004, 170–181.

    Article  Google Scholar 

  65. Boley, B. A. and Weiner, J. H., Theory of Thermal Stresses, Wiley, New York, 1960.

    Google Scholar 

  66. Leissa, A. W., Vibration of Plates, NASA, Washington, DC, 1969.

    Google Scholar 

  67. Nayfeh, A. H. and Pai, F. P., Linear and Nonlinear Structural Mechanics, Wiley, New York, 2004.

    Google Scholar 

  68. Veijola, T., Kuisma, H., Lahdenperä, J., and Ryhänen, T., ‘Equivalent-circuit model of the squeezed gas film in a silicon accelerometer’, Sensors and Actuators A 48, 1995, 239–248.

    Article  Google Scholar 

  69. Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973.

    Google Scholar 

  70. Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981.

    Google Scholar 

  71. Legtenberg, R. and Tilmans, H. A., ‘Electrostatically driven vacuum-encapsulated polysilicon resonators. Part I. Design and fabrication’, Sensors and Actuators A 45, 1994, 57–66.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, U.S.A.

    ALI H. NAYFEH & EIHAB M. ABDEL-RAHMAN

  2. Department of Mechanical Engineering, State University of New York at Binghamton, Binghamton, NY, 13902, U.S.A.

    MOHAMMAD I. YOUNIS

Authors
  1. ALI H. NAYFEH
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. MOHAMMAD I. YOUNIS
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. EIHAB M. ABDEL-RAHMAN
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ALI H. NAYFEH.

Rights and permissions

Reprints and permissions

About this article

Cite this article

NAYFEH, A.H., YOUNIS, M.I. & ABDEL-RAHMAN, E.M. Reduced-Order Models for MEMS Applications. Nonlinear Dyn 41, 211–236 (2005). https://doi.org/10.1007/s11071-005-2809-9

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-005-2809-9

Key Words

Search

Navigation