Abstract
Microbeams and microcantilevers are the main part of many MEMS. There are several body and contact forces affecting a vibrating microbeam. Among them, there are some forces appearing to be more significant in micro and nanosize scales. Accepting an analytical approach, we present the mathematical modeling of microresonators dynamics and develop effective equations to be utilized to study the electrically actuated microresonators. The presented nonlinear model includes the initial deflection due to polarization voltage, mid-plane stretching, and axial loads as well as the nonlinear displacement coupling of electric force. It also includes the thermal and squeeze-film phenomena. The equations are nondimensionalized and the design parameters are developed. In order to have a set of equations, depending on depth of accuracy and difficulty, we present equations of motion for linearized and different level of nonlinearity. The simulation method makes it easy for investigators to pick the appropriate equation depending on their design and application. It is shown that the equation of motion for microresonators is highly nonlinear, parametric, and externally excited. The most important phenomena affecting the motion of microbeam-based and microcantilever-based microresonators are reviewed in this chapter and the corresponding forces are introduced. The mechanical and electrical forces are the primary forces that cause the microresonators work. There are also two specific phenomena: squeeze-film and thermal damping, that their effects on MEMS dynamics are considered secondary compared to mechanical and electrical forces. Some tertiary phenomena such as van der Waals, Casimir, and fringing field effects are also introduced. There are a few reported investigations on secondary phenomena, and their effects are defined. However, based on some reported theoretical and experimental results, we qualitatively analyze them and present two nonlinear functions to define the restoring and damping behavior of squeeze-film. In addition, we use two Lorentzian functions to describe the restoring and damping forces caused by thermal phenomena.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Blech JJ (1983), On isothermal squeeze-films. J Lubric Technol 105:615–620.
Bordag M, Kilmchitskaya GL, Mostepanenko VM (1995) Corrections to the Casimir force between plates with stochastic surfaces. Phys Lett 200:95–102.
Casimir HBG (1948) On the attraction between two perfectly conducting plates. Proc K Ned Akad Wet 51:793–795.
Ding JN, Meng YG, Wen SZ (2000) Mechanical stability and sticking in model microelectromechaincal systems (MEMS) under Casimir force. Int J Nonlinear Sci Numer Simul 1:373–378.
Golnaraghi MF, Jazar RN (2001) Development and analysis of a simplified nonlinear model of a hydraulic engine mount. J Vibrat Contr 7(4):495–526.
Jazar RN (2006) Mathematical modeling and simulation of thermoelastic effects in flexural microcantilever resonators dynamics. J Vibrat Contr 12(2):139–163.
Jazar RN, Golnaraghi MF (2002) Nonlinear modeling, experimental verification, and theoretical analysis of a hydraulic engine mount. J Vibrat Contr 8(1):87–116.
Jazar RN, Mahinfalah M, Khazaei A, Alimi MH (2006) Squeeze-film phenomena in microresonators dynamics: a mathematical modeling point of view. ASME international mechanical engineering congress and exposition, Chicago, IL, November 5–10.
Jazar RN, Mahinfalah M (2006) Squeeze-film damping in microresonators dynamics. Nonlinear science and complexity conference, Beijing, China, August 07–12.
Jazar RN, Mahinfalah M, Mahmoudian N, Aagaah MR (2009) Effects of nonlinearities on the steady state dynamic behavior of electric actuated microcantilever-based resonators. J Vibrat Contr 15(9):1283–1306.
Jazar RN, Mahinfalah M, Rastgaar Aagaah M, Mahmoudian N, Khazaei A, Alimi MH (2005) Mathematical modeling of thermal effects in steady state dynamics of microresonators using Lorentzian function: part 1 – thermal damping. ASME international mechanical engineering congress and exposition, Orlando, FL, November 5–11.
Kaajakari V, Mattila T, Oja A, Seppä H (2004) Nonlinear limits for single-crystal silicon microresonators. IEEE J Microelectromech Syst 13(5):715–724.
Khaled ARA, Vafai K, Yang M, Zhang X, Ozkan CS (2003) Analysis, control and augmentation of microcantilever deflections in bio-sensing systems. Sens Actuat B 94:103–115.
Laumoreaux SK (1999) Calculation of the Casimir force between imperfectly conducting plates. Phys Rev A 59:3149–3153.
Laumoreaux SK (1997) Determination of the Casimir force in the 0.6 to 6 m range. Phys Rev Lett 78:5–8.
Leus V, Elata D (2004) Fringing field effect in electrostatic actuators. Technical report ETR-2004-2, TECHNION, Israel Institute of Technology, Faculty of Mechanical Engineering.
Lifshitz R, Roukes ML (2000) Thermoelastic damping in micro- and nanomechanical systems. Phys Rev 61(8):5600–5609.
Mahmoudian N, Rastgaar Aagaah M, Jazar RN, Mahinfalah M (2004) Dynamics of a micro electro mechanical system (MEMS), The 2004 international conference on MEMS, NANO, and smart systems, Banff, Alberta, Canada, August 25–27.
Meirovitch L (1996) Principles and technologies of vibrations, Prentice Hall, New Jersey.
Nayfeh AH, Younis MI (2004) A new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping. J Micromech Microeng 14:170–181.
Nguyen CTC (1995) Micromechanical resonators for oscillators and filters. Proc IEEE international ultrasonic symposium, Seattle, WA, Nov. 7–10 pp 489–499.
Rastgaar AM, Mahmoudian N, Jazar RN, Mahinfalah M, Khazaei A, Alimi MH (2005), Mathematical modeling of thermal effects in steady state dynamics of microresonators using Lorentzian function: part 2 - temperature relaxation. ASME international mechanical engineering congress and exposition, Orlando, FL, November 5–11.
Tadayon MA, Sayyaadi H, Jazar RN (2006) Nonlinear modeling and simulation of thermal effects in microcantilever resonators dynamic. Int MEMS Conf 2006 (iMEMS2006), Singapore, 9–12 May.
Yang J, Ono T, Esashi M (2002) Energy dissipation in submicrometer thick single-cristal silicon cantilevers. J Microelectromech Syst 11(6):775–783.
Younis MI (2004) Modeling and simulation of micrielectromecanical system in multi-physics fields. Ph.D. thesis, Mechanical Engineering, Virginia Polytechnic Institute and State University.
Younis MI, Abdel-Rahman EM, Nayfeh A (2003) A reduced-order model for electrically actuated microbeam-based MEMS. J Microelectromech Syst 12(5):672–680.
Younis MI, Nayfeh AH (2003) A study of the nonlinear response of a resonant microbeam to electric actuation. J Nonlin Dyn 31:91–117.
Zhang C, Xu G, Jiang Q (2004) Characterization of the squeeze-film damping effect on the quality factor of a microbeam resonator. J Micromech Microeng 14:1302–1306.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Jazar, R.N. (2012). Nonlinear Mathematical Modeling of Microbeam MEMS. In: Dai, L., Jazar, R. (eds) Nonlinear Approaches in Engineering Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1469-8_3
Download citation
DOI: https://doi.org/10.1007/978-1-4614-1469-8_3
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-1468-1
Online ISBN: 978-1-4614-1469-8
eBook Packages: EngineeringEngineering (R0)