Abstract
The present article studies the mechanical behavior of a FGM micro-beam subjected to a nonlinear electrostatic pressure. The FGM micro-beam is made of metal and ceramic and material properties vary continuously along the beam thickness according to a power-law. The nonlinear equation of dynamic motion of the FGM micro-beam is derived. By solving the equation of the static deflection, equilibrium positions of the micro-beam are determined and shown in the state control space. To study the stability of the fixed points, the trajectories of the beam motion are illustrated in the phase plane for different initial conditions. In order to find the response of the micro-beam to a step DC applied voltage, the nonlinear equation of motion is solved using a Galerkin based reduced order model. Moreover, time histories and phase portraits for different applied voltages are illustrated. The effect of different power law exponent on the stability of the micro-beam is studied.
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Ahangar, S., Rezazadeh, G., Shabani, R., Ahmadi, G., Toloei, A.: On the stability of a microbeam conveying fluid considering modified couple stress theory. Int. J. Mech. Mater. Des. 7, 327–342 (2011)
Baker, C., Porteous, H.L.: Linear algebra and differential equation. Ellis Horwood, New York (1990)
Bao, M., Wang, W.: Future of microelectromechanical systems (MEMS). Sens. Actuators A 56, 135–141 (1996)
Craciunescu, C.M., Wuttig, M.: New ferromagnetic and functionally grade shape memory alloys. J. Optoelectron. Adv. Mater. 5(1), 139–146 (2003)
Das, K. Batra, R.C.: Symmetry breaking, snap-through, and pull-in instabilities under dynamic loading of microelectromechanical shallow arch. Smart Mater. Struct. 18 (2009). doi: 10.1088/0964-1726/18/11/115008
Desai, P., Kant, T.: A mixed semi analytical solution for functionally graded (FG) finite length cylinders of orthotropic materials subjected to thermal load. Int. J. Mech. Mater. Des. 8(1), 89–100 (2012)
Filipich, C.P., Piovan, M.T.: The dynamics of thick curved beams constructed with functionally graded materials. Mech. Res. Commun. 37(6), 565–570 (2010)
Fu, Y.Q., Du, H.J., Huang, W.M., Zhang, S., Hu, M.: TiNi-based thin films in MEMS applications: a review. Sens. Actuators A 112(2–3), 395–408 (2004)
Gharib, A., Salehi, M., Fazeli, S.: Deflection control of functionally graded material beams with bonded piezoelectric sensors and actuators. Mater. Sci. Eng. A 498, 110–114 (2008)
Hasanyan, D.J., Batra, R.C., Harutyunyan, S.: Pull-in instabilities in functionally graded microthermoelectromechanical systems. J. Thermal Stress. 31, 1006–1021 (2008)
Jia, X.L., Yang, J., Kitipornchai, S.: Characterization of FGM micro-switches under electrostatic and Casimir forces. Mater. Sci. Eng. A 10, 012178 (2010)
Kang, Y.-A., Li, X.-F.: Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force. Int. J. Non-Linear Mech. 44, 696–703 (2009)
Kiani, Y., Eslami, M.R.: Thermal buckling analysis of functionally graded material beams. Int. J. Mech. Mater. Des. 6(3), 229–238 (2010)
Kiani, Y., Rezaei, M., Taheri, S., Eslami, M.R.: Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams. Int. J. Mech. Mater. Des. 7(3), 185–197 (2011)
Lee, Z., Ophus, C., Fischer, L.M., et al.: Metallic NEMS components fabricated from nanocomposite Al-Mo films. Nanotechnology 17(12), 3063–3070 (2006)
Mehdaoui, A., Pisani, M.B., Tsamados, D., Casset, F., Ancey, P., Ionescu, A.M.: MEMS tunable capacitors with fragmented electrodes and rotational electro-thermal drive. Microsyst. Technol. 13(11), 1589–1594 (2007)
Naderi, A., Saidi, A.R.: On pre-buckling configuration of functionally graded Mindlin rectangular plates. Mech. Res. Commun. 37(6), 535–538 (2010)
Nguyen, C.T.C., Katehi, L.P.B., Rebeiz, G.M.: Micromachined devices for wireless communications. Proc. IEEE 86, 1756–1768 (1998)
Pendhari, S., Kant, T., Desai, Y., Subbaiah, C.V.: Static solutions for functionally graded simply supported plates. Int. J. Mech. Mater. Des. 8(1), 51–69 (2012)
Piovan, T., Sampaio, R.: Vibrations of axially moving flexible beams made of functionally graded materials. J. Thin-Walled Struct. 46, 112–121 (2008)
Rahaeifard, M. Kahrobaiyan, M.H. Ahmadian, M.T.: Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials, In: DETC2009-86254, 3rd international conference on micro- and nanosystems (MNS3), August 30–September 2, San Diego, CA, USA (2009)
Rezazadeh, G., Tahmasebi, A., Zubtsov, M.: Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage. Microsyst. Technol. 12(12), 1163–1170 (2006)
Rezazadeh, G., Khatami, F., Tahmasebi, A.: Investigation of the torsion and bending effects on static stability of electrostatic torsional micromirrors. Microsyst. Technol. 13(7), 715–722 (2007)
Rezazadeh, G., Fathalilou, M., Shabani, R., Tarverdilou, S., Talebian, S.: Dynamic characteristics and forced response of an electrostatically-actuated micro-beam subjected to fluid loading. Microsyst. Technol. 15(9), 1355–1363 (2009)
Sadeghian, H., Rezazadeh, G., Osterberg, P.: Application of the generalized differential quadrature method to the study of pull-in phenomena of MEMS switches. J. Microelectromech. Syst. 16(6), 1334–1340 (2007)
Sankar, B.V.: An elasticity solution for functionally graded beams. Compos. Sci. Technol. 61, 689–696 (2001)
Sazonova, V.: A tunable carbon nanotube resonator. Ph.D. Thesis, Cornell University (2006)
Senturia, S.: Microsystem design. Kluwer, Norwell (2001)
Seydel, R.: Practical bifurcation and stability analysis, 3rd edn. Springer, New York (2010)
Simsek, M., Kocatürk, T.: Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos. Struct. 90, 465–473 (2009)
Simsek, M.: Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuclear Eng. Des. 240, 697–705 (2010)
Sina, S.A., Navazi, H.M., Haddadpour, H.: An analytical method for free vibration analysis of functionally graded beams. Mater. Des. 30, 741–747 (2009)
Sofiyev, A.H.: Buckling analysis of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation. Mech. Res. Commun. 37(6), 539–544 (2010)
Witvrouw, A. Mehta, A.: The use of functionally graded poly-SiGe layers for MEMS applications. Mater. Sci. Forum. 492–493, 255–260 (2005)
Younis, M., Abdel-Rahman, E.M., Nayfeh, A.: A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst. 12(5), 672–680 (2003)
Zhang, Y., Zhao, Y.: Numerical and analytical study on the pull-in instability of micro- structure under electrostatic loading. Sens. Actuators A 127, 366–367 (2006)
Zhong, Z., Yu, T.: Analytical solution of a cantilever functionally graded beam. Compos. Sci. Technol. 67(3–4), 481–488 (2007)
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Abbasnejad, B., Rezazadeh, G. Mechanical behavior of a FGM micro-beam subjected to a nonlinear electrostatic pressure. Int J Mech Mater Des 8, 381–392 (2012). https://doi.org/10.1007/s10999-012-9202-x
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DOI: https://doi.org/10.1007/s10999-012-9202-x