Pull-in behavior analysis of vibrating functionally graded micro-cantilevers under suddenly DC voltage

Document Type: Research Paper


National Iranian South Oil Company (NISOC), Ahvaz, Iran


The present research attempts to explain dynamic pull-in instability of functionally graded micro-cantilevers actuated by step DC voltage while the fringing-field effect is taken into account in the vibrational equation of motion. By employing modern asymptotic approach namely Homotopy Perturbation Method with an auxiliary term, high-order frequency-amplitude relation is obtained, then the influences of material properties and actuation voltage on dynamic pull-in behavior are investigated. It is demonstrated that the auxiliary term in the homotopy perturbation method is extremely effective for higher order approximation and two terms in series expansions are sufficient to produce an acceptable solution. The strength of this analytical procedure is verified through comparison with numerical results.


[1] Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H., Rahaeifard, M., “On the size-dependent behavior of functionally graded micro-beams”, Materials and Design, Vol. 31, pp. 2324–2329, 2010.

[2] Lü, C.F., Chen, W.Q., Lim, C.W., “Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies”, Composites Science and Technology, Vol. 69, pp. 1124–1130, 2009.

[3] Craciunescu, C.M., Wuttig, M., “New ferromagnetic and functionally grade shape memory alloys”, J Optoelectron Adv Mater, Vol. 5, No. 1, pp. 139–46, 2003.

[4] Fu, Y.Q., Du, H.J., Zhang, S., “Functionally graded TiN/TiNi shape memory alloy films”, Mater Lett, Vol. 57, No. 20, pp. 2995–9, 2003.

[5] Fu, Y.Q., Du, H.J., “Huang WM, Zhang S, Hu M. TiNi-based thin films in MEMS applications: a review”, Sensors Actuat A, Vol. 112, No. (2–3), pp. 395-408, 2004.

[6] Witvrouw, A., Mehta, A., “The use of functionally graded poly-SiGe layers for MEMS applications, Functionally Graded Mater, Vol. 8, pp. 255–60, 2005.

[7] Lee, Z., Ophus, C., Fischer, L.M., et al. “Metallic NEMS components fabricated from nanocomposite Al-Mo films”, Nanotechnology, Vol. 17, No. 12, pp. 3063–70, 2006.

[8] Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T., “The modified couple stress functionally graded Timoshenko beam formulation”, Materials and Design, Vol. 32, pp. 1435–1443, 2011.

[9] Jafar, I., Sadeghi-Pournaki, Zamanzadeh, M.R., Shabani, R., Rezazadeh, G., “Mechanical Behavior of a FGM Capacitive Micro-Beam Subjected to a Heat Source”, Journal of Solid Mechanics, Vol. 3, No. 2, pp. 158-171, 2011.

[10] Ke, L.L., Wang, Y.S., “Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory”, Composite Structures, Vol. 93, pp. 342–350, 2011.

[11] Sharafkhani, N., Rezazadeh, G., Shabani, R., “Study of mechanical behavior of circular FGM micro-plates under nonlinear electrostatic and mechanical shock loadings”, Acta Mech, Vol. 223, pp. 579–591, 2012.

[12] Das, K., Batra, R.C., “Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems,” J. Micromech. Microeng., Vol. 19, 035008, 2009. doi:10.1088/0960-1317/19/3/035008.

[13] Fu, Y., Zhang, J., “Size-dependent pull-in phenomena in electrically actuated nano beams incorporating surface energies,” Applied Mathematical Modelling, Vol. 35, pp. 941-951, 2011.

[14] Wang, Y.G., Lin, W.H., Feng, Z.J., Li, X.M., “Characterization of extensional multi-layer microbeams in pull-in phenomenon and vibrations,” International Journal of Mechanical Sciences, Vol. 54, pp. 225–233, 2012.

[15] Jia, X.L., Yang, J., Kitipornchai, S., “Pull-in instability of geometrically nonlinear micro-switches under electrostatic and Casimir forces,” Acta Mech., Vol. 218, pp. 161-174, 2011. doi: 10.1007/s00707-010-0412-8.

[16] Sedighi, H.M., Shirazi, K.H., “Vibrations of micro-beams actuated by an electric field via Parameter Expansion Method,” Acta Astronautica, Vol. 85, pp. 19-24, 2013.

[17] Rahaeifard, M., Ahmadian, M.T., Firoozbakhsh, K., “Size-dependent dynamic behavior of microcantilevers under suddenly applied DC voltage,” Proc IMechE Part C: J Mechanical Engineering Science, DOI: 10.1177/0954406213490376.

[18] Rajabi, F., Ramezani, S., “A nonlinear microbeam model based on strain gradient elasticity theory,” Acta Mechanica Solida Sinica,Vol. 26, No. 1, 2013, doi: 10.1016/S0894-9166(13)60003-8.

[19] Towfighian, S., Heppler, G.R., Abdel-Rahman, E.M., “Analysis of a Chaotic Electrostatic Micro-Oscillator,” Journal of Computational and Nonlinear Dynamics, Vol. 6, No. 1, 011001, 2011.

[20] He, J.H., “Max-Min approach to nonlinear oscillators”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 9, pp. 207-210.

[21] Sedighi, H.M., Shirazi, K.H., Noghrehabadi, A., “Application of Recent Powerful Analytical Approaches on the Non-Linear Vibration of Cantilever Beams”, Int. J. Nonlinear Sci. Numer. Simul., Vol. 13, No. 7–8, pp. 487–494, 2012.

[22] Ghadimi, M., Barari, A., Kaliji, H.D., Domairry, G., “Periodic solutions for highly nonlinear oscillation systems” Archives of Civil and Mechanical Engineering, Vol. 12, No. 3, pp. 389-395, 2012.

[23] Sedighi, H.M., Shirazi, K.H., Zare, J., “An analytic solution of transversal oscillation of quintic nonlinear beam with homotopy analysis method”, International Journal of Non-Linear Mechanics, Vol. 47, pp. 777-784, 2012.

[24] Noghrehabadi, A., Ghalambaz, M., Ghanbarzadeh, A., “A new approach to the electrostatic pull-in instability of nanocantilever actuators using the ADM–Padé technique”, Computers & Mathematics with Applications, Vol. 64, No. 9, pp. 2806–2815, 2012.

[25] Kaliji, H.D., Ghadimi, M., Pashaei, M.H., “Study the behavior of an electrically exciting nanotube using optimal homotopy asymptotic method”, Int. J. Appl. Mechanics, Vol. 04, 1250004, 2012, DOI: 10.1142/S1758825112001336.

[26] Shou, D.H., He, J.H., “Application of parameter-expanding method to strongly nonlinear oscillators”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. (1), pp. 121-124, 2007.

[27] Sedighi, H.M., Shirazi, K.H., Noghrehabadi, A.R., Yildirim, A., “Asymptotic Investigation of Buckled Beam Nonlinear Vibration,” Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, Vol. 36, No. (M2), pp. 107-116, 2012.

[28] He, J. H., “Hamiltonian approach to nonlinear oscillators”, Physics Letters A, Vol. 374, No. (23), pp. 2312-2314, 2010.

[29] Sedighi, H.M., Shirazi, K.H., Attarzadeh, M.A., “A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches”, Acta Astronautica, Vol. 91, pp. 245-250, 2013.

[30] He, J.H., “Homotopy Perturbation Method with an Auxiliary Term”, Abstract and Applied Analysis, Vol. 2012, 857612, doi:10.1155/2012/857612.

[31] Batra, R.C., Porfiri, M., Spinello, D., “Vibrations of narrow microbeams predeformed by an electric field”, Journal of Sound and Vibration, Vol. 309, pp. 600-612, 2008.

[32] He, J.H., “Homotopy perturbation method with two expanding parameters,” Ind. J. Phys., Vol. 88, No. 2, pp. 193-196, 2014.