Abstract
The nonlinear vibration of a nanobeam under electrostatic force is investigated through the nonlocal strain gradient theory. Using Galerkin method, the partial differential equation of motion is reduced to an ordinary nonlinear differential one. The equivalent linearization method with a weighted averaging and a variational approach are used independently to establish the frequency–amplitude relationship under closed-forms for comparison purpose. Effects of material and operational parameters on the frequency ratio (the ratio of nonlinear frequency to linear frequency), on the nonlinear frequency, and on the stable configuration of the nanobeam are studied and discussed.
Similar content being viewed by others
References
Abdi, J., Koochi, A., Kazemi, A.S., Abadyan, M.: Modeling the effects of size dependence and dispersion forces on the pull-in instability of electrostatic cantilever NEMS using modified couple stress theory. Smart Mater. Struct. 20, 055011 (2011). https://doi.org/10.1088/0964-1726/20/5/055011
Akgöz, B., Civalek, Ö.: Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory. Arch. Appl. Mech. 82, 423–443 (2012). https://doi.org/10.1007/s00419-011-0565-5
Akgöz, B., Civalek, Ö.: A size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int. J. Eng. Sci. 70, 1–14 (2013). https://doi.org/10.1016/j.ijengsci.2013.04.004
Anh, N.D.: Short Communication Dual approach to averaged values of functions: a form for weighting coefficient. Vietnam J. Mech. 37(2), 145–150 (2015). https://doi.org/10.15625/0866-7136/37/2/6206
Anh, N.D., Hai, N.Q., Hieu, D.V.: The equivalent linearization method with a weighted averaging for analyzing of nonlinear vibrating systems. Latin Am. J. Solids Struct. 14, 1723–1740 (2017). https://doi.org/10.1590/1679-78253488
Aranda-Ruiz, J., Loya, J., Fernández-Sáez, J.: Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory. Compos. Struct. 4(9), 2990–3001 (2012). https://doi.org/10.1016/j.compstruct.2012.03.033
Batra, R.C., Porfiri, M., Spinello, D.: Review of modeling electrostatically actuated microelectromechanical systems. Smart Mater. Struct. 16(6), R23 (2007). https://doi.org/10.1088/0964-1726/16/6/R01
Chaterjee, S., Pohit, G.: A large deflection model for the pull-in analysis of electrostatically actuated microcantilever beams. J. Sound Vib. 322(4–5), 969–986 (2009). https://doi.org/10.1016/j.jsv.2008.11.046
Chong, A.C.M., Yang, F., Lam, D.C.C., Tong, P.: Torsion and bending of micron-scaled structures. J. Mater. Res. 16(04), 1052–1058 (2001). https://doi.org/10.1557/JMR.2001.0146
Chuang, W.C., Lee, H.L., Chang, P.Z., Hu, Y.C.: Review on the modeling of electrostatic MEMS. Sensors (Basel) 10(6), 6149–6171 (2010). https://doi.org/10.3390/s100606149. Epub 2010
Dean, R.N., Luque, A.: Applications of microelectromechanical systems in industrial processes and services. IEEE Trans. Ind. Electron. 56(4), 913–925 (2009). https://doi.org/10.1109/tie.2009.2013691
Duan, J.S., Rach, R.: A pull-in parameter analysis for the cantilever NEMS actuator model including surface energy, fringing field and Casimir effects. Int. J. Solids Struct. 50(22–23), 3511–3518 (2013). https://doi.org/10.1016/j.ijsolstr.2013.06.012
Eom, K., Park, H.S., Yoon, D.S., Kwon, T.: Nanomechanical resonators and their applications in biological/chemical detection: nanomechanics principles. Phys. Rep. 503(4–5), 115–163 (2011). https://doi.org/10.1016/j.physrep.2011.03.002
Eringen, A.C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10(5), 425–435 (1972a). https://doi.org/10.1016/0020-7225(72)90050-X
Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972b). https://doi.org/10.1016/0020-7225(72)90070-5
Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972). https://doi.org/10.1016/0020-7225(72)90039-0
Fakhrabadi, M.M.S., Khorasani, P.K., Rastgoo, A., Ahmadian, M.T.: Molecular dynamics simulation of pull-in phenomena in carbon nanotubes with Stone–Wales defects. Solid State Commun. 157, 38–44 (2013). https://doi.org/10.1016/j.ssc.2012.12.016
Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42(2), 475–487 (1994). https://doi.org/10.1016/0956-7151(94)90502-9
Fu, Y., Zhang, J.: Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies. Appl. Math. Model. 35, 941–951 (2011). https://doi.org/10.1016/j.apm.2010.07.051
He, J.-H.: Variational approach for nonlinear oscillators. Chaos, Solitons Fractals 34, 1430–1439 (2007). https://doi.org/10.1016/j.chaos.2006.10.026
Hieu, D.V., Hai, N.Q.: Analyzing of nonlinear generalized duffing oscillators using the equivalent linearization method with a weighted averaging. Asian Res. J. Math. 9(1), 1–14 (2018). https://doi.org/10.9734/ARJOM/2018/40684
Hieu, D.V. Hai, N.Q., Hung, D.T.: Analytical approximate solutions for oscillators with fractional order restoring force and relativistic oscillators. Int. J. Innov. Sci. Eng. Technol. 4(12), 28–35 (2017)
Hieu, D.V., Hai, N.Q., Hung, D.T.: The equivalent linearization method with a weighted averaging for solving undamped nonlinear oscillators. J. Appl. Math. 2018, Article ID 7487851 (2018). https://doi.org/10.1155/2018/7487851
Koiter, W.T.: Couple-stresses in the theory of elasticity: I and II. Philos. Trans. R. Soc. Lond. B 67, 17–44 (1964)
Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequency of Bernoulli-Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008). https://doi.org/10.1016/j.ijengsci.2007.10.002
Kroner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3(5), 731–742 (1967). https://doi.org/10.1016/0020-7683(67)90049-2
Krylov, S.: Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures. Int. J. Non-Linear Mech. 42(4), 626–642 (2007). https://doi.org/10.1016/j.ijnonlinmec.2007.01.004
Krylov, N., Bogoliubov, N.: Introduction to Nonlinear Mechanics. Princenton University Press, New York (1943)
Li, L., Hu, Y.: Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int. J. Eng. Sci. 97, 84–94 (2015). https://doi.org/10.1016/j.ijengsci.2015.08.013
Li, L., Hu, Y., Ling, L.: Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory. Compos. Struct. 133, 1079–1092 (2015). https://doi.org/10.1016/j.compstruct.2015.08.014
Lim, C.W., Zhang, G., Reddy, J.N.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015). https://doi.org/10.1016/j.jmps.2015.02.001
Loh, O.Y., Espinosa, H.D.: Nanoelectromechanical contact switches. Nat. Nanotechnol. 7, 283–295 (2012). https://doi.org/10.1038/nnano.2012.40
Lu, L., Guo, X., Zhao, J.: Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. Int. J. Eng. Sci. 116, 12–24 (2017). https://doi.org/10.1016/j.ijengsci.2017.03.006
Luo, A.C.J., Wang, F.Y.: Chaotic motion in a micro-electro-mechanical system with non-linearity from capacitors. Commun. Nonlinear Sci. Numer. Simul. 7, 31–49 (2002). https://doi.org/10.1016/S1007-5704(02)00005-9
Ma, H.M., Gao, X.L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008). https://doi.org/10.1016/j.jmps.2008.09.007
Ma, J.B., Jiang, L., Asokanthan, S.F.: Influence of surface effects on the pull-in instability of NEMS electrostatic switches. Nanotechnology 21(50), 505708 (2010a). https://doi.org/10.1088/0957-4484/21/50/505708
Ma, H.M., Gao, X.L., Reddy, J.N.: A nonclassical Reddy-Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010b). https://doi.org/10.1615/IntJMultCompEng.v8.i2.30
Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N.: Nonlocal and strain gradient based model for electrostatically actuated silicon nanobeams. Microsyst. Technol. 21, 457–464 (2015). https://doi.org/10.1007/s00542-014-2110-2
Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3(1), 1–7 (1963). https://doi.org/10.1007/BF02327219
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964). https://doi.org/10.1007/BF00248490
Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965). https://doi.org/10.1016/0020-7683(65)90006-5
Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415 (1962). https://doi.org/10.1007/BF00253946
Nejad, M.Z., Hadib, A., Rastgoo, A.: Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nanobeams based on nonlocal elasticity theory. Int. J. Eng. Sci. 103, 1–10 (2016). https://doi.org/10.1016/j.ijengsci.2016.03.001
Oh, K.W., Ahn, C.H.: A review of microvalves. J. Micromech. Microeng. 16(5), R13 (2006). https://doi.org/10.1088/0960-1317/16/5/R01
Park, S.K., Gao, X.L.: Bernoulli-Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16(11), 2355 (2006). https://doi.org/10.1088/0960-1317/16/11/015
Qian, Y.H., Ren, D.X., Lai, S.K., Chen, S.M.: Analytical approximations to nonlinear vibration of an electrostatically actuated microbeam. Commun. Nonlinear Sci. Numer. Simul. 17, 1947–1955 (2012). https://doi.org/10.1016/j.cnsns.2011.09.018
Sadeghian, H., Yang, C.K., Goosen, J.F.L., Van der Drift, E., Bossche, A., French, P.J., Van Keulen, F.: Characterizing size-dependent effective elastic modulus of silicon nanocantilevers using electrostatic pull-in instability. Appl. Phys. Lett. 94, 221903 (2009). https://doi.org/10.1063/1.3148774
Sadeghzadeh, S., Kabiri, A.: Application of higher order Hamiltonian approach to the nonlinear vibration of micro electro mechanical systems. Latin Am. J. Solids Struct. 13, 478–497 (2016). https://doi.org/10.1590/1679-78252557
Sedighi, H.M.: Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory. Acta Astronaut. 95, 111–123 (2014). https://doi.org/10.1016/j.actaastro.2013.10.020
Şimşek, M.: Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int. J. Eng. Sci. 105, 12–27 (2016). https://doi.org/10.1016/j.ijengsci.2016.04.013
Stolken, J.S., Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46(14), 5109–5115 (1998). https://doi.org/10.1016/S1359-6454(98)00153-0
Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)
Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur. J. Mech. A. Solids 29, 591–599 (2010). https://doi.org/10.1016/j.euromechsol.2009.12.005
Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002). https://doi.org/10.1016/S0020-7683(02)00152-X
Yiming, F., Zhang, J., Wan, L.: Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Curr. Appl. Phys. 11, 482–485 (2011). https://doi.org/10.1016/j.cap.2010.08.037
Younis, M.I., Abdel-Rahman, E.M., Nayfeh, A.: A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst. 12(5), 672–680 (2003). https://doi.org/10.1109/JMEMS.2003.818069
Zhang, W.-M., Yan, H., Peng, Z.-K., Meng, G.: Electrostatic pull-in instability in MEMS/NEMS: a review. Sens. Actuators A Phys. 214, 187–218 (2014). https://doi.org/10.1016/j.sna.2014.04.025
Acknowledgements
Funding was provided by Vietnam National Foundation for Science and Technology Development (NAFOSTED) (Grant No. 107.04-2018.12).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dang, VH., Nguyen, DA., Le, MQ. et al. Nonlinear vibration of nanobeams under electrostatic force based on the nonlocal strain gradient theory. Int J Mech Mater Des 16, 289–308 (2020). https://doi.org/10.1007/s10999-019-09468-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10999-019-09468-8