Abstract
The presented paper investigates the nonlinear vibration of a nanobeam with a piezoelectric layer bounded to its top surface considering the nonlocal piezoelectricity theory. To do this, Hamilton’s principle is implemented to derive the governing nonlinear vibration equations of the nanobeam by assumption of nonlocal piezoelectricity and a nonlinear strain–displacement relation. Then, the Galerkin separation method is applied to transform and simplify the partial differential equation of the nonlinear oscillation to an ordinary one with quadratic and cubic nonlinearities in the time domain. By implementing the multiple-scale perturbation method, an analytical relation for the nonlinear natural frequencies is obtained as a function of the oscillation amplitude and the nonlocal size scale parameter. Then, the nonlinear vibration characteristics of the nanobeam are investigated at higher modes of vibration and the size scale effects are reviewed comprehensively. It is observed that the nonlocal parameter decreases the nonlinear natural frequencies and becomes noticeable at higher modes of vibration. Moreover, by increasing the amplitude ratio, the nonlocal effects are decreased and the nonlocal nonlinear frequency approaches the local one. Also, the amplitude ratio has increasing effects on the nonlinear frequencies.
Similar content being viewed by others
References
Bakhtiari-Nejad, F., Nazemizadeh, M.: Size-dependent dynamic modeling and vibration analysis of MEMS/NEMS-based nanomechanical beam based on the nonlocal elasticity theory. Acta Mech. 227(5), 1363–1379 (2016)
Giannopoulos, G.I., Georgantzinos, S.K.: Establishing detection maps for carbon nanotube mass sensors: molecular versus continuum mechanics. Acta Mech. 228(6), 2377–2390 (2017)
Goldstein, R.V., Gorodtsov, V.A., Lisovenko, D.S.: Chiral elasticity of nano/microtubes from hexagonal crystals. Acta Mech. 229(5), 2189–2201 (2018)
Piekarski, B., DeVoe, D., Dubey, M., Kaul, R., Conrad, J.: Surface micromachined piezoelectric resonant beam filters. Sens. Actuators, A 91(3), 313–320 (2001)
Lazarus, A., Thomas, O., Deü, J.F.: Finite element reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS. Finite Elem. Anal. Des. 49(1), 35–51 (2012)
Piekarski, B., Dubey, M., Zakar, E., Polcawich, R., Devoe, D., Wickenden, D.: Sol-gel PZT for MEMS applications. Integr. Ferroelectr. 42(1), 25–37 (2002)
Korayem, M.H., Badkoobeh, H.H., Taheri, M.: Dynamic modeling and simulation of rough cylindrical micro/nanoparticle manipulation with atomic force microscopy. Microscopy and microanalysis: the official journal of Microscopy Society of America, Microbeam Analysis Society, Microscopical Society of Canada 1–16 (2014)
Demir, Ç., Civalek, Ö., Akgöz, B.: Free vibration analysis of carbon nanotubes based on shear deformable beam theory by discrete singular convolution technique. Math. Comput. Appl. 15(1), 57–65 (2010)
Alibeigi, B., Beni, Y.T., Mehralian, F.: On the thermal buckling of magneto-electro-elastic piezoelectric nanobeams. Eur. Phys. J. Plus 133(3), 133 (2018)
Abdollahi, M., Saidi, A.R., Mohammadi, M.: Buckling analysis of thick functionally graded piezoelectric plates based on the higher-order shear and normal deformable theory. Acta Mech. 226(8), 2497–2510 (2015)
Lumentut, M.F., Shu, Y.C.: A unified electromechanical finite element dynamic analysis of multiple segmented smart plate energy harvesters: circuit connection patterns. Acta Mech. 229(11), 4575–4604 (2018)
Firdaus, S.M., Azid, I.A., Sidek, O., Ibrahim, K., Hussien, M.: Enhancing the sensitivity of a mass-based piezoresistive micro-electro-mechanical systems cantilever sensor. Micro Nano Lett. IET 5(2), 85–90 (2010)
Zhang, W., Meng, G., Li, H.: Adaptive vibration control of micro-cantilever beam with piezoelectric actuator in MEMS. Int. J. Adv. Manuf. Technol. 28(3), 321–327 (2006)
Souayeh, S., Kacem, N., Najar, F., & Foltête, E.: Nonlinear dynamics of parametrically excited carbon nanotubes for mass sensing applications. In: ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Crete Island, Greece, 25–27 May (2015)
SoltanRezaee, M., Bodaghi, M., Farrokhabadi, A., Hedayati, R.: Nonlinear stability analysis of piecewise actuated piezoelectric microstructures. Int. J. Mech. Sci. 160, 200–208 (2019)
Zhu, H.T., Zbib, H.M., Aifantis, E.C.: Strain gradients and continuum modeling of size effect in metal matrix composites. Acta Mech. 121(1–4), 165–176 (1997)
Agrawal, R., Peng, B., Gdoutos, E.E., Espinosa, H.D.: Elasticity size effects in ZnO nanowires\(-\) a combined experimental-computational approach. Nano Lett. 8(11), 3668–3674 (2008)
Mohammed, Y., Hassan, M.K., El-Ainin, H.A., Hashem, A.M.: Size effect analysis of open-hole glass fiber composite laminate using two-parameter cohesive laws. Acta Mech. 226(4), 1027–1044 (2015)
Giannopoulos, G.I.: Fullerenes as mass sensors: a numerical investigation. Physica E 56, 36–42 (2014)
Feng, C., Jiang, L.Y.: Molecular dynamics simulation of squeeze-film damping effect on nano resonators in the free molecular regime. Physica E 43(9), 1605–1609 (2011)
Akgöz, B., Civalek, Ö.: Free vibration analysis of axially functionally graded tapered Bernoulli-Euler micro beams based on the modified couple stress theory. Compos. Struct. 98, 314–322 (2013)
Akgöz, B., Civalek, Ö.: Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int. J. Eng. Sci. 49(11), 1268–1280 (2011)
Beni, Y.T.: A nonlinear electro-mechanical analysis of nanobeams based on the size-dependent piezoelectricity theory. J. Mech. 33(3), 289–301 (2017)
Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972)
Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41(3), 305–312 (2003)
Civalek, Ö., Demir, Ç.: Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory. Appl. Math. Model. 35(5), 2053–2067 (2011)
Demir, Ç., Civalek, Ö.: Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models. Appl. Math. Model. 37(22), 9355–9367 (2013)
Asemi, S.R., Farajpour, A., Mohammadi, M.: Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory. Compos. Struct. 116, 703–712 (2014)
Nazemizadeh, M., Bakhtiari-Nejad, F.: Size-dependent free vibration of nano/microbeams with piezo-layered actuators. Micro Nano Lett. 10(2), 93–98 (2015)
Nazemizadeh, M., Bakhtiari-Nejad, F.: A general formulation of quality factor for composite micro/nano beams in the air environment based on the nonlocal elasticity theory. Compos. Struct. 132(15), 772–783 (2015)
Kiani, K., Pakdaman, H.: Nonlocal vibrations and potential instability of monolayers from double-walled carbon nanotubes subjected to temperature gradients. Int. J. Mech. Sci. 144, 576–599 (2018)
Mercan, K., Numanoglu, H.M., Akgöz, B., Demir, C., Civalek, Ö.: Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix. Arch. Appl. Mech. 87(11), 1797–1814 (2017)
Demir, C., Mercan, K., Numanoglu, H.M., Civalek, O.: Bending response of nanobeams resting on elastic foundation. J. Appl. Comput. Mech. 4(2), 105–114 (2018)
Zhou, Z.G., Wu, L.Z., Du, S.Y.: Non-local theory solution for a Mode I crack in piezoelectric materials. Eur. J. Mech. A/Solids 25(5), 793–807 (2006)
Li, H., Preidikman, S., Balachandran, B., Mote Jr., C.D.: Nonlinear free and forced oscillations of piezoelectric microresonators. J. Micromech. Microeng. 16(2), 356 (2006)
Li, C., Lim, C.W., Yu, J.L.: Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load. Smart Mater. Struct. 20(1), 015023 (2011)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)
Lestari, W., Hanagud, S.: Nonlinear vibration of buckled beams: some exact solutions. Int. J. Solids Struct. 38(26), 4741–4757 (2001)
Rao, G.V., Raju, K.K., Raju, I.S.: Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates. Comput. Struct. 6(3), 169–172 (1976)
Author information
Authors and Affiliations
Corresponding author
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
Nazemizadeh, M., Bakhtiari-Nejad, F., Assadi, A. et al. Nonlinear vibration of piezoelectric laminated nanobeams at higher modes based on nonlocal piezoelectric theory. Acta Mech 231, 4259–4274 (2020). https://doi.org/10.1007/s00707-020-02736-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-020-02736-1