×

On the dynamics of imperfect shear deformable microplates. (English) Zbl 1423.74542

Summary: This paper investigates the nonlinear forced dynamical behaviour of a geometrically imperfect viscoelastic shear-deformable microplate. The third-order shear deformation plate theory and the Kelvin-Voigt viscoelastic model are utilised in the framework of the modified version of the couple-stress theory to develop a model for the microsystem. The developed model is in the form of partial differential equations (PDEs) and accounts for geometric nonlinearities, damping nonlinearities, micro-scale size effects, and initial imperfection. Five coupled PDEs are derived for the five independent displacements and rotations. These PDEs are truncated to a set of nonlinearly coupled ordinary differential equations via application of a two-dimensional modal decomposition based on the Galerkin technique. The final set of equations consists of quadratic and cubic nonlinear terms for both damping and stiffness. An efficient numerical algorithm based on a continuation scheme is utilised to analyse the nonlinear forced vibration characteristics of such complicated system. The effects imperfection amplitude, damping nonlinearities, and micro-scale size on forced resonant vibration response are highlighted.

MSC:

74K20 Plates
74H50 Random vibrations in dynamical problems in solid mechanics
70K30 Nonlinear resonances for nonlinear problems in mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ashoori Movassagh, A.; Mahmoodi, M. J., A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory, European Journal of Mechanics - A/Solids, 40, 50-59, (2013) · Zbl 1406.74067
[2] Ashoori, A.; Mahmoodi, M. J., A nonlinear thick plate formulation based on the modified strain gradient theory, Mechanics of Advanced Materials and Structures, 1-7, (2017)
[3] Attia, M. A.; Abdel Rahman, A. A., On vibrations of functionally graded viscoelastic nanobeams with surface effects, International Journal of Engineering Science, 127, 1-32, (2018) · Zbl 1423.74383
[4] Bahaadini, R.; Saidi, A. R.; Hosseini, M., On dynamics of nanotubes conveying nanoflow, International Journal of Engineering Science, 123, 181-196, (2018) · Zbl 1423.74456
[5] Bakhshi Khaniki, H.; Hosseini-Hashemi, S., Dynamic response of biaxially loaded double-layer viscoelastic orthotropic nanoplate system under a moving nanoparticle, International Journal of Engineering Science, 115, 51-72, (2017) · Zbl 1423.74534
[6] Chen, S. H.; Feng, B., Size effect in micro-scale cantilever beam bending, Acta Mechanica, 219, 291-307, (2011) · Zbl 1333.74057
[7] Dehrouyeh-Semnani, A. M.; BehboodiJouybari, M.; Dehrouyeh, M., On size-dependent lead-lag vibration of rotating microcantilevers, International Journal of Engineering Science, 101, 50-63, (2016) · Zbl 1423.74385
[8] Farokhi, H.; Ghayesh, M. H., Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory, International Journal of Mechanical Sciences, 90, 133-144, (2015)
[9] Farokhi, H.; Ghayesh, M. H., Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams, International Journal of Engineering Science, 91, 12-33, (2015) · Zbl 1423.74694
[10] Farokhi, H.; Ghayesh, M. H., Size-dependent parametric dynamics of imperfect microbeams, International Journal of Engineering Science, 99, 39-55, (2016) · Zbl 1423.74472
[11] Farokhi, H.; Ghayesh, M. H., Nonlinear resonant response of imperfect extensible Timoshenko microbeams, International Journal of Mechanics and Materials in Design, 13, 43-55, (2017)
[12] Farokhi, H.; Ghayesh, M. H., Supercritical nonlinear parametric dynamics of Timoshenko microbeams, Communications in Nonlinear Science and Numerical Simulation, 59, 592-605, (2018)
[13] Farokhi, H.; Ghayesh, M. H., Nonlinear mechanics of electrically actuated microplates, International Journal of Engineering Science, 123, 197-213, (2018) · Zbl 1423.74541
[14] Farokhi, H.; Ghayesh, M. H.; Amabili, M., Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory, International Journal of Engineering Science, 68, 11-23, (2013) · Zbl 1423.74473
[15] Farokhi, H.; Ghayesh, M. H.; Gholipour, A.; Hussain, S., Motion characteristics of bilayered extensible Timoshenko microbeams, International Journal of Engineering Science, 112, 1-17, (2017) · Zbl 1423.74475
[16] Farokhi, H.; Ghayesh, M. H.; Hussain, S., Large-amplitude dynamical behaviour of microcantilevers, International Journal of Engineering Science, 106, 29-41, (2016) · Zbl 06984713
[17] Fleck, N. A.; Muller, G. M.; Ashby, M. F.; Hutchinson, J. W., Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia, 42, 475-487, (1994)
[18] Ghayesh, M. H., Stability characteristics of an axially accelerating string supported by an elastic foundation, Mechanism and Machine Theory, 44, 1964-1979, (2009) · Zbl 1178.70026
[19] Ghayesh, M. H., On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: parametric study, Acta Mechanica Solida Sinica, 24, 4, 373-382, (2011)
[20] Ghayesh, M. H., Dynamics of functionally graded viscoelastic microbeams, International Journal of Engineering Science, 124, 115-131, (2018) · Zbl 1423.74186
[21] Ghayesh, M. H., Functionally graded microbeams: simultaneous presence of imperfection and viscoelasticity, International Journal of Mechanical Science, 140, 339-350, (2018)
[22] Ghayesh, M. H., Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams, Applied Mathematical Modelling, 59, 583-596, (2018)
[23] Ghayesh, M. H., Mechanics of tapered AFG shear-deformable microbeams, Microsystem Technologies, 24, 1743-1754, (2018)
[24] Ghayesh, M. H., Nonlinear vibrations of axially functionally graded Timoshenko tapered beams, Journal of Computational and Nonlinear Dynamics, 13, 4, (2018)
[25] Ghayesh, M. H., Nonlinear dynamics of multilayered microplates, Journal of Computational and Nonlinear Dynamics, 13, 2, (2018)
[26] Ghayesh, M. H.; Amabili, M.; Farokhi, H., Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory, International Journal of Engineering Science, 63, 52-60, (2013) · Zbl 1423.74392
[27] Ghayesh, M. H.; Amabili, M.; Farokhi, H., Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams, International Journal of Engineering Science, 71, 1-14, (2013) · Zbl 1423.74479
[28] Ghayesh, M. H.; Amabili, M.; Paidoussis, M. P., Nonlinear vibrations and stability of an axially moving beam with an intermediate spring-support: two-dimensional analysis, Nonlinear Dynamics, 70, 1, 335-354, (2012)
[29] Ghayesh, M. H.; Farokhi, H., Nonlinear dynamics of microplates, International Journal of Engineering Science, 86, 60-73, (2015) · Zbl 1423.74543
[30] Ghayesh, M. H.; Farokhi, H., Chaotic motion of a parametrically excited microbeam, International Journal of Engineering Science, 96, 34-45, (2015) · Zbl 1423.74480
[31] Ghayesh, M. H.; Farokhi, H., On the viscoelastic dynamics of fluid-conveying microtubes, International Journal of Engineering Science, 127, 186-200, (2018) · Zbl 1423.76034
[32] Ghayesh, M. H.; Paidoussis, M. P.; Amabili, M., Nonlinear dynamics of cantilevered extensible pipes conveying fluid, Journal of Sound and Vibration, 332, 6405-6418, (2013)
[33] Ghayesh, M. H.; Amabili, M., Nonlinear dynamics of axially moving viscoelastic beams over the buckled state, Computers and Structures, 112-113, 406-421, (2012)
[34] Ghayesh, M. H.; Amabili, M., Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam, Composites Part B: engineering, 60, 371-377, (2014)
[35] Ghayesh, M. H.; Farokhi, H.; Alici, G., Size-dependent performance of microgyroscopes, International Journal of Engineering Science, 100, 99-111, (2016) · Zbl 1423.70010
[36] Ghayesh, M. H.; Farokhi, H.; Amabili, M., Nonlinear dynamics of a microscale beam based on the modified couple stress theory, Composites Part B: Engineering, 50, 318-324, (2013)
[37] Ghayesh, M. H.; Farokhi, H.; Amabili, M., Nonlinear behaviour of electrically actuated MEMS resonators, International Journal of Engineering Science, 71, 137-155, (2013) · Zbl 06976164
[38] Ghayesh, M. H.; Farokhi, H.; Amabili, M., In-plane and out-of-plane motion characteristics of microbeams with modal interactions, Composites Part B: Engineering, 60, 423-439, (2014)
[39] Ghayesh, M. H.; Farokhi, H.; Gholipour, A., Oscillations of functionally graded microbeams, International Journal of Engineering Science, 110, 35-53, (2017) · Zbl 1406.74291
[40] Ghayesh, M. H.; Farokhi, H.; Gholipour, A., Vibration analysis of geometrically imperfect three-layered shear-deformable microbeams, International Journal of Mechanical Sciences, 122, 370-383, (2017)
[41] Ghayesh, M. H.; Farokhi, H.; Gholipour, A.; Tavallaeinejad, M., Nonlinear oscillations of functionally graded microplates, International Journal of Engineering Science, 122, 56-72, (2018) · Zbl 06985843
[42] Ghayesh, M. H.; Farokhi, H.; Hussain, S., Viscoelastically coupled size-dependent dynamics of microbeams, International Journal of Engineering Science, 109, 243-255, (2016) · Zbl 1423.74188
[43] Gholipour, A.; Farokhi, H.; Ghayesh, M. H., In-plane and out-of-plane nonlinear size-dependent dynamics of microplates, Nonlinear Dynamics, 79, 1771-1785, (2015)
[44] Ghorbani Shenas, A.; Malekzadeh, P., Free vibration of functionally graded quadrilateral microplates in thermal environment, Thin-Walled Structures, 106, 294-315, (2016)
[45] Hadi, A.; Nejad, M. Z.; Hosseini, M., Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, 128, 12-23, (2018)
[46] Haque, M. A.; Saif, M. T.A., Strain gradient effect in nanoscale thin films, Acta Materialia, 51, 3053-3061, (2003)
[47] Hashemi, S. H.; Samaei, A. T., Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory, Physica E: Low-dimensional Systems and Nanostructures, 43, 1400-1404, (2011)
[48] Hosseini, M.; Bahaadini, R., Size dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory, International Journal of Engineering Science, 101, 1-13, (2016) · Zbl 06984666
[49] Jomehzadeh, E.; Noori, H. R.; Saidi, A. R., The size-dependent vibration analysis of micro-plates based on a modified couple stress theory, Physica E: Low-dimensional Systems and Nanostructures, 43, 877-883, (2011)
[50] Khaniki, H. B., On vibrations of nanobeam systems, International Journal of Engineering Science, 124, 85-103, (2018) · Zbl 1423.74396
[51] Kiani, K., Thermo-elasto-dynamic analysis of axially functionally graded non-uniform nanobeams with surface energy, International Journal of Engineering Science, 106, 57-76, (2016) · Zbl 1423.74491
[52] Lam, D. C.C.; Yang, F.; Chong, A. C.M.; Wang, J.; Tong, P., Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, 51, 1477-1508, (2003) · Zbl 1077.74517
[53] Li, L.; Tang, H.; Hu, Y., The effect of thickness on the mechanics of nanobeams, International Journal of Engineering Science, 123, 81-91, (2018) · Zbl 1423.74497
[54] Li, Y. S.; Pan, E., Static bending and free vibration of a functionally graded piezoelectric microplate based on the modified couple-stress theory, International Journal of Engineering Science, 97, 40-59, (2015) · Zbl 1423.74401
[55] McFarland, A. W.; Colton, J. S., Role of material microstructure in plate stiffness with relevance to microcantilever sensors, Journal of Micromechanics and Microengineering, 15, 1060, (2005)
[56] Medina, L.; Gilat, R.; Krylov, S., Latching in bistable electrostatically actuated curved micro beams, International Journal of Engineering Science, 110, 15-34, (2017) · Zbl 1423.74313
[57] Mojahedi, M., Size dependent dynamic behavior of electrostatically actuated microbridges, International Journal of Engineering Science, 111, 74-85, (2017) · Zbl 1423.74640
[58] Mojahedi, M.; Rahaeifard, M., A size-dependent model for coupled 3D deformations of nonlinear microbridges, International Journal of Engineering Science, 100, 171-182, (2016) · Zbl 1423.74504
[59] Nabian, A.; Rezazadeh, G.; Almassi, M.; Borgheei, A.-M., On the stability of a functionally graded rectangular micro-plate subjected to hydrostatic and nonlinear electrostatic pressures, Acta Mechanica Solida Sinica, 26, 205-220, (2013)
[60] Qi, L.; Huang, S.; Fu, G.; Zhou, S.; Jiang, X., On the mechanics of curved flexoelectric microbeams, International Journal of Engineering Science, 124, 1-15, (2018) · Zbl 1423.74317
[61] Shahverdi, H.; Barati, M. R., Vibration analysis of porous functionally graded nanoplates, International Journal of Engineering Science, 120, 82-99, (2017) · Zbl 1423.74408
[62] She, G.-L.; Yuan, F.-G.; Ren, Y.-R.; Xiao, W.-S., On buckling and postbuckling behavior of nanotubes, International Journal of Engineering Science, 121, 130-142, (2017) · Zbl 1423.74353
[63] Taati, E., Analytical solutions for the size dependent buckling and postbuckling behavior of functionally graded micro-plates, International Journal of Engineering Science, 100, 45-60, (2016) · Zbl 1423.74354
[64] Wang, B.; Zhou, S.; Zhao, J.; Chen, X., A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory, European Journal of Mechanics - A/Solids, 30, 517-524, (2011) · Zbl 1278.74103
[65] Wang, Y.-G.; Lin, W.-H.; Zhou, C.-L., Nonlinear bending of size-dependent circular microplates based on the modified couple stress theory, Archive of Applied Mechanics, 84, 391-400, (2014) · Zbl 1355.74054
[66] Zaitsev, S.; Shtempluck, O.; Buks, E.; Gottlieb, O., Nonlinear damping in a micromechanical oscillator, Nonlinear Dynamics, 67, 859-883, (2012) · Zbl 1314.70027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.