Abstract
This paper aims at investigating the size-dependent nonlinear behaviour of a viscoelastic imperfect extensible microbeam taking into account both transverse and longitudinal displacements and inertia. The size-dependent potential energy is formulated in the framework of the modified couple stress theory. The works due to the viscous parts of the stress tensor and the deviatoric part of the couple stress tensor are obtained in terms of system parameters. The kinetic energy as well as the work of external dynamic loading is obtained as functions of the displacement field. Hamilton’s principle is employed in order to balance the work and energy terms which results in the coupled nonlinear equations of motion for the longitudinal and transverse directions. A high-dimensional weighted-residual technique is employed so as to discretise the coupled equations of longitudinal and transverse motions and hence the continuous system with infinite number of degrees of freedom is truncated into a reduced-order model with sufficient degrees of freedom for accurate results capable of capturing almost all modal interactions. This high-dimensional nonlinear coupled reduced-order model is solved for the fundamental coupled nonlinear resonant response via use of a continuation method as well as direct time integration for response characterisation with special consideration to the coupled effect of the viscousity, initial imperfection, and length-scale parameter on the system response in the longitudinal and transverse directions. It is shown that the deviation between the response of viscoelastic and elastic systems is substantial for fairly large excitation forces.
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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)
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)
Antonello, R., Oboe, R., Prandi, L., Biganzoli, F.: Automatic mode matching in MEMS vibrating gyroscopes using extremum-seeking control. IEEE Trans. Ind. Electron. 56, 3880–3891 (2009)
Asghari, M., Kahrobaiyan, M.H., Ahmadian, M.T.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1749–1761 (2010)
Azizi, S., Ghazavi, M.R., Rezazadeh, G., Khadem, S.E.: Thermo-elastic damping in a functionally graded piezoelectric micro-resonator. Int. J. Mech. Mater. Des. 11, 357–369 (2014)
Bethe, K., Baumgarten, D., Frank, J.: Creep of sensor’s elastic elements: metals versus non-metals. Sens. Actuators, A 23, 844–849 (1990)
Chen, S.H., Feng, B.: Size effect in micro-scale cantilever beam bending. Acta Mech. 219, 291–307 (2011)
Chorsi, M.T., Azizi, S., Bakhtiari-Nejad, F.: Application of quadratic controller to control the pull-in instability of a micro-resonator. Int. J. Mech. Mater. Des. 11, 111–123 (2015)
Elwenspoek, M., Jansen, H.V.: Silicon Micromachining. Cambridge University Press, Cambridge (2004)
Farokhi, H., Ghayesh, M.: Nonlinear resonant response of imperfect extensible Timoshenko microbeams. Int. J. Mech. Mater. Des. 1–13 (2015a). doi:10.1007/s10999-015-9316-z
Farokhi, H., Ghayesh, M.: Size-dependent behaviour of electrically actuated microcantilever-based MEMS. Int. J. Mech. Mater. Desi. 12, 301–315 (2015b)
Farokhi, H., Ghayesh, M.H.: Size-dependent behaviour of electrically actuated microcantilever-based MEMS. Int. J. Mech. Mater. Des. (2015c)
Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W.: Mechanism-based strain gradient plasticity— I. Theory. J. Mech. Phys. Solids 47, 1239–1263 (1999)
Ghayesh, M.H., Farokhi, H., Amabili, M.: In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Compos. B Eng. 60, 423–439 (2014)
Haque, M.A., Saif, M.T.A.: Strain gradient effect in nanoscale thin films. Acta Mater. 51, 3053–3061 (2003)
Koutsawa, Y., Haberman, M.R., Daya, E.M., Cherkaoui, M.: Multiscale design of a rectangular sandwich plate with viscoelastic core and supported at extents by viscoelastic materials. Int. J. Mech. Mater. Des. 5, 29–44 (2009)
Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)
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)
McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15, 1060 (2005)
Mohamed, K.T., Ata, A.A., El-Souhily, B.M.: Dynamic analysis algorithm for a micro-robot for surgical applications. Int. J. Mech. Mater. Des. 7, 17–28 (2011)
Esfahani, A. M., Bahrami, M.: Vibration analysis of a circular thin polymeric piezoelectric diaphragm with fluid interaction. Int. J. Mech. Mater. Des. 12, 401–411 (2016)
Nateghi, A., Salamat-talab, M., Rezapour, J., Daneshian, B.: Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory. Appl. Math. Model. 36, 4971–4987 (2012)
Ramezani, S.: A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory. Int. J. Non-Linear Mech. 47, 863–873 (2012)
Salamat-talab, M., Nateghi, A., Torabi, J.: Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory. Int. J. Mech. Sci. 57, 63–73 (2012)
Teh, K.S., Lin, L.: Time-dependent buckling phenomena of polysilicon micro beams. Microelectron. J. 30, 1169–1172 (1999)
Tuck, K., Jungen, A., Geisberger, A., Ellis, M., Skidmore, G.: A study of creep in polysilicon MEMS devices. J. Eng. Mater. Technol. 127, 90–96 (2005)
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)
Zhang, J., Fu, Y.: Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory. Meccanica 47, 1649–1658 (2012)
Zhang, W.M., Meng, G.: Nonlinear dynamic analysis of electrostatically actuated resonant MEMS sensors under parametric excitation. IEEE Sens. J. 7, 370–380 (2007)
Zhang, D., Gao, Z., Fassi, I.: Design optimization of a spatial hybrid mechanism for micromanipulation. Int. J. Mech. Mater. Des. 7, 55–70 (2011)
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The financial support to this research by the start-up grant of the University of Adelaide is gratefully acknowledged.
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Ghayesh, M.H., Farokhi, H. Viscoelastically coupled size-dependent behaviour of imperfect extensible microbeams. Int J Mech Mater Des 13, 569–581 (2017). https://doi.org/10.1007/s10999-016-9356-z
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DOI: https://doi.org/10.1007/s10999-016-9356-z