Forced Oscillation of Simply-Supported Microbeams Considering Nonlinear Effects

Year 2011, Volume: 3 Issue: 1, 27 - 41, 01.03.2011

M. Rafiee A. Nezamabadi

Abstract

Nonlinear free and forced oscillation of microscale simply supported beams is investigated in this paper. Introducing a material length scale parameter, the nonlinear model is conducted within the context of non-classical continuum mechanics. By using a combination of the modified couple stress theory and Hamilton’s principle the nonlinear equation of motion is derived. The nonlinear frequencies of a beam with initial lateral displacement are discussed. Equations have been solved using an exact method for free vibration and multiple times scales (MTS) method for forced vibration and some analytical relations have been obtained for natural frequency of oscillations. The results have been compared with previous work and good agreement has been obtained. Also forced vibrations of system in primary resonance have been studied and the effects of different parameters on the frequency-response have been investigated. It is shown that the size effect is significant when the ratio of characteristic thickness to internal material length scale parameter is approximately equal to one, but is diminishing with the increase of the ratio. Our results also indicate that the nonlinearity has a great effect on the vibration behavior of microscale beams

Keywords

Microbeam, Free and forced vibration, nonlinear vibration

References

• M. Moghimi Zand, M.T. Ahmadian, Characterization of coupled-domain multi-layer microplates in pull-in phenomenon, International Journal of Mechanical Sciences 49 (2007) –1237.
• C.W. Lim, L.H. He, Size-dependent nonlinear response of thin elastic films with nano- scale thickness, International Journal of Mechanical Sciences 46 (2004) 1715-1726.
• J. Pei, F. Tian, T. Thundat, Glucose biosensor based on the microcantilever, Analytical Chemistry 76 (2004) 292–297.
• M. Li, H.X. Tang, M.L. Roukes, Ultra-sensitive NEMS-based cantilevers for sensing, scanned probe and very high-frequency applications, Nature Nanotechnology 2 (2007) 114–120.
• N.A. Stelmashenko, M.G. Walls, L.M. Brown, Y.V. Milman, Microindentations on W and Mo oriented single crystals: an STM study, Acta Metallurgica et Materialia 41 (1993) 2855–2865.
• W.J. Poole, M.F. Ashby, N.A. Fleck, Micro-hardness of annealed and work-hardened copper polycrystals, Scripta Materialia 34 (1996) 559–564.
• S.K. Park, X.-L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering 16 (2006) 2355–2359.
• M. Asghari, M.T. Ahmadian, M.H. Kahrobaiyan, M. Rahaeifard, On the size-dependent behavior of functionally graded micro beams, Materials and Design 31 (2010) 2324–2329.
• L. Wang, Size-dependent vibration characteristics of microtubes conveying fluid, Journal of Fluids and Structures, in press. doi:10.1016/ j.jfluidstructs.2010.02.005.
• L. Wang, Dynamical behaviors of double-walled carbon nanotubes conveying fluid accounting for the role of small length scale, Computational Materials Science 45 (2009) 584–588.
• N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia 42 (1994) 475–487.
• Q. Ma, D.R. Clarke, Size dependent hardness of silver single crystals, Journal of Materials Research 10 (1995) 853–863.
• J.S. Stolken, A.G. Evans, Microbend test method for measuring the plasticity length scale, Acta Materialia 46 (1998) 5109–5115.
• A.C.M. Chong, D.C.C. Lam, Strain gradient plasticity effect in indentation hardness of polymers, Journal of Materials Research 14 (1999) 4103–4110.
• D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids (2003) 1477–1508.
• A.W. McFarland, J.S. Colton, Role of material microstructure in plate stiffness with relevance to microcantilever sensors, Journal of Micromechanics and Microengineering 15 (2005) 1060–1067.
• R.D. Mindlin, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis 16 (1964) 51–78.
• R.D. Mindlin, H.F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11 (1962) 415–448.
• R.A. Toupin, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11 (1962) 385–414.
• F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39 (2002) 2731–2743.
• S.L. Kong, S.J. Zhou, Z.F. Nie, K. Wang, The size-dependent natural frequency of Bernoulli–Euler micro-beams, International Journal of Engineering Science 46 (2008) 427–
• H. Sadeghian, G. Rezazadeh, P.M. Osterberg, Application of the generalized differential quadrature method to the study of pull-In phenomena of MEMS switches, Journal of Microelectromechanical Systems 16 (2007) 1334–1340.
• C.W. Bert, M. Malik, Differential quadrature method in computational mechanics: a review, Applied Mechanics Reviews 49 (1996) 1–27.
• A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, John Wiley, New York, 1979.
• M.A. Foda, On non-linear free vibrations of a beam with pinned ends, Journal of King Saud University 7 (1995) 93–107.
• Byrd, P.F., Friedman, M.D., 1971, handbook of Elliptic Integrals for Engineers and Scientis, Berlin.
• Zhou SJ, Li ZQ. Length scales in the static and dynamic torsion of a circular cylindrical micro-bar. J Shandong Univ Technol 2001;31(5):401–7.
• Kang X, Xi XF. Size effect on the dynamic characteristic of a micro beam based on Cosserat theory. J Mech Strength 2007;29(1):1–4.