In this paper, the nonlinear forced-vibration of Euler-Bernoulli beams with large deflections is investigated based on the modified couple stress theory, a non-classical theory capable of capturing size effects. The classical theory is unable to predict the size effects. In systems with the dimensions in order of microns and sub-microns the size effects are very significant. For some specific beams subjected to a concentrated force at its middle as the harmonic exciter, the size-dependent responses are investigated for primary, super-harmonic and sub-harmonic resonances. The results show that the frequency-responses of the system are highly size-dependent.
AsghariMKahrobaiyanMHAhmadianMT (2010a) A nonlinear Timoshenko beam formulation based on the modified couple stress theory. International Journal of Engineering Science48: 1749–1761.
2.
AsghariMKahrobaiyanMHRahaeifardMAhmadianMT (2010b) Investigation of the size effect in Timoshenko beams based on the couple stress theory. Archive of Applied Mechanics[in press]. doi:10.1007/s00419-010-0452-5.
3.
BanksHInmanD (1991) On damping mechanism in beams. ASME Journal of Applied Mechanics58: 716–723.
4.
BatraRCPorfiriMSpinelloD (2008a) Vibrations and pull-in instabilities of microelectromechanical von Kármán elliptic plates incorporating the Casimir force. Journal of Sound and Vibration315: 939–960. doi: 10.1016/j.jsv.2008.02.008.
5.
BatraRCPorfiriMSpinelloD (2008b) Vibrations of narrow microbeams predeformed by an electric field. Journal of Sound and Vibration309: 600–612. doi: 10.1016/j.jsv.2007.07.030.
6.
BauchauOBottassoC (1994) Space-time perturbation modes for non-linear dynamic analysis of beams. Non-linear Dynamics6: 21–35.
7.
BegleyMRHutchinsonJW (1998) The mechanics of size-dependent indentation. Journal of the Mechanics and Physics of Solids46: 2049–2068. doi: 10.1016/S0022-5096((98)00018-0.
8.
ChaterjeeSPohitG (2009) A large deflection model for the pull-in analysis of electrostatically actuated microcantilever beams. Journal of Sound and Vibration322: 969–986. doi: 10.1016/j.jsv.2008.11.046.
9.
ChongACMLamDCC (1999) Strain gradient plasticity effect in indentation hardness of polymers. Materials Research14(10): 4103–4110.
10.
FleckNAMullerGMAshbyMFHutchinsonJW (1992) Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia42(2): 475–487.
11.
GaoYBrennanMJJosephPFMuggletonJMHunaidiO (2005) On the selection of acoustic/vibration sensors for leak detection in plastic water pipes. Journal of Sound and Vibration283: 927–941. doi: 10.1016/j.jsv.2004.05.004.
12.
HamdanMNAbdallahSAl-QaisiaA (2010) Modeling and study of dynamic performance of a valveless micropump. Journal of Sound and Vibration329: 3121–3136. doi: 10.1016/j.jsv.2010.01.035.
13.
HaoZ (2008) Thermoelastic damping in the contour-mode vibrations of micro- and nano-electromechanical circular thin-plate resonators. Journal of Sound and Vibration313: 77–96. doi: 10.1016/j.jsv.2007.11.035.
14.
HassanpourPAEsmailzadehECleghornWLMillsJK (2010) Nonlinear vibration of micromachined asymmetric resonators. Journal of Sound and Vibration329: 2547–2564. doi: 10.1016/j.jsv.2009.10.033.
15.
HinoJYoshimuraTAnanthanarayanaN (1985) Vibration analysis of non-linear beams subjected to a moving load using the finite element method. Journal of Sound and Vibration100(4): 477–491.
16.
KahrobaiyanMHAsghariMRahaeifardMAhmadianMT (2010) Investigation of the size dependent dynamic characteristics of AFM microcantilevers. International Journal of Engineering Science[in press]. doi:10.1016/j.ijengsci.2010.06.003.
17.
KeL-LWangY-SYangJKitipornchaiS (2011) Nonlinear free vibration of size dependent functionally graded microbeams. International Journal of Engineering Science[in press]. doi:10.1016/j.ijengsci.2010.12.008.
18.
KhademSERezaeeM (2002) Non-linear free vibration analysis of a string under bending moment effects using the perturbation method. Journal of Sound and Vibration254(4): 677–691.
19.
KoiterWT (1964) Couple-stresses in the theory of elasticity: I and II Proc. K. Ned. Akad. Wet. B 6717–44.
20.
KongSZhouSNieZWangK (2008) The size-dependent natural frequency of Bernoulli–Euler micro-beams. International Journal of Engineering Science46: 427–437. doi: 10.1016/S0020-7683((02)00152-X.
21.
LamDCCYangFChongACMWangJTongP (2003) Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids51: 1477–1508. doi: 10.1016/S0022-5096((03)00053-X.
22.
MaHMGaoXLReddyJN (2008) A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids56: 3379–3391. doi: 10.1016/j.jmps.2008.09.007.
23.
MahdaviMHFarshidianfarATahaniMMahdaviSDalirH (2008) A more comprehensive modeling of atomic force microscope cantilever. Ultramicroscopy109: 54–60. doi: 10.1016/j.ultramic.2008.08.003.
24.
McFarlandAWColtonJS (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. Journal of Micromechanics and Microengineering15(5): 1060–1067.
25.
MindlinRDTierstenHF (1962) Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis11(1): 415–448.
26.
Moghimi ZandMAhmadianMT (2009a) Application of homotopy analysis method in studying dynamic pull-in instability of microsystems. Mechanics Research Communications36: 851–858. doi: 10.1016/j.mechrescom. 2009.03.004.
27.
Moghimi ZandMAhmadianMT (2009b) Vibrational analysis of electrostatically actuated microstructures considering nonlinear effects. Communications in Nonlinear Science and Numerical Simulation14: 1664–1678. doi: 10.1016/j.cnsns.2008.05.009.
MookDTQuangNHPlautRH (1986) The influence of an internal resonance on non-linear structural vibrations under combination resonance conditions. Journal of Sound and Vibration104(2): 229–241.
30.
NayfehAH (1979) Nonlinear Oscillation. New York: John Wiley and Sons.
31.
NixWDGaoH (1998) Indentation size effects in crystalline materials: a law for strain gradient plasticity. Journal of the Mechanics and Physics of Solids46: 411–425. doi: 10.1016/S0022-5096((97)00086-0.
32.
ParkSKGaoXL (2006) Bernoulli–Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering16: 2355–2359. doi: 10.1088/0960-1317/16/11/015.
33.
PorfiriM (2008) Vibrations of parallel arrays of electrostatically actuated microplates. Journal of Sound and Vibration315: 1071–1085. doi: 10.1016/j.jsv.2008.02.007.
34.
RaoSS (2007) Vibration of Continuous Systems. Hoboken, NJ: John Wiley and Sons, Inc.
35.
SinhaA (2005) Nonlinear dynamics of atomic force microscope with PI feedback. Journal of Sound and Vibration288: 387–394. doi: 10.1016/j.jsv.2005.03.026.
36.
StolkenJSEvansAG (1998) Microbend test method for measuring the plasticity length scale. Acta Materialia46(14): 5109–5115.
37.
ToupinRA (1962) Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis11(1): 385–414.
38.
VoglaAWangDTStorasPBakkeTTakloMMVThomsonA (2009) Design, process and characterisation of a high-performance vibration sensor for wireless condition monitoring. Sensors and Actuators A153: 155–161. doi: 10.1016/j.sna.2009.04.011.
39.
WangL (2010) Size-dependent vibration characteristics of fluid-conveying microtubes. Journal of Fluids and Structures26: 675–684. doi: 10.1016/j.jfluidstructs.2010.02.005.
40.
XiaWWangLYinL (2010) Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration. International Journal of Engineering Science[in press]. doi:10.1016/j.ijengsci.2010.04.010.
41.
YangFChongACMLamDCCTongP (2002) Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures39(10): 2731–2743.
42.
YimingFZhangJ (2010) Modeling and analysis of microtubules based on a modified couple stress theory. Physica E42: 1741–1745. doi: 10.1016/j.physe.2010. 01.033.
43.
ZhouSJLiZQ (2001) Length scales in the static and dynamic torsion of a circular cylindrical micro-bar. Journal of Shandong University of Technology31(5): 401–407.
