25Analysis of Forced Vibration of Viscoelastic Nanocomposite Plate Reinforced Via Carbon Nanotubes


Nanocomposite plates are one of the best options for reducing free and forced vibrations. The present study has investigated the forced vibrations of viscoelastic nanocomposite plates based on equations of free motion in a field with external force. This analysis was performed on primary resonance and in the state where only one single mode of vibration is stimulated. Besides, frequency equations were obtained without using Galerkin segregation method, and by these equations, the impact of thickness, Poisson coefficient, damping coefficient, fraction volume of carbon nanotubes, and nanotube arrangement on the forced state response were determined. The calculations highlight that an increase in damping factor results in a decrease in the maximum amplitude. It was also found that an increase in the ratio of plate’s thickness to its length causes an increase in the deviation of plate’s resonant frequency from its normal frequency. In addition, as the arrangement of carbon nanotubes alters, system’s hardening behavior in FGX arrangement state is less solid compared to uniform distribution; yet, it enjoys more hardness in FGO arrangement state.  Moreover, change in the arrangement partially altered the resonant frequency. A change in the plate’s ratio of width and length was also found to cause a change in the frequency response curve.

Keywords: Carbon nanotubes, forced vibration, nanocomposite, viscoelastic


  • Rayleigh L., “Theory of sound”, 2nd ed., London: Macmillan Press, (1945).

  • Leissa A W., Clausen W. E., Hulbert L.E, Hopper AT., “A comparison of Approximate methods for the solution of plate bending problems”. AIAA Journal, Vol. 7, pp. 920-8, (1969).

  • Lanra PAA, Duran R., “A note on forced vibrations of clamped rectangular plate”, Journal of sound and vibration, Vol. 42, pp. 129-35, (1975).

  • Srinivasan AV., “Large-amplitude free oscillations of beams and plates”, AIAA J, Vol. 3(10), pp. 1951-3, (1965).

  • Shaw S.W, Pierre C. “Non-linear modes and invarant manifolds”, Journal of sound and vibration, pp. 170-73, (1991).

  • Kim T.W, Kim J.H., “Nonlinear Vibration of viscoelastic laminated composite plates, International Journal of solids and structures”, Vol. 39, pp. 2857-70, (2002).

  • Nayfeh A.H. “Perturbation Methods”, Wiley, (1973).

  • Nayfeh A.H, Mook D.T, “Nonlinear Oscillations” , Wiley, (1979).

  • Nayfeh A.H. “Introduction to perturbation Techniques”, Wiley, (1981).

  • Nayfeh A.H, Chin C., “Non linear Normal Modes of a Cantilever beams”, Journal of Vibration and Acoustics ; Vol. 117, pp. 474-81, (1995).

  • V. G. Karnaukhov and Ya. V. Tkachenko, “Damping the vibrations of a rectangular plate with piezoelectric actuators”, International Applied Mechanics, Vol. 44, No. 2, (2008).

  • I.D. Breslavsky, “Stress distribution over plates vibrating at large amplitudes“, Journal of Sound and Vibration, Vol. 331, pp. 2901–2910, (2012).

  • Yan Wang, ” Transverse vibration of viscoelastic rectangular plate with linearly varying thickness and multiple cracks”, Journal of Sound and Vibration, Vol. 318, pp. 1005–1023 (2008).

  • M. Amabili, “Geometrically nonlinear vibrations of rectangular plates carrying a concentrated mass”, Journal of Sound and Vibration, Vol. 329, pp. 4501– 4514, (2010).

  • Gounaris G.D., Anifanits N.K., “Structural Damping Determination by finite element Approuch”, Computers and Structure, Vol. 73, pp. 445-452, (1999).

  • Judge J., Pratap R., “Asymptotic States of a Bilinear Hysteretic Oscillator in a Fully Dissipative Phase Space”, Journal of Sound and Vibration, Vol. 278, No. 3, pp. 548-557, (1998).

  • Giovanni F., Walter Lacarbonara, Roberto Alessi; “Vibrations of carbon nanotube-reinforced composites”, Journal of Sound and Vibration, Vol. 329, pp. 1875–1889, (2010).

  • Bandarian, M., et al, “Thermal, mechanical and acoustic damping properties of flexible open-cell polyurethane/multi-walled carbon nanotube foams: effect of surface functionality of nanotubes”  Polym. Int Vol. 60, pp. 475-482,  (2011).

  • Ajayan, P. M., et al,  “Utilizing interfaces in carbon nanotube reinforced polymer composites for structural damping” Journal of Materials Science. Vol. 41, pp.7824-29, (2006).

  • Linn, J., Lang, H., Tuganov, A., “Derivation of a viscoelastic constitutive model of Kelvin-Voigt type for Cosser at rods” IMSD  conference, (2013).