Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


Non-classical Natural Frequency Analysis of Piezoelectric Cylindrical Nano-shell with Surface Energy Effects

Journal of Vcibration Testing and System Dynamics 3(3) (2019) 237--258 | DOI:10.5890/JVTSD.2019.09.001

Sayyid H. Hashemi Kachapi

Department of Mechanical Engineering, Babol Noshirvani University of Technology, P.O. Box 484, Shariati Street, Babol, Mazandaran 47148-71167, Iran

Download Full Text PDF



In this paper, the size-dependent effect on the free vibration analysis of cylindrical piezoelectric nano-shell with arbitrary boundary conditions is studied. The cylindrical piezoelectric nano-shell is modeled based on Gurtin-Murdoch surface elasticity theory and the linear Von-Karman- Donnell’s strain-displacement. Also, the assumed mode method is used for changing the partial differential equations into ordinary differential equations. A variety of new vibration results including frequencies and mode shapes for piezoelectric cylindrical nano-shell with non-classical restraints as well as different material parameters are presented, which may serve as benchmark solution for future researches. The convergence, accuracy and reliability of the current formulation are validated by comparisons with existing experimental and numerical results published in the literature, with excellent agreements achieved.


‘This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors’.


  1. [1]  Jalili, N. (2010), Piezoelectric-Based Vibration Control: From Macro to Micro/Nano Scale Systems, Springer, New York.
  2. [2]  Mindlin, R.D. and Tiersten, H.F. (1962), Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11(1), 415-48.
  3. [3]  Eringen, A.C. (1972), Nonlocal polar elastic continua, International Journal of Engineering Science, 10(1), 1-16.
  4. [4]  Mindlin, R.D. and Eshel, N.N. (1968), On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, 4(1), 109-24.
  5. [5]  Mindlin, R.D. (1965), Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, 1(4), 417-38.
  6. [6]  Gurtin, M.E. and Murdoch, A.I. (1975), A continuum theory of elastic material surface, Archive for Rational Mechanics and Analysis, 57(4), 291-323.
  7. [7]  Gurtin, M.E. and Murdoch, A.I. (1978), Surface stress in solids, International Journal of Solids and Structures, 14(6), 431-40.
  8. [8]  Lu, P., He, L.H., Lee, H.P., and Lu, C. (2006), Thin plate theory including surface effects, International Journal of Solids and Structures, 43(16), 4631-47.
  9. [9]  Rouhi, H., Ansari, R., and Darvizeh, M. (2016), Analytical treatment of the nonlinear free vibration of cylindrical nanoshells based on a first-order shear deformable continuum model including surface influences, Acta Mechanic, DOI 10.1007/s00707-016-1595-4.
  10. [10]  Sahmani, S., Aghdam, M.M., and Akbarzadeh, A.H. (2016), Size-dependent buckling and postbuckling behavior of piezoelectric cylindrical nanoshells subjected to compression and electrical load, Materials & Design, 105, 341-51.
  11. [11]  Zhu, C.S., Fang, X.Q., and Liu, J.X. (2017), Surface energy effect on buckling behavior of the functionally graded nano-shell covered with piezoelectric nano-layers under torque, International Journal of Mechanical Sciences, 133, 662-673.
  12. [12]  Gheshlaghi, B. and Hasheminejad, S.M. (2011), Surface effects on nonlinear free vibration of nanobeams, Composites Part B: Engineering, 42(4), 934-7.
  13. [13]  Wang, K.F. and Wang, B.L. (2013), Effect of surface energy on the non-linear postbuckling behavior of nanoplates, International Journal of Non-Linear Mechanics, 55, 19-24.
  14. [14]  Ghorbanpour, A.A., Kolahchi, R., and Hashemian, M. (2014), Nonlocal surface piezoelasticity theory for dynamic stability of double-walled boron nitride nanotube conveying viscose fluid based on different theories, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, DOI: 10.1177/0954406214527270.
  15. [15]  Leissa, A.W. (1993), Vibration of Shells, NASA SP 288, US Government Printing Office, 1973, Reprinted by the Acoustical Society of America.
  16. [16]  Liew, K.M., Lim, C.W., and Kitipornchai, S. (1997), Vibration of shallow shells: a review with bibliography, Applied Mechanics Reviews, 50(8), 431-444
  17. [17]  Qatu, M.S., Sullivan, R.W., and Wang, W. (2010), Recent research advances on the dynamic analysis of composite shells: 2000-2009”, Composite Structures, 93, 14-31.
  18. [18]  Ye, T., Jin, G., Chen, Y., and Shi, S. (2014), A unified formulation for vibration analysis of open shells with arbitrary boundary conditions, International Journal of Mechanical Sciences, 81, 42-59.
  19. [19]  Tornabene, F., Fantuzzi, N., and Bacciocchi, M. (2016), The GDQ method for the free vibration analysis of arbitrarily shaped laminated composite shells using a NURBS-based isogeometric approach, Composite Structures, 154, 190-218.
  20. [20]  Ye, T., Jin, G., Chen, Y., Ma, X., and Su, Z. (2013), Free vibration analysis of laminated composite shallow shells with general elastic boundaries, Composite Structures, 106, 470-490.
  21. [21]  Fazzolari, F.A. (2014), A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells, Composite Structures, Part B, 62, 143-158.
  22. [22]  Hirvani, C.H., Patil, R.K., Panda, S.K., Mahapatra, S.S., Mandal, S.K., Srivastava, S., and Buragohain, M.K. (2016), Experimental and numerical analysis of free vibration of de-laminated curved panel, Aerospace Science and Technology, 54, 253-370.
  23. [23]  Mirza, S. and Alizadeh, Y. (1995), Free vibration of partially supported cylindrical shells, Shock and Vibration, 2(4), 297-306.
  24. [24]  Zhou, J. and Yang, B. (1995), Distributed transfer function method for analysis of cylindrical shells, AIAA Journal, 33(9), 1698-1708.
  25. [25]  Loy, C.T., Lam, K.Y., and Shu, C. (1997), Analysis of cylindrical shells using generalized differential quadrature, Shock and Vibration, 4(3), 193-198.
  26. [26]  Yim, J.S., Sohn, D.S., and Lee, Y.S. (1998), Free vibration of clamped free circular cylindrical shell with a plate attached at an arbitrary axial position, Journal of Sound and Vibration, 13(1), 75-88.
  27. [27]  Naeem, M.N. and Sharma, C.B. (2000), Prediction of natural frequencies for thin circular cylindrical shells, Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science, 214(10), 1313-1328.
  28. [28]  Donnell, L.H. (1976), Beam, Plates and Shells, McGraw-Hill, New York, NY, USA.
  29. [29]  Amabili, M. (2008), Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, New York, NY, USA.
  30. [30]  Sabzikar Boroujerdy, M. and Eslami, M.R. (2014), Axisymmetric snap-through behavior of Piezo-FGM shallow clamped spherical shells under thermo-electro-mechanical loading, International Journal of Pressure Vessels and Piping, 120-121, 1926.
  31. [31]  Mirzavand, B. and Eslami, M.R. (2011), A closed-form solution for thermal buckling of piezoelectric FGM hybrid cylindrical shells with temperature dependent properties, Mechanics of Advanced Materials and Structures, 18, 185-193.
  32. [32]  Jafari, A.A., Khalili, S.M.R., and Tavakolian, M. (2014), Nonlinear vibration of functionally graded cylindrical shells embedded with a piezoelectric layer, Thin Walled Structures, 79, 8-15.
  33. [33]  Rao, S.S. (2007), Vibration of Continuous Systems, John Wiley & Sons, New Jersey.