Skip to main content
SpringerLink
Log in

On the solution of nonlinear vibration of truncated conical shells covered by functionally graded coatings

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

This paper deals with the linear and nonlinear vibrations of a truncated conical shell; both internal and external surfaces are covered by functionally graded coatings (FGCs). The theoretical formulation is based on the von Karman–Donnell-type nonlinear kinematics. The material properties of FGCs are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The fundamental relations, the modified Donnell-type nonlinear motion, and compatibility equations of the truncated conical shell with FGCs are derived. The basic equations are reduced to the ordinary differential equation depending on time with geometric nonlinearity using the Superposition and Galerkin methods. By applying the homotopy perturbation method to the foregoing equation, the relation between nonlinear frequency parameters with the dimensionless amplitude of a truncated conical shell with FGCs is obtained. Parametric studies are performed to illustrate the effect of different values of thickness and material composition of the FGCs on the frequency-amplitude relationships. The validity of the present solution is demonstrated by comparison with solutions available in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Niino A., Maeda S.: Recent development status of functionally gradient materials. ISIJ Int. 30, 699–703 (1990)

    Article  Google Scholar 

  2. Miyamoto Y., Kaysser W.A., Rabin B.H., Kaeasaki A., Ford R.G.: Functionally Graded Materials: Design, Processing and Applications. Kluwer, Dordrecht (1999)

    Book  Google Scholar 

  3. Shults U., Peters M., Bach Fr.-W., Tegeder G.: Graded coatings for thermal, wear and corrosion barriers. Mater. Sci. Eng. 362, 61–80 (2003)

    Article  Google Scholar 

  4. Weng G.J.: Effective bulk moduli of two functionally graded composites. Acta Mech. 166, 57–67 (2003)

    Article  MATH  Google Scholar 

  5. Tsukamoto H.: Design of functionally graded thermal barrier coatings based on a nonlinear micromechanical approach. Comput. Mater. Sci. 50, 429–436 (2010)

    Article  Google Scholar 

  6. Vinson J.R.: Sandwich structures. Appl. Mech. Rev. 54, 201–214 (2001)

    Article  Google Scholar 

  7. Zenkour A.M.: A comprehensive analysis of functionally graded sandwich plates: part 2-buckling and free vibration. Int. J. Solid Struct. 42, 5243–5258 (2005)

    Article  MATH  Google Scholar 

  8. Sofiyev A.H., Deniz A., Akcay I.H., Yusufoglu E.: The vibration and stability of a three-layered conical shell containing a FGM layer subjected to axial compressive load. Acta Mechanica 183, 129–144 (2006)

    Article  MATH  Google Scholar 

  9. Shodja H.M., Haftbaradaran H., Asghari M.: A thermoelasticity solution of sandwich structures with functionally graded coating. Compos. Sci. Tech. 67, 1073–1080 (2007)

    Article  Google Scholar 

  10. Li Q., Iu V.P., Kou K.P.: Three-dimensional vibration analysis of functionally graded material sandwich plates. J. Sound Vib. 311, 498–515 (2008)

    Article  Google Scholar 

  11. Zenkour A.M., Alghamdi N.A.: Bending analysis of functionally graded sandwich plates under the effect of mechanical and thermal loads. Mech. Adv. Mater. Struct. 17, 419–432 (2010)

    Article  Google Scholar 

  12. Sobhy M.: Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Compos. Struct. 99, 76–87 (2013)

    Article  Google Scholar 

  13. Zaki M., Tarlochan F., Ramesh S.: Two dimensional elastic deformations of functionally graded coated plates with clamped edges. Compos. B Eng. 45, 1010–1022 (2013)

    Article  Google Scholar 

  14. Sofiyev A.H., Kuruoglu N.: Torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium. Compos. B Eng. 45, 1133–1142 (2013)

    Article  Google Scholar 

  15. Wu C.P., Kuo C.H.: A unified formulation of pvd-based finite cylindrical layer methods for functionally graded material sandwich cylinders. Appl. Math. Model. 37, 916–938 (2013)

    Article  MathSciNet  Google Scholar 

  16. Alijani F., Amabili M., Karagiozis K., Bakhtiari-Nejad F.: Nonlinear vibrations of functionally graded doubly curved shallow shells. J. Sound Vib. 330, 1432–1454 (2011)

    Article  Google Scholar 

  17. Arciniega R.A., Reddy J.N.: Large deformation analysis of functionally graded shells. Int. J. Solid Struct. 44, 2036–2052 (2007)

    Article  MATH  Google Scholar 

  18. Chorfi S.M., Houmat A.: Non-linear free vibration of a functionally graded doubly-curved shallow shell of elliptical plan-form. Compos. Struct. 92, 2573–2581 (2010)

    Article  Google Scholar 

  19. Darabi M., Darvizeh M., Darvizeh A.: Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading. Compos. Struct. 83, 201–211 (2008)

    Article  Google Scholar 

  20. Liew K.M., Yang J., Wu Y.F.: Nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a temperature gradient. Comput. Meth. Appl. Mech. Eng. 195, 1007–1026 (2006)

    Article  MATH  Google Scholar 

  21. Zhao X., Liew K.M.: Geometrically nonlinear analysis of functionally graded shells. Int. J. Mech. Sci. 51, 131–144 (2009)

    Article  MATH  Google Scholar 

  22. Shen H.S.: Functionally Graded Materials, Nonlinear Analysis of Plates and Shells. CRC Press, Florida (2009)

    Book  Google Scholar 

  23. Shen H.S.: Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium. Compos. Struct. 94, 1144–1154 (2012)

    Article  Google Scholar 

  24. Zhang J., Li S.: Dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load. Compos. Struct. 92, 2979–2983 (2010)

    Article  Google Scholar 

  25. Sofiyev A.H.: Non-linear buckling behavior of FGM truncated conical shells subjected to axial load. Int. J. Non linear Mech. 46, 711–719 (2011)

    Article  MathSciNet  Google Scholar 

  26. Sofiyev A.H.: The non-linear vibration of FGM truncated conical shells. Compos. Struct. 94, 2237–2245 (2012)

    Article  Google Scholar 

  27. Sofiyev, A.H., Kuruoglu, N.: Large-amplitude vibration of the geometrically imperfect FGM truncated conical shell. J. Vib. Contr. (2013). doi:10.1177/1077546313480998

  28. Kitipornchai S., Yang J., Liew K.M.: Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections. Int. J. Solid Struct. 41, 2235–2257 (2004)

    Article  MATH  Google Scholar 

  29. Xia X.K., Shen H.S.: Vibration of post-buckled sandwich plates with FGM face sheets in a thermal environment. J. Sound Vib. 314, 254–274 (2008)

    Article  Google Scholar 

  30. Shen H.S., Li S.R.: Postbuckling of sandwich plates with FGM face sheets and temperature-dependent properties. Compos. B Eng. 39, 332–334 (2008)

    Article  Google Scholar 

  31. Wang Z.Z., Shen H.S.: Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations. Compos. Struct. 93, 2521–2532 (2011)

    Article  Google Scholar 

  32. Wang Z.X., Shen H.S.: Nonlinear dynamic response of sandwich plates with FGM face sheets resting on elastic foundations in thermal environments. Ocean Eng. 57, 99–110 (2013)

    Article  MathSciNet  Google Scholar 

  33. Natarajan S., Ganapathi M.: Bending and vibration of functionally graded material sandwich plates using an accurate theory. Finite Elem. Anal. Des. 57, 32–42 (2012)

    Article  Google Scholar 

  34. Leissa, A.W.: Vibration of Shells. NASA SP-288 (1973)

  35. Agamirov, V.L.: Dynamic Problems of Nonlinear Shells Theory. Nauka, Moscow (1990); (in Russian)

  36. Amabili M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, New York (2008)

    Book  MATH  Google Scholar 

  37. Nayfeh A.H.: Introduction to Perturbation Methods. Wiley, New York (1981)

    Google Scholar 

  38. Liao S.J., Chwang A.T.: Application of homotopy analysis method in nonlinear oscillations. ASME J. Appl. Mech. 65, 914–922 (1998)

    Article  MathSciNet  Google Scholar 

  39. He J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)

    Article  MATH  Google Scholar 

  40. He J.H.: A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int. J. Non linear Mech. 35, 37–43 (2000)

    Article  MATH  Google Scholar 

  41. He J.H.: Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Non linear Sci. Num. Simul. 6, 207 (2005)

    Google Scholar 

  42. Li F.M., Kishimoto K., Huang W.H.: The calculations of natural frequencies and forced vibration responses of conical shell using the Rayleigh–Ritz method. Mech. Res. Comm. 36, 595–602 (2009)

    Article  MATH  Google Scholar 

  43. Kerboua Y., Lakis A.A., Hmila M.: Vibration analysis of truncated conical shells subjected to flowing fluid. Appl. Math. Model. 34, 791–809 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  44. Prado Z., Goncalves P.B., Paidoussis M.P.: Non-linear vibrations and instabilities of orthotropic cylindrical shells with internal flowing fluid. Int. J. Mech. Sci. 52, 1437–1457 (2010)

    Article  Google Scholar 

  45. Pradhan S.C., Loy C.T., Lam K.Y., Reddy J.N.: Vibration characteristics of functionally graded cylindrical shells under various boundary conditions. Appl. Acoust. 61, 111–129 (2000)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Machine Elements and Lifting-and-Shifting Machines of Azerbaijan Technical University, Baku, Azerbaijan

    A. M. Najafov

  2. Department of Civil Engineering, Suleyman Demirel University, 32260, Isparta, Turkey

    A. H. Sofiyev

  3. Department of Mathematics and Computer Science, Bahcesehir University, Istanbul, Turkey

    N. Kuruoglu

Authors
  1. A. M. Najafov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. H. Sofiyev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. Kuruoglu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. H. Sofiyev.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Najafov, A.M., Sofiyev, A.H. & Kuruoglu, N. On the solution of nonlinear vibration of truncated conical shells covered by functionally graded coatings. Acta Mech 225, 563–580 (2014). https://doi.org/10.1007/s00707-013-0980-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-013-0980-5

Keywords

Search

Navigation