Skip to main content
SpringerLink
Log in

Nonaxisymmetric vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

The nonaxisymmetric problem of natural vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials is solved. The properties of the material change continuously along a radial coordinate according to an exponential law. The lateral surfaces of the cylinder are free of external tractions and short circuited by electrodes. After separation of variables and representation of the components of the displacement vector in the form of standing circumferential waves, the initially three-dimensional problem is reduced to a two-dimensional partial differential equation problem. By using the method of spline-collocations with respect to a longitudinal coordinate, this two-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed in terms of ordinary differential equations. This problem is solved by the stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Results were obtained numerically and subsequently analyzed in this paper.

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. Baily T., Hubbard E.: Distributed piezoelectric polymer active vibration control of cantiliver beam. J. Guid. Control Dyn. 8(5), 605–611 (1985)

    Article  Google Scholar 

  2. Baz A., Poh S.: Performance on an active control system with piezoelectric actuators. J. Sound. Vib. 126(2), 327–343 (1988)

    Article  ADS  Google Scholar 

  3. Berlincourt, D.: Piezoelectric crystals and ceramics. In: Mattiat, O.E. (ed.) Ultrasonic Transducer Materials, chap. 2, pp. 62–124. Plenum Press, New York (1971)

  4. Birman V., Byrd L.W.: Modeling and analysis of functionally graded materials and structures. ASME Appl. Mech. Rev. 60, 195–216 (2007)

    Article  ADS  Google Scholar 

  5. Chen W.Q., Ding H.J.: On free vibrations of a functionally graded piezoelectric rectangular plate. Acta Mech. 153, 207–216 (2002)

    Article  MATH  Google Scholar 

  6. Chen W.Q., Lu Y., Ye J.R., Cai J.B.: 3D electroelastic fields in a functionally graded piezoceramic hollow sphere under mechanical and electric loading. Arch. Appl. Mech. 72, 39–51 (2002)

    Article  ADS  MATH  Google Scholar 

  7. Crawly E.F., de Luis J.: Use of piezoelectric actuators as elements of intelligent structures. AIAA J. 25(10), 1373–1385 (1987)

    Article  ADS  Google Scholar 

  8. Dai H.L., Wang X.: Transient wave propagation in piezoelectric hollow spheres subjected to thermal shock and electric excitation. Struct. Eng. Mech. 19(4), 441–457 (2005)

    Google Scholar 

  9. Dai H.L., Fu Y.M., Yang J.H.: Electromagnetoelastic behaviors of functionally graded piezoelectric solid cylinder and sphere. Acta Mech. Sin. 23, 55–63 (2007)

    Article  ADS  MATH  Google Scholar 

  10. Dai H.L., Hong L., Fu M.Y., Xiao X.: Analytical solution for electromagnetothermoelastic behaviors of functionally graded piezoelectric hollow cylinder. Appl. Math. Model. 34, 343–357 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ghorbanpour A.A., Salari M., Khademizadeh H., Arefmanesh A.: Magnetothermo-elastic stress and perturbation of magnetic field vector in a functionally graded hollow sphere. Arch. Appl. Mech. 80, 189–200 (2010)

    Article  MATH  Google Scholar 

  12. Grigorenko, Y.M., Grigorenko, A.Y., Vlaikov, G.G.: Problems of Mechanics for Anisotropic Inhomogeneous Shells on Basic of Different Models. Monograph, Kiev, Akademperiodika (2009a)

  13. Grigorenko A.Y., Efimova T.L, Loza I.A.: An approach to the investigation of vibrations of hollow piezoceramic cylinders of finite length [in Russian]. Rep. NASU 25(6), 61–67 (2009b)

    Google Scholar 

  14. Grigorenko A.Y., Efimova T.L, Loza I.A.: Free vibrations of axially polarized piezoceramic hollow cylinders of finite length. Int. Appl. Mech. 46(6), 625–633 (2010)

    Article  ADS  Google Scholar 

  15. Grigorenko A., Müller W.H., Wille R., Yaremchenko S.: Numerical solution of stress-strain state in hollow cylinder by means of spline approximation. J. Math. Sci. 180(2), 135–145 (2012)

    Article  MathSciNet  Google Scholar 

  16. Heyliger P.R.: A note on the static behavior of simply-supported laminated piezoelectric cylinder. Int. J. Solids Struct. 34(29), 3781–3794 (1997)

    Article  MATH  Google Scholar 

  17. Loza I.A.: Propagation of nonaxisymmetric waves in hollow piezoceramic cylinder with radial polarization. Sov. Appl. Mech. 21(1), 22–27 (1985)

    Article  ADS  Google Scholar 

  18. Saravanos D.A., Heyliger P.R.: Mechanics and computational models for laminated piezoelectric beams, plates, and shells. ASME Appl. Mech. Rev. 52, 305–320 (1999)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Numerical Methods, S.P. Timoshenko Institute of Mechanics, NAS of Ukraine, 3, P. Nesterov Street, Kiev, 03057, Ukraine

    A. Ya. Grigorenko & I. A. Loza

  2. Fakultät V, Institut für Mechanik, LKM, Technische Universität Berlin, Einsteinufer 5, 13591, Berlin, Germany

    W. H. Müller & R. Wille

Authors
  1. A. Ya. Grigorenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. H. Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Wille
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. I. A. Loza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. H. Müller.

Additional information

Communicated by Andreas Öchsner.

Dedicated to Prof. I. Müller in honor of his 75th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grigorenko, A.Y., Müller, W.H., Wille, R. et al. Nonaxisymmetric vibrations of radially polarized hollow cylinders made of functionally gradient piezoelectric materials. Continuum Mech. Thermodyn. 24, 515–524 (2012). https://doi.org/10.1007/s00161-012-0239-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-012-0239-8

Mathematics Subject Classification

Keywords

Search

Navigation