Advertisement

Applied Mathematics and Mechanics

, Volume 30, Issue 8, pp 969–982 | Cite as

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

  • Shi-rong Li (李世荣)
  • Hou-de Su (苏厚德)
  • Chang-jun Cheng (程昌钧)
Article

Abstract

Free vibration of statically thermal postbuckled functionally graded material (FGM) beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electromechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

Key words

functionally graded material laminated beams with piezoelectric layers thermal buckling free vibration natural frequency 

Chinese Library Classification

O343 

2000 Mathematics Subject Classification

74H45 74F05 74H15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Coffin, D. W. and Bloom, F. Elastica solution for the hygrothermal buckling of a beam. International Journal of Non-Linear Mechanics 34(5), 935–947 (1999)CrossRefMathSciNetGoogle Scholar
  2. [2]
    Vaz, M. A. and Solano, R. F. Post-buckling analysis of slender elastic rods subjected to uniform thermal loads. Journal of Thermal Stresses 26(9), 847–860 (2003)CrossRefGoogle Scholar
  3. [3]
    Vaz, M. A. and Solano, R. F. Thermal postbuckling of slender elastic rods with hinged ends constrained by a linear spring. Journal of Thermal Stresses 27(4), 367–380 (2004)CrossRefGoogle Scholar
  4. [4]
    Li, Shirong and Cheng, Changjun. Thermal postbuckling analysis of heated elastic rods. Applied Mathematics and Mechanics (English Edition) 21(5), 133–140 (2000) DOI:10.007/BF02458513MATHGoogle Scholar
  5. [5]
    Li, S. R., Zhou, Y. H., and Zheng, X. J. Thermal postbuckling of a heated elastic rod with pinned-fixed ends. Journal of Thermal Stresses 25(1), 45–56 (2002)CrossRefGoogle Scholar
  6. [6]
    Li, S. R. and Batra, R. C. Thermal buckling and postbuckling of Euler-Bernoulli beams supported on nonlinear elastic foundations. AIAA Journal 45(3), 711–720 (2007)CrossRefGoogle Scholar
  7. [7]
    Li, S. R. and Zhou, Y. H. Geometrically nonlinear analysis of Timoshenko beams under thermomechanical loadings. Journal of Thermal Stresses 26(9), 867–872 (2003)CrossRefMathSciNetGoogle Scholar
  8. [8]
    Li, S. R., Teng, Z. C., and Zhou, Y. H. Free vibration of heated Euler-Bernoulli beams with thermal postbuckling deformations. Journal of Thermal Stresses 27(9), 843–856 (2004)CrossRefGoogle Scholar
  9. [9]
    Sankar, B. V. An elasticity solution for functionally graded beams. Composites Science and Technology 61(5), 689–696 (2001)CrossRefGoogle Scholar
  10. [10]
    Chakraborty, A., Gopalakrishnan, S., and Reddy, J. N. A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences 45(3), 519–539 (2003)MATHCrossRefGoogle Scholar
  11. [11]
    Bhangale, R. K. and Ganesan, N. Thermoelastic buckling and vibration behavior of a functionally graded sandwich beam with constrained viscoelastic core. Journal of Sound and Vibration 295(1–2), 294–316 (2006)CrossRefGoogle Scholar
  12. [12]
    Xia, X. K. and Shen, H. S. Vibration of postbuckled FGM hybrid laminated plates in thermal environment. Engineering Structures 30(9), 2420–2435 (2008)CrossRefGoogle Scholar
  13. [13]
    Xia, X. K. and Shen, H. S. Nonlinear vibration of postbuckled FGM plates with shear deformation. Journal of Vibration Engineering 21(2), 120–125 (2008)Google Scholar
  14. [14]
    Li, S. R., Zhang, J. H., and Zhao, Y. G. Thermal postbuckling of functionally graded material Timoshenko beams. Applied Mathematics and Mechanics (English Edition) 27(6), 803–810 (2006) DOI:10.1007/s10483-006-0611-yMATHCrossRefGoogle Scholar
  15. [15]
    Crawley, E. F. and de Luis, J. Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal 25(10), 1373–1385 (1987)CrossRefGoogle Scholar
  16. [16]
    Zhou, Y. H. and Wang, J. Z. Vibration control of piezoelectric beam-type-plates with geometrical nonlinear deformation. International Journal of Non-Linear Mechanics 39(6), 909–920 (2004)MATHCrossRefGoogle Scholar
  17. [17]
    Zhou, Y. H., Wang, J. Z., Zheng, X. J., and Jiang, Q. Vibration control of variable thickness plates with piezoelectric sensors and actuators based on wavelet theory. Journal of Sound and Vibration 237(3), 395–410 (2000)CrossRefGoogle Scholar
  18. [18]
    Lin, Q. R., Liu, Z. X., and Wang, Z. G. Analysis of beams with piezoelectric actuators. Applied Mathematics and Mechanics (English Edition) 22(9), 969–975 (2001) DOI:10.1023/A:1016320611141CrossRefGoogle Scholar
  19. [19]
    Fridman, Y. and Abramovich, H. Enhanced structural behavior of flexible laminated composite beams. Composite Structures 82(1), 140–154 (2008)CrossRefGoogle Scholar
  20. [20]
    Yu, T., and Zhong, Z. A general solution of a clamped functionally graded cantilever-beam under uniform loadings (in Chinese). Acta Mechanica Solida Sinica 27(1), 15–20 (2006)MathSciNetGoogle Scholar
  21. [21]
    Huang, X. L. and Shen, H. S. Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. Journal of Sound and Vibration, 289(1–2), 25–53 (2006)CrossRefMathSciNetGoogle Scholar
  22. [22]
    Li, Shirong, Batra, Romesh C, and Ma, Lansheng. Vibration of thermally postbuckled orthotropic circular plate. Journal of Thermal Stresses 30(1), 43–57 (2007)CrossRefGoogle Scholar
  23. [23]
    William, H. P., Brain, P. F., San, A. T., and William, T. V. Numerical Recipes-The Art of Scientific Computing, Cambridge University Press, London (1986)MATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH 2009

Authors and Affiliations

  • Shi-rong Li (李世荣)
    • 1
  • Hou-de Su (苏厚德)
    • 1
  • Chang-jun Cheng (程昌钧)
    • 2
  1. 1.Department of Engineering MechanicsLanzhou University of TechnologyLanzhouP. R. China
  2. 2.Department of MechanicsShanghai UniversityShanghaiP. R. China

Personalised recommendations