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Benchmark solution for free vibration of functionally graded moderately thick annular sector plates

Abstract

In the present article, an exact analytical solution for free vibration analysis of a moderately thick functionally graded (FG) annular sector plate is presented. Based on the first-order shear deformation plate theory, five coupled partial differential equations of motion are obtained without any simplification. Doing some mathematical manipulations, these highly coupled equations are converted into a sixth-order and a fourth-order decoupled partial differential equation. The decoupled equation are solved analytically for an FG annular sector plate with simply supported radial edges. The accurate natural frequencies of the FG annular sector plates with nine different boundary conditions are presented for several aspect ratios, some thickness/length ratios, different sector angles, and various power law indices. The results show that variations of the thickness, aspect ratio, sector angle, and boundary condition of the FG annular sector plates can change the vibration wave number. Also for an FG annular sector plate with one free edge, in opposite to the other boundary conditions, the natural frequency decreases with increasing the aspect ratio for small aspect ratios. Moreover, the mode shape contour plots are depicted for an FG annular sector plate with various boundary conditions. The accurate natural frequencies of FG annular sector plates are presented for the first time and can serve as a benchmark solution.

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References

  1. Koizumi M.: FGM activities in Japan. Compos. Part B. 28, 1–4 (1997)

    Article  Google Scholar 

  2. Reddy J.N.: Analysis of functionally graded plates. Int. J. Num. Meth. Eng. 47, 663–684 (2000)

    MATH  Article  Google Scholar 

  3. Srinivasan R.S., Thiruvekatachari V.: Free vibration of transverse isotropic annular sector Mindlin plates. J. Sound Vib. 101, 193–201 (1985)

    Article  Google Scholar 

  4. Cheung Y.K., Kwok W.L.: Dynamic analysis of circular and sector thick, layered plates. J. Sound Vib. 42, 147–158 (1975)

    MATH  Article  Google Scholar 

  5. Mizusawa T.: Vibration of thick annular sector plates using semi-analytical methods. J. Sound Vib. 150, 245–259 (1991)

    Article  Google Scholar 

  6. Xiang Y., Liew K.M., Kitipornchai S.: Transverse vibration of thick annular sector plates. ASCE, J. Eng. Mech. 119, 1579–1599 (1993)

    Article  Google Scholar 

  7. McGee O.G., Leissa A.W., Huang C.S.: Vibration of completely free sectorial plates. J. Sound Vib. 164, 565–569 (1993)

    Article  Google Scholar 

  8. McGee O.G., Huang C.S., Leissa A.W.: Comprehensive exact solutions for free vibrations of thick annular sectorial plates with simply supported radial edges. Int. J. Mech. Sci. 37, 537–566 (1995)

    MATH  Article  Google Scholar 

  9. Huang C.S., McGee O.G., Leissa A.W.: Exact analytical solutions for free vibrations of thick sectorial plates with simply supported radial edges. Int. J. Solids Struct. 31, 1609–1631 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  10. Liew K.M., Liu F.L.: Differential quadrature method for vibration analysis of shear deformable annular sector plates. J. Sound vib. 230, 335–356 (2000)

    Article  Google Scholar 

  11. Wang X., Wang Y.: Free vibration analyses of thin sector plates by the new version of differential quadrature method. Com. Meth. Appl. Mech. Eng. 193, 3957–3971 (2004)

    MATH  Article  Google Scholar 

  12. Yongqiang L., Jian L.: Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier p-element. J. Sound Vib. 305, 457–466 (2007)

    Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  14. Zhou D., Lo S.H., Cheung Y.K.: 3-D vibration analysis of annular sector plates using the Chebyshev–Ritz method. J. Sound Vib. 320, 421–437 (2009)

    Article  Google Scholar 

  15. Jomehzadeh E., Saidi A.R.: Analytical solution for free vibration of transversely isotropic sector plates using a boundary layer function. Thin-Walled Struct. 47, 82–88 (2009)

    Article  Google Scholar 

  16. Jomehzadeh E., Saidi A.R.: Accurate natural frequencies of transversely isotropic moderately thick annular sector plates. J. Mech. Eng. Sci. Proc. IMechE, Part C. 223, 307–317 (2009)

    Article  Google Scholar 

  17. Srinivasan R.S., Thiruvekatachari V.: Free vibration analysis of laminated annular sector plates. J. Sound Vib. 109, 89–96 (1986)

    Article  Google Scholar 

  18. Atashipour S.R., Saidi A.R., Jomehzadeh E.: On the boundary layer phenomenon in bending of thick annular sector plates using third-order shear deformation theory. Acta Mech. 211, 89–99 (2010)

    MATH  Article  Google Scholar 

  19. Huang C.S., Ho K.H.: An analytical solution for vibrations of a polarly orthotropic Mindlin sectorial plate with simply supported radial edges. J. Sound Vib. 273, 277–294 (2004)

    Article  Google Scholar 

  20. Sharma A., Sharda H.B., Nath Y.: Stability and vibration of thick laminated composite sector plates. J. Sound Vib. 287, 1–23 (2005)

    Article  Google Scholar 

  21. Nosier A., Yavari A., Sarkani S.: On a boundary layer phenomenon in Mindlin-Reissner plate theory for laminated circular sector plates. Acta Mech. 151, 149–161 (2001)

    MATH  Article  Google Scholar 

  22. Malekzadeh P.: Three-dimensional free vibration analysis of thick laminated annular sector plates using a hybrid method. Compos. Struct. 90, 428–437 (2009)

    Article  Google Scholar 

  23. Malekzadeh P., Golbahar Haghighi M.R., Gholami M.: Dynamic response of thick laminated annular sector plates subjected to moving load. Compos. Struct. 92, 155–163 (2010)

    Article  Google Scholar 

  24. Malekzadeh P., Afsari A., Zahedinejad P., Bahadori R.: Three-dimensional layer wise-finite element free vibration analysis of thick laminated annular plates on elastic foundation. Appl. Math. Model. 34, 776–790 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  25. Nie G.J., Zhong Z.: Vibration analysis of functionally graded annular sectorial plates with simply supported radial edges. Compos. Struct. 84, 167–176 (2008)

    Article  Google Scholar 

  26. Hosseini-Hashemi Sh., Akhavan H., Rokni Damavandi Taher H., Daemi N., Alibeigloo A.: Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation. Mat. Des. 31, 1871–1880 (2010)

    Article  Google Scholar 

  27. Hosseini-Hashemi Sh., Rokni Damavandi Taher H., Akhavan H.: Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations. Compos. Struct. 92, 1734–1743 (2010)

    Article  Google Scholar 

  28. Reddy J.N.: Theory and Analysis of Elastic Plates and Shells. 2nd edn. CRC Press, Philadelphia (2007)

    Google Scholar 

  29. Hasani Baferani, A., Saidi, A.R., Jomehzadeh, E.: An exact solution for free vibration of thin functionally graded rectangular plates. IMechE, J. Mech. Eng. Sci. Part: C. doi:10.1243/09544062JMES2171 (in press)

  30. Nosier A., Fallah F.: Reformulation of Mindlin-Reissner governing equations of functionally graded circular plates. Acta Mech. 198, 209–233 (2008)

    MATH  Article  Google Scholar 

  31. Irschik H.: On vibrations of layered beams and plates. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 73, 34–45 (1993)

    MathSciNet  Google Scholar 

  32. Naderi A., Saidi A.R.: On pre-buckling configuration of functionally graded Mindlin rectangular plates. Mech. Res. Commun. 37, 535–538 (2010)

    Article  Google Scholar 

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Correspondence to A. R. Saidi.

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Saidi, A.R., Baferani, A.H. & Jomehzadeh, E. Benchmark solution for free vibration of functionally graded moderately thick annular sector plates. Acta Mech 219, 309–335 (2011). https://doi.org/10.1007/s00707-011-0459-1

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  • DOI: https://doi.org/10.1007/s00707-011-0459-1

Keywords

  • Free Vibration
  • Elastic Foundation
  • Functionally Grade
  • Free Vibration Analysis
  • Differential Quadrature Method