KSME Journal

, Volume 10, Issue 2, pp 146–157 | Cite as

Vibrational characteristics of annular plates and rings of radially varying thinkness

  • Seung -Ho Jang


In this paper, annular plates having thickness variation are studied by deriving the equations of motion on the basis of the Mindlin plate theory. The Chebyshev collocation method is employed to solve the differential equation governing the transverse motion of such plates. The dimensionless frequencies are evaluated for different values of taper constant (α), thickness ratio (h u). radii ratio (ε) and power (n). The results of an experimental investigation are also presented, and the agreement between these findings and the predicted values in theory is remarkably good. As a result of this study, it is found that the effects of rotatory inertia and transverse shear deformation reduce the natural frequencies for all boundary conditions and for all values ofn. h o, ∈, a ands (mode number). This study also showed that the natural frequencies of annular plates with thickness expressed by the nth power function are higher than those by the (n−1)th power function for positive values of α, and vice versa for negative values ofe for all three boundary conditions. Moreover, there is a proof that the natural frequencies of annular plates tend to be higher as the taper constant decrease and/or as the radii ratio increase for all three boundary conditions and for all values ofn, s andh o.

Key Words

Forced Vibration Rofor Natural Frequency Annular Plate Inertia Shear Deformation Boundary Condition 



Flexurai rigidity of plate *−Eh 3/12(1ν 2))


Young’s modutus of annular plate


Shear modulus of annular plate


Dimensionless variable (−h/a)


Unknown constants

Mr, Mθ

Bending moments per unit length


Twisting moment per unit length

Qr, Qθ

Radial and tangential shearing forces per unit length


Chebyshes polynomials


Chebyshes polynomials with superseript meaning integration with respecty


Outer radius


Inner radius


Thickness of plate defined by Eq. (5)


Thickness ratio


Number of Chebyshev collocation point




External force per unit area


Mode number




Dimensionless variable (−r/a)


Chebyshev constant defined by Eq. (7b)

Greek Letters


Taper constant


Ratio of the inner and outer radin (b/a)


Averaging shear coefficient (−π 2/12)


Poisson’s ratio


Density (mass per unit velume)


Circular frequency


Dimensionless frequency parameter including the effects of rotatory inertia


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  1. Bathe, K. J., 1982,Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, N. J.Google Scholar
  2. Bathe, K. J., Brezzi, F. and Cho, S. W., 1989, “The MITC7 and MITC9 Plate Bending Elements.”Comp. Struct., Vol. 32, pp. 797–814.zbMATHCrossRefGoogle Scholar
  3. Conway, H. D., 1958, “Some Special Solution for the Flexural Vibration of Disc of Varying Thickness,”Ingeniuer Archiv. Vol. 26, pp. 408–410.zbMATHCrossRefMathSciNetGoogle Scholar
  4. Dhatt, G. and Touzot, G., 1984,The Finite Element Method Displayed, John Wiley & Sons, New York.zbMATHGoogle Scholar
  5. Gorman, D. G., 1983, “Natural Frequency of Transverse Vibration of Polar Orthotropic Variable Thickness Annular Plates,”J. Sound. Vibration, Vol. 86, pp. 47–60.CrossRefGoogle Scholar
  6. Harris, G. W., 1968, “The Normal Modes of a Circular Plate of Variable Thickness,”Quarterly Journal of Mechanics and Applied Mathematics, Vol. 21, pp. 322–327.CrossRefGoogle Scholar
  7. Hinton, E. and Huang, H. C., 1986, “A Family of Quadrilateral Mindlin Plate Elements with Substitute Shear Strain Fields”Comp. Struct., Vol. 23, pp. 409–431.CrossRefGoogle Scholar
  8. Kang, L. C., 1992, “A Fourier Series Method for Polygonal Domains; Large Element Computation for Plates,” Ph. D. Dissertation, Stanford University.Google Scholar
  9. Lenox, T. A. and Conway, H. D., 1980, “An Exact Closed Form Solution for the Flexural Vibration of a Thin Annular Plate Having a Parabolic Thickness Variation,”J. Sound. Vibration., Vol. 68, pp. 231–239.zbMATHCrossRefGoogle Scholar
  10. Leissa, A. W., 1969,Vibration of Plates, NASA, sp-100.Google Scholar
  11. Mindlin, R. D., 1951, “Influence of Rotary Inertia on Flexural Motions of Isotropic, Elastic Plates,”J. Appl. Mech., Vol. 18, pp. 31–38.zbMATHGoogle Scholar
  12. Prathap, G. and Babu, C. R., 1986, “A Field Consistent Three-noded Quadratic Curved Axisymmetric Shell Element,”Int. J. Numer. Methods Eng., Vol. 23, pp. 711–723.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Ral, R. and Gupta, U. S., 1982, “Axisymmetric Vibration of Polar Orthotoropic Annular Plate of Variable Thickness,”J. Sound. Vib., Vol. 83, pp. 229–240.CrossRefGoogle Scholar
  14. Ramaiah, G. K. and Vijayakumar, K., 1985, “Vibration of Annular Plates with Linear Thickness Profiles,”J. Sound. Vib., Vol. 38, pp. 322–327.Google Scholar
  15. Soni, S. R. and Rao, L. A., 1975, “Axisymmetric Vibration of Annular Plates of Variable Thickness,”J. Sound. Vib., Vol. 38, pp. 465–473.zbMATHCrossRefGoogle Scholar
  16. Vogel, S. M. and Skinner, D. W., 1965, “Natural Frequency of Transversely Vibrating Uniform Annular Plate,”J. Appl. Mech., Vol. 32 Dec. pp. 926–931.Google Scholar
  17. Zienkiewicz, O. C. and Taylor, R. L., 1991,The Finite Element Method, 4th edn, Vol. 2, McGraw-Hill.Google Scholar

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© The Korean Society of Mechanical Engineers (KSME) 1996

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  • Seung -Ho Jang

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