A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular rings with isosceles trapezoidal and triangular cross-sections. Displacement components $us,$$uz,$ and $uθ$ in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the ϕ and z directions. Potential (strain) and kinetic energies of the circular ring are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for the circular rings with isosceles trapezoidal and equilateral triangular cross-sections having completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. The method is applicable to thin rings, as well as thick and very thick ones. [S0739-3717(00)00702-9]

1.
Endo
,
M.
,
1972
, “
Flexural Vibrations of a Ring with Arbitrary Cross Section
,”
Bull. JSME
,
15
, pp.
446
454
.
2.
Singal
,
R. K.
, and
Williams
,
K.
,
1988
, “
A Theoretical and Experimental Study of Vibrations of Thick Circular Cylindrical Shells and Rings
,”
J. Vibr. Acoust. Stress, Reliability Design
,
110
, pp.
533
537
.
3.
Leissa
,
A. W.
, and
So
,
J.
,
1995
, “
Three-Dimensional Vibration of Truncated Hollow Cones
,”
J. Vibr. Control
,
1
, pp.
145
158
.
4.
Tsui, Edward, Y. W., 1968, Stresses in Shells of Revolution, Pacific Coast Publishers.
5.
Kang, J. H., 1997, “Three-Dimensional Vibration Analysis of Thick Shells ofRevolution with Arbitrary Curvature and Variable Thickness,” Ph.D. Dissertation, The Ohio State University.
6.
Kantorovich, L. V., and Krylov, V. I., 1958, Approximate Methods in Higher Analysis, Noordhoff, Groningen.
7.
Ritz
,
W.
,
1909
, “
U¨ber eine neue Methode zur Lo¨sung gewisser Variationsprobleme der mathematischen Physik
,”
J. fu¨r die Reine und Angewandte Mathematik
,
135
, pp.
1
61
.
8.
Abramowitz, M., and Stegun, I. A., 1964, Handbook of Mathematical Functions, U.S. Government Printing Office, Washington, DC.
9.
Leissa
,
A. W.
, and
So
,
J.
,
1995
, “
Comparisons of Vibration Frequencies for rods and beams from One-Dimensional and Three-Dimensional Analyses
,”
J. Acoust. Soc. Am.
,
98
, No.
4
, pp.
2122
2135
.
10.
So
,
J.
, and
Leissa
,
A. W.
,
1997
, “
Free Vibrations of Thick Hollow Circular Cylinders from Three-Dimensional Analysis
,”
ASME J. Vibr. Acoust.
,
119
, pp.
89
95
.