A three-dimensional (3-D) method of analysis is developed for the free vibration frequencies of hollow circular cylinders of elastic material. The method is based upon local coordinates whose origin is attached to the center of cylindrical wall. It assumes for the three displacement components a Fourier series in the circumferential (θ) direction and algebraic polynomials in the radial (q) and axial (z) directions. Convergence studies for completely free cylinders show that the analysis can yield frequencies which are exact to five or six significant figures. These accurate frequencies are compared with those from other 3-D analyses available for free hollow circular cylinders having various length-to-outside diameter (L/Do) and inside-to-outside diameter (Di/Do) ratios. Extensive, accurate data are presented for the first 10 frequencies of each circumferential wave number 0 through 5 for hollow circular cylinders having Di/Do of 0.1, 0.5, and 0.9, with L/Do = 0.2, 1 and 5 and a Poisson’s ratio (v) = 0.3.

Issue Section:
Research Papers
Topics:
Circular cylinders, Free vibrations, Algebra, Cylinders, Displacement, Fourier series, Poisson ratio, Polynomials, Waves
1.
Armenakas, A. E., Gazis, D. C., and Herrmann, G., 1969, Free Vibrations of Circular Cylindrical Shells, Pergamon Press, New York.
2.
Chree
C.
,
1886
, “
Longitudinal Waves of a Solid Bar
,”
Quart. J. Math.
, Vol.
21
, p.
287
287
.
3.
Endo
M.
,
1972
, “
Flexural Vibrations of a Ring with Arbitrary Cross Section
,”
Bulletin of JSME
, Vol.
15
, pp.
446
–
454
.
4.
Gazis
D. C.
,
1959
, “
Three Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders
,”
Journal of the Acoustical Society of America
, Vol.
31
, pp.
568
–
578
.
5.
Girgis
R. S.
, and
Verma
S. P.
,
1981
, “
Method for Accurate Determination of Resonant Frequencies and Vibration Behaviour of Stators of Electrical Machines
,”
IEE Proc. B. Electr. Power Appl.
, Vol.
128
, pp.
1
–
11
.
6.
Gladwell
G. M. L.
, and
Vijay
D. K.
,
1975
, “
Natural Frequencies of Free Finite Length Circular Cylinders
,”
Journal of Sound and Vibration
, Vol.
42
, pp.
387
–
397
.
7.
Hughes, W. F. and Gaylord, E. W., 1964, Basic Equations of Engineering Science, Schaum Publishing Company, p. 71.
8.
Hutchinson
J. R.
, and
El-Azhari
S. A.
,
1986
a, “
Vibrations of Free Hollow Circular Cylinders
,”
ASME Journal of Applied Mechanics
, Vol.
53
, pp.
641
–
646
.
9.
Hutchinson, J. R., and El-Azhari, S. A., 1986b, “On the Vibration of Thick Annular Plates,” Refined Dynamical Theories of Beams, Plates and Shells and Their Applications, Proceedings of the Euromech-Colloquium 219, pp. 102–111.
10.
Kantorovich, L. V., and Krylov, V. I., 1958, Approximate Methods in Higher Analysis, Noordhoff, Groningen.
11.
Leissa, A. W., 1969, Vibration of Plates, U.S. Govt. Printing Office, Washington, D.C., 353 pp. (reprinted by the Acoustical Society of America, 1993).
12.
Leissa, A. W., 1973, Vibration of Shells, U.S. Govt. Printing Office, Washington, D.C., 428 pp. (reprinted by the Acoustical Society of America, 1993).
13.
Leissa
A. W.
, and
So
J.
,
1995
, “
Comparisons of Vibration Frequencies for Rods and Beams from 1-D and 3-D Analyses
,”
Journal of the Acoustical Society of America
, Vol.
181
, pp.
2122
–
2135
.
14.
Pochhammer
L.
,
1876
, “
U¨ber die Fortpflanzungsgeschwindigkeiten kleiner Schwingungen in einem unbegretzten isotropen Kreiszylinder
,”
J. Peine Agnew. Math.
, Vol.
81
, pp.
324
–
326
.
15.
Singal
R. K.
, and
Williams
K.
,
1988
, “
A Theoretical and Experimental Study of Vibrations of Thick Circular Cylindrical Shells and Rings
,”
ASME JOURNAL OF VIBRATION, ACOUSTICS, STRESS, AND RELIABILITY IN DESIGN
, Vol.
110
, pp.
533
–
537
.
16.
So, J., 1993, “Three Dimensional Vibration Analysis of Elastic Bodies of Revolution,” Ph.D. Dissertation, The Ohio State University.
This content is only available via PDF.
Copyright © 1997
 by The American Society of Mechanical Engineers
You do not currently have access to this content.