By introducing the application of the differential quadrature method (DQM) to the dynamic analysis of thin circular cylindrical shells, the work of this paper makes a step forward in furthering the potential of the DQM in the area of structural mechanics. The problem is identified by an eighth-order system of coupled partial differential equations in terms of the three displacement components. The proposed differential quadrature solution is semi-analytical in that Flu¨gge’s representation of the displacement components by trigonometric sine and cosine functions of the circumferential coordinate is employed. The results of the differential quadrature solutions of the natural frequencies of various shell cases are compared and shown to be in excellent agreement with the published, and also some recalculated, results of exact solutions for freely supported, clamped-clamped, clamped-free, and free-free shells. Comparisons are also made with the published experimental data of clamped-clamped and clamped-free shells.

1.
Arnold
R. N.
, and
Warburton
G. B.
,
1953
, “
The Flexural Vibrations of Thin Cylinders
,”
Proceedings of the Institution of Mechanical Engineers
, Vol.
167
, pp.
62
80
.
2.
Bathe, K-J., 1982, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ.
3.
Bellman
R.
, and
Casti
J.
,
1971
, “
Differential Quadrature and Long Term Integration
,”
Journal of Mathematical Analysis and Applications
, Vol.
34
, pp.
235
238
.
4.
Bellman
R.
,
Kashef
B. G.
, and
Casti
J.
,
1972
, “
Differential Quadrature: A Technique for the Rapid Solution of Nonlinear Partial Differential Differential Equations
,”
Journal of Computational Physics
, Vol.
10
, pp.
40
52
.
5.
Bert
C. W.
,
Jang
S. K.
, and
Striz
A. G.
,
1988
, “
Two New Approximate Methods for Analyzing Free Vibration of Structural Components
,”
AIAA Journal
, Vol.
26
, pp.
612
618
.
6.
Bert
C. W.
,
Jang
S. K.
, and
Striz
A. G.
,
1989
, “
Nonlinear Bending Analysis of Orthotropic Rectangular Plates by the Method of Differential Quadrature
,”
Computational Mechanics
, Vol.
5
, pp.
217
226
.
7.
Bert
C. W.
,
Wang
X.
, and
Striz
A. G.
,
1993
, “
Differential Quadrature for Static and Free Vibration Analysis of Anisotropic Plates
,”
International Journal of Solids and Structures
, Vol.
30
, pp.
1737
1744
.
8.
Bjo¨rck
A˚.
, and
Pereyra
V.
,
1970
, “
Solution of Vandermonde System of Equations
,”
Mathematics of Computation
, Vol.
24
, pp.
893
903
.
9.
Cheung, Y. K., 1976, Finite Strip Method in Structural Analysi, Pergamon Press, New York, NY.
10.
Civan
F.
, and
Sliepcevich
C. M.
,
1983
, “
Application of Differential Quadrature to Transport Processes
,”
Journal of Mathematical Analysis and Applications
, Vol.
93
, pp.
206
221
.
11.
Civan
F.
, and
Sliepcevich
C. M.
,
1984
, “
On the Solution of the Thomas-Fermi Equation by Differential Quadrature
,”
Journal of Computational Physics
, Vol.
56
, pp.
343
348
.
12.
Civan, F., and Sliepcevich, C. M., 1986, “Solving Integro-Differential Equations by the Quadrature Method,” Integral Methods in Science and Engineering-86, ed., A. Haji-Shaikh, Hemisphere Publishing Co., New York, NY, pp. 106–113.
13.
Flu¨gge, W., 1960, Stresses in Shells, Springer-Verlag, Berlin, Germany.
14.
Forsberg
K.
,
1964
, “
Influence of Boundary Conditions on the Modal Characteristics of Thin Cylindrical Shells
,”
AIAA Journal
, Vol.
2
, pp.
2150
2157
.
15.
Hamming, R. W., 1973, Numerical Methods for Scientists and Engineers, McGraw-Hill, New York, NY.
16.
Hinton, E., and Owen, D. R. J., 1979, An Introduction to Finite Element Computations, Pine-Ridge Press Limited, Swansea, U.K.
17.
Huebner, K. H., 1975, The Finite Element Method for Engineers, John Wiley & Sons, New York, NY.
18.
Jang
S. K.
,
Bert
C. W.
, and
Striz
A. G.
,
1989
, “
Application of Differential Quadrature to Static Analysis of Structural Components
,”
International Journal for Numerical Methods in Engineering
, Vol.
28
, pp.
561
577
.
19.
Koval, L. R., and Cranch, E. T., 1962, “On the Free Vibrations of Thin Cylindrical Shells Subjected to an Initial Static Torques,” Proceedings of the Fourth US National Congress of Applied Mechanics, Vol. 1, pp. 107–117.
20.
Leissa, A. W., 1973, Vibration of Shells, NASA SP-288, US Government Printing Office, Washington, D.C.
21.
Malik
M.
, and
Bert
C. W.
,
1994
, “
Differential Quadrature Solution for Steady State Incompressible and Compressible Lubrication Problems
,”
ASME Journal of Tribology
, Vol.
116
, pp.
296
302
.
22.
Malik, M., Bert, C. W., and Kukreti, A. R., 1993, “Differential Quadrature Solution of Uniformly Loaded Circular Plate Resting on Elastic Half-Space,” Contact Mechanics Computational Techniques, eds., M. H. Aliabadi and C. A. Brebbia, Computational Mechanics Publications, Southampton, UK, pp. 385–396.
23.
Malik
M.
, and
Civan
F.
,
1995
, “
A Comparative Study of Differential Quadrature and Cubature Methods Vis-a`-Vis Some Conventional Techniques in Context of Convection-Diffusion-Reaction Problems
,”
Chemical Engineering Science
, Vol.
50
, pp.
531
547
.
24.
Mingle
J. O.
,
1973
, “
Computational Considerations in Nonlinear Diffusion
,”
International Journal for Numerical Methods in Engineering
, Vol.
7
, pp.
103
116
.
25.
Press, W. H., Flannery, S. A., Teukolsky, S. A., and Vetterling, W. T., 1988, Numerical Recipes in C, Cambridge University Press, Cambridge, UK.
26.
Quan
J. R.
, and
Chang
C. T.
,
1989
a, “
New Insights in Solving Distributed System Equations by the Quadrature Method—I. Analysis
,”
Computers in Chemical Engineering
, Vol.
13
, pp.
779
788
.
27.
Quan
J. R.
, and
Chang
C. T.
,
1989
b, “
New Insights in Solving Distributed System Equations by the Quadrature Method—II. Numerical Experiments
,”
Computers in Chemical Engineering
, Vol.
13
, pp.
1017
1024
.
28.
Resnick, B. S., and Dugundji, J., 1966, “Effects of Orthotropy, Boundary Conditions, and Eccentricity on the Vibrations of Cylindrical Shells,” Massachusetts Institute of Technology, Aeroelastic and Structures Research Laboratory, ASRL Technical Report 134–2, NTIS Document AD-648077.
29.
Sherbourne
A. N.
, and
Pandey
M. D.
,
1991
, “
Differential Quadrature Method in the Buckling Analysis of Beams and Composite Plates
,”
Computers and Structures
, Vol.
40
, pp.
903
913
.
30.
Shu
C.
, and
Richards
B. E.
,
1992
, “
Application of Generalized Differential Quadrature to Solve Two-Dimensional Incompressible Navier-Stokes Equations
,”
International Journal of Numerical Methods in Fluids
, Vol.
15
, pp.
791
798
.
31.
Wang
X.
, and
Bert
C. W.
,
1993
, “
A New Approach in Applying Differential Quadrature to Static and Free Vibrational Analyses of Beams and Plates
,”
Journal of Sound and Vibration
, Vol.
162
, pp.
566
572
.
32.
Warburton
G. B.
,
1965
, “
Vibration of Thin Cylindrical Shells
,”
Journal of Mechanical Engineering Science
, Vol.
7
, pp.
399
407
.
33.
Warburton
G. B.
, and
Higgs
J.
,
1970
, “
Natural Frequencies of Thin Cantilever Cylindrical Shells
,”
Journal of Sound and Vibration
, Vol.
11
, pp.
335
338
.
34.
Weingarten
V. I.
,
1964
, “
Free Vibration of Thin Cylindrical Shells
,”
AIAA Journal
, Vol.
2
, pp.
717
722
.
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