The free-vibration behavior of rectangular plates constitutes an important field in applied mechanics, and the natural frequencies are known to be primarily affected by the boundary conditions as well as aspect and thickness ratios. Any one of the three classical edge conditions, i.e., free, simply supported, and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations the present paper introduces the Polya counting theory in combinatorial mathematics. Formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three classical edge conditions and is used to numerically verify the numbers. In this numerical study the number of combinations in the free-vibration behavior is determined for some plate models by using the derived formulas. Results are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the modified Ritz method. [S0021-8936(00)02203-0]

Issue Section:
Technical Papers
Keywords:
vibrations, combinatorial mathematics, structural engineering
Topics:
Anisotropy, Free vibrations, Plates (structures), Boundary-value problems, Mathematics
1.
Leissa, A. W., 1969, “Vibration of Plates,” NASA-160, U.S. Government Printing Office, Washington, D.C.
2.
Blevins, R. D., 1979, Formulas for Natural Frequency and Mode Shape, Van Nostrand Reinhold, New York.
3.
Gorman, D. J., 1982, Free Vibration Analysis of Rectangular Plates, Elsevier, New York.
4.
Sekiya, S., Hamada, M., and Sumi, S., 1982, Handbook for Strength and Design for Plate Structures, Asakura Publishing Co., Tokyo (in Japanese).
5.
Liu, C. L., 1968, Introduction of Combinatorial Mathematics, McGraw-Hill, New York, pp. 126–166.
6.
Dornhoff, L. L., and Hohn, F. E., 1978, Applied Modern Algebra, Macmillan, New York, pp. 242–255.
7.
Leissa
,
A. W.
,
1973
, “
The Free Vibration of Rectangular Plates
,”
J. Sound Vib.
,
31
, pp.
257
–
293
.
8.
Slomson, A., An Introduction to Combinatorics, Chapman and Hall, London, pp. 109–112.
9.
Cohen, H., 1992, Mathematics for Scientists and Engineers, Prentice-Hall, Englewood Cliffs, NJ, pp. 515–521.
10.
Jones, R. M., 1975, Mechanics of Composite Materials, Scripta, Washington D.C.
11.
Vinson, J. R., and Sierakowski, R. L., 1986, The Behavior of Structures Composed of Composite Materials, Martinus Nijhoff, Dordrecht.
12.
Narita
,
Y.
,
Ohta
,
Y.
,
Yamada
,
G.
, and
Kobayashi
,
Y.
,
1992
, “
Analytical Method for Vibration of Angle-Ply Cylindrical Shells Having Arbitrary Edges
,”
Am. Inst. Aeronaut. Astronaut. J.
,
30
, pp.
790
–
796
.
13.
Narita, Y., 1995, “Series and Ritz-Type Buckling Analysis,” Buckling and Postbuckling of Composite Plates, edited by G. J. Turvey and I. H. Marshall, eds., Chapman and Hall, London, pp. 33–57.
14.
Gorman
,
D. J.
,
1976
, “
Free Vibration Analysis of Cantilever Plates by the Method of Superposition
,”
J. Sound Vib.
,
49
, pp.
453
–
467
.
15.
Gorman
,
D. J.
,
1977
, “
Free-Vibration Analysis of Rectangular Plates With Clamped-Simply Supported Edge Conditions by the Method of Superposition
,”
ASME J. Appl. Mech.
,
44
, pp.
743
–
749
.
16.
Gorman
,
D. J.
,
1978
, “
Free Vibration Analysis of the Completely Free Rectangular Plate by the Method of Superposition
,”
J. Sound Vib.
,
57
, pp.
437
–
447
.
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