The so-called non-dimensional influence function method that was developed by the authors is extended to free vibration analysis of arbitrarily shaped plates with the free boundary condition. A method proposed in this paper can be applied to plates with only smoothly varying boundary shapes. In the proposed method, a local polar coordinate system has been employed at each boundary node to effectively consider the free boundary condition, which is much more complex than the simply supported or clamped boundary condition. The local coordinates system devised allowed us to successfully deal with the radius of curvature involved in the free boundary condition, and, as a result, the accuracy of the proposed method has been reinforced. Finally, verification examples showed that the natural frequency and mode shapes obtained by the proposed method agree excellently with those given by other analytical or numerical methods.
Issue Section:
Research Papers
1.
Kang
, S. W.
, Lee
, J. M.
, and Kang
, Y. J.
, 1999, “Vibration Analysis of Arbitrarily Shaped Membranes Using Non-dimensional Dynamic Influence Function
.” J. Sound Vib.
0022-460X, 221
, pp. 117
–132
.2.
Kang
, S. W.
, and Lee
, J. M.
, 2000, “Eigenmode Analysis of Arbitrarily Shaped Two-dimensional Cavities by the Method of Point-Matching
.” J. Acoust. Soc. Am.
0001-4966, 107
(3
), pp. 1153
–1160
.3.
Kang
, S. W.
, and Lee
, J. M.
, 2000, “Application of Free Vibration Analysis of Membranes Using the Non-dimensional Dynamic Influence Function
.” J. Sound Vib.
0022-460X, 234
(3
), pp. 455
–470
.4.
Kang
, S. W.
, and Lee
, J. M.
, 2002, “Free Vibration Analysis of Arbitrarily Shaped Plates With Clamped Edges Using Wave-type Functions
.” J. Sound Vib.
0022-460X, 242
(1
), pp. 9
–26
.5.
Kang
, S. W.
, 2002, “Free Vibration Analysis of Arbitrarily Shaped Plates With a Mixed Boundary Condition Using Non-dimensional Dynamic Influence Functions
.” J. Sound Vib.
0022-460X, 256
(3
), pp. 533
–549
.6.
Hughes
, T. J. R.
, 1987, The Finite Element Method
, Prentice-Hall
, Englewood Cliffs, NJ.7.
Brebbia
, C. A.
, 1978, The Boundary Element Method for Engineers
, John Wiley & Sons
, New York.8.
Brebbia
, C. A.
, Telles
, J. C. F.
, and Wrobel
, L. C.
, 1984, Boundary Element Techniques
, Springer-Verlag
, New York.9.
Conway
, H. D.
, 1961, “The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching
.” ASME J. Appl. Mech.
0021-8936 28
, pp. 288
–291
.10.
Singh
, B.
, and Chakraverty
, S.
, 1992, “Transverse Vibration of Simply Supported Elliptical and Circular Plates Using Boundary Characteristic Orthogonal Polynomials in Two Variables
.” J. Sound Vib.
0022-460X, 152
(1
), pp. 149
–155
.11.
Conway
, H. D.
, and Karnham
, K. A.
, 1965, “The Free Flexural Vibration of Triangular, Rhombic and Parallelogram Plates and Some Anologies
.” Int. J. Mech. Sci.
0020-7403, 7
, pp. 811
–816
.12.
Mazumdar
, J.
, 1973, “Transverse Vibration of Membranes of Arbitrary Shape by the Method of Constant-Deflection Contours
.” J. Sound Vib.
0022-460X, 27
, pp. 47
–57
.13.
Nagaya
, K.
, 1978, “Vibrations and Dynamic Response of Membranes With Arbitrary Shape
.” ASME J. Appl. Mech.
0021-8936, 45
, pp. 153
–158
.14.
Durvasula
, S.
, 1969, “Free Vibration of Simply Supported Parallelogrammic Plates
.” J. Aircr.
0021-8669, 6
, pp. 66
–68
.15.
Chopra
, I.
, and Durvasula
, S.
, 1972, “Vibration of Simply Supported Trapezoidal Plates, I. Symmetric Trapezoids
.” J. Sound Vib.
0022-460X, 19
, pp. 379
–392
.16.
Chopra
, I.
, and Durvasula
, S.
, 1972, “Vibration of Simply Supported Trapezoidal Plates, II. Unsymmetric Trapezoids
.” J. Sound Vib.
0022-460X, 20
, pp. 125
–134
.17.
Durvasula
, S.
, 1968, “Natural Frequencies and Modes of Skew Membranes
.” J. Acoust. Soc. Am.
0001-4966, 44
, pp. 1636
–1646
.18.
Durvasula
, S.
, 1969, “Natural Frequencies and Modes of Clamped Skew Plates
.” AIAA J.
0001-1452, 7
, pp. 1164
–1167
.19.
Hasegawa
, M.
, 1957, “Vibration of Clamped Parallelogrammic Isotropic Flat Plates
.” J. Aeronaut. Sci.
0095-9812, 24
, pp. 145
–146
.20.
Hamada
, M.
, 1959, “Compressive or Shear Buckling Load and Fundamental Frequency of a Rhomboidal Plate with All Edges Clamped
.” Bull. JSME
0021-3764, 2
, pp. 520
–526
.21.
Nair
, P. S.
, and Durvasula
, S.
, 1973, “Vibration of Skew Plates
.” J. Sound Vib.
0022-460X, 26
, pp. 1
–19
.22.
Dickinson
, S. M.
, 1978, “The Buckling and Frequency of Flexural Vibration of Rectangular, Isotropic and Orthotropic Plates Using Rayleigh’s Method
.” J. Sound Vib.
0022-460X, 61
, pp. 1
–8
.23.
Leissa
, A. W.
, 1973, “The Free Vibration of Rectangular Plates
.” J. Sound Vib.
0022-460X, 31
, pp. 257
–293
.24.
Sato
, K.
, 1973, “Free-Flexural Vibrations of an Elliptical Plate With Free Edges
.” J. Acoust. Soc. Am.
0001-4966, 54
, pp. 547
–550
.25.
Zhang
, X.
, Song
, K. Z.
, Lu
, M. W.
, and Liu
, X.
, 2000, “Meshless Methods Based on Collocation With Radial Basis Functions
.” Comput. Mech.
0178-7675, 26
, pp. 333
–343
.26.
Alves
, C. J. S.
, and Antunes
, P. R. S.
, 2005, “The Method of Fundamental Solutions Applied to the Calculation of Eigenfrequencies and Eigenmodes of 2D Simply Connected Shapes
.” Comput., Mater., Continua
1546-2218, 2
(4
), pp. 251
–265
.27.
Chen
, J. T.
, Chen
, I. L.
, and Lee
, Y. T.
, 2005, “Eigensolutions of Multiply-Connected Membranes Using Method of Fundamental Solution, Engineering Analysis With Boundary Elements
.” Eng. Anal. Boundary Elem.
0955-7997, 29
(2
), pp. 166
–174
.28.
Reutskiy
, S. Y.
, 2005, “The Method of Fundamental Solutions for Eigenproblems With Laplace and Biharmonic Operators
.” Comput., Mater., Continua
1546-2218, 2
(3
), pp. 177
–188
.29.
Chen
, J. T.
, Lee
, Y. T.
, Chen
, I. L.
, and Chen
, K. H.
, 2004, “Mathematical Analysis and Treatment for the True and Spurious Eigenequations of Circular Plates in the Meshless Method Using Radial Basis Function
.” J. Chin. Inst. Eng.
0253-3839, 27
(4
), pp. 547
–561
.30.
Meirovitch
, L.
, 1967, Analytical Methods in Vibrations
, Macmillan
, New York, pp. 179
–189
.31.
Chen
, J. T.
, Chen
, I. L.
, Chen
, K. H.
, and Lee
, Y. T.
, 2003, “Comments on Free Vibration Analysis of Arbitrarily Shaped Plates With Clamped Edges Using Wave-type Functions
.” J. Sound Vib.
0022-460X, 262
(2
), pp. 370
-378
.32.
Chen
, J. T.
, Chang
, M. H.
, Chen
, K. H.
, and Lin
, S. R.
, 2002, “The Boundary Collocation Method With Meshless Concept for Acoustic Eigenanalysis of Two-dimensional Cavities Using Radial Basis Function
.” J. Sound Vib.
0022-460X, 257
(4
), pp. 667
–711
.Copyright © 2008
by American Society of Mechanical Engineers
You do not currently have access to this content.
