A new formulation for the NDIF method (the nondimensional dynamic influence function method) is introduced to efficiently extract eigenvalues and mode shapes of arbitrarily shaped, homogeneous membranes with the fixed boundary. The NDIF method, which was developed by the authors for the accurate free vibration analysis of arbitrarily shaped membranes and plates including acoustic cavities, has the feature that it yields highly accurate solutions compared with other analytical methods or numerical methods (the finite element method and the boundary element method). However, the NDIF method has the weak point that the system matrix of the method is not independent of the frequency parameter and as a result the method needs the inefficient procedure of searching eigenvalues by plotting the values of the determinant of the system matrix in the frequency parameter range of interest. An improved formulation presented in the paper does not require the above-mentioned inefficient procedure because a newly developed system matrix is independent of the frequency parameter. Finally, the validity of the proposed method is shown in several case studies, which indicate that eigenvalues and mode shapes obtained by the proposed method are very accurate compared to those calculated by exact, analytica, or numerical methods.

References

References
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.
,
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.
,
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.
,
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.
,
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.
,
256
(
3
), pp.
533
549
.
6.
Kang
,
S. W.
,
Kim
,
I. S.
, and
Lee
,
J. M.
, 2008, “
Free Vibration Analysis of Arbitrarily Shaped Plates With Smoothly Varying Free Edges Using NDIF Method
,”
ASME J. Vib. Acoust.
,
130
(
4
), p.
041010
.
7.
Conway
,
H. D.
, and
Farnham
,
K. A.
, 1965, “
The Free Flexural Vibration of Triangular, Rhombic and Parallelogram Plates and Some Analogies
,”
Int. J. Mech. Sci.
,
7
, pp.
811
816
.
8.
Durvasula
,
S.
, 1968, “
Natural Frequencies and Modes of Skew Membranes
,”
J. Acoust. Soc. Am.
,
44
, pp.
1636
1646
.
9.
Mazumdar
,
J.
, 1973, “
Transverse Vibration of Membranes of Arbitrary Shape by the Method of Constant-Deflection Contours
,”
J. Sound Vib.
,
27
, pp.
47
57
.
10.
Nagaya
,
K.
, 1978, “
Vibrations and Dynamic Response of Membranes With Arbitrary Shape
,”
ASME J. Appl. Mech.
,
45
, pp.
153
158
.
11.
Chopra
,
I.
, and
Durvasula
,
S.
, 1971, “
Vibration of Simply Supported Trapezoidal Plates: I. Symmetric Trapezoids
,”
J. Sound Vib.
,
19
(4)
, pp.
379
392
.
12.
Hamada
,
M.
, 1959, “
Compressive or Shear Buckling Load and Fundamental Frequency of a Rhomboidal Plate With All Edges Clamped
,”
Bull. Jpn. Soc. Mech. Eng.
,
2
, pp.
520
526
.
13.
Conway
,
H. D.
, 1961, “
The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching
,”
ASME J. Appl. Mech.
,
28
, pp.
288
291
.
14.
Durvasula
,
S.
, 1969, “
Free Vibration of Simply Supported Parallelogrammic Plates
,”
J. Aircraft
,
6
, pp.
66
68
.
15.
Chopra
,
I.
, and
Durvasula
,
S.
, 1972, “
Vibration of Simply Supported Trapezoidal Plates, II. Unsymmetric Trapezoids
,”
J. Sound Vib.
,
20
, pp.
125
134
.
16.
Leissa
,
A. W.
, 1973, “
The Free Vibration of Rectangular Plates
,”
J. Sound Vib.
,
31
, pp.
257
293
.
17.
Sato
,
K.
, 1973, “
Free-Flexural Vibrations of an Elliptical Plate With Free Edges
,”
J. Acoust. Soc. Am.
,
54
, pp.
547
550
.
18.
Nair
,
P. S.
, and
Durvasula
,
S.
, 1973, “
Vibration of Skew Plates
,”
J. Sound Vib.
,
26
, pp.
1
19
.
19.
Dickinson
,
S. M.
, 1978, “
The Buckling and Frequency of Flexural Vibration of Rectangular, Isotropic and Orthotropic Plates Using Rayleigh’s Method
,”
J. Sound Vib.
,
61
, pp.
1
8
.
20.
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.
,
152
(
1
), pp.
149
155
.
21.
Hughes
,
T. J. R.
, 1987,
The Finite Element Method
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
22.
Brebbia
,
C. A.
,
Telles
,
J. C. F.
, and
Wrobel
,
L. C.
, 1984,
Boundary Element Techniques
,
Springer
,
New York
.
23.
Nardini
,
D.
, and
Brebbia
,
C. A.
, 1982, “
A New Approach to Free Vibration Analysis Using Boundary Elements
,”
Appl. Math.l Model.
,
7
(
3
), pp.
157
162
.
24.
Wilkinson
,
J. H.
, 1965,
The Algebraic Eigenvalue Problem
,
Oxford University Press
,
New York
.
25.
Spiegel
,
M. R.
, 1983,
Advanced Mathematics
,
McGraw-Hill
,
Singapore
.
26.
Gohberg
,
I.
,
Lancaster
,
P.
, and
Rodman
,
L.
, 1982,
Matrix Polynomials
,
Academic
,
New York
.
27.
Blevins
,
R. D.
, 1979,
Formulas for Natural Frequency and Mode Shape
,
Litton Educational
,
New York
.
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