In order to compare numerical and analytical results for the free vibration analysis of Kirchhoff plates with both partially and completely damaged boundaries, the Chebyshev collocation and perturbation methods are utilized in this paper, where the damaged boundaries are represented by distributed translational and torsional springs. In the Chebyshev collocation method, the convergence studies are performed to determine the sufficient number of the grid points used. In the perturbation method, the small perturbation parameter is defined in terms of the damage parameter of the plate, and a sequence of recurrent linear boundary value problems is obtained which is further solved by the separation of variables technique. The results of the two methods are in good agreement for small values of the damage parameter as well as with the results in the literature for the undamaged case. The case of mixed damaged boundary conditions is also treated by the Chebyshev collocation method.

References

References
1.
Leissa
,
A.
, 1973, “
The Free Vibration of Rectangular Plates
,”
J. Sound Vib.
31
, pp.
257
293
.
2.
Bakhtiari-Nejad
,
F.
, and
Nazari
,
M.
, 2008, “
Nonlinear Vibration Analysis of Isotropic Cantilever Plate With Viscoelastic Laminate
,”
Nonlinear Dyn.
56
, pp.
325
356
.
3.
Dickinson
,
S.
, and
Li
,
E.
, (1982), “
On the Use of Simply Supported Plate Functions in the Rayleigh-Ritz Method Applied to the Vibration of Rectangular Plates
,”
J. Sound Vib.
80
, pp.
292
297
.
4.
Cupial
,
P.
, 1997, “
Calculation of the Natural Frequencies of Composite Plates by the Rayleigh-Ritz Method With Orthogonal Polynomials
,”
J. Sound Vib.
201
, pp.
385
387
.
5.
Bhat
, 1985, “
Natural Frequencies of Rectangular Plates Using Characteristic Orthogonal Polynomials in the Rayleigh-Ritz Method
,”
J. Appl. Mech.
102
, pp.
493
499
.
6.
Li
,
W.
,
Daniels
,
M.
, and
Fourier
,
A.
, 2002, “
Series Method for the Vibrations of Elastically Restrained Plates Arbitrarily Loaded With Springs and Masses
,”
J. Sound Vib.
252
, pp.
768
781
.
7.
Li
,
W.
,
Zhang
,
X.
,
Du
,
J.
, and
Liu
,
Z.
, 2009, “
An Exact Series Solution for the Transverse Vibration of Rectangular Plates With General Elastic Boundary Supports
,”
J. Sound Vib.
321
, pp.
254
269
.
8.
Shu
,
C.
, and
Du
,
H.
, 1997, “
A Generalized Approach for Implementing General Boundary Conditions in the GDQ Free Vibration Analysis of Plates
,”
Int. J. Solids Struct.
34
, pp.
837
846
.
9.
Bert
,
C.
,
Wang
,
X.
, and
Striz
,
A.
, 1993, “
Differential Quadrature for Static and Free Vibrational Analysis of Anisotropic Plates
,”
Int. J. Solids Struct.
30
, pp.
1737
1744
.
10.
Butcher
,
E.
,
Sevostianov
,
I.
,
Sari
,
M.
, and
Al-Shudeifat
,
M.
, “
Use of Nonlinear Vibration Frequencies and Electrical Conductivity Measurements in the Separation of Internal and Boundary Damage in Structures
,”
Proceedings of IMEC2008 ASME International Mechanical Engineering Congress and Exposition
, Boston, MA.
11.
Butcher
,
E.
,
Sevostianov
,
I.
, and
Burton
,
T.
, “
On the Separation of Internal and Boundary Damage From Combined Measurements of Electrical Conductivity and Vibration Frequencies
,”
Int. J. Eng. Sci.
46
, pp.
986
975
.
12.
Sari
,
M.
, and
Butcher
,
E.
, 2010, “
Natural Frequencies and Critical Loads of Beams and Columns with Damaged Boundaries Using Chebyshev Polynomials
,”
Int. J. Eng. Sci.
48
, pp.
862
873
.
13.
Yagci
,
B.
,
Filiz
,
S.
,
Romero
,
L.
, and
Ozdoganlar
,
O.
, 2009, “
A Spectral-Tchebychev Technique for Solving Linear and Nonlinear Beam Equations
,”
J. Sound Vib.
321
, pp.
375
404
.
14.
Lin
,
C.
, and
Jen
,
M.
, 2004, “
Analysis of Non-Rectangular Laminated Anisotropic Plates by Chebyshev Collocation Method
,”
JSME Int. J.
47
, pp.
146
156
.
15.
Lin
,
C.
, and
Jen
,
M.
, 2005, “
Analysis of a Laminated Anisotropic Plates By Chebyshev Collocation Method
,”
Composites, Part B
36
, pp.
155
169
.
16.
Trefethen
,
L.
, 2000,
Spectral Methods in Matlab, Software, Environments, and Tools
, (
SIAM
,
Philadelphia, PA
).
17.
Andrianov
,
I.
,
Gristchak
,
V.
, and
Ivankov
,
A.
, 1994, “
New Asymptotic Method for the Natural, Free and Forced Oscillations of Rectangular Plates with Mixed Boundary Conditions
,”
Technische Mechanik
14
, pp.
185
192
.
18.
Andrianov
,
I.
,
Awrejcewicz
,
J.
, and
Ivankov
,
A.
, 2005, “
Artificial Small Parameter Method - Solving Mixed Boundary Value Problems
,”
Math. Probl. Eng.
3
, pp.
325
340
.
19.
Liu
,
F.
, and
Liew
,
K.
, 1999, “
Analysis of Vibrating Thick Rectangular Plates With Mixed Boundary Constraints Using Differential Quadrature Element Method
,”
J. Sound Vib.
225
, pp.
915
934
.
20.
Shu
,
C.
, and
Wang
,
C.
, 1999, “
Treatment of Mixed and Nonuniform Boundary Conditions in GDQ Vibration Analysis of Rectangular Plates
,”
Eng. Struct.
21
, pp.
125
134
.
21.
Wei
,
G.
,
Zhao
,
Y.
, and
Xiang
,
Y.
, 2001, “
The Determination of Natural Frequencies of Rectangular Plates With Mixed Boundary Conditions by Discrete Singular Convolution
,”
Int. J. Mech. Sci.
43
, pp.
1731
1746
.
22.
Leissa
,
A.
,
Laura
,
P.
, and
Gutierrez
,
R.
, 1979, “
Vibrations of Rectangular Plates With Non-Uniform Elastic Edge Supports
,”
J. Appl. Mech.
47
, pp.
891
895
.
23.
Saito
,
A.
,
Castanier
,
M.
, and
Pierre
,
C.
, 2009, “
Estimation and Veering Analysis of Nonlinear Resonant Frequencies of Cracked Plates
,”
J. Sound Vib.
326
, pp.
725
739
.
24.
Plaut
,
R.
, and
Virgin
,
L.
, “
Vibration and Snap-Through of Bent Elastica Strips Subjected to End Rotations
,”
J. Appl. Mech.
76
, p.
041011.1
.
25.
Plaut
,
R.
,
Murphy
,
K.
, and
Virgin
,
L.
, “
Curve and Surface Veering for a Braced Column
,”
J. Sound Vib.
187
, pp.
879
885
.
26.
Laura
,
P.
, and
Gutierrez
,
R.
, “
Analysis of Vibrating Rectangular Plates With Non-Uniform Boundary Conditions by Using the Differential Quadrature Method
,”
J. Sound Vib.
173
, pp.
702
706
.
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