A modified Fourier–Ritz approach is developed in this study to analyze the free in-plane vibration of orthotropic annular sector plates with general boundary conditions. In this approach, two auxiliary sine functions are added to the standard Fourier cosine series to obtain a robust function set. The introduction of a logarithmic radial variable simplifies the expressions of total energy and the Lagrangian function. The improved Fourier expansion based on the new variable eliminates all the potential discontinuities of the original displacement function and its derivatives in the entire domain and effectively improves the convergence of the results. The radial and circumferential displacements are formulated with the modified Fourier series expansion, and the arbitrary boundary conditions are simulated by the artificial boundary spring technique. The number of terms in the truncated Fourier series and the appropriate value of the boundary spring retraining stiffness are discussed. The developed Ritz procedure is used to obtain accurate solution with adequately smooth displacement field in the entire solution domain. Numerical examples involving plates with various boundary conditions demonstrate the robustness, precision, and versatility of this method. The method developed here is found to be computationally economic compared with the previous method that does not adopt the logarithmic radial variable.

References

References
1.
Leissa
,
A. W.
,
McGee
,
O. G.
, and
Huang
,
C. S.
,
1993
, “
Vibrations of Circular Plates Having V-Notches or Sharp Radial Cracks
,”
J. Sound Vib.
,
161
(
2
), pp.
227
239
.
2.
Huang
,
C. S.
,
Leissa
,
A. W.
, and
McGee
,
O. G.
,
1993
, “
Exact Analytical Solutions for the Vibrations of Sectorial Plates With Simply Supported Radial Edges
,”
ASME J. Appl. Mech.
,
60
(
2
), pp.
478
483
.
3.
Jomehzadeh
,
E.
, and
Saidi
,
A. R.
,
2009
, “
Analytical Solution for Free Vibration of Transversely Isotropic Sector Plates Using a Boundary Layer Function
,”
Thin Walled Struct.
,
47
(
1
), pp.
82
88
.
4.
Cheung
,
Y. K.
, and
Kwok
,
W. L.
,
1975
, “
Dynamic Analysis of Circular and Sector Thick, Layered Plates
,”
J. Sound Vib.
,
42
(
2
), pp.
147
158
.
5.
Liew
,
K. M.
,
Xiang
,
Y.
, and
Kitipornchai
,
S.
,
1995
, “
Research on Thick Plate Vibration: A Literature Survey
,”
J. Sound Vib.
,
180
(
1
), pp.
163
176
.
6.
Onoe
,
M.
,
1956
, “
Contour Vibrations of Isotropic Circular Plates
,”
J. Acoust. Soc. Am.
,
28
(
6
), pp.
1158
1162
.
7.
Holland
,
R.
,
1966
, “
Numerical Studies of Elastic-Disk Contour Modes Lacking Axial Symmetry
,”
J. Acoust. Soc. Am.
,
40
(
5
), pp.
1051
1057
.
8.
Irie
,
T.
,
Yamada
,
G.
, and
Muramoto
,
Y.
,
1984
, “
Natural Frequencies of In-Plane Vibration of Annular Plates
,”
J. Sound Vib.
,
97
(
1
), pp.
171
175
.
9.
Leung
,
A. Y. T.
,
Zhu
,
B.
,
Zheng
,
J.
, and
Yang
,
H.
,
2004
, “
Analytic Trapezoidal Fourier p-Element for Vibrating Plane Problems
,”
J. Sound Vib.
,
271
(1–2), pp.
67
81
.
10.
Lyon
,
R. H.
,
1986
, “
In-Plane Contribution to Structural Noise Transmission
,”
Noise Control Eng. J.
,
26
(
1
), pp.
22
27
.
11.
Farag
,
N.
, and
Pan
,
J.
,
2003
, “
Modal Characteristics of In-Plane Vibration of Circular Plates Clamped at the Outer Edge
,”
J. Acoust. Soc. Am.
,
113
(
4
), pp.
1935
1946
.
12.
Chen
,
S. S.
, and
Liu
,
T. M.
,
1975
, “
Extensional Vibration of Thin Plates of Various Shapes
,”
J. Acoust. Soc. Am.
,
58
(
4
), pp.
828
831
.
13.
Bashmal
,
S.
,
Bhat
,
R.
, and
Rakheja
,
S.
,
2009
, “
In-Plane Free Vibration of Circular Annular Disks
,”
J. Sound Vib.
,
322
(
1
), pp.
216
226
.
14.
Park
,
C. I.
,
2008
, “
Frequency Equation for the In-Plane Vibration of a Clamped Circular Plate
,”
J. Sound Vib.
,
313
(
1–2
), pp.
325
333
.
15.
Seok
,
J.
, and
Tiersten
,
H.
,
2004
, “
Free Vibrations of Annular Sector Cantilever Plates—Part 2: In-Plane Motion
,”
J. Sound Vib.
,
271
(
3
), pp.
773
787
.
16.
Ravari
,
M. K.
, and
Forouzan
,
M.
,
2011
, “
Frequency Equations for the In-Plane Vibration of Orthotropic Circular Annular Plate
,”
Arch. Appl. Mech.
,
81
(
9
), pp.
1307
1322
.
17.
Vladimir
,
N.
,
Hadžić
,
N.
,
Senjanović
,
I.
, and
Xing
,
Y.
,
2014
, “
Potential Theory of In-Plane Vibrations of Rectangular and Circular Plates
,”
Int. J. Eng. Modell.
,
27
(3–4), pp.
69
84
.
18.
Kim
,
C.-B.
,
Cho
,
H. S.
, and
Beom
,
H. G.
,
2012
, “
Exact Solutions of In-Plane Natural Vibration of a Circular Plate With Outer Edge Restrained Elastically
,”
J. Sound Vib.
,
331
(
9
), pp.
2173
2189
.
19.
Singh
,
A.
, and
Muhammad
,
T.
,
2004
, “
Free In-Plane Vibration of Isotropic Non-Rectangular Plates
,”
J. Sound Vib.
,
273
(
1
), pp.
219
231
.
20.
Wang
,
Q.
,
Shi
,
D.
,
Liang
,
Q.
, and
Fazl e Ahad
,
2016
, “
A Unified Solution for Free In-Plane Vibration of Orthotropic Circular, Annular and Sector Plates With General Boundary Conditions
,”
Appl. Math. Modell.
,
40
(
21–22
), pp.
9228
9253
.
21.
Li
,
W. L.
,
2000
, “
Free Vibrations of Beams With General Boundary Conditions
,”
J. Sound Vib.
,
237
(
4
), pp.
709
725
.
22.
Li
,
W. L.
,
2002
, “
Comparison of Fourier Sine and Cosine Series Expansions for Beams With Arbitrary Boundary Conditions
,”
J. Sound Vib.
,
255
(
1
), pp.
185
194
.
23.
Shi
,
X.
,
Li
,
W.
, and
Shi
,
D.
,
2014
, “
Free In-Plane Vibrations of Annular Sector Plates With Elastic Boundary Supports
,”
Meetings on Acoustics Acoustical Society of America
(
ASA
), Indianapolis, IN, Oct. 27–31.
24.
Zhang
,
K.
,
Shi
,
D.
,
Teng
,
X.
,
Zhao
,
Y.
, and
Liang
,
Q.
,
2015
, “
A Series Solution for the In-Plane Vibration of Sector Plates With Arbitrary Inclusion Angles and Boundary Conditions
,”
J. Vibroeng.
,
17
(
2
), pp.
870
882
.
25.
Du
,
J.
,
Li
,
W. L.
,
Jin
,
G.
,
Yang
,
T.
, and
Liu
,
Z.
,
2007
, “
An Analytical Method for the In-Plane Vibration Analysis of Rectangular Plates With Elastically Restrained Edges
,”
J. Sound Vib.
,
306
(
3–5
), pp.
908
927
.
26.
Jin
,
G.
,
Ye
,
T.
,
Chen
,
Y.
,
Su
,
Z.
, and
Yan
,
Y.
,
2013
, “
An Exact Solution for the Free Vibration Analysis of Laminated Composite Cylindrical Shells With General Elastic Boundary Conditions
,”
Compos. Struct.
,
106
, pp.
114
127
.
27.
Chen
,
Y.
,
Jin
,
G.
, and
Liu
,
Z.
,
2014
, “
Flexural and In-Plane Vibration Analysis of Elastically Restrained Thin Rectangular Plate With Cutout Using Chebyshev–Lagrangian Method
,”
Int. J. Mech. Sci.
,
89
, pp.
264
278
.
28.
Wang
,
Q.
,
Shi
,
D.
,
Liang
,
Q.
, and
Shi
,
X.
,
2016
, “
A Unified Solution for Vibration Analysis of Functionally Graded Circular, Annular and Sector Plates With General Boundary Conditions
,”
Composites, Part B
,
88
, pp.
264
294
.
29.
Wang
,
Q.
,
Shi
,
D.
, and
Shi
,
X.
,
2016
, “
A Modified Solution for the Free Vibration Analysis of Moderately Thick Orthotropic Rectangular Plates With General Boundary Conditions, Internal Line Supports and Resting on Elastic Foundation
,”
Meccanica
,
51
(
8
), pp.
1985
2017
.
30.
Yao
,
W.
,
Zhong
,
W.
, and
Lim
,
C. W.
,
2009
,
Symplectic Elasticity
,
World Scientific
,
Singapore
.
31.
Kim
,
K.
, and
Yoo
,
C. H.
,
2010
, “
Analytical Solution to Flexural Responses of Annular Sector Thin-Plates
,”
Thin-Walled Struct.
,
48
(
12
), pp.
879
887
.
32.
Dozio
,
L.
,
2010
, “
Free In-Plane Vibration Analysis of Rectangular Plates With Arbitrary Elastic Boundaries
,”
Mech. Res. Commun.
,
37
(
7
), pp.
627
635
.
33.
Gorman
,
D. J.
,
2006
, “
Exact Solutions for the Free In-Plane Vibration of Rectangular Plates With Two Opposite Edges Simply Supported
,”
J. Sound Vib.
,
294
(1–2), pp.
131
161
.
34.
Budiansky
,
B.
, and
Hu
,
P. C.
,
1946
, “
The Lagrangian Multiplier Method of Finding Upper and Lower Limits to Critical Stresses of Clamped Plates
,” NASA Langley Research Center, Hampton, VA, Technical Report No.
848
.https://ntrs.nasa.gov/search.jsp?R=19960017539
35.
Ilanko
,
S.
,
Monterrubio
,
L.
, and
Mochida
,
Y.
,
2014
,
The Rayleigh-Ritz Method for Structural Analysis
,
Wiley
, Hoboken, NJ.
36.
Monterrubio
,
L. E.
, and
Ilanko
,
S.
,
2015
, “
Proof of Convergence for a Set of Admissible Functions for the Rayleigh–Ritz Analysis of Beams and Plates and Shells of Rectangular Planform
,”
Comput. Struct.
,
147
, pp.
236
243
.
You do not currently have access to this content.