A modified Fourier spectral element method (SEM) is developed to study vibration behaviors of general built-up systems that consist of various rotationally symmetric structures, including conical, cylindrical, and spherical shells, and annular plates. The substructures are formulated as spectral elements based on the type of elementary structure. These spectral elements can be easily assembled using three-dimensional elastic couplers at the interfaces to coordinate the forces and moments of adjacent substructures with respect to a cylindrical coordinate system. A modified Fourier series that can adapt general coupling and boundary conditions is used to describe the displacement components of each spectral element. The generalized variation of the expansion coefficients yields the vibration equations of the built-up system. The convergence and accuracy of the method are validated through several numerical examples for various shell and plate combinations. The dynamic behaviors of the rotationally symmetric built-up structures are investigated by modal and forced vibration analysis. The method can predict the vibration responses of rotationally symmetric, built-up structures with low computational cost.

References

References
1.
Leissa
,
A. W.
,
1993
,
Vibrations of Shells
,
Acoustical Society of America
,
New York
.
2.
Arafat
,
H. N.
,
Nayfeh
,
A. H.
, and
Faris
,
W.
,
2004
, “
Natural Frequencies of Heated Annular and Circular Plates
,”
Int. J. Solid Struct.
,
41
(
11–12
), pp.
3031
3051
.
3.
Cheng
,
L.
,
Li
,
Y. Y.
, and
Yam
,
L. H.
,
2003
, “
Vibration Analysis of Annular-Like Plates
,”
J. Sound Vib.
,
262
(
5
), pp.
1153
1170
.
4.
Jin
,
G. Y.
,
Xie
,
X.
, and
Liu
,
Z. G.
,
2014
, “
The Haar Wavelet Method for Free Vibration Analysis of Functionally Graded Cylindrical Shells Based on the Shear Deformation Theory
,”
Compos. Struct.
,
108
, pp.
435
448
.
5.
Ming
,
F. R.
,
Zhang
,
A. M.
, and
Yao
,
X. L.
,
2013
, “
Static and Dynamic Analysis of Elastic Shell Structures With Smoothed Particle Method
,”
Acta Phys. Sin.
,
62
(
11
), p.
110203
.
6.
Kildishev
,
A. V.
,
Salon
,
S.
,
Chari
,
M. V. K.
, and
Kwon
,
O. M.
,
2003
, “
Spatial Harmonic Analysis of FEM Results in Magnetostatics
,”
IEEE Trans. Mag.
,
39
(5), pp.
2024
3036
.
7.
Huang
,
D. T.
, and
Soedel
,
W.
,
1993
, “
Natural Frequencies and Modes of a Circular Plate Welded to a Circular Cylindrical Shell at Arbitrary Axial Position
,”
J. Sound Vib.
,
162
(
3
), pp.
403
427
.
8.
Yim
,
J. S.
,
Sohn
,
D. S.
, and
Lee
,
Y. S.
,
1998
, “
Free Vibration of Clamped-Free Circular Cylindrical Shell With a Plate Attached at an Arbitrary Axial Position
,”
J. Sound Vib.
,
213
(
1
), pp.
75
88
.
9.
Irie
,
T.
,
Yamada
,
G.
, and
Muramoto
,
Y.
,
1984
, “
Free Vibration of Joined Conical-Cylindrical Shells
,”
J. Sound Vib.
,
95
(
1
), pp.
31
39
.
10.
Liang
,
S.
, and
Chen
,
H. L.
,
2006
, “
The Natural Vibration of a Conical Shell With an Annular end Plate
,”
J. Sound Vib.
,
294
(
4–5
), pp.
927
943
.
11.
Caresta
,
M.
, and
Kessissoglou
,
N. J.
,
2010
, “
Free Vibrational Characteristics of Isotropic Coupled Cylindrical-Conical Shells
,”
J. Sound Vib.
,
329
(
6
), pp.
733
751
.
12.
Wei
,
J. H.
,
Chen
,
M. X.
,
Hou
,
G. X.
,
Xie
,
K.
, and
Deng
,
N. Q.
,
2013
, “
Wave Based Method for Free Vibration Analysis of Cylindrical Shell With Nonuniform Stiffener Distribution
,”
ASME J. Vib. Acoust.
,
135
(
6
), p.
061011
.
13.
Chen
,
M. X.
,
Zhang
,
C.
,
Tao
,
X. F.
, and
Deng
,
N. Q.
,
2014
, “
Structural and Acoustic Responses of a Submerged Stiffened Conical Shell
,”
Shock Vib.
,
2014
, pp.
1
14
.
14.
Ma
,
Y. B.
,
Zhang
,
Y. H.
, and
Kennedy
,
D.
,
2015
, “
A Hybrid Wave Propagation and Statistical Energy Analysis on the Mid-Frequency Vibration of Built-Up Plate Systems
,”
J. Sound Vib.
,
352
, pp.
63
79
.
15.
Song
,
X. Y.
,
Han
,
Q. K.
, and
Zhai
,
J. Y.
,
2015
, “
Vibration Analyses of Symmetrically Laminated Composite Cylindrical Shells With Arbitrary Boundaries Conditions Via Rayleigh–Ritz Method
,”
Compos. Struct.
,
134
, pp.
820
830
.
16.
Lee
,
Y. S.
,
Yang
,
M. S.
,
Kim
,
H. S.
, and
Kim
,
J. H.
,
2002
, “
A Study on the Free Vibration of the Joined Cylindrical-Spherical Shell Structures
,”
Comput. Struct.
,
80
(
27–30
), pp.
2405
2414
.
17.
Monterrubio
,
L. E.
,
2009
, “
Free Vibration of Shallow Shells Using the Rayleigh-Ritz Method and Penalty Parameters
,”
Proc. Inst. Mech. Eng. Part C
,
223
(
10
), pp.
2263
2272
.
18.
Cheng
,
L.
, and
Nicolas
,
J.
,
1992
, “
Free Vibration Analysis of a Cylindrical Shell-Circular Plate System With General Coupling and Various Boundary Conditions
,”
J. Sound Vib.
,
155
(
2
), pp.
231
247
.
19.
Qu
,
Y. G.
,
Chen
,
Y.
,
Long
,
X. H.
,
Hua
,
H. X.
, and
Meng
,
G.
,
2013
, “
A Modified Variational Approach for Vibration Analysis of Ring-Stiffened Conical-Cylindrical Shell Combinations
,”
Eur. J. Mech. A Solids
,
37
, pp.
200
215
.
20.
Jafari
,
A. A.
, and
Bagheri
,
M.
,
2006
, “
Free Vibration of Non-Uniformly Ring Stiffened Cylindrical Shells Using Analytical, Experimental and Numerical Method
,”
Thin-Walled Structures
,
44
(
1
), pp.
82
90
.
21.
Damatty
,
A. A. E.
,
Saafan
,
M. S.
, and
Sweedan
,
A. M. I.
,
2005
, “
Dynamic Characteristics of Combined Conical-Cylindrical Shells
,”
Thin-Walled Structures
,
43
(
9
), pp.
1380
1397
.
22.
Tavakoli
,
M. S.
, and
Singh
,
R.
,
1989
, “
Eigen Solution of Joined/Hermitic Shell Structures Using the State Space Method
,”
J. Sound Vib.
,
130
(1), pp.
97
123
.
23.
Sivadas
,
K. R.
, and
Ganesan
,
N.
,
1993
, “
Free Vibration Analysis of Combined and Stiffened Shells
,”
Comput. Struct.
,
46
(
3
), pp.
537
546
.
24.
Stanley
,
A. J.
, and
Fanesan
,
N.
,
1996
, “
Frequency Response of Shell-Plate Combinations
,”
Comput. Struct.
,
59
(6), pp.
1083
1094
.
25.
Benjedou
,
A.
,
2000
, “
Vibrations of Complex Shells of Revolution Using B-Spline Finite Elements
,”
Comput. Struct.
,
74
(
4
), pp.
429
440
.
26.
Lee
,
U.
,
2009
,
Spectral Element Method in Structural Dynamics
,
Wiley
,
New York
.
27.
Ahmida
,
K. M.
, and
Arruda
,
J. R. F.
,
2001
, “
Spectral Element-Based Prediction of Active Power Flow in Timoshenko Beams
,”
Int. J. Solids Struct.
,
38
(
10–13
), pp.
1669
1679
.
28.
Nanda
,
N.
, and
Kapuria
,
S.
,
2015
, “
Spectral Finite Element for Wave Propagation in Curved Beams
,”
ASME J. Vib. Acoust.
,
137
(
4
), p.
041005
.
29.
Beli
,
D.
,
Silva
,
P. B.
, and
Arruda
,
J. R. D. F.
,
2015
, “
Vibration Analysis of Flexible Rotating Rings Using a Spectral Element Formulation
,”
ASME J. Vib. Acoust.
,
137
(
4
), p.
041003
.
30.
Kudela
,
P.
,
Żak
,
A.
,
Krawczuk
,
M.
, and
Ostachowicz
,
W.
,
2007
, “
Modelling of Wave Propagation in Composite Plates Using the Time Domain Spectral Element Method
,”
J. Sound Vib.
,
302
(
4–5
), pp.
728
745
.
31.
Igawa
,
H.
,
Komatru
,
K.
,
Yamaguchi
,
I.
, and
Kasai
,
T.
,
2004
, “
Wave Propagation Analysis of Frame Structures Using the Spectral Element Method
,”
J. Sound Vib.
,
277
(
4–5
), pp.
1071
1081
.
32.
Wu
,
Z. J.
,
Li
,
F. M.
, and
Wang
,
Y. Z.
,
2013
, “
Vibration Band Gap Behaviors of Sandwich Panels With Corrugated Cores
,”
Comput. Struct.
,
129
, pp.
30
39
.
33.
Li
,
W. L.
,
2000
, “
Free Vibration of Beams With General Boundary Conditions
,”
J. Sound Vib.
,
237
(4), pp.
207
228
.
34.
Ma
,
X. L.
,
Jin
,
G. Y.
,
Shi
,
S. X.
,
Ye
,
T. G.
, and
Liu
,
Z. G.
,
2015
, “
An Analytical Method for Vibration Analysis of Cylindrical Shells Coupled With Annular Plate Under General Elastic Boundary and Coupling Conditions
,”
J. Vib. Control
, (epub).
35.
Zhang
,
X. F.
, and
Li
,
W. L.
,
2009
, “
Vibration Analysis of Rectangular Plates With Arbitrary Non-Uniform Elastic Edge Restraints
,”
J. Sound Vib.
,
326
(
1–2
), pp.
221
234
.
36.
Jin
,
G. Y.
,
Ma
,
X. L.
,
Shi
,
S. X.
,
Ye
,
T. G.
, and
Liu
,
Z. G.
,
2014
, “
A Modified Fourier Series Solution for Vibration Analysis of Truncated Conical Shells With General Boundary Conditions
,”
Appl. Acoust.
,
85
, pp.
82
96
.
37.
Jin
,
G. Y.
,
Ye
,
T. G.
,
Ma
,
X. L.
,
Chen
,
Y. H.
,
Su
,
Z.
, and
Xie
,
X.
,
2013
, “
A Unified Approach for the Vibration Analysis of Moderately Thick Composite Laminated Cylindrical Shells With Arbitrary Boundary Conditions
,”
Int. J. Mech. Sci.
,
75
, pp.
357
376
.
38.
Ye
,
T. G.
,
Jin
,
G. Y.
, and
Su
,
Z.
,
2016
, “
Three-Dimensional Vibration Analysis of Functionally Graded Sandwich Deep Open Spherical and Cylindrical Shells With General Restraints
,”
J. Vib. Control
,
22
(
15
), pp.
3326
3354
.
39.
Chen
,
Y. H.
,
Jin
,
G. Y.
,
Zhu
,
M. G.
,
Liu
,
Z. G.
, and
Li
,
W. L.
,
2012
, “
Vibration Behaviors of a Box-Type Structure Built Up By Plates and Energy Transmission Through the Structure
,”
J. Sound Vib.
,
331
(
4
), pp.
849
867
.
40.
Jin
,
G. Y.
,
Ye
,
T. G.
,
Jia
,
X.
, and
Gao
,
S. Y.
,
2014
, “
A General Fourier Solution for the Vibration Analysis of Composite Laminated Structure Elements of Revolution With General Elastic Restraints
,”
Compos. Struct.
,
109
, pp.
150
168
.
41.
Jin
,
G. Y.
,
Su
,
Z.
,
Shi
,
S. X.
,
Ye
,
T. G.
, and
Gao
,
S. Y.
,
2014
, “
Three-Dimensional Exact Solution for the Free Vibration of Arbitrarily Thick Functionally Graded Rectangular Plates With General Boundary Conditions
,”
Compos. Struct.
,
108
, pp.
565
577
.
42.
Jin
,
G. Y.
,
Ye
,
T. G.
,
Wang
,
X. R.
, and
Miao
,
X. H.
,
2016
, “
A Unified Solution for the Vibration Analysis of FGM Doubly-Curved Shells of Revolution With Arbitrary Boundary Conditions
,”
Compos. Part B
,
89
, pp.
230
252
.
43.
Su
,
Z.
,
Jin
,
G. Y.
,
Wang
,
X. R.
, and
Miao
,
X. H.
,
2015
, “
Modified Fourier–Ritz Approximation for the Free Vibration Analysis of Laminated Functionally Graded Plates With Elastic Restraints
,”
Int. J. Appl. Mech.
,
7
(
5
), p.
1550073
.
44.
Xu
,
H. A.
,
Li
,
W. L.
, and
Du
,
J.
,
2014
, “
Modal Analysis of General Plate Structures
,”
ASME J. Vib. Acoust.
,
136
(2), p.
0210021
.
45.
Li
,
W. Y.
,
Wang
,
G.
, and
Du
,
J.
,
2016
, “
Vibration Analysis of Conical Shells by the Improved Fourier Expansion-Based Differential Quadrature Method
,”
Shock Vib.
,
2016
, pp.
1
10
.
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