A semi-analytic method is presented to analyze free and forced vibrations of combined conical–cylindrical–spherical shells with ring stiffeners and bulkheads. First, according to locations of discontinuity, the combined shell is divided into one opened spherical shell and a number of conical segments, cylindrical segments, stiffeners, and bulkheads. Meanwhile, a semi-analytic approach is proposed to analyze the opened spherical shell. The opened spherical shell is divided into narrow strips, which are approximately treated as conical shells. Then, Flügge theory is adopted to describe motions of conical and cylindrical segments, and stiffeners with rectangular cross section are modeled as annular plates. Displacement functions of conical segments, cylindrical segments, and annular plates are expanded as power series, wave functions, and Bessel functions, respectively. To analyze arbitrary boundary conditions, artificial springs are employed to restrain displacements at boundaries. Last, continuity and boundary conditions are synthesized to the final governing equation. In vibration characteristics analysis, high accuracy of the present method is first demonstrated by comparing results of the present method with ones in literature and calculated by ansys. Further, axial displacement of boundaries and open angle of spherical shell have significant influence on the first two modes, and forced vibrations are easily affected by bulkheads and external force.

References

References
1.
Benjeddou
,
A.
,
2000
, “
Vibration of Complex Shells of Revolution Using B-Spline Finite Elements
,”
Comput. Struct.
,
74
(
4
), pp.
429
440
.
2.
Lee
,
Y. S.
, and
Choi
,
M. H.
,
2001
, “
Free Vibrations of Circular Cylindrical Shells With an Interior Plate Using the Receptance Method
,”
J. Sound Vib.
,
248
(
3
), pp.
477
497
.
3.
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
.
4.
Hu
,
W. C. L.
, and
Raney
,
J. P.
,
1967
, “
Experimental and Analytical Study of Vibrations of Joined Shells
,”
AIAA J.
,
5
(
5
), pp.
976
980
.
5.
Galletly
,
G. D.
, and
Mistry
,
J.
,
1974
, “
The Free Vibrations of Cylindrical Shells With Various End Closures
,”
Nucl. Eng. Des.
,
30
(
2
), pp.
249
268
.
6.
Irie
,
T.
,
Yamada
,
G.
, and
Muramoto
,
Y.
,
1984
, “
Free Vibration of Joined Conical-Cylindrical Shells
,”
J. Sound Vib.
,
95
(
1
), pp.
31
39
.
7.
Tavakoli
,
M. S.
, and
Singh
,
R.
,
1989
, “
Eigensolutions of Joined/Hermetic Shell Structures Using the State Space Method
,”
J. Sound Vib.
,
130
(
1
), pp.
97
123
.
8.
Sivadas
,
K. R.
, and
Ganesan
,
N.
,
1993
, “
Free Vibration Analysis of Combined and Stiffened Shells
,”
Comput. Struct.
,
46
(
3
), pp.
537
546
.
9.
Patel
,
B. P.
,
Ganapathi
,
M.
, and
Kamat
,
S.
,
2000
, “
Free Vibration Characteristics of Laminated Composite Joined Conical-Cylindrical Shells
,”
J. Sound Vib.
,
237
(
5
), pp.
920
930
.
10.
El Damatty
,
A. A.
,
Saafan
,
M. S.
, and
Sweedan
,
A. M. I.
,
2005
, “
Dynamic Characteristics of Combined Conical-Cylindrical Shells
,”
Thin-Walled Struct.
,
43
(
9
), pp.
1380
1397
.
11.
Efraim
,
E.
, and
Eisenberger
,
M.
,
2006
, “
Exact Vibration Frequencies of Segmented Axisymmetric Shells
,”
Thin-Walled Struct.
,
44
(
3
), pp.
281
289
.
12.
Caresta
,
M.
,
Kessissoglou
,
N. J.
, and
Tso
,
Y.
,
2008
, “
Low Frequency Structural and Acoustic Responses of a Submarine Hull
,”
Acoust. Aust.
,
36
(
2
), pp.
47
52
.
13.
Caresta
,
M.
, and
Kessissoglou
,
N. J.
,
2010
, “
Free Vibrational Characteristics of Isotropic Coupled Cylindrical-Conical Shells
,”
J. Sound Vib.
,
329
(
6
), pp.
733
751
.
14.
Caresta
,
M.
, and
Kessissoglou
,
N. J.
,
2010
, “
Acoustic Signature of a Submarine Hull Under Harmonic Excitation
,”
Appl. Acoust.
,
71
(
1
), pp.
17
31
.
15.
Qu
,
Y. G.
,
Chen
,
Y.
,
Long
,
X. H.
,
Hua
,
H. X.
, and
Meng
,
G.
,
2013
, “
A Variational Method for Free Vibration Analysis of Joined Cylindrical-Conical Shells
,”
J. Vib. Control
,
19
(
16
), pp.
2319
2334
.
16.
Ma
,
X. L.
,
Jin
,
G. Y.
,
Xiong
,
Y. P.
, and
Liu
,
Z. G.
,
2014
, “
Free and Forced Vibration Analysis of Coupled Conical-Cylindrical Shells With Arbitrary Boundary Conditions
,”
Int. J. Mech. Sci.
,
88
, pp.
122
137
.
17.
Chen
,
M. X.
,
Xie
,
K.
,
Jia
,
W. C.
, and
Xu
,
K.
,
2015
, “
Free and Forced Vibration of Ring-Stiffened Conical-Cylindrical Shells With Arbitrary Boundary Conditions
,”
Ocean Eng.
,
108
, pp.
241
256
.
18.
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
.
19.
Pan
,
X.
,
Tso
,
Y.
, and
Juniper
,
R.
,
2008
, “
Active Control of Radiated Pressure of a Submarine Hull
,”
J. Sound Vib.
,
311
(
1–2
), pp.
224
242
.
20.
Qu
,
Y. G.
,
Wu
,
S. H.
,
Chen
,
Y.
, and
Hua
,
H. X.
,
2013
, “
Vibration Analysis of Ring-Stiffened Conical-Cylindrical-Spherical Shells Based on a Modified Vibrational Approach
,”
Int. J. Mech. Sci.
,
69
, pp.
72
84
.
21.
Kalnins
,
A.
, and
Naghdi
,
P. M.
,
1960
, “
Axisymmetric Vibrations of Shallow Elastic Spherical Shells
,”
J. Acoust. Soc. Am.
,
32
(
3
), pp.
342
347
.
22.
Kalnins
,
A.
,
1961
, “
On Vibration of Shallow Spherical Shells
,”
J. Acoust. Soc. Am.
,
33
(
8
), pp.
1102
1107
.
23.
Naghdi
,
P. M.
, and
Kalnins
,
A.
,
1962
, “
On Vibration of Elastic Spherical Shells
,”
ASME J. Appl. Mech.
,
29
(
1
), pp.
65
72
.
24.
Kalnins
,
A.
,
1964
, “
Effect of Bending on Vibrations of Spherical Shells
,”
J. Acoust. Soc. Am.
,
36
(
1
), pp.
74
81
.
25.
Prasad
,
C.
,
1964
, “
On Vibrations of Spherical Shells
,”
J. Acoust. Soc. Am.
,
36
(
3
), pp.
489
494
.
26.
Wilkinson
,
J. P.
, and
Kalnins
,
A.
,
1965
, “
On Nonsymmetric Dynamic Problems of Elastic Shells
,”
ASME J. Appl. Mech.
,
32
(
3
), pp.
525
532
.
27.
Ramakrishnan
,
C. V.
, and
Shah
,
A. H.
,
1970
, “
Vibration of an Aelotropic Spherical Shell
,”
J. Acoust. Soc. Am.
,
47
(
5
), pp.
1366
1374
.
28.
Niordson
,
F. I.
,
1984
, “
Free Vibrations of Thin Elastic Spherical Shells
,”
Int. J. Solids Struct.
,
20
(
7
), pp.
667
687
.
29.
Niordson
,
F. I.
,
1988
, “
The Spectrum of Free Vibrations of a Thin Elastic Spherical Shell
,”
Int. J. Solids Struct.
,
24
(
9
), pp.
947
961
.
30.
Leissa
,
A. W.
,
1993
,
Vibration of Shells
,
Acoustical Society of America
,
New York
.
31.
Qatu
,
M. S.
,
2002
, “
Recent Research Advances in the Dynamic Behavior of Shells: 1989–2000—Part 1: Laminated Composite Shells
,”
ASME Appl. Mech. Rev.
,
55
(
4
), pp.
325
349
.
32.
Qatu
,
M. S.
,
2002
, “
Recent Research Advances in the Dynamic Behavior of Shells: 1989–2000—Part 2: Homogeneous Shells
,”
ASME Appl. Mech. Rev.
,
55
(
4
), pp.
415
434
.
33.
Artioli
,
E.
, and
Viola
,
E.
,
2006
, “
Free Vibration Analysis of Spherical Caps Using a G.D.Q. Numerical Solution
,”
ASME J. Pressure Vessel Technol.
,
128
(
3
), pp.
370
378
.
34.
Tong
,
L. Y.
,
1993
, “
Free Vibration of Orthotropic Conical Shells
,”
Int. J. Eng. Sci.
,
31
(
5
), pp.
719
733
.
35.
Hoppmann
,
W. H.
, II
,
1958
, “
Some Characteristics of the Flexural Vibrations of Orthogonally Stiffened Cylindrical Shells
,”
J. Acoust. Soc. Am.
,
30
(
1
), pp.
77
82
.
36.
Thein
,
W.
, and
Hu
,
W. C. L.
,
1968
, “
Vibration Analysis of Stiffened Cylinders Including Inter-Ring Motion
,”
J. Acoust. Soc. Am.
,
43
(
5
), pp.
1005
1016
.
37.
Basdekas
,
N. L.
, and
Chi
,
M.
,
1971
, “
Response of Oddly Stiffened Circular Cylindrical Shell
,”
J. Sound Vib.
,
17
(
2
), pp.
187
206
.
38.
Jafari
,
A. A.
, and
Bagheri
,
M.
,
2006
, “
Free Vibration of Non-Uniformly Ring Stiffened Cylindrical Shells Using Analytical, Experimental and Numerical Methods
,”
Thin-Walled Struct.
,
44
(
1
), pp.
82
90
.
39.
Chen
,
M. X.
,
Wei
,
J. H.
,
Xie
,
K.
,
Deng
,
N. Q.
, and
Hou
,
G. X.
,
2013
, “
Wave Based Method for Free Vibration Analysis of Ring Stiffened Cylindrical Shells With Intermediate Large Frame Ribs
,”
Shock Vib.
,
20
(
3
), pp.
459
479
.
40.
Wei
,
J. H.
,
Chen
,
M. X.
,
Hou
,
G. X.
,
Xie
,
K.
, and
Deng
,
N. Q.
,
2013
, “
Wave Based Method for Free Vibration Analysis of Ring Stiffened Cylindrical Shells With Non-Uniform Stiffener Distribution
,”
ASME J. Vib. Acoust.
,
135
(
6
), p.
061011
.
41.
Xie
,
K.
,
Chen
,
M. X.
,
Deng
,
N. Q.
, and
Jia
,
W. C.
,
2015
, “
Free and Forced Vibration of Submerged Ring-Stiffened Conical Shells With Arbitrary Boundary Conditions
,”
Thin-Walled Struct.
,
96
, pp.
240
255
.
42.
Chen
,
M. X.
,
Xie
,
K.
,
Xu
,
K.
, and
Yu
,
P.
,
2015
, “
Wave Based Method for Free and Forced Vibration Analysis of Cylindrical Shells With Discontinuity in Thickness
,”
ASME J. Vib. Acoust.
,
137
(
5
), p.
051004
.
43.
Tso
,
Y. K.
, and
Hansen
,
C. H.
,
1995
, “
Wave Propagation Through Cylinder/Plate Junctions
,”
J. Sound Vib.
,
186
(
3
), pp.
447
461
.
You do not currently have access to this content.