Nonlinear axisymmetric free vibration analysis of liquid-filled spherical shells with volume constraint condition using membrane theory is presented in this paper. The energy functional of the shell and contained liquid can be expressed based on the principle of virtual work using surface fundamental form and is written in the appropriate forms. Natural frequencies and the corresponding mode shapes for specified axisymmetric vibration amplitude of liquid-filled spherical shells can be calculated by finite element method (FEM). A nonlinear numerical solution can be obtained by the modified direct iteration technique. The results indicate that the Lagrange multiplier is a parameter for adapting the internal pressure in order to sustain the shell in equilibrium state for each mode of vibration with the volume constraint condition. The axisymmetric mode shapes of the liquid-filled spherical shells under volume constraint condition were found to be in close agreement with those in existing literature for an empty spherical shell. Finally, the effects of support condition, thickness, initial internal pressure, bulk modulus of internal liquid, and elastic modulus on the nonlinear axisymmetric free vibration and change of pressure of the liquid-filled spherical shells with volume constraint condition were demonstrated. The parametric studies showed that the change of pressure has a major impact on the fundamental vibration mode when compared with the higher vibration modes.

References

References
1.
Grossman
,
P. L.
,
Koplik
,
B.
, and
Yu
,
Y. Y.
,
1969
, “
Nonlinear Vibrations of Shallow Spherical Shells
,”
ASME J. Appl. Mech.
,
36
(
3
), pp.
451
458
.
2.
Hwang
,
Y. F.
,
1985
, “
A Numerical Approach for Vibration Analysis of an Axisymmetric Shell Structure With a Nonuniform Edge Constraint
,”
ASME J. Vib. Acoust.
,
107
(
2
), pp.
203
209
.
3.
Phadke
,
A.
, and
Cheung
,
K.
,
2003
, “
Nonlinear Response of Fluid-Filled Membrane in Gravity Waves
,”
ASCE J. Eng. Mech.
,
129
(
7
), pp.
739
750
.
4.
Wang
,
C. M.
,
Vo
,
K. K.
, and
Chai
,
Y. H.
,
2006
, “
Membrane Analysis and Minimum Weight Design of Submerged Spherical Domes
,”
ASCE J. Struct. Eng.
,
132
(
2
), pp.
253
259
.
5.
Duffey
,
T. A.
,
Pepin
,
J. E.
,
Robertson
,
A. N.
,
Steinzig
,
M. L.
, and
Coleman
,
K.
,
2007
, “
Vibrations of Complete Spherical Shells With Imperfections
,”
ASME J. Vib. Acoust.
,
129
(
3
), pp.
363
370
.
6.
Hu
,
J.
,
Qiu
,
Z.
, and
Su
,
T. C.
,
2010
, “
Axisymmetric Vibrations of a Piezoelectric Spherical Shell Submerged in a Compressible Viscous Fluid Medium
,”
ASME J. Vib. Acoust.
,
132
(
6
), p.
061002
.
7.
Jiammeepreecha
,
W.
,
Chucheepsakul
,
S.
, and
Huang
,
T.
,
2014
, “
Nonlinear Static Analysis of an Axisymmetric Shell Storage Container in Spherical Polar Coordinates With Constraint Volume
,”
Eng. Struct.
,
68
, pp.
111
120
.
8.
Jiammeepreecha
,
W.
,
Chucheepsakul
,
S.
, and
Huang
,
T.
,
2015
, “
Parametric Study of an Equatorially Anchored Deepwater Fluid-Filled Periodic Symmetric Shell With Constraint Volume
,”
ASCE J. Eng. Mech.
,
141
(
8
), p.
04015019
.
9.
Karamanos
,
S. A.
,
Patkas
,
L. A.
, and
Platyrrachos
,
M. A.
,
2005
, “
Sloshing Effects on the Seismic Design of Horizontal-Cylindrical and Spherical Industrial Vessels
,”
ASME J. Pressure Vessel Technol.
,
128
(
3
), pp.
328
340
.
10.
Utsumi
,
M.
, and
Tazuke
,
H.
,
2015
, “
Analysis of Vibration of a Tank Caused by an Explosion
,”
ASME J. Vib. Acoust.
,
137
(
5
), p.
051006
.
11.
Langhaar
,
H. L.
,
1964
,
Foundations of Practical Shell Analysis
, University of Illinois at Urbana-Champaign, Champaign, IL.
12.
Lamb
,
H.
,
1882
, “
On the Vibrations of a Spherical Shell
,”
Proc. London Math. Soc.
,
s1–14
(1), pp.
50
56
.
13.
Kalnins
,
A.
,
1964
, “
Effect of Bending on Vibrations of Spherical Shells
,”
J. Acoust. Soc. Am.
,
36
(
1
), pp.
74
81
.
14.
Niordson
,
F. I.
,
1984
, “
Free Vibrations of Thin Elastic Spherical Shells
,”
Int. J. Solids Struct.
,
20
(
7
), pp.
667
687
.
15.
Ross
,
E. W.
, Jr.
,
1965
, “
Natural Frequencies and Mode Shapes for Axisymmetric Vibration of Deep Spherical Shells
,”
ASME J. Appl. Mech.
,
32
(
3
), pp.
553
561
.
16.
Ross
,
E. W.
, Jr.
, and
Matthews
,
W. T.
,
1967
, “
Frequencies and Mode Shapes for Axisymmetric Vibration of Shells
,”
ASME J. Appl. Mech.
,
34
(
1
), pp.
73
80
.
17.
Kunieda
,
H.
,
1984
, “
Flexural Axisymmetric Free Vibrations of a Spherical Dome: Exact Results and Approximate Solutions
,”
J. Sound Vib.
,
92
(
1
), pp.
1
10
.
18.
Kim
,
J. G.
,
1998
, “
A Higher-Order Harmonic Element for Shells of Revolution Based on the Modified Mixed Formulation
,” Ph.D. thesis, Seoul National University, Seoul, South Korea.
19.
Singh
,
A. V.
, and
Mirza
,
S.
,
1985
, “
Asymmetric Modes and Associated Eigenvalues for Spherical Shells
,”
ASME J. Pressure Vessel Technol.
,
107
(
1
), pp.
77
82
.
20.
Ram
,
K. S. S.
, and
Babu
,
T. S.
,
2002
, “
Free Vibration of Composite Spherical Shell Cap With and Without a Cutout
,”
Comput. Struct.
,
80
(
23
), pp.
1749
1756
.
21.
Panda
,
S. K.
, and
Singh
,
B. N.
,
2009
, “
Nonlinear Free Vibration of Spherical Shell Panel Using Higher Order Shear Deformation Theory—A Finite Element Approach
,”
Int. J. Pressure Vessels Piping
,
86
(
6
), pp.
373
383
.
22.
Wang
,
X. H.
, and
Redekop
,
D.
,
2005
, “
Natural Frequencies and Mode Shapes of an Orthotropic Thin Shell of Revolution
,”
Thin-Walled Struct.
,
43
(
5
), pp.
735
750
.
23.
Artioli
,
E.
, and
Viola
,
E.
,
2006
, “
Free Vibration Analysis of Spherical SHELLS Using a G.D.Q. Numerical Solution
,”
ASME J. Pressure Vessel Technol.
,
128
(
3
), pp.
370
378
.
24.
Kang
,
J. H.
, and
Leissa
,
A. W.
,
2005
, “
Free Vibration Analysis of Complete Paraboloidal Shells of Revolution With Variable Thickness and Solid Paraboloids From a Three-Dimensional Theory
,”
Comput. Struct.
,
83
(
31–32
), pp.
2594
2608
.
25.
Lee
,
J.
,
2009
, “
Free Vibration Analysis of Spherical Caps by the Pseudospectral Method
,”
J. Mech. Sci. Technol.
,
23
(
1
), pp.
221
228
.
26.
Kang
,
S. W.
,
Atluri
,
S. N.
, and
Kim
,
S. H.
,
2012
, “
Application of the Nondimensional Dynamic Influence Function Method for Free Vibration Analysis of Arbitrarily Shaped Membranes
,”
ASME J. Vib. Acoust.
,
134
(
4
), p.
041008
.
27.
Menaa
,
M.
, and
Lakis
,
A. A.
,
2013
, “
Numerical Investigation of the Flutter of a Spherical Shell
,”
ASME J. Vib. Acoust.
,
136
(
2
), p.
021010
.
28.
Engin
,
A. E.
, and
Liu
,
Y. K.
,
1970
, “
Axisymmetric Response of a Fluid-Filled Spherical Shell in Free Vibrations
,”
J. Biomech.
,
3
(
1
), pp.
11
22
.
29.
Ventsel
,
E. S.
,
Naumenko
,
V.
,
Strelnikova
,
E.
, and
Yeseleva
,
E.
,
2010
, “
Free Vibrations of Shells of Revolution Filled With a Fluid
,”
Eng. Anal. Boundary Elem.
,
34
(
10
), pp.
856
862
.
30.
Rajasekaran
,
S.
, and
Murray
,
D. W.
,
1973
, “
Incremental Finite Element Matrices
,”
ASCE J. Struct. Div.
,
99
(
12
), pp.
2423
2438
http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0020396.
31.
Chen
,
J. S.
, and
Huang
,
T.
,
1985
, “
Appropriate Forms in Nonlinear Analysis
,”
ASCE J. Eng. Mech.
,
110
(
10
), pp.
1215
1226
.
32.
Langhaar
,
H. L.
,
1962
,
Energy Methods in Applied Mechanics
,
Wiley
,
New York
.
33.
Cook
,
R. D.
,
Malkus
,
D. S.
,
Plesha
,
M. E.
, and
Witt
,
R. J.
,
2002
,
Concepts and Applications of Finite Element Analysis
,
4th ed.
,
Wiley
,
New York
.
34.
Flügge
,
W.
,
1960
,
Stresses in Shells
,
Springer-Verlag
,
Berlin
.
35.
Prathap
,
G.
, and
Varadan
,
T. K.
,
1978
, “
The Large Amplitude Vibration of Hinged Beams
,”
Comput. Struct.
,
9
(
2
), pp.
219
222
.
36.
ABAQUS
,
2016
, “
Abaqus Analysis User’s Manual, Version 2016
,” Hibbitt, Karlsson and Sorensen, Pawtucket, RI.
You do not currently have access to this content.