In this paper, a finite strip for vibration analysis of rotating toroidal shells subjected to internal pressure is developed. The expressions for strain and kinetic energies are formulated in a previous paper in which vibrations of a toroidal shell with a closed cross section are analyzed using the Rayleigh–Ritz method (RRM) and Fourier series. In this paper, however, the variation of displacements u, v, and w with the meridional coordinate is modeled through a discretization with a number of finite strips. The variation of the displacements with the circumferential coordinate is taken into account exactly by using simple sine and cosine functions of the circumferential coordinate. A unique argument nφ+ωt is used in order to be able to capture traveling modes due to the shell rotation. The finite strip properties, i.e., the stiffness matrix, the geometric stiffness matrix, and the mass matrices, are defined by employing bar and beam shape functions, and by minimizing the strain and kinetic energies. In order to improve the convergence of the results, also a strip of a higher-order is developed. The application of the finite strip method is illustrated in cases of toroidal shells with closed and open cross sections. The obtained results are compared with those determined by the RRM and the finite element method (FEM).

References

1.
Love
,
A. E. H.
,
1944
,
Treatise on the Mathematic Theory of Elasticity
, 4th ed.,
Dover Publications, Mineola, NY
.
2.
Vlasov
,
V. Z.
,
1949
,
The General Theory of Shells and Its Industrial Applications
,
Gostekhizdat
,
Moscow, Russia
(in Russian).
3.
Goldenveizer
,
A. L.
,
1961
,
Theory of Thin Shells
,
Pergamon Press
,
Elmsford, NY
.
4.
Novozhilov
,
V. V.
,
1964
,
Thin Shell Theory
,
P. Noordhoff
,
Groningen, The Netherlands
.
5.
Soedel
,
W.
,
2004
,
Vibrations of Shells and Plates
, 3rd ed.,
Marcel Dekker
,
New York
.
6.
Alujević
,
N.
,
Campillo-Davo
,
N.
,
Kindt
,
P.
,
Desmet
,
W.
,
Pluymers
,
B.
, and
Vercammen
,
S.
,
2017
, “
Analytical Solution for Free Vibrations of Rotating Cylindrical Shells Having Free Boundary Conditions
,”
Eng. Struct.
,
132
, pp.
152
171
.
7.
Sun
,
S.
,
Chu
,
S.
, and
Cao
,
D.
,
2012
, “
Vibration Characteristics of Thin Rotating Cylindrical Shells With Various Boundary Conditions
,”
J. Sound Vib.
,
331
(
4
), pp.
170
186
.
8.
Sun
,
S.
,
Cao
,
D.
, and
Han
,
Q.
,
2013
, “
Vibration Studies of Rotating Cylindrical Shells With Arbitrary Edges Using Characteristic Orthogonal Polynomials in the Rayleigh-Ritz Method
,”
Int. J. Mech. Sci.
,
68
, pp.
180
189
.
9.
Kim
,
Y. J.
, and
Bolton
,
J. S.
,
2004
, “
Effects of Rotation on the Dynamics of a Circular Cylindrical Shell With Application to Tire Vibration
,”
J. Sound Vib.
,
275
(
3–5
), pp.
605
621
.
10.
Lecomte
,
C.
,
Graham
,
W. R.
, and
Dale
,
M.
,
2010
, “
A Shell Model for Tyre Belt Vibrations
,”
J. Sound Vib.
,
329
(
10
), pp.
1717
1742
.
11.
Senjanović
,
I.
,
1972
,
Theory of Shells of Revolution
,
Ship Research Institute
,
Zagreb, Croatia
.
12.
Burcher
,
R.
, and
Rydill
,
L.
,
1998
,
Concepts in Submarine Design
,
Cambridge University Press
,
Cambridge, UK
.
13.
Ross
,
C. T. F.
,
2011
,
Pressure Vessels, External Pressure Technology
,
Woodhead Publishing
,
Oxford, UK
.
14.
Senjanović
,
I.
,
Rudan
,
S.
,
Tomić
,
M.
, and
Vladimir
,
N.
,
2008
, “
Some Structural Aspects of LPG Cargo Tank Design and Construction
,”
International Conference on Design and Operation of LPG Ships
, London, Jan. 30–31, pp.
79
96
.
15.
Harte
,
R.
, and
Eckstein
,
U.
,
1986
, “
Derivation of Geometrically Nonlinear Finite Shell Elements Via Tensor Notation
,”
Int. J. Numer. Methods Eng.
,
23
(
3
), pp.
367
384
.
16.
Eckstein
,
U.
,
1983
, “
Nichtlineare Stabilitätsberechnung elastischer Schalentragwerke
,” TWM 83, Ruhr-Universität Bochum, Bochum, Germany.
17.
Bathe
,
K. J.
,
1996
,
Finite Element Procedures
,
Prentice Hall
,
Englewood Cliffs, NJ
.
18.
Gavrić
,
L.
,
1994
, “
Finite Element Computation of Dispersion Properties of Thin-Walled Waveguides
,”
J. Sound Vib.
,
173
(
1
), pp.
113
124
.
19.
Finnveden
,
S.
, and
Fraggstedt
,
M.
,
2008
, “
Waveguide Finite Elements for Curved Structures
,”
J. Sound Vib.
,
312
(
4–5
), pp.
644
671
.
20.
Waki
,
Y.
,
Mace
,
B. R.
, and
Brennan
,
M. J.
,
2009
, “
Free and Forced Vibrations of a Tyre Using a Wave/Finite Element Approach
,”
J. Sound Vib.
,
323
(
3–5
), pp.
737
756
.
21.
Sabiniarz
,
P.
, and
Kropp
,
W.
,
2010
, “
A Waveguide Finite Element Aided Analysis of the Wave Field on a Stationary Tyre, Not in Contact With the Ground
,”
J. Sound Vib.
,
329
(
15
), pp.
3041
3064
.
22.
Hoever
,
C.
,
Tsotras
,
A.
,
Saemann
,
E. U.
, and
Kropp
,
W.
,
2013
, “
A Comparison Between Finite Element and Waveguide Finite Element Methods for the Simulation of Tyre/Road Interaction
,”
Inter Noise Conference
, Innsbruck, Austria, Sept. 15–18, pp.
1
10
.
23.
Senjanović
,
I.
,
Áatipović
,
I.
,
Alujević
,
N.
,
Vladimir
,
N.
, and
Čakmak
,
D.
,
2018
, “
A Finite Strip for the Vibration Analysis of Rotating Cylindrical Shells
,”
Thin-Walled Struct.
,
122
, pp.
158
172
.
24.
Cheung
,
Y. K.
,
1976
,
Finite Strip Method in Structural Analysis
,
Pergamon Press
,
Oxford, UK
.
25.
Gould
,
P. L.
,
1985
,
Finite Element Analysis of Shells of Revolution
,
Pitman Advanced Publishing Program
,
Boston, MA
.
26.
Senjanović
,
I.
,
Alujević
,
N.
,
Áatipović
,
I.
,
Čakmak
,
D.
, and
Vladimir
,
N.
,
2018
, “
Vibration Analysis of Rotating Toroidal Shell by Rayleigh-Ritz Method and Fourier Series
,”
Eng. Struct.
,
173
, pp.
870
891
.
27.
Szilard
,
R.
,
2004
,
Theories and Applications of Plate Analysis, Classical, Numerical and Engineering Methods
,
Wiley
,
Hoboken, NJ
.
28.
Belytschko
,
T.
,
Liu
,
W. K.
, and
Moran
,
B.
,
2000
,
Nonlinear Finite Elements for Continua and Structures
,
Wiley
,
New York
.
29.
Dassault Systèmes
,
2009
, “
ABAQUS 6.9 User's Guide and Theoretical Manual
,”
Hibbitt, Karlsson & Sorensen, Inc., Providence, RI
.
30.
Kim
,
W.
, and
Chung
,
J.
,
2002
, “
Free Non-Linear Vibrations of a Rotating Thin Ring With the In-Plane and Out-of-Plane Motions
,”
J. Sound Vib.
,
258
(
1
), pp.
167
178
.
31.
Tisseur
,
F.
, and
Meerbergen
,
K.
,
2001
, “
The Quadratic Eigenvalue Problem
,”
SIAM Rev.
,
43
(
2
), pp.
235
286
.
You do not currently have access to this content.