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ARTICLE IN PRESS
JOURNAL OFSOUND ANDVIBRATION
0022-460X/$ - s
doi:10.1016/j.js
�CorrespondE-mail addr
Journal of Sound and Vibration 314 (2008) 330–342
www.elsevier.com/locate/jsvi
A study on free vibration of a ring-stiffened thin circularcylindrical shell with arbitrary boundary conditions
Zhi Pan�, Xuebin Li, Janjun Ma
Wuhan 2nd Ship Design & Research Institute, Hubei Province, PR China
Received 1 July 2007; received in revised form 18 December 2007; accepted 4 January 2008
Handling Editor: L.G. Tham
Available online 4 March 2008
Abstract
The vibration of ring-stiffened cylinders associated with arbitrary boundary conditions is investigated. Displacements of
cylinders can be easily described by trigonometric functions when the cylinders are shear diaphragms supported at both
ends. As to other boundary conditions, exponential functions are used and axial factors are introduced. An eighth-order
algebraic equation for this axial factor is derived. The physical meaning of the axial factor is studied. Both analytical and
numerical studies prove that, when the axial factor is a pure imaginary number, the cylinder appears to have a certain
length with shear diaphragm boundary conditions. The effects of shell parameters and hydrostatic pressure on the axial
factor are determined in the analysis.
r 2008 Elsevier Ltd. All rights reserved.
1. Introduction
The use of cylindrical shells is common in many fields, such as aerospace, mechanical, civil and marineengineering structures. To effectively enhance the flexural and axial stiffness, stiffeners are frequently used.Many engineers have considerable interest in the free vibration of thin circular cylindrical shells and numerousinvestigations have been devoted to this, e.g. Ref. [1–6]. The methods used to study the vibrational behavior ofthin shells range from energy methods based on the Rayleigh–Ritz procedure to analytical methods in which,respectively, closed–form solutions of the governing equations and iterative solution approaches were used.The circular cylindrical shell supported at both ends by shear diaphragms (SD–SD) has received the mostattention in the literatures. This is due to the fact that one simple form of the solutions to the eighth-orderdifferential equations of motion is capable of satisfying the SD–SD boundary conditions exactly. Thefrequency equation is a polynomial of order six in the shell frequency parameter. When solving problems witharbitrary boundary conditions, many authors [7–11] select general solutions of shell equations. They havechosen exponential functions for the modal displacements along the axial direction, substituted them into theequations of motion and then enforced the eighth-order frequency determinants which are coupled together.
ee front matter r 2008 Elsevier Ltd. All rights reserved.
v.2008.01.008
ing author.
ess: pan_zhi@yahoo.com.cn (Z. Pan).
ARTICLE IN PRESS
Nomenclature
A2 sectional area of the stiffenerB stretching stiffness of the shell, B ¼ Eh/
1�m2
br width of the rectangular stiffenerD bending stiffness of the shell, D ¼ Eh3/
12(1�m2)d2 stiffener spacingE modulus of Young’s elasticitye2 eccentricity of the centroid of the ring
stiffener sectionG shear modulush thickness of the shellh h+A2/d2hr height of the rectangular stiffenerI0 sectional moment of inertia of the stiffen-
er about the centroid of the stiffenerI2 sectional moment of inertia of the stiffen-
er about the middle surface of the shell
iffiffiffiffiffiffiffi�1p
J2 torsion constant of stiffener cross-sectionk h2/12R2
L length of the shell between endsm axial mode numbern number of circumferential wavesQ hydrostatic pressureR radius of the mean surface of the shellt timex, y, z cylindrical coordinates (Fig. 1)Un, Vn, Wn amplitudes of the displacement in the
x, y and z directionsu, v, w components of the displacement in the x,
y and z directionsl axial factorm Poisson’s ratior mass densityo circular frequency
Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 331
Recently, many new approaches were developed for cylinder shell for free vibration analysis ofcylindrical and conical shells. Xiang et al. [12] use the state-space technique to derive the homogenousdifferential equation system for a shell segment and a domain decomposition approach is developed tocater to the continuity requirements between shell segments. Bingen Yang and Jianping Zhout [13] haveused the distributed transfer functions of the structural components, and then various static and dynamicproblems of stiffened shells have been systematically formulated. With this transfer function formulation,the static and dynamic response, natural frequencies and mode shapes and buckling loads of generalstiffened cylindrical shells under arbitrary external excitations and boundary conditions can bedetermined in exact and closed forms. Civalek Omer [14] proposes a discrete singular convolution methodfor the free vibration analysis of rotating conical shells. Frequency parameters of the forward modesare obtained for different types of boundary conditions, rotating velocity and geometric parameters.A wave propagation approach [15–18] is also developed to predict the natural frequency of thecylinders. This method combines an exact frequency wavenumber characteristic formula with appropriatebeam functions in the axial direction to give predictions of natural frequencies of circular cylindricalshells.
In the general solutions of shell equations, axial factors are usually introduced. But few literatureshave discussed the physical meanings of the axial factors. It has been numerically proved that the modesassociated with pure imaginary axial factors conform to the shear diaphragm end conditions of a finiteshell of length [19]. The analysis in this paper complements the author’s previous work [19] in the sensethat it investigates the physical meaning of the axial factors both numerically and analytically, and the effectsof the shells’ parameters and hydrostatic pressure are also studied. In this paper, a general solution usingexponential functions for the modal displacements along both the axial and the circumferential directions ispresented.
The investigation into the problem of the cylinder stiffened by ring stiffeners uses a ‘‘smeared frame’’approach, and represents the discretely reinforced cylinder by a uniformly orthotropic cylinder. When thestiffener spacing increases or becomes irregular or if the wavelength of the eigenmode becomes smaller thanthe stiffener spacing, this approach is no longer useful. The classical linear shell theory of Flugge [3] is adoptedin the present analysis.
ARTICLE IN PRESSZ. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342332
2. Exact solution of motion equations
The stiffeners are not considered as discrete members, but their effects are averaged or ‘‘smearedout’’ over the shell. Shell equations of motion for a thin-walled circular cylinder, stiffened by evenlyspaced uniform rings (Fig. 1) under hydrostatic pressure, are given below according to Flugge’s shelltheory:
qNx
qxþ
qNyx
Rqy�Q
q2u
Rqy2þ
qw
qx
� ��
QR
2
q2u
qx2¼ rh
q2u
qt2,
qNxy
qxþ
qNy
Rqy�
1
R
qMy
Rqyþ
qMxy
qx
� ��Q
q2v
Rqy2�
qw
Rqy
� ��
QR
2
q2v
qx2¼ rh
q2vqt2
,
q2Mx
qx2þ
q2Mxy
Rqxqyþ
q2Myx
Rqxqyþ
q2My
R2qy2þ
Ny
RþQ
qu
qx�
qv
Rqy�
q2w
Rqy2
� ��
QR
2
q2w
qx2
� �¼ rh
q2w
qt2. (1)
The forces and moments are given in the Appendix.The general solution for the equations can be written in the following form:
ðunðx; yÞ; vnðx; yÞ; wnðx; yÞÞ ¼ ðUn; V n; W nÞelx=Reinyeiot; (2)
where the axial factor l is a complex number.Substituting Eq. (2) into Eq. (1), it can be written as
½Ljk�fUn; V n; W ngT ¼ f0g; j; k ¼ 1; 2; 3. (3)
The elements of the coefficient matrix Ljk are given in the Appendix.The nondimensional circular frequency and hydrostatic pressure used in Eq. (3) are given as
O2 ¼r 1� m2� �
R2
Eo2, (4)
Q0 ¼QR 1� m2� �Eh
. (5)
For the unstiffened cylinder case, the coefficient matrix of Eq. (3) is symmetrical. The coefficient matrix thatcontains the influence of hydrostatic pressure is always antisymmetric.
For nontrivial solutions of Eq. (3), one requires
detðLjkÞ ¼ 0; j; k ¼ 1; 2; 3. (6)
The determinant can be expressed as
F ðo; n; l; Q0Þ ¼ 0. (7)
Assuming a known vibration frequency and a number of circumferential waves n, we can obtain an eighth-order algebraic equation for l:
P8l8þ P6l
6þ P4l
4þ P2l
2þ P0 ¼ 0. (8)
hr
br
Rh
e2
d2
z
x
z yx R
L
Fig. 1. Ring-stiffened circular cylindrical shell.
ARTICLE IN PRESSZ. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 333
The roots of Eq. (8) usually have the form [6]:
l ¼ �k1 � ik2; �k3;�ik4 (9)
where kj (j ¼ 1, 2, 3, 4) are real quantities.Analytical solutions for Eq. (8) can be written as
ðl1;2Þ2¼ G16 �
ffiffiffiffiffiffiL1
p
2�
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 �
L3
4ffiffiffiffiffiffiL1
p
s;
ðl3;4Þ2¼ G16 �
ffiffiffiffiffiffiL1
p
2þ
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 �
L3
4ffiffiffiffiffiffiL1
p
s,
ðl5;6Þ2¼ G16 þ
ffiffiffiffiffiffiL1
p
2�
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ
L3
4ffiffiffiffiffiffiL1
p
s;
ðl7;8Þ2¼ G16 þ
ffiffiffiffiffiffiL1
p
2þ
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ
L3
4ffiffiffiffiffiffiL1
p
s, (10)
where
G1 ¼ P24 � 3P2P6 þ 12P0P8,
G2 ¼ 2P34 � 9P2P4P6 þ 27P0P
26 þ 27P2
2P8 � 72P0P4P8,
G3 ¼ ð�4G31 þ G2
2Þ1=2; G4 ¼ G2 þ G3,
G5 ¼ G1=34 ; G6 ¼ 21=3G1,
G7 ¼ P26=4P2
8; G8 ¼ G5=3P821=3,
G9 ¼ G6=3G1=34 P8; G10 ¼ �8P2=P8,
G11 ¼ P36=P3
8; G12 ¼ 4P4P6=P28,
G13 ¼ � 2P4=3P8; G14 ¼ P26=2P2
8,
G15 ¼ � 1=3P821=3; G16 ¼ �P6=4P8,
G17 ¼ � 4P4=3P8; G18 ¼ �G6=3G1=34 P8,
L1 ¼ G13 þ G7 þ G8 þ G9; L2 ¼ G14 þ G17 þ G18 þ G15G5,
L3 ¼ G10 � G11 þ G12.
3. Numerical examples
An examples is studied in the present analysis. The assumptions for the basic parameters of shells andstiffeners are [20]: h ¼ 0.119� 10�2m, R ¼ 10.37� 10�2m, d2 ¼ 3.14� 10�2m, br ¼ 0.218� 10�2m, hr ¼
0.291� 10�2m, and m ¼ 0.3, E ¼ 2.06� 1011N/m2.The roots of Eq. (8) are computed for the following five cases:
(1)
an unstiffened shell with O variation for Q0 ¼ 0 and n ¼ 3. (2) an unstiffened shell with O variation for Q0 ¼ 0 and n ¼ 9. (3) an internal-ring stiffened shell with O variation for Q0 ¼ 0.008 and n ¼ 3. (4) an unstiffened shell with O variation for Q0 ¼ 0.008 and n ¼ 9. (5) an internal-ring stiffened shell with O variation for Q0 ¼ 0.008 and n ¼ 9.The results from Eq. (10) are shown in Figs. 2–6 for nondimensional frequencies. At any frequency, fourdistinctly different values of l are seen to exist, although only two may be seen on the figures. This is due to
ARTICLE IN PRESS
41,2
Ω2
-3
3
6
9
12
15
18
21
0-π/2
1,21
2
3
4
3
1,2,4
A
B
Am
plitu
deP
hase
1 2 3 4 5 6
Fig. 3. Roots of Eq. (8) for unstiffened cylinders (n ¼ 9, Q ¼ 0).
0.5
Am
plitu
de
4
31,2
-3
3
6
9
12
15
18
0-π/2
Pha
se
A
1,2
1
23
4
1,2,4
Ω2
1.0 1.5 2.0
Fig. 2. Roots of Eq. (8) for unstiffened cylinders (n ¼ 3, Q ¼ 0).
Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342334
curves 1, 2 and 3, 4 representing coincident complex conjugate pairs. Each curve has its negative counterpart,meaning that eight l exist altogether at any frequency. The phrases are �p/2 and 0 indicate that l is pureimaginary or real, respectively.
It can be seen in the figures that curves 1 and 2 always translate to two curves representing pure real roots atpoint A from coincident complex conjugate pairs. Curves 3 and 4 always represent pure imaginary roots andreal roots, respectively, over the frequency ranges calculated, except in Fig. 3. Fig. 3 shows that curves 3 and 4represent coincident complex conjugate pairs in the very low frequency range, and translate to curves 3 and 4at point B.
A further comparison made between Figs. 3 and 5 shows that hydrostatic pressure delays the appearance ofbifucation point A, and increases the amplitude of the pure imaginary roots markedly. Hydrostatic pressuredoes not effect curves 1, 2, 4 very much. Examination of Figs. 2 and 3 shows that point A appears much laterin high-order vibration than in low-order vibration. The stiffeners affect mainly the initial values of the pureimaginary roots on comparing Fig. 5 with 6.
ARTICLE IN PRESS
1,2 1,23
4
-3
3
6
9
12
15
18
0-π/2
A1,2
1
23
4
1,2,4
Am
plitu
deP
hase
0.5
Ω2
1.0 1.5 2.0
Fig. 4. Roots of Eq. (8) for internal-ring cylinders (n ¼ 3, Q0 ¼ 0.008).
Am
plitu
de
1,2,44
31,2
A
-3
3
6
9
12
15
18
21
0-π/2
Pha
se
1,2
1
2
3
4
Ω2
1 2 3 4 5 6
Fig. 5. Roots of Eq. (8) for unstiffened cylinders (n ¼ 9, Q0 ¼ 0.008).
Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 335
3.1. The physical meaning of pure imaginary roots
The above calculations show that Eq. (8) has the pure imaginary roots as 7ils for certain combinations of(o, n, Q0). Substituting l ¼7ils into Eq. (8), Eq. (8) can be rewritten as
P8l8s � P6l
6s þ P4l
4s � P2l
2s þ P0 ¼ 0. (11)
In order to investigate the physical meaning of the pure imaginary roots, the classical separation of variables,method for determining the natural frequencies of cylinders with shear diaphragm boundary conditions isdeduced here. A comparison will be made between two approaches. Suppose that the cylinders supported atboth ends by shear diaphragms (SD–SD) have a certain length L for classical analysis. The shear diaphragmboundary conditions can be closely approximated in physical application simply by means of rigidly attachinga thin, flat, circular cover plate at each end [6].
ARTICLE IN PRESS
4
1,2
A
-3
3
6
9
12
15
18
21
0-π/2
Pha
se
1,2
1
2
4
3
1,2,4
3
Am
plitu
de
Ω2
1 2 3 4 5 6
Fig. 6. Roots of Eq. (8) for internal-ring cylinders (n ¼ 9, Q0 ¼ 0.008).
Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342336
Consider the closed circular cylindrical shell of finite length L which satisfies the boundary conditions
v ¼ w ¼ Nx ¼Mx ¼ 0 at ends. (12)
Generally, deformation functions for the boundary conditions are given as follows:
umn x; yð Þ ¼ Umn cos nyð Þcos lclxð Þcosoclt,
vmn x; yð Þ ¼ V mn sin nyð Þsin lclxð Þcosoclt,
wmn x; yð Þ ¼W mn cos nyð Þsin lclxð Þcosoclt, (13)
where lcl ¼ mpR/L.
Substituting Eq. (13) into Eq. (1), a set of equations similar to Eq. (3) is obtained. For nontrivial solution ofequations, the determinant of the coefficient matrix must be zero. This leads to an equation with variablesðocl;m; n; lcl;Q0clÞ as follows:
F 2 ocl;m; n; lcl;Q0cl� �
¼ 0. (14)
Eq. (14) is a cubic equation with variable vibration frequency o2cl when lcl ¼ mpR/L is given. Each
combination of m and n has three roots of frequency. The minimum root among the roots for all combinationsof m and n is namely the basic frequency of the cylinder.
Eq. (14) can also be written as an eighth-order algebraic equation for lcl:
P08l8cl þ P06l
6cl þ P04l
4cl þ P02l
2cl þ P00 ¼ 0. (15)
A comparison was made between fP8;P6;P4;P2;P0g and P08;P06;P04;P02;P00
� �; it is interesting to note that:
P08 ¼ P8; P06 ¼ �P6; P04 ¼ P4; P02 ¼ P2; P00 ¼ P0. (16)
These findings appear to prove that Eqs. (11) and (15) are the same. This reasoning certainly seems to offer amathematical explanation for the conclusion that, when the axial factor is a pure imaginary number, thegeneral solution, Eq. (2), for the shell equations appears to describe the vibration mode of a cylinder with acertain length and shear diaphragm boundary conditions. The length of the cylinder can be obtained from
lsj j ¼mpR
L. (17)
After a pure imaginary root is obtained from Eq. (8), the length (L/R) of the cylinder with shear diaphragmssupported at ends is determined according to Eq. (17). Then the, corresponding natural frequency ocl fromEq. (14) is compared with the given frequency o, which is given before Eq. (8), and is resolved. Incidentally,
ARTICLE IN PRESSZ. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 337
if the cylinder is shear diaphragm supported, the length of the cylinder equals to m multiplies the half-wavelength in the axial direction; hence, only m ¼ 1 is discussed in the paper.
A numerical example is given for comparison in Table 1. Good agreement can be seen between the approachpresented and the classical method for cylinders with shear diaphragm boundary conditions on comparing thesecond column with the fifth column.
3.2. Effects of Stiffeners on ls
The effects of stiffeners on pure imaginary roots are studied here. The symmetrical-ring stiffened cylinder isintroduced here to make a comparison. The eccentricity of the centroid of ring stiffener section e2 equals tozero when the ring is symmetrical.
Fig. 7 shows that the values of |ls| increase along with an increase of frequency. The starting points of thepure imaginary roots in Fig. 7 are the bifurcation points of the complex conjugate pairs. In order to comparethe difference of the curves in Fig. 7 clearly, curves of differences based on internal-ring cylinder are shown inFig. 8. The difference is defined as
Diffll2;3s
� l1s
l1s
� 100%, (18)
Table 1
Comparison with classical method
Present approach (Eq. (7)) Classical method (Eq. (14))
O |ls| Lcl(� 10�2) Ocl
n ¼ 3
0.1 1.0362 31.44018 0.10000
0.5 3.2444 10.04137 0.50000
1.0 11.3558 2.86886 1.00000
1.5 18.6236 1.75129 1.49792
Unstiffened shell, Q0 ¼ 0, h ¼ 0.119� 10�2m, R ¼ 10.37� 10�2m, E ¼ 2.06� 1011N/m2, m ¼ 0.3.
0.20
2
4
6
8
10
hr = 0.582×10-2mn = 2
Ω
internal-ringexternal-ring symmetrical-ring
| λ s
|
0.4 0.6 0.8 1.0
Fig. 7. Effects of stiffeners’ locations (Q ¼ 0).
ARTICLE IN PRESS
-30
-20
-10
0
10
20external-ring symmetrical-ring
hr = 0.582×10-2mn = 2
Diff
λ
0.2Ω
0.4 0.6 0.8 1.0
Fig. 8. Curves of difference in stiffeners’ locations.
0
2
4
6
8
| λ s
|
hr= 0.291×10-2mhr= 0.582×10-2m
external-ringn=2
0.2Ω
0.4 0.6 0.8 1.0
Fig. 9. Effects of stiffeners’ parameters (Q ¼ 0).
Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342338
where l1s corresponds to the internal ring; l2;3s corresponds to the external ring and the symmetrical ring,respectively.
It can be seen in Fig. 8 that |ls| obtained from the external-ring cylinder are greater than thoseobtained from the internal-ring cylinder when Oo0.235, but smaller when O40.235. |ls| obtained from thesymmetrical-ring cylinder are always the smallest in the frequency ranges. This indicates that the naturalfrequency of the symmetrical-ring cylinder is always the highest in the three location forms.
The effects on |ls| of stiffeners’ parameters with no initial pressure(Q0 ¼ 0) are shown in Fig. 9. It can beseen that the stronger the ring, the smaller the |ls|, meaning that the increase of stiffeners’ stiffness heightensthe cylinder’s natural frequency. Fig. 10 shows the difference between |ls| obtained from the stiffeners of twodifferent sizes, and the difference is defined as Eq. (18). It can be seen that the difference fluctuates from largeto small along with a reduction in shell length.
ARTICLE IN PRESS
-20
-15
-10
-5
0
Diff
λ external-ringn=2
0.2Ω
0.4 0.6 0.8 1.0
Fig. 10. Curves of difference in stiffeners’ parameters.
0
5
10
15
20hr= 0.291×10-2mhr= 0.582×10-2m
external-ringn=2
| λ s
|
0.2Ω
0.4 0.6 0.8 1.0
Fig. 11. Effects of stiffeners’ parameters(Q0 ¼ 0.008).
Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 339
When the hydrostatic pressure exists(Q0 ¼ 0.008), the effects of stiffeners’ parameters are presented inFig. 11. The hydrostatic pressure increases the value of |ls|, and the increase of stiffeners’ stiffness increase thecylinder’s natural frequency as we found with no hydrostatic pressure. As multiple values appear withincreasing frequency, we cannot obtain the difference line between the two lines in the whole frequency range.Before multiple values appear, the hydrostatic pressure reduces the difference between the two lines in the low-frequency range, but heighten the difference in the high-frequency range.
3.3. Effects of initial stress on ls
In this paper, hydrostatic pressure is considered as the initial stress. Figs. 12 and 13 show theeffects of hydrostatic pressure on the pure imaginary roots. It can be seen that hydrostatic pressure reducesthe natural frequency of cylinders with a certain length. Hydrostatic pressure has much more effectson the vibration behavior when the vibration order is higher. Curves start at zero frequency, which
ARTICLE IN PRESS
0
2
4
6
8
10
| λ s
|
Q′= 0Q′=1.0×10-3
n=3internal-ring
0.2Ω
0.4 0.6 0.8 1.0
Fig. 12. Effects on high-order vibration of hydrostatic pressure (n ¼ 3) (internal-ring, br ¼ 0.218� 10�2m, hr ¼ 0.291� 10�2m).
0.0
2
4
6
8
10
| λ s
|
Ω
Q′= 0Q′=1.0×10-3
n=9internal-ring
0.2 0.4 0.6 0.8 1.0
Fig. 13. Effects on low-order vibration of hydrostatic pressure (n ¼ 9) (internal-ring, br ¼ 0.218� 10�2m, hr ¼ 0.291� 10�2m).
Z. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342340
indicates that the hydrostatic pressure is just the buckling pressure of the cylinder with shear diaphragmends.
4. Conclusion
When solving the vibration problem of ring-stiffened cylinders associated with arbitrary boundaryconditions, exponential functions are used and axial factors are introduced here. An eighth-order algebraicequation for this axial factor is derived. The physical meaning of the axial factor is studied. Both analyticaland numerical researches prove that when the axial factor is a pure imaginary number, the cylinder appears tohave a certain length with shear diaphragm boundary conditions. The effects of stiffeners’ parameters andhydrostatic pressure on the axial factor are studied.
Hydrostatic pressure increases the module of the pure imaginary roots markedly, but does not affect theother curves very much. The stiffeners affect mainly the initial value of the pure imaginary roots. The natural
ARTICLE IN PRESSZ. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342 341
frequency of the symmetrical-ring stiffened cylinder is always the highest in the three disposal forms.Increasing the stiffeners’ stiffness can increase the cylinder’s natural frequency.
Appendix A. Forces and moments
Eight internal forces including membrane forces Nx, Ny, Nxy and Nyx, bending moments Mx and My andtorsioning moments Mxy and Myx are given
Nx ¼ Bqu
qxþ m
qv
Rqy�
w
R
� � �þ
D
R
q2w
qx2,
Ny ¼ B 1þ m2� � qv
Rqy�
w
R
� �þ m
qu
qx� w2
w
Rþ
q2w
Rqy2
� � ��
D
R
q2w
R2qy2þ
w
R2
� �,
Nyx ¼1� m2
Bqu
Rqyþ
qv
qx
� �þð1� mÞD
2R
qu
R2qy�
q2wRqxqy
�,
Nxy ¼1� m2
Bqu
Rqyþ
qv
qx
� �þð1� mÞD
2R
q2wRqxqy
þqv
Rqx
�,
Mx ¼ D �q2w
qx2� m
q2w
R2qy2�
qu
Rqx� m
qv
R2qy
�,
My ¼ D � 1þ Z2� � w
R2þ
q2w
R2qy2
� �� m
q2wqx2þ z2
qv
R2qy�
w
R2
� � �,
Mxy ¼ 1� mð ÞD �q2w
Rqxqy�
qv
Rqx
� �,
Myx ¼ D 1� mð Þ �q2w
Rqxqyþ
1
2R
qu
Rqy�
qv
qx
� �� �� Zt2
q2wRqxqy
�.
Appendix B. The elements of the coefficient matrix
L11 ¼ l2 þð1� mÞ
2ð1þ kÞðinÞ2 �Q0 ðinÞ2 þ
l2
2
� �þ O2,
L12 ¼ð1� mÞ
2lin,
L13 ¼ � lmþ kl3 �ð1� mÞk
2lðinÞ2 �Q0l,
L21 ¼ð1� mÞ
2lin,
L22 ¼ ð1þ m2ÞðinÞ2� kz2ðinÞ
2þ
1� m2ð1þ 3kÞl2 �Q0 ðinÞ2 þ
l2
2
� �þ O2,
L23 ¼ � ð1þ m2Þinþð3� mÞ
2l2ink þ Z2 inþ ðinÞ3
� k � z2ðinÞ
3k þQ0ðinÞ,
L31 ¼ lm� l3k þð1� mÞ
2lðinÞ2k þQ0l,
L32 ¼ ð1þ m2Þin�ð3� mÞ
2l2ink þ z2ðinÞ
3k �Q0in,
L33 ¼ k �l4 � ð2þ Zt2Þl2ðinÞ2 � ð1þ Z2ÞðinÞ
4� ð2þ 2z2 þ Z2ÞðinÞ
2� ð1þ z2Þ
� ��ð1þ m2Þ �Q0 ðinÞ2 þ
l2
2
� �þ O2,
ARTICLE IN PRESSZ. Pan et al. / Journal of Sound and Vibration 314 (2008) 330–342342
where
m2 ¼EA2
Bd2; w2 ¼
EA2e2
Rd2B; Z2 ¼
EI2
Dd2; Zt2
¼GJ2
Dd2; z2 ¼
12R2w2h2
.
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