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Jianhui Wei

Meixia Chen

1

e-mail: chemx26@hust.edu.cn

Guoxiang Hou

Kun Xie

Naiqi Deng

School of Naval Architecture

and Ocean Engineering,

Huazhong University of

Science and Technology,

Wuhan 430074, China

Wave Based Method for Free

Vibration Analysis of Cylindrical

Shells With Nonuniform

Stiffener Distribution

Wave based method (WBM) is presented to analysis the free vibration characteristics of

cylindrical shells with nonuniform stiffener distributions for arbitrary boundary condi-

tions. The stiffeners are treated as discrete elements. The equations of motion of annular

circular plate are used to describe the motion of stiffeners. Instead of expanding the

dynamic ﬁeld variables in terms of polynomial approximation in element based method

(ﬁnite element method etc), the ring-stiffened cylindrical shell is divided into several sub-

structures and the dynamic ﬁeld variables in each substructure are expressed as wave

function expansions. Boundary conditions and continuity conditions between adjacent

substructures are used to form the ﬁnal matrix to be solved. Natural frequencies of cylin-

drical shells with uniform rings spacing and eccentricity distributions for shear

diaphragm-shear diaphragm boundary conditions have been calculated by WBM model

which shows good agreement with the experimental results and the analytical results of

other researchers. Natural frequencies of cylindrical shells with other boundary condi-

tions have also been calculated and the results are compared with the ﬁnite element

method which also shows good agreement. Effects of the nonuniform rings spacing and

nonuniform eccentricity and effects of boundary conditions on the fundamental frequen-

cies and the beam mode frequencies have been studied. Different stiffener distributions

are needed to increase the fundamental frequencies and beam mode frequencies for

different boundary conditions. WBM model presented in this paper can be recognized as

a semianalytical and seminumerical method which is quite useful in analyzing the vibra-

tion characteristics of cylindrical shells with nonuniform rings spacing and eccentricity

distributions. [DOI: 10.1115/1.4024055]

Keywords: wave based method, cylindrical shell with nonuniform stiffener distribution,

free vibration analysis

1 Introduction

Ring stiffened cylindrical shells are widely used in many struc-

ture applications such as aeronautic, aerospace, underwater vessel

and so on. The free vibration characteristics of cylindrical shells

are quite important for use of on-board equipments and so on.

Most research focus on cylindrical shells with uniform stiffener

distribution because of the convenience of the establishment of

mathematical model of the problem, especially convenient for

dealing with stiffeners. But in practical engineering applications,

stiffeners are sometimes unequally spaced and often with different

sizes, such as the existence of intermediate large frame ribs, so a

more general model is needed to deal with cylindrical shells with

nonuniform stiffener distribution, such as nonuniform eccentric-

ity, unequally spaced and different sizes for ring stiffeners.

Many investigations have been developed to analyze the vibra-

tion characteristics of cylindrical shells, e.g., Refs. [1–10]. Energy

methods based on the Rayleigh–Ritz procedure and analytical

methods are most widely used. Most often cylindrical shells with

shear diaphragm-shear diaphragm boundary condition have been

investigated because the solution to the governing equations

which satisﬁes boundary conditions can be easily expressed by

trigonometric functions. Exponential functions have been chosen

for the modal displacements along the axial direction, which

are substituted into the equations of motions and then the eight

speciﬁed boundary conditions are enforced to calculate the natural

frequency of cylindrical shells with arbitrary boundary conditions.

In order to effectively enhance the ﬂexural and axial stiffness

of cylindrical shells, stiffeners are frequently used. The even

spaced rings or stringers are either “smeared out” over the surface

of the cylindrical shells or treated as discrete members [11–28].

With rings or stringers “smeared out,” the cylindrical shells are

treated as orthotropic cylindrical shells [14–18]. The stiffeners

must be equally and closely distributed and also with the same

depth and width and large errors will be introduced if the stiff-

eners are large. And also this method is not applicable at mid and

high frequencies if the wavelength and the rib spacing are in the

same order for neglecting wave transmission and reﬂection at

the discontinuities. Analytical method for the determination of the

vibration characteristics of even spaced ring-stiffened shells with

shear diaphragm–shear diaphragm (SD-SD) was presented in

Refs. [19,20], where rings or stringers are treated as discrete num-

bers and equations of motions of beams are used to describe the

motion of stiffeners. Assuming that the stiffeners uniformly

spaced, the displacement at the position of stiffeners are expanded

as trigonometric functions which satisfy simply supported bound-

ary conditions to calculate the natural frequencies in Ref. [19].

The method is conﬁned to solve the problem of even spaced

ring-stiffened cylindrical shells and simply supported boundary

condition. C.H. Hodges et al. [21,22] studied the vibration charac-

teristics of periodic ring-stiffened cylindrical shells with circular

T-section ribs according to Bloch’s or Floquet’s theorem [23].

Recently, Wave propagation method is given in Refs. [24–27]to

1

Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the

JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 11, 2012; ﬁnal

manuscript received February 25, 2013; published online June 19, 2013. Assoc.

Editor: Marco Amabili.

Journal of Vibration and Acoustics DECEMBER 2013, Vol. 135 / 061011-1Copyright V

C2013 by ASME

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study the vibration characteristics of cylindrical shell, with the

effects of ring stiffeners “smeared out” over the surface of the

shell in Refs. [25,27]. Beam function with shear diaphragm-shear

diaphragm (SD-SD), clamped-clamped (C-C) and clamped-shear

diaphragm (C-SD) boundary condition is used to approximate the

axial mode function of ring-stiffened cylindrical shell. With the

stiffeners treated as discrete elements, Ritz method is applied to

analyze the free vibration characteristics of simply supported cy-

lindrical shells with nonuniform rings spacing and eccentricity in

Ref. [28] and some conclusions have been made.

A wave based method (WBM) was ﬁrst proposed in Ref. [29]

for prediction of the steady-state dynamics analysis of coupled

vibro-acoustic systems which enables accurate predictions in the

midfrequency range. Wave based method can be understood as a

semianalytical and seminumerical method which is computation-

ally less demanding than corresponding element based models. In

contrast with the ﬁnite element method (FEM), in which the

dynamic ﬁeld variables within each element are expanded in

terms of local, nonexact shape functions, usually polynomial

approximation, the dynamic ﬁeld variables in WBM are expressed

as wave function expansions. Modeling of the vibro-acoustic

coupling between the pressure ﬁeld in an acoustic cavity with

arbitrary shape and the out-of-plane displacement of a ﬂat plate

with arbitrary shape was discussed in Ref. [30] and the unbounded

domain was discussed in Ref. [31]. The application of the wave

based method for the particular case where stress singularities

appear in one or more corners of a polygonal plate domain is dis-

cussed in Ref. [32].

In this paper, WBM method is extended to analyze free vibra-

tion characteristics of cylindrical shells with nonuniform rings

spacing and eccentricity distributions for arbitrary boundary

conditions. The cylindrical shell is divided into several substruc-

tures the motions of which are described by the equations of

Donnell–Mushtari theory and the stiffeners are treated as sepa-

rate substructures the motions of which are described by the

equations of annular circular plates. The displacements within

each substructure of cylindrical shells are expanded as a set of

wave functions obtained in Ref. [19], and those within the stiff-

ener substructure are expanded as a set of wave functions in Ref.

[33]. Then boundary conditions and continuity conditions

between substructures are used to form the ﬁnal matrix to calcu-

late the natural frequencies. Compared with traditional wave

propagation method in analyzing free vibration characteristics of

ring stiffened cylindrical shells, WBM method has a great

advantage in dealing with stiffener of arbitrary sizes and arbi-

trary distributions for arbitrary boundary conditions. Compared

with ﬁnite element method possessing a signiﬁcantly greater

generality, a better convergence can be obtained by WBM

method and also the size of the ﬁnal matrix formed to calculate

the natural frequencies is much smaller than that formed by ﬁ-

nite element method. As a semianalytical and seminumerical

method, the shape function chosen in WBM method is more

convenient to explain the mechanism of wave propagation

including vibration transmission and reﬂection. The ﬁrst part of

this paper reviews the work already done about free vibration of

cylindrical shells. WBM model of cylindrical shells with nonuni-

form rings spacing and eccentricity distribution is described in

the second part. Numerical results are discussed in the third part

and some conclusions are made in the fourth part.

2 Basic Concept of Wave Based Model

Wave based method (WBM) is developed for steady-state

dynamic analysis of cylindrical shells with nonuniform rings

spacing and stiffener distribution. In contrast with the ﬁnite

element method (FEM), in which the dynamic ﬁeld variables

within each element are expanded in terms of local, nonexact

shape functions, usually polynomial approximation, the dynamic

ﬁeld variables within each substructure in WBM model are

expressed as wave function expansions, which exactly satisfy

the governing dynamic equations of the substructure, and then

boundary conditions and continuity conditions are utilized to

form the ﬁnal matrix to calculate the natural frequencies. After

that, wave function contribution factors used for postprocessing

can be determined.

Similar to the ﬁnite element method, several steps are needed to

construct a WBM model as follows:

(1) Divide the whole model into different substructures. Differ-

ent governing equations are used for different types of

structure, such as beam, plate and shell. Also different

physical properties and the positions of the discontinuities

where coupling effects occur need to be considered.

(2) Select suitable wave functions for each substructure. The

dynamic ﬁeld variables, such as the displacement, the

velocity and so on, can be expanded by the wave functions

which can exactly satisfy the governing equations of the

substructures.

(3) Form the ﬁnal matrix to be solved according to boundary

conditions and continuity conditions. Eight speciﬁed

boundary conditions and eight continuity conditions

between different substructures, including forces and dis-

placements, are used to form the ﬁnal matrix.

(4) Solve the ﬁnal matrix to get the natural frequencies and the

wave function contribution factors. The size of the matrix

depends on the total number of substructures and the num-

ber of wave function contribution factors in each substruc-

ture. Compared with traditional element based model, more

unknowns exist in the shape functions but the number of

substructures is far more less than the number of elements

so the ﬁnal size of matrix formed is much smaller than

FEM.

(5) Postprocessing. The dynamic ﬁeld variables can be

obtained with the wave function contribution factors, such

as the displacement, stress, strain and so on.

2.1 Substructure Division of Ring-Stiffened Cylindrical

Shells. Cylindrical shells and annular circular plates are main

types of structures composing the ring-stiffened cylindrical shells

and the stiffeners appear as a discontinuity dividing the cylindrical

shell into different substructures. As a ring-stiffened cylindrical

shell shown in Fig. 1, the total number of stiffeners is N, and the

cylindrical shell is divided into (Nþ1) bays. The cylindrical shell

and the stiffeners are made of the same material, so the total num-

ber of substructures is (2Nþ1) with Nsubstructures describing

the motions of stiffeners and (Nþ1) substructures describing the

motions of cylindrical shells. Compared with element based

method, this method of substructure division has much less

elements.

According to Donnell–Mushtari theory, the following equations

describe the motion of thin cylindrical shells [1,2]:

Fig. 1 Ring stiffened cylindrical shells

061011-2 / Vol. 135, DECEMBER 2013 Transactions of the ASME

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L11uþL12 vþL13w¼0

L21uþL22 vþL23w¼0

L31uþL32 vþL33w¼0

9

>

>

=

>

>

;

(1)

Differential operators Lij i;j¼1;2;3ðÞare given in Appendix A.

u, v, w are the axial, circumferential and outward normal

displacements.

The stiffeners are treated as discrete members the motions of

which are described by the equations of motions of annular circular

plates. Internal stiffening type is considered here which can be

applied to external stiffening type easily. The vibrations of annular

circular plates with inner radius a

1

and outer radius a(also the ra-

dius of the cylindrical shell) with bending and in-plane motions are

described in Eq. (2). The axial displacement w

p

, radial displace-

ment u

p

and circumferential displacement v

p

of the plate in polar

coordinates are shown in Fig. 2.hp¼@wp=@ris the twist angle.

@

@r

@up

@rþup

rþ1

r

@vp

@h

1p

2r

@

@h

@vp

@rþvp

r1

r

@up

@h

¼

qp12

p

Ep

@2up

@t2

1

r

@

@h

@up

@rþup

rþ1

r

@vp

@h

þ1p

2

@

@r

@vp

@rþvp

r1

r

@up

@h

¼

qp12

p

Ep

@2vp

@t2

r4wpqpx2hp

Dp

wp¼0

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

(2)

h

p

is the plate thickness. E

p

,r

p

and pare respectively the Young’s

modulus, density and Poisson’s ratio of the annular circular plate.

ris the radial coordinate. xis the angular frequency. Dp¼Eph3

p=

12ð12

pÞis the ﬂexural rigidity.

2.2 Selection of Wave Functions. For a modal vibration, the

axial, circumferential and outward normal displacements of the

cylindrical shells and the annular circular plates are usually

expressed as a solution expansion

w¼X

ns

i¼1

AiWwiðx;/Þsinðn/Þejxt

v¼X

ns

i¼1

BiWviðx;/Þcosðn/Þejxt

u¼X

ns

i¼1

CiWuiðx;/Þsinðn/Þejxt

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

(3)

Wwiðx;/Þ;Wvi ðx;/Þ;Wuiðx;/Þare the structure wave functions

which satisfy Eqs. (1) and (2) for a speciﬁed circumferential wave

number n,A

i

,B

i

,C

i

are the wave contribution factors. n

s

is the

number of wave functions designating the wave propagation in

the axial direction for the cylindrical shell and the radial direction

for the annular circular plate.

To insure the WBM model to converge towards the exact solu-

tion, the selection of suitable wave function is quite important.

According to analysis in Ref. [19], a set of wave functions

describing the free vibration of cylindrical shell is selected as

follows:

Ww1¼Wv1¼Wu1¼ek1x;Ww2¼Wv2¼Wu2¼ek1x

Ww3¼Wv3¼Wu3¼cosk2x;Ww4¼Wv4¼Wu4¼sin k2x

Ww5¼Wv5¼Wu5ek3xcosk4x;Ww6¼Wv6¼Wu6¼ek3xsin k4x

Ww7¼Wv7¼Wu7¼ek3xcosk4x;Ww8¼Wv8¼Wu8¼ek3xsin k4x

(4)

The wave contribution factors are as follows:

B1¼n1A1;B2¼n1A2;B3¼n2A3;B4¼n2A4

B5¼n3A5þn4A6;B6¼n4A5þn3A6

B7¼n3A7n4A8;B8¼n4A7þn3A8

(5)

C1¼g1A1;C2¼g1A2;C3¼g2A3;C4¼g2A4

C5¼g3A5þg4A6;C6¼g4A5þg3A6

C7¼g3A7þg4A8;C8¼g4A7g3A8

(6)

In Eqs. (4)–(6),k1;6ik1;6ðk36ik4Þare character roots to be

determined which are given in Appendix B.n

1

n

4

and g

1

g

4

are constant coefﬁcients related to character roots which are also

given in Appendix B.

According to Ref. [33], a set of wave functions describing the

free vibration of annular circular plate are selected as follows:

wwp1¼JnkpBr

;wwp2¼YnkpBr

;

wwp3¼InkpBr

;wwp4¼KnkpBr

wvp1¼nJnkpLr

=r;wvp2¼dJnkpTr

=dr;

wvp3¼nYnkpLr

=r;wvp4¼dYnkpTr

=dr

wup1¼dJnkpLr

=dr;wup2¼nJnkpT r

=r;

wup3¼dYnkpLr

=dr;wup4¼nYnkpT r

=r

(7)

The wave contribution factors are as follows:

A1¼B1¼B1n;A2¼B2¼B2n;A3¼B3¼B3n;A4¼B4¼B4n

C1¼A1n;C2¼A2n;C3¼A3n;C4¼A4n(8)

Fig. 2 Displacement of annular circular plate

Journal of Vibration and Acoustics DECEMBER 2013, Vol. 135 / 061011-3

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n

s

¼8 is the number of wave functions. J

n

,Y

n

are, respectively,

Bessel functions of the ﬁrst and the second kind. I

n

,K

n

are,

respectively, modiﬁed Bessel functions of the ﬁrst and second

kind. kpB ¼ðqpx2hp=DpÞ1=4is the plate bending wave number,

kpL ¼x½qpð1t2

pÞ=Ep1=2;kpT ¼x½2qpð1þtpÞ=Ep1=2are the

wave numbers for in-plane waves in the circular plate. The coefﬁ-

cients Ai;n;Bi;ni¼1;2;3;4ðÞare determined from the boundary

conditions of the annular circular plates.

2.3 Matrix Formed for Calculating Natural Frequencies.

As shown in Fig. 3, the cylindrical shells with arbitrary boundary

conditions have four displacement constraints (u,v,w,h) and four

force or moment constraints (M,S,T,N), i.e.,

u¼0orM¼0

v¼0orS¼0

w¼0orT¼0

h¼0orN¼0

9

>

>

>

=

>

>

>

;

(9)

where hdesignate the twisting angle and M,S,T,Ndesignate

bending moment, transverse shear, tangential shear and axial force

per unit length of the cylindrical shell and the expressions for M,

S,T,Nare expressed by Eq. (C6) in Appendix C. Combination of

these eight boundary conditions can present arbitrary boundary

conditions.

The cylindrical shell with inside stiffeners is divided into two

bays by the stiffeners as shown in Fig. 4where continuity equa-

tions must be satisﬁed.

The displacements of the connection of the adjacent cylindrical

shells must be equal, which can lead to the following relationship:

uL

k¼uR

k;vL

k¼vR

k;wL

k¼wR

k;hL

k¼hR

k(10)

At the outer radius of the annular circular plate which is shown

in Fig. 4, the continuity conditions of displacements and forces

can be expressed as follows:

up;kr¼a¼wR

k;vp;kr¼a¼vR

k;wp;kr¼a¼uR

k;hp;kr¼a¼hR

k

(11)

NL

k¼Npx;kþNR

k

SL

k¼Npr;kþSR

k

TL

k¼TR

kNph;k

ML

k¼MR

kMp;k

8

>

>

>

>

<

>

>

>

>

:

(12)

The annular circular plates have one free edge at the inner

radius (outer radius for the external stiffening type) where bound-

ary conditions are applied as follows:

Npxr¼a1¼0;Npr r¼a1¼0;Nphr¼a1¼0;Mpr¼a1¼0

(13)

uL

k;vL

k;wL

k;hL

kand uR

k;vR

k;wR

k;hR

kdenote the axial,

circumferential, radial displacement and the twist angle of the left

bay and the right bay of the cylindrical shell of the kth stiffener

respectively. up;kr¼a;vp;kr¼a;wp;kr¼a;hp;kr¼adenote the

radial, circumferential, axial displacement and the twist angle of

the kth stiffener at the outer radius of the circular plate.

ML

k;SL

k;TL

k;NL

kand MR

k;SR

k;TR

k;NR

kdesignate the

bending moment, transverse shear, tangential shear and axial force

per unit length of the right and the left bay of the cylindrical shell,

respectively. ML

k;Npr;k;TL

k;Npx;kand Mpr¼a1;Npr r¼a1;

Nphr¼a1;Npx r¼a1designate the bending moment, transverse

shear, tangential shear and axial force per unit length of the kth

annular circular plate at the outer and inner radius of the cylindri-

cal shell, respectively.

Combing Eqs. (11) and (13), the eight wave function contribu-

tors of the annular circular plate can be expressed by the eight

wave function contributors of the adjacent cylindrical shells. The

cylindrical shell is divided into (Nþ1) subsystems which means

that there are totally 8(Nþ1) wave contribution factors to be

solved. The boundary conditions at both ends of the cylindrical

shell and the continuity conditions between the stiffeners and ad-

jacent bays of cylindrical shells can be written and assembled in

matrix form for each circumferential mode number n. Omitting

the subindex n, the assembled matrix is

½KfAg¼0(14)

where {A} is the 8(Nþ1) unknown coefﬁcient vectors and

½K¼

½B1ð0Þ

½D1ðb1Þ ½D2ð0Þ

½F1ðb1Þ ½F2ð0Þ

½D2ðb2Þ ½D3ð0Þ

½F2ðb2Þ ½F3ð0Þ

½DNðbNÞ ½DNþ1ð0Þ

½FNðbNÞ ½FNþ1ð0Þ

½BNþ1ðbNþ1Þ

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

(15)

For the kth substructure of the cylindrical shell, the 4 8 matrix

blocks [D

k

] and [F

k

] designating displacement and force continu-

ity conditions used in Eq. (15) are as follows. For the continuity

equations between stiffeners and adjacent bays of cylindrical

shells, blocks [D

k

] and blocks [F

k

] are given by

½DkðxÞ48¼

wk;1ðxÞwk;8ðxÞ

vk;1ðxÞvk;8ðxÞ

uk;1ðxÞuk;8ðxÞ

hk;1ðxÞhk;8ðxÞ

2

6

6

6

4

3

7

7

7

5

;k¼1;2;…;Nþ1(16)

Fig. 4 Interaction forces between stiffener and adjacent bays

of cylindrical shells

Fig. 3 Displacement constraints and force constraints

061011-4 / Vol. 135, DECEMBER 2013 Transactions of the ASME

½FkðxÞ48¼

Sk;1ðxÞSk;8ðxÞ

Tk;1ðxÞTk;8ðxÞ

Nk;1ðxÞNk;8ðxÞ

Mk;1ðxÞMk;8ðxÞ

2

6

6

6

4

3

7

7

7

5

;k¼1;2;…;Nþ1(17)

The initial and ﬁnal blocks [B

1

] and [B

Nþ1

] are expressed in terms

of displacement and/or forces, depending on the boundary condi-

tions at each end of the cylindrical shell. Combination of eight

boundary conditions in Eq. (9) can present arbitrary boundary

conditions. Free, clamped and shear-diaphragm boundary condi-

tions are considered here and are given by

Free end (F)

½BkðxÞ48¼

Sk;1ðxÞSk;8ðxÞ

Tk;1ðxÞTk;8ðxÞ

Nk;1ðxÞNk;8ðxÞ

Mk;1ðxÞMk;8ðxÞ

2

6

6

6

4

3

7

7

7

5

;k¼1;Nþ1(18)

Clamped end (C)

½BkðxÞ48¼

wk;1ðxÞwk;8ðxÞ

vk;1ðxÞvk;8ðxÞ

uk;1ðxÞuk;8ðxÞ

hk;1ðxÞhk;8ðxÞ

2

6

6

6

4

3

7

7

7

5

;k¼1;Nþ1(19)

Shear-diaphragm (SD)

½BkðxÞ48¼

wk;1ðxÞwk;8ðxÞ

vk;1ðxÞvk;8ðxÞ

Nk;1ðxÞNk;8ðxÞ

Mk;1ðxÞMk;8ðxÞ

2

6

6

6

4

3

7

7

7

5

;k¼1;Nþ1(20)

B1

½;BNþ1½;Dk

½;Fk

½ðk¼1Nþ1Þare given in Appendix C.

Uk;1ðU¼w;v;u;h;S;T;N;MÞdesignate the values of ﬁeld

variables at the position xof the kth bay of cylindrical shell. b

k

designates the length of the kth bay.

2.4 Solution of the Matrix and Postprocessing. When ana-

lyzing the free vibration characteristic of ring-stiffened cylindrical

shell, the value of the determinant of [K] is calculated for a

sequence of trial values of frequency until a sign change is met,

then the zero of the determinant is located by iterative interpola-

tion. After a natural frequency has been accurately calculated, the

solution to the homogeneous equations (Eq. (14)), normalized by

taking ANþ1;8¼1, can be calculated and the mode shape can be

obtained.

3 Numerical Results and Discussion

3.1 Cylindrical Shells With Uniform Ring Spacing and

Eccentricity. The WBM model is ﬁrst utilized to calculate the

natural frequencies of two ring-stiffened cylindrical shells with

C-C, C-SD, SD-SD, C-F, SD-F, F-F boundary conditions. The

geometry and material properties of the two shells denoted as

model M1 and model M2 which are given in Table 1. Both of the

two shells are with external stiffeners of rectangular cross section.

Three kinds of shells with stiffeners of different depths are calcu-

lated in model M1. The cylindrical shells and the stiffeners have

the same materials in both model M1 and model M2. The WBM

results are compared with the analytical results already done in

relevant references and also FEM results.

The ﬁnite element package ANSYS is used to calculate natural

frequencies of model M1 and model M2. Both the cylindrical

shells and the stiffeners are modeled with shell63 element. In

order to ensure the convergence of results calculated by FEM, M1

model with d

R

¼0.00291 is modeled with three different meshes

as shown in Table 2and the natural frequencies of vibrations

modes m¼1, n¼16(m,ndenote the axial half wave number

and the circumferential wave number respectively) with SD-SD

boundary conditions have been calculated with the results shown

in Table 3. The two numbers in Table 2for cylindrical shell

Table 1 Geometry dimensions and material properties of

stiffened cylindrical shells

Geometry and material properties

Characteristics M1 Model M2 Model

Radius, a(m) 0.1037 0.049759

Length, l(m) 0.4709 0.3945

Thickness, h(m) 0.00119 0.001651

Density, q(kg/m

3

) 7850 2762

Young’s modulus, E(Pa) 2.06 10

11

6.89510

10

Poission’s ratio, 0.3 0.3

Ring width, b

R

(m) 0.00218 0.003175

Ring depth, d

R

(m) 0.00291,0.00582,0.00873 0.005334

Number of rings, N14 19

Stiffening type External External

Table 2 FEM models with different meshes

Number of elements

Mesh 1 Mesh 2 Mesh 3

Cylindrical shell 100 90 160 120 200 150

Stiffeners 100 114 160 114 200 114

Total number 10400 21440 32800

Table 3 Natural frequencies of M1 model calculated by three

different meshes

Natural frequencies (Hz)

Mode number Mesh 1 Mesh 2 Mesh 3

n¼1 1610 1610.3 1610

n¼2 690.87 690.85 690.84

n¼3 555.15 555.04 555.04

n¼4 875.46 874.94 874.91

n¼5 1372 1371 1370

n¼6 1971 1967.5 1967

Table 4 Comparison of circular frequencies of M1 model

(rad/s). (a) Basdekas and Chi [20]; (b) Galletly [14]; (c) Wah and

Hu [19]; (d) Bosor (smeared rings) [12] and (e) Gan and Li [27].

d

R

(m) n(a)(b)(c)(d)(e) FEM WBM

0.00291 2 4550 4470 4314 4420 4409 4341 4370

3 3870 3655 3173 3680 3674 3487 3560

4 6550 5950 4565 6000 6000 5497 5590

5 10000 9510 7058 9620 9604 8614 8715

0.00582 2 4580 4450 4235 4481 4357 4386

3 6710 6235 4615 6492 5717 5782

4 12830 11790 7982 12149 10154 10248

5 20120 19020 12660 19694 15639 15767

0.00871 2 5040 4885 4378 4954 4659 4689

3 10330 9500 6651 9873 8193 8269.8

4 20200 18010 12154 18884 14608 14735

5 31800 25570 19221 30606 21803 21992

Journal of Vibration and Acoustics DECEMBER 2013, Vol. 135 / 061011-5

denote the number of elements in the circumferential direction

and the axial direction respectively and the three numbers for

stiffeners denote the number of elements in the circumferential

direction, the radial direction and the number of stiffeners in the

axial direction, respectively. 10,500, 21,600 and 33,000 nodes are

used in mesh 1, mesh 2 and mesh 3, respectively, with each node

having six degrees of freedom UX, UY, UZ, ROTX, ROTY and

ROTZ designating three translation degrees and three rotational

degrees of freedom. Mesh 2 which can achieve both high compu-

tation efﬁciency and adequate converged results is used in the

following analysis.

Table 4shows the comparison of natural frequencies of model

M1 between WBM results and the analytical results of Basdekas

and Chi [20], Galletly [14], Wah and Hu [19], Bosor [12] and L.

Gan and X. B. Li [27] and also with FEM results. From the table

we can see that good agreement can be achieved between different

methods for most cases while for other cases large differences

have been introduced. The reason why large differences exist is

mainly due to some simpliﬁcations dealing with the stiffeners in

different methods. In method (a), a modiﬁed variation method is

adopted and the motions of the stiffeners are described by the

equations of motions of beams. In method (b), the stiffeners are

treated as discrete members but the inter-ring displacement is

neglected. In method (c), the stiffeners are also treated as discrete

members but the eccentricity of the stiffeners with respect to the

shell midsurface is neglected. In method (d), the stiffeners are

treated by “smeared method” which can only present an accurate

solution when the wavelength is small and the stiffeners are with

small sizes. In method (e), the stiffeners are also treated by

“smeared method” while beam function is used to approximate

the axial mode function of ring stiffened cylindrical shells.

Table 5shows the comparison of natural frequencies of model

M2 between WBM results and the experimental results and

the analytical results of Hoppmann [11], the analytical results of

Mustafa and Ali [13] and the analytical results of Jafari and

Bagheri [28] for various modes of vibrations with SD-SD bound-

ary conditions. As we can see from the tables, when the wave

number is small, quite good agreement can be achieved between

WBM methods and other methods. The differences increase with

the increment of the wave number, while in all the cases WBM

results and the experimental results agree quite well. The reason is

that the wave length decreases with the increment of the wave-

number and either the stiffener is treated with “smeared method”

or treated as discrete members with the motions of stiffener

described by the equations of motions of beams can both intro-

duced errors while WBM model is much more exact in such

cases.

Table 6shows comparison between WBM results of M1 model

for d

R

¼0.00291m with C-C, C-SD, SD-SD, C-F, SD-F, F-F

boundary conditions. Good agreements have been achieved which

shows that WBM model has adequate accuracy to determine the

natural frequencies of ring stiffened cylindrical shells with

arbitrary boundary conditions.

3.2 Cylindrical Shells With Nonuniform Ring Spacing and

Eccentricity. Numerical calculations have been performed here

to study the effects of nonuniform rings spacing and nonuniform

stiffeners eccentricity distribution, separately and simultaneously.

Table 5 Comparison of natural frequencies of M2 model (Hz).

(e) Gan and Li [27]; (f) Jafari and Bagheri [28]; (g) Mustafa and

Ali [13]; (h) Hoppmann [11]; and (i) Hoppmann [11].

nm (e)(f)(g)(h) Experiment (i) WBM

1 1 1216 1199.58 1204 1154.67

2 3536 3493.59 3498 3325.82

3 5907 5839.89 5844 5533.53

4 7538.00

5 9249.70

2 1 1635 1564.47 1587 1530 1530 1585.07

2 2176 2113.84 2129 2100 2040 2112.62

3 3430 3378.17 3386 3330 3200 3298.39

4 4860 4440 4712.78

5 6480 6200 6121.56

3 1 4578 4387.59 4462 4230 4080 4156.73

2 4573 4400.58 4437 4320 4090 4174.51

3 4788 4595.79 4627 4500 4520 4407.83

4 5040 5000 4937.64

5 5760 5700 5707.83

4 1 8781 8377.75 8559 8100 7424.87

2 8731 8392.63 8482 8100 7403.14

3 8728 8449.89 8438 8190 7520 7435.08

4 8280 7800 7580.76

5 7920 7873.03

5 1 14172 13490.7 13780 13050 11059.28

2 14119 13508.9 13695 13100

3 14069 13555.4 13595 13140

4 13230 11400 11087.96

5 11207.63

Table 6 Comparison of natural frequencies of M1 model

C-C(Hz) SD-SD(Hz) C-SD(Hz)

Mode FEM WBM Error FEM WBM Error FEM WBM Error

m¼1n¼1 2005.8 2010.3 0.22% 1610.3 1614.1 0.24% 1786.2 1790.2 0.22%

n¼2 1105.5 1109.8 0.39% 690.9 695.5 0.67% 900.8 905.2 0.49%

n¼3 784.7 793.7 1.15% 555.0 566.5 2.07% 658.9 669.0 1.53%

n¼4 950.6 964.5 1.46% 874.9 889.6 1.68% 904.3 918.7 1.59%

n¼5 1396.0 1412.1 1.15% 1371.0 1387.1 1.17% 1379.7 1396.1 1.19%

n¼6 1977.9 1995.7 0.90% 1967.5 1985.3 0.90% 1971.1 1989.0 0.91%

F-F(Hz) SD-F(Hz) C-F(Hz)

Mode FEM WBM Error FEM WBM Error FEM WBM Error

m¼1n¼1 3155.0 3122.2 1.04% 2249.1 2254.2 0.23% 2282.0 2287.4 0.24%

n¼2 1483.0 1488.6 0.38% 1044.0 1048.8 0.46% 1221.8 1226.5 0.38%

n¼3 882.7 892.7 1.13% 682.7 693.5 1.58% 818.7 820.2 0.18%

n¼4 925.5 941.1 1.69% 888.8 903.8 1.69% 935.6 950.3 1.57%

n¼5 1374.0 1390.2 1.18% 1371.8 1388.3 1.20% 1383.0 1400.0 1.23%

n¼6 1968.7 1986.5 0.90% 1968.0 1985.9 0.91% 1972.4 1990.2 0.90%

061011-6 / Vol. 135, DECEMBER 2013 Transactions of the ASME

M1 model with d

R

¼0.00291 m with uniform stiffener distribution

is considered here, whose properties are shown in Table 1. Some

cases of nonuniform rings spacing and nonuniform stiffener

distribution are shown in Fig. 5, for which cases the total stiffener

mass maintain constant with that of M1 model with evenly rings

spacing and equally depth for all stiffeners.

Equating the mass of stiffeners in uniform and nonuniform

distributions, a second-order equation corresponding to d

Rk

’ can

be obtained as follows [28]:

1

N

1

aþdR1þh=2

2

X

N

k¼1

d02

Rk þ2

N

1

aþdR1þh=2

X

N

k¼1

d0

Rk

2ðaþh=2Þðd0dR1Þ

ðaþdR1þh=2Þ2ðd2

0d2

R1Þ

ðaþdR1þh=2Þ2¼0(21)

Here the difference d0

Rbetween the depths of adjacent stiffeners is

considered to maintain constant and the nonuniform eccentricity

is symmetrically distributed about the midsection of the shell.

Select a value of d

R1

and a distribution function of d

Rk

,d0

Rk can be

determined by solving Eq. (21). The depth of each stiffener can be

obtained as follows:

d0

Rk ¼ðk1Þd0

RkN=2

ðNkÞd0

Rk>N=2

(22)

dRk ¼dR1þd0

Rk (23)

Similarly, the nonuniform spacing is considered to be symmet-

rical distributed about the midsection of the shell and the differen-

ces between the lengths of adjacent bays of cylindrical shell

maintain constant, the nonuniform distribution can be obtained by

selecting a value of d

1

. The spacing of the kth bay of cylindrical

shell can be obtained as follows:

d0¼LðNþ1Þd1

N2=4(24)

dk¼d1þðk1Þd0kN=2

d1þðNþ1kÞd0k>N=2

(25)

In the above equations, d

1

denotes the ﬁrst ring spacing at the

left end and d

R1

denotes the depth of the ﬁrst stiffener at the left

end. d0denotes the differences between the lengths of adjacent

bays of cylindrical shells and d0

Rdenotes the differences between

the depths of adjacent stiffeners. d0

Rk denotes the differences

between the depths of the kth stiffener and the ﬁrst stiffener. d

Rk

and d

k

denote the depth of the kth stiffener and the length of kth

bay of cylindrical shell, respectively. d

0

denotes the depth of the

stiffeners of uniform eccentricity distribution.

For d

1

<0.00314, the rings are closely spaced at the two

ends of the cylindrical shell and on the other hand for

d

1

>0.00314, the rings are compressed in the midsection of the

cylindrical shell. For d

R1

>0.00291, the eccentricities of the

stiffeners at the two ends are larger than those in the midsection

and for d

R1

<0.00291, the eccentricities of the stiffeners at the

two ends are smaller than those in the midsection. For

d

1

¼0.00314 and d

R1

¼0.00291, it denotes uniformly rings spac-

ing and eccentricity distribution. In the following calculation,

d1¼0:00114 0:00514 and dR1 ¼0:00091 0:00491.

Figures 6(a)–6(c)shows the variations of natural frequencies

ðm¼1;n¼16Þof cylindrical shells with respect to the depth

(d

R1

) of the ﬁrst stiffener with SD-SD, F-F, SD-F boundary

conditions. Here the rings spacing is uniform and only the effect

of nonuniform eccentricity is considered. It should be noted that

decrement of the depth of the ﬁrst stiffener increase the mass and

stiffness in the midsection of the cylindrical shell and decrease

them at the two ends.

For the three kinds of boundary conditions, the fundamental

frequencies all occur at mode m¼1, n¼3 except when

d

R1

¼0.00491 for SD-F boundary condition for which case the

fundament frequency mode changes from m¼1, n¼3tom¼1,

n¼4. Increment of the depth of the ﬁrst stiffener lowers the fun-

damental frequencies for SD-SD boundary conditions and for

other two boundary conditions the fundament frequencies ﬁrst

decrease and then increase with the increment of the depth of the

ﬁrst stiffener and the lowest fundamental frequencies both occur

when d

1

¼0.00314 and d

R1

¼0.00291, which denotes uniformly

rings spacing and eccentricity distribution.

From Fig. 6, we can see that for SD-SD and SD-F boundary

conditions, increment of the depth of the ﬁrst stiffener leads to

increment of natural frequencies of the beam mode (m¼1, n¼1)

while for F-F boundary conditions it leads to the decrement of nat-

ural frequencies of the beam mode. Also the natural frequencies

of beam mode with F-F boundary conditions are the highest

among the three kinds of boundary conditions and those with SD-

SD boundary conditions are the lowest.

Figures 7(a)–7(c)shows the variations of fundamental frequen-

cies with respect to the depth of the ﬁrst stiffener for given d

1

with SD-SD, F-F, SD-F boundary conditions. For SD-SD bound-

ary condition, increment of the depth of the ﬁrst stiffener

lower the fundamental frequencies and increment of the ﬁrst ring

spacing increase the fundamental frequencies from which we can

make a conclusion that the cylindrical shells get the highest funda-

mental frequency when d

R1

¼0.00091 and d

1

¼0.0514, in another

word, more mass concentrated in the midsection of the cylindrical

shells, the higher the fundamental frequencies are. For the other

two kinds of boundary conditions, the fundamental frequencies

ﬁrst increase and then decrease with respect to the increment of

the depth of the ﬁrst stiffener but the fundamental frequencies not

always get the lowest value when rings spacing and eccentricity

are uniformly distributed which depends on the value of d

1

.

Natural frequency curve crossing is observed which shows that

the cylindrical shells get the lowest fundamental frequencies

when d

1

¼0.0114 before the intersection point and get the lowest

fundamental frequencies when d

1

¼0.0514 after the intersection

point.

Figures 8(a)–8(c)shows the variations of beam mode frequen-

cies with respect to the depth of the ﬁrst stiffener for given d

1

with SD-SD, F-F, SD-F boundary conditions. We can see from

Fig. 5 Uniform and nonuniform rings spacing and eccentricity distribution

Journal of Vibration and Acoustics DECEMBER 2013, Vol. 135 / 061011-7

Fig. 7that for SD-SD boundary conditions the beam mode fre-

quencies increase with the increment of depth of the ﬁrst stiffener

and decrease with the increment of the ﬁrst ring spacing from

which we can make a conclusion that the cylindrical shells get

the highest beam mode frequency when d

R1

¼0.00491 and

d

1

¼0.0114 which means that the less the mass distributed in the

Fig. 6 Natural frequency variations versus the depth of the

ﬁrst stiffener of equally rings spacing (d

1

50.00314) and nonun-

iformly eccentricity distribution with SD-SD, SD-F, F-F boundary

conditions: (a) SD-SD (b) F-F (c) SD-F —n—n51——n52—

~—n53—!—n54—3—n55—"—n56

Fig. 7 Fundamental frequency variations versus the depth

of the ﬁrst stiffener for given d

1

with SD-SD, F-F, SD-F boundary

conditions: (a)SD-SD(b)F-F(c)SD-F—n—d

1

50.0114 ——

d

1

50.0164 —~—d

1

50.0214 —!—d

1

50.0264 —3—d

1

50.0314

—"—d

1

50.0364 —^—d

1

50.0414 —$—d

1

50.0464 —

^

—

d

1

50.0514

061011-8 / Vol. 135, DECEMBER 2013 Transactions of the ASME

midsection of the cylindrical shell, the higher the beam mode fre-

quencies are, which is opposite to the variations of the fundamen-

tal frequencies with respect to the mass distribution. This

conclusion is the same with Jafari and Bagheri [28] where the rea-

son is explained by energy theory. In the beam mode, the shape of

stiffeners remain circular and the strain energy of stiffeners does

not affect the total energy of the system and only the kinetic

energy of stiffeners contributes in the total energy; therefore dec-

rement of mass distribution in the midsection of the shell reduces

the kinetic energy and raises the beam mode natural frequency.

For F-F boundary condition, the effects of nonuniform ring spac-

ing and eccentricity distribution is rather complicated but a con-

clusion can be made that the cylindrical shells get the lowest

beam mode frequency when d

R1

¼0.00491 and d

1

¼0.0114 and

get the highest beam mode frequency when d

R1

¼0.00491 and

d

1

¼0.0514. For SD-F boundary condition, beam mode frequen-

cies increase with the increment of the depth of the ﬁrst stiffener

for given d

1

and before the ﬁrst intersection point d

R1

¼0.00241

the beam mode frequencies decrease with the increment of the

ﬁrst spacing and after that point the beam mode frequencies ﬁrst

decrease then increase. The cylindrical shells get the highest

beam mode frequency when d

R1

¼0.00491 and d

1

¼0.0214 and

get the lowest beam mode frequency when d

R1

¼0.00091 and

d

1

¼0.0414.

3.3 Effects of Boundary Conditions on Cylindrical Shells

With Nonuniform Rings Spacing and Eccentricity. From the

above analysis we can see that boundary conditions have an

important effect on the beam mode frequencies and fundamental

frequencies of cylindrical shells with nonuniform rings spacing

and eccentricity distribution. Here C-C boundary condition is ﬁrst

considered, and then the boundary conditions are released simul-

taneously at the two ends gradually to consider the effects of

boundary conditions on the vibration characteristics of cylindrical

shells. The natural frequencies of three models are calculated here

which is shown in Fig. 5and they are denoted as M1 Model 1, M1

Model 2 and M1 Model 3, respectively. The fundamental frequen-

cies are shown in Table 7and the beam mode frequencies are

shown in Table 8. The two numbers in the bracket in Table 7

denote the axial half wave number and the circumferential wave

number, respectively.

We can see from Table 7that M1 Model 2 always gets the

highest fundamental frequency and M1 Model 1 always gets the

lowest fundamental frequency except when u¼T¼S¼M¼0.

For M1 Model 2, m¼1, n¼3 always appears as the fundamental

frequency mode except when N¼v¼S¼M¼0 and m¼1, n¼4

appears as the fundamental frequency mode in some cases for M1

Model 1.

The axial displacement constraint has the largest effects on the

fundamental frequencies. Release of the axial displacement con-

straint decreases the fundamental frequencies sharply and does

not change the vibration modes most of which still appear as

m¼1, n¼3. Release of other three kinds of boundary conditions

decrease the fundamental frequencies much less and lead to the

fundamental frequency mode switching from m¼1, n¼3to

m¼1, n¼2orm¼1, n¼4 for some cases.

From Table 8we can see that the effects of boundary conditions

on beam mode frequencies are rather complicated. Different rings

spacing and eccentricity distributions are needed to increase the

beam mode frequency for different boundary conditions. M1

Model 1 always gets the highest beam mode frequency when the

circumferential displacement constraint is imposed except for C-C

boundary condition.

Release of the circumferential displacement constraint can

increase the beam mode frequencies rapidly. Release of the other

three kinds of displacement constraints will decrease the beam

mode and release of the axial displacement constraint lower the

beam mode frequency more than the other two. A conclusion can

be made that the circumferential displacement constraint has the

largest effects on the beam mode frequencies of the cylindrical

shells, and the effects of the axial displacement constraint is larger

than the other two.

Fig. 8 Beam mode frequency variations versus the depth

of the ﬁrst stiffener for given d

1

with SD-SD, F-F, SD-F boundary

conditions: (a)SD-SD(b)F-F(c)SD-F—n—d

1

50.0114 ——

d

1

50.0164 —~—d

1

50.0214 —!—d

1

50.0264 —3—d

1

50.0314

—"—d

1

50.0364 —^—d

1

50.0414 —$—d

1

50.0464 —

^

—

d

1

50.0514

Journal of Vibration and Acoustics DECEMBER 2013, Vol. 135 / 061011-9

4 Conclusions

Wave based method (WBM) which can be recognized as a

semianalytical and seminumerical method presented in this paper

is quite useful in analyzing the free vibration characteristics of

cylindrical shells with nonuniform rings spacing and stiffener

distribution for arbitrary boundary conditions. The ring-stiffened

cylindrical shells can be divided into different substructures

according to the type of the structure. The motion of each bay of

cylindrical shell is described by the equations of Donnell–

Mushtari theory and the motions of the stiffeners are described by

the equations of motion of annular circular plate. In contrast with

the ﬁnite element method (FEM), in which the dynamic ﬁeld vari-

ables within each element are expanded in terms of local, non-

exact shape functions, usually polynomial approximation, the

dynamic ﬁeld variables within each substructure in WBM are

expressed as wave function expansions, which exactly satisfy the

governing dynamic equations of the substructure. Numerical

results show good agreement with experimental results and

analytical results of other researchers for shear diaphragm-shear

diaphragm boundary condition and also show good agreement

with FEM results for other boundary conditions.

Effects of the nonuniform rings spacing and eccentricity distri-

bution on fundamental frequencies and beam mode frequencies of

ring-stiffened cylindrical shells have been studied. For SD-SD

boundary conditions, more mass concentrated in the midsection of

the cylindrical shells, the higher the fundamental frequencies are

while the lower the beam mode frequencies are. For F-F and SD-F

boundary conditions, the effects are a little complicated. Numeri-

cal results of the effects of boundary conditions show that the

axial displacement constraint has the largest effects on fundamen-

tal frequencies and the circumferential displacement constraint

has the largest effects on beam mode frequencies. Release of the

axial displacement constraint decreases the fundamental frequen-

cies sharply and release of the circumferential displacement con-

straint increases the beam mode frequencies rapidly.

Acknowledgment

All the work in this paper obtains great support from the

National Natural Science Foundation of China (51179071) and

the Fundamental Research Funds for the Central Universities,

HUST: 2012QN056.

Table 7 Fundamental frequencies with different boundary conditions

Fundamental frequency (Hz)

Boundary condition M1 model 1 M1 model 2 M1 model 3

C-C u¼v¼w¼h¼0 710.369(1,4) 869.084(1,3) 793.693(1,3)

One boundary condition released N¼v¼w¼h¼0 489.112(1,3) 674.216(1,3) 571.517(1,3)

u¼T¼w¼h¼0 710.160(1,4) 842.905(1,3) 770.544(1,3)

u¼v¼S¼h¼0 709.458(1,4) 865.559(1,3) 789.784(1,3)

u¼v¼w¼M¼0 709.833(1,4) 866.113(1,3) 790.518(1,3)

Two boundary conditions released N¼T¼w¼h¼0 488.939(1,3) 672.183(1,3) 570.852(1,3)

N¼v¼S¼h¼0 485.590(1,3) 671.791(1,3) 568.789(1,3)

N¼v¼w¼M¼0 (SD-SD) 483.323(1,3) 669.668(1,3) 566.477(1,3)

u¼T¼S¼h¼0 701.333(1,4) 1305.736(1,3) 852.117(1,4)

u¼T¼w¼M¼0 690.939(1,3) 798.868(1,3) 730.671(1,3)

u¼v¼S¼M¼0 709.452(1,4) 865.205(1,3) 789.447(1,3)

Three boundary conditions released N¼T¼S¼h¼0 624.969(1,4) 928.643(1,3) 849.354(1,4)

N¼T¼w¼M¼0 476.919(1,3) 655.691(1,3) 557.706(1,3)

N¼v¼S¼M¼0 483.280(1,3) 570.729(1,2) 566.456(1,3)

u¼T¼S¼M¼0 1309.268(1,3) 1305.412(1,3) 1121.270(1,4)

Four boundary conditions released N¼T¼S¼M¼0 (F-F) 622.777(1,4) 921.218(1,3) 847.124(1,4)

Table 8 Beam mode frequencies with different boundary conditions

Beam mode natural frequency (Hz)

Boundary condition M1 model 1 M1 model 2 M1 model 3

C-C u¼v¼w¼h¼0 2109.405 1923.436 2010.241

One boundary condition released N¼v¼w¼h¼0 1685.649 1553.840 1614.918

u¼T¼w¼h¼0 4053.764 4114.167 4100.927

u¼v¼S¼h¼0 2106.321 1920.270 2006.712

u¼v¼w¼M¼0 2107.066 1920.901 2007.418

Two boundary conditions released N¼T¼w¼h¼0 3398.049 3520.205 3497.146

N¼v¼S¼h¼0 1685.143 1553.434 1614.483

N¼v¼w¼M¼0 (SD-SD) 1684.613 1553.038 1614.050

u¼T¼S¼h¼0 3823.196 3804.443 3879.559

u¼T¼w¼M¼0 3925.962 3931.617 3991.735

u¼v¼S¼M¼0 2106.192 1920.045 2006.462

Three boundary conditions released N¼T¼S¼h¼0 3027.103 3106.542 3122.503

N¼T¼w¼M¼0 3207.836 3325.436 3320.550

N¼v¼S¼M¼0 1684.610 1553.038 1614.050

u¼T¼S¼M¼0 3823.178 3804.428 3879.551

Four boundary conditions released N¼T¼S¼M¼0 (F-F) 3026.607 3106.211 3122.168

061011-10 / Vol. 135, DECEMBER 2013 Transactions of the ASME

Appendix A

The differential operators in Eq. (1) are as follows:

L11 ¼@2

@x2þ1

2

@2

@/2qð12Þa2

E

@2

@t2

L22 ¼1

2

@2

@x2þ@2

@/2qð12Þa2

E

@2

@t2

L33 ¼1þb@4

@x4þ2@4

@x2@/2þ@4

@/4

þqð12Þa2

E

@2

@t2

L12 ¼L21 ¼1þ

2

@2

@x@h;L13 ¼L31 ¼@

@x;L23 ¼L32 ¼@

@/

The radius of the cylindrical shell is designated by a, and the

thickness by h.b¼h2=12a2, The axial and circumferential coor-

dinates are x;/,x¼

x=a. The mass density of the shell’s material

is designated by q, Young’s modulus by Eand Poission’s ratio

by .

Appendix B

For modal vibration of cylindrical shell with a speciﬁed circum-

ferential wave number n, the general solution to Eq. (1) can be

written as

w¼Aekxsin n/cos xt

v¼Bekxcos n/cos xt

u¼Cekxsin n/cos xt

9

>

=

>

;

(B1)

where kis a characteristic root to be determined, xis circular fre-

quency and /is circumferential coordinate angle.

Substituting Eq. (B1) to Eq. (1) leads to three homogenous

equations for the three constants A,B,C. For nontrivial solution,

the determinant of their coefﬁcients must vanish, which give the

following fourth-degree equation in k

2

.

k8þg6k6þg4k4þg2k2þg0¼0(B2)

where

g6¼3

1

X24n2

g4¼6n43ð3Þ

1n2X2þ2

1v

X4þ1

bð12X2Þ

g2¼n2X2

1½3ð3Þn24X24n6

þX2

b3þ2þ2n23

1X2

g0¼½1=bð1Þ½ð1Þn22X2½bn6X2ð1þbn4þn2X2Þ

where Xis a dimensionless frequency parameter:

X2¼x2a2ð12Þðq=EÞ(B3)

For the usual range of parameters and n1, the roots of

Eq. (B2) have the form

k¼6k1;6ik2;6ðk36k4Þ(B4)

where k

1

,k

2

,k

3

,k

4

are real quantities. For each root, the ratios

C/A,B/Acan be determined by Eq. (1), so the general solution of

u,v,wcan be expressed by eight real constants A

1

A

8

.

w¼fA1ek1xþA2ek1xþA3cos k2xþA4sin k2x

þA5ek3xcos k4xþA6ek3xsin k4xþA7ek3xcos k4x

þA8ek3xsin k4xgsin n/cos xt(B5)

v¼fA1n1ek1xþA2n1ek1xþA3n2cos k2xþA4n2sin k2x

þA5ek3xðn3cos k4xn4sin k4xÞþA6ek3xðn4cos k4x

þn3sin k4xÞþA7ek3xðn3cos k4xþn4sin k4xÞ

A8ek3xðn4cos k4xn3sin k4xÞg cos n/cos xt(B6)

u¼fA1g1ek1xA2g1ek1xA3g2sin k2xþA4g2cos k2x

þA5ek3xðg3cos k4xg4sin k4xÞþA6ek3xðg4cos k4x

þg3sin k4xÞA7ek3xðg3cos k4xþg4sin k4xÞ

þA8ek3xðg4cos k4xg3sin k4xÞg sin n/cos xt(B7)

where

n1¼G1=D1;g1¼H1=D1

n2¼G2=D2;g2¼H2=D2

n3¼R1Q1þR2Q2

Q2

1þQ2

2

;g3¼S1Q1þS2Q2

Q2

1þQ2

2

n4¼R2Q1R1Q2

Q2

1þQ2

2

;g4¼S2Q1S1Q2

Q2

1þQ2

2

and

D1¼ð1Þk4

1þk2

1½2n2ð1Þþð3ÞX2

þðn2X2Þ½ð1Þn22X2

G1¼n½k2

1ð2þ2Þþð1Þn22X2

H1¼k1½k2

1ð1Þþ2ðX2n2Þþn2ð1þÞ

D2¼ð1Þk4

2k2

2½2n2ð1Þþð3ÞX2

þðn2X2Þ½ð1Þn22X2

G2¼n½k2

2ð2þ2Þþð1Þn22X2

H2¼k2½k2

2ð1Þ2ðX2n2Þn2ð1þÞ

Q1¼ð1Þfðk2

3k2

4Þ24k2

3k2

4gþðk2

3k2

4Þf2n2ð1Þ

þð3ÞX2gþðn2X2Þfð1Þn22X2g

Q2¼4k3k4ðk2

3k2

4Þð1Þþ2k3k4f2n2ð1Þþð3ÞX2g

R1¼nfðk2

3k2

4Þð2þ2Þþð1Þn22X2g

R2¼2nk3k4ð2þ2Þ

S1¼k3fð1Þðk2

33k2

4Þþ2ðX2n2Þþn2ð1þÞg

S2¼k4fð1Þð3k2

3k2

4Þþ2ðX2n2Þþn2ð1þÞg

Appendix C

Equations (B5)–(B7) can be written as follows:

wðxÞ¼wðxÞA;vðxÞ¼vðxÞA;uðxÞ¼uðxÞA(C1)

where

wðxÞ¼½w1ðxÞw2ðxÞw7ðxÞw8ðxÞ (C2)

vðxÞ¼½v1ðxÞv2ðxÞ v7ðxÞv8ðxÞ (C3)

uðxÞ¼½u1ðxÞu2ðxÞ u7ðxÞu8ðxÞ (C4)

A¼A1A2 A7A8

½(C5)

Journal of Vibration and Acoustics DECEMBER 2013, Vol. 135 / 061011-11

The twisting angle, forces and moments are given in Ref. [1],

h¼@w

@x

M¼D

a2

@2w

@x2þ@2w

@/2

S¼D

a2

@3w

@x3þ2ðÞ

@3w

@x@/2

T¼Eh

2a1þtðÞ

@u

@/þ@v

@x

N¼Eh

a1t2

ðÞ

@u

@xþ@v

@/þw

8

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

:

(C6)

Substituting Eqs. (B5)–(B7) to Eq. (C6)

w1¼ek1x;w2¼ek1x;w3¼cos k2x;w4¼sin k2x

w5¼ek3xcos k4x;w6¼ek3xsin k4x;w7¼ek3xcos k4x;w8¼ek3xsin k4x

v1¼n1ek1x;v2¼n1ek1x;v3¼n2cos k2x;v4¼n2sin k2x

v5¼G1ek3x;v6¼G2ek3x;v7¼G3ek3x;v8¼G4ek3x

u1¼g1ek1x;u2¼g1ek1x;u3¼g2sin k2x;u4¼g2cos k2x

u5¼F1ek3x;u6¼F2ek3x;u7¼F3ek3x;u8¼F4ek3x

h1¼k1ek1x;h2¼k1ek1x;h3¼k2sin k2x;h4¼k2cos k2x

h5¼H1ek3x;h6¼H2ek3x;h7¼H3ek3x;h8¼H4ek3x

M1¼D

a2d1ek1x;M2¼D

a2d1ek1x;M3¼D

a2d2cos k2x;M4¼D

a2d2sin k2x

M5¼D

a2ek3xðd4sin k4xd3cos k4xÞ;M6¼D

a2ek3xðd3sin k4xþd4cos k4xÞ

M7¼D

a2ek3xðd3cos k4xþd4sin k4xÞ;M8¼D

a2ek3xðd4cos k4xd3sin k4xÞ

S1¼D

a3c1ek1x;S2¼D

a3c1ek1x;S3¼D

a3c2sin k2x;S4¼D

a3c2cos k2x

S5¼D

a3ek3xðc3cos k4xc4sin k4xÞ;S6¼D

a3ek3xðc3sin k4xþc4cos k4xÞ

S7¼D

a3ek3xðc3cos k4xþc4sin k4xÞ;S8¼D

a3ek3xðc3sin k4xc4cos k4xÞ

T1¼D

a3v1ek1x;T2¼D

a3v1ek1x;T3¼D

a3v2sin k2x;T4¼D

a3v2cos k2x

T5¼D

a3ek3xðv4sin k4xv3cos k4xÞ;T6¼D

a3ek3xðv4cos k4xþv3sin k4xÞ

T7¼D

a3ek3xðv3cos k4xþv4sin k4xÞ;T8¼D

a3ek3xðv3sin k4xv4cos k4xÞ

N1¼D

a3l1ek1x;N2¼D

a3l1ek1x;N3¼D

a3l2cos k2x;N4¼D

a3l2sin k2x

N5¼D

a3ek3xðl4sin k4xl3cos k4xÞ;N6¼D

a3ek3xðl4cos k4xþl3sin k4xÞ

N7¼D

a3ek3xðl3cos k4xþl4sin k4xÞ;N8¼D

a3ek3xðl4cos k4xl3sin k4xÞ

F1¼g3cos k4xg4sin k4x;F2¼g4cos k4xþg3sin k4x

F3¼g3cos k4xþg4sin k4x;F4¼g4cos k4xg3sin k4x

061011-12 / Vol. 135, DECEMBER 2013 Transactions of the ASME

G1¼n3cos k4xn4sin k4x;G2¼n4cos k4xþn3sin k4x

G3¼n3cos k4xþn4sin k4x;G4¼n4cos k4xþn3sin k4x

H1¼k3cos k4xk4sin k4x;H2¼k4cos k4xþk3sin k4x

H3¼k3cos k4xþk4sin k4x;H4¼k4cos k4xk3sin k4x

d1¼k2

1n2;d2¼k2

2þn2;d3¼k2

3k2

4n2;d4¼2k3k4

c1¼k1fk2

1ð2Þn2g;c2¼k2fk2

2þð2Þn2g

c3¼k3fk2

33k2

4ð2Þn2g;c4¼k4f3k2

3k2

4ð2Þn2g

v1¼½ð1Þ=2bðng1þn1k1Þ;v2¼½ð1Þ=2kðng2þn2k2Þ

v3¼½ð1Þ=2bðng3þn3k3n4k4Þ;v4¼½ð1Þ=2kðng4þn4k3þn3k4Þ

l1¼ð1=bÞfg1k1þð1nn1Þg;l2¼ð1=bÞfg2k2þð1nn2Þg

l3¼ð1=bÞfg3k3g4k4þð1nn3Þg;l4¼ð1=bÞðg4k3þg3k4nn4Þ

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Journal of Vibration and Acoustics DECEMBER 2013, Vol. 135 / 061011-13