Characteristics of the vibrational power flow propagation in a ...

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Applied Acoustics 67 (2006) 550–569

www.elsevier.com/locate/apacoust

Characteristics of the vibrational powerflow propagation in a submerged periodic

ring-stiffened cylindrical shell

J. Yan, T.Y. Li *, T.G. Liu, J.X. Liu

Department of Naval Architecture and Ocean Engineering, Huazhong University of Science

and Technology, Wuhan 430074, P.R. China

Received 7 February 2005; received in revised form 2 June 2005; accepted 18 August 2005Available online 25 October 2005

Abstract

By using space-harmonic analysis method, the characteristics of the vibrational power flow prop-agation in an infinite periodic ring-stiffened cylindrical shell immersed in water are studied. The har-monic motion of the shell and the sound pressure field in the fluid are described by Flugge shellequations and Helmholtz equation, respectively, and four kinds of the rings� forces and momentsare considered. Along the shell axial direction, the propagation of the power flow carried by differentinternal forces (moments) of the shell wall can be obtained, thus the total power flow in the shell walland the ratios of the component power flow carried by different shell internal forces (moments) to thetotal power flow are also studied. It is found that characteristics of the vibrational power flow prop-agation vary with different circumferential modes order and different frequencies. Moreover, thepresence of the stiffeners and structural damping will greatly influence the results.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Periodic ring-stiffened cylindrical shell; Space-harmonic analysis method; Vibrational power flowpropagation

1. Introduction

Thin cylindrical shells stiffened by periodic ring stiffeners are commonly used as primarystructural components in aircraft and marine structures. The determination of the dynamic

0003-682X/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apacoust.2005.08.006

* Corresponding author.E-mail address: [email protected] (T.Y. Li).

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J. Yan et al. / Applied Acoustics 67 (2006) 550–569 551

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properties of these cylindrical shells is thus an important subject in the area of noise andvibration control. Since Goyder and White [1–3] first definitely presented the concept ofpower flow, the method of vibrational power flow has become a useful tool in this area.

Among the extensive researches on the power flow of the coupled system of cylindricalshell and fluid, most concentrated on the thin cylindrical elastic shells filled with fluid. Ful-ler and Fahy [4] first characterized the fluid–structure interaction in terms of power flowand energy distribution. Their work was concerned with the solution and physical inter-pretation of the dispersion equation for the coupled system. Both dispersion behaviorand energy distribution of free waves were investigated. The distribution of vibrationalenergy between the shell wall and the fluid in the shell for free modes of propagationwas obtained and its variation with frequency and material parameters was studied. Brev-art and Fuller [5] investigated on the distribution of vibrational energy between the fluidfield and the shell by considering the effect of an internal uniform flow. They gave the effectof Mach number on the dispersion curves and the distribution of vibrational energy, butthe analysis of the characteristics of the wave propagation was not very clear. Pavic [6]gave the expressions of the axial and the circumferential components of the unit energyflow along the shell wall in terms of the characteristic shell�s components of the motion.He analyzed the influence of the three different terms contributing to the total energy flowand pointed out that the three terms exhibit considerable variations in contributing the to-tal energy flow. Pavic [7] also investigated the wave motion and vibro-acoustical energyflow through a fluid-filled pipe. The expressions of the energy flow in the wall and the fluidwere given from the surface vibration and can be simplified at lower frequencies. Using themethod of residues, Xu and Zhang [8] investigated the vibrational power flow input froman external force and the transmission along the shell axial direction by different internalforces of the shell wall. The results showed that the input power flow and the power flowtransmission depend mainly upon the characteristics of the free propagating waves in thiscoupled system. Sorokin and Ershova [9] used Green�s matrix and the boundary integralequation to study free plane wave propagation in infinitely long periodic elastic structureswith and without heavy fluid loading. They also presented an efficient algorithm to com-pute Bloch parameters (propagation constants) and locate the band gaps. Sorokin et al.[10] formulated an energy flow through arbitrary cross-section of an infinitely long shellat each circumferential mode number. An inspection into the energy redistribution be-tween several transmission paths in a near field and the influence of excitation conditionson steady fluctuations of the overall energy flow in a far field was also performed. Both ofthese studies and the earlier work of Fuller and Fahy [4] provided information on theenergy flow in the time-average sense or on the energy distribution within the system atvarious frequencies. Brevart and Fuller [11] used an analytical approach to study theinstantaneous exchange of energy appearing along an infinite fluid-filled cylindrical elasticshell excited by an impulsive radial point force.

In the case of the shell immersed in fluid, its dispersion equation is similar to that of theshell filled with fluid, but the term of fluid loading differs. Scott [12] has presented a de-tailed study of the dispersion curves for the fluid-loaded cylindrical shells. Guo [13] pre-sented an approximate analytical solution for the dispersion equation of fluid-loadedcylindrical shells in the intermediate and high-frequency region. Zuo et al. [14] were inter-ested in studying the vibrational energy flow in fluid-loaded shells. They presented anumerical method for searching the propagation waves on the real wave number axis.Borgiotti and Rosen [15] investigated the power flow analysis of surface waves on a

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adddur-e-

ld-efense0hl

552 J. Yan et al. / Applied Acoustics 67 (2006) 550–569


fluid-loaded cylinder by using a shell theory that includes the effects of shear deformatioand rotary inertia.

While cylindrical shell structures with periodic ring stiffeners vibrate in fluid, they tranmit vibrational energy by combinations of wave motion, such as flexural waves, longitudinal waves and torsional waves. When these different types of waves encounterdiscontinuity in the periodicity, they interact and are converted from one type into another. This is an important part of the wave propagation process in periodic structureBut when stiffened cylindrical shell is studied, it becomes very complicated because othe effect of the fluid–structure interaction and the presence of stiffeners. Harari [16,17investigated the wave propagation in shells with a wall joint, and the related problem otravelling waves encountering a ring stiffener attached to the shell wall. The structural dicontinuity considered in the papers consists of a spring-type insert and the results shohigh power reflection coefficients at the cut-on frequencies of various torsional waves. Fuler [18] has investigated the effects of discontinuities in the wall of a cylindrical shell otravelling flexural waves, and both the geometry and material discontinuities have beeconsidered. Xu and Zhang [19] studied the transmission loss of vibrational power flothrough the wall joint in fluid-filled shells and presented an analysis of power flow tranmission and reflection at the joint. They found material stiffness of the joint and frequencare two important factors that strongly influence the results. Zhang and Zhang [20] studiethe input vibrational power flow from an external force into a shell with periodic stiffenerin vacuo. The power transmitted by different internal forces of the shell wall was deriveand its variation with frequency and the shell axial distance was also studied. By idealizinthe coupled system, Lee and Kim [21] first obtained the exact solution for this type oproblem. Characteristics of the system responses and effects of important design parameters were also studied using the transmission losses calculated from the analysis.

Space-harmonic analysis method is an effective method to analyze this coupled systemof periodic structure and fluid. By using space-harmonic analysis, the surface motion toconverted harmonic pressure can be regarded as an infinite sum of space harmonics anthe effects of fluid loading on the forced harmonic response can easily be included. Meaand his collaborators made a great deal contributions on space-harmonic analysis methoby analyzing periodic beam, plate and shell, and their work is well summarized in [22]. Xand Zhang [23] used it to study the input power flow from a cosine harmonic circumfeential line force into an infinite cylindrical fluid-filled shell with periodic stiffeners, but thvibrational power flow propagation in this periodically stiffened structure was not mentioned in their article.

In this paper, space-harmonic analysis method is extended to analyze the vibrationapower flow propagation in an infinite periodic ring-stiffened cylindrical shell immersein fluid. The harmonic motion of the shell and the sound pressure field in the fluid are described by Flugge shell equations and Helmholtz equation, respectively. First, the responsof the coupled system to a convected harmonic pressure is derived. Secondly, four kinds oshell internal forces (moments) can be obtained according to the relationships between thstress and the strain of the shell. At last, the forced vibrational power flow propagatioalong the shell axial direction is given. The total power flow in the shell wall and the ratioof the component power flow carried by different shell internal forces (moments) to thtotal power are also studied. The discussions focus on the circumferential modes n =and n = 1 at three kinds of representative frequencies (that is low, middle and higfrequencies, respectively). Utilizing the analysis results, characteristics of the vibrationa



power flow propagation through a periodic ring-stiffened shell are interpreted with specialattention to the role of the periodic ring stiffeners.

2. Forced vibration of the coupled system

An infinite thin-walled periodic ring-stiffened cylindrical shell immersed in fluid shownin Fig. 1 is considered. Supposing the density of the fluid outside is qf and the sound veloc-ity in it is Cf. The shell is characterized by its mean radius a, wall thickness h, mass densityqs, Young�s modulus E and the Poisson�s ratio l. The ring stiffeners have uniform rectan-gular section with width b and height d, attached at x = mL (L is the stiffener spacing,m = 0, ± 1, ± 2, . . .). The connections are rigid, so that, at each line of attachment, theshell and the stiffener have the same linear velocity and angular velocity. The insider stiff-eners may exert axial force, shear and moments on the shell.

2.1. Forces and moments of a ring stiffener acting on the shell

When the external force is applied on the shell-fluid coupled system, the attached ringstiffener will vibrate together with the shell. Omitting shear deformation and rotary inertia,the motion equations of a ring stiffener are [23]

E1I

R41

o4w

oh4� o3v

oh3

� �þ E1A

R21

wþ ovoh

� �þ q1A

o2wot2

¼ F wðh; tÞ; ð1aÞ

E1I

R41

o3w

oh3� o2v

oh2

� �� E1A

R21

owoh

þ o2v

oh2

� �þ q1A

o2vot2

¼ F vðh; tÞ; ð1bÞ

E1I

R41

o4u

oh4� o

2ðR1uÞoh2

� �� G1J

R41

o2u

oh2þ o

2ðR1uÞoh2

� �þ q1A

o2uot2

¼ F uðh; tÞ; ð1cÞ

E1I

R21

R1u� o2u

oh2

� �� G1J

R21

o2u

oh2þ o

2ðR1uÞoh2

� �þ q1Ip

o2ðR1uÞot2

¼ R1F uðh; tÞ; ð1dÞ

where R1 is the mean radius of a stiffener, A is the area of stiffener cross-section, I and �I arethe principal moments of inertia paralleling the axial and the radial, respectively,Ip ¼ I þ �I is the polar moments of inertia, J is the constant of twist of cross-section, q1is mass density of stiffener material, E1 and G1 are Young�s modulus and shear modulusof stiffener material, respectively, Fw, Fv, Fu and M are the sideward forces and momentsof the stiffener acting on the shell, respectively, u = ow/ox.

Fig. 1. Periodical ring-stiffened shell immersed in fluid.

t

Þ

r.

e

ÞÞ

-

s

ÞÞÞÞ

1,

,

,

Þ

d

Þ

Þ

j

i-



In order to consider the influence of eccentricity, one may express the displacements athe center of the ring in terms of displacements at middle surface

u� ¼ u� eu; v� ¼ v 1þ eR

� �� eRowoh

; w� ¼ w; u� ¼ u; ð2

where e is the distance from the middle surface of shell to the geometric center of stiffeneWhen e � a, the second equation is deduced to v� = v.

When the vibration of the shell is in the nth mode, the ring stiffener vibrates in the sammode, then

½ u� v w u�T ¼ eixt½ u�a cos nh va sin nh wa cos nh ua cos nh �T; ð3a½ F u F v F w M �T ¼ eixt½ F ua cos nh F va sin nh F wa cos nh Ma cos nh �T; ð3b

where x is circular frequency, i ¼ffiffiffiffiffiffiffi�1

p, subscript a means displacements and forces (mo

ments) at middle surface of shell.Introducing Eqs. (2), (3) into (1) with eixt, cosnh and sinnh omitted, the reaction force

and moments in the line circumferential stiffener at x = 0 can be expressed as follows:

F w;0 ¼ f1wð0Þ=D1 þ f2vð0Þ=D1; ð4aF v;0 ¼ f3wð0Þ=D1 þ f4vð0Þ=D1; ð4b� iM0 ¼ f5uð0Þ=D1 þ f6wð0Þ=D1; ð4cF u;0 ¼ f7uð0Þ=D1 þ f8wð0Þ=D1; ð4d

where f1 = �K1D1, f2 = �K2D1, f3 = K2D1, f4 = K3D1, f5 = �iK5D1, f6 ¼ �kxðK6 � ea K5ÞD

f7 = K4D1, f8 ¼ �ikxðK5 � ea K4ÞD1, D1 = a2(1 � l2)/(Eh), K1 ¼ R1

aE1IR41

n4 þ E1AR21

� q1Ax2

� �K2 ¼ R1

aE1IR41

n3 þ E1AR21

n� �

, K3 ¼ R1

aE1IR41

n2 þ E1AR21

n2 � q1Ax2

� �, K4 ¼ R1

aE1IR41

n4 þ G1JR41

n2 � q1Ax2

� �K5 ¼ 1

aE1IR21

n2 þ G1JR21

n2� �

and K6 ¼ 1a

E1IR21

þ G1JR21

n2��

q1x2IpÞ.

2.2. The responses of the coupled system

The shell is excited by a harmonic line pressure F, acting on x = 0, expressed as

F ðh; tÞ ¼ F 0 cosðnhÞdð0Þ expðixtÞ; ð5where d is the Dirac delta function.

Then the forces exerted on the shell included the external load F, the fluid loading anradial force, circumference force, axial force and moments of the stiffener, respectively.

The wave motions in the shell wall and the sound field in the fluid of the coupled systemare described by Flugge shell equations and Helmholtz equation, respectively [24,25]

X3

i¼1

Lijðu; v;wÞ þ F j ¼ 0; j ¼ 1; 2; 3; ð6

Dpf þ k20pf ¼ 0; ð7where Lij denote differential operators, k0 = x/cf is the free wave number in the fluid. Fdenote terms about the external loads exerted on the shell in the directions of the coordnate x-, r- and h-axes and are expressed as:



F 1 ¼ D1

X1m¼�1

F u;mdðx� mLÞ; ð8aÞ

F 2 ¼ D1

X1m¼�1

F v;mdðx� mLÞ; ð8bÞ

F 3 ¼ D1 ðF � pfÞ þX1

m¼�1F w;mdðx� mLÞ þ

X1m¼�1

Mmd0ðx� mLÞ

" #; ð8cÞ

where Fw,m, Fv,m, Fu,m and Mm are the values of Fw, Fv, Fu and M of the mth stiffener,respectively.

The normal mode shapes assumed for the displacement of the shell wall, associated withan axial wave number kns, are given by:

u ¼P1n¼0

P1s¼1

Uns cosðnhÞ expðixt � iknsxÞ;

v ¼P1n¼0

P1s¼1

V ns sinðnhÞ expðixt � iknsxÞ;

w ¼P1n¼0

P1s¼1

W ns cosðnhÞ expðixt � iknsxÞ.

8>>>>>>><>>>>>>>:

ð9Þ

The wave radiated from the shell to the outside will have the same periodic character-istics as the structural wave in the shell, at r = a, pf can be represented as

pf ¼X1n¼0

X1s¼1

Pns cosðnhÞH ð2Þn ðkrrÞ expðixt � iknsxÞ; ð10Þ

where H ð2Þn ð Þ is the second kind Hankel function of order n, kr is the radial wave number.

Application of the fluid momentum equation at the shell wall, r = a, produces

Pns ¼ x2qf=½ðkraÞH ð2Þ0n ðkraÞ�W ns. ð11Þ

Introducing Eq. (11) into (10), acoustic pressure can be obtained as

pf ¼X1n¼0

X1s¼1

x2qfHð2Þn ðkraÞ

ðkraÞH ð2Þ0n ðkraÞ

W ns cosðnhÞ expðixt � iknsxÞ. ð12Þ

Introducing Eqs. (9), (12) into (6) and taking the Fourier transform, the following equa-tions are obtained:

L3�3½ � ~U ~V ~W� �T ¼ ~F u

~F v~F þ ~F w þ ~M

� �T; ð13Þ

where ~U , ~V and ~W are the spectral displacements:

~F ¼ D1

Z 1

�1F 0 e

ikx e�ikxx dx ¼ D1F 0dðk � kxÞ; ð14aÞ

~F u ¼ D1

Z 1

�1

X1m¼�1

F u;mdðx� mLÞeikx dx ¼ D1

X1m¼�1

F u;m eikmL; ð14bÞ

~F v ¼ D1

Z 1

�1

X1m¼�1

F v;mdðx� mLÞeikx dx ¼ D1

X1m¼�1

F v;m eikmL; ð14cÞ

Þ

Þ

oe

Þf

-

Þ.

y

Þo

Þ

Þ

Þ

Þ



~F w ¼ D1

Z 1

�1

X1m¼�1

F w;mdðx� mLÞeikx dx ¼ D1

X1m¼�1

F w;m eikmL; ð14d

~M ¼ D1

Z 1

�1

X1m¼�1

Mmd0ðx� mLÞeikx dx ¼ �iD1kx

X1m¼�1

Mm eikmL; ð14e

where kx is the axial wave number.The equations governing the motion of this coupled system differ from the in vacu

shell equations [20] because of the presence of the fluid loading term, FL, which can bobtained from the boundary condition of the shell wall and is given by [14]

FL ¼ X2ðqf=qsÞða=hÞðH ð2Þn ðkraÞÞ=ðkraÞH ð2Þ0

n ðkraÞ; ð15where X ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqsa2x2ð1� l2Þ=E

pis the non-dimensional frequency, which is the ratio o

excitation frequency to ring frequency.Expansion of the determinant of the amplitude coefficient in Eq. (13) provides the char

acteristic equation of this coupled system, expressed as

P 1ðkÞ þ P 2ðkÞFL ¼ 0; ð16where both P1(k) and P2(k) are polynomials of the non-dimensional axial wave number kFor the sake of brevity, the coefficients are not given here.

Since the structure is periodic in x-axes direction, the responses of the shell wall satisfthe periodicity condition,

uðxþ LÞ vðxþ LÞ wðxþ LÞ½ �T ¼ e�ikxL uðxÞ vðxÞ wðxÞ½ �T. ð17Similarly, the forces and the moments in the stiffeners acting on the outer shell wall alssatisfy the periodicity condition,

F u;m F v;m F w;m Mm½ �T ¼ e�ikxmL F u;0 F v;0 F w;0 M0½ �T. ð18Introducing Eq. (18) into (14b)–(14e), the following equations are obtained:

~F u~F v~F w~M

264

375 ¼ D1

F u;0P1

m¼�1e�ikxmL eikmL

F u;0P1


F u;0P1


�ikM0

P1m¼�1

e�ikxmL eikmL

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;. ð19

By using the Poisson sum formula [26] in Eq. (19),X1m¼�1

e�ikxmL eikmL ¼ 2pX1

m¼�1d 2mpþ ðkx � kÞL½ �. ð20

The following equations can be obtained

~F u~F v~F w~M

264

375 ¼ 2pD1

F u;0P1

m¼�1d 2mpþ ðkx � kÞL½ �

F v;0P1


F w;0P1


�ikM 0

P1m¼�1

d 2mpþ ðkx � kÞL½ �

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;. ð21



Let matrix I be the inverse of matrix L, then the spectral displacements can be obtainedfrom Eq. (13) as follows:

~U~V~W

8><>:

9>=>; ¼

I11 I12 I13I21 I22 I23I31 I32 I33

264

375

~F u

~F v

~F þ ~F w þ ~M

8><>:

9>=>;. ð22Þ

Taking the inverse Fourier transform of Eq. (22), the shell displacements are obtainedas

uðxÞ vðxÞ wðxÞ½ �T ¼ D1 up1 þ uF 1 vp1 þ vF 1 wp1 þ wF 1½ �T; ð23Þwhere

up1 ¼F 0

2pI13ðkxÞe�ikxx; ð24aÞ

vp1 ¼F 0

2pI23ðkxÞe�ikxx; ð24bÞ

wp1 ¼F 0

2pI33ðkxÞe�ikxx; ð24cÞ

uF 1 ¼X1

m¼�1e�ikxx½F u;0I11ðkmÞ þ F v;0I12ðkmÞ þ F w;0I13ðkmÞ � iM0kmI13ðkmÞ�; ð24dÞ

vF 1 ¼X1

m¼�1e�ikxx½F u;0I21ðkmÞ þ F v;0I22ðkmÞ þ F w;0I23ðkmÞ � iM0kmI23ðkmÞ�; ð24eÞ

wF 1 ¼X1

m¼�1e�ikxx½F u;0I31ðkmÞ þ F v;0I32ðkmÞ þ F w;0I33ðkmÞ � iM0kmI33ðkmÞ�; ð24fÞ

where km = kx + 2mp/L.Then the space displacements at x = 0 can be obtained as

uð0Þ ¼ D1

F 0

2pI13ðkxÞ þ D1

X1m¼�1

½F u;0I11ðkmÞ þ F v;0I12ðkmÞ þ F w;0I13ðkmÞ

� iM0kmI13ðkmÞ�; ð25aÞ

vð0Þ ¼ D1

F 0

2pI23ðkxÞ þ D1

X1m¼�1


� iM0kmI23ðkmÞ�; ð25bÞ

wð0Þ ¼ D1

F 0

2pI33ðkxÞ þ D1

X1m¼�1


� iM0kmI33ðkmÞ�. ð25cÞ

Notice that once the forces and the moments of the 0th stiffener in Eqs. (25) are given,the response of the shell to the applied pressure load F can be solved. Introdu- cing Eqs. (4)into (25), the shell displacements at x = 0 can be obtained in matrix form as

l11 l12 l13l21 l22 l23l31 l32 l33

264

375

uð0Þvð0Þwð0Þ

8><>:

9>=>; ¼ D1

pI1pI2pI3

8><>:

9>=>;; ð26Þ



where l11 = 1 � f7J11 � f5K13, l12 = �f4J12 � f2J13, l13 = �f8J11 � f3J12 � f1J13 � f6K13,l21 = �f7J12 � f5K23, l23 = �f8J12 � f3J22 � f1J23 � f6K23, l31 = �f7J13 � f5K33, l32 =�f4J23 � f2J33, l33 = 1 � f8J13 � f3J23 � f1J33 � f6K33, pI1 ¼ F 0

2p I13, pI2 ¼F 0

2p I23, pI3 ¼F 0

2p I33,J 11 ¼

P1m¼�1I11ðkmÞ, J 12 ¼

P1m¼�1I12ðkmÞ, J 13 ¼

P1m¼�1I13ðkmÞ, J 22 ¼

P1m¼�1I22ðkmÞ,

J 23 ¼P1

m¼�1I23ðkmÞ, J 33 ¼P1

m¼�1I33ðkmÞ, K13 ¼P1

m¼�1kmI13ðkmÞ,K23 ¼P1

m¼�1kmI23ðkmÞ, K33 ¼

P1m¼�1kmI33ðkmÞ.

Thus, the displacements of the stiffened shell at x = 0 are obtained, then the reactionforces and moments of the 0th stiffener can be obtained from Eq. (4), and so the responsesof arbitrary cross-section of the stiffened shell (that is u(x), v(x) and w(x)) can be obtainedfrom Eq. (23).

3. Power flow propagation of the coupled system

The noise control and vibration reduction methods depend upon the nature of thevibration, and thus it is of prime importance to predict the characteristics of vibrationalpower flow. In this section, the forced vibrational power flow propagation along the shellaxial direction will be discussed.

When an external harmonic line force is applied radially on the wall of the fluid-loadedshell, the forced vibration waves will propagate in the shell-fluid coupled system, and thusthe power flow will also be transmitted along the shell axial direction. According to Eq.(23), u(x), v(x) and w(x) can be obtained. Meanwhile, there will be four internal forces(moments) of the shell wall along the axial direction which can be easily derived fromFlugge shell theory [24]. These internal forces (moments) are shown in Fig. 2.

Nx ¼Da

u0 þ lv� þ lwð Þ � Ka3

w00; ð27aÞ

Nxh ¼Da1� l2

u� þ v0ð Þ þ Ka3

1� l2

v0 � w0�ð Þ; ð27bÞ

Mx ¼Ka2

w00 þ lw�� u0 � lv�ð Þ; ð27cÞ

Fig. 2. The internal forces and moments in the shell.



Mxh ¼Ka2

1� lð Þ w0� � v0ð Þ; ð27dÞ

Qx ¼ oMx=ox; ð27eÞ

where D ¼ Eh1�l2 is the tensile stiffness of the shell, K ¼ Eh3

12ð1�l2Þ is the bending stiffness of theshell, and ðÞ0 ¼ a oðÞ

ox , ðÞ� ¼ oðÞ

oh .Then we can obtained the axial force NpF, the effective torsional shear force TpF, the

effective transverse shear force SpF and bending moment MpF

NpF ¼ Nx; ð28aÞ

T pF ¼ Nxh �Mxh

a; ð28bÞ

SpF ¼ Qx þ1

aoMxh

oh; ð28cÞ

MpF ¼ Mx. ð28dÞ

Introducing Eqs. (23) and (27) into (28), the internal forces (moments) can be obtained

NpF ¼ DD1

a½að�ikxup1 þ uF 2Þ þ lnðvp1 þ vF 1Þ þ lðwp1 þ wF 1Þ

� b2a2ð�k2xwp1 þ wF 3Þ�; ð29aÞ

T pF ¼ DD1

a1� l2

½�nðup1 þ uF 1Þ þ ð1þ 3b2Það�ikxvp1 þ vF 2Þ

þ 3b2anð�ikxwp1 þ wF 2Þ�; ð29bÞ

SpF ¼ DD1

ab2½a3ðik3xwp1 þ wF 4Þ � an2ð�ikxwp1 þ wF 2Þ � anð�ikxvp1 þ vF 2Þ

� a2ð�k2xup1 þ uF 3Þ�; ð29cÞMpF ¼ DD1b

2½a2ð�k2xwp1 þ wF 3Þ � ln2ðwp1 þ wF 1Þ � að�ikxup1 þ uF 2Þ� lnðvp1 þ vF 1Þ�; ð29dÞ

where b2 = h2/(12a2) is thickness factor of the shell,

uF 2 ¼X1

m¼�1ð�ikmÞe�ikmx F u;0I11ðkmÞ þ F v;0I12ðkmÞ þ F w;0I13ðkmÞ � iM0kmI13ðkmÞ½ �; ð30aÞ

uF 3 ¼X1

m¼�1ð�k2mÞe�ikmx F u;0I11ðkmÞ þ F v;0I12ðkmÞ þ F w;0I13ðkmÞ � iM0kmI13ðkmÞ½ �; ð30bÞ

vF 2 ¼X1

m¼�1ð�ikmÞe�ikmx F u;0I21ðkmÞ þ F v;0I22ðkmÞ þ F w;0I23ðkmÞ � iM0kmI23ðkmÞ½ �; ð30cÞ

wF 2 ¼X1

m¼�1ð�ikmÞe�ikmx F u;0I31ðkmÞ þ F v;0I32ðkmÞ þ F w;0I33ðkmÞ � iM0kmI33ðkmÞ½ �; ð30dÞ

wF 3 ¼X1

m¼�1ð�k2mÞe�ikmx F u;0I31ðkmÞ þ F v;0I32ðkmÞ þ F w;0I33ðkmÞ � iM0kmI33ðkmÞ½ �; ð30eÞ

wF 4 ¼X1

m¼�1ð�ik3mÞe�ikmx F u;0I31ðkmÞ þ F v;0I32ðkmÞ þ F w;0I33ðkmÞ � iM0kmI33ðkmÞ½ �. ð30fÞ



Then, the total internal forces (moments) can be obtained from Eqs. (29) by integralrepresentation

NðxÞ ¼ 1

2p

Z 1

�1NpF cosðnhÞeixt dkx; ð31aÞ

T ðxÞ ¼ 1

2p

Z 1

�1T pF sinðnhÞeixt dkx; ð31bÞ

SðxÞ ¼ 1

2p

Z 1

�1SpF cosðnhÞeixt dkx; ð31cÞ

MðxÞ ¼ 1

2p

Z 1

�1MpF cosðnhÞeixt dkx. ð31dÞ

Eq. (23) can also be expressed as the integral form

uðxÞ vðxÞ wðxÞ½ �T ¼ 1

2pD1

R1�1ðup1 þ uF 1Þ dkxR1�1ðvp1 þ vF 1Þ dkxR1�1ðwp1 þ wF 1Þ dkx

8><>:

9>=>;. ð32Þ

According to the definition of the power flow [8], at the interface x of the shell wall, thevibrational power flow transmitted by these forces (moments) are, respectively, expressedas

PNðxÞ ¼1

2

Z 2p

0

Re NðxÞ � ðixÞuðxÞ�½ �a dh

¼ D1

8pnn�Re

Z 1

�1NpF dkx

� ðixÞ

Z 1

�1ðup1 þ uF 1Þ dkx

� � �; ð33aÞ

PT ðxÞ ¼1

2

Z 2p

0

Re T ðxÞ � ðixÞvðxÞ�½ �a dh

¼ D1

8pnn�Re

Z 1

�1T pF dkx

� ðixÞ

Z 1

�1ðvp1 þ vF 1Þ dkx

� � �; ð33bÞ

PSðxÞ ¼1

2

Z 2p

0

Re SðxÞ � ð�ixÞwðxÞ�½ �a dh

¼ D1

8pnn�Re

Z 1

�1SpF dkx

� ð�ixÞ

Z 1

�1ðwp1 þ wF 1Þ dkx

� � �; ð33cÞ

PMðxÞ ¼1

2

Z 2p

0

Re MðxÞ � ðixÞðowðxÞ=oxÞ�½ �a dh

¼ D1

8pnn�Re

Z 1

�1NpF dkx

� ðixÞ

Z 1

�1ð�ikxwp1 þ wF 2Þ dkx

� � �; ð33dÞ

where Æ denotes vector product, the asterisk denotes the complex conjugate, and

nn ¼0:5; n ¼ 0;

1; n 6¼ 0.

ð34Þ

The total vibrational power flow in the shell wall is the sum of the component power flowcarried by internal forces (moments) and expressed as



P shellðxÞ ¼ PNðxÞ þ PT ðxÞ þ PSðxÞ þ PMðxÞ. ð35Þ

4. Numerical computation and results discussion

Some numerical examples will now be given to illustrate the features of the theoreticalresults. In this article, an integrated numerical method discussed in [23] is employed to cal-culate the integration. This method is to integrate numerically along the pure imaginaryaxis of the complex wave number domain. Structural damping is introduced into the shellmaterial in order to avoid singularities in the integrand function along the integrationpath. Damping is introduced into the shell material by modifying the Young�s modulesE to be complex such as E 0 = E(1 � ig). Thus, the solution of free wave numbers of thissystem is unnecessary.

The parameters considered in this paper are shown as following, a thickness to radiusratio of h/a = 0.02, stiffener spacing L = 0.4a, stiffener width and height b = d = 2h, thematerial parameters of the stiffeners are the same as those of the shell (see Table 1 formaterial properties). The discussions will focus on the circumferential modes n = 0 andn = 1, as these two modes represent typical behavior of the system. The following calcu-lations all adopt these parameters if it is not mentioned especially.

Once the external harmonic line force inputs power flow into the coupled system, thepower flow in the shell transmitted along the axial direction can be known. Accordingto symmetry, half of the input power will be transmitted in the positive direction of theshell axial while the other half will be transmitted in the negative direction. Thus,P 0shellðxÞ ¼ P shellðxÞ=ð0:5P shellð0ÞÞ ðx > 0Þ is defined to describe the characteristics of the power

flow in the shell transmitted along the axial direction. In the same time, the ratios of thepower flow carried by different shell internal forces (moments) (those are P 0

NðxÞ ¼ PNðxÞ=P shellðxÞ, P 0

T ðxÞ ¼ PT ðxÞ=P shellðxÞ, P 0SðxÞ ¼ PSðxÞ=P shellðxÞ and P 0

MðxÞ ¼ PMðxÞ=P shellðxÞ, respectively)to the total power in the shell wall also can be obtained.

Here, the variation of P 0shellðxÞ and the variations of P 0

NðxÞ, P0T ðxÞ, P

0SðxÞ and P 0

MðxÞ againstnon-dimensional distance x/a will be investigated for different circumferential mode ordersn and different non-dimensional frequencies X. Thus, the characteristics of shell motionswill be interpreted in the view of energy.

4.1. Power flow transmitted by the shell

Fig. 3 shows the power flow P 0shellðxÞ transmitted by the shell in the coupled system. The

numerical results focus on the circumferential modes order n = 0 and n = 1 at three rep-resentative frequencies, respectively. In order to investigate the influence of the stiffeners,the results of a shell without stiffeners are also plotted. The following conclusions can bedrawn.

Table 1Material properties

Young�s modulus (N/m2) Poisson�s ratio Density (kg/m3) Free wave speed (m/s) Damping factor

Steel 2.07 · 1011 0.3 7850 5200 0.1Water – – 1000 1500 –

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

llehseht

ybdetti

msnartre

woP

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(b)

(3) Non-dimensional distance x/a

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0llehs

ehtyb

dettimsnartre

woP

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(b)


0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

llehseht

ybdetti

msnartre

woP

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(b)


Fig. 3. Power transmitted by the shell. (The stiffeners� positions are displayed by vertical dash lines). (a) n = 0, (b)n = 1: (1) X = 0.3, (2) X = 1.2, (3) X = 3.0. –––, unstiffened shell and ——, stiffened shell.





At the driving point, x/a = 0, there is P 0shellðxÞ ¼ 1 for any circumferential mode order n

and non-dimensional frequency X. When x/a increases, P 0shellðxÞ will couple with the fluid

and gradually attenuates because of the structural damping and sound radiation.At low frequencies (X = 0.3, which is below ring frequency and just exceeds the first

cut-on frequency, as shown in Fig. 3(1)), the variations of P 0shellðxÞ has a periodic wave mo-

tion, which means a strong shell-fluid coupling. For circumferential mode order n = 0(breathing mode), the effect of shell–fluid coupling is violence and wave motion phenom-enon can be seen. The reason is that the only important propagating wave in the coupledsystem is fluid type. For circumferential mode order n = 1 (bending mode), the effect ofshell–fluid coupling becomes weaker than that of circumferential mode order n = 0. Com-paring with the unstiffened shell in the fluid, the results of periodic ring-stiffened shell al-most resemble except several peaks and troughs because of the influence of the stiffeners,and the appearances of peaks and troughs correspond to the positions of the stiffeners(those are x/a = 0.4, 0.8, 1.2, . . ., respectively, stiffener spacing L = 0.4a). In some casespredominantly wave motion can propagate in the positive axial direction while the energypropagates in the reverse direction, thus ‘‘negative power flow’’ may be generated.

At middle frequencies (X = 1.2, which is slightly higher than ring frequency, as shownin Fig. 3(2)), the effect of shell–fluid coupling is weaker but wave motion phenomenon canstill be seen, which means the exchange of energy between the shell and the fluid is stillexisting. For circumferential mode order n = 0, because the torsional motion is uncoupledwith other motions, P 0

shellðxÞ in the stiffened shell is almost the same as that in the unstiffenedshell. For circumferential mode order n = 1, P 0

shellðxÞ in the stiffened shell is quite differentfrom that in the unstiffened shell. This is because the stiffener spacing and shell wavelengthalong axial direction are comparable at middle frequencies, so the presence of the period-ically spaced stiffeners has a strong influence on the vibrational power flow propagation.

At high frequencies (X = 3.0, which corresponds to a relatively much higher frequency,as shown in Fig. 3(3)), the effect of shell–fluid coupling is further weaker and wave motionphenomenon is hard to observe. For circumferential mode order n = 0 and n = 1, P 0

shellðxÞonly fluctuates at close range of the driving point, then quickly attenuates as an exponentfunction due to the structural damping with the increasing of x/a for both the stiffenedshell and the unstiffened shell.

4.2. Power flow transmitted by different internal forces (moments) of the shell

Fig. 4 shows the power flow transmitted by different internal forces (moments) of theshell wall, P 0

NðxÞ, P0T ðxÞ, P

0SðxÞ and P 0

MðxÞ plotted against non-dimensional axial distance x/afor the coupled system. (The positions of the stiffeners are x/a = 0.4, 0.8, 1.2, . . . , respec-tively.) The numerical results vary with different circumferential modes order and differentfrequencies, and the results of a shell without stiffeners are also plotted. The following con-clusions can be drawn.

For circumferential mode order n = 0, because the torsional motion is uncoupled withother motions, so P 0

shellðxÞ ¼ P 0NðxÞ þ P 0

SðxÞ þ P 0MðxÞ. While for circumferential mode order

n = 1, P 0shellðxÞ ¼ P 0

NðxÞ þ P 0T ðxÞ þ P 0

SðxÞ þ P 0MðxÞ.

At the driving point x/a = 0, for any circumferential mode order n and non-dimen-sional frequencies X, there is P 0

shellðxÞ ¼ 1. At close range of the driving point,P 0shellðxÞ P 0

SðxÞ, and the total energy is almost concentrated on flexural motion. As x/aincreases, P 0

shellðxÞ changes into other types of power flow.

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(b)


)stnemo

m(secrofllehstnereffid

ybdetti

msnartre

woP

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(b)


)stnemo


ybdetti

msnartre

woP

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(b)


)stnemo


ybdetti

msnartrewoP

Fig. 4. Power transmitted by different shell forces (moments). (The stiffeners� positions are displayed by verticaldash lines). (a) unstiffened shell, (b) stiffened shell. (1) n = 0, X = 0.3, (2) n = 1, X = 0.3, (3) n = 0, X = 1.2, (4)n = 1, X = 1.2, (5) n = 0, X = 3.0 and (6) n = 1, X = 3.0. –Æ–Æ–, P 0

NðxÞ; –––, P0T ðxÞ; , P 0

SðxÞ and ——, P 0MðxÞ.



0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8-0.2

0.0

0.2

0.4

0.6

0.8

1.0

(b)


)stnemo


ybdetti

msnartre

woP

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(b)


)stnemo


ybdetti

msnartrewo

P

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(b)


)stnemo


ybdetti

msnartre

woP

Fig. 4 (continued)





At low frequencies (X = 0.3, as shown in Fig. 4(1) and (2)), for circumferential modeorder n = 0, P 0

SðxÞ þ P 0MðxÞ � P 0

NðxÞ, the shell motion is predominately extensional, flexuralmotion only account for a small energy flow that is even in reverse direction. For circum-ferential mode order n = 1, P 0

SðxÞ þ P 0MðxÞ � P 0

NðxÞ þ P 0T ðxÞ, the motion is mainly in the tor-

sional and extensional directions. The trend in the stiffened shell is similar to that in theunstiffened shell.

At middle frequencies (X = 1.2, as shown in Fig. 4(3) and (4)), for any circumferentialmode order n, at close range of the driving point P 0

SðxÞ þ P 0MðxÞ P 0

NðxÞ þ P 0T ðxÞ, the motion

of the shell wall is mainly in the radial direction. Apart from the driving point, P 0SðxÞ and

P 0MðxÞ gradually attenuate while P 0

NðxÞ gradually increases. Which means the dominatingmotion begins to change from flexural motion to extensional motion along the axial direc-tion. The results of the periodic ring-stiffened shell differ from those of the unstiffened shelllargely. This is because the stiffener spacing and shell wavelength along axial direction arecomparable at middle frequencies, so the presence of the periodically spaced stiffeners hasa strong influence on the different internal forces (moments) of the shell wall.

At high frequencies (X = 3.0, as shown in Fig. 4(5) and (6)), the regularities are similarto those at middle frequencies. At close range of the driving point, the flexural wave stilldominates and P 0

NðxÞ 0, which is because there exists mainly flexural motion in the shellat high frequencies and the axial force hardly transmits power flow under a radial excitingforce. Apart from the driving point, P 0

NðxÞ increases but the amplitude of the increase isslower than that at middle frequencies. Comparing with the unstiffened shell, P 0

SðxÞ andP 0MðxÞ descend quicker while P 0

NðxÞ ascends quicker, which means the dominating motionchanges from flexural motion to extensional motion quickly in stiffened shell. The powerflow carried by different forces (moments) in the stiffened shell also changes along the axialdistance, but the variation amplitudes are much smaller than that in the unstiffened shell.

For all frequencies X and circumferential mode orders n, P 0SðxÞ P 0

MðxÞ, which means thepower transmitted by the transverse shear force equals that transmitted by the bendingmoments.

For an unstiffened shell immersed in fluid, owing to the effect of coupling between theshell and the fluid, at middle or high frequencies, the power flow carried by different shellforces (moments) is the ‘‘periodic’’ function of distance x/a. But this variation amplitude ismuch weaker in the case of stiffened shell.

4.3. Influence of structural damping

Fig. 5 shows the influence of structural damping on the power flow transmitted by theshell for circumferential mode order n = 0 at middle frequencies (X = 1.2). It can be seenthe power flow transmitted by the shell attenuates obviously along the axial direction withthe increasing of the structural damping. When structural damping is small (g = 0.02), theinfluence of the periodically spaced stiffeners is quite apparent. (The positions of the stiff-eners are x/a = 0.4, 0.8, 1.2, . . ., respectively.) When the structural damping is rather large(g = 0.2), the curve becomes smoother and attenuates rapidly.

For the same circumferential mode order and frequencies, Fig. 6 shows the influence ofstructural damping on the power flow transmitted by different internal forces (moments)of the shell wall, (The positions of the stiffeners are x/a = 0.4, 0.8, 1.2, . . . respectively.)When the structural damping is small (g = 0.02), the ratios of each internal forces(moments) of the shell wall remain steady after a certain distance. This means structural

dWa

5

fla

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0Po

wer

tran

smitt

ed b

y th

e sh

ell

(a)

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.80.0

0.2

0.4

0.6

0.8

1.0

(b)

Non-dimensional distance x/a

Fig. 5. Influence of structural damping on the power flow transmitted by the shell for n = 0 and X = 1.2. (Thestiffeners� positions are displayed by vertical dash lines). (a) g = 0.02, (b) g = 0.2.

0 .0 0 .4 0 .8 1 .2 1 .6 2 .0 2 .4 2 .8 3 .2 3 .6 4 .0 4 .4 4 .80 .0

0 .2

0 .4

0 .6

0 .8

1 .0

0 .0 0 .4 0 .8 1 .2 1 .6 2 .0 2 .4 2 .8 3 .2 3 .6 4 .0 4 .4 4 .80 .0

0 .2

0 .4

0 .6

0 .8

1 .0

Non-dimensional distance x/a

(a) (b)

)stnemo


ybdetti

msnartrewoP

Fig. 6. Influence of structural damping on the power flow transmitted by different shell forces (moments) forn = 0 and X = 1.2. (The stiffeners� positions are displayed by vertical dash lines). (a) g = 0.02, (b) g = 0.2. –Æ–Æ–,P 0NðxÞ; –––, P

0T ðxÞ; , P 0

SðxÞ and ——, P 0MðxÞ



amping has little influence on the values of P 0NðxÞ, P

0T ðxÞ, P

0SðxÞ and P 0

MðxÞ when it is small.hen the structural damping is rather large (g = 0.2), P 0

SðxÞ and P 0MðxÞ are greatly influenced

nd attenuated as an exponent function while P 0NðxÞ increases reversely.

. Conclusions

By using space-harmonic analysis method, the characteristics of the vibrational powerow propagation in an infinite periodic ring-stiffened cylindrical shell immersed in waterre studied analytically. The numerical method is adopted to compute the power flow



propagation and some comparisons are made with those in the unstiffened shell. As areshown in the paper, the energy propagation and the energy redistribution between variouskinds of waves in a shell of given parameters are very sensitive to the excitation frequencyand circumferential modes order.

At low and high frequencies, the vibrational power flow transmitted along the shell�saxial direction depends mainly upon the circumferential mode orders n and differentnon-dimensional frequencies X, whether the stiffeners exist has little influence on the re-sults. Since the stiffener spacing and shell wavelength along axial direction are comparableat middle frequencies, the presence of the stiffeners has a strong influence on the vibra-tional power flow propagation.

When power flow transmits along the shell�s axial direction in the stiffened shell, theappearances of peaks and troughs correspond to the positions of the stiffeners.

For all frequencies X and circumferential mode orders n, P 0SðxÞ P 0

MðxÞ, which means thepower transmitted by the transverse shear force equals that transmitted by the bendingmoments.

For circumferential mode order n = 0, the power flow in the shell wall is predominantlycarried by the motion in the axial direction. For circumferential mode order n = 1, thepower in the shell is mainly carried by the motion in the axial and circumferentialdirections.

Structural damping will have a great influence on the power flow transmitted by theshell. With the increasing of the structural damping, the power flow transmitted by theshell gradually attenuates along the axial direction while the ratios of the different internalforces (moments) of the shell will be greatly changed.

Acknowledgements

The authors are grateful for the financial assistance provided by the National NaturalScience Foundation of China (Contract No. 50375059).

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[1

[1

[1

[1

[1

[1[1

[1

[1

[2

[2

[2

[2

[2[2[2



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