Research ArticleStructural and Acoustic Responses ofa Submerged Stiffened Conical Shell

Meixia Chen Cong Zhang Xiangfan Tao and Naiqi Deng

School of Naval Architecture and Ocean Engineering Huazhong University of Science and Technology Wuhan Hubei 430074 China

Correspondence should be addressed to Cong Zhang zc celia163com

Received 5 February 2013 Accepted 3 December 2013 Published 4 March 2014

Academic Editor Anindya Ghoshal

Copyright copy 2014 Meixia Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies the vibrational behavior and far-field sound radiation of a submerged stiffened conical shell at low frequenciesThe solution for the dynamic response of the conical shell is presented in the form of a power series A smeared approach is used tomodel the ring stiffeners Fluid loading is taken into account by dividing the conical shell into narrow strips which are consideredto be local cylindrical shells The far-field sound pressure is solved by the Element Radiation Superposition Method Excitationsin two directions are considered to simulate the loading on the surface of the conical shell These excitations are applied alongthe generator and normal to the surface of the conical shell The contributions from the individual circumferential modes on thestructural responses of the conical shell are studiedThe effects of the external fluid loading and stiffeners are discussedThe resultsfrom the analytical models are validated by numerical results from a fully coupled finite elementboundary element model

1 Introduction

The reduction of noise emitted by maritime platforms is animportant topic in naval research A truncated cone is usedto support the aft-shaft-bearing of the submarine propellerThe structural and acoustic responses of a conical shell arenot as widely reported in literature as in the case of cylindricalshells This is due to the increased mathematical complexityassociated with the effect of the variation of the radius alongthe length of the cone on the elastic wavesMuch of the earlierwork on the equations of motion for conical shells and formsof solution with different boundary conditions has beensummarized by Leissa [1] The approximate locations of thenatural frequencies have been found using the Rayleigh-Ritzmethod by several authors [2 3] Tong presented a procedurefor the free vibration analysis of isotropic and orthotropicconical shells in the form of a power series [4] Guo studiedthe propagation and radiation properties of elastic waves ina cone in vacuo [5] For fluid loaded conical shells Guoinvestigated fluid loading effects of waves on conical shells inwhich a perturbation technique is applied by introducing asmall vertex angle parameter [6] Caresta and Kessissoglouconsidered the fluid loading by dividing the conical shell intonarrow strips and each of these strips is treated as a locally

cylindrical shell [7] Donnell-Mushtari and Flugge equationsof motion were compared by Caresta and Kessissoglou [8] Itwas shown that in order to use the Flugge equations ofmotionwith the power series solution an approximation of the shearforce is required

Many researchers have investigated approximate meth-ods to solve the acoustic radiation from vibrating elasticstructures The boundary integral equation formulation isa popular method A good review on the evolution ofboundary element techniques was given by Chien et al[9] Coupled finite elementboundary element methods havebeen presented by Jeans and Mathews [10] Caresta andKessissoglou used the Helmholtz integral to calculate thefar-field radiated sound pressure of a submerged cylindri-cal shell closed by two truncated conical shells [11] Sincethe boundary element method (BEM) requires significantcomputational time and memory many researchers havebeen working on improving the efficiency of BEM [12] orexploring other approximate methods [13 14] Wang et alproposed an approximate method named Element RadiationSuperposition Method (ERSM) to predict the radiated pres-sure based on the surface velocity [15] The total radiationpressure equals the superposition of the radiated pressurefrom each surface element which can be obtained through

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 954253 14 pageshttpdxdoiorg1011552014954253

2 Shock and Vibration

120572 R1R(x) R0

wcuc

xR2

c

120579

L

Figure 1 Coordinate system and displacements for a thin conical shell

weighting the vibration velocity using the transfer quantityThe proposed method was found to be superior in terms ofcomputational efficiency ERSM can be implemented to solvevarious models by adopting regular baffles such as sphericalinfinite cylindrical and spherical baffles

In this work the structural and acoustic responses ofa stiffened conical shell submerged in a heavy fluid arepresented Section 2 describes the solution for the dynamicresponse of a conical shell which is presented in the formof a power series Fluid loading is taken into account bydividing the shell into narrow strips similar to the procedurespresented by Caresta and Kessissoglou [7] The stiffening ofthe structure is considered by smearing the stiffeners overthe conical shell surface which results in orthotropicmaterialbehavior of the conical shell In Section 3 the ElementRadiation Superposition Method (ERSM) is introduced tosolve the far-field pressure of a conical shell based on thetheory presented in [15]The total acoustic pressure of conicalshell is assumed to be the superposition of the radiatedpressure of all divided elements which is approximatedusing the analytical solution of the radiated pressure from apiston with a cylindrical baffle In Section 4 the validity andconvergence of the method to solve the structural responsesand the far-field acoustic pressure of a stiffened conicalshell are discussed The contributions from the individualcircumferential modes for different load cases are examinedThe effects of the external fluid loading and the ring stiffenersare described

2 Dynamic Model of a SubmergedStiffened Conical Shell

21 Dynamic Response of a Conical Shell For a conical shellthe coordinate system (119909 120579) is defined in Figure 1 119906

119888and V119888

are the orthogonal components of the displacement in the119909 and 120579 directions respectively and 119908

119888is the displacement

normal to the shell surface 120572 is the semivertex angle of theconical shell119877

0is themean radius of the shell119877

1is the radius

of the smaller end and 1198772is the radius of the larger end 119877(119909)

is the radius of the cone at location 119909 119871 is the length of the

cone along its generator According to the Flugge theory theequations of motion are given by [1]

120597 (119877 (119909) 119873119909)

120597119909+

120597119873120579119909

120597120579minus 119873120579sin120572 minus 119877 (119909) 120588ℎ

1205972

119906119888

1205971199052= 0

120597119873120579

120597120579+

120597 (119877 (119909) 119873119909120579

)

120597119909+ 119873120579119909sin120572 + 119876

120579cos120572

minus 119877 (119909) 120588ℎ1205972V119888

1205971199052= 0

minus119873120579cos120572 +

120597 (119877 (119909) 119876119909)

120597119909+

120597119876120579

120597120579minus 119877 (119909) 120588ℎ

1205972119908119888

1205971199052= 0

(1)

where 120588 and ℎ are the density and the thickness of the conicalshell respectively 119873

119909 119873120579 119873120579119909 119873119909120579 119876119909 and 119876

120579are the

forces for a conical shell and are given in the Appendix For aconical shell with stiffeners the effect of the stiffeners can beconsidered by averaging their properties over the surface ofthe conicalshellThe smeared approach can be used when theshells are reinforced by closely spaced uniform stiffeners [16]Figure 2 shows a schematic diagram of the ring stiffeners 119864

119903

is the modulus of elasticity of the stiffeners 119887119903and ℎ

119903are

the thickness and height of the ring stiffeners respectively119860 = 119887

119903ℎ119903is the cross-sectional area of the ring stiffeners 119890

is the distance from the centroid of the stiffener to the shell119868 is the moment of inertia of the stiffeners about line of thereference (themiddle line of the shell) 119889 and 119889

119871= 119889cos120572 are

the distances between the ring stiffeners along the axial andgenerator direction of the conical shell respectivelyUsing thesmeared approach the thickness of the conical shell ℎ in (21)is replaced by ℎ where ℎ = ℎ + 119860119889

119871 119873120579and 119872

120579are replaced

by 1198731015840120579and 1198721015840

120579 where [17]

1198731015840

120579= 119873120579

+ 120583120576120579

+ 120594119896120579

1198721015840

120579= 119872120579

+ 120594120576120579

+ 120578119896120579

(2)

and 120576120579is the strain in the shell and 119896

120579is the variation of

the curvature Consider 120583 = 119864119903119860119889119871 = 119864119903119860119890119889119871 and 120578 =

119864119903119868119889119871 The validity of the smeared approach is discussed in

Section 41

Shock and Vibration 3

d

hr

dL

br

e h

Figure 2 The parameters of the stiffeners

When the fluid loading on the surface of the stiffenedconical shell is considered the equations of motion can beexpressed as

11987111

119906119888

+ 11987112V119888

+ 11987113

119908119888

minus 120588ℎ1205972119906119888

1205971199052= 0

11987121

119906119888

+ 11987122V119888

+ 11987123

119908119888

minus 120588ℎ1205972V119888

1205971199052= 0

11987131

119906119888

+ 11987132V119888

+ 11987133

119908119888

minus 120588ℎ1205972119908119888

1205971199052+ 119901119886

= 0

(3)

With the power series solution the displacements of theconical shell are given by [7]

119906119888

(119909 120579 119905) = 119906119888

(119909) cos (119899120579) 119890minus119895120596119905

=

infin

sum119899=0

u119888119899

sdot x119899cos (119899120579) 119890

minus119895120596119905

V119888

(119909 120579 119905) = V119888

(119909) sin (119899120579) 119890minus119895120596119905

=

infin

sum119899=0

v119888119899

sdot x119899sin (119899120579) 119890

minus119895120596119905

119908119888

(119909 120579 119905) = 119908119888

(119909) cos (119899120579) 119890minus119895120596119905

=

infin

sum119899=0

w119888119899

sdot x119899cos (119899120579) 119890

minus119895120596119905

(4)

where the vectors u119888119899 k119888119899 and w

119888119899are

u119888119899

= [1199061198881

(119909) sdot sdot sdot 1199061198888

(119909)]

k119888119899

= [V1198881

(119909) sdot sdot sdot V1198888

(119909)]

w119888119899

= [1199081198881

(119909) sdot sdot sdot 1199081198888

(119909)]

(5)

x119899is the vector of the eight unknown coefficients

x119899

= [1198860

1198861

1198870

1198871

1198880

1198881

1198882

1198883]119879

(6)

The fluid loading effects on the conical shell are taken intoaccount using an approximatemethod by dividing the conicalshell into 119873 narrow strips and each of these strips is treatedas a cylindrical shell The low frequency range is considered

here where the variation of the real wave number that is usedto calculate the fluid loading parameter is approximately alinear function of119877

0[7] A qualitative analysis on the number

of cylindrical subdivisions required to represent the conicalshell is given in [7] A tolerance 119879 is introduced due to theassumption that the wave number does not change along thelength of the cone [7]Thus the length of each segment of thecone is determined by 119871

119894le 21198770119894

119879(sin120572) 119879 depends onthe geometry of the model and the frequency range whichis analyzed in Section 41 The 119873 parts are connected by 8continuity equations at each interface

The external pressure 119901119886on every part of the conical shell

is

119901119886

=1199011015840119886

cos120572 (7)

where 1199011015840119886is the external pressure loading on the cylindrical

shell which can be approximated using an infinite model asshown in [18] and is given by

1199011015840

119886=

120588ℎ1198882119871

1198772

0

119865119871119908119888

119865119871

=

minusΩ2

1198770

ℎ

120588119891

120588

119867119899

(120573119891

)

120573119891

1198671015840119899

(120573119891

) 119896119891

gt 119896119899

minusΩ21198770

ℎ

120588119891

120588

119870119899

(120573119891

)

120573119891

1198701015840119899

(120573119891

) 119896119891

lt 119896119899

120573119891

=

1198770radic1198962119891

minus 1198962119899 119896119891

gt 119896119899

1198770radic1198962119899

minus 1198962119891

119896119891

lt 119896119899

(8)

where Ω = 1205961198770119888119871is the nondimensional ring frequency

119896119891

= 120596119888119891is the fluid wave number and 120588

119891 119888119891are the density

and speed of sound in the fluid respectively 119896119899is the hull

axial wave number and 120596 is the radian frequency 1198770is the

mean radius of the conical shell segment 119867119899is the Hankel

function of order 119899 119870119899is the modified Bessel function of

order 119899 1198671015840119899and 1198701015840

119899are their derivatives with respect to the

argument

22 Boundary Conditions and Excitation The dynamicresponse of the conical shell for each circumferentialmode number 119899 is expressed in terms of x

119899=

[1198860

1198861

1198870

1198871

1198880

1198881

1198882

1198883]119879 for each segment of the

conical shell which can be solved by continuity conditionsbetween the segments and boundary conditions at the endsThe various boundary conditions for the conical shell are asfollows [8]Free end

119873119909

= 0 (9)

119873119909120579

+119872119909120579

1198770

= 0 (10)

119881119909

= 0 (11)

119872119909

= 0 (12)

4 Shock and Vibration

F2 F1

x

Figure 3 Excitation along the generator and normal to the surfaceat the smaller end of the conical shell

clamped end

119906119888

= 0 V119888

= 0 119908119888

= 0120597119908119888

120597119909= 0 (13)

shear-diaphragm (SD) end

119909

= 0 V119888

= 0 119908119888

= 0 119872119909

= 0 (14)

where for the SD boundary condition the change fromthe conical coordinate system defined in Figure 1 to thecylindrical coordinate system is considered

119909and 119908

119888are

expressed as

119909

= 119873119909cos120572 minus 119881

119909sin120572

119908119888

= 119906119888sin120572 + 119908

119888cos120572

(15)

The forces and moments of a conical shell are given in theAppendix The four boundary conditions at each end ofthe conical shell with the eight continuity equations at eachjunction of two conical segments can be arranged in matrixform AX = 0 where X = [x

1 x2 x

119899]

Two excitation cases as shown in Figure 3 are considered1198651is the excitation along the 119909 direction For the free

end boundary condition at the end of the conical shellthe external force results in a modification of the forceequilibrium at the end given by (9) which now becomes [11]

119873119909

=1

1198771

1198650120575 (119909 minus 119909

0) 120575 (120579 minus 120579

0) 119890minus119895120596119905

(16)

Excluding the harmonic time dependency multiplying (16)by cos(119899120579) and integrating from minus120587 to 120587 yielded

119873119909

= 1205761198650cos (119899120579) (17)

where 120576 = 12120587119886 if 119899 = 0 and 120576 = 1120587119886 if 119899 = 0 Similarly 119888

is the excitation normal to the shell surface In this case (11)becomes

119881119909

= 1205761198650cos (119899120579) (18)

Point excitation in any direction can be expressed by thesetwo excitations The FRF can be calculated for the case ofa point force located at any junction The axial and radial

displacements are obtained from the components of thedisplacements

1199060

= 119906119888cos120572 minus 119908

119888sin120572

1199080

= 119908119888cos120572 + 119906

119888sin120572

(19)

3 Far-Field Sound Pressure

In this section the theory for the boundary element methodis used to demonstrate that the far-field acoustic pressure ofthe shell can be solved by superposing the pressure of theindividual elements An analytical solution of the acousticpressure radiated by a cylindrical piston in a cylindrical baffleis used to obtain the pressure of conical elements

31 Acoustic Transfer Vector According to the Helmholtzboundary integral equation the acoustic pressure radiated bythe arbitrary surface can be expressed as [19]

119862 (r) 119901 (r)

= int1198780

(119901 (r119904)

120597

120597119899119866 (r r

119904) + 119895120588

119891120596119866 (r r

119904) V (r119904)) 119889119878

(20)

where r119904is the position vector on the vibrating surface and r

is the position of the observation field point 119901(r119904) and V(r

119904)

are the surface pressure and normal velocity respectively119866(r r119904) = 119890minus1198941198961198911198774120587119877 is free-space Greenrsquos function and 119877 =

|r minus r119904|

By discretizing the surface into 119872 elements (20) can beexpressed as

119862 (r) 119901 (r) =

119872

sum119894=1

int119878119894

(119901 (r119904)

120597

120597119899119866 (r r

119904) + 119895120588120596119866 (r r

119904) V) 119889119878

(21)

Equation (21) can be rearranged as [15]

119901 (r) = [C]119879

119872times1[p119904] + [D]

119879

119872times1[v] (22)

where C and D are the vectors independent of the surfacevelocityThe relationship between the surface pressure vectorp119904and the normal velocity k can be deduced as

[119860]119872times119872

p119904 = [119861]

119872times119872v (23)

where the matrices 119860 119861 are independent of the surfacevelocity but depend on the shape of the structure andfrequency Equation (22) can be expressed as

119901 (r) = ([C]119879

119872times1[119860]minus1

119872times119872[119861]119872times119872

+ [D]119879

119872times1) [v]

= [G]119879

119872times1[v]

(24)

The vector G is called Acoustic Transfer Vector (ATV)which is an array of transfer functions between the surfacenormal velocities and the pressure at the field pointTheATVdepends on frequency geometry acoustical parameters of thefluid and position of the observation point

Shock and Vibration 5

z998400

21205720

2L0

L1

y998400

x9984001205721

(a)

z998400

y998400

x998400

120572

o998400

(b)

Figure 4 The cylindrical baffle and piston

80

75

70

65

60

55

50

45

400 50 100 150 200 250 300

Frequency (Hz)

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

AnalyticalFEA

Figure 5 Range of validity for the stiffeners smeared over theconical shell

32 Element Radiation Superposition Method The conicalshell is divided into 119872 elements [G]

119879

119872times1= [1198921 1198922 119892

119872]

Each element of the ATV 119892119894 is equal to the acoustic pressure

radiated by the corresponding surface element vibrating withunity surface velocity while the other elements are at rest Forthe case where only one element vibrates with the velocity V

119894

the radiated pressure of the field point at r is given by

119901119894(r) = [119892

1 1198922 119892

119872] [0 0 V

119894 0]

119879

(25)

Therefore given the velocities [V1 V2 V

119894 V

119872]119879 the

radiated pressure is

119901 (r) = 1199011

(r) + 1199012

(r) + sdot sdot sdot + 119901119872

(r) (26)

Hence the total radiation pressure is the superposition of theacoustic pressures radiated by the surface elements vibratingwith a rigid baffle Since the velocities [V

1 V2 V

119894 V

119872]

can be obtained from the solution of dynamic response ofthe conical shell the main task is to find the acoustic transfervalues [119892

1 1198922 119892

119872]

According to Wang [15] the ATV of the element on acylindrical shell is equal to the acoustic pressure radiatedby a baffled piston vibrating with unity velocity The far-field acoustic pressure radiated by a cylindrical piston in acylindrical baffle can be obtained by solving the Helmholtzdifferential equation in closed form which is given by [20]

119901 (119877 120579 120601) =212058811988812057201198710

1205872119877 sin 1205791198950

(1198961198710cos 120579) 119890

119895119896(119877minus1198711 cos 120579)

times

infin

sum119899=minusinfin

(minus119895)119899

1198950

(1198991205720)

119867(1)

119899

1015840

(119896119886 sin 120579)119890119895119899(120601minus1205721) sdot V

(27)

where 21198710is the axial length of the piston 2120572

0is the radial

angle of the piston 1198711is the axial distance between the center

of the piston and the origin and 1205721is the azimuth angle of

the center of the piston All of these parameters are shownin Figure 4(a) V is the velocity of the baffled piston For avelocity equal to unity the ATV of the each cylindrical shellelement 119892

119894can be obtained from 119892

119894= 119901119894(119877 120579 120601)

The radiated pressure from each element of the conicalshell can be approximated by the analytical solution of theradiated pressure of a piston in a cylindrical baffle as shown

6 Shock and Vibration

90

80

70

60

50

40

30

200 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(a)

90

80

70

60

50

400 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(b)

Figure 6 Comparison with FEA results when the space or size of stiffeners changes (a) Results when the space between stiffeners 119889 = 042(b) results when cross-section of the stiffeners is 0028m times 004m

90

80

70

60

50

40

30

20

10

00 20 40 60 80

10minus12

m)

32

19

10

19

10

(T = 3)

(T = 5)

(T = 10)

Figure 7 Comparison for different divisions for the conical shell

in Figure 4(b) The radius of this cylindrical baffle is equalto the mean radius of the conical strip which includes thiselement The ATV of the piston in the cylindrical baffle issolved using its local coordinate The angle between the axisof the conical shell and the axis of the cylindrical baffle is thesemivertex angle of the conical shell In this way the ATVelements of the conical shell [119892

1 1198922 119892

119872] can be obtained

and the total radiated pressure of the conical shell is solved bysumming up the products of the individual transfer quantitiesand the corresponding velocities

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

BEM5 parts per conical segment

3 parts per conical segment1 part per conical segment

Figure 8 Validity and convergence of solving far-field pressure

4 Results for Structural and AcousticResponses for a Conical Shell

The structural and acoustic responses for the conical shellunder harmonic excitation are presented using the followingparameters 119871 = 89m 119877

1= 05m 119877

2= 325m and

ℎ = 0014m The stiffeners have a rectangular cross-sectionof 0014m times 004m (119887

119903= 0014m ℎ

119903= 004m) and are

evenly spaced by 119889 = 021mThematerial properties for steelare density 120588 = 7800 kgmminus3 Youngrsquos modulus 119864 = 21 times

1011Nmminus2 and Poissonrsquos ratio 120583 = 03 Structural damping

Shock and Vibration 7

90

80

70

60

50

40

300 20 40 60 80

10minus12

m)

2

2

22

(a)

90

80

70

60

500 20 40 60 80

10minus12

m)

3

1 2 2

2

(b)

80

60

40

20

0

minus20

minus400 20 40 60 80

10minus6

(c)

Figure 9 Structural and acoustic responses due to 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

is introduced using complex Youngrsquos modulus 119864(1 minus 119895120578119904)

where 120578119904

= 002 is the structural loss factor The surroundingfluid has a density of 120588

119891= 1000 kgmminus3 and speed of sound

119888119891

= 1500msminus1 The structural responses are predicted ata point equal to the drive point of the excitation as shownin Figure 3 The distance between the far-field point and theaxis of the conical shell is 100mThe circumferential angle ofthe observation point is 1205872 and its axial position is equal tothe axial position of the drive pointThe boundary conditionsconsidered here are free at the smaller end of the conical shelland clamped at the larger end

A finite element model is developed in ANSYS to validatethe results obtained from analytical model presented hereThe conical shell model and the ring stiffeners are built usingShell 63 elements and Beam 188 elements respectively Fluid

30 and Fluid 130 elements are used to simulate the externalfluid and the absorbing boundaries around the fluid domainANSYS is used to obtain the structural responses SYSNOISEis then used to calculate the far-field pressure using the directboundary element method

41 Validity and Convergence The validity of smearedapproach for the stiffened conical shell and the convergenceof dividing the conical shell are discussed in what follows Amodel of the stiffened conical shell in vacuo is built to validatethe smeared approach It is observed in Figure 5 that theanalytical results for the structural responses due to 119865

1match

the finite element analysis (FEA) results in the low frequencyrange However the accuracy decreases rapidly when thefrequency is above 180Hz where the structural wavelength

8 Shock and Vibration

100

95

90

85

80

75

70

65

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

2

2

2

3

(a)

110

105

100

95

90

85

80

75

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

3

1

2

2

2

(b)

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

FEBEAnalytical

(c)

Figure 10 Structural and acoustic responses due to 1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

is of the same order as the distance between the stiffenersIt is further observed that the increase of the spacing andwidth of the stiffeners affects the accuracy of this method asshown in Figures 6(a) and 6(b) respectively To conclude thesmeared approach is only suitable when the conical shells arereinforced by closely spaced stiffeners that is 119889119871 is smalland sufficiently small stiffeners that is 119887

119903119871 is small in the

low frequency range where the wavelength is larger than thestiffener spacing

As discussed in Section 2 the conical shell is dividedinto 119873 segments and the length of each segment is 119871

119894le

21198770119894

119879(sin120572) For the model in this paper 119879 = 5 Theresult for the structural responses due to 119865

1for 119879 = 5 is

compared with the results for 119879 = 3 and 119879 = 10 inFigure 7 It can be seen that the result converged for 119879 = 5

Two additional cases of dividing the conical shell into partswith equal lengths are also considered in Figure 7 It showsthat segments with fixed 119871

1198941198770119894yield faster convergence than

fixed length segments The FEA result is also presented inFigure 7 to validate the converged analytical result

The validity and convergence of the Element Radia-tion Superposition Method (ERSM) for solving the far-fieldacoustic pressure of a conical shell are discussed in whatfollows Each conical segment with the length of 119871

119894le

21198770119894

119879(sin120572) is divided into 1 3 and 5 parts in its axialdirection corresponding to 19 37 and 95 parts respectivelyfor 119879 = 5 and 19 conical segments The total number ofelements for the conical shell is 1140 2220 and 5700 respec-tively for a circumferential angle of 6∘ for each element Eachelement can be regarded as a piston in a cylindrical baffle

Shock and Vibration 9

100

90

80

70

60

50

40

30

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

0 20 40 60 80 100

Frequency (Hz)

(a)

120

100

80

60

40

20

00 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

80

60

40

20

0

minus20

minus40

minus60

minus800 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

In waterIn air

(c)

Figure 11 Comparison of conical shells in water and in air under 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

The results in Figure 8 show that the acoustic pressure dueto 1198652converges when there are 3 parts per conical segment

The BEM result is also shown in Figure 8 As discussed inSection 3 the pressure is obtained from the velocity andATV In this part only for both methods the velocities areanalytical solutions of the structural responsesThedifferencebetween ERSM and BEM lies in the solution of the ATVTheATV for ERSM is obtained from the analytical solution ofthe Helmholtz differential equation for a piston baffled in acylindrical shell which is used to approximate a piston baffledin a conical shell while the ATV for BEM is obtained bythe numerical solution of the Helmholtz boundary integralformulationThe results in Figure 8 indicate that the locationsof the peaks are well matched but the errors of amplitudesof the peaks are less than 5 at lower frequencies and lessthan 10 at higher frequencies However ERSM has a muchquicker solution thanBEM due to its avoidance of solving theHelmholtz boundary integral formulation Therefore ERSM

is an effectivemethod to solve the far-field pressure of conicalshell

42 Effect of Excitation on a Conical Shell The structural andacoustic responses of the conical shell due to the excitationshown in Figure 3 are presented in Figures 9 and 10 Thecomparison between the analytical results and the FEAresults indicates that the axial and radial displacements usingthese two methods agree well except for some errors athigher frequencies These errors arise due to approximatesolutions for the stiffeners and the external fluid loadingThestiffeners are considered by averaging their properties overthe surface of the conical shell The fluid loading is takeninto account by dividing the conical shell into narrow stripswhich are treated as cylindrical shells and the external fluidloading is approximated using an infinite modelThe far-fieldacoustic pressures obtained using the analytical and FEBEmethods are also in good agreement Errors are caused by

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

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2 Shock and Vibration

120572 R1R(x) R0

wcuc

xR2

c

120579

L

Figure 1 Coordinate system and displacements for a thin conical shell

weighting the vibration velocity using the transfer quantityThe proposed method was found to be superior in terms ofcomputational efficiency ERSM can be implemented to solvevarious models by adopting regular baffles such as sphericalinfinite cylindrical and spherical baffles

In this work the structural and acoustic responses ofa stiffened conical shell submerged in a heavy fluid arepresented Section 2 describes the solution for the dynamicresponse of a conical shell which is presented in the formof a power series Fluid loading is taken into account bydividing the shell into narrow strips similar to the procedurespresented by Caresta and Kessissoglou [7] The stiffening ofthe structure is considered by smearing the stiffeners overthe conical shell surface which results in orthotropicmaterialbehavior of the conical shell In Section 3 the ElementRadiation Superposition Method (ERSM) is introduced tosolve the far-field pressure of a conical shell based on thetheory presented in [15]The total acoustic pressure of conicalshell is assumed to be the superposition of the radiatedpressure of all divided elements which is approximatedusing the analytical solution of the radiated pressure from apiston with a cylindrical baffle In Section 4 the validity andconvergence of the method to solve the structural responsesand the far-field acoustic pressure of a stiffened conicalshell are discussed The contributions from the individualcircumferential modes for different load cases are examinedThe effects of the external fluid loading and the ring stiffenersare described

2 Dynamic Model of a SubmergedStiffened Conical Shell

21 Dynamic Response of a Conical Shell For a conical shellthe coordinate system (119909 120579) is defined in Figure 1 119906

119888and V119888

are the orthogonal components of the displacement in the119909 and 120579 directions respectively and 119908

119888is the displacement

normal to the shell surface 120572 is the semivertex angle of theconical shell119877

0is themean radius of the shell119877

1is the radius

of the smaller end and 1198772is the radius of the larger end 119877(119909)

is the radius of the cone at location 119909 119871 is the length of the

cone along its generator According to the Flugge theory theequations of motion are given by [1]

120597 (119877 (119909) 119873119909)

120597119909+

120597119873120579119909

120597120579minus 119873120579sin120572 minus 119877 (119909) 120588ℎ

1205972

119906119888

1205971199052= 0

120597119873120579

120597120579+

120597 (119877 (119909) 119873119909120579

)

120597119909+ 119873120579119909sin120572 + 119876

120579cos120572

minus 119877 (119909) 120588ℎ1205972V119888

1205971199052= 0

minus119873120579cos120572 +

120597 (119877 (119909) 119876119909)

120597119909+

120597119876120579

120597120579minus 119877 (119909) 120588ℎ

1205972119908119888

1205971199052= 0

(1)

where 120588 and ℎ are the density and the thickness of the conicalshell respectively 119873

119909 119873120579 119873120579119909 119873119909120579 119876119909 and 119876

120579are the

forces for a conical shell and are given in the Appendix For aconical shell with stiffeners the effect of the stiffeners can beconsidered by averaging their properties over the surface ofthe conicalshellThe smeared approach can be used when theshells are reinforced by closely spaced uniform stiffeners [16]Figure 2 shows a schematic diagram of the ring stiffeners 119864

119903

is the modulus of elasticity of the stiffeners 119887119903and ℎ

119903are

the thickness and height of the ring stiffeners respectively119860 = 119887

119903ℎ119903is the cross-sectional area of the ring stiffeners 119890

is the distance from the centroid of the stiffener to the shell119868 is the moment of inertia of the stiffeners about line of thereference (themiddle line of the shell) 119889 and 119889

119871= 119889cos120572 are

the distances between the ring stiffeners along the axial andgenerator direction of the conical shell respectivelyUsing thesmeared approach the thickness of the conical shell ℎ in (21)is replaced by ℎ where ℎ = ℎ + 119860119889

119871 119873120579and 119872

120579are replaced

by 1198731015840120579and 1198721015840

120579 where [17]

1198731015840

120579= 119873120579

+ 120583120576120579

+ 120594119896120579

1198721015840

120579= 119872120579

+ 120594120576120579

+ 120578119896120579

(2)

and 120576120579is the strain in the shell and 119896

120579is the variation of

the curvature Consider 120583 = 119864119903119860119889119871 = 119864119903119860119890119889119871 and 120578 =

119864119903119868119889119871 The validity of the smeared approach is discussed in

Section 41

Shock and Vibration 3

d

hr

dL

br

e h

Figure 2 The parameters of the stiffeners

When the fluid loading on the surface of the stiffenedconical shell is considered the equations of motion can beexpressed as

11987111

119906119888

+ 11987112V119888

+ 11987113

119908119888

minus 120588ℎ1205972119906119888

1205971199052= 0

11987121

119906119888

+ 11987122V119888

+ 11987123

119908119888

minus 120588ℎ1205972V119888

1205971199052= 0

11987131

119906119888

+ 11987132V119888

+ 11987133

119908119888

minus 120588ℎ1205972119908119888

1205971199052+ 119901119886

= 0

(3)

With the power series solution the displacements of theconical shell are given by [7]

119906119888

(119909 120579 119905) = 119906119888

(119909) cos (119899120579) 119890minus119895120596119905

=

infin

sum119899=0

u119888119899

sdot x119899cos (119899120579) 119890

minus119895120596119905

V119888

(119909 120579 119905) = V119888

(119909) sin (119899120579) 119890minus119895120596119905

=

infin

sum119899=0

v119888119899

sdot x119899sin (119899120579) 119890

minus119895120596119905

119908119888

(119909 120579 119905) = 119908119888

(119909) cos (119899120579) 119890minus119895120596119905

=

infin

sum119899=0

w119888119899

sdot x119899cos (119899120579) 119890

minus119895120596119905

(4)

where the vectors u119888119899 k119888119899 and w

119888119899are

u119888119899

= [1199061198881

(119909) sdot sdot sdot 1199061198888

(119909)]

k119888119899

= [V1198881

(119909) sdot sdot sdot V1198888

(119909)]

w119888119899

= [1199081198881

(119909) sdot sdot sdot 1199081198888

(119909)]

(5)

x119899is the vector of the eight unknown coefficients

x119899

= [1198860

1198861

1198870

1198871

1198880

1198881

1198882

1198883]119879

(6)

The fluid loading effects on the conical shell are taken intoaccount using an approximatemethod by dividing the conicalshell into 119873 narrow strips and each of these strips is treatedas a cylindrical shell The low frequency range is considered

here where the variation of the real wave number that is usedto calculate the fluid loading parameter is approximately alinear function of119877

0[7] A qualitative analysis on the number

of cylindrical subdivisions required to represent the conicalshell is given in [7] A tolerance 119879 is introduced due to theassumption that the wave number does not change along thelength of the cone [7]Thus the length of each segment of thecone is determined by 119871

119894le 21198770119894

119879(sin120572) 119879 depends onthe geometry of the model and the frequency range whichis analyzed in Section 41 The 119873 parts are connected by 8continuity equations at each interface

The external pressure 119901119886on every part of the conical shell

is

119901119886

=1199011015840119886

cos120572 (7)

where 1199011015840119886is the external pressure loading on the cylindrical

shell which can be approximated using an infinite model asshown in [18] and is given by

1199011015840

119886=

120588ℎ1198882119871

1198772

0

119865119871119908119888

119865119871

=

minusΩ2

1198770

ℎ

120588119891

120588

119867119899

(120573119891

)

120573119891

1198671015840119899

(120573119891

) 119896119891

gt 119896119899

minusΩ21198770

ℎ

120588119891

120588

119870119899

(120573119891

)

120573119891

1198701015840119899

(120573119891

) 119896119891

lt 119896119899

120573119891

=

1198770radic1198962119891

minus 1198962119899 119896119891

gt 119896119899

1198770radic1198962119899

minus 1198962119891

119896119891

lt 119896119899

(8)

where Ω = 1205961198770119888119871is the nondimensional ring frequency

119896119891

= 120596119888119891is the fluid wave number and 120588

119891 119888119891are the density

and speed of sound in the fluid respectively 119896119899is the hull

axial wave number and 120596 is the radian frequency 1198770is the

mean radius of the conical shell segment 119867119899is the Hankel

function of order 119899 119870119899is the modified Bessel function of

order 119899 1198671015840119899and 1198701015840

119899are their derivatives with respect to the

argument

22 Boundary Conditions and Excitation The dynamicresponse of the conical shell for each circumferentialmode number 119899 is expressed in terms of x

119899=

[1198860

1198861

1198870

1198871

1198880

1198881

1198882

1198883]119879 for each segment of the

conical shell which can be solved by continuity conditionsbetween the segments and boundary conditions at the endsThe various boundary conditions for the conical shell are asfollows [8]Free end

119873119909

= 0 (9)

119873119909120579

+119872119909120579

1198770

= 0 (10)

119881119909

= 0 (11)

119872119909

= 0 (12)

4 Shock and Vibration

F2 F1

x

Figure 3 Excitation along the generator and normal to the surfaceat the smaller end of the conical shell

clamped end

119906119888

= 0 V119888

= 0 119908119888

= 0120597119908119888

120597119909= 0 (13)

shear-diaphragm (SD) end

119909

= 0 V119888

= 0 119908119888

= 0 119872119909

= 0 (14)

where for the SD boundary condition the change fromthe conical coordinate system defined in Figure 1 to thecylindrical coordinate system is considered

119909and 119908

119888are

expressed as

119909

= 119873119909cos120572 minus 119881

119909sin120572

119908119888

= 119906119888sin120572 + 119908

119888cos120572

(15)

The forces and moments of a conical shell are given in theAppendix The four boundary conditions at each end ofthe conical shell with the eight continuity equations at eachjunction of two conical segments can be arranged in matrixform AX = 0 where X = [x

1 x2 x

119899]

Two excitation cases as shown in Figure 3 are considered1198651is the excitation along the 119909 direction For the free

end boundary condition at the end of the conical shellthe external force results in a modification of the forceequilibrium at the end given by (9) which now becomes [11]

119873119909

=1

1198771

1198650120575 (119909 minus 119909

0) 120575 (120579 minus 120579

0) 119890minus119895120596119905

(16)

Excluding the harmonic time dependency multiplying (16)by cos(119899120579) and integrating from minus120587 to 120587 yielded

119873119909

= 1205761198650cos (119899120579) (17)

where 120576 = 12120587119886 if 119899 = 0 and 120576 = 1120587119886 if 119899 = 0 Similarly 119888

is the excitation normal to the shell surface In this case (11)becomes

119881119909

= 1205761198650cos (119899120579) (18)

Point excitation in any direction can be expressed by thesetwo excitations The FRF can be calculated for the case ofa point force located at any junction The axial and radial

displacements are obtained from the components of thedisplacements

1199060

= 119906119888cos120572 minus 119908

119888sin120572

1199080

= 119908119888cos120572 + 119906

119888sin120572

(19)

3 Far-Field Sound Pressure

In this section the theory for the boundary element methodis used to demonstrate that the far-field acoustic pressure ofthe shell can be solved by superposing the pressure of theindividual elements An analytical solution of the acousticpressure radiated by a cylindrical piston in a cylindrical baffleis used to obtain the pressure of conical elements

31 Acoustic Transfer Vector According to the Helmholtzboundary integral equation the acoustic pressure radiated bythe arbitrary surface can be expressed as [19]

119862 (r) 119901 (r)

= int1198780

(119901 (r119904)

120597

120597119899119866 (r r

119904) + 119895120588

119891120596119866 (r r

119904) V (r119904)) 119889119878

(20)

where r119904is the position vector on the vibrating surface and r

is the position of the observation field point 119901(r119904) and V(r

119904)

are the surface pressure and normal velocity respectively119866(r r119904) = 119890minus1198941198961198911198774120587119877 is free-space Greenrsquos function and 119877 =

|r minus r119904|

By discretizing the surface into 119872 elements (20) can beexpressed as

119862 (r) 119901 (r) =

119872

sum119894=1

int119878119894

(119901 (r119904)

120597

120597119899119866 (r r

119904) + 119895120588120596119866 (r r

119904) V) 119889119878

(21)

Equation (21) can be rearranged as [15]

119901 (r) = [C]119879

119872times1[p119904] + [D]

119879

119872times1[v] (22)

where C and D are the vectors independent of the surfacevelocityThe relationship between the surface pressure vectorp119904and the normal velocity k can be deduced as

[119860]119872times119872

p119904 = [119861]

119872times119872v (23)

where the matrices 119860 119861 are independent of the surfacevelocity but depend on the shape of the structure andfrequency Equation (22) can be expressed as

119901 (r) = ([C]119879

119872times1[119860]minus1

119872times119872[119861]119872times119872

+ [D]119879

119872times1) [v]

= [G]119879

119872times1[v]

(24)

The vector G is called Acoustic Transfer Vector (ATV)which is an array of transfer functions between the surfacenormal velocities and the pressure at the field pointTheATVdepends on frequency geometry acoustical parameters of thefluid and position of the observation point

Shock and Vibration 5

z998400

21205720

2L0

L1

y998400

x9984001205721

(a)

z998400

y998400

x998400

120572

o998400

(b)

Figure 4 The cylindrical baffle and piston

80

75

70

65

60

55

50

45

400 50 100 150 200 250 300

Frequency (Hz)

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

AnalyticalFEA

Figure 5 Range of validity for the stiffeners smeared over theconical shell

32 Element Radiation Superposition Method The conicalshell is divided into 119872 elements [G]

119879

119872times1= [1198921 1198922 119892

119872]

Each element of the ATV 119892119894 is equal to the acoustic pressure

radiated by the corresponding surface element vibrating withunity surface velocity while the other elements are at rest Forthe case where only one element vibrates with the velocity V

119894

the radiated pressure of the field point at r is given by

119901119894(r) = [119892

1 1198922 119892

119872] [0 0 V

119894 0]

119879

(25)

Therefore given the velocities [V1 V2 V

119894 V

119872]119879 the

radiated pressure is

119901 (r) = 1199011

(r) + 1199012

(r) + sdot sdot sdot + 119901119872

(r) (26)

Hence the total radiation pressure is the superposition of theacoustic pressures radiated by the surface elements vibratingwith a rigid baffle Since the velocities [V

1 V2 V

119894 V

119872]

can be obtained from the solution of dynamic response ofthe conical shell the main task is to find the acoustic transfervalues [119892

1 1198922 119892

119872]

According to Wang [15] the ATV of the element on acylindrical shell is equal to the acoustic pressure radiatedby a baffled piston vibrating with unity velocity The far-field acoustic pressure radiated by a cylindrical piston in acylindrical baffle can be obtained by solving the Helmholtzdifferential equation in closed form which is given by [20]

119901 (119877 120579 120601) =212058811988812057201198710

1205872119877 sin 1205791198950

(1198961198710cos 120579) 119890

119895119896(119877minus1198711 cos 120579)

times

infin

sum119899=minusinfin

(minus119895)119899

1198950

(1198991205720)

119867(1)

119899

1015840

(119896119886 sin 120579)119890119895119899(120601minus1205721) sdot V

(27)

where 21198710is the axial length of the piston 2120572

0is the radial

angle of the piston 1198711is the axial distance between the center

of the piston and the origin and 1205721is the azimuth angle of

the center of the piston All of these parameters are shownin Figure 4(a) V is the velocity of the baffled piston For avelocity equal to unity the ATV of the each cylindrical shellelement 119892

119894can be obtained from 119892

119894= 119901119894(119877 120579 120601)

The radiated pressure from each element of the conicalshell can be approximated by the analytical solution of theradiated pressure of a piston in a cylindrical baffle as shown

6 Shock and Vibration

90

80

70

60

50

40

30

200 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(a)

90

80

70

60

50

400 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(b)

Figure 6 Comparison with FEA results when the space or size of stiffeners changes (a) Results when the space between stiffeners 119889 = 042(b) results when cross-section of the stiffeners is 0028m times 004m

90

80

70

60

50

40

30

20

10

00 20 40 60 80

10minus12

m)

32

19

10

19

10

(T = 3)

(T = 5)

(T = 10)

Figure 7 Comparison for different divisions for the conical shell

in Figure 4(b) The radius of this cylindrical baffle is equalto the mean radius of the conical strip which includes thiselement The ATV of the piston in the cylindrical baffle issolved using its local coordinate The angle between the axisof the conical shell and the axis of the cylindrical baffle is thesemivertex angle of the conical shell In this way the ATVelements of the conical shell [119892

1 1198922 119892

119872] can be obtained

and the total radiated pressure of the conical shell is solved bysumming up the products of the individual transfer quantitiesand the corresponding velocities

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

BEM5 parts per conical segment

3 parts per conical segment1 part per conical segment

Figure 8 Validity and convergence of solving far-field pressure

4 Results for Structural and AcousticResponses for a Conical Shell

The structural and acoustic responses for the conical shellunder harmonic excitation are presented using the followingparameters 119871 = 89m 119877

1= 05m 119877

2= 325m and

ℎ = 0014m The stiffeners have a rectangular cross-sectionof 0014m times 004m (119887

119903= 0014m ℎ

119903= 004m) and are

evenly spaced by 119889 = 021mThematerial properties for steelare density 120588 = 7800 kgmminus3 Youngrsquos modulus 119864 = 21 times

1011Nmminus2 and Poissonrsquos ratio 120583 = 03 Structural damping

Shock and Vibration 7

90

80

70

60

50

40

300 20 40 60 80

10minus12

m)

2

2

22

(a)

90

80

70

60

500 20 40 60 80

10minus12

m)

3

1 2 2

2

(b)

80

60

40

20

0

minus20

minus400 20 40 60 80

10minus6

(c)

Figure 9 Structural and acoustic responses due to 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

is introduced using complex Youngrsquos modulus 119864(1 minus 119895120578119904)

where 120578119904

= 002 is the structural loss factor The surroundingfluid has a density of 120588

119891= 1000 kgmminus3 and speed of sound

119888119891

= 1500msminus1 The structural responses are predicted ata point equal to the drive point of the excitation as shownin Figure 3 The distance between the far-field point and theaxis of the conical shell is 100mThe circumferential angle ofthe observation point is 1205872 and its axial position is equal tothe axial position of the drive pointThe boundary conditionsconsidered here are free at the smaller end of the conical shelland clamped at the larger end

A finite element model is developed in ANSYS to validatethe results obtained from analytical model presented hereThe conical shell model and the ring stiffeners are built usingShell 63 elements and Beam 188 elements respectively Fluid

30 and Fluid 130 elements are used to simulate the externalfluid and the absorbing boundaries around the fluid domainANSYS is used to obtain the structural responses SYSNOISEis then used to calculate the far-field pressure using the directboundary element method

41 Validity and Convergence The validity of smearedapproach for the stiffened conical shell and the convergenceof dividing the conical shell are discussed in what follows Amodel of the stiffened conical shell in vacuo is built to validatethe smeared approach It is observed in Figure 5 that theanalytical results for the structural responses due to 119865

1match

the finite element analysis (FEA) results in the low frequencyrange However the accuracy decreases rapidly when thefrequency is above 180Hz where the structural wavelength

8 Shock and Vibration

100

95

90

85

80

75

70

65

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

2

2

2

3

(a)

110

105

100

95

90

85

80

75

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

3

1

2

2

2

(b)

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

FEBEAnalytical

(c)

Figure 10 Structural and acoustic responses due to 1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

is of the same order as the distance between the stiffenersIt is further observed that the increase of the spacing andwidth of the stiffeners affects the accuracy of this method asshown in Figures 6(a) and 6(b) respectively To conclude thesmeared approach is only suitable when the conical shells arereinforced by closely spaced stiffeners that is 119889119871 is smalland sufficiently small stiffeners that is 119887

119903119871 is small in the

low frequency range where the wavelength is larger than thestiffener spacing

As discussed in Section 2 the conical shell is dividedinto 119873 segments and the length of each segment is 119871

119894le

21198770119894

119879(sin120572) For the model in this paper 119879 = 5 Theresult for the structural responses due to 119865

1for 119879 = 5 is

compared with the results for 119879 = 3 and 119879 = 10 inFigure 7 It can be seen that the result converged for 119879 = 5

Two additional cases of dividing the conical shell into partswith equal lengths are also considered in Figure 7 It showsthat segments with fixed 119871

1198941198770119894yield faster convergence than

fixed length segments The FEA result is also presented inFigure 7 to validate the converged analytical result

The validity and convergence of the Element Radia-tion Superposition Method (ERSM) for solving the far-fieldacoustic pressure of a conical shell are discussed in whatfollows Each conical segment with the length of 119871

119894le

21198770119894

119879(sin120572) is divided into 1 3 and 5 parts in its axialdirection corresponding to 19 37 and 95 parts respectivelyfor 119879 = 5 and 19 conical segments The total number ofelements for the conical shell is 1140 2220 and 5700 respec-tively for a circumferential angle of 6∘ for each element Eachelement can be regarded as a piston in a cylindrical baffle

Shock and Vibration 9

100

90

80

70

60

50

40

30

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

0 20 40 60 80 100

Frequency (Hz)

(a)

120

100

80

60

40

20

00 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

80

60

40

20

0

minus20

minus40

minus60

minus800 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

In waterIn air

(c)

Figure 11 Comparison of conical shells in water and in air under 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

The results in Figure 8 show that the acoustic pressure dueto 1198652converges when there are 3 parts per conical segment

The BEM result is also shown in Figure 8 As discussed inSection 3 the pressure is obtained from the velocity andATV In this part only for both methods the velocities areanalytical solutions of the structural responsesThedifferencebetween ERSM and BEM lies in the solution of the ATVTheATV for ERSM is obtained from the analytical solution ofthe Helmholtz differential equation for a piston baffled in acylindrical shell which is used to approximate a piston baffledin a conical shell while the ATV for BEM is obtained bythe numerical solution of the Helmholtz boundary integralformulationThe results in Figure 8 indicate that the locationsof the peaks are well matched but the errors of amplitudesof the peaks are less than 5 at lower frequencies and lessthan 10 at higher frequencies However ERSM has a muchquicker solution thanBEM due to its avoidance of solving theHelmholtz boundary integral formulation Therefore ERSM

is an effectivemethod to solve the far-field pressure of conicalshell

42 Effect of Excitation on a Conical Shell The structural andacoustic responses of the conical shell due to the excitationshown in Figure 3 are presented in Figures 9 and 10 Thecomparison between the analytical results and the FEAresults indicates that the axial and radial displacements usingthese two methods agree well except for some errors athigher frequencies These errors arise due to approximatesolutions for the stiffeners and the external fluid loadingThestiffeners are considered by averaging their properties overthe surface of the conical shell The fluid loading is takeninto account by dividing the conical shell into narrow stripswhich are treated as cylindrical shells and the external fluidloading is approximated using an infinite modelThe far-fieldacoustic pressures obtained using the analytical and FEBEmethods are also in good agreement Errors are caused by

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

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International Journal of

Shock and Vibration 3

d

hr

dL

br

e h

Figure 2 The parameters of the stiffeners

When the fluid loading on the surface of the stiffenedconical shell is considered the equations of motion can beexpressed as

11987111

119906119888

+ 11987112V119888

+ 11987113

119908119888

minus 120588ℎ1205972119906119888

1205971199052= 0

11987121

119906119888

+ 11987122V119888

+ 11987123

119908119888

minus 120588ℎ1205972V119888

1205971199052= 0

11987131

119906119888

+ 11987132V119888

+ 11987133

119908119888

minus 120588ℎ1205972119908119888

1205971199052+ 119901119886

= 0

(3)

With the power series solution the displacements of theconical shell are given by [7]

119906119888

(119909 120579 119905) = 119906119888

(119909) cos (119899120579) 119890minus119895120596119905

=

infin

sum119899=0

u119888119899

sdot x119899cos (119899120579) 119890

minus119895120596119905

V119888

(119909 120579 119905) = V119888

(119909) sin (119899120579) 119890minus119895120596119905

=

infin

sum119899=0

v119888119899

sdot x119899sin (119899120579) 119890

minus119895120596119905

119908119888

(119909 120579 119905) = 119908119888

(119909) cos (119899120579) 119890minus119895120596119905

=

infin

sum119899=0

w119888119899

sdot x119899cos (119899120579) 119890

minus119895120596119905

(4)

where the vectors u119888119899 k119888119899 and w

119888119899are

u119888119899

= [1199061198881

(119909) sdot sdot sdot 1199061198888

(119909)]

k119888119899

= [V1198881

(119909) sdot sdot sdot V1198888

(119909)]

w119888119899

= [1199081198881

(119909) sdot sdot sdot 1199081198888

(119909)]

(5)

x119899is the vector of the eight unknown coefficients

x119899

= [1198860

1198861

1198870

1198871

1198880

1198881

1198882

1198883]119879

(6)

The fluid loading effects on the conical shell are taken intoaccount using an approximatemethod by dividing the conicalshell into 119873 narrow strips and each of these strips is treatedas a cylindrical shell The low frequency range is considered

here where the variation of the real wave number that is usedto calculate the fluid loading parameter is approximately alinear function of119877

0[7] A qualitative analysis on the number

of cylindrical subdivisions required to represent the conicalshell is given in [7] A tolerance 119879 is introduced due to theassumption that the wave number does not change along thelength of the cone [7]Thus the length of each segment of thecone is determined by 119871

119894le 21198770119894

119879(sin120572) 119879 depends onthe geometry of the model and the frequency range whichis analyzed in Section 41 The 119873 parts are connected by 8continuity equations at each interface

The external pressure 119901119886on every part of the conical shell

is

119901119886

=1199011015840119886

cos120572 (7)

where 1199011015840119886is the external pressure loading on the cylindrical

shell which can be approximated using an infinite model asshown in [18] and is given by

1199011015840

119886=

120588ℎ1198882119871

1198772

0

119865119871119908119888

119865119871

=

minusΩ2

1198770

ℎ

120588119891

120588

119867119899

(120573119891

)

120573119891

1198671015840119899

(120573119891

) 119896119891

gt 119896119899

minusΩ21198770

ℎ

120588119891

120588

119870119899

(120573119891

)

120573119891

1198701015840119899

(120573119891

) 119896119891

lt 119896119899

120573119891

=

1198770radic1198962119891

minus 1198962119899 119896119891

gt 119896119899

1198770radic1198962119899

minus 1198962119891

119896119891

lt 119896119899

(8)

where Ω = 1205961198770119888119871is the nondimensional ring frequency

119896119891

= 120596119888119891is the fluid wave number and 120588

119891 119888119891are the density

and speed of sound in the fluid respectively 119896119899is the hull

axial wave number and 120596 is the radian frequency 1198770is the

mean radius of the conical shell segment 119867119899is the Hankel

function of order 119899 119870119899is the modified Bessel function of

order 119899 1198671015840119899and 1198701015840

119899are their derivatives with respect to the

argument

22 Boundary Conditions and Excitation The dynamicresponse of the conical shell for each circumferentialmode number 119899 is expressed in terms of x

119899=

[1198860

1198861

1198870

1198871

1198880

1198881

1198882

1198883]119879 for each segment of the

conical shell which can be solved by continuity conditionsbetween the segments and boundary conditions at the endsThe various boundary conditions for the conical shell are asfollows [8]Free end

119873119909

= 0 (9)

119873119909120579

+119872119909120579

1198770

= 0 (10)

119881119909

= 0 (11)

119872119909

= 0 (12)

4 Shock and Vibration

F2 F1

x

Figure 3 Excitation along the generator and normal to the surfaceat the smaller end of the conical shell

clamped end

119906119888

= 0 V119888

= 0 119908119888

= 0120597119908119888

120597119909= 0 (13)

shear-diaphragm (SD) end

119909

= 0 V119888

= 0 119908119888

= 0 119872119909

= 0 (14)

where for the SD boundary condition the change fromthe conical coordinate system defined in Figure 1 to thecylindrical coordinate system is considered

119909and 119908

119888are

expressed as

119909

= 119873119909cos120572 minus 119881

119909sin120572

119908119888

= 119906119888sin120572 + 119908

119888cos120572

(15)

The forces and moments of a conical shell are given in theAppendix The four boundary conditions at each end ofthe conical shell with the eight continuity equations at eachjunction of two conical segments can be arranged in matrixform AX = 0 where X = [x

1 x2 x

119899]

Two excitation cases as shown in Figure 3 are considered1198651is the excitation along the 119909 direction For the free

end boundary condition at the end of the conical shellthe external force results in a modification of the forceequilibrium at the end given by (9) which now becomes [11]

119873119909

=1

1198771

1198650120575 (119909 minus 119909

0) 120575 (120579 minus 120579

0) 119890minus119895120596119905

(16)

Excluding the harmonic time dependency multiplying (16)by cos(119899120579) and integrating from minus120587 to 120587 yielded

119873119909

= 1205761198650cos (119899120579) (17)

where 120576 = 12120587119886 if 119899 = 0 and 120576 = 1120587119886 if 119899 = 0 Similarly 119888

is the excitation normal to the shell surface In this case (11)becomes

119881119909

= 1205761198650cos (119899120579) (18)

Point excitation in any direction can be expressed by thesetwo excitations The FRF can be calculated for the case ofa point force located at any junction The axial and radial

displacements are obtained from the components of thedisplacements

1199060

= 119906119888cos120572 minus 119908

119888sin120572

1199080

= 119908119888cos120572 + 119906

119888sin120572

(19)

3 Far-Field Sound Pressure

In this section the theory for the boundary element methodis used to demonstrate that the far-field acoustic pressure ofthe shell can be solved by superposing the pressure of theindividual elements An analytical solution of the acousticpressure radiated by a cylindrical piston in a cylindrical baffleis used to obtain the pressure of conical elements

31 Acoustic Transfer Vector According to the Helmholtzboundary integral equation the acoustic pressure radiated bythe arbitrary surface can be expressed as [19]

119862 (r) 119901 (r)

= int1198780

(119901 (r119904)

120597

120597119899119866 (r r

119904) + 119895120588

119891120596119866 (r r

119904) V (r119904)) 119889119878

(20)

where r119904is the position vector on the vibrating surface and r

is the position of the observation field point 119901(r119904) and V(r

119904)

are the surface pressure and normal velocity respectively119866(r r119904) = 119890minus1198941198961198911198774120587119877 is free-space Greenrsquos function and 119877 =

|r minus r119904|

By discretizing the surface into 119872 elements (20) can beexpressed as

119862 (r) 119901 (r) =

119872

sum119894=1

int119878119894

(119901 (r119904)

120597

120597119899119866 (r r

119904) + 119895120588120596119866 (r r

119904) V) 119889119878

(21)

Equation (21) can be rearranged as [15]

119901 (r) = [C]119879

119872times1[p119904] + [D]

119879

119872times1[v] (22)

where C and D are the vectors independent of the surfacevelocityThe relationship between the surface pressure vectorp119904and the normal velocity k can be deduced as

[119860]119872times119872

p119904 = [119861]

119872times119872v (23)

where the matrices 119860 119861 are independent of the surfacevelocity but depend on the shape of the structure andfrequency Equation (22) can be expressed as

119901 (r) = ([C]119879

119872times1[119860]minus1

119872times119872[119861]119872times119872

+ [D]119879

119872times1) [v]

= [G]119879

119872times1[v]

(24)

The vector G is called Acoustic Transfer Vector (ATV)which is an array of transfer functions between the surfacenormal velocities and the pressure at the field pointTheATVdepends on frequency geometry acoustical parameters of thefluid and position of the observation point

Shock and Vibration 5

z998400

21205720

2L0

L1

y998400

x9984001205721

(a)

z998400

y998400

x998400

120572

o998400

(b)

Figure 4 The cylindrical baffle and piston

80

75

70

65

60

55

50

45

400 50 100 150 200 250 300

Frequency (Hz)

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

AnalyticalFEA

Figure 5 Range of validity for the stiffeners smeared over theconical shell

32 Element Radiation Superposition Method The conicalshell is divided into 119872 elements [G]

119879

119872times1= [1198921 1198922 119892

119872]

Each element of the ATV 119892119894 is equal to the acoustic pressure

radiated by the corresponding surface element vibrating withunity surface velocity while the other elements are at rest Forthe case where only one element vibrates with the velocity V

119894

the radiated pressure of the field point at r is given by

119901119894(r) = [119892

1 1198922 119892

119872] [0 0 V

119894 0]

119879

(25)

Therefore given the velocities [V1 V2 V

119894 V

119872]119879 the

radiated pressure is

119901 (r) = 1199011

(r) + 1199012

(r) + sdot sdot sdot + 119901119872

(r) (26)

Hence the total radiation pressure is the superposition of theacoustic pressures radiated by the surface elements vibratingwith a rigid baffle Since the velocities [V

1 V2 V

119894 V

119872]

can be obtained from the solution of dynamic response ofthe conical shell the main task is to find the acoustic transfervalues [119892

1 1198922 119892

119872]

According to Wang [15] the ATV of the element on acylindrical shell is equal to the acoustic pressure radiatedby a baffled piston vibrating with unity velocity The far-field acoustic pressure radiated by a cylindrical piston in acylindrical baffle can be obtained by solving the Helmholtzdifferential equation in closed form which is given by [20]

119901 (119877 120579 120601) =212058811988812057201198710

1205872119877 sin 1205791198950

(1198961198710cos 120579) 119890

119895119896(119877minus1198711 cos 120579)

times

infin

sum119899=minusinfin

(minus119895)119899

1198950

(1198991205720)

119867(1)

119899

1015840

(119896119886 sin 120579)119890119895119899(120601minus1205721) sdot V

(27)

where 21198710is the axial length of the piston 2120572

0is the radial

angle of the piston 1198711is the axial distance between the center

of the piston and the origin and 1205721is the azimuth angle of

the center of the piston All of these parameters are shownin Figure 4(a) V is the velocity of the baffled piston For avelocity equal to unity the ATV of the each cylindrical shellelement 119892

119894can be obtained from 119892

119894= 119901119894(119877 120579 120601)

The radiated pressure from each element of the conicalshell can be approximated by the analytical solution of theradiated pressure of a piston in a cylindrical baffle as shown

6 Shock and Vibration

90

80

70

60

50

40

30

200 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(a)

90

80

70

60

50

400 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(b)

Figure 6 Comparison with FEA results when the space or size of stiffeners changes (a) Results when the space between stiffeners 119889 = 042(b) results when cross-section of the stiffeners is 0028m times 004m

90

80

70

60

50

40

30

20

10

00 20 40 60 80

10minus12

m)

32

19

10

19

10

(T = 3)

(T = 5)

(T = 10)

Figure 7 Comparison for different divisions for the conical shell

in Figure 4(b) The radius of this cylindrical baffle is equalto the mean radius of the conical strip which includes thiselement The ATV of the piston in the cylindrical baffle issolved using its local coordinate The angle between the axisof the conical shell and the axis of the cylindrical baffle is thesemivertex angle of the conical shell In this way the ATVelements of the conical shell [119892

1 1198922 119892

119872] can be obtained

and the total radiated pressure of the conical shell is solved bysumming up the products of the individual transfer quantitiesand the corresponding velocities

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

BEM5 parts per conical segment

3 parts per conical segment1 part per conical segment

Figure 8 Validity and convergence of solving far-field pressure

4 Results for Structural and AcousticResponses for a Conical Shell

The structural and acoustic responses for the conical shellunder harmonic excitation are presented using the followingparameters 119871 = 89m 119877

1= 05m 119877

2= 325m and

ℎ = 0014m The stiffeners have a rectangular cross-sectionof 0014m times 004m (119887

119903= 0014m ℎ

119903= 004m) and are

evenly spaced by 119889 = 021mThematerial properties for steelare density 120588 = 7800 kgmminus3 Youngrsquos modulus 119864 = 21 times

1011Nmminus2 and Poissonrsquos ratio 120583 = 03 Structural damping

Shock and Vibration 7

90

80

70

60

50

40

300 20 40 60 80

10minus12

m)

2

2

22

(a)

90

80

70

60

500 20 40 60 80

10minus12

m)

3

1 2 2

2

(b)

80

60

40

20

0

minus20

minus400 20 40 60 80

10minus6

(c)

Figure 9 Structural and acoustic responses due to 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

is introduced using complex Youngrsquos modulus 119864(1 minus 119895120578119904)

where 120578119904

= 002 is the structural loss factor The surroundingfluid has a density of 120588

119891= 1000 kgmminus3 and speed of sound

119888119891

= 1500msminus1 The structural responses are predicted ata point equal to the drive point of the excitation as shownin Figure 3 The distance between the far-field point and theaxis of the conical shell is 100mThe circumferential angle ofthe observation point is 1205872 and its axial position is equal tothe axial position of the drive pointThe boundary conditionsconsidered here are free at the smaller end of the conical shelland clamped at the larger end

A finite element model is developed in ANSYS to validatethe results obtained from analytical model presented hereThe conical shell model and the ring stiffeners are built usingShell 63 elements and Beam 188 elements respectively Fluid

30 and Fluid 130 elements are used to simulate the externalfluid and the absorbing boundaries around the fluid domainANSYS is used to obtain the structural responses SYSNOISEis then used to calculate the far-field pressure using the directboundary element method

41 Validity and Convergence The validity of smearedapproach for the stiffened conical shell and the convergenceof dividing the conical shell are discussed in what follows Amodel of the stiffened conical shell in vacuo is built to validatethe smeared approach It is observed in Figure 5 that theanalytical results for the structural responses due to 119865

1match

the finite element analysis (FEA) results in the low frequencyrange However the accuracy decreases rapidly when thefrequency is above 180Hz where the structural wavelength

8 Shock and Vibration

100

95

90

85

80

75

70

65

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

2

2

2

3

(a)

110

105

100

95

90

85

80

75

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

3

1

2

2

2

(b)

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

FEBEAnalytical

(c)

Figure 10 Structural and acoustic responses due to 1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

is of the same order as the distance between the stiffenersIt is further observed that the increase of the spacing andwidth of the stiffeners affects the accuracy of this method asshown in Figures 6(a) and 6(b) respectively To conclude thesmeared approach is only suitable when the conical shells arereinforced by closely spaced stiffeners that is 119889119871 is smalland sufficiently small stiffeners that is 119887

119903119871 is small in the

low frequency range where the wavelength is larger than thestiffener spacing

As discussed in Section 2 the conical shell is dividedinto 119873 segments and the length of each segment is 119871

119894le

21198770119894

119879(sin120572) For the model in this paper 119879 = 5 Theresult for the structural responses due to 119865

1for 119879 = 5 is

compared with the results for 119879 = 3 and 119879 = 10 inFigure 7 It can be seen that the result converged for 119879 = 5

Two additional cases of dividing the conical shell into partswith equal lengths are also considered in Figure 7 It showsthat segments with fixed 119871

1198941198770119894yield faster convergence than

fixed length segments The FEA result is also presented inFigure 7 to validate the converged analytical result

The validity and convergence of the Element Radia-tion Superposition Method (ERSM) for solving the far-fieldacoustic pressure of a conical shell are discussed in whatfollows Each conical segment with the length of 119871

119894le

21198770119894

119879(sin120572) is divided into 1 3 and 5 parts in its axialdirection corresponding to 19 37 and 95 parts respectivelyfor 119879 = 5 and 19 conical segments The total number ofelements for the conical shell is 1140 2220 and 5700 respec-tively for a circumferential angle of 6∘ for each element Eachelement can be regarded as a piston in a cylindrical baffle

Shock and Vibration 9

100

90

80

70

60

50

40

30

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

0 20 40 60 80 100

Frequency (Hz)

(a)

120

100

80

60

40

20

00 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

80

60

40

20

0

minus20

minus40

minus60

minus800 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

In waterIn air

(c)

Figure 11 Comparison of conical shells in water and in air under 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

The results in Figure 8 show that the acoustic pressure dueto 1198652converges when there are 3 parts per conical segment

The BEM result is also shown in Figure 8 As discussed inSection 3 the pressure is obtained from the velocity andATV In this part only for both methods the velocities areanalytical solutions of the structural responsesThedifferencebetween ERSM and BEM lies in the solution of the ATVTheATV for ERSM is obtained from the analytical solution ofthe Helmholtz differential equation for a piston baffled in acylindrical shell which is used to approximate a piston baffledin a conical shell while the ATV for BEM is obtained bythe numerical solution of the Helmholtz boundary integralformulationThe results in Figure 8 indicate that the locationsof the peaks are well matched but the errors of amplitudesof the peaks are less than 5 at lower frequencies and lessthan 10 at higher frequencies However ERSM has a muchquicker solution thanBEM due to its avoidance of solving theHelmholtz boundary integral formulation Therefore ERSM

is an effectivemethod to solve the far-field pressure of conicalshell

42 Effect of Excitation on a Conical Shell The structural andacoustic responses of the conical shell due to the excitationshown in Figure 3 are presented in Figures 9 and 10 Thecomparison between the analytical results and the FEAresults indicates that the axial and radial displacements usingthese two methods agree well except for some errors athigher frequencies These errors arise due to approximatesolutions for the stiffeners and the external fluid loadingThestiffeners are considered by averaging their properties overthe surface of the conical shell The fluid loading is takeninto account by dividing the conical shell into narrow stripswhich are treated as cylindrical shells and the external fluidloading is approximated using an infinite modelThe far-fieldacoustic pressures obtained using the analytical and FEBEmethods are also in good agreement Errors are caused by

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

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Shock and Vibration

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International Journal of

4 Shock and Vibration

F2 F1

x

Figure 3 Excitation along the generator and normal to the surfaceat the smaller end of the conical shell

clamped end

119906119888

= 0 V119888

= 0 119908119888

= 0120597119908119888

120597119909= 0 (13)

shear-diaphragm (SD) end

119909

= 0 V119888

= 0 119908119888

= 0 119872119909

= 0 (14)

where for the SD boundary condition the change fromthe conical coordinate system defined in Figure 1 to thecylindrical coordinate system is considered

119909and 119908

119888are

expressed as

119909

= 119873119909cos120572 minus 119881

119909sin120572

119908119888

= 119906119888sin120572 + 119908

119888cos120572

(15)

The forces and moments of a conical shell are given in theAppendix The four boundary conditions at each end ofthe conical shell with the eight continuity equations at eachjunction of two conical segments can be arranged in matrixform AX = 0 where X = [x

1 x2 x

119899]

Two excitation cases as shown in Figure 3 are considered1198651is the excitation along the 119909 direction For the free

end boundary condition at the end of the conical shellthe external force results in a modification of the forceequilibrium at the end given by (9) which now becomes [11]

119873119909

=1

1198771

1198650120575 (119909 minus 119909

0) 120575 (120579 minus 120579

0) 119890minus119895120596119905

(16)

Excluding the harmonic time dependency multiplying (16)by cos(119899120579) and integrating from minus120587 to 120587 yielded

119873119909

= 1205761198650cos (119899120579) (17)

where 120576 = 12120587119886 if 119899 = 0 and 120576 = 1120587119886 if 119899 = 0 Similarly 119888

is the excitation normal to the shell surface In this case (11)becomes

119881119909

= 1205761198650cos (119899120579) (18)

Point excitation in any direction can be expressed by thesetwo excitations The FRF can be calculated for the case ofa point force located at any junction The axial and radial

displacements are obtained from the components of thedisplacements

1199060

= 119906119888cos120572 minus 119908

119888sin120572

1199080

= 119908119888cos120572 + 119906

119888sin120572

(19)

3 Far-Field Sound Pressure

In this section the theory for the boundary element methodis used to demonstrate that the far-field acoustic pressure ofthe shell can be solved by superposing the pressure of theindividual elements An analytical solution of the acousticpressure radiated by a cylindrical piston in a cylindrical baffleis used to obtain the pressure of conical elements

31 Acoustic Transfer Vector According to the Helmholtzboundary integral equation the acoustic pressure radiated bythe arbitrary surface can be expressed as [19]

119862 (r) 119901 (r)

= int1198780

(119901 (r119904)

120597

120597119899119866 (r r

119904) + 119895120588

119891120596119866 (r r

119904) V (r119904)) 119889119878

(20)

where r119904is the position vector on the vibrating surface and r

is the position of the observation field point 119901(r119904) and V(r

119904)

are the surface pressure and normal velocity respectively119866(r r119904) = 119890minus1198941198961198911198774120587119877 is free-space Greenrsquos function and 119877 =

|r minus r119904|

By discretizing the surface into 119872 elements (20) can beexpressed as

119862 (r) 119901 (r) =

119872

sum119894=1

int119878119894

(119901 (r119904)

120597

120597119899119866 (r r

119904) + 119895120588120596119866 (r r

119904) V) 119889119878

(21)

Equation (21) can be rearranged as [15]

119901 (r) = [C]119879

119872times1[p119904] + [D]

119879

119872times1[v] (22)

where C and D are the vectors independent of the surfacevelocityThe relationship between the surface pressure vectorp119904and the normal velocity k can be deduced as

[119860]119872times119872

p119904 = [119861]

119872times119872v (23)

where the matrices 119860 119861 are independent of the surfacevelocity but depend on the shape of the structure andfrequency Equation (22) can be expressed as

119901 (r) = ([C]119879

119872times1[119860]minus1

119872times119872[119861]119872times119872

+ [D]119879

119872times1) [v]

= [G]119879

119872times1[v]

(24)

The vector G is called Acoustic Transfer Vector (ATV)which is an array of transfer functions between the surfacenormal velocities and the pressure at the field pointTheATVdepends on frequency geometry acoustical parameters of thefluid and position of the observation point

Shock and Vibration 5

z998400

21205720

2L0

L1

y998400

x9984001205721

(a)

z998400

y998400

x998400

120572

o998400

(b)

Figure 4 The cylindrical baffle and piston

80

75

70

65

60

55

50

45

400 50 100 150 200 250 300

Frequency (Hz)

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

AnalyticalFEA

Figure 5 Range of validity for the stiffeners smeared over theconical shell

32 Element Radiation Superposition Method The conicalshell is divided into 119872 elements [G]

119879

119872times1= [1198921 1198922 119892

119872]

Each element of the ATV 119892119894 is equal to the acoustic pressure

radiated by the corresponding surface element vibrating withunity surface velocity while the other elements are at rest Forthe case where only one element vibrates with the velocity V

119894

the radiated pressure of the field point at r is given by

119901119894(r) = [119892

1 1198922 119892

119872] [0 0 V

119894 0]

119879

(25)

Therefore given the velocities [V1 V2 V

119894 V

119872]119879 the

radiated pressure is

119901 (r) = 1199011

(r) + 1199012

(r) + sdot sdot sdot + 119901119872

(r) (26)

Hence the total radiation pressure is the superposition of theacoustic pressures radiated by the surface elements vibratingwith a rigid baffle Since the velocities [V

1 V2 V

119894 V

119872]

can be obtained from the solution of dynamic response ofthe conical shell the main task is to find the acoustic transfervalues [119892

1 1198922 119892

119872]

According to Wang [15] the ATV of the element on acylindrical shell is equal to the acoustic pressure radiatedby a baffled piston vibrating with unity velocity The far-field acoustic pressure radiated by a cylindrical piston in acylindrical baffle can be obtained by solving the Helmholtzdifferential equation in closed form which is given by [20]

119901 (119877 120579 120601) =212058811988812057201198710

1205872119877 sin 1205791198950

(1198961198710cos 120579) 119890

119895119896(119877minus1198711 cos 120579)

times

infin

sum119899=minusinfin

(minus119895)119899

1198950

(1198991205720)

119867(1)

119899

1015840

(119896119886 sin 120579)119890119895119899(120601minus1205721) sdot V

(27)

where 21198710is the axial length of the piston 2120572

0is the radial

angle of the piston 1198711is the axial distance between the center

of the piston and the origin and 1205721is the azimuth angle of

the center of the piston All of these parameters are shownin Figure 4(a) V is the velocity of the baffled piston For avelocity equal to unity the ATV of the each cylindrical shellelement 119892

119894can be obtained from 119892

119894= 119901119894(119877 120579 120601)

The radiated pressure from each element of the conicalshell can be approximated by the analytical solution of theradiated pressure of a piston in a cylindrical baffle as shown

6 Shock and Vibration

90

80

70

60

50

40

30

200 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(a)

90

80

70

60

50

400 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(b)

Figure 6 Comparison with FEA results when the space or size of stiffeners changes (a) Results when the space between stiffeners 119889 = 042(b) results when cross-section of the stiffeners is 0028m times 004m

90

80

70

60

50

40

30

20

10

00 20 40 60 80

10minus12

m)

32

19

10

19

10

(T = 3)

(T = 5)

(T = 10)

Figure 7 Comparison for different divisions for the conical shell

in Figure 4(b) The radius of this cylindrical baffle is equalto the mean radius of the conical strip which includes thiselement The ATV of the piston in the cylindrical baffle issolved using its local coordinate The angle between the axisof the conical shell and the axis of the cylindrical baffle is thesemivertex angle of the conical shell In this way the ATVelements of the conical shell [119892

1 1198922 119892

119872] can be obtained

and the total radiated pressure of the conical shell is solved bysumming up the products of the individual transfer quantitiesand the corresponding velocities

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

BEM5 parts per conical segment

3 parts per conical segment1 part per conical segment

Figure 8 Validity and convergence of solving far-field pressure

4 Results for Structural and AcousticResponses for a Conical Shell

The structural and acoustic responses for the conical shellunder harmonic excitation are presented using the followingparameters 119871 = 89m 119877

1= 05m 119877

2= 325m and

ℎ = 0014m The stiffeners have a rectangular cross-sectionof 0014m times 004m (119887

119903= 0014m ℎ

119903= 004m) and are

evenly spaced by 119889 = 021mThematerial properties for steelare density 120588 = 7800 kgmminus3 Youngrsquos modulus 119864 = 21 times

1011Nmminus2 and Poissonrsquos ratio 120583 = 03 Structural damping

Shock and Vibration 7

90

80

70

60

50

40

300 20 40 60 80

10minus12

m)

2

2

22

(a)

90

80

70

60

500 20 40 60 80

10minus12

m)

3

1 2 2

2

(b)

80

60

40

20

0

minus20

minus400 20 40 60 80

10minus6

(c)

Figure 9 Structural and acoustic responses due to 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

is introduced using complex Youngrsquos modulus 119864(1 minus 119895120578119904)

where 120578119904

= 002 is the structural loss factor The surroundingfluid has a density of 120588

119891= 1000 kgmminus3 and speed of sound

119888119891

= 1500msminus1 The structural responses are predicted ata point equal to the drive point of the excitation as shownin Figure 3 The distance between the far-field point and theaxis of the conical shell is 100mThe circumferential angle ofthe observation point is 1205872 and its axial position is equal tothe axial position of the drive pointThe boundary conditionsconsidered here are free at the smaller end of the conical shelland clamped at the larger end

A finite element model is developed in ANSYS to validatethe results obtained from analytical model presented hereThe conical shell model and the ring stiffeners are built usingShell 63 elements and Beam 188 elements respectively Fluid

30 and Fluid 130 elements are used to simulate the externalfluid and the absorbing boundaries around the fluid domainANSYS is used to obtain the structural responses SYSNOISEis then used to calculate the far-field pressure using the directboundary element method

41 Validity and Convergence The validity of smearedapproach for the stiffened conical shell and the convergenceof dividing the conical shell are discussed in what follows Amodel of the stiffened conical shell in vacuo is built to validatethe smeared approach It is observed in Figure 5 that theanalytical results for the structural responses due to 119865

1match

the finite element analysis (FEA) results in the low frequencyrange However the accuracy decreases rapidly when thefrequency is above 180Hz where the structural wavelength

8 Shock and Vibration

100

95

90

85

80

75

70

65

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

2

2

2

3

(a)

110

105

100

95

90

85

80

75

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

3

1

2

2

2

(b)

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

FEBEAnalytical

(c)

Figure 10 Structural and acoustic responses due to 1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

is of the same order as the distance between the stiffenersIt is further observed that the increase of the spacing andwidth of the stiffeners affects the accuracy of this method asshown in Figures 6(a) and 6(b) respectively To conclude thesmeared approach is only suitable when the conical shells arereinforced by closely spaced stiffeners that is 119889119871 is smalland sufficiently small stiffeners that is 119887

119903119871 is small in the

low frequency range where the wavelength is larger than thestiffener spacing

As discussed in Section 2 the conical shell is dividedinto 119873 segments and the length of each segment is 119871

119894le

21198770119894

119879(sin120572) For the model in this paper 119879 = 5 Theresult for the structural responses due to 119865

1for 119879 = 5 is

compared with the results for 119879 = 3 and 119879 = 10 inFigure 7 It can be seen that the result converged for 119879 = 5

Two additional cases of dividing the conical shell into partswith equal lengths are also considered in Figure 7 It showsthat segments with fixed 119871

1198941198770119894yield faster convergence than

fixed length segments The FEA result is also presented inFigure 7 to validate the converged analytical result

The validity and convergence of the Element Radia-tion Superposition Method (ERSM) for solving the far-fieldacoustic pressure of a conical shell are discussed in whatfollows Each conical segment with the length of 119871

119894le

21198770119894

119879(sin120572) is divided into 1 3 and 5 parts in its axialdirection corresponding to 19 37 and 95 parts respectivelyfor 119879 = 5 and 19 conical segments The total number ofelements for the conical shell is 1140 2220 and 5700 respec-tively for a circumferential angle of 6∘ for each element Eachelement can be regarded as a piston in a cylindrical baffle

Shock and Vibration 9

100

90

80

70

60

50

40

30

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

0 20 40 60 80 100

Frequency (Hz)

(a)

120

100

80

60

40

20

00 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

80

60

40

20

0

minus20

minus40

minus60

minus800 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

In waterIn air

(c)

Figure 11 Comparison of conical shells in water and in air under 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

The results in Figure 8 show that the acoustic pressure dueto 1198652converges when there are 3 parts per conical segment

The BEM result is also shown in Figure 8 As discussed inSection 3 the pressure is obtained from the velocity andATV In this part only for both methods the velocities areanalytical solutions of the structural responsesThedifferencebetween ERSM and BEM lies in the solution of the ATVTheATV for ERSM is obtained from the analytical solution ofthe Helmholtz differential equation for a piston baffled in acylindrical shell which is used to approximate a piston baffledin a conical shell while the ATV for BEM is obtained bythe numerical solution of the Helmholtz boundary integralformulationThe results in Figure 8 indicate that the locationsof the peaks are well matched but the errors of amplitudesof the peaks are less than 5 at lower frequencies and lessthan 10 at higher frequencies However ERSM has a muchquicker solution thanBEM due to its avoidance of solving theHelmholtz boundary integral formulation Therefore ERSM

is an effectivemethod to solve the far-field pressure of conicalshell

42 Effect of Excitation on a Conical Shell The structural andacoustic responses of the conical shell due to the excitationshown in Figure 3 are presented in Figures 9 and 10 Thecomparison between the analytical results and the FEAresults indicates that the axial and radial displacements usingthese two methods agree well except for some errors athigher frequencies These errors arise due to approximatesolutions for the stiffeners and the external fluid loadingThestiffeners are considered by averaging their properties overthe surface of the conical shell The fluid loading is takeninto account by dividing the conical shell into narrow stripswhich are treated as cylindrical shells and the external fluidloading is approximated using an infinite modelThe far-fieldacoustic pressures obtained using the analytical and FEBEmethods are also in good agreement Errors are caused by

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

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International Journal of

Shock and Vibration 5

z998400

21205720

2L0

L1

y998400

x9984001205721

(a)

z998400

y998400

x998400

120572

o998400

(b)

Figure 4 The cylindrical baffle and piston

80

75

70

65

60

55

50

45

400 50 100 150 200 250 300

Frequency (Hz)

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

AnalyticalFEA

Figure 5 Range of validity for the stiffeners smeared over theconical shell

32 Element Radiation Superposition Method The conicalshell is divided into 119872 elements [G]

119879

119872times1= [1198921 1198922 119892

119872]

Each element of the ATV 119892119894 is equal to the acoustic pressure

radiated by the corresponding surface element vibrating withunity surface velocity while the other elements are at rest Forthe case where only one element vibrates with the velocity V

119894

the radiated pressure of the field point at r is given by

119901119894(r) = [119892

1 1198922 119892

119872] [0 0 V

119894 0]

119879

(25)

Therefore given the velocities [V1 V2 V

119894 V

119872]119879 the

radiated pressure is

119901 (r) = 1199011

(r) + 1199012

(r) + sdot sdot sdot + 119901119872

(r) (26)

Hence the total radiation pressure is the superposition of theacoustic pressures radiated by the surface elements vibratingwith a rigid baffle Since the velocities [V

1 V2 V

119894 V

119872]

can be obtained from the solution of dynamic response ofthe conical shell the main task is to find the acoustic transfervalues [119892

1 1198922 119892

119872]

According to Wang [15] the ATV of the element on acylindrical shell is equal to the acoustic pressure radiatedby a baffled piston vibrating with unity velocity The far-field acoustic pressure radiated by a cylindrical piston in acylindrical baffle can be obtained by solving the Helmholtzdifferential equation in closed form which is given by [20]

119901 (119877 120579 120601) =212058811988812057201198710

1205872119877 sin 1205791198950

(1198961198710cos 120579) 119890

119895119896(119877minus1198711 cos 120579)

times

infin

sum119899=minusinfin

(minus119895)119899

1198950

(1198991205720)

119867(1)

119899

1015840

(119896119886 sin 120579)119890119895119899(120601minus1205721) sdot V

(27)

where 21198710is the axial length of the piston 2120572

0is the radial

angle of the piston 1198711is the axial distance between the center

of the piston and the origin and 1205721is the azimuth angle of

the center of the piston All of these parameters are shownin Figure 4(a) V is the velocity of the baffled piston For avelocity equal to unity the ATV of the each cylindrical shellelement 119892

119894can be obtained from 119892

119894= 119901119894(119877 120579 120601)

The radiated pressure from each element of the conicalshell can be approximated by the analytical solution of theradiated pressure of a piston in a cylindrical baffle as shown

6 Shock and Vibration

90

80

70

60

50

40

30

200 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(a)

90

80

70

60

50

400 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(b)

Figure 6 Comparison with FEA results when the space or size of stiffeners changes (a) Results when the space between stiffeners 119889 = 042(b) results when cross-section of the stiffeners is 0028m times 004m

90

80

70

60

50

40

30

20

10

00 20 40 60 80

10minus12

m)

32

19

10

19

10

(T = 3)

(T = 5)

(T = 10)

Figure 7 Comparison for different divisions for the conical shell

in Figure 4(b) The radius of this cylindrical baffle is equalto the mean radius of the conical strip which includes thiselement The ATV of the piston in the cylindrical baffle issolved using its local coordinate The angle between the axisof the conical shell and the axis of the cylindrical baffle is thesemivertex angle of the conical shell In this way the ATVelements of the conical shell [119892

1 1198922 119892

119872] can be obtained

and the total radiated pressure of the conical shell is solved bysumming up the products of the individual transfer quantitiesand the corresponding velocities

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

BEM5 parts per conical segment

3 parts per conical segment1 part per conical segment

Figure 8 Validity and convergence of solving far-field pressure

4 Results for Structural and AcousticResponses for a Conical Shell

The structural and acoustic responses for the conical shellunder harmonic excitation are presented using the followingparameters 119871 = 89m 119877

1= 05m 119877

2= 325m and

ℎ = 0014m The stiffeners have a rectangular cross-sectionof 0014m times 004m (119887

119903= 0014m ℎ

119903= 004m) and are

evenly spaced by 119889 = 021mThematerial properties for steelare density 120588 = 7800 kgmminus3 Youngrsquos modulus 119864 = 21 times

1011Nmminus2 and Poissonrsquos ratio 120583 = 03 Structural damping

Shock and Vibration 7

90

80

70

60

50

40

300 20 40 60 80

10minus12

m)

2

2

22

(a)

90

80

70

60

500 20 40 60 80

10minus12

m)

3

1 2 2

2

(b)

80

60

40

20

0

minus20

minus400 20 40 60 80

10minus6

(c)

Figure 9 Structural and acoustic responses due to 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

is introduced using complex Youngrsquos modulus 119864(1 minus 119895120578119904)

where 120578119904

= 002 is the structural loss factor The surroundingfluid has a density of 120588

119891= 1000 kgmminus3 and speed of sound

119888119891

= 1500msminus1 The structural responses are predicted ata point equal to the drive point of the excitation as shownin Figure 3 The distance between the far-field point and theaxis of the conical shell is 100mThe circumferential angle ofthe observation point is 1205872 and its axial position is equal tothe axial position of the drive pointThe boundary conditionsconsidered here are free at the smaller end of the conical shelland clamped at the larger end

A finite element model is developed in ANSYS to validatethe results obtained from analytical model presented hereThe conical shell model and the ring stiffeners are built usingShell 63 elements and Beam 188 elements respectively Fluid

30 and Fluid 130 elements are used to simulate the externalfluid and the absorbing boundaries around the fluid domainANSYS is used to obtain the structural responses SYSNOISEis then used to calculate the far-field pressure using the directboundary element method

41 Validity and Convergence The validity of smearedapproach for the stiffened conical shell and the convergenceof dividing the conical shell are discussed in what follows Amodel of the stiffened conical shell in vacuo is built to validatethe smeared approach It is observed in Figure 5 that theanalytical results for the structural responses due to 119865

1match

the finite element analysis (FEA) results in the low frequencyrange However the accuracy decreases rapidly when thefrequency is above 180Hz where the structural wavelength

8 Shock and Vibration

100

95

90

85

80

75

70

65

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

2

2

2

3

(a)

110

105

100

95

90

85

80

75

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

3

1

2

2

2

(b)

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

FEBEAnalytical

(c)

Figure 10 Structural and acoustic responses due to 1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

is of the same order as the distance between the stiffenersIt is further observed that the increase of the spacing andwidth of the stiffeners affects the accuracy of this method asshown in Figures 6(a) and 6(b) respectively To conclude thesmeared approach is only suitable when the conical shells arereinforced by closely spaced stiffeners that is 119889119871 is smalland sufficiently small stiffeners that is 119887

119903119871 is small in the

low frequency range where the wavelength is larger than thestiffener spacing

As discussed in Section 2 the conical shell is dividedinto 119873 segments and the length of each segment is 119871

119894le

21198770119894

119879(sin120572) For the model in this paper 119879 = 5 Theresult for the structural responses due to 119865

1for 119879 = 5 is

compared with the results for 119879 = 3 and 119879 = 10 inFigure 7 It can be seen that the result converged for 119879 = 5

Two additional cases of dividing the conical shell into partswith equal lengths are also considered in Figure 7 It showsthat segments with fixed 119871

1198941198770119894yield faster convergence than

fixed length segments The FEA result is also presented inFigure 7 to validate the converged analytical result

The validity and convergence of the Element Radia-tion Superposition Method (ERSM) for solving the far-fieldacoustic pressure of a conical shell are discussed in whatfollows Each conical segment with the length of 119871

119894le

21198770119894

119879(sin120572) is divided into 1 3 and 5 parts in its axialdirection corresponding to 19 37 and 95 parts respectivelyfor 119879 = 5 and 19 conical segments The total number ofelements for the conical shell is 1140 2220 and 5700 respec-tively for a circumferential angle of 6∘ for each element Eachelement can be regarded as a piston in a cylindrical baffle

Shock and Vibration 9

100

90

80

70

60

50

40

30

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

0 20 40 60 80 100

Frequency (Hz)

(a)

120

100

80

60

40

20

00 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

80

60

40

20

0

minus20

minus40

minus60

minus800 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

In waterIn air

(c)

Figure 11 Comparison of conical shells in water and in air under 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

The results in Figure 8 show that the acoustic pressure dueto 1198652converges when there are 3 parts per conical segment

The BEM result is also shown in Figure 8 As discussed inSection 3 the pressure is obtained from the velocity andATV In this part only for both methods the velocities areanalytical solutions of the structural responsesThedifferencebetween ERSM and BEM lies in the solution of the ATVTheATV for ERSM is obtained from the analytical solution ofthe Helmholtz differential equation for a piston baffled in acylindrical shell which is used to approximate a piston baffledin a conical shell while the ATV for BEM is obtained bythe numerical solution of the Helmholtz boundary integralformulationThe results in Figure 8 indicate that the locationsof the peaks are well matched but the errors of amplitudesof the peaks are less than 5 at lower frequencies and lessthan 10 at higher frequencies However ERSM has a muchquicker solution thanBEM due to its avoidance of solving theHelmholtz boundary integral formulation Therefore ERSM

is an effectivemethod to solve the far-field pressure of conicalshell

42 Effect of Excitation on a Conical Shell The structural andacoustic responses of the conical shell due to the excitationshown in Figure 3 are presented in Figures 9 and 10 Thecomparison between the analytical results and the FEAresults indicates that the axial and radial displacements usingthese two methods agree well except for some errors athigher frequencies These errors arise due to approximatesolutions for the stiffeners and the external fluid loadingThestiffeners are considered by averaging their properties overthe surface of the conical shell The fluid loading is takeninto account by dividing the conical shell into narrow stripswhich are treated as cylindrical shells and the external fluidloading is approximated using an infinite modelThe far-fieldacoustic pressures obtained using the analytical and FEBEmethods are also in good agreement Errors are caused by

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

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International Journal of

6 Shock and Vibration

90

80

70

60

50

40

30

200 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(a)

90

80

70

60

50

400 50 100 150

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

(b)

Figure 6 Comparison with FEA results when the space or size of stiffeners changes (a) Results when the space between stiffeners 119889 = 042(b) results when cross-section of the stiffeners is 0028m times 004m

90

80

70

60

50

40

30

20

10

00 20 40 60 80

10minus12

m)

32

19

10

19

10

(T = 3)

(T = 5)

(T = 10)

Figure 7 Comparison for different divisions for the conical shell

in Figure 4(b) The radius of this cylindrical baffle is equalto the mean radius of the conical strip which includes thiselement The ATV of the piston in the cylindrical baffle issolved using its local coordinate The angle between the axisof the conical shell and the axis of the cylindrical baffle is thesemivertex angle of the conical shell In this way the ATVelements of the conical shell [119892

1 1198922 119892

119872] can be obtained

and the total radiated pressure of the conical shell is solved bysumming up the products of the individual transfer quantitiesand the corresponding velocities

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

BEM5 parts per conical segment

3 parts per conical segment1 part per conical segment

Figure 8 Validity and convergence of solving far-field pressure

4 Results for Structural and AcousticResponses for a Conical Shell

The structural and acoustic responses for the conical shellunder harmonic excitation are presented using the followingparameters 119871 = 89m 119877

1= 05m 119877

2= 325m and

ℎ = 0014m The stiffeners have a rectangular cross-sectionof 0014m times 004m (119887

119903= 0014m ℎ

119903= 004m) and are

evenly spaced by 119889 = 021mThematerial properties for steelare density 120588 = 7800 kgmminus3 Youngrsquos modulus 119864 = 21 times

1011Nmminus2 and Poissonrsquos ratio 120583 = 03 Structural damping

Shock and Vibration 7

90

80

70

60

50

40

300 20 40 60 80

10minus12

m)

2

2

22

(a)

90

80

70

60

500 20 40 60 80

10minus12

m)

3

1 2 2

2

(b)

80

60

40

20

0

minus20

minus400 20 40 60 80

10minus6

(c)

Figure 9 Structural and acoustic responses due to 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

is introduced using complex Youngrsquos modulus 119864(1 minus 119895120578119904)

where 120578119904

= 002 is the structural loss factor The surroundingfluid has a density of 120588

119891= 1000 kgmminus3 and speed of sound

119888119891

= 1500msminus1 The structural responses are predicted ata point equal to the drive point of the excitation as shownin Figure 3 The distance between the far-field point and theaxis of the conical shell is 100mThe circumferential angle ofthe observation point is 1205872 and its axial position is equal tothe axial position of the drive pointThe boundary conditionsconsidered here are free at the smaller end of the conical shelland clamped at the larger end

A finite element model is developed in ANSYS to validatethe results obtained from analytical model presented hereThe conical shell model and the ring stiffeners are built usingShell 63 elements and Beam 188 elements respectively Fluid

30 and Fluid 130 elements are used to simulate the externalfluid and the absorbing boundaries around the fluid domainANSYS is used to obtain the structural responses SYSNOISEis then used to calculate the far-field pressure using the directboundary element method

41 Validity and Convergence The validity of smearedapproach for the stiffened conical shell and the convergenceof dividing the conical shell are discussed in what follows Amodel of the stiffened conical shell in vacuo is built to validatethe smeared approach It is observed in Figure 5 that theanalytical results for the structural responses due to 119865

1match

the finite element analysis (FEA) results in the low frequencyrange However the accuracy decreases rapidly when thefrequency is above 180Hz where the structural wavelength

8 Shock and Vibration

100

95

90

85

80

75

70

65

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

2

2

2

3

(a)

110

105

100

95

90

85

80

75

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

3

1

2

2

2

(b)

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

FEBEAnalytical

(c)

Figure 10 Structural and acoustic responses due to 1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

is of the same order as the distance between the stiffenersIt is further observed that the increase of the spacing andwidth of the stiffeners affects the accuracy of this method asshown in Figures 6(a) and 6(b) respectively To conclude thesmeared approach is only suitable when the conical shells arereinforced by closely spaced stiffeners that is 119889119871 is smalland sufficiently small stiffeners that is 119887

119903119871 is small in the

low frequency range where the wavelength is larger than thestiffener spacing

As discussed in Section 2 the conical shell is dividedinto 119873 segments and the length of each segment is 119871

119894le

21198770119894

119879(sin120572) For the model in this paper 119879 = 5 Theresult for the structural responses due to 119865

1for 119879 = 5 is

compared with the results for 119879 = 3 and 119879 = 10 inFigure 7 It can be seen that the result converged for 119879 = 5

Two additional cases of dividing the conical shell into partswith equal lengths are also considered in Figure 7 It showsthat segments with fixed 119871

1198941198770119894yield faster convergence than

fixed length segments The FEA result is also presented inFigure 7 to validate the converged analytical result

The validity and convergence of the Element Radia-tion Superposition Method (ERSM) for solving the far-fieldacoustic pressure of a conical shell are discussed in whatfollows Each conical segment with the length of 119871

119894le

21198770119894

119879(sin120572) is divided into 1 3 and 5 parts in its axialdirection corresponding to 19 37 and 95 parts respectivelyfor 119879 = 5 and 19 conical segments The total number ofelements for the conical shell is 1140 2220 and 5700 respec-tively for a circumferential angle of 6∘ for each element Eachelement can be regarded as a piston in a cylindrical baffle

Shock and Vibration 9

100

90

80

70

60

50

40

30

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

0 20 40 60 80 100

Frequency (Hz)

(a)

120

100

80

60

40

20

00 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

80

60

40

20

0

minus20

minus40

minus60

minus800 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

In waterIn air

(c)

Figure 11 Comparison of conical shells in water and in air under 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

The results in Figure 8 show that the acoustic pressure dueto 1198652converges when there are 3 parts per conical segment

The BEM result is also shown in Figure 8 As discussed inSection 3 the pressure is obtained from the velocity andATV In this part only for both methods the velocities areanalytical solutions of the structural responsesThedifferencebetween ERSM and BEM lies in the solution of the ATVTheATV for ERSM is obtained from the analytical solution ofthe Helmholtz differential equation for a piston baffled in acylindrical shell which is used to approximate a piston baffledin a conical shell while the ATV for BEM is obtained bythe numerical solution of the Helmholtz boundary integralformulationThe results in Figure 8 indicate that the locationsof the peaks are well matched but the errors of amplitudesof the peaks are less than 5 at lower frequencies and lessthan 10 at higher frequencies However ERSM has a muchquicker solution thanBEM due to its avoidance of solving theHelmholtz boundary integral formulation Therefore ERSM

is an effectivemethod to solve the far-field pressure of conicalshell

42 Effect of Excitation on a Conical Shell The structural andacoustic responses of the conical shell due to the excitationshown in Figure 3 are presented in Figures 9 and 10 Thecomparison between the analytical results and the FEAresults indicates that the axial and radial displacements usingthese two methods agree well except for some errors athigher frequencies These errors arise due to approximatesolutions for the stiffeners and the external fluid loadingThestiffeners are considered by averaging their properties overthe surface of the conical shell The fluid loading is takeninto account by dividing the conical shell into narrow stripswhich are treated as cylindrical shells and the external fluidloading is approximated using an infinite modelThe far-fieldacoustic pressures obtained using the analytical and FEBEmethods are also in good agreement Errors are caused by

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

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DistributedSensor Networks

International Journal of

Shock and Vibration 7

90

80

70

60

50

40

300 20 40 60 80

10minus12

m)

2

2

22

(a)

90

80

70

60

500 20 40 60 80

10minus12

m)

3

1 2 2

2

(b)

80

60

40

20

0

minus20

minus400 20 40 60 80

10minus6

(c)

Figure 9 Structural and acoustic responses due to 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

is introduced using complex Youngrsquos modulus 119864(1 minus 119895120578119904)

where 120578119904

= 002 is the structural loss factor The surroundingfluid has a density of 120588

119891= 1000 kgmminus3 and speed of sound

119888119891

= 1500msminus1 The structural responses are predicted ata point equal to the drive point of the excitation as shownin Figure 3 The distance between the far-field point and theaxis of the conical shell is 100mThe circumferential angle ofthe observation point is 1205872 and its axial position is equal tothe axial position of the drive pointThe boundary conditionsconsidered here are free at the smaller end of the conical shelland clamped at the larger end

A finite element model is developed in ANSYS to validatethe results obtained from analytical model presented hereThe conical shell model and the ring stiffeners are built usingShell 63 elements and Beam 188 elements respectively Fluid

30 and Fluid 130 elements are used to simulate the externalfluid and the absorbing boundaries around the fluid domainANSYS is used to obtain the structural responses SYSNOISEis then used to calculate the far-field pressure using the directboundary element method

41 Validity and Convergence The validity of smearedapproach for the stiffened conical shell and the convergenceof dividing the conical shell are discussed in what follows Amodel of the stiffened conical shell in vacuo is built to validatethe smeared approach It is observed in Figure 5 that theanalytical results for the structural responses due to 119865

1match

the finite element analysis (FEA) results in the low frequencyrange However the accuracy decreases rapidly when thefrequency is above 180Hz where the structural wavelength

8 Shock and Vibration

100

95

90

85

80

75

70

65

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

2

2

2

3

(a)

110

105

100

95

90

85

80

75

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

3

1

2

2

2

(b)

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

FEBEAnalytical

(c)

Figure 10 Structural and acoustic responses due to 1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

is of the same order as the distance between the stiffenersIt is further observed that the increase of the spacing andwidth of the stiffeners affects the accuracy of this method asshown in Figures 6(a) and 6(b) respectively To conclude thesmeared approach is only suitable when the conical shells arereinforced by closely spaced stiffeners that is 119889119871 is smalland sufficiently small stiffeners that is 119887

119903119871 is small in the

low frequency range where the wavelength is larger than thestiffener spacing

As discussed in Section 2 the conical shell is dividedinto 119873 segments and the length of each segment is 119871

119894le

21198770119894

119879(sin120572) For the model in this paper 119879 = 5 Theresult for the structural responses due to 119865

1for 119879 = 5 is

compared with the results for 119879 = 3 and 119879 = 10 inFigure 7 It can be seen that the result converged for 119879 = 5

Two additional cases of dividing the conical shell into partswith equal lengths are also considered in Figure 7 It showsthat segments with fixed 119871

1198941198770119894yield faster convergence than

fixed length segments The FEA result is also presented inFigure 7 to validate the converged analytical result

The validity and convergence of the Element Radia-tion Superposition Method (ERSM) for solving the far-fieldacoustic pressure of a conical shell are discussed in whatfollows Each conical segment with the length of 119871

119894le

21198770119894

119879(sin120572) is divided into 1 3 and 5 parts in its axialdirection corresponding to 19 37 and 95 parts respectivelyfor 119879 = 5 and 19 conical segments The total number ofelements for the conical shell is 1140 2220 and 5700 respec-tively for a circumferential angle of 6∘ for each element Eachelement can be regarded as a piston in a cylindrical baffle

Shock and Vibration 9

100

90

80

70

60

50

40

30

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

0 20 40 60 80 100

Frequency (Hz)

(a)

120

100

80

60

40

20

00 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

80

60

40

20

0

minus20

minus40

minus60

minus800 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

In waterIn air

(c)

Figure 11 Comparison of conical shells in water and in air under 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

The results in Figure 8 show that the acoustic pressure dueto 1198652converges when there are 3 parts per conical segment

The BEM result is also shown in Figure 8 As discussed inSection 3 the pressure is obtained from the velocity andATV In this part only for both methods the velocities areanalytical solutions of the structural responsesThedifferencebetween ERSM and BEM lies in the solution of the ATVTheATV for ERSM is obtained from the analytical solution ofthe Helmholtz differential equation for a piston baffled in acylindrical shell which is used to approximate a piston baffledin a conical shell while the ATV for BEM is obtained bythe numerical solution of the Helmholtz boundary integralformulationThe results in Figure 8 indicate that the locationsof the peaks are well matched but the errors of amplitudesof the peaks are less than 5 at lower frequencies and lessthan 10 at higher frequencies However ERSM has a muchquicker solution thanBEM due to its avoidance of solving theHelmholtz boundary integral formulation Therefore ERSM

is an effectivemethod to solve the far-field pressure of conicalshell

42 Effect of Excitation on a Conical Shell The structural andacoustic responses of the conical shell due to the excitationshown in Figure 3 are presented in Figures 9 and 10 Thecomparison between the analytical results and the FEAresults indicates that the axial and radial displacements usingthese two methods agree well except for some errors athigher frequencies These errors arise due to approximatesolutions for the stiffeners and the external fluid loadingThestiffeners are considered by averaging their properties overthe surface of the conical shell The fluid loading is takeninto account by dividing the conical shell into narrow stripswhich are treated as cylindrical shells and the external fluidloading is approximated using an infinite modelThe far-fieldacoustic pressures obtained using the analytical and FEBEmethods are also in good agreement Errors are caused by

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

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Shock and Vibration

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DistributedSensor Networks

International Journal of

8 Shock and Vibration

100

95

90

85

80

75

70

65

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

2

2

2

3

(a)

110

105

100

95

90

85

80

75

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

AnalyticalFEA

2

3

1

2

2

2

(b)

100

80

60

40

20

0

minus20

minus400 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

FEBEAnalytical

(c)

Figure 10 Structural and acoustic responses due to 1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

is of the same order as the distance between the stiffenersIt is further observed that the increase of the spacing andwidth of the stiffeners affects the accuracy of this method asshown in Figures 6(a) and 6(b) respectively To conclude thesmeared approach is only suitable when the conical shells arereinforced by closely spaced stiffeners that is 119889119871 is smalland sufficiently small stiffeners that is 119887

119903119871 is small in the

low frequency range where the wavelength is larger than thestiffener spacing

As discussed in Section 2 the conical shell is dividedinto 119873 segments and the length of each segment is 119871

119894le

21198770119894

119879(sin120572) For the model in this paper 119879 = 5 Theresult for the structural responses due to 119865

1for 119879 = 5 is

compared with the results for 119879 = 3 and 119879 = 10 inFigure 7 It can be seen that the result converged for 119879 = 5

Two additional cases of dividing the conical shell into partswith equal lengths are also considered in Figure 7 It showsthat segments with fixed 119871

1198941198770119894yield faster convergence than

fixed length segments The FEA result is also presented inFigure 7 to validate the converged analytical result

The validity and convergence of the Element Radia-tion Superposition Method (ERSM) for solving the far-fieldacoustic pressure of a conical shell are discussed in whatfollows Each conical segment with the length of 119871

119894le

21198770119894

119879(sin120572) is divided into 1 3 and 5 parts in its axialdirection corresponding to 19 37 and 95 parts respectivelyfor 119879 = 5 and 19 conical segments The total number ofelements for the conical shell is 1140 2220 and 5700 respec-tively for a circumferential angle of 6∘ for each element Eachelement can be regarded as a piston in a cylindrical baffle

Shock and Vibration 9

100

90

80

70

60

50

40

30

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

0 20 40 60 80 100

Frequency (Hz)

(a)

120

100

80

60

40

20

00 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

80

60

40

20

0

minus20

minus40

minus60

minus800 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

In waterIn air

(c)

Figure 11 Comparison of conical shells in water and in air under 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

The results in Figure 8 show that the acoustic pressure dueto 1198652converges when there are 3 parts per conical segment

The BEM result is also shown in Figure 8 As discussed inSection 3 the pressure is obtained from the velocity andATV In this part only for both methods the velocities areanalytical solutions of the structural responsesThedifferencebetween ERSM and BEM lies in the solution of the ATVTheATV for ERSM is obtained from the analytical solution ofthe Helmholtz differential equation for a piston baffled in acylindrical shell which is used to approximate a piston baffledin a conical shell while the ATV for BEM is obtained bythe numerical solution of the Helmholtz boundary integralformulationThe results in Figure 8 indicate that the locationsof the peaks are well matched but the errors of amplitudesof the peaks are less than 5 at lower frequencies and lessthan 10 at higher frequencies However ERSM has a muchquicker solution thanBEM due to its avoidance of solving theHelmholtz boundary integral formulation Therefore ERSM

is an effectivemethod to solve the far-field pressure of conicalshell

42 Effect of Excitation on a Conical Shell The structural andacoustic responses of the conical shell due to the excitationshown in Figure 3 are presented in Figures 9 and 10 Thecomparison between the analytical results and the FEAresults indicates that the axial and radial displacements usingthese two methods agree well except for some errors athigher frequencies These errors arise due to approximatesolutions for the stiffeners and the external fluid loadingThestiffeners are considered by averaging their properties overthe surface of the conical shell The fluid loading is takeninto account by dividing the conical shell into narrow stripswhich are treated as cylindrical shells and the external fluidloading is approximated using an infinite modelThe far-fieldacoustic pressures obtained using the analytical and FEBEmethods are also in good agreement Errors are caused by

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

International Journal of

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Acoustics and VibrationAdvances in

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Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

100

90

80

70

60

50

40

30

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

0 20 40 60 80 100

Frequency (Hz)

(a)

120

100

80

60

40

20

00 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

80

60

40

20

0

minus20

minus40

minus60

minus800 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

In waterIn air

(c)

Figure 11 Comparison of conical shells in water and in air under 1198651(a) axial displacement (b) radial displacement and (c) far-field pressure

The results in Figure 8 show that the acoustic pressure dueto 1198652converges when there are 3 parts per conical segment

The BEM result is also shown in Figure 8 As discussed inSection 3 the pressure is obtained from the velocity andATV In this part only for both methods the velocities areanalytical solutions of the structural responsesThedifferencebetween ERSM and BEM lies in the solution of the ATVTheATV for ERSM is obtained from the analytical solution ofthe Helmholtz differential equation for a piston baffled in acylindrical shell which is used to approximate a piston baffledin a conical shell while the ATV for BEM is obtained bythe numerical solution of the Helmholtz boundary integralformulationThe results in Figure 8 indicate that the locationsof the peaks are well matched but the errors of amplitudesof the peaks are less than 5 at lower frequencies and lessthan 10 at higher frequencies However ERSM has a muchquicker solution thanBEM due to its avoidance of solving theHelmholtz boundary integral formulation Therefore ERSM

is an effectivemethod to solve the far-field pressure of conicalshell

42 Effect of Excitation on a Conical Shell The structural andacoustic responses of the conical shell due to the excitationshown in Figure 3 are presented in Figures 9 and 10 Thecomparison between the analytical results and the FEAresults indicates that the axial and radial displacements usingthese two methods agree well except for some errors athigher frequencies These errors arise due to approximatesolutions for the stiffeners and the external fluid loadingThestiffeners are considered by averaging their properties overthe surface of the conical shell The fluid loading is takeninto account by dividing the conical shell into narrow stripswhich are treated as cylindrical shells and the external fluidloading is approximated using an infinite modelThe far-fieldacoustic pressures obtained using the analytical and FEBEmethods are also in good agreement Errors are caused by

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

110

100

90

80

70

60

50

400 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

130

120

110

100

90

80

70

600 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

In waterIn air

100

50

0

minus50

minus1000 20 40 60 80 100

Pres

sure

(dB

ref10

minus6

Pa)

Frequency (Hz)

(c)

Figure 12 Comparison of conical shells in water and in air under1198652(a) axial displacement (b) radial displacement and (c) far-field pressure

the velocities which are gained from the results of structuralresponses and the ERSM as it has a different solution of theATV from the BEM Thus it can be found that some errorsexist especially at higher frequencies due to complexity andapproximations of the pressure solution fromERSM But theyare acceptable and applicable in the preliminary research inthe case that the analytical methods are able to save muchtime of the calculations

The contributions of the independent circumferentialmodes with mode order 119899 are marked in the figures of theaxial and radial displacements It is observed that in bothcases the 119899 = 2modes are predominantly excited followed by119899 = 1 and 119899 = 3 modes Higher order circumferential modesdo not significantly contribute to the structural responses Itis also found that both structural and acoustic responses dueto the 119865

2excitation are significantly higher than those due to

the 1198651excitation

43 Effect of External Fluid Loading The effect of the fluidloading on the structural and acoustic responses for thetwo excitation cases is presented for the conical shell inair in Figures 11 and 12 It is observed that the effects ofthe external fluid for the two cases are similar The fluidloading reduces the peaks of the axial and radial resonantfrequencies due to damping Moreover the resonant peaksshift to lower frequencies because of themass-like effect of thefluid loading The amplitude of the far-field pressure for theconical shell in water is significantly higher than the pressurefor the conical shell in air which indicates that the sound istransmitted more efficiently in water than in air

44 Effect of Ring Stiffeners The effect of the ring stiffenerson the structural and acoustic responses of the conical shellresponses is presented in Figures 13 and 14The ring stiffenersare observed to have a stiffening effect resulting in an increase

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

80

70

60

50

40

300 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

0 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

100

90

80

70

60

50

(b)

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

Pa)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 13 Comparison of conical shells with and without stiffeners under 1198651(a) axial displacement (b) radial displacement and (c) far-field

pressure

of the resonant frequencies for both excitation cases Theshift of the resonant peaks of the far-field pressure is not assignificant as that for the displacements which shows thatthe added stiffness has little effect on the radiated pressure Itis further observed that the amplitudes of the pressure fromconical shells with and without ring stiffeners are similarwhichmeans that the ring stiffeners do not significantly affectthe transmission of sound into the acoustic far-field

5 Conclusion

An analytical method to predict the structural and acousticresponses of a stiffened conical shell immersed in fluid hasbeen presented The solution for the dynamic response of aconical shell was presented in the formof a power series Fluidloading on the shell was taken into account by dividing the

shell into narrow strips and stiffeners were modeled usinga smeared approach The Element Radiation SuperpositionMethod was used to solve the far-field pressure of a conicalshell Good agreement between the analytical and FEMBEMresults was observed Two cases of excitation by consideringforces acting in different directions at the smaller end ofthe conical shell were examined in which the forces actedalong the generator and normal to the surface of the shellIn both cases 119899 = 2 modes were predominantly excitedfollowed by 119899 = 1 and 119899 = 3 modes Both the structuraland acoustic responses of the conical shell under excitationin normal direction were greater than for excitation along thegeneratorThe effects of fluid loading and ring stiffeners wereexamined The results showed that external fluid loading hasboth damping and mass-like effects while stiffeners mainlyincrease the stiffness of the conical shell

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

110

100

90

80

70

600 20 40 60 80 100

Axi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(a)

120

110

100

90

80

700 20 40 60 80 100

Radi

al d

ispla

cem

ent (

dB re

f10

minus12

m)

Frequency (Hz)

(b)

100

80

60

40

20

0

minus20

minus40

Pres

sure

(dB

ref10

minus6

m)

0 20 40 60 80 100

Frequency (Hz)

With stiffenersWithout stiffeners

(c)

Figure 14 Comparison of conical shells with and without stiffeners under 1198652(a) axial displacement (b) radial displacement and (c) far-field

pressure

Appendix

The forces and moments for a conical shell are given by [1]

119873119909

=119864ℎ

1 minus 1205832(

120597119906119888

120597119909+ 120583 (

1

119877 (119909)

120597V119888

120597120579+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888)

minusℎ2

12

cos120572

119877 (119909)

1205972119908119888

1205971199092)

119873120579

=119864ℎ

1 minus 1205832[

1

119877 (119909)

120597V119888

120597120579+ 120583

120597119906119888

120597119909+sin120572

119877 (119909)119906119888

+cos120572

119877 (119909)119908119888

+ℎ2

12(cos120572

1198773 (119909)

1205972119908119888

1205971205792+sin120572 cos120572

1198772 (119909)

120597119908119888

120597119909

+sin120572cos2120572

1198773 (119909)119906119888

+cos31205721198773 (119909)

119908119888)]

119873119909120579

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12

times (minuscos120572

1198772 (119909)

1205972119908119888

120597120579120597119909+

cos21205721198772 (119909)

120597V119888

120597119909

+cos120572 sin120572

1198773 (119909)

120597119908119888

120597120579minuscos2120572 sin120572

1198773V119888)]

119873120579119909

=119864ℎ

2 (1 + 120583)[

1

119877 (119909)

120597119906119888

120597120579+

120597V119888

120597119909minussin120572

119877 (119909)V119888

+ℎ2

12(cos120572

1198772 (119909)

1205972

119908119888

120597120579120597119909minuscos120572 sin120572

1198773 (119909)

120597119908119888

120597120579

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

+cos21205721198773 (119909)

120597119906119888

120597120579+cos2120572 sin120572

1198773 (119909)V119888)]

119872119909

=119864ℎ3

12 (1 minus 1205832)

times [minus1205972119908119888

1205971199092+ 120583

times (cos120572

1198772 (119909)

120597V119888

120597120579minus

1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909)

+cos120572

119877 (119909)

120597119906119888

120597119909]

119872120579

=119864ℎ3

12 (1 minus 1205832)

times [minus1

1198772 (119909)

1205972119908119888

1205971205792minussin120572

119877 (119909)

120597119908119888

120597119909minus 120583

1205972119908119888

1205971199092

minuscos120572 sin120572

1198772 (119909)119906119888

minuscos21205721198772 (119909)

119908119888]

119872119909120579

=119864ℎ3

12 (1 + 120583)

times (minus1

119877 (119909)

1205972

119908119888

120597119909120597120579+cos120572

119877 (119909)

120597V119888

120597119909+

sin120572

1198772 (119909)

120597119908119888

120597120579

minussin120572 cos120572

1198772V119888)

119872120579119909

=119864ℎ3

24 (1 + 120583)

times (minus2

119877 (119909)

1205972119908119888

120597119909120597120579+

2 sin120572

1198772 (119909)

120597119908119888

120597120579+cos120572

119877 (119909)

120597V119888

120597119909

minuscos120572

1198772 (119909)

120597119906119888

120597120579minuscos120572 sin120572

1198772 (119909)V119888)

119876119909

=1

119877 (119909)

120597 (119877 (119909) 119872119909)

120597119909+

1

119877 (119909)

120597 (119872120579119909

)

120597120579minussin120572

119877 (119909)119872120579

119876120579

=1

119877 (119909)

120597 (119872120579)

120597120579+

1

119877 (119909)

120597 (119877119872119909120579

)

120597119909+sin120572

119877 (119909)119872120579119909

119881119909

= 119876119909

+1

119877

120597119872119909120579

120597120579

(A1)

where119864120583 and ℎ respectively are Youngrsquosmodulus Poissonrsquosratio and shell thickness

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

All the work in this paper obtains great support from theNationalNatural Science Foundation of China (51179071) andthe Fundamental Research Funds for the Central Universi-ties HUST 2012QN056

References

[1] A W Leissa Vibration of Shells American Institue of PhysicsNew York NY USA 1993

[2] H Saunders E J Wisniewski and P R Paslay ldquoVibrations ofconical shellsrdquo Journal of the Acoustical Society of America vol32 pp 765ndash772 1960

[3] H Garnet and J Kempner ldquoAxisymmetric free vibration ofconical shellsrdquo Journal of Applied Mechanics vol 31 pp 458ndash466 1964

[4] L Tong ldquoFree vibration of orthotropic conical shellsrdquo Interna-tional Journal of Engineering Science vol 31 no 5 pp 719ndash7331993

[5] Y P Guo ldquoNormalmode propagation on conical shellsrdquo Journalof the Acoustical Society of America vol 96 no 1 pp 256ndash2641994

[6] Y P Guo ldquoFluid-loading effects on waves on conical shellsrdquoJournal of the Acoustical Society of America vol 97 no 2 pp1061ndash1066 1995

[7] M Caresta and N J Kessissoglou ldquoVibration of fluid loadedconical shellsrdquo Journal of the Acoustical Society of America vol124 no 4 pp 2068ndash2077 2008

[8] M Caresta and N J Kessissoglou ldquoFree vibrational character-istics of isotropic coupled cylindrical-conical shellsrdquo Journal ofSound and Vibration vol 329 no 6 pp 733ndash751 2010

[9] C C Chien H Rajiyah and S N Atluri ldquoAn effectivemethod for solving the hypersingular integral equations in 3-D acousticsrdquo Journal of the Acoustical Society of America vol88 no 2 pp 918ndash937 1990

[10] R A Jeans and I CMathews ldquoSolution of fluid-structure inter-action problems using a coupled finite element and variationalboundary element techniquerdquo Journal of the Acoustical Societyof America vol 88 no 5 pp 2459ndash2466 1990

[11] M Caresta and N J Kessissoglou ldquoAcoustic signature of asubmarine hull under harmonic excitationrdquo Applied Acousticsvol 71 no 1 pp 17ndash31 2010

[12] O Von Estorff ldquoEfforts to reduce computation time in numer-ical acoustics - An overviewrdquo Acta Acustica vol 89 no 1 pp1ndash13 2003

[13] M T Van Nhieu ldquoAn approximate solution to sound radiationfrom slender bodiesrdquo Journal of the Acoustical Society of Amer-ica vol 96 no 2 pp 1070ndash1079 1994

[14] M Utschig J D Achenbach and T Igusa ldquoReduction toparts a semianalytical approach to the structural acoustics ofa cylindrical shell with hemispherical endcapsrdquo Journal of theAcoustical Society of America vol 100 no 2 pp 871ndash881 1996

[15] BWangW Tang and F Jun ldquoApproximate method to acousticradiation problems element radiation superposition methodrdquoChinese Journal of Acoustics vol 27 pp 350ndash357 2008

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

[16] A Rosen and J Singer ldquoVibrations of axially loaded stiffenedcylindrical shellsrdquo Journal of Sound and Vibration vol 34 no 3pp 357ndash378 1974

[17] M Baruch J Arbocz and G Q Zhang ldquoLaminated conicalshells considerations for the variations of the stiffness coeffi-cientsrdquo in Proceedings of the 35th AIAAASMEASCEAHSASCStructures Structural Dynamics and Materials Conference pp2505ndash2516 April 1994

[18] M Caresta and N J Kessissoglou ldquoStructural and acousticresponses of a fluid-loaded cylindrical hull with structuraldiscontinuitiesrdquo Applied Acoustics vol 70 no 7 pp 954ndash9632009

[19] E A Skelton and J H JamesTheoretical Acoustics of Underwa-ter Structures Imperial College Press London UK 1997

[20] MC Junger andD Feit Sound Structure andTheir InteractionMIT Press Cambridge UK 1986

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

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RotatingMachinery

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VLSI Design

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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Acoustics and VibrationAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

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DistributedSensor Networks

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