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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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Shock and Vibration
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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
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Shock and Vibration
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Navigation and Observation
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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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