vibro-acoustic behavior of a hollow fgm cylinder excited by on-surface

Composite Structures 92 (2010) 86–96

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Composite Structures

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Vibro-acoustic behavior of a hollow FGM cylinder excited by on-surfacemechanical drives

Seyyed M. Hasheminejad *, Ali Ahamdi-SavadkoohiAcoustics Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran

a r t i c l e i n f o

Article history:Available online 3 July 2009

Keywords:Sound radiationInhomogeneous cylinderHelical wavesShadow regionCut-off frequency

0263-8223/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.compstruct.2009.06.014

* Corresponding author.E-mail address: [email protected] (S.M. Hashemi

a b s t r a c t

The linear three-dimensional elasticity theory in conjunction with the powerful transfer matrix solutiontechnique is employed to investigate the steady-state nonaxisymmetric sound radiation characteristics ofan arbitrarily thick functionally graded hollow cylinder of infinite length subjected to arbitrary time-har-monic on-surface concentrated mechanical drives. A formal integral expression for the radiated pressurefield in the frequency domain is obtained by utilizing the spatial Fourier transform along the shell axisand Fourier series expansion in the circumferential direction. The method of stationary phase is subse-quently employed to evaluate the integral for an observation point in the far-field. The analytical resultsare illustrated with numerical examples in which water-submerged metal-ceramic FGM cylinders aredriven by harmonic concentrated radial/transverse surface forces and circumferential moment. Thefar-field radiated pressure amplitudes and directivities are calculated and compared with those of equiv-alent bi-laminate hollow cylinders with comparable volume fractions of constituent materials. Theeffects of FGM material profile, cylinder thickness, excitation frequency and type on the radiated far-fieldare examined. Limiting cases are considered and the validity of results is established by comparison withthe data in the existing literature as well as with the aid of a commercial finite element package.

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1. Introduction

Among various shell geometries, the dynamic behavior of cylin-drical shell structure is the subject of most of the recent research,since it has extensively been used as the basic element in manytypes of engineering structures [1]. In many practical engineeringapplications, cylindrical shells are coupled with internal and/orexternal fluids. Therefore, a lot of attention has also been paid tothe vibration of the shell-fluid coupled systems [2–4]. In particular,numerous authors have studied the vibro-acoustic characteristicsof fluid-loaded cylindrical shells due to their extensive applicationsin various engineering disciplines such as the marine, petrochem-ical, nuclear, power generation and aerospace industries (e.g., radi-ation from cylindrical machinery casings, elastic ducts, wind musicinstruments, aircraft fuselages, submarine hulls, industrial pipesand tubes). Only the most important contributions relevant tothe present study shall be briefly reviewed here. Burroughs andHallander [5] derived analytical expressions for the far field acous-tic radiation from an infinite fluid-loaded circular cylindrical shellreinforced with two sets of parallel periodic ribs and excited byvarious types of concentrated mechanical point drives. Ricks andSchmidt [6] described a global matrix method for modeling layered

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nejad).

cylindrical shells (with multiple viscoelastic layers) subjected totime-harmonic ring forces that can push on the shell in the radial,circumferential, and axial directions. Guo [7] used an asymptoticanalysis to study sound radiation from thin cylindrical shells dri-ven by on-surface forces in terms of the shell-borne helical wavesexcited in the shell. He found that tangential forces acting on theshell cause more acoustic radiation than normal forces, as the for-mer excite radiating shell waves more easily than the latter. Pathakand Stepanishen [8] used the classical integral transform techniquealong with the standard stationary phase method to address theproblem of acoustic harmonic radiation from a fluid-loaded infinitethick-walled cylindrical elastic shell subjected to arbitrary spatialloading. Constans et al. [9] used a material tailoring approach,based on finite element method to present a numerical design toolfor minimizing radiated sound power from a vibrating shell struc-ture driven with a point excitation harmonic force input. Ruzzeneand Baz [10] developed a finite element model to study the effectof stiffening rings and damping treatments on both the structuralvibration and noise radiation of fluid-loaded axisymmetric cylin-drical shells. Ko et al. [11] used the theory of elasticity to developa theoretical model for evaluating the reduction of structure-bornenoise generated by an axially symmetric ring force which is ap-plied on the interior of a coated cylindrical shell submerged inwater. Lecable et al. [12] employed the spatial Fourier transformin conjunction with the stationary phase method to develop a

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Nomenclature

aj outer radius of the jth layer in the FGM shellc speed of sound in the surrounding fluid mediumcp,cs velocities of dilatational and distortional waves in the

elastic mediumc½j�p ; c

½j�s dilatational and distortional wave velocities in the jth

layer of FGM shellfr ; fh radial and transverse driving forcesh total thickness of the shellhj thickness of the jth layer in the FGM shellj layer number in the FGM shellk = x/c acoustic wave number in the outer fluid mediumkp, ks compressional and shear elastic wave numbers in the

elastic mediumk½j�p ; k

½j�s compressional and shear elastic wave numbers in the

jth layer of FGM shelln mode numberp acoustic pressure in the outer fluid mediumq total number of layers in the FGM shellr radial coordinate in the cylindrical coordinate system�rj mean radius of the jth layer in FGM shellu displacement vector in the elastic mediumur, uh, uz radial, tangential and axial displacements in the elastic

mediumu½j�r ;u

½j�h ;u

½j�z radial, tangential and axial displacements in the jth

layer of FGM shellu½j�r;n;u

½j�h;n;u

½j�z;n modal amplitudes of the radial, tangential and axialdisplacement components in the jth layer of FGM shell

x, y, z Cartesian coordinate systemAn modal coefficient of the out-going radiated sound waveBn, Cn, Dn, En, Fn, Gn sound transmission coefficients in the elastic

mediumB½j�n ;C

½j�n ;D

½j�n ; E

½j�n ; F

½j�n ;G

½j�n sound transmission coefficients in the jth

layer of FGM shellFr,h radial and transverse applied load amplitudesHð1Þn cylindrical Hankel functions of the first kind of order nJn cylindrical Bessel function of the first kind of order nM½j�n local transfer matrix for the jth layer of FGM shellM0 applied circumferential moment amplitude

Mz circumferential moment drive

Q ½j�n coefficient matrix for the jth layer of FGM shellR observation point distance parameterTn global transfer matrix for the FGM shellU displacement vector in the surrounding fluid mediumVF volume fraction function of the metallic constituentW½j�

n field variable vector for the jth layer of FGM shell

X½j�n amplitude vector for the jth layer of FGM shellYn cylindrical Bessel function of the second kind of order na ceramic volume fraction ratioc power law variation exponent coefficientd() Dirac delta functionh azimuthal coordinate in the cylindrical coordinate sys-

temk;l Lame parameters of the elastic mediumk1;2;l1;2 Lame parameters for the inner and outer constituting

materialsk½j�;l½j� Lame parameters for the jth layer of FGM shelln Fourier transform parameterq fluid mass densityqs mass density of the elastic mediumq1,2 mass densities for the inner and outer constituting

materialsq½j�s mass density of the jth layer in FGM shellrrr, rrh, rrz radial and shear stress components in the elastic

medium

r½j�rr ;r½j�rh;r

½j�rz radial and shear stress components in the jth layer ofFGM shell

r½j�rr;n;r½j�rh;n;r

½j�rz;n modal amplitudes of radial and shear stress com-

ponents in the jth layer of FGM shellu polar angle in the cylindrical coordinate system/ scalar displacement potential for the elastic compres-

sional wavew vector displacement potential for the elastic shear wavex circular frequency

S.M. Hasheminejad, A. Ahamdi-Savadkoohi / Composite Structures 92 (2010) 86–96 87

normal-mode series solution for acoustic radiation from a thickcylindrical elastic shell when its external surface is subjected to apoint harmonic source. Skelton [13] presented a method for evalu-ating the sum of circumferential harmonics expressed in the formof infinite series of trigonometric functions for the radial, axial andcircumferential displacement components of an infinite elasticcylindrical shell immersed in compressible fluid and excited by atime-harmonic line force, in the asymptotic limit of heavy exteriorfluid-loading. More recently, Cuschieri [14] presented numericalresults for the far-field acoustic radiation as a function of frequencyand radiation angle for a fluid-loaded cylindrical shell with a com-pliant layer on its external surface, and excited by an internal ra-dial ring force, using a normally reacting impedance layer modelfor the compliant layer. Yan et al. [15] developed an analyticalmethod to study radiated sound power characteristics of (thevibrational power flow propagation in) an infinite, submerged,periodically stiffened, and structurally damped thin cylindricalshell, excited by ring forces and moments. Ramachandran andNarayanan [16] presented a simplified analytical method to predictthe modal density and the radiation efficiency of a longitudinallystiffened cylinder, by solving an eigenvalue problem based onstrain energies and kinetic energies of the total structure, for usein statistical energy analysis (SEA).

In recent years, the study of functionally graded materials(FGMs) has attracted a lot of attention. FGMs are advanced com-posites, microscopically engineered to have a smooth spatial vari-ation of material properties in order to improve overallperformance. This is achieved by fabricating the composite mate-rial to have a gradual spatial variation in the constituent materials’relative volume fractions and microstructure, thus tailoring itsmaterial composition based on functional performance require-ments [17]. The concept of FGM was first introduced in 1984 bya group of material scientists in Japan [17] as an alternative to lam-inated composite materials which show a mismatch in propertiesat material interfaces. FGM shell structures offer great promise inapplications where the operating conditions are severe, includingspacecraft heat shields, heat exchanger tubes, fusion reactors, stor-age tanks, pressure vessels, and general wear and corrosion resis-tant coatings in aerospace, automobile, marine, nuclear, anddefense industries. For example, thermal barrier shell or pipelinestructures may form from a mixture of ceramic and a metal [18].The composition is varied from a ceramic-rich surface to a metal-rich surface, with a desired variation of the volume fractions ofthe two materials in between the two surfaces in order to relaxthe residual stresses which may give rise to fracture and failureof components during their fabrications or service processes [19].

88 S.M. Hasheminejad, A. Ahamdi-Savadkoohi / Composite Structures 92 (2010) 86–96

The ceramic constituent of the material provides superior resis-tance to temperature, corrosion, oxidation and/or wear, while themetallic part possess ductility and mechanical shock resistance re-quired as a structural component [20]. A review on various aspectsof FGMs can be found in the monograph by Miyamoto et al. [21].Also, for a recent review on dynamic behavior of FGM and inhomo-geneous cylindrical shells the reader is referred to Ref. [22].

The above review clearly indicates that while there exist a nota-ble body of literature on acoustic radiation from submerged homo-geneous cylindrical shells subject to external loads, rigorousanalytic or numerical solutions involving a functionally graded cyl-inder seem to be nonexistent. The primary purpose of the currentwork is to fill this gap. Accordingly, we employ the spatial Fouriertransform along the shell axis and Fourier series expansion in thecircumferential direction in conjunction with the powerful transfermatrix solution technique to carry out an accurate analysis for radi-ation of acoustic waves by a thick-walled inhomogeneous hollowcylinder under the action of arbitrary time-harmonic radial/trans-verse driving forces and circumferential moment. The method ofstationary phase is used to evaluate the formal integral expressionobtained for the radiated pressure at an observation point in thefar-field. Particular attention is paid to assessment of the effectsof material compositional gradient, shell thickness, loading typeand excitation frequency on the radiated sound field. The proposedmodel is of academic interest basically due to its inherent value as acanonical problem in structural acoustics. It is of practical value forstructural acoustic engineers involved in the dynamic analysis anddesign of fluid-loaded thick-walled FGM cylindrical shells, tubes, orpipelines with possible applications in chemical industries, nuclearpower plants, submarine and offshore installations [23–25]. Thepresented accurate solution can serve as the benchmark for com-parison to solutions obtained by strictly numerical or asymptoticapproaches. It may be complemented by other techniques such asFourier synthesis method [26], series of resonance modes [27], ordirect time-domain approach [28,29], to solve the correspondingtime-dependent problems (i.e., the transient fluid–structure inter-action of submerged cylindrical shell structures experiencingtime-dependent mechanical forces or shock loads).

2. Formulation

2.1. Basic field equations

We consider the problem of sound radiation from an evacuatedinfinite elastic hollow cylinder, immersed in an acoustic medium ofmean density q and sound speed c, subjected to radial/transverseexternal loads and circumferential moment at its outer boundary,as shown in Fig. 1. The observation point is located at a distanceR, a polar angle u from the centerline of the shell (z-axis), and anazimuthal angle h with respect to the x-axis. The pressure fluctua-tion in the surrounding acoustic medium, p(r, h, z) is governed bythe linear wave equation [30]

r2pþ k2p ¼ 0; ð1Þ

in the domain defined by r P aq, �p 6 h 6 p, and �1 6 z 6 +1,where k = x/c is the acoustic wave number, x is the angular fre-quency, and the harmonic time dependence of exp(�ixt) is sup-pressed here and henceforth. Also, the fluid displacement vector iswritten as [30]

U ¼ ð1=qx2Þrp: ð2Þ

The wave motion in the isotropic elastic shell material is gov-erned by the classical Navier’s equation [31]

qs@2u@t2 ¼ lr2uþ ðkþ lÞrðr � uÞ; ð3Þ

where qs is the solid material density, (k;l) are the Lame parame-ters, and u is the vector solid material displacement that can advan-tageously be expressed as sum of the gradient of a scalar potentialand the curl of a vector potential:

u ¼ r/þr� w; ð4Þ

with the condition r � w = 0. Therefore, the displacement compo-nents of the elastic medium in the cylindrical coordinate systemmay be expressed in terms of appropriate scalar potentials as [31]

urðr; h; z;xÞ ¼@/@rþ 1

r@wz

@h� @wh

@z;

uhðr; h; z;xÞ ¼1r@/@hþ @wr

@z� @wz

@r;

uzðr; h; z;xÞ ¼@/@zþ @wh

@rþ 1

rwh �

1r@wr

@h;

ð5Þ

where /(r, h, z) and w(r, h, z), respectively, satisfy Helmholtz-typeequations:

ðr2 þ k2pÞ/ ¼ 0;

ðr2 þ k2s Þw ¼ 0;

ð6Þ

where kp,s = x/cp,s are the elastic wave numbers, in whichcp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ 2lÞ=qs

pand cs ¼

ffiffiffiffiffiffiffiffiffiffiffil=qs

pare the velocities of dilatational

and distortional waves, respectively. Also, the pertinent stress com-ponents are [31]

rrrðr; h; z;xÞ ¼ kr2/þ 2l @2/@r2 þ

1r@2wz

@r@h� 1

r2

@wz

@h� @

2wh

@r@z

!;

rrhðr; h; z;xÞ ¼ l 2r@2/@r@h

� 2r2

@/@hþ @

2wr

@r@z� 1

r@wr

@z� @

2wz

@r2

þ1r@wz

@r� 1

r@2wh

@z@h

!;

rrzðr; h; z;xÞ ¼ l 2@2/@r@z

� 1r@2wr

@r@hþ 1

r2

@wr

@hþ 1

r@wh

@r� 1

r2 wh

þ @2wh

@r2 �@2wh

@z2 þ1r@2wz

@z@h

!:

ð7Þ

The solution of the problem may be obtained by the introduc-tion of the spatial Fourier transform along the shell axis [32]

~f ðr; h; n;xÞ ¼Z 1

�1f ðr; h; z;xÞe�inzdz; ð8aÞ

f ðr; h; z;xÞ ¼ 12p

Z 1

�1

~f ðr; h; n;xÞeinzdn; ð8bÞ

where n is the Fourier transform parameter. After application of (8a)to the Helmholtz Eqs. (1) and (6), the resulting partial differentialequations can readily be solved by the standard method of variableseparation [31]. Accordingly, the transformed acoustic pressure andelastic displacement potentials may respectively be expanded as

~pðr;h;n;xÞ¼Xþ1

n¼�1~pnðr;n;xÞeinh;

~/ðr;h;n;xÞ¼ 1qsx2

Xþ1n¼�1

½Bnðn;xÞJnðgprÞþCnðn;xÞYnðgprÞ�einh;

~wrðr;h;n;xÞ¼1

qsx2

Xþ1n¼�1

½Dnðn;xÞJnþ1ðgsrÞþEnðn;xÞYnþ1ðgsrÞ�einh;

~whðr;h;n;xÞ¼�1

qsx2

Xþ1n¼�1

½Dnðn;xÞJnþ1ðgsrÞþEnðn;xÞYnþ1ðgsrÞ�einh;

~wzðr;h;n;xÞ¼1

qsx2

Xþ1n¼�1

½Fnðn;xÞJnðgsrÞþGnðn;xÞYnðgsrÞ�einh;

ð9Þ

where ~pnðr; n;xÞ ¼ Anðn;xÞHð1Þn ðgrÞ; An through Gn areunknown Fourier coefficients, g ¼ ðk2 � n2Þ1=2

;gp ¼ ðk2p � n2Þ1=2

;

qa

rf θθf

zM

z

y

x

FGM Cylinder

Fluid Medium

R

P

0aϕθ

h

0a 1a qa

γ = 5

γ = 0.2

r

Fig. 1. Problem configuration.


gs ¼ ðk2s � n2Þ1=2

; Hð1Þn is the Hankel function of the first kind, and Jn

and Yn are the cylindrical Bessel functions of first and second kind,respectively [33]. Also, it is assumed that the Fourier transform ofthe radiated acoustic pressure in the fluid satisfies the classicalSommerfeld radiation condition [2].

2.2. T-matrix approach and boundary conditions

At this point we consider the cylindrical shell to be of variablematerial properties (functionally graded) along the radial direction,and of uniform thickness h, inner radius a0 and outer radius aq (seeFig. 1). The specific steps taken in constructing the model for mak-ing use of the transfer matrix solution technique are well describedin Ref. [22]. Adopting a laminate model, the functionally gradedshell is assumed to be composed of q layers of homogeneous iso-tropic materials which are perfectly bonded at their interfacesand lined up such that their axes of symmetry coincide with eachother. The Lame constants, ðk½j�;l½j�Þ; and mass density, q½j�s ; withinthe jth layer of inner radius aj�1 (j = 1, . . . , q), outer radius aj, anduniform thickness hj = aj � aj�1 may described by the simple ruleof mixture as [34]

k½j� ¼ VFð�rjÞk1 þ ½1� VFð�rjÞ�k2;

l½j� ¼ VFð�rjÞl1 þ ½1� VFð�rjÞ�l2;

q½j�s ¼ VFð�rjÞq1 þ ½1� VFð�rjÞ�q2;

ð10Þ

where �rj ¼ ðaj þ aj�1Þ=2;VFð�rjÞ is the volume fraction of the ‘‘inner”material in the jth layer of the multilayered shell, and ðk1;2;l1;2Þ,and q1,2 are the Lame parameters and mass density of inner andouter constituting materials, respectively.

Substitution of the scalar potentials Eq. (9) into the relations (5)and (7), leads to the matrix form of the formal solutions for thetransformed field variables associated with the jth layer [35], i.e.,

~W½j�n ¼ Q ½j�n X½j�n ; ð11Þ

where

~W½j�n ¼ ½~u½j�r;n; ~u

½j�h;n; ~u

½j�z;n; ~r½j�rr;n; ~r½j�rh;n; ~r½j�rz;n�

T;

X½j�n ¼ ½B½j�n ;C

½j�n ;D

½j�n ; E

½j�n ; F

½j�n ;G

½j�n �

T;

and the elements of the coefficient matrix Q ½j�n are given in theAppendix. The above matrix relation can advantageously be special-ized at the inner and outer radii of the jth layer as

~W½jþ�n ¼ Q ½jþ�n X½j�n ;

~W½j��n ¼ Q ½j��n X½j�n ;

ð12Þ

where the superscripts [j+] and [j�] denote the quantities evaluatedat r = aj and r = aj�1, respectively. The above two equations can besupplemented with the continuity conditions between each inter-face layer, i.e., ~W½jþ�

n ¼ ~W½ðjþ1Þ��n at r ¼ aj. Thus, by eliminating the

common amplitude vector X½j�n in (12), the field vector ~W½jþ�n may

be related to ~W½j��n by

~W½jþ�n ¼M½j�

n~W½j��

n ; ð13Þ

where M½j�n ¼ Q ½jþ�n ½Q

½j��n �

�1 is the local transfer matrix for jth layer,which relates the field variables at its outer surface to those atthe inner surface. Subsequently, by invoking the continuity condi-tions between all interface layers, the field variables at the outer ra-dius of the layered shell (i.e., at r = aq) is related to those at the innerradius (i.e., at r = a0) via a 6 � 6 global transfer matrix, Tn by

~W½qþ�n ¼ Tn

~W½1��n ; ð14Þ

where Tn ¼M½q�n M½q�1�

n � � �M½1�n :

The unknown coefficients An through Gn, and the elements ofthe field variable vectors, ~W½1��

n and ~W½qþ�n ; can be determined from

the appropriate boundary conditions imposed at the inner (r = a0)and the outer (r = aq) surfaces of the multilayered shell. These con-ditions are explicitly written as [8,12]:

– vanishing of the radial and tangential stresses at the innersurface


~rrrðr ¼ a0; h; n;xÞ ¼ ~rrhðr ¼ a0; h; n;xÞ ¼ ~rrzðr ¼ a0; h; n;xÞ ¼ 0;

ð15aÞ

– continuity of the normal fluid and solid displacements at theouter surface
~urðr ¼ aq; h; n;xÞ ¼ ~Urðr ¼ aq; h; n;xÞ; ð15bÞ
– equilibrium of the radial stress with the applied external radialload, circumferential moment (or radial couple) and fluid pres-sure at the outer surface

~rrrðr ¼ aq; h; n;xÞ ¼ ~f rðh; n;xÞ þ ~Mzðh; n;xÞ � ~pðr ¼ aq; h; n;xÞ;ð15cÞ

– equality of the tangential stress and applied external transverseload at the outer surface

~rrhðr ¼ aq; h; n;xÞ ¼ ~f hðh; n;xÞ; ð15dÞ

– vanishing of the tangential stress at the outer surface

~rrzðr ¼ aq; h; n;xÞ ¼ 0; ð15eÞ

Finally, by making use of the global transfer relation (14), appli-cation of boundary conditions (15) leads to the following impor-tant matrix equation:

S1;n T11;n T12;n T13;n T14;n T15;n T16;n

S2;n T41;n T42;n T43;n T44;n T45;n T46;n

0 T51;n T52;n T53;n T54;n T55;n T56;n

0 T61;n T62;n T63;n T64;n T65;n T66;n

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

26666666666666664

37777777777777775

An

~u½1��r;n

~u½1��h;n

~u½1��z;n

~r½1��rr;n

~r½1��rh;n

~r½1��rz;n

2666666666666666664

3777777777777777775

¼

0

~f r;n þ ~Mz;n þ ~pn

~f h;n

0

0

0

0

266666666666666664

377777777777777775

; ð16Þ

where Tij,n(i, j = 1, . . . , 6) are elements of the global transfer matrix,

S1;n ¼ kinHð1Þ0

n ðkaqÞ; S2;n ¼ Hð1Þn ðkaqÞ; and ~f r;nðn;xÞ;~f h;nðn;xÞ; and~Mz;nðn;xÞ are, respectively, the transformed modal components ofthe applied radial force, transverse force and circumferential mo-ment, which will be defined below. Clearly, the transformed vari-

ables ~W½j�n and the Fourier coefficients X½j�n can readily be

determined from relations (14) and (11), once the global transfermatrix is computed.

2.3. External forcing and far-field pressure

At this point, we consider concentrated radial and transverseforces and circumferential moment acting at the external surfacepoint (r = aq; h = h0; z = z0) of the inhomogeneous shell, whichmay respectively be expanded in the form (note that h0 = z0 = 0 isassumed in Fig. 1) [5]

frðh;z;xÞ ¼ FrðxÞdðz� z0ÞXþ1

m¼�1d½aqðh�ðh0þ2pmÞÞ�

fhðh;z;xÞ ¼ FhðxÞdðz� z0ÞXþ1


Mzðh;z;xÞ ¼ FrðxÞdðz� z0ÞXþ1


n�Xþ1

m¼�1d½aqðh�ðh0� dhþ2pmÞÞ�

oð17Þ

where Fr(x) and Fh(x) are the radial and transverse load ampli-tudes, d() is the Dirac delta function, and one notes that, the circum-ferential point moment drive at (h0, z0) is constructed by using twoout-of-phase radial point forces of magnitude Fr; one at (h0, z0) andthe other at (h0 � dh, z0), where it is assumed that the separation be-tween the two point drives that form the moment is infinitesimal[5]. Fourier transformation of above expressions with respect to z,with subsequent application of the Poisson summation formula[36]:

Xþ1m¼�1

gðxþ 2pmÞ ¼ 12p

Xþ1n¼�1

gðnÞeinx; ð18Þ

where gðnÞ ¼R1�1 gðxÞe�inxdx; and after some straight forward

manipulations, Eq. (17) reduce to

~f rðh; n;xÞ ¼Xþ1

n¼�1

~f r;nðn;xÞeinh;

~f hðh; n;xÞ ¼Xþ1

n¼�1

~f h;nðn;xÞeinh;

~Mzðh; n;xÞ ¼Xþ1

n¼�1

~Mz;nðn;xÞeinh;

ð19Þ

where ~f r;nðn;xÞ ¼ ð2paqÞ�1e�iðnz0þnh0ÞFrðxÞ; ~f h;nðn;xÞ ¼ ð2paqÞ�1

e�iðnz0þnh0ÞFrðxÞ; and ~Mz;nðn;xÞ ¼ �inð2pa2qÞ�1e�iðnz0þnh0ÞM0ðxÞ; in

which M0(x) = Fr(x)aqdh is the magnitude of the applied moment.Finally, by applying the inverse Fourier transform (8b) to first of

(9), the radiated acoustic pressure may be written as

pðr; h; z;xÞ ¼ 12p

Z þ1

�1~pðr; h; n;xÞeinzdn

¼ 12p

Z þ1

�1

Xþ1n¼0

Anðn;xÞHð1Þn ðgrÞ cosðnhÞeinzdn: ð20Þ

In the near field, analytic evaluation of the above integral is verycomplicated. However, it can always be approximated in the far-field by the method of stationary phase [2]. This technique is baseon the fact that the resultant contribution of ranges of integrationwhere the modulus of the integrand varies slowly with n while thephase fluctuates rapidly is relatively small, because of cancellationbetween neighboring regions of opposite phase and nearly equalamplitude. The main contribution to the above integral arises fromthe region of the integration where the phase of the integrandchanges slowly with n, thus minimizing the cancellation. The cor-responding value of n is the point of stationary phase defined bythe condition n = kcos u [2]. Following the standard procedure out-lined in Ref [2], the stationary phase far-field approximation tointegral (20) is obtained as (see also [8])

limR!1

pðR;h;u;xÞ¼ 1p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiksinu

p Xþ1n¼0

i�ðnþ1ÞAnðkcosu;xÞcosðnhÞ" #

eikR1

R1:

ð21Þ

This completes the necessary background required for the anal-ysis of the problem. Next we consider some numerical examples.


3. Numerical results

To illustrate the nature and general behavior of the solution, weconsider some numerical examples in this section. Realizing thenumber of parameters involved here and our computing limita-tions, we confine our attention to a particular model. The inhomo-geneous shell (total thickness h = h1 + h2 + � � � + hq) is assumed to beevacuated, submerged in water (c = 1480 m/s, q = 1000 kg/m3),and mechanically excited at the point h = h0 = z = z0 = 0 on its outersurface (r = aq). The mechanical properties of the shell are assumedto vary smoothly and continuously with the change of volume con-centrations of the constituting materials (zirconia and aluminum)in the radial direction according to a power law distribution. Thephysical properties of the constituents are given in Table 1 [37].The volume fraction of aluminum in the FGM shell may be variedfrom 100% on the inner interface (at r = a0) to zero on the outerinterface with water (at r = a0 + h = aq). Two distinct material gradi-ent profiles with the following assumptions for volume fraction ofinner material (aluminum) across the shell thickness, are studied:

VFðrÞ ¼ 1� r � a0

h

� �c; ðc ¼ 0:2;5Þ; ð22Þ

where a0 6 r 6 aq (see Fig. 1). It is clear that c = 0.2 simulates a me-tal-rich composition (MR), and c = 5 denotes a ceramic-rich profile(CR) in the radial direction. Also considered are two equivalent bi-laminate ZrO2–Al shells containing equal volume fractions of con-stituent materials as in cases of the ceramic rich (CR) and metal rich(MR) graded shells, which are denoted henceforth by ‘‘BLCR” and‘‘BLMR”, respectively. The calculated physical and geometric datafor the two constituents present in the equivalent bi-laminate shellsare given in Table 2.

A general Mathematica code was constructed for computing thetransfer matrix, Tn, treating the system of Eq. (16), and to numeri-cally evaluate the stationary phase far-field approximation to inte-gral (20) via summation (21). The computations were performedon a network of personal computers, and the convergence ofnumerical solutions was secured in a simple trial and error man-ner, by increasing the number of modes, n, while looking for stea-diness or stability in the numerical values of the solutions. It wasfound that by using a truncation constant of nmax = 20, uniformconvergence can be obtained in all loading situations. Also, thefar-field value of the radial coordinate for calculating the far-fieldradiated pressure amplitude, |p(r = R1, h, u, x)|, was simply chosenby making several computer runs while seeking for the conver-

Table 1The physical properties of the constituents [37].

Material qs (kg/m3) k (GPa) l (GPa)

Aluminium (Al) 2706 57.09 26.69Zirconia (ZrO2) 5700 129 94.8

Table 2The calculated physical and geometric data for the two constituents present in theequivalent bi-laminate shells.

Shell type a a0/aq

hZrO2 =aq hAl/aq uscðdegreeÞ up

c ðdegreeÞ

BLMR(c = 0.2)

h = 0.1aq 0.171 0.9 0.0163 0.0837 64.37 77.25h = 0.25aq 0.177 0.75 0.0396 0.2104 64.44 77.27h = 0.5aq 0.192 0.5 0.0748 0.4252 64.60 77.31

BLCR (c = 5) h = 0.1aq 0.840 0.9 0.0832 0.0168 68.18 78.40h = 0.25aq 0.850 0.75 0.2076 0.0424 68.28 78.41h = 0.5aq 0.873 0.5 0.4124 0.0876 68.30 78.44

gence of the results. The choice of R1 = 10aq was found to be ade-quate for all cases considered.

Before presenting the main numerical results, we shall establishthe overall validity of the work. Accordingly, we first computed thenormalized far-field pressure amplitude, |p(R1 = 10aq, h = 0, u, x)/Fr(x)|, versus dimensionless frequency (kaq) and axial parameter(z/aq) for a radially-driven (h = h0 = p; z = z0 = 0) water-submergedhomogeneous steel shell, by greatly increasing the number of lay-ers (i.e., using q = 100) and setting VF ? 1, h/aq = 0.02, q[j] =7900 kg/m3, k½j� ¼ 113 GPa; l½j� ¼ 75:9 GPa ðfor j ¼ 1;2; . . . ;100Þ,in our general Mathematica code. The outcome, as shown inFig. 2a shows excellent agreement with that displayed in Fig. 4 ofRef. [12]. As a further check, we used our general code to computethe total surface displacement amplitude, |u(r = aq; h = p/2; x)|(m),versus frequency (f = x/2p, Hz) for the very thick bi-laminateZrO2–Al shells of Table 2 (h = 0.5aq), submerged in air (c = 340 m/s, q = 1.2 kg/m3), under the action of a pair of two-dimensionalconstant amplitude (Fr = 1 MPa) diametrical distributed loads(�#0 < h < #0 and p � #0 < h < p + #0; #0 = p/6), which may be repre-sented by a Fourier series expansion of the form (see Fig. 2b):

Fig. 2. (a) The normalized far-field pressure amplitude, |p(R1 = 10aq, h = 0, u, x)/Fr(x)|, versus dimensionless frequency and axial parameter for a radially-driven(h = h0 = p; z = z0 = 0) water-submerged homogeneous steel shell. (b). Comparison ofthe computed total surface displacement amplitude, |u(r = aq; h = p/2; x)|(m), forthe thick bi-laminate ZrO2–Al shells of Table 2 (h = 0.5aq), submerged in air, andunder the action of a pair of two-dimensional constant amplitude (Fr = 1 MPa)diametrical distributed loads (#0 = p/6) with the numerical results calculated byusing the commercial finite element code ABAQUS [38].


fr;nðxÞ ¼2FrðxÞ#0=p ðn ¼ 0Þ2½1þ ð�1Þn�FrðxÞ sinðn#0Þ=np ðn > 0Þ:

�ð23Þ

The results, as displayed in Fig. 2b, exhibit excellent agreementswith those calculated by using the commercial finite element codeABAQUS [38].

Fig. 3 displays the normalized far-field pressure amplitude|p(R1, h, u, x)/C(x)|, at the observation point ðR1 ¼ 10aq; h ¼0;u ¼ 80�Þ versus dimensionless frequency (kaq) for the radially-driven (C = Fr), transversely-driven (C = Fh), and moment-driven(C = M0) FGM shell for selected thickness parameters (h/aq = 0.1,0.25, 0.5), and material gradient profiles (MR: metal rich, CR: cera-mic rich, BLMR: bilaminate metal rich, BLCR: bilaminate ceramicrich). Here, it should be noted that the polar angle of observationpoint is chosen such that u > up;s

c ; where up;sc ¼ cos�1ðc=cp;sÞ de-

notes the cutoff angle of the longitudinal and shear waves [2],which is determined by the ratio of the corresponding elastic wavespeed to that of the sound speed. The last column in Table 2 dis-

Fig. 3. The normalized far-field pressure amplitude jpðR1 ¼ 10aq; h ¼ 0;u ¼ 80� ;xÞ=Cðdriven (C = Fh), and moment-driven (C = M0) FGM cylinders for selected thickness pabilaminate metal rich, BLCR: bilaminate ceramic rich).

plays the cutoff angles for the equivalent ZrO2–Al FGM shell. It isclear that by choosing u ¼ 80� > up;s

c ; the purely reactive standingwaves are avoided, and contributions from both shear and com-pressional waves are expected to appear (i.e., we remain withinthe primarily propagating conical wave regime). The most impor-tant observations are as follows. The far-field radiated pressuremagnitudes are very small at low to intermediate frequencies,nearly regardless of the thickness parameter. Increasing the shellthickness leads to an overall decrease of the pressure magnitudesin an average sense, as well as a noticeable rightward shift in res-onance peaks appearing in the intermediate to high frequencyrange (kaq > 3). Furthermore, while there is nearly no distinctionbetween the bilaminate and FGM radiated pressure curves forshells of h/aq = 0.1 wall thickness, the difference between the pres-sure curves becomes gradually noticeable as the thickness param-eter increases, especially for the thickest shell (h/aq = 0.5) withmetal rich (MR) profile at high frequencies. In particular, in con-trast with the ceramic rich (CR) case, increasing the shell wall

xÞj; versus dimensionless frequency for the radially-driven (C = Fr), transversely-rameters and material gradient profiles (MR: metal rich, CR: ceramic rich, BLMR:


thickness seems to cause a noticeable drop in the overall radiatedpressure amplitudes associated with the BLMR cylinder relative to

Fig. 4. The normalized polar far-field pressure directivity pattern, |p(R1 = 10aq; h, u, x)/Cwall thickness h/aq = 0.1, at selected dimensionless frequencies.

the MR cylinder. This may be linked to the somewhat larger changein the ceramic volume fraction a ¼ VZrO2=ðVZrO2 þ VAlÞ with shell

(x)|, for the radially-driven, transversely-driven, and moment-driven FGM shells of


wall thickness in the BLMR cylinder, in comparison with that in theBLCR cylinder (see second column in Table 2). The above situationwill be reversed, if the inner and outer materials were inter-changed. In the latter case, there will be notable increase in thepressure amplitudes of the BLCR cylinder in comparison with theCR cylinder (numerical results are not shown for briefness).

It is clear from the subplots that the shell-borne elastic wavescontribute significantly to the far-field. In all cases, the frequencydependence of the far-field pressure response is dictated by contri-butions from resonances of helical waves, which are seen in thesefigures as pressure maxima/minima. In particular, as the shell-borne waves diffract many times when they propagate aroundthe shell, the multiple diffractions coherently enforce each other,which is the cause of the rapid variations (resonances) in the pres-sure spectra shown in the figure. These shell-borne waves carrywith them information about the shell properties [7], their radia-tion covers a wide spatial domain, unaffected by the shadowing ef-fects of the shell, and they may overwhelm in amplitude the directradiation from the forces in some directions, the latter following adirect fluid-borne propagation path from the driving forces to thefar field. The radiated pressure spectrums associated with the ra-dial loading appear to be very similar to those of the circumferen-tial moment drive, while the transverse loading case exhibits acharacteristically different behavior. In particular, the overallfar-field pressure magnitudes associated with the radial drive areconsiderably smaller than those of the transverse drive (i.e., forces

Fig. 5. The normalized azimuthal far-field pressure directivity pattern, |p(R1 = 10aq; htransversely-driven, and moment-driven FGM shells of wall thickness h/aq = 0.1.

lying tangential to the shell surface cause more radiation thanforces acting in the normal direction). This may be explained bythe fact that, although normal forces energize more transversevibrations, but most of this vibrational energy is acoustically harm-less; it is in the form of flexural waves that, at frequencies belowthe coincidence frequency of the shell plating, are subsonic inphase speed and, hence, are not radiating [2]. This way, more ofthe energy imparted to the shell goes into formation of evanescentwaves in the structural near field, making this type of drive an inef-ficient acoustic radiator. Furthermore, the relatively broad oscilla-tory pressure spectra associated with the radial and moment drives(first and third columns in Fig. 3), appear to primarily contain con-tributions from the directly radiating waves. On the other hand, therelatively sharp and high amplitude spectral peaks appearing inthe far-field pressure response of the transversely excited shell(second column in Fig. 3) indicate that the radiating shell-bornewaves associated with various modes of propagation on theshell-fluid boundary are primarily energized. In particular, forcestangential to the shell surface appear to most efficiently exciteshear and compressional waves in the shell, which in turn leadto most efficient sound radiation [7]. Also, it is clear that increasingthe shell thickness causes an increase in the sharpness of the far-field pressure curves associated with a radial or moment drive(see first and third columns in Fig. 3). This implies that the directradiation effects gradually diminish as the shell thickness is in-creased. Lastly, it appears that the difference between the far-field

; u = p/2; x)/C(x)|, at selected dimensionless frequencies for the radially-driven,


pressure curves of the ceramic rich (CR) and metal rich (MR) shellsin case of transverse excitation is considerably more that those forthe radial or moment excitation. This may be linked to the propa-gation of the highly property-dependent shell borne waves in caseof transverse excitation.

Fig. 4 displays the normalized far-field pressure directivitypattern, |p(R1 = 10aq; h, u, x)/C(x)|, in the x–z plane (�p/2 <u < p/2), for the radially-driven, transversely-driven, and mo-ment-driven FGM shells of wall thickness h/aq = 0.1, at selecteddimensionless frequencies (kaq = 0.1, 1, 10). Far-field directivityis useful and instructive not only because it shows the spatialdistribution of radiated acoustic energy, but also because it re-flects the nature of that which produces the radiation. Commentssimilar to the above remarks can readily be made. The mostimportant distinctions are as follow. At the lowest excitation fre-quency (kaq = 0.1), the pressure distributions show the lowestdirectionality with a dipole-like pattern, while they nearly en-tirely overlap for the radially-excited and moment-excited shells.The difference between the metal rich and ceramic rich patternsfor the transversely-driven shell may be linked to the excitationof highly property-dependent shell borne (helical) waves men-tioned earlier. As the excitation frequency increases (kaq = 1,10), the directionality of dipole-like patterns gets severely dis-torted (i.e., it cannot be described by any simple pattern), andthe strong overlapping effect formerly observed in the pressuredirectivity of the radially-excited and moment-excited shells(i.e., for kaq = 0.1) no longer exists. This can be explained bythe emergence of the property-dependent shear and compres-sional shell-borne waves at intermediate and high excitation fre-quencies. Furthermore, the nearly symmetric far-field directivityplots for the transversely-excited shell (second column inFig. 4), implies that direct radiations from the tangential forceare negligible in comparison with those from helical waves. Also,one can readily note the absence of any noticeable far-field radi-ation below the cut-off angle of shell-borne waves in the trans-versely-excited shell ðus

c � 64�Þ; even at kaq = 10, and regardlessof FGM material profile. As for the radially- or moment-drivenshells, on other hand, no helical wave radiation can reach thefar-field below the cut-off angle within the insonified regionð0 < u < us

cÞ; and there is a notable primary lobe observed inthis region. Keeping in mind that the shell-borne (helical) wavesare not affected by the shadowing effects of the shell [7], the for-mation of this primary lobe is linked to the direct radiation ef-fects discussed earlier (also, note that these effects completelydisappear below the cut-off angle within the shadow region).Thus, within the insonified region, all involving mechanisms areexpected to contribute to the far field, the relative importanceof direct and helical wave radiation depends on both the obser-vation angle and the orientation of the excitation force.

Fig. 5 displays the normalized far-field pressure directivitypattern, |p(R1 = 10aq; h; u = p/2; x)/C(x)|, in the x–y plane(0 < h < p) at selected dimensionless frequencies (kaq = 0.1, 1,10) for the radially-driven, transversely-driven, and moment-dri-ven FGM shells of wall thickness h/aq = 0.1. Many of the previ-ously made comments also apply here. The only otherobservation is perhaps that the far-field directivity of the basi-cally softer metal rich radially- and moment-driven FGM shellsat the highest excitation frequency (kaq = 10) in the insonified re-gion (0 < h < p/2) is perceptibly lower than that of the harderceramic rich shell. This point to the importance of direct radia-tion effects in the former case (e.g., note the smoother directivitypattern in the insonified region of the metal rich shell). On theother hand, the essential invariance of pressure directivities inthe insonified region with respect to the material gradient profileof the transversely-excited shell points to the insignificance of di-rect radiation effects in this case.

4. Conclusions

An analytical vibro-acoustic model based on the three-dimen-sional theory of elasticity is developed and exercised for the acous-tic radiation from a fluid-loaded functionally graded infinitehollow cylinder driven by concentrated harmonic loads. The meth-od of stationary phase is used to numerically evaluate the formalexpression obtained for the radiated pressure in the far field. Themost important observations are summarized as follows:

� Increasing the cylinder thickness leads to an overall decrease inthe far-field pressure amplitudes, as well as a noticeable right-ward shift in the spectral (resonance) peaks. The pressure spec-trum of the FGM cylinder nearly overlaps with that of theequivalent bilaminate cylinder, even for cylinders of moderatewall thickness. As the cylinder thickness is greatly increased,the difference becomes detectable, especially for the very thickmetal rich cylinder at high excitation frequencies.

� The moderately broad oscillatory pressure spectra computed forthe radial and circumferential moment drives are found to con-tain contributions from the directly radiating waves. On theother hand, the relatively sharp and high amplitude spectralpeaks corresponding to the transversely excited cylinder pointto the excitation of radiating shell-borne (helical) waves. Theradiation of these waves, which carry with them informationabout the material properties, is unaffected by the shadowingeffects of the cylinder, and overwhelm the direct radiationeffects in some directions, making the transversely excited cyl-inder a more efficient acoustic radiator. Also, increasing the cyl-inder thickness causes a notable increase in the sharpness of thefar-field pressure spectrum curves associated with the radial/moment drives, implying the fall of direct radiation effects withshell thickness.

� At very low excitation frequencies, the pressure directivity dis-tributions show the lowest directionality with a dipole-like pat-tern, and the metal rich patterns for the radial and momentexcitations nearly entirely overlap with the ceramic rich ones.Conversely, the clear distinction between the metal rich andceramic rich patterns for the transversely-driven cylinder in awide range of frequencies is linked to the excitation of highlyproperty-dependent shell borne waves.

� The nearly symmetric far-field polar (u�) directivity plots with aweak shadowing effect for the transversely-excited cylinder fur-ther indicate that direct radiation from the tangential force isnegligible in comparison with the radiation due to helical waves.Also, lack of any noticeable far-field radiation below the cut-offangle of shell-borne waves regardless of FGM material profile,even at high excitation frequencies, is noted for the trans-versely-excited cylinder. On the contrary, a notable primary lobeis observed at the highest excitation frequency for the radially-or moment-driven cylinder below the cut-off angle within theinsonified region, which is linked to the direct radiation effects.

� The smoother (lower directionality) and generally higher ampli-tude far-field azimuthal (h�) pressure distributions in the inson-ified region of the basically softer metal rich radially- andmoment-driven FGM cylinders at the highest excitation fre-quency, in comparison with those of the harder ceramic rich cyl-inder, is linked to the relative significance of direct radiationeffects for the metal rich cylinder.

� Finally, it may be concluded that, except for the very thick cylin-ders at high excitation frequencies (h/aq > 0.5, kaq > 10), the far-field acoustic radiation response of relatively complex FGM cyl-inders may rather accurately be approximated with those ofequivalent bi-laminate cylinders, especially for the radially-and moment-driven cylinders.


Appendix A

Elements of the Q ½j�n matrix are as follows:

Q ½j�11;n ¼12gp½Jn�1ðg½j�p rÞ � Jnþ1ðg½j�p rÞ�;

Q ½j�12;n ¼12gp½Yn�1ðg½j�p rÞ � Ynþ1ðg½j�p rÞ�;

Q ½j�13;n ¼ nJnþ1ðg½j�s rÞ; Q ½j�14;n ¼ nYnþ1ðg½j�s rÞ;

Q ½j�15;n ¼inr

Jnðg½j�s rÞ; Q ½j�16;n ¼inr

Ynðg½j�s rÞ;

Q ½j�21;n ¼ �inr

Jnðg½j�p rÞ; Q ½j�22;n ¼ �inr

Ynðg½j�p rÞ;

Q ½j�23;n ¼ nJnþ1ðg½j�s rÞ; Q ½j�24;n ¼ nYnþ1ðg½j�s rÞ;

Q ½j�25;n ¼ �12g½j�s ½Jn�1ðg½j�s rÞ � Jnþ1ðg½j�s rÞ�;

Q ½j�26;n ¼ �12g½j�s ½Yn�1ðg½j�s rÞ � Ynþ1ðg½j�s rÞ�;

Q ½j�31;n ¼ inJnðg½j�p rÞ; Q ½j�32;n ¼ inYnðg½j�p rÞ;Q ½j�33;n ¼ ig½j�s Jnðg½j�s rÞ; Q ½j�34;n ¼ ig½j�s Ynðg½j�s rÞ;Q ½j�35;n ¼ 0; Q ½j�36;n ¼ 0;

Q ½j�41;n ¼ �ðk½j�k2½j�

p þ 2l½j�g2½j�p ÞJnðg½j�p rÞ

þ 1rl½j�g½j�p ðn� 1ÞJn�1ðg½j�p rÞ þ ðnþ 1ÞJnþ1ðg½j�p rÞ

h i;

Q ½j�42;n ¼ �ðk½j�k2½j�

p þ 2l½j�g2½j�p ÞYnðg½j�p rÞ

þ 1rl½j�g½j�p ðn� 1ÞYn�1ðg½j�p rÞ þ ðnþ 1ÞYnþ1ðg½j�p rÞ

h i;

Q ½j�43;n ¼ 2l½j�n g½j�s Jnðg½j�s rÞ � 1rðnþ 1ÞJnþ1ðg½j�s rÞ

� �;

Q ½j�44;n ¼ 2l½j�n g½j�s Ynðg½j�s rÞ � 1rðnþ 1ÞYnþ1ðg½j�s rÞ

� �;

Q ½j�45;n ¼irl½j�n g½j�s ðJn�1ðg½j�s rÞ � Jnþ1ðg½j�s rÞÞ � 2

rJnðg½j�s rÞ

� �;

Q ½j�46;n ¼irl½j�n g½j�s ðYn�1ðg½j�s rÞ � Ynþ1ðg½j�s rÞÞ � 2

rYnðg½j�s rÞ

� �;

Q ½j�51;n ¼irl½j�n 2

rJnðg½j�p rÞ � g½j�p ðJn�1ðg½j�p rÞ � Jnþ1ðg½j�p rÞÞ

� �;

Q ½j�52;n ¼irl½j�n 2

rYnðg½j�p rÞ � g½j�p ðYn�1ðg½j�p rÞ � Ynþ1ðg½j�p rÞÞ

� �;

Q ½j�53;n ¼ l½j�n g½j�s Jnðg½j�s rÞ � 2rðnþ 1ÞJnþ1ðg½j�s rÞ

� �;

Q ½j�54;n ¼ l½j�n g½j�s Ynðg½j�s rÞ � 2rðnþ 1ÞYnþ1ðg½j�s rÞ

� �;

Q ½j�55;n ¼1rg½j�s l½j� g½j�s rJnðg½j�s rÞ þ ðnþ 2ÞJnþ1ðg½j�s rÞ þ ðn� 2ÞJn�1ðg½j�s rÞ

� ;

Q ½j�56;n ¼1rg½j�s l½j� g½j�s rYnðg½j�s rÞ þ ðnþ 2ÞYnþ1ðg½j�s rÞ þ ðn� 2ÞYn�1ðg½j�s rÞ

� ;

Q ½j�61;n ¼ g½j�p l½j�n Jn�1ðg½j�p rÞ � Jnþ1ðg½j�p rÞh i

;

Q ½j�62;n ¼ g½j�p l½j�n Yn�1ðg½j�p rÞ � Ynþ1ðg½j�p rÞh i

;

Q ½j�63;n ¼ l½j� ðg½j�s � nÞðg½j�s þ nÞJn�1ðg½j�s rÞ � 1

rg½j�s

nðg2½j�s � 2n2ÞJnðg½j�s rÞ

" #;

Q ½j�64;n ¼ l½j� ðg½j�s � nÞðg½j�s þ nÞYn�1ðg½j�s rÞ � 1

rg½j�s

nðg2½j�s � 2n2ÞYnðg½j�s rÞ

" #;

Q ½j�65;n ¼1rl½j�nnJnðg½j�s rÞ; Q ½j�66;n ¼

1rl½j�nnYnðg½j�s rÞ:

References

[1] Leissa AV. Vibration of shells. NASA SP-288, Washington, DC; 1973.[2] Junger MC, Feit D. Sound, structures and their interaction. 2nd ed. MA,

Cambridge: MIT Press; 1986.

[3] Qatu MS. Recent research advances in the dynamic behavior of shells: 1989–2000, part 2: homogeneous shells. Appl Mech Rev 2002;55:415–34.

[4] Amabili M, Paidoussis MP. Review of studies on geometrically nonlinearvibrations and dynamics of circular cylindrical shells and panels, with andwithout fluid–structure interaction. Appl Mech Rev 2003;56:349–56.

[5] Burroughs CB, Hallander JE. Acoustic radiation from fluid-loaded, ribbedcylindrical shells excited by different types of concentrated mechanical drives.J Acoust Soc Am 1992;91:2721–39.

[6] Ricks DC, Schmidt H. A numerically stable global matrix method forcylindrically layered shells excited by ring forces. J Acoust Soc Am1994;95:3339–49.

[7] Guo YP. Radiation from cylindrical shells driven by on-surface forces. J AcoustSoc Am 1994;95:2014–21.

[8] Pathak AG, Stepanishen PR. Acoustic harmonic radiation from fluid-loadedinfinite cylindrical elastic shells using elasticity theory. J Acoust Soc Am1994;96:573–82.

[9] Constans EW, Koopmann GH, Belegundu AD. The use of modal tailoring tominimize the radiated sound power of vibrating shells: theory andexperiment. J Sound Vib 1998;217:335–50.

[10] Ruzzene M, Baz A. Finite element modeling of vibration and sound radiationfrom fluid-loaded damped shells. Thin-Wall Struct 2000;36:21–46.

[11] Ko SH, Seong W, Pyo S. Structure-borne noise reduction for an infinite, elasticcylindrical shell. J Acoust Soc Am 2001;109:1483–95.

[12] Lecable C, Conoir J, Lenoir O. Acoustic radiation of cylindrical elastic shellssubjected to a point source: investigation in terms of helical acoustic rays. JAcoust Soc Am 2001;110:1783–91.

[13] Skelton EA. Line force receptance of an elastic cylindrical shell with heavyexterior fluid loading. J Sound Vib 2002;256:131–53.

[14] Cuschieri JM. The modeling of the radiation and response Green’s function of afluid-loaded cylindrical shell with an external compliant layer. J Acoust SocAm 2006;119:2150–69.

[15] Yan J, Li TY, Liu TG. Characteristics of the vibrational power flow propagationin a submerged periodic ring-stiffened cylindrical shell. Appl Acoust2006;67:550–69.

[16] Ramachandran P, Narayanan S. Evaluation of modal density, radiationefficiency and acoustic response of longitudinally stiffened cylindrical shell. JSound Vib 2007;304:154–74.

[17] Holt JB, Koizumi M, Hirai T, Munir ZA, editors. The concept of FGM. Ceramictransaction: functionally gradient materials, vol. 34. The American CeramicSociety; 1993. p. 3–10.

[18] Afsar AM, Sekine H. Optimum material distributions for prescribed apparentfracture toughness in thick-walled FGM circular pipes. Int J Pres Ves Pip2001;78:471–84.

[19] Niino M, Hirai T, Watanabe R. Functionally gradient materials aimed at heatresisting materials for space plant. J Jpn Soc Compos Mater 1987;13:257–64.

[20] Sofiyev AH. The stability of compositionally graded ceramic–metal cylindricalshells under a periodic axial impulsive loading. Compos Struct2005;69:247–57.

[21] Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG. Functionally gradedmaterials: design, processing and applications. Chapman Hall; 1999.

[22] Hasheminejad SM, Rajabi M. Acoustic resonance scattering from a submergedfunctionally graded cylindrical shell. J Sound Vib 2007;302:208–28.

[23] Hopkins P. Transmission pipelines: how to improve their integrity and preventfailures. Pipeline technology. Proceedings of the second international pipelinetechnology conference, vol. 1. Amsterdam: Elsevier; 1995. p. 683–706.

[24] Stress corrosion cracking on Canadian oil and gas pipelines. Report of theinquiry, National Energy Board, MH-2-95, Calgary; 1996.

[25] Chen Z, Ikeda K, Murakami T, Takeda T, Xie JX. Fabrication of composite pipesby multi-billet extrusion technique. J Mater Process Technol 2003;137:10–6.

[26] Landau LA, Lifshitz EM. Fluid mechanics. Oxford: Pergamon Press; 1959.[27] Habault D, Filippi PJT. A numerical method for the computation of the

resonance frequencies and modes of a fluid-loaded plate: application to thetransient response of the system. J Sound Vib 2004;270:207–31.

[28] Hassan SE, Stepanishen PR. Response of force excited elastic solids withinternal fluid loading. J Acoust Soc Am 2004;116:891–9.

[29] Iakovlev S. Interaction between an external shock wave and a cylindrical shellfilled with and submerged into different fluids. J Sound Vib 2009;322:401–37.

[30] Pierce AD. Acoustics, an introduction to its physical principles andapplications. New York: American Institute of Physics; 1991.

[31] Veksler ND. Resonance acoustic spectroscopy. Berlin: Springer; 1993.[32] Hilderbrand FB. Methods of applied mathematics. Prentice Hall of India; 1961.[33] Abramowitz M, Stegun IA. Handbook of mathematical functions. Washington

DC: National Bureau of Standards; 1964.[34] Chen WQ, Bian ZG, Lv DF, Ding HJ. 3D free vibration analysis of a functionally

graded piezoelectric hollow cylinder filled with compressible fluid. Int J SolidsStruct 2004;41:947–64.

[35] Nayfeh AH, Nagy PB. General study of axisymmetric waves in layeredanisotropic fibers and their composites. J Acoust Soc Am 1996;99:931–41.

[36] Wikipedia, The free encyclopedia. Poisson summation formula. <http://en.wikipedia.org/wiki/Poisson_summation_formula>; 2009 [accessed January2009].

[37] Parveen GN, Chin CD, Reddy JN. Thermoelastic analysis of functionally gradedceramic–metal cylinder. J Eng Mech 1999;125:1259–67.

[38] ABAQUS, Analysis User’s Manual Version 6.6. On-line Documentation.

http://en.wikipedia.org/wiki/Poisson_summation_formula

http://en.wikipedia.org/wiki/Poisson_summation_formula

vibro-acoustic behavior of a hollow fgm cylinder excited by on-surface

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