Acoustic radiation from shear deformable stiffened laminated cylindrical shells

Journal of Sound and Vibration 331 (2012) 651–670

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Acoustic radiation from shear deformable stiffened laminated cylindrical shells Xiongtao Cao n, Hongxing Hua, Chao Ma State Key Laboratory of Mechanical System and Vibration, Shanghai Jiaotong University, Dongchuan Road 800, China

a r t i c l e i n f o

abstract

Article history: Received 21 April 2011 Received in revised form 8 August 2011 Accepted 3 October 2011 Handling Editor: G. Degrande Available online 19 October 2011

An analytical model of acoustic radiation from shear deformable laminated cylindrical shells with initial axial loadings and doubly periodic rings is presented. The shear deformation and rotary inertia of the rings are taken into account and the rings interact with the cylindrical shell only through the normal forces. The far-field sound pressure is found by using the Fourier wavenumber transform and stationary phase method. High frequency limitation issues of the first-order shear deformation theory are discussed and the effects of the second set of rings, axial initial loadings and multiple external loadings on the far-field acoustic radiation are explored. Further, the helical wave spectra of the radial displacement and sound pressure are used to study the vibrational and acoustic characteristics of the laminated shells. Above the ring frequency, the profile of the helical wave spectra of the far-field sound pressure induced by the cylindrical shell is an ellipse and the patterns of the helical wave spectra of the far-field sound pressure keep unchanged. Moreover, the ellipse distinguishes the supersonic wavenumbers and subsonic wavenumbers from the helical wave spectra of the radial displacement and surface sound pressure in the wavenumber domain. The bright spots and highlights of the helical wave spectra show that the corresponding waves are dominant. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Stiffened composite laminated cylindrical shells have important applications in aerospace, mechanical and marine engineering. The cylindrical shells reinforced by doubly periodic rings are the ideal analytical models of the submarine hulls, e.g., the first set of stiffeners is the rings and the second set of stiffeners is the bulkheads. Since the aircraft fuselages and submarine hulls are giant and the profiles of these structures are similar to the cylinders, many investigators would like to use the infinite cylindrical shells to study the vibrational and acoustic characteristics of these structures [1–3]. Kamran et al. explored sound transmission through an infinite composite laminated cylindrical shell by using the classical thin shell theory [1] and the first-order shear deformation shell theory [2] in the context of the transmission of airborne sound into the aircraft interior. They pointed out that the first-order shear deformation theory was strongly recommended to obtain transmission losses for a composite shell, especially in the high frequency range. The classical thin shell theory is not suited to describe the high frequency characteristics of the laminated cylindrical shells [4]. Therefore, it is necessary to use the shear deformation shell theory to investigate sound radiation from thick laminated shells in the high frequency range.

n

Corresponding author. E-mail addresses: [email protected], [email protected] (X. Cao).

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.10.006

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X. Cao et al. / Journal of Sound and Vibration 331 (2012) 651–670

Up to now, a good many researchers have already developed lots of models to study vibroacoustic behavior of stiffened cylindrical shells and plates by using the classical shell (plate) theories. Mace [5] presented a method for sound radiation from a point-excited infinite isotropic plate reinforced by two sets of parallel stiffeners. Mace revised the method adopted by Greenspon [6] to find the far-field sound pressure. Cray [7] derived the far-field and near-field acoustic pressure of a line-driven thin elastic plate with ribs. Portions of the plate are configured with periodic repeating sections that have aperiodic rib spacing. Numerical integration techniques are used to obtain the near-field acoustic pressure. Scattering from an infinite elastic plate stiffened by two sets of periodic ribs subjected to an incident wave had been studied by Belinskii [8]. Mace [9] examined sound radiation from an infinite plate reinforced by orthogonal stiffeners. On the basis of the space harmonic expansion [10] and Fourier transform, the far-field sound pressure is found by solving the simultaneous equations. The responses of infinite periodically stiffened plates with fluid loadings subjected to convected harmonic pressure and point forces were explored by Mace [11,12]. Mead [13] studied free wave motion of a fluid-loaded plate with two sets of orthogonal stiffeners by using the periodic structure theory. It has been pointed out that subsonic harmonics can radiate sound in the periodic structure. Rumerman [14] derived the expressions for the forced responses and free modes of a periodically stiffened infinite plate. General solutions are obtained without restrictions on frequency for the spectral velocity responses of a plate subjected to a harmonic excitation. Yin et al. [15] explored acoustic radiation from composite laminated plates reinforced by doubly periodic parallel stiffeners. A model of acoustic radiation from an isotropic plate with one set of stiffeners was presented by Maxit [16]. The discrete Fourier transform is used to derive the displacement in the physical space from that in the wavenumber domain. Recently, Cao et al. [17] further studied acoustic radiation from shear deformable laminated plates with two sets of stiffeners by using the far-field sound pressure and the transverse displacement spectra. They deduced that the transverse point force exciting the second set of stiffeners yielded much lower far-field sound pressure in the medium and high frequency range due to the larger dynamic stiffness of the second set of stiffeners. Hull and Welch [18] developed a three-dimensional analytical model of a rib-stiffened plate affixed by an acoustic coating, which was excited by an incoming acoustic wave. By using the three-dimensional elastic theory, motion for the backing plate and acoustic coating is governed by the Navier–Cauchy equations of motion. The ribs have significant effects on the dynamic responses of the structure in the high frequency range. Lee and Kim [19] explored sound transmission through the periodically stiffened cylindrical shells subjected to an incident plane wave. Solutions are presented in terms of the space harmonic expansion. The stiffener is modeled by a combination of the lumped mass in conjunction with translational and rotational springs. Yan et al. [20] investigated the radiated sound power characteristics of an infinite periodically stiffened cylindrical shell excited by a radial cosine harmonic line force. By using the space harmonic expansion, the responses of the periodic structure are obtained. Efimtsov and Lazarev [21,22] proposed an effective method for predicting the vibrational and acoustic characteristics of an orthogonally stiffened cylindrical shell. The prediction method is on the basis of the space harmonic expansion. The displacements and forces are expressed in the form of special double trigonometric series. Laulagnet and Guyader [23] studied sound radiation from a finite stiffened cylindrical shell by using the modal superposition. The effects of the stiffeners on vibroacoustic behavior of cylindrical shells are discussed. Burroughs [24] derived analytical expressions for the far-field acoustic radiation from a point-driven circular cylindrical shell reinforced by doubly periodic rings. Recently, Yin et al. [25] investigated acoustic radiation from a point-driven infinite laminated shell stiffened by doubly periodic rings. The thin composite laminated shell is modeled by the classical shell theory and the effects of two sets of rings on the radial displacement of laminated cylindrical shells are identified by using the helical wave spectra. Vibroacoustic characteristics of shear deformable laminated cylindrical shells reinforced by two sets of rings have been rarely investigated in the literature. The present study extends Yin et al.’s work [25] to sound radiation from stiffened thick laminated cylindrical shells with initial axial loadings. Although the helical wave spectra of the radial displacement have been used to identify the effects of rings on bending motion of the laminated cylindrical shells in Yin et al.’s work [25], the physical meanings of the helical wave spectra of sound pressure have not studied. In the present study, the helical wave spectra of the radial displacement, the surface acoustic pressure and the far-field sound pressure are explored in detail. The shear deformation of the rings is considered in order to describe motion for the rings in the high frequency range. The dynamic stiffness of a shear deformable ring is significantly different from that of a ring modeled by the classical beam. The effects of the spacing, force location, initial axial loadings and multiple loadings on the far-field sound pressure are studied. For the harmonic vibration, a time dependent factor eiot will be suppressed throughout. 2. The mathematical model An infinite laminated circular cylindrical shell with two sets of rings immersed in the fluid is shown in Fig. 1. Two sets of periodically spaced rings are attached to the inside surface of the shell. The first set of rings is the small rings with spacing l, and the second set of rings is the large ones that replace every qth ring in the first set. 2.1. Equations of motion for shear deformable laminated cylindrical shells The displacement field of a composite cylindrical shell can be described by U 1 ða1 , a2 , a3 Þ ¼ u1 ða1 , a2 Þ þ a3 b1 ða1 , a2 Þ,

(1)

X. Cao et al. / Journal of Sound and Vibration 331 (2012) 651–670

α

653

α α

Fig. 1. An infinite laminated circular cylindrical shell with two sets of rings.

U 2 ða1 , a2 , a3 Þ ¼ u2 ða1 , a2 Þ þ a3 b2 ða1 , a2 Þ,

(2)

U 3 ða1 , a2 Þ ¼ u3 ða1 , a2 Þ,

(3)

where U1, U2 and U3 are the displacements of the composite cylindrical shell in the a1 , a2 and a3 directions. u1, u2 and u3 are the displacements of a point at the middle surface. b1 and b2 are the rotations of a transverse normal about the a2 - and a1 -axes, respectively. The strain–displacement relations of a circular cylindrical shell are

e ¼ ½e11 e22 e12 T ¼ e0 þ a3 j, e0 ¼



qu1 u3 1 qu2 1 qu1 qu2 þ þ a qa2 a qa2 qa1 qa1 a

j¼



qb1 1 qb2 1 qb1 qb2 þ qa1 a qa2 a qa2 qa1

s ¼ ½e23 e13 T ¼

T

T

,

,

  1 qu3 u2 qu T þ b2  b1 þ 3 , a qa2 a qa1

(4)

where a is the radius of curvature and e0 is the membrane strain vector of the middle surface. j and s are the curvature and transverse shear strain vectors of the middle surface, respectively. For the composite laminated cylindrical shell, the force resultants are 2 3 2 3 N11 M11      N A B e0 6 7 6 7 , N ¼ 4 N22 5, M ¼ 4 M22 5, ¼ j M B D N12 M12 "

Q 23 Q 13

#

" ¼k

A44

A45

A45

A55

#

s,

(5)

where k is the shear correction factor, taking into account the non-uniformity of the shear strain distribution through the thickness of the shell. k is 5/6 given by Reissner [26] and p2 =12 presented by Mindlin [27]. The stiffness matrices A, B and D are 2 3 2 3 2 3 A11 A12 A16 B11 B12 B16 D11 D12 D16 6A 7 6 7 6 7 A ¼ 4 12 A22 A26 5, B ¼ 4 B12 B22 B26 5, D ¼ 4 D12 D22 D26 5, A16 A26 A66 B16 B26 B66 D16 D26 D66 where the elements Aij ,Bij ,Dij are given by Aij ¼

N X

ðmÞ

Q ij ðhm hm1 Þ,

m¼1

Dij ¼

Bij ¼

N 1 X ðmÞ 2 2 Q ðhm hm1 Þ, 2 m ¼ 1 ij

N 1 X ðmÞ 3 3 Q ðhm hm1 Þ 3 m ¼ 1 ij

Aij ¼

N X m¼1

ðmÞ

Q ij ðhm hm1 Þ

ði,j ¼ 1; 2,6Þ,

ði,j ¼ 4; 5Þ,

(6)

654

X. Cao et al. / Journal of Sound and Vibration 331 (2012) 651–670 ðmÞ

and Q ij are the reduced stiffness coefficients of the mth layer. hm and hm1 are the coordinates of the top and bottom ðmÞ surfaces of the mth lamina in the a3 direction. Q ij are expressed by ðmÞ

2 4 ðmÞ ðmÞ ðmÞ 4 2 Q 11 ¼ Q ðmÞ 11 cos a þ 2ðQ 12 þ2Q 66 Þsin a cos a þ Q 22 sin a, ðmÞ

2 ðmÞ ðmÞ ðmÞ ðmÞ 2 Q 12 ¼ Q ðmÞ 12 þ ðQ 11 þQ 22 2Q 12 4Q 66 Þsin a cos a, ðmÞ

2 4 ðmÞ ðmÞ ðmÞ 4 2 Q 22 ¼ Q ðmÞ 22 cos a þ 2ðQ 12 þ2Q 66 Þsin a cos a þ Q 11 sin a, ðmÞ

2 ðmÞ ðmÞ ðmÞ ðmÞ 2 Q 66 ¼ Q ðmÞ 66 þ ðQ 11 þQ 22 2Q 12 4Q 66 Þsin a cos a, ðmÞ

3 ðmÞ ðmÞ ðmÞ ðmÞ 3 Q 16 ¼ ðQ ðmÞ Q 12 2Q ðmÞ 11 Q 12 2Q 66 Þsin a cos aðQ 22 66 Þsin a cos a, ðmÞ

3 ðmÞ ðmÞ ðmÞ ðmÞ 3 Q 26 ¼ ðQ ðmÞ Q 12 2Q ðmÞ 11 Q 12 2Q 66 Þsin a cos aðQ 22 66 Þsin a cos a, ðmÞ

2 ðmÞ 2 Q 44 ¼ Q ðmÞ 44 cos a þQ 55 sin a, ðmÞ

ðmÞ Q 45 ¼ ðQ ðmÞ 55 Q 44 Þcos a sin a, ðmÞ

2 ðmÞ 2 Q 55 ¼ Q ðmÞ 55 cos a þQ 44 sin a,

(7)

where a is the fiber orientation (the anti-clockwise direction is assumed to be positive). For the mth orthotropic layer, the reduced stiffness Q ijðmÞ of the mth lamina are defined as Q ðmÞ 11 ¼

E1 , 1n12 n21

ðmÞ Q 12 ¼

ðmÞ ¼ G12 , Q 66

E1 n21 , 1n12 n21

Q ðmÞ 44 ¼ G23 ,

Q ðmÞ 22 ¼

E2 , 1n12 n21

ðmÞ Q 55 ¼ G13 ,

(8)

where E1 and E2 are the elastic moduli, G12, G23 and G13 are the shear moduli, and n12 , n21 are Poisson’s ratios. The potential energy V N0 due to the initial axial loadings N0 is given by "    # Z Z 1 1 2p qu3 2 qu2 2 V N0 ¼ N0 þ (9) a da1 da2 , 2 1 0 qa1 qa1 where the sign of N0 is positive for the tensional axial loadings and negative for the compressional axial loadings. By using Hamilton’s principle, the equations of motion for the shear deformable laminated cylindrical shell with two sets of rings are derived and given by 2 3 f1 2 32 3 u1 L11 L12 L13 L14 L15 6 7 f2 6L 76 7 6 7 6 21 L22 L23 L24 L25 76 u2 7 6 7 6 76 7 6 pe pa pr pb 7 6 L31 L32 L33 L34 L35 76 u3 7 ¼ 6 7, (10) 6 76 7 6 7 m1 76 7 6 6L 7 4 41 L42 L43 L44 L45 54 b1 5 6 7 4 5 m2 b2 L51 L52 L53 L54 L55 where f1, f2 and pe are the external forces in the a1 , a2 and a3 directions. m1 and m2 are the external moments in the a2 and a1 directions. pa is the acoustic pressure acting on the cylindrical shell. pr and pb are the normal reactive forces of the first and the second sets of rings, respectively. The differential operators Lij are defined as L11 ¼ A11

q2 A66 q2 2A16 q2 q2  2  þ I1 2 , 2 2 a q a q a a qa1 qa2 qt 1 2

L12 ¼ L21 ¼ 

A12 þA66 q2 A26 q2 q2  A16 2 , a qa1 qa2 a2 qa22 qa1

L13 ¼ L31 ¼ 

L14 ¼ L41 ¼ B11

L15 ¼ L51 ¼ 

A12 q A26 q  , a qa1 a2 qa2

q2 B66 q2 2B16 q2 q2  2  þ I2 2 , 2 2 a q a q a a qa1 qa2 qt 1 2

B12 þ B66 q2 B26 q2 q2  B16 2 , a qa1 qa2 a2 qa22 qa1

X. Cao et al. / Journal of Sound and Vibration 331 (2012) 651–670

L22 ¼ A66

q2 A22 q2 A44 2A26 q2 q2 q2  þ 2  N0 2 þ I1 2 , a qa1 qa2 a qa21 a2 qa22 qa1 qt

L23 ¼ L32 ¼ 

L24 ¼ L42 ¼ 

A22 þA44 q A45 þ A26 q  , qa2 qa1 a a2

B66 þ B12 q2 A45 B26 q2 q2  2  B16 2 , a qa1 qa2 a a qa22 qa1

L25 ¼ L52 ¼ B66

L33 ¼ A55

655

q2 B22 q2 A44 2B26 q2 q2    þ I2 2 , a a qa1 qa2 qa21 a2 qa22 qt

q2 A44 q2 2A45 q2 A22 q2 q2  2  þ 2 N0 2 þ I1 2 , 2 2 a qa1 qa2 a qa1 a qa2 qa1 qt

L34 ¼ L43 ¼

    B12 q B26 A45 q A55 þ  , qa1 a a qa2 a2

L35 ¼ L53 ¼

    B22 A44 q B26 q A45  þ , 2 qa1 a qa2 a a

L44 ¼ D11

q2 D66 q2 2D16 q2 q2  2  þ A55 þ I3 2 , 2 2 a q a q a a qa1 qa2 qt 1 2

L45 ¼ L54 ¼ 

L55 ¼ 

D12 þ D66 q2 D26 q2 q2 þ A45  2 D16 2 , a qa1 qa2 a qa22 qa1

D22 q2 q2 2D26 q2 q2 D66 2  þ A44 þ I3 2 , a qa1 qa2 a2 qa22 qa1 qt

(11)

in which ðI1 ,I2 ,I3 Þ ¼

Z

h=2 h=2

ð1, a3 , a23 ÞrðmÞ da3 ,

(12)

where rðmÞ is the mass density of the mth layer and h is the thickness of the composite cylindrical shell. The displacements as well as all the general forces due to the external forces and moments, fluid loadings, reactive forces of two sets of rings can be decomposed into the following harmonic series: 1 X gða1 , a2 Þ ¼ g n ða1 Þeina2 : (13) n ¼ 1

gða1 , a2 Þ will be replaced with u1, u2, u3, b1 , b2 , f1, f2, pe , m1, m2, pa, pr and pb in the following derivation. The Fourier transform is defined by Z þ1 1 f~ ðkÞ ¼ f ða1 Þeika1 da1 , (14) 2p 1 where k is the wavenumber. The inverse Fourier transform is given by Z þ1 f~ ðkÞeika1 dk: f ða1 Þ ¼

(15)

1

2.2. External point force The external point force pe ða1 , a2 Þ with the amplitude f3 located at the point ða1j , a2j Þ radially interacts with the cylindrical shell, which is described by pe ða1 , a2 Þ ¼

f3 dða1 a1j Þdða2 a2j Þ: a

(16)

Taking the Fourier transform of Eq. (16) with respect to a1 and the periodic Fourier transform with respect to a2 , one obtains p~ e ¼

1 X f 3 eiðka1j þ na2j Þ n ¼ 1

að2pÞ2

eina2 :

(17)

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X. Cao et al. / Journal of Sound and Vibration 331 (2012) 651–670

Therefore, p~ e can be decomposed into the harmonic component in the following form: p~ en ðkÞ ¼

f 3 eiðka1j þ na2j Þ að2pÞ2

(18)

:

2.3. The fluid loadings The acoustic pressure pða1 , a2 , a3 Þ in the fluid satisfies the Helmholtz equation in the cylindrical coordinates

r2 pða1 , a2 , a3 Þ þ k20 pða1 , a2 , a3 Þ ¼ 0,

(19)

where the wavenumber k0 in the fluid is o=c, and c is the speed of sound in the fluid. The Laplacian operator r2 is defined by   1 q2 1 q q q2 r2 ¼ 2 2 þ a3 (20) þ 2, qa3 a3 qa2 a3 qa3 qa1 where a1 , a2 and a3 denote the coordinates of the axial, circumferential and radial directions, respectively. The boundary condition at the fluid–shell interface is  qpða1 , a2 , a3 Þ ¼ o2 ra u3 ða1 , a2 Þ, (21)  qa3 a3 ¼ a where ra is the mass density of the fluid. Taking the Fourier transform of Eqs. (19) and (21) with respect to a1 , one obtains the acoustic pressure, satisfying the Sommerfeld radiation condition at the infinity qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 X o2 ra u~ 3n ðkÞHð1Þ n ð k0 k a3 Þ ina2 ~ a2 , a3 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi e , (22) pðk, 0 2 2 2 2 n ¼ 1 k0 k Hð1Þ n ð k0 k aÞ 2

2

ð1Þ where Hð1Þ n ðzÞ is the nth-order Hankel function of the first kind. If k0 is less than k , H n ðzÞ is replaced with K n ðzÞ, which is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 the nth-order modified Bessel function of the second kind, and k0 k is replaced with k k0 . Therefore, the harmonic decomposition p~ an of the fluid loadings acting on the cylindrical shell can be written as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 o2 ra u~ 3n ðkÞHð1Þ n ð k0 k aÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi p~ an ðk,aÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (23) ¼ Z n ðkÞu~ 3n ðkÞ, 0 2 2 2 2 k0 k Hð1Þ n ð k0 k aÞ

where Z n ðkÞ is the impedance of the fluid loadings. 2.4. Reactive forces by two sets of rings The equations of motion for the isotropic circular ring are derived on the basis of the first-order shear deformation theory and only in-plane motion is considered. It should be noted that the governing equations of motion for the circular beam can also be obtained from the equations of motion for the thick isotropic cylindrical shell by neglecting axial motion and the rotation b1 in the a2 direction in Eq. (10). The equations of motion for the rings are described by Er Ar q2 ur2 kGr Ar r ðkGr þEr ÞAr qur3 kGr Ar r r þ u2 þ rr Ar u€ 2   b2 ¼ 0, qa2 ar a2r qa22 a2r a2r

(24)

ðkGr þ Er ÞAr qur2 kGr Ar q2 ur3 Er Ar ur3 kGr Ar qbr2 r  þ þ rr Ar u€ 3  ¼ qr3 , 2 2 2 2 qa2 ar qa2 ar ar qa2 ar

(25)





kGr Ar ar

ur2 þ

r r kGr Ar qur3 Er Ir q2 b2 r þ kGr Ar b2  2 þ rr Ir b€ 2 ¼ 0, 2 ar qa2 ar qa2

(26)

where Er, Gr, rr , Ar and Ir are the elastic modulus, shear modulus, mass density, area of beam section, and inertia moment, respectively. k is the shear correction factor. ar is the radius of the center line. qr3 are the normal reactive forces acting on the ring due to the cylindrical shell. ur2 and ur3 are the translational displacements in the circumferential and radial r directions, respectively. b2 is the rotation angle in the axial direction. The solutions of Eqs. (24)–(26) can be written as ur2 ¼

1 X n ¼ 1

ur2n eina2 ,

ur3 ¼

1 X n ¼ 1

ur3n eina2 ,

br2 ¼

1 X n ¼ 1

br2n eina2 , qr3 ¼

1 X n ¼ 1

qr3n eina2 :

(27)

X. Cao et al. / Journal of Sound and Vibration 331 (2012) 651–670

657

By using Eqs. (27), (24) and (26), one obtains ur2n ¼

   in 1 Er ur3n ¼ Dr1 ur3n , 1C r1 þ r r ar kar Gr 1C 2 C 1

(28)

  in 1 Er ur ¼ Dr2 ur3n , C r2   r r ar kar Gr 3n 1C 2 C 1

(29)

br2n ¼

where the coefficients C r1 , C r2 , Dr1 and Dr2 are defined by C r1 ¼ ar þ

C r2 ¼

Dr1 ¼

n2 Er Ir ar rr Ir o2  , ar kGr Ar kGr Ar

Er n2

kGr ar

þ

1 ar rr o2  , ar kGr

(30)

(31)

   in 1 Er , 1C r1 þ r r ar kar Gr 1C 2 C 1

(32)

  in 1 Er r C   : 2 ar kar Gr 1C r2 C r1

(33)

Dr2 ¼ By using Eqs. (25), (27)–(29), one obtains

qr3n ¼ K rn ur3n ,

(34)

where the dynamic stiffness K rn of the small ring is defined by K rn ¼

inðkGr þEr ÞAr r kn2 Gr Ar Er Ar inkGr Ar Dr2 D1 þ þ 2 rr Ar o2  : 2 2 ar ar ar ar

(35)

The transverse displacement of the ring is equal to that of the cylindrical shell along the line a1 ¼ ml, there holds ur3n ¼ u3n ðmlÞ:

(36)

Therefore, the forces pr induced by the first set of rings can be expressed by 1 X

1 X

pr ¼

K rn u3n ðmlÞeina2 dða1 mlÞ:

(37)

n ¼ 1 m ¼ 1

Taking the Fourier transform of Eq. (37) with respect to a1 , one obtains p~ r ¼

1 1 X 1 X K rn u3n ðmlÞeina2 eikml : 2p n ¼ 1 m ¼ 1

According to the inverse Fourier transform, one obtains Z u3n ðmlÞ ¼

þ1

u~ 3n ðk1 Þeik1 ml dk1 :

(38)

(39)

1

Substituting Eqs. (39) into (38), one obtains p~ r ¼

Z þ1 1 1 X 1 X K rn eina2 eiðk1 kÞml dk1 : u~ 3n ðk1 Þ 2p n ¼ 1 1 m ¼ 1

(40)

Poisson’s summation formula can be used to show 1 X m ¼ 1

eimlk1 ¼ 2p

1 X

dðlk1 2mpÞ:

(41)

m ¼ 1

Substituting Eqs. (41) into (40), one obtains the harmonic decomposition p~ rn   1 K rn X 2mp : p~ rn ðkÞ ¼ u~ 3n k l m ¼ 1 l Similarly, the harmonic decomposition p~ bn due to the second set of rings is described by   1 K K rn X 2mp , u~ 3n k p~ bn ðkÞ ¼ bn ql ql m ¼ 1

(42)

(43)

658

X. Cao et al. / Journal of Sound and Vibration 331 (2012) 651–670

where the dynamic stiffness K bn of the large rings are given by K bn ¼

inðkGb þEb ÞAb b kn2 Gb Ab Eb Ab inkGb Ab Db2 D1 þ þ 2 rb Ab o2  : 2 2 ab ab ab ab

(44)

3. Solutions in the wavenumber domain Taking the Fourier transform of Eq. (10) obtains 2~ L 11 L~ 12 6~ 6 L 21 L~ 22 6 6 L~ 31 L~ 32 6 6~ 4 L 41 L~ 42 L~ 51 L~ 52

with respect to a1 and substituting Eqs. (13), (23), (42) and (43) into (10), one L~ 13 L~ 23

L~ 14 L~ 24

L~ 33 L~ 43 L~ 53

L~ 34 L~ 44 L~ 54

3 32 ~ 3 2 u 1n f~ 1n L~ 15 6 7 76 7 7 L~ 25 76 u~ 2n 7 6 f~ 2n 7 76 ~ 7 6 7 ~L 35 76 u 3n 7 ¼ 6 p~ en p~ rn p~ bn p~ sn 7, 76 7 6 6 7 6 b~ 7 6 ~L 45 7 7 ~ 1n 54 1n 5 4 m 5 ~ ~L 55 b 2n ~ 2n m

(45)

where the elements L~ ij of the matrix L~ are the transformed operators, and given as 2

A66 n 2A16 kn 2 I1 o2 , L~ 11 ¼ A11 k þ þ a a2 ðA12 þ A66 Þkn A26 n2 2 þ þ A16 k , L~ 12 ¼ L~ 21 ¼ a a2 ikA12 inA26  2 , L~ 13 ¼ L~ 31 ¼  a a

2

B66 n 2B16 kn 2 I2 o2 , L~ 14 ¼ L~ 41 ¼ B11 k þ þ a a2

ðB12 þ B66 Þkn B26 n2 2 þ þ B16 k , L~ 15 ¼ L~ 51 ¼ a a2 A22 n2 A44 2A26 kn 2 2 þ N0 k I1 o2 , þ 2 þ L~ 22 ¼ A66 k þ a a2 a inðA22 þA44 Þ ikðA45 þ A26 Þ , L~ 23 ¼ L~ 32 ¼   a a2 knðB66 þB12 Þ A45 B26 n2 2 L~ 24 ¼ L~ 42 ¼  þ þ B16 k , a a a2 B22 n2 A44 2knB26 2 þ I2 o2 ,  L~ 25 ¼ L~ 52 ¼ B66 k þ a a a2 A44 n2 2A45 kn A22 2 2 þ 2 þN 0 k þ Z n ðkÞI1 o2 , L~ 33 ¼ A55 k þ þ a a2 a     B12 B26 A45 A55 þ in ,  L~ 34 ¼ L~ 43 ¼ ik a a a2     B22 A44 B26 L~ 35 ¼ L~ 53 ¼ in þ ik A  , 45 a a a2 D66 n2 2D16 kn 2 þA55 I3 o2 , L~ 44 ¼ D11 k þ þ a a2 ðD12 þ D66 Þkn D26 n2 2 þ A45 þ þ D16 k , L~ 45 ¼ L~ 54 ¼ a a2 D22 n2 2D26 kn 2 þ A44 I3 o2 : þ D66 k þ L~ 55 ¼ a a2

(46)

The solution of Eq. (45) can be written as U ¼ F1 F2

    1 1 K rn X 2mp K K rn X 2mp F2 bn , u~ 3n k u~ 3n k l m ¼ 1 l ql ql m ¼ 1

(47)

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659

where U ¼ ½u~ 1n u~ 2n u~ 3n b~ 1n b~ 2n T ,

F1 ¼ L~

1

~ 1n m ~ 2n T , ½f~ 1n f~ 2n p~ en m

1 F2 ¼ L~ ½0 0 1 0 0T :

(48)

Extracting u~ 3n from Eq. (47), one obtains u~ 3n ðkÞ ¼ F13 ðkÞF23 ðkÞ

    1 1 K rn X 2mp K K rn X 2mp F23 ðkÞ bn , u~ 3n k u~ 3n k l m ¼ 1 l ql ql m ¼ 1

(49)

where F13 and F23 are the third element of the vectors F1 and F2 , respectively. A similar procedure of solution as given by Mace [5] will be done in the following derivation. The following notations are defined: G¼

K rn , l

K bn K rn : ql

H¼

(50)

Therefore, in Eq. (49), one obtains u~ 3n ðkÞ ¼ F13 ðkÞF23 ðkÞG

    q1 1 X X 2mp 2g p 2mp F23 ðkÞH  : u~ 3n k u~ 3n k l ql l m ¼ 1 m ¼ 1 g ¼ 0 1 X

Substituting k ¼ k2r p=ql2dp=l into Eq. (51), and summing the resulting equation over all d, one obtains       q1 1 1 1 X X X X 2r p 2dp 2r p 2mp 2g p 2mp  ¼ Pr ðkÞGY r ðkÞ  HY r ðkÞ  , u~ 3n k u~ 3n k u~ 3n k ql l ql l ql l m ¼ 1 m ¼ 1 g ¼ 0 d ¼ 1

(51)

(52)

where P r ðkÞ ¼

  2r p 2dp  , F13 k ql l d ¼ 1 1 X

Y r ðkÞ ¼

  2r p 2dp  : F23 k ql l d ¼ 1 1 X

The following relations exist when the resulting equation is simplified:     1 1 X X 2r p 2mp 2dp 2r p 2mp   ¼  , u~ 3n k u~ 3n k ql l l ql l m ¼ 1 m ¼ 1   2g p 2r p 2mp 2dp    ¼ u~ 3n k ql ql l l m ¼ 1 g ¼ 0 1 X

q1 X

  2g p 2mp  : u~ 3n k ql l m ¼ 1 g ¼ 0

In Eq. (52), one obtains   Pr ðkÞHY r ðkÞ 2r p 2mp  ¼ u~ 3n k ql l m ¼ 1 1 X

P1

q1 X

1 X

  2g p 2mp  u~ 3n k ql l : 1 þ GY r ðkÞ

m ¼ 1

(53)

(54)

(55)

Pq1

g¼0

(56)

The following formula can be obtained in Eq. (56) when r is equal to zero: 1 X m ¼ 1



u~ 3n k



2mp ¼ l

P 0 ðkÞHY 0 ðkÞ

P1

  2g p 2mp  u~ 3n k ql l : 1 þGY 0 ðkÞ

m ¼ 1

Pq1

g¼0

Summing both sides of Eq. (56) over r from 0 to q  1, one obtains !, !   q1 q1 q1 1 X X X X 2r p 2mp Pr ðkÞ Y r ðkÞ ~  ¼ u 3n k 1 þH : 1þ GY r ðkÞ 1þ GY r ðkÞ ql l m ¼ 1 r ¼ 0 r¼0 r¼0

(57)

(58)

Substituting Eqs. (57) and (58) into (51), one obtains 2

3 P P r ðkÞ H q1 r¼0 6 GP0 ðkÞ 7 1 þGY r ðkÞ 7:  u~ 3n ðkÞ ¼ F13 ðkÞF23 ðkÞ6 41 þGY ðkÞ þ  5 P Y ðkÞ r 0 q1 ð1 þGY 0 ðkÞÞ 1 þH r ¼ 0 1 þGY r ðkÞ

The energy rule of the Fourier integral can be used to show Z Z 1 Z 2p 1 X un3 u3 da1 da2 ¼ 4p2 1

0

n ¼ 1

1

2

9u~ 3n 9 dk,

(59)

(60)

1

where the symbol n denotes the complex conjugate. In Eq. (60), it is also indicated that u~ 3n is crucial to the mean squared radial displacement and kinetic energy. The acoustic pressure in the far field can be derived from Eq. (22) by using the

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inverse Fourier transform qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 X o2 ru~ 3n ðkÞHð1Þ n ð k0 k a3 Þ ina2 ika1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi e e dk: (61) pða1 , a2 , a3 Þ ¼ 2 2 2 2 1 n ¼ 1 k0 k Hð1Þ0 aÞ ð k k 0 n qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 By using large parameters approximation for Hð1Þ ð k k a Þ and the stationary phase method [28], one obtains (the 3 0 n stationary phase point is k ¼ k0 cos y) Z

1

pðR, y, jÞ ¼ 

1 2io2 reik0 R X u~ 3n ðk0 cos yÞ inj e ðiÞn , k0 R sin y n ¼ 1 Hð1Þ0 n ðk0 a sin yÞ

(62)

where y and j are the polar and azimuthal angles in the spherical coordinates. In Eq. (62), the far-field sound pressure propagates like the spherical wave. The sound pressure level (SPL) is defined by SPL ¼ 20 logðp=p0 Þ, where p0 is the reference sound pressure 1  10

6

(63)

Pa.

4. Numerical results The structural dampings of the rings and cylindrical shell take effect by using the complex elastic modulus Eð1iZÞ, where Z is the loss factor. Zr , Zb and Zc are the loss factors of the first set of rings, the second set of rings, and the laminated cylindrical shell, respectively. Geometric and material parameters of the rings with rectangle section and cylindrical shell are listed in Table 1. The external fluid is water. The sound speed c in the fluid is 1500 m/s and the mass density ra of the fluid is 1000 kg/m3. A unit harmonic radial point force is located at (0,0, a) and the sound field point is located at Q 1 ðR ¼ 50 m, y ¼ p=4, j ¼ p=4Þ in the spherical coordinates. The initial axial loading N0 is zero. These parameters keep unchanged without special explanation in the discussion. 4.1. Validity of the present method Yin’s method [25] is used to verify the present model in the low to medium frequency range and the restrictions of Yin’s method will be shown as well. Yin’s model is on the basis of the classical beam and shell theories. The material and geometric parameters of two sets of rings listed in Table 1 are used in the calculation, but the heights hb of the second set of rings are replaced with 0.16 m (in order to satisfy the condition of thin beam). The widths of the large rings still keep unchanged. The thickness h of an isotropic cylindrical shell is 0.004 and the material parameters of the cylindrical shell are the same as those of rings. SPL of the cylindrical shell with two sets of rings presented by Yin’s approach and our method is shown in Fig. 2. The results given by the two methods are in good agreement when the frequency is less than 3 kHz. Great differences can be observed when the frequency is more than 10 kHz. The dynamic stiffness of a ring in the first set given by the shear deformable beam and classical beam theories at 1 kHz is shown in Fig. 3. The results agree well only in the low circumferential wavenumber n and great discrepancies can be observed in the high circumferential wavenumber n. For the thin beams and shells, good results can be obtained in the low frequency range by using the classical beam and shell theories. However, the shear deformable beam and shell theories are suited to investigate acoustic radiation from stiffened cylindrical shells in the high frequency range. Since the first-order shear deformation shell theory is not yet a three-dimensional elastic theory, there are high frequency limitation issues for this shell theory. Hayek and Boisvert [29] studied the vibration of elliptic cylindrical shells by using a higher order shell theory in order to characterize the high frequency responses of shells composed of low wave speed elastomeric type materials, e.g., rubber. In this context, high frequency implies that wavelengths are comparable to the thickness of the shell. For a given frequency range of interest, a shell made of a low wave speed elastomeric material could have the thickness stretch modes within that range. However, for the high wave speed material, such as steel, the corresponding thickness stretch modes will occur in the extremely high frequency range. The following discussions will explore the high frequency limitation issues for the first-order shear deformation theory. By using the three-dimensional elastic theory, the equations of motion for the isotropic elastic solids (hollow cylinders) in the orthogonal curvilinear coordinates are given by [30] u ¼ rf þ r  w,

(64)

Table 1 Parameters of the rings with rectangle section and cylindrical shell (in SI units). Er

rr Ir h

Zr

2.1  1011 7.8  103 4.27  10  7 0.03 0.02

Eb

rb Ib a

Zb

2.1  1011 7.8  103 8.53  10  4 1.5 0.02

Gr Ar hr

8.08  1010 8.0  10  4 0.08

k Zp

p2 =12 0.02

Gb Ab hb l q

8.08  1010 1.6  10  2 0.8 0.3 5

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661

85 80 75 70 SPL (dB)

65 60 55 50 45 40 35 30

1: Present method, first order shear deformation theories 2: Yin’s method, classical beam and shell theories

0

2

4

6

8 10 12 Frequency (kHz)

14

16

18

20

Fig. 2. SPL of the isotropic cylindrical shell with two sets of rings.

1013 1: Shear deformable beam theory, 1kHz 2: Classical beam theory, 1kHz

1012

1011

|Krn|

1010

109

108

100

90

80

60

70

50

40

30

20

10

0

−10

−20

−30

−40

−50

−60

−70

−90

−100

106

−80

107

Circumferential wave number n Fig. 3. Dynamic stiffness of a ring in the first set given by the shear deformable and the classical beam theories.

r2 f ¼

1 q2 f , c2d qt 2

r2 w ¼

1 q2 w , c2s qt 2

rw ¼ 0,

(65)

where u is the displacement vector, f is the scalar potential and w is the vector potential. cd and cs are the dilatational and shear wave speeds, respectively, and given by cd ¼ ½ðl þ 2mÞ=rs 1=2 ,

cs ¼ ðm=rs Þ1=2 ,

(66)

where rs is the mass density. l and m are the Lame´ constants. There is a special frequency point f h defined by f h ¼ cs =h, which means that the shear wave wavelength in the cylinder is just equal to the thickness of the cylinder. For a steel isotropic cylindrical shell with thickness h equal to 0.03 m, f h is equal to 107 kHz. Therefore, the high frequency limitation for the first-order shear deformation and higher order shear deformation theories [31] is determined by f h =lh , where lh is a

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permitted coefficient and denotes the ratio of the permitted wavelength to the thickness h. In general, lh could be chosen as 5. Thus, for the isotropic cylindrical shell, the high frequency limitation for the first-order shear deformation theory is about 20 kHz. The characteristics of wave propagation in the composite laminated cylindrical shells are complicated. In Eq. (10), it can be found that there are five different kinds of wave speeds corresponding to five equations of motion for the symmetric and antisymmetric laminated cylindrical shells. The five wave speeds are ðA11 =I1 Þ1=2 , ðA66 =I1 Þ1=2 , ðA55 =I1 Þ1=2 , ðD11 =I3 Þ1=2 and ðD66 =I3 Þ1=2 , respectively. One can find the lowest wave speed denoted by cl among the five wave speeds. Therefore, the high frequency limitation of the first-order shear deformation theory for the symmetric and antisymmetric laminated cylindrical shells is defined by cl =ðhlh Þ. Thus, for the isotropic moderately thick cylindrical shells, the high frequency limitation can be written as ðA55 =I1 Þ1=2 =ðhlh Þ ¼ cs k=ðhlh Þ. If the shear strain distribution through the thickness of the isotropic shell is uniform (k ¼ 1), the high frequency limitations given by the above two methods are the same.

4.2. Acoustic radiation from stiffened composite cylindrical shells Sound pressure and the helical wave spectra of stiffened composite laminated cylindrical shells are studied on the basis of the first-order shear deformation shell theory. Material parameters of the orthotropic layers are listed in Table 2 and lamination schemes are depicted in Table 3. The parameters listed in Table 1 are still used in the calculation. Two lamination schemes, i.e., the symmetric ð751=451=151=151=451=751Þ and antisymmetric ð751=451=151=151=451=751Þ plies are used for the numerical calculation. The convergence check of the numerical solution to SPL of the antisymmetric laminated cylindrical shell with two sets of rings is shown in Fig. 4. In the model, a unit harmonic radial point force is located at the point ð0; 0,aÞ. The far-field sound pressure is convergent until n is equal to 90 in the full frequency range. In the high frequency range, n should be chosen as a large integer in order to obtain the convergent results and the effects of the large circumferential wavenumbers n on the far-field sound pressure must be considered. SPL of the antisymmetric laminated cylindrical shell with two sets of rings is illustrated in Fig. 5 when the radial point force is located at the points ð0:75,0,aÞ and ð0; 0,aÞ, respectively. The point ð0:75,0,aÞ is just in the middle between a small ring and large ring for q¼3. Sound pressure induced by the radial point force at the point ð0:75,0,aÞ is larger than that caused by the radial point force located at the large ring ð0; 0,aÞ. It is due to the fact that the large ring has much larger dynamic stiffness than the small ring. Cao et al. [17] had explained this phenomenon for acoustic radiation from laminated plates with two sets of stiffeners in detail. The same reasons can be applied to the shells with two sets of rings. In Fig. 6, SPL of the antisymmetric laminated cylindrical shell with two sets of rings is illustrated for q¼3,9. The second set of rings causes intense wavenumber conversion and fluctuation of SPL. The larger the q is, the stronger the fluctuation of SPL. This phenomenon was firstly observed by Mace [5], later by Burroughs [24]. The radial point force is moved at a new point P 1 ð0:75,0,aÞ, which is just in the middle of two adjacent large rings. SPL of the symmetric and antisymmetric laminated cylindrical shells with two sets of rings is illustrated in Fig. 7. It can be observed that two laminated cylindrical shells with two sets of rings have different sound pressure in the far field. For a vibrating cylindrical shell in the fluid, free waves may travel in the axial and circumferential directions, which can be regarded as resultant helical waves traveling in helical paths specified by the circumferential and axial wavenumbers. Williams et al. [32] defined the helical wave spectra of the radial displacement, which was in terms of the inverse Fourier

Table 2 Material parameters of composite laminated cylindrical shells. Material ID

E1 (GPa)

E2 (GPa)

G12 (GPa)

G13 (GPa)

G23 (GPa)

n12

r (kg/m3)

1 2 3

138 131 207

8.96 10.3 20.7

7.1 6.9 6.9

7.1 6.2 6.9

6.2 6.2 4.1

0.3 0.22 0.3

7800 7800 7800

Table 3 Lamination schemes of composite cylindrical shells. Ply No.

Material ID

Thickness (m)

Ply angle (deg.)

1 2 3 4 5 6

1 2 3 3 2 1

0.005 0.005 0.005 0.005 0.005 0.005

75 45 15 7 15 7 45 7 75

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80 75

2

3

70

SPL (dB)

65 60

1

55 50 1: n = 40 2: n = 90 3: n = 100

45 40 35

0

2

4

6

8 10 12 Frequency (kHz)

14

16

18

20

Fig. 4. Convergence check of the numerical solution to SPL of the antisymmetric cylindrical shell with two sets of rings.

100 90

SPL (dB)

80 70 60 50 1: q = 3, unit radial point force at (0.75, 0, a) 2: q = 3, unit radial point force at (0, 0, a)

40 30

0

2

4

6

8 10 12 Frequency (kHz)

14

16

18

20

Fig. 5. SPL of the antisymmetric laminated cylindrical shell with two sets of rings at different force reactive points.

transform u3 ða1 , a2 Þ ¼

Z

þ1

1

1 X

u~ 3n ðkÞeina2 eika1 dk:

(67)

n ¼ 1

The integral and summation represent a decomposition of the radial displacement in terms of helical waves, given by eiðka1 þ na2 otÞ with the complex amplitudes u~ 3n ðkÞ, which is called the helical wave spectra of the radial displacement. Similarly, Choi et al. [33] defined the helical wave spectra of sound pressure Z þ1 X 1 pða1 , a2 , a3 Þ ¼ p~ n ðk, a3 Þeina2 eika1 dk, (68) 1

n ¼ 1

where p~ n ðk, a3 Þ are the helical wave spectra of sound pressure. Distinctly, bending waves and sound pressure are the summation of the infinite helical waves whose circumferential wavenumbers and axial wavenumbers are given by n and k. In the following discussion, bending waves and sound pressure will be studied in the form of the helical wave spectra defined in Eqs. (67) and (68). Above the ring frequency, a cylindrical shell behaves like a plate [28,34,35]. In the present model, the ring frequency of the laminated cylindrical shell is 159.1 Hz. In Fig. 8, the helical wave spectra of the radial displacement for the symmetric and antisymmetric laminated cylindrical shells with two sets of rings are shown at 1 kHz. In Fig. 9, the helical wave spectra of surface sound pressure for the symmetric and antisymmetric laminated cylindrical

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95 90 85

SPL (dB)

80 75 70 65 60 1: q = 9, unit radial point force at (0.75 ,0, a) 2: q = 3, unit radial point force at (0.75 ,0, a)

55 50

0

2

4

6

8 10 12 Frequency (kHz)

14

16

18

20

Fig. 6. SPL of the antisymmetric laminated cylindrical shell with two sets of rings, different spacing of two adjacent large rings.

95 90 85 80

SPL (dB)

75 70 65 60 55 50 1: Stiffened symmetric laminated cylindrical shell 2: Stiffened antisymmetric laminated cylindrical shell

45 40

0

2

4

6

8

10

12

14

16

18

20

Frequency (kHz) Fig. 7. SPL of the symmetric and antisymmetric cylindrical shells with two sets of rings.

shells with two sets of rings are shown at 1 kHz. In Fig. 10, the helical wave spectra of the far-field sound pressure for the symmetric and antisymmetric laminated cylindrical shells with two sets of rings are illustrated at 1 kHz. In Figs. 8 and 9, according to Eq. (60), it is shown that contribution of the circumferential wavenumbers n ¼ 725 to the radial displacement and the surface sound pressure is predominant. In Fig. 10, it is indicated that the circumferential wavenumbers n¼ 725 in Figs. 8 and 9 cannot effectively radiate sound into the far field. Although the circumferential wavenumber harmonics can radiate sound into the far field [35], the effects of the high circumferential wavenumber harmonics on the far-field sound pressure can be neglected in the low frequency range. The wavenumber k0 in the fluid is 4.189 rad/m at 1 kHz. Above the ring frequency, acoustic radiation from a cylindrical shell is similar to that from a plate qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 [28,34,35]. Therefore, if the resultant wavenumber k þ n2 =a2 is less than k0 , only these wavenumber harmonics can radiate sound into the far field above the ring frequency. Ellipses can be drawn in Figs. 8–10 according to the definition 2

2

2

k =k0 þ n2 =ðk0 a2 Þ o1. Moreover, the helical wave spectra of the far-field sound pressure are just enclosed by the ellipses in Fig. 10. Wave propagation features and different patterns of the helical wave spectra for the stiffened symmetric and antisymmetric cylindrical shells can be identified in Fig. 10. In Figs. 8 and 9, the wavenumbers out of the ellipses correspond to the subsonic waves that can only cause the near-field sound. The flexural waves in the shell are subsonic and only produce exponentially decaying evanescent waves in the fluid [28]. The wavenumbers within the ellipse correspond to the supersonic waves that induce sound radiation into the far field. The spectral amplitudes within the

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665

Fig. 8. (a) Helical wave spectra of the radial displacement for the symmetric laminated cylindrical shell with two sets of rings, 1 kHz; (b) helical wave spectra of the radial displacement for the antisymmetric laminated cylindrical shell with two sets of rings, 1 kHz.

Fig. 9. (a) Helical wave spectra of surface sound pressure for the symmetric laminated cylindrical shell with two sets of rings, 1 kHz, a3 ¼ a; (b) helical wave spectra of surface sound pressure for the antisymmetric laminated cylindrical shell with two sets of rings, 1 kHz, a3 ¼ a.

ellipse are crucial to evaluating the quantity of the far-field sound pressure. In Figs. 9 and 10, according to Eq. (60), it can be found that the mean squared sound pressure in the near field is larger than that in the far field. In Fig. 11, the helical wave spectra of the radial displacement for the symmetric and antisymmetric laminated cylindrical shells with two sets of rings are shown at 5 kHz. In Fig. 12, the helical wave spectra of surface sound pressure for the symmetric and antisymmetric laminated cylindrical shells with two sets of rings are shown at 5 kHz. In Fig. 13, the helical wave spectra of the far-field sound pressure for the symmetric and antisymmetric laminated cylindrical shells with two sets of rings are illustrated at 5 kHz. The wavenumber k0 in the fluid is 20.49 rad/m at 5 kHz. The harmonic waves that correspond to the circumferential wavenumbers n¼ 774, 763 and 740 have great influence on the radial displacement and surface sound pressure for the stiffened symmetric laminated cylindrical shell in Figs. 11(a) and 12(a). Similarly, the harmonic waves that correspond to the circumferential wavenumbers n ¼ 769, 761 and 740 play important roles in the radial displacement and surface sound pressure for the stiffened antisymmetric laminated cylindrical shell in Figs. 11(b) and 12(b). Angle-plies of the laminated shells have marked effects on acoustic radiation and the radial displacement, which can be concluded in Figs. 11–13. The maximum of the helical wave spectra in Fig. 8(a) is much larger than that in Fig. 11(a). According to Eq. (60), it is shown that the mean squared radial displacement of the stiffened symmetric laminated cylindrical shell at 1 kHz is larger than that of the shell at 5 kHz. For the stiffened antisymmetric laminated cylindrical shell, similar conclusions can also be drawn. There are two clear profiles of ellipses in Fig. 13. The semimajor

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Fig. 10. (a) Helical wave spectra of the far-field sound pressure for the symmetric laminated cylindrical shell with two sets of rings, 1 kHz, a3 ¼ 10 m; (b) helical wave spectra of the far-field sound pressure for the antisymmetric laminated cylindrical shell with two sets of rings, 1 kHz, a3 ¼ 10 m.

Fig. 11. (a) Helical wave spectra of the radial displacement for the symmetric laminated cylindrical shell with two sets of rings, 5 kHz; (b) helical wave spectra of the radial displacement for the antisymmetric laminated cylindrical shell with two sets of rings, 5 kHz.

and semiminor axes of the ellipses are ak0 and k0 , respectively. In the far field, the decaying evanescent waves disappear and only the supersonic waves propagate. Therefore, the patterns of the helical spectra of the far-field sound pressure keep unchanged, but the spectral amplitudes change. Moreover, the patterns in Fig. 13(a) and (b) coincide exactly with those within the ellipses in Fig. 12(a) and (b), respectively. The waves specified by the circumferential and axial wavenumbers at the bright spots and highlights of the helical wave spectra are predominant. The far-field sound pressure of the symmetric laminated cylindrical shell with two sets of rings under different initial axial loadings in combination with a unit radial point force is shown in Fig. 14. The initial axial loadings have trivial influence on the far-field sound pressure. It is indicated that sound pressure induced by the initial axial loadings due to the propeller of the submarine can be neglected in comparison with that caused by the radial force. SPL of the symmetric laminated cylindrical shell with two sets of rings, induced by two groups of point force loadings is shown in Fig. 15. All the loadings with unit amplitude are located at the point P 1 . Sound pressure induced by the radial point force in combination with the axial and circumferential point forces has the same characteristics as that caused by only the radial point force. Contribution of the axial and circumferential loadings to sound pressure is trivial. It is due to the fact that the axial and circumferential stiffnesses of the cylindrical shell are much larger than the radial stiffness. Sound pressure induced by the

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667

Fig. 12. (a) Helical wave spectra of surface sound pressure for the symmetric laminated cylindrical shell with two sets of rings, 5 kHz, a3 ¼ a; (b) helical wave spectra of surface sound pressure for the antisymmetric laminated cylindrical shell with two sets of rings, 5 kHz, a3 ¼ a.

Fig. 13. (a) Helical wave spectra of the far-field sound pressure for the symmetric laminated cylindrical shell with two sets of rings, 5 kHz, a3 ¼ 10 m; (b) helical wave spectra of the far-field sound pressure for the antisymmetric laminated cylindrical shell with two sets of rings, 5 kHz, a3 ¼ 10 m.

radial point force, and the point moments in combination with the radial point force is illustrated in Fig. 16. Sound pressure induced by the radial point force (f 3 ¼ 1) is much less than that caused by the multiple loadings (f 3 ¼ 1, m1 ¼ 1, m2 ¼ 1). The bending waves are sensitive to the moments and change the far-field sound pressure. In Figs. 15 and 16, it can be concluded that only the radial forces as well as the moments need to be taken into account for sound radiation from cylindrical shells.

5. Conclusions Acoustic radiation from shear deformable stiffened laminated cylindrical shells is investigated in terms of sound pressure and the helical wave spectra. The far-field sound pressure is derived by using the Fourier wavenumber transform and stationary phase method. The physical meanings of the helical wave spectra are pointed out. Through the numerical results, it is shown that the dynamic stiffness of a ring taken into account the shear deformation is much lower than that of a ring modeled by the classical beam theory in the large circumferential wavenumbers. The shear deformation of the rings must be considered in the high frequency range. Large circumferential wavenumbers have significant influence on the far-field sound pressure and many circumferential wavenumber items are needed in order to obtain a convergent solution to sound pressure in the high frequency range. The far-field sound pressure induced by the

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95 90 85

SPL (dB)

80 75 70 65 60 55 50 45

0

2

4

6

8

10

12

14

16

18

20

Frequency (kHz) Fig. 14. SPL of the symmetric laminated cylindrical shell with two sets of rings under different initial axial loadings in combination with the unit radial point force.

100

90

SPL (dB)

80

70

60

50

40

0

2

4

6

8

10

12

14

16

18

20

Frequency (kHz) Fig. 15. SPL of the symmetric laminated cylindrical shell with two sets of rings, multiple point forces acting on the shell.

forces acting on the large rings is much lower than that caused by the forces exerting on the shell or the small rings. The larger the q is, the stronger the fluctuation of sound pressure. The helical wave spectra of the radial displacement and surface sound pressure identify the dominant circumferential wavenumbers and show the wave propagation characteristics of the symmetric and antisymmetric laminated shells with two sets of rings. The helical wave spectra can be used to assess the kinetic energy in conjunction with the mean squared radial displacement and sound pressure. Above the ring frequency, the helical wave spectra of the far-field sound pressure 2 2 2 are confined by an ellipse, which is defined by k =k0 þ n2 =ðk0 a2 Þ o 1. The ellipse is the boundary distinguishing the subsonic waves and the supersonic waves from the wavenumber domain. The bright spots and highlights of the helical wave spectra show that the corresponding waves are dominant. Moreover, the ellipse provides a convergent criterion that n should be larger than k0 a in order to obtain exact sound pressure. Therefore, for a long cylindrical shell with large radius, e.g., submarine hull, it seems to be difficult to study acoustic radiation from this finite cylindrical shell by using the modal superposition. The effects of the initial axial loadings on the far-field sound pressure can be neglected. It is indicated that the initial axial propulsive loadings have trivial contribution to acoustic radiation from submarine hull due to the propeller. The farfield sound pressure induced by the axial and circumferential point forces can be neglected in comparison with that caused by the radial point force with the same amplitude. Bending moments contribute significantly to the acoustic field as well as the radial forces do. The influence of moments on sound pressure must be considered.

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669

130 120 110

SPL (dB)

100 90 80 70 60 50 40

0

2

4

6

8

10

12

14

16

18

20

Frequency (kHz) Fig. 16. SPL of the symmetric laminated cylindrical shell with two sets of rings, multiple loadings (the radial point force and point moments) acting on the shell.

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