Acoustic Radiation from a Finite-Length Shell with Non ... - CiteSeerX

Journal of Sound and Vibration (1996) 197(3), 329–350

ACOUSTIC RADIATION FROM A FINITE-LENGTH SHELL WITH NON-AXISYMMETRIC SUBSTRUCTURES USING A SURFACE VARIATIONAL PRINCIPLE S.-H. C Core Technology Research Center, Samsung Advanced Institute of Technology, Suwon 440-600, Korea

 T. I  J. D. A Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208-3109, U.S.A. (Received 18 April 1995, and in final form 8 April 1996) A modal-based method is developed to analyze the acoustic radiation of axisymmetric submerged shells of finite length with non-axisymmetric internal substructures, subjected to time-harmonic loads. In this method, a variational principle is used to derive impedance relations between the surface pressure and the surface velocity of the shell. These impedance relations are combined with the structural equations of motions based on a Lagrange energy formulation to form a complete set of equations for the fluid–structure interaction problem. Fourier series expansions are used to represent the circumferential dependence of the surface pressure and velocity. The method is demonstrated for two different configurations of substructures: circular ribs supporting length-wise beams and a spatially and modally dense array of oscillators. Since the substructures couple the circumferential modes, a large system of equations must be inverted. A matrix decomposition technique is used to reduce the size of the system of equations of the first example. For the second example, the asymptotic limit for an infinite number of substructures is developed using an integral form for the substructure impedance. It is shown that the Monte Carlo simulations for the oscillator substructures converge to the asymptotic results. It is also shown that, below the ring frequency, oscillators can induce damping that is more effective in reducing far-field radiation than high loss factors in the shell. 7 1996 Academic Press Limited

1. INTRODUCTION

Acoustic radiation from elastic structures with internal substructures has been of great interest for many years. The radiated field can be treated analytically for structures with an outer geometry conforming to a separable co-ordinate system and with substructures attached along co-ordinate lines. Two important examples are infinite plates reinforced by beams, which were examined by Crighton and Maidanik [1] and by Photidias [2], and infinitely long cylindrical shells with circular ribs, which were studied by Bernblit [3], Burroughs [4] and Burroughs and Hallander [5]. If the outer geometry does not conform to a separable co-ordinate system, a numerical approach, based on boundary or finite 329 0022–460X/96/430329 + 22 $25.00/0

7 1996 Academic Press Limited

330

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element techniques, is commonly used. Here the surface response is typically expressed in terms of local shape functions. Williams et al. [6] and Sandman [7] used the boundary element method to determine the acoustic pressure field radiated from finite-length shells. Ettouney et al. [8] used the boundary integral equation approach to account for the fluid loading on a finite-length shell containing internal systems. An alternative method using global shape functions, developed by Ginsberg et al. [9], Wu et al. [10] and Wu [11], uses a variational method to determine the relationship between surface pressure and surface velocity for axisymmetric bodies. This method has been used to solve a fluid–structure interaction problem for plates by Ginsberg and McDaniel [12] and Ginsberg and Chu [13] and for axisymmetric shells by Chen and Ginsberg [14]. More recently, Bjarnason et al. [15] and Choi et al. [16] applied this method for a fluid– structure–substructure interaction problem; i.e., a submerged shell with internal substructures subjected to axisymmetric and non-axisymmetric harmonic excitations. In these papers, Lagrange multipliers are used to account for the force–velocity relationships at the connections between the shell and substructures. To reduce the complexity of the problem, only axisymmetric substructures were considered, so that the problem could be decoupled for each circumferential harmonic. The present paper is an extension of reference [16] which allows for non-axisymmetric substructures. This extension results in a totally coupled system. The analysis consists of the following steps. First, the unknown surface response of the shell is expanded into a series of global shape functions that span the entire surface of the axisymmetric body. The variational principle is used to derive impedance relations between the surface pressure and the surface velocity responses. These impedance relations are combined with the structural equations of motion based on a Lagrange energy formulation to form a complete set of equations for the fluid–structure interaction problem. The computationally intensive task of finding the shell surface impedance is separated from the vibration analysis of the internal substructures using a Lagrange formulation. Since only the outer surface of the shell is exposed to the surrounding fluid, the same set of surface pressure–velocity impedance relations can be used repeatedly for any configuration of internal substructures and applied loads. The coupling of substructures, the influence of substructure vibration on the main structure and the subsequent acoustic radiation in the fluid can be effectively studied using this approach. The method is demonstrated for two examples. For the first example, a submerged cylindrical shell with end caps is considered. The substructures consist of circular ribs supporting two beams in the axial direction. Since the beams couple the circumferential modes, a large system of equations must be inverted. In this paper, a matrix decomposition technique [17] is used systematically to reduce the size of the system of equations. For the second example, the effect of a large number of internal substructures is considered. Soize [18], Pierce et al. [19] and Igusa and Xu [20] have demonstrated that if the number and modal density of substructures is high, their aggregate dynamic effect can be described by surprisingly simple analytic expressions. It is shown that this result applies to substructures in fluid-loaded shells. The shell of the first example is used again and substructures are modeled as spring–mass oscillators attached to the interior of the shell. Several hundred Monte Carlo simulations are used in this study. In each simulation, the masses and attachment locations of the substructures are the same, but the natural frequencies are chosen at random, so that they are uniformly distributed over a frequency range. For comparison, the asymptotic limit for an infinite number of substructures is developed using an integral form for the substructure impedance. This integral form was first developed by Skudrzyk [21] in his mean value theory, and was applied to multiple substructures by Igusa and Xu [20]. The asymptotic limit is based on the criteria that the

   

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spacing of the natural frequencies and the spatial spacing of the substructures approach zero. It is shown that, in the limit, the acoustic response of the system depends only on the total mass, the damping and the natural frequency range of the substructures. Furthermore, the substructure impedance does not couple any of the shell modes. The Monte Carlo simulations for the oscillator substructures converge to the asymptotic results. The computational effort for the asymptotic analysis is less than 1% of that required for each Monte Carlo simulation. An analysis of power flow is included, as previous studies have shown that it provides useful insight into the behavior of submerged systems [22, 23]. The results indicate that most of the power which is put in by the applied load returns to the shell, with only a few percent of the power radiating into the far field. This is in agreement with recent experiments on a related problem conducted by Mann et al. [23]. Furthermore, it is shown that for frequencies less than the ring frequency, a large number of lightly damped oscillators reduces acoustic radiation substantially more than high loss factors in the shell. This gives further confirmation to the above-mentioned studies on substructures with high modal density [18–20] as well as a fundamental study by Strasberg and Feit [24].

2. VARIATIONAL EQUATIONS FOR THE SURFACE PRESSURE

The procedure for implementing the variational principle for structural acoustic problems was outlined in references [9–16]. In this section the basic equations for axisymmetric shells with non-axisymmetric loads are briefly reviewed. First, the arc length parameter, s, is defined such that s = −sm and s = sm correspond to the ends of the shell, as shown in Figure 1. The shape of the axisymmetric shell is described in parametric form by the functions r(s) and z(s), which are the co-ordinates of the generating curve in the radial and axial directions, normalized with respect to a length parameter, a, which is usually the cylindrical shell radius. The position vector of each point on the shell is described by x = x(s, u), where u is the azimuthal angle. Next, an expansion of the surface pressure and displacement in terms of global shape functions is used. For an axisymmetric shell geometry one can use Fourier series expansions to represent the u dependence of all variables. The s dependence is expanded in a series of basis functions. To develop a non-dimensional representation, we scale the displacement by the radius a and the pressure by rc2, where r and c are the fluid mass density and sound speed, respectively. Thus the

n

s=0

Shell

ϕ

θ

es s =–sm

r

z

s=sm

Figure 1. The co-ordinate system for the axisymmetric shell.

.-   .

332

pressure p and displacement w are written as Nc

Np

p(x, t) = rc2 s s pmn cmn (s) cos nu e−ivt, n=0 m=1 Nc

Nw

w(x, t) = a s s wmn fmn (s) cos (nu − us )e−ivt,

(1, 2)

n=0 m=1

where pmn and wmn are modal amplitudes and cmn (s) and fmn (s) are the pressure and displacement basis functions. The displacement basis functions have three components which correspond to the displacements tangential and normal to the shell surface. Herein, bold lower case letters are used to denote such three-component vectors. Furtheremore, the e−ivt term will be omitted for convenience, with the understanding that all harmonic response expressions are to be multiplied by this term. The number of circumferential modes, Nc , and the numbers of basis functions, Np and Nw , are chosen to be sufficiently large such that the contribution of the last term in the series is several orders of magnitude smaller than the entire sum. One of the most important virtues of the variational principle is that, once a sufficient number of modes is included in the expansions for the surface pressure and displacement fields, convergence is extremely rapid [15]. In the present paper, the natural modes of the dry shell will be used for the displacement basis functions fmn (s) and a Fourier series will be used for the pressure basis functions cmn (s). Also, only driving forces symmetric about u = 0 will be considered. Hence, the cosine series expansion is used for the pressure and displacement fields which are symmetric with respect to u = 0, and the sine series expansion is used for the circumferential displacement field which is antisymmetric. In equation (2), the phase angle us is used to accommodate the two circumferential expansions. The value of us is p/2 for the circumferential displacement and zero otherwise. The variational principle formulation [9–16] will be used to obtain a relationship between the surface pressure and displacement. The relation between the unknown surface pressure and displacement is given for each circumferential harmonic, n, as An pn = Bn wn ,

(3)

where pn = {p1n p2n · · · pNp n }T and wn = {w1n w2n · · · wNw n }T are vectors of surface pressure and displacement coefficients. The elements of matrices An and Bn are given in Appendix A. 3. STRUCTURAL DYNAMICS EQUATIONS

In this section, it is shown how the equations of motion for non-axisymmetric internal substructures can be combined with the preceding equations for the fluid-loaded shell. Two very different configurations of substructures are considered: circular ribs supporting length-wise beams and a spatially and modally dense array of oscillators. 3.1.  1 Consider the system shown in Figure 2, in which circular ribs are rigidly connected to the shell at x = xk (sk , u), for k = 1, 2, . . . , K, where K is the total number of the ribs. Two longitudinally configured beams are rigidly connected to the ribs at u = 2ub . The ribs are modelled as in reference [25], with four types of motions: radial–flexural motion, tangential–extensional motion, out-of-plane flexural motion and torsional motion. The first three types of motion are equated with the shell’s radial, circumferential and

   

333

r Arc length, s Load

ϕ z

sm

Ring stiffener Beam stiffener 12a Figure 2. Example 1: rib-stiffened shell with internal beams.

longitudinal displacements, and the torsion is equated with the derivative of the shell radial displacement with respect to the arc length s. The beams are modelled using Timoshenko theory. At the rigid beam/rib connections, the continuity of the displacement and rotation are enforced. The displacements of the kth rib and the beam are expanded as Nc

4

u(k) (u) = s s uin(k) h(k) in cos (nu − us ), n=0 i=1

Nr

v(z) = s nr jr (z),

(4, 5)

r=1

respectively, where uin(k) and nr are modal amplitudes and h(k) in and jr (z) are the normal modes of the rib and beam with free boundary conditions; z denotes the axial co-ordinate. The normal modes are four-component vectors with three displacement components and a rotation in the r–z plane. Next, we will use Lagrange’s equation with the modal amplitudes as generalized co-ordinates. The kinetic energy T and potential energy V of the combined system are given by Nc Nw K Nc 4 Nr T 1 2 ˙ mn + 2 s s s 12 (u˙in(k) )2 + 2 s 12 n˙ r2 , 5= s s 2 w ra k=1 n=0 i=1 n=0 m=1 r=1

(6a)

Nc Nw K Nc 4 Nr V 1 1 1 2 2 (k) (k) 2 ˜ r2 nr2 , 5 = s s 2 vmn wmn + 2 s s s 2 (vin uin ) + 2 s 2 v ra k=1 n=0 i=1 n=0 m=1 r=1

(6b)

where vmn , vin(k) and v˜ r are the natural frequencies of the dry shell, the kth rib and the beam, respectively. Each mode shape is normalized with respect to the corresponding mass so that the modal mass term is omitted. The work done on the shell by external forces consists of two parts: the work U e done by an external harmonic excitation qe(s, u) on the shell and q˜(z) on the beam and the work Uf done by the fluid pressure. The work quantities are given by Nc Nw Nr Ue e 2 3 = s s qmn wmn + s q˜r wr , rc a n=0 m=1 r=1

Nc Nw Uf f 2 3 = s s q mn wmn . rc a n=0 m=1

(7, 8)

e f Here qmn , q˜r and q mn are the generalized forces for the external loading on the shell, the

.-   .

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external loading on the beam and the fluid loading, respectively. They are given by e qmn =

gg p

sm

−p

−sm

q˜r =

g

[qe(s, u) · fmn (s) cos (nu − us )r(s)] ds du,

(9)

Np

L

[q˜ (z) · jr (z)] dz,

f q mn = − s Rml,n pln ,

(10, 11)

l=1

−L

where the Rml,n and displacement–pressure coupling coefficients, defined by Rml,n = pen

g

sm

cln (s)[n(s) · fmn (s)]r(s) ds.

(12)

−sm

Here, en = 1 + dn0 in which dij denotes the Kronecker delta. Next, Lagrange multipliers are used to account for the reaction forces at the shell/rib connections and rib/beam connections. The constraints are specified by equating the displacements and rotations of the shell and substructures at their connections. Since the ribs are axisymmetric, the connections are circular arcs described by points x = xk (sk , u). The constraint conditions for the shell/rib connection are given by Nc

f (k) = s n=0

6

Nw

s wmn m=1

$

%

7

4 fmn (sk ) − s uin(k) h(k) cos (nu − us ) = 0, in −n(sk ) · f'm n (sk ) i=1

(13)

for k = 1, 2, . . . , K. Equation (13) holds for every circumferential mode number n; hence one can rewrite it as (k) (k) f (k) (k) n = Cn wn − C n u n = 0,

(14)

T for k = 1, 2, . . . , K. Here u (k) n = {u1n u2n u3n u4n } is a vector of displacement coefficients of the kth rib. The components of the 4 × Nw matrix C(k) (k) n and the 4 × 4 matrix C n are given as

C(k) n =

$

f1n (sk ) −n(sk ) · f'1n (sk )

··· ···

%

fNw n (sk ) , −n(sk ) · f'Nw n (sk )

C (k) n = [h1n h2n h3n h4n ].

(15) (16)

The constraint conditions for the rib/beam connections are given by Nc

(k) g(k) = s E(k) (k)v, n un − E

(17)

n=0

for k = 1, 2, , . . . , K, where v = {n1 n2 · · · nNr }T is a vector of displacement coefficients of the beam. The components of the 4 × 4 matrix E(k) (k) are given n and the 4 × Nr matrix E as E(k) n = [h1n cos (nub − us ) E (k) = [j1 (zk )

· · · h4n cos (nub − us )],

(18)

· · · jNr (zk )].

(19)

By substituting the expressions for the kinetic and potential energies, the generalized forces

   

335

and the constraint conditions into Lagrange’s equations, one obtains the following equations: K

(k) e Rn pn + Dn wn − 2 s C(k) n ln = qn ,

(20)

k=1 (k) (k) (k) (k) D  (k) (k) = 0, n un + 2Cn ln − 2En l T

D v + 2E (k)  l(k) = q˜ .

T

T

(21, 22)

(k) (k) (k) (k) T (k) (k) (k) (k) T (k) = {l 1n l 2n l 3n l 4n } are vectors of Lagrange Here the vectors l(k) n = {l1n l2n l3n l4n } and l multipliers. The components are the interaction forces and moments at the shell/rib and rib/beam connections, respectively. The matrices Dn , D  (k) are diagonal, with elements n and D defined as

Dij,n =

a2 2 (v − v 2 )dij , c 2 in

D (k) ij,n =

a 2 (k)2 (v − v 2 )dij , c 2 in

D ij =

a2 2 (v˜ − v 2 )dij . c2 i

(23–25)

The equations of motions (20)–(22), the fluid–structure coupling equation (3) and the constraint equations (14) and (17) yield a complete set of simultaneous equations for the combined shell–fluid–substructure system. 3.2.  2 Consider the system shown in Figure 3, in which the substructures, modelled as spring–mass oscillators, are connected to the shell at x = xk (sk , uk ), for k = 1, 2, . . . , K, where K is the total number of substructures attached in the region 0 E s E sm , 0 E u E p. The locations and properties of the spring–mass oscillators are assumed to be symmetric about the s = 0 and u = 0 planes. A spring–mass system is characterized by a mass, ra 3mk , and a complex-valued natural frequency, vk . A complex-valued natural frequency is introduced to account for damping in the spring–mass system. The kinetic and potential energy of the combined system are given by Nc Nw K T 1 2 ˙ mn + 4 s 12 mk u˙k2 , 5= s s 2 w ra k=1 n=0 m=1

(26)

0

1

2

Nc Nw K Nc Nw V 1 1 2 2 2 = s s v w + 4 s m v u − s s cmn,k wmn , k k mn mn k 2 ra 5 n = 0 m = 1 2 k=1 n=0 m=1

(27)

where uk is the non-dimensional displacement of the kth mass and cmn,k is defined as cmn,k = [n(sk ) · fmn (sk )] cos (nuk ).

(28)

Fluid n(s )

r s=0 x 1

ω1 m1

xk (sk ,θk )

x2

ω2 m2

uk ···

xK

ωk mk

s=sc

ωK ···

mK

Shell s=sm

Figure 3. Example 2: shell with internal spring–mass oscillators.

z

.-   .

336

The work quantities are now given by Nc Nw K Ue e 2 3 = s s qmn wmn + s qk uk , rc a n=0 m=1 k=1

Nc Nw Uf f 2 3 = s s q mn wmn . rc a n=0 m=1

(29, 30)

e f and q mn are the generalized forces defined as in equations (9) and (11), and qk Here qmn is the external loading on the kth substructure. By substituting the expressions for the kinetic and potential energies and the generalized forces into Lagrange’s equations, one obtains the following equations:

Np

s Rml,n pln + l=1

−4

a2 a 2 Nc Nw 2 2 2 (−v + vmn )wmn + 4 2 s s c c q=0 p=1

$

%

K

s mk vk2 cmn,k cpq,k wpq k=1

a2 K e s (m v 2 c )u = qmn , c 2 k = 1 k k mn,k k

(31)

Nc

Nw

mk (−v 2 + vk2 )uk − mk vk2 s s cpq,k wpq = q=0 p=1

c2 q . a2 k

(32)

Solving equation (32) for uk yields uk =

Nc Nw 1 qk . 2 s s cpq,k wpq + 2 1 − v /vk q = 0 p = 1 mk (−v 2 + vk2 )a 2/c 2

(33)

Using the variational principle, equation (3), one can rewrite the first term in equation (31) as Np

Nw

l=1

l=1

s Rml,n pln = s Qml,n wln ,

(34)

where Np

Np

Qml,n = s s Rmi,n A−1 ij,n Bjl,n .

(35)

i=1 j=1

By substituting equations (33) and (34) into equation (31), the following system of equations for the displacement coefficients is obtained: Nc

Nw

s s Smpnq wpq = qmn

(36)

q=0 p=1

for m = 1, 2, . . . , Nw and n = 1, . . . , Nc , where K

2 Smpnq = Qmp,n dnq + (−v 2a 2/c 2 + vmn a 2/c 2 )dmp dnq + 4 s k=1

$

mk vk2 a 2/c 2 c c 1 − vk2 /v 2 mn,k pq,k

%

(37)

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337

4000 Matrix size

(a) 3000 2000 1000 0

CPU minutes

500 400

(b)

300 200 100 0

CPU minutes

6 (c) 4 2

0

10 20 30 40 100 Excitation frequency, ka

Figure 4. Computational effort versus excitation frequency for Example 1: (a) matrix size; (b) time to evaluate the matrix elements; (c) time to invert the matrix (HP 715 workstation, 31 Mflops double precision).

and K

e qmn = qmn +4 s k=1

cmn,k qk . 1 − v 2/vk2

(38)

Equation (36) is solved to yield the displacement coefficients, and the pressure coefficients are obtained using equation (3). 4. COMMENTS ON THE COMPUTATIONAL EFFORT

Non-axisymmetric substructures couple the circumferential modes; hence the size of the matrix may exceed several thousand for problems such as those illustrated in Figures 2 and 3. However, the solution of the matrix equations is not unduly difficult because the matrix is sparse. Nevertheless, the size of the system of equations can be significantly reduced using a matrix decomposition technique. First, equations (3) and (20) are combined to eliminate pn . The resulting equation is then combined with equation (14) to eliminate wn . By eliminating pn and wn one can obtain a simultaneous equation for the substructure displacements and the Lagrange multipliers. The size of the system of equations can be further reduced by following the same steps to eliminate the other variables. The surface pressure and displacement are obtained by back-substituting the Lagrange multipliers into equation (20) and combining the results with equation (3). An indication of the computational effort required to solve the acoustic problems of interest here is given in Figure 4. The graphs in this figure show, for Example 1, the size

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338

of the system matrix, the time to calculate the elements of the matrix and the time to invert the matrix. These figures show that most of the computational effort is used to evaluate the system matrix and that little effort is needed for its inversion. In the matrix evaluation step, over 99·9% of the effort is in calculating the submatrices An and Bn . These submatrices, which correspond to the acoustic properties of the fluid domain, are in terms of the double surface integrals from the surface variational principle. The computational profile in Figure 4 is in sharp contrast with those of spatial discretization methods, such as the boundary element method. In the discretization methods, the system matrices are relatively easy to evaluate, yet computationally time-consuming to invert. The modal-based method developed herein is particularly suited to studying substructure effects. This is because the computationally intensive submatrices, An and Bn , are independent of the substructures. Thus, it is possible to use these submatrices to examine the effects of different substructure geometries, connections and loading conditions. After the one-time cost of computing An and Bn , most of the additional cost for solving each substructure problem is in the relatively simple task of inverting the system matrices, as shown in Figure 4. When a range of excitation frequencies is of interest, the computational effort can be reduced by using a spline approximation for each element of the submatrices An and Bn . In the example studies, the surface variational principle integrals in these submatrices were evaluated at only 50 excitation frequencies. These evaluations were more than sufficient to determine accurate spline approximations for all frequencies in the interval 0 Q ka Q 10. 5. EXAMPLE STUDY

To demonstrate the method, a finite-length shell with a cylindrical mid-section and hemispherical endcaps is considered (Figures 2 and 3). The shell is modelled as a steel structure submerged in water. The ratio of the overall shell length to the radius is 12:1 and the ratio of the cylinder radius to the wall thickness is 50:1. Structural damping is introduced by a complex-valued elastic modulus E = E0 (1 − ih), where E0 is the nominal real-valued modulus and h is the loss factor. Unless noted otherwise, a loss factor of 0·04 is used for the shell. The free vibration modes of the shell are determined by an analysis similar to that of Klosner et al. [26], whereby the equations of motion for the cylindrical and hemispherical portions of the shell are treated separately. Equations due to Sanders are used for both the cylindrical and hemispherical portions of the shell. The mode shapes for the combined shell/end-cap system are then derived by enforcing compatibility conditions at the shell/end-cap interfaces. A set of sinusoidal functions given by cm 0 = cos [(m − 1)ps/sm ],

cmn = cos [(2m − 1)ps/2sm ],

n e 1,

(39)

is employed as the pressure basis functions. 5.1.  1 The substructures used in the first example have the following physical and geometrical properties. Circular ribs are rigidly connected to the shell at z = 22·0a and z = 25·0a. The ribs have a square cross-section of dimension 0·2a, with the same material properties as the shell. Two beams are rigidly connected to the ribs at ub = 22p/3. They have the same material properties and square cross-section as the ribs. Point loads of amplitude rc 2a 2 are applied at the mid-points of both beams. The surface response is computed at ka = 1, 2, 4 and 8, and the amplitude of the surface pressure is plotted in Figure 5 as a function of arc length, s, and circumferential angle, u. It can be seen in Figures 5(c) and

Figure 5. The surface pressure of the shell–ring–beam system: (a) ka = 1·0; (b) ka = 2·0; (c) ka = 4·0; (d) ka = 8·0.

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340

.-   .

Figure 6. Helical wave spectra of the surface pressures shown in Figure 5: (a) ka = 1·0; (b) ka = 2·0; (c) ka = 4·0; (d) ka = 8·0. Open circles correspond to the flexural wavenumbers of the infinite cylindrical shell.

(d) that the substructures cause an increase in the surface pressure at the location of the ribs. To investigate how the substructures alter the vibrating behavior of the shell, helical wave spectra are used. Details of the theory and interpretation of helical wave spectra can be found in Williams et al. [27]. Briefly, the amplitude of a wave with two wavenumbers (a circumferential wavenumber ku = n/a and an axial wavenumber kz ) is displayed as a

   

341

T 1 The natural frequencies of a free–free circular ring Natural frequencies, ka ZXXXXXXXXXXCXXXXXXXXXXV Flexural motion Extensional motion

Circumferential mode, n 0 1 2 3 4 5 6 7 8 9

3·539 0 0·363 1·026 1·966 3·179 4·663 6·418 8·442 10·74

0 5·009 7·929 11·22 14·63 18·10 21·59 25·10 28·62 32·15

density plot. Each point on the helical wave spectra is characterized by a wavenumber kcs = zkz2 + ku2 and a propagation direction tan−1 (kz /ku ). The helical wave spectra corresponding to the surface pressures in Figure 5 are shown in Figures 6(a)–(d). For comparison, the flexural wavenumbers of an equivalent, empty infinite cylindrical shell are plotted as circles. These benchmark results are obtained by solving the frequency equations of an infinite cylindrical shell [28]. It can be seen that some of the helical spectra for the finite shell coincide with the infinite shell spectra, particularly at lower circumferential wavenumbers. Since the shell is excited only through the ribs, the dynamic characteristics of the circular ribs are important in understanding the results. For this purpose, the natural frequencies of the unsupported circular ribs are listed in Table 1. The table shows that for the excitation frequency range 0 Q ka Q 10, only the first eight circumferential flexural rib modes would be in resonance. Consequently, the flexural wavenumbers corresponding to higher circumferential modes (n e 9) would not appear on the shell. This behavior is clearly shown in Figure 6. –1

Pressure amplitudes

10

10–2

10–3

0

20

40 60 80 Number of simulations

100

120

Figure 7. The average pressure amplitude versus the number of simulations for ka = 2·0; —, (m, n) = (3, 0); – · – · –, (m, n) = (3, 1); – – –, (m, n) = (8, 4); · · · · · , (m, n) = (6, 8).

.-   .

342

In Figure 6(a), it is shown that the n = 3 circumferential mode dominates the responses. There are two reasons for this. First, the beams are connected to the ribs at u = 22p/3. Second, as shown in Table 1, the excitation frequency, ka = 1, is very close to the rib’s third flexural natural frequency. The interaction between the rib resonant modes and the shell circumferential modes is also evident in the remaining plots in Figure 6. At ka = 2, 4 and 8, Table 1 indicates resonances with rib flexural modes 4, 6 and 8. The corresponding helical spectra in Figures 6(b)–(d) show significant shell surface responses in circumferential modes 4, 6 and 8. In addition, the n = 2 circumferential mode in Figure 6(d) is enhanced due to resonance of the rib extensional mode at ka = 8. Finally, in Figures 6(a) and (b) it is shown that the n = 1 circumferential mode is important for ka = 1 and ka = 2. This is due to the fact that at such low frequencies, there is a significant rigid body response to the load. In Figures 6(c) and (d), it can be seen that the flexural wave locus is repeated with equal spacing in the horizontal direction. Similar results have been reported in reference [14] for a shell with bulkheads. This phenomenon was qualitatively explained in terms of the dispersion relation of the infinite cylindrical shell. It was shown that the spacing between the repeating patterns is given by kd = 2p/d, where d is the distance between substructures. In Figures 6(c) and (d), the value of kd is approximately p/2, which corresponds to the distance between the first set of circular ribs. 5.2.  2 In the next example, shown in Figure 3, the substructures are simple oscillators distributed at equal spacing over the inner surface of the cylindrical shell. The positions of the oscillators are given by the arc length co-ordinate si = (2i − 1)sc /2K1 and the circumferential angle uj = (2j − 1)p/2K2 , where i = 1, 2, . . . , K1 and j = 1, 2, . . . , K2 . The total number of oscillators is K = K1 × K2 . All oscillators have equal mass and the mass of oscillators per unit area is denoted by raMs . The natural frequencies of the oscillators are chosen at random, so that they are uniformly distributed over a frequency range vL E v E vU . The natural frequencies are multiplied by (1 − ihs )1/2 to impose damping on the oscillators. A loss factor of hs = 0·05 is used for the oscillators. For all but the last study in this example, a loss factor of h = 0·04 is used for the shell. A point load of amplitude rc 2a 2 is applied to the mid-point of the shell. For the Monte Carlo simulations the following values are used: vL a/c = 1, vU a/c = 5, Ms = 0·2, K1 = 100 and K2 = 63. It was found after numerous simulations that only two parameters, the oscillator mass per unit area and the natural frequency range, are important if the number of the oscillators is sufficiently large. This is in agreement with recent theoretical studies on the dynamic properties of large numbers of substructures [18–20, 24]. In Figure 7 the pressure amplitudes for several representative shell modes (m, n) are shown for ka = 2. The pressure amplitudes are averaged over the number of simulations. Convergence to stable ‘‘mean values’’ is observed for all cases. In Figures 8(a) and (b) the surface pressure and velocity are shown for the shell with oscillators at ka = 2, averaged over 120 Monte Carlo simulations. For comparison, the surface responses for an empty shell subjected to the same excitation are presented in Figure 9. It can be seen that the primary effect of the oscillators is to confine the surface responses to a small area near the excitation. Following the theory developed in reference [20], an asymptotic limit for an infinite number of oscillators will be developed. The collective oscillator force on the (m, n) mode of the shell can be written as a

s qmn =s j=1

g

1S

[Zj (v)w˙ (s, u)d(x − xj )] · fmn (s) cos (nu − us ) dS,

(40)

Figure 8. The surface response of the shell/spring–mass system at ka = 2·0 using 120 Monte Carlo simulations: (a) pressure; (b) velocity.

    343

Figure 9. The surface response of the empty shell at ka = 2·0: (a) presure; (b) velocity.

344 .-   .

Figure 10. The surface response of the shell/spring–mass system at ka = 2·0 using the approximate impedance in equation (47): (a) pressure; (b) velocity.

    345

.-   .

346 –1

Radiated power/input power

10

10–2

10–3

10–4

10–5

0

1

2 Frequency, ka

3

4

Figure 11. The ratio of the radiated and input power versus frequency: a comparison between the empty shell with varying loss factors and the shell with oscillators. —, Empty shell, hs = 0·04; - - - , empty shell, hs = 0·10; — – —, empty shell, hs = 0·20; – – –. shell with oscillators.

where Zj (v) and xj denote the impedance and the location of the jth oscillator and 1S represents the surface of the shell. Although the oscillator reaction force on the shell has only a radial component, all three components are used in equation (40). This is an approximation, which is needed later in the analysis to take advantage of the orthogonality of the displacement shape functions. This approximation has insignificant effects on the results, because the radial component governs the near and far field acoustic responses. Using equation (40), the modal expansions for the shell velocity are Nc

Nw

a

s qmn = s s w˙pq s Zj (v)fmn (sj ) · fpq (sj ) cos (nuj − us ) cos (quj − us ). q=0 p=1

(41)

j=1

The ensemble average for many shells with randomly generated oscillators is Nc

Nw

g

s qmn 1 s s w˙pq Z (v) q=0 p=1

=

fmn (s) · fpq (s) cos (nu − us ) cos (qu − us ) dS

1S

ra Z (v)w˙mn . rh

(42)

Here, Z (v) is the asymptotic limit for the average oscillator impedance as the number of oscillators grows large [20]: Z (v) 1 −iMs

v vU − vL

g

vU

vL

m(m − ivhs ) dm. m 2 − v 2 − imvhs

(43)

Using equation (42) to represent the modal reaction forces due to the oscillators, the following system of equations is obtained: Nw

e s S mpn wpn = qmn p=1

(44)

   

347

for m = 1, 2, . . . , Nw and n = 1, 2, . . . , Nc , where

0

1

2 v 2a 2 vmn a2 va ra Z (v) dmp . −i 2 + 2 c c c rs h

S mpn = Qmp,n + −

(45)

The asymptotic oscillator impedance in equation (43) can be easily evaluated, either numerically or analytically. For the example in this section the asymptotic impedance is found to be 1·533 − 2·126i at ka = 2. The real and imaginary parts of the impedance can be interpreted as the damping and inertia forces associated with the oscillators. Since equation (44) is decoupled with respect to the circumferential mode numbers, the shell surface response can be obtained with only a small fraction of the computational effort required to evaluate the ‘‘exact’’ expressions in equation (36). Shown in Figure 10 are the surface pressure and velocity fields obtained using the asymptotic impedance. They are in good agreement with the results in Figure 8, which is based on equation (36). The distribution of surface pressures in Figures 8–10 indicates that a significant amount of power has been dissipated due to the oscillators. This is in agreement with the relatively high effective oscillator damping, quantified by the imaginary component of the asymptotic impedance. To investigate the dissipative effects of the oscillators further, the ratio of the input and radiated power is examined. This ratio is plotted as a function of excitation frequency in Figure 11. The power ratio for the shell with oscillators is compared with those of three empty shells with the following loss factors: the original h = 0·04 as well as h = 0·10 and 0·20. The main result is that the oscillators are more effective in dissipating structural energy than an empty shell with relatively high loss factors. This behavior is particularly prominent for frequencies less than ka = 2·5, which is close to the shell ring frequency. In all cases, the amount of power radiating into the fluid is only a few percent of the input power at frequencies higher than the ring frequency, and decreases to 1/10 000th of the input power at lower frequencies. As stated in the introduction, this power flow behavior gives further confirmation of earlier, more fundamental studies on large numbers of substructures [18–20, 24] and is in agreement with related experimental work [23].

6. CONCLUSIONS

A modal-based method has been developed to analyze the acoustic radiation of submerged shells of finite length with non-axisymmetric internal substructures. Global shape functions are used for the displacement and pressure fields over the surface of the shell, and the relationship between these two fields is established by a variational principle. This method is powerful in that it is possible to analyze varying configurations of internal substructures without the need to re-evaluate the complex variational principle integrals. The method was applied to a cylindrical shell with end caps. Two different substructure configurations were considered: circular ribs supporting length-wise beams and a spatially and modally dense array of oscillators. Helical wave spectra were presented for the first example, and the results were explained on the basis of the resonance characteristics of the shell and the internal substructures. For the second example, the asymptotic limit for an infinite number of substructures was developed using an integral form of the substructure impedance. It was shown that Monte Carlo simulations for the oscillators converge to the asymptotic results. It was also shown that oscillators can induce damping that is more effective in reducing far field power radiation than high loss factors in the shell.

348

.-   . ACKNOWLEDGMENT

This research was supported by the Office of Naval Research under contract No. 91-J-1980 whose support is gratefully acknowledged.

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22. T. S. MC and J. M. C 1994 Proceedings of the National Conference on Noise Control Engineering, Ft. Lauderdale, FL 1–4 May, 563–568. Mobility power flow (MPF) approach applied to a fluid-loaded shell with a plate bulkhead. 23. J. A. M, III, E. W, K. W and K. G 1991 Journal of the Acoustical Society of America 90, 1656–1664. Time-domain analysis of the energy exchange between structural vibrations and acoustic radiation using near-field acoustical holography measurements. 24. M. S and D. F 1993 Journal of the Acoustical Society of America 94, 1814(A). Vibration damping of large structures induced by attached small resonant structures. 25. T. W and W. C. L. H 1968 Journal of the Acoustical Society of America 43, 1005–1016. Vibration analysis of stiffened cylinders including inter-ring motion. 26. J. M. K, M. L. P and Y. N. C 1976 American Institute of Aeronautics and Astronautics Journal 14, 833–834. Vibrations of a stiffened capped cylinder. 27. E. G. W, B. H. H and J. A. B 1990 Journal of the Acoustical Society of America 87, 513–522. Experimental investigation of the wave propagation on a point-driven, submerged capped cylinder using K-space analysis. 28. M. C. J and D. F 1986 Sound, Structures, and their Interaction. Cambridge, MA: The MIT Press; second edition.

APPENDIX A: SURFACE VARIATIONAL PRINCIPLE MATRICES

The elements of matrices An and Bn in equation (3) are given as Aml,n =

g g sm

sm

−sm

−sm

{(ka)2cln (s)cmn (s')H2n (s, s')−c'ln (s)c'm n (s')H3n (s, s')

− n 2cln (s)cmn (s')H4n (s, s')+n[c'l n (s)cmn (s') cos 8'r(s) + cln (s)c'm n (s') cos 8r(s')]H5n (s, s')} ds' ds,

g

sm

Bml,n = 2p(ka)2

(A1)

cmn (s)[n(s) · fln (s)]r(s) ds

−sm

+ (ka)2

g g sm

sm

−sm

−sm

cmn (s)[n(s') · fln (s')]H1n (s, s') ds' ds,

(A2)

where the indices m and l denote row and column numbers, respectively. The functions Hin , i = 1, . . . , 5, appearing in the preceding integrals, are given as

g6 p

H1n (s, s') =

[r(s)−r(s') cos U]

−p

0

× ika −

H2n (s, s') =

g

p

g

p

dz(s) dr(s) − [z(s)−z(s')] ds ds

1

1 G(xs =x's ) cos (nU)r(s)r(s') dU, R R

[sin 8 sin 8' cos U + cos 8 cos 8']G(xs =x's ) cos (nU)r(s)r(s') dU,

−p

H3n (s, s') =

−p

7

cos UG(xs =x's ) cos(nU)r(s)r(s') dU,

.-   .

350 H4n (s, s') =

g

p

g

p

[cos 8 cos 8' cos U + sin 8 sin 8']G(xs =x's ) cos (nU) dU,

−p

H5n (s, s') =

sin UG(xs =x's ) sin(nU) dU,

(A3)

−p

where U = u' − u is a relative azimuthal angle and the function G(xs =x's ) is the free space Green function, G(xs =x's ) = eikaR/R,

R = =xs − x's =/a.

(A4)

All of the integrands in equations (A3) become singular when R = 0, i.e., at s = s' and U = 0. Numerical integration of equations (A1) and (A2) is carried out to avoid these singularities. The procedure was described in detail by Wu et al. [10].