acoustic radiation from a shell with internal structures

Acoustic radiation from a shell with internal structures

M. EI-Raheb and P. Wagner Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109

(Received 20 April 1988; accepted for publication 5 lanuary 1989)

A method is developed to compute frequency response and acoustic radiation of a complex shell. The axisymmetric geometry of the shell includes cylindrical, conical, and spherical segments stiffen.ed by discrete rings and bulkheads. The shell is coupled to internal masses and elastic frames. Shell segments are treated by transfer matrices. Rings, bulkheads, frames, and concentrated masses are treated by impedances at junctions of segments. The shell is coupled to an external acoustic fluid treated by Green's function and curved surface elements. A major issue facing the method's treatment of the fluid would be lack of existence or uniqueness encountered in the uncoupled, external acoustic problem at characteristic wavenumbers. By using a simple spherical shell, without internal structures, this potential hindrance is shown not to arise. A fuller application of the method awaits subsequent results.

PACS numbers: 43.40.Ey, 43.40.At

INTRODUCTION

Acoustic radiation of elastic shells has been treated ex-

tensively in the past 20 years. In the early 1960's before general purpose computer programs were developed, exact analysis was necessary for studying elastoacoustic interaction. Today the ubiquitous finite element and boundary element methods are used routinely for complex geometries. Despite their generality, these methods are limited duc to the costs incurred and also because they yield little physical in- sight. Analysis is advantageous for shells of revolution because the size of the problem increases only with a circumferential wavenumber while in a finite element model the size

increases with longitudinal and radial wavenumbers as well. A finite element approach would be useful when determin- ing local reactions and equivalent stiffnesses to combine in a faster analytical model.

In the past, analysis used series solutions to the clasto- acoustic equations for simple geometries as typified in the work of Junget and Feit. 1 In the case of a sphere, 2 eigenfunctions of the Helmholtz equation in spherical coordinates were used as trial functions in the elastic equations. A difficulty is that this representation converges slowly. Addi- tional difficulties arise when adopted to the configuration of a finite cylinder with end caps. To apply this method, the configuration would be approximated by a closed and continuous surface over which the Helmholtz equation yields to exact analysis. An obvious candidate is an ellipsoid of revolution that fits best. Prolate spheroid angle functions would then serve as a truncated set of trial functions for the elastic

displacements? This basic assumption bears strongly on the validity and limitations of this method for the following reasons. Since the elastic and acoustic fields are governed by different types of equations, the respective sets of eigenfunctions are not mutually orthogonal. In fact, each elastic mode of vibration of the shell will excite an infinite set of acoustic

modes. Imposing orthogonality on both sets by assuming that the acoustic set is also valid for the elastic problem elim- inates important coupling terms in the resulting coefficient matrix. Furthermore, the prolate spheroid angle functions of

the acoustic field are alien to the elastic field and distort the

shell's elastodynamics, especially at locations where stiffness and mass change abruptly.

An attempt to redress some of these difficulties was re- ported in Ref. 4. In oacuo eigenfunctions of the shell were determined using BOSOR4, a finite difference energy program for shells of revolution? These eigenfunctions genera- ted mobilities of the shell that were then coupled to the external acoustic medium by discrete sources in a Green's function formulation. This modal solution succeeds when

modeling lightly stiflened shells for which smearing circumferential rings is a valid approximation. Experience with heavily stiflened shells including bulkheads has shown that a large number of modes is necessary for convergence of the modal expansion. The added modes are excited by local stiff- ener and bulkhead motions. Furthermore, approximation of the acoustics by discrete sources at centroids of flat surface elements converges slowly with number of elements. 6

A very different approach that might be mentioned is typified by Ref. 7. There, the shell is idealized as an infinite cylinder, permitting integration instead of summing over shell eigenfunctions. Some aspects, such as the effects offing stiffeners, which neglected both twisting and longitudinal reactions, however, can be conveniently investigated there- by. Still, neither the nearfield nor the farfield will reproduce the radiation from finite shells.

The analytical procedures described below yield an improved method for computing the acoustic radiation from elastic shells of revolution, stiflened by discrete rings and bulkheads. The shell is excited by harmonic mechanical forces acting on internal elastic frames attached at multiple terminations. The shell includes masses and inertias concen-

trated at discrete points on its surface. Two coupling mecha- nisms are provided for: the interaction between the shell and (1) the external acoustic medium and (2) the internal frame. The method consists of four distinct phases.

(i) The first phase solves the elastic field equations to derive transfer matrices. The configuration is divided into segments with cylindrical, conical, and spherical generators. Discrete rings, bulkheads, and masses are described by im-

2452 J. Acoust. Soc. Am. 85 (6), June 1989 0001-4966/89/062452-13500.80 ¸ 1989 Acoustical Society of America 2452

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perlances at junctions between segments. (ii) The second phase treats the elastic frame as a sys-

tem of beams governed by Timoshenko equations. An impedance matrix is computed by imposing unit displacements and rotations at each termination independently while fixing all other terminations. This impedance matrix relates forces and moments to displacements and rotations at terminations.

(iii) The third phase solves the acoustic field equations in terms of pressure and velocity influence coefficients. The surface of the shell is subdivided into rings. A source density varying harmonically over the circumference is distributed on each ring.

(iv) The fourth phase couples the shell to the internal frame by means of the impedance matrix. It also couples the shell to the external acoustic medium by equilibrium as the acoustic pressure forces the shell, and by compatibility as elastic and acoustic displacements normal to the shell wall are made continuous.

I. SHELL AND $TIFFENERS

Figure I shows the three basic elements of the axisymmetric geometry: a cylindrical shell between a ring and a bulkhead. This section derives transfer matrices of the segment of shell and impedances of the ring and bulkhead. Throughout this work, harmonic time dependence will be assumed. Furthermore, in Sees. I and II, dependence on circumferential angle 0 will be expressed by the harmonic wavenumber n.

A. Transfer matrix of shell segment

Consider a segment of an axisymmetric shell. Let fs = {N•x, N•o, Qx•, M• }r be a vector of forces and moment resultants as shown in Fig. 1 and gs = {u,o,w,X} r be the corresponding vector of displacements and rotation. The shell equations can be written as a system of 12 first-order ordinary differential equations uncoupled in each circumferential harmonic n

dS - B-S, (1)

dx

where S r = {fT, gr} s and B is a matrix of coefficients derived in Reft 8 that depends on geometry, material properties, and /•.

For a cylindrical segment, set (1) has constant coefficients and a solution exists in terms of exponentials e x•. The )[ 's are determined from the eigenvalue problem 9

(B -- ,t l).S = O, (2)

where I is the unit matrix. The general solution of S can now be expressed as

S(x) = A.eA•-A-•.S(0), (3)

where A is the matrix of eigenvectors of (2), A is the diagonal matrix with coefficients (/l•e[2 ..... 2 8), and the axial coordinate x is zero at the start of a segment. Therefore,

S(I) = T(I)-S(0), T(I) = A.e'•t.A -•, (4)

where T(I) is the transfer matrix relating shell variables at x = I to those at x = 0.

For conical and spherical segments, B is a variable coefficient matrix and (1) does not yield to exact analysis. In- stead, T(I) is the solution of set ( 1 ) after numerical integration in the range 0<x<l with the unit matrix I for initial conditions.

B. Impedances of ring and bulkhead

The circul ar ring is an element reinforcing the junction between segments of shell. Details of the derivation of the Z matrix are omitted here since they can be found in Reft 9. Only the final result is mentioned for completeness:

fs = Z,-gs, (5)

where ZR is the impedance of the ring for a fixed n. The bulkhead has a function similar to the ring's and is

modeled as a circular plate with a concentric hole. Since motions in the plane and out of the plane of the plate are

FIG. 1. Basic structural elements of shell.

2453 d. Acoust. Sec. Am., Vol. 85, No. 6, June 1989 M. EI-Rahob and P. Wagner: Acoustic radiation from a shell 2453


uncoupled, they will be derived separately and then com- bined in the final impedance matrix. For motions in the plane of the plate, the equilibrium equations are

l d ( rN,, ) -- n N.o -- Nø + phco2u = O, r dr r r

(6a)

1 d (rN, o) +nN• + -- -- + phco2v = O, rdr r r

in the domain ai<r<ao, where

2v,, = l _ 7 v - + , -- - • u , (6b) N• 2(1+v) •--r r

1 --• r r dr I '

where (E,v,h&) are Young's modulus, Poisson's ratio, plate thickness, and density, r is radial coordinate, o is radian frequency and the remaining vafiabl• are shown in Fig. 1. Eliminating •e stress resultants produces ordina• differential •uations in the two v•tom of displacements

ß g• = 0, (7)

n a +• D•=•4 r dr 2 • 2 ' •0 ao

Dn = -- n 2 r dr 2 ' '

v d a 1 d

+ n 2

•2 + a•,

•o •o

where g•" = {u,o}• • = El[ p(1 -- •)a• ], and ao is the outer radius. The solution to (7) is

t• •-- -- (l//•l) [ ClJ •, ([•l,r) 4' CzY ; ([5',r) ]

-- (n/i9•r)[ GJ,(13zr) + C4Y,( /•zr) ], o = -- (n/l•,Zr) [C,J. (/•,r) + CaY.(/•,r) ]

-- (1/fta) [ C3J ; ( •2 !') 4' C4Y ; (l•r) ],

l•, =o/rOoao, /]z = [42/(1 --v)/•,],

(8)

where Ck are constant coefficients, (Jn, Y, ) are Bessel functions and ( )' is derivative with respect to the argument. Substituting (8) in (6b), then casting in matrix form gives

f•'• (r) = B.? ( r)-C • i), ( 9 ) g(d• (r) •o . (o =B• (r) C , (10)

where •a = {N•,, N•o} r and C •'• = {Ck} r, k = 1,4. If the inner rim r = ai is free of tractions, then

B)ø (a•).C •i) = 0. ( 11 ) Evaluating (10) at r = ao then eliminating C •o by means of ( 11 ) determines the impedance matrix for inplane motions:

a7'(.o)-[ a:"(.o)] -' ' 1'01' for a• y•0, (12)

where I and 0 are the unit and null matrices of order 2. If

a• = 0, then Yn and Y• drop from (8) and Z• • follows from evaluating (9) and (10) at r =- a o and eliminating Cm. This yields

Z•' =B}•)(ao).(Bgø•)(ao)) -•, forai =0, (13) where in (13) B}• • and 1t {'• are reduced 2X2 matrices formed by columns 1 and 3 of B• and

For bending motions out of the plane of the plate, the classical plate equation is

(FV 4 o 2 ) da I d n a - w = 0, V z = q • • dr 2 r dr r 2' (o o a o

(14) h

1• ao The solution to (14) is

w = C•J. (Ar) + C6Y. (Ar) + C71 n (Ar) + C8K . (Ar),

(15)

where A = (ro/rooao z •)"2. Substituting (15) in the expres- sions for dw/dr, Q., and M•. produc• the relations

•f(r) = B}ø• (r)'C •ø•, (16a) g•O•(r ) (o• . =Bg (r) C , (16b)

where •o• = {Q•, M•}• gtO• = {w, dw/dr}• and C (ø• = {C•}• k = 5,8. Now, by the steps that led to the impedance matrices (12 ) and (13 ) with supermfipt (o) replacing (i),

}oq a, ) ] ' = B• (ao) ] for a• = 0. (17b)

Reordedng {[g}• for inpl•e and out of plane motion to conform with variables of the shell by means era reordefing matfix M,v produc• the 8 X 8 im•ce matrix of•e bulk- h•d:

fs = M,v.f•, gs = M,v.g•, fs = Z•'gs,

Z. M•'[ z•ø O] = .Mj ' o ' (18)

C. Conditions at a junction

Now that the transfer matrices and junction iml•dances have been derived in,4 and B, equilibrium of f and compatibility of g at the interface of conjoined segments of shell are needed. First, express T as

T=[ tff t•] t•r t• ' thus

f2., = tff.f,.i + tfs-g.. ,, (19)

g2. i: tgFf•.i + tgg'gL•,

2454 d. Acoust. Sec. Am., Vol. 85, No. 6, June 1989 M. EI-Rahob and P. Wagner: Acoustic radiation from a shell 2454


where {fl,i, gl.i} and {f2.. g2.•) are the vectors at ends 1 and 2 of the ith segment. Express equilibrium of forces and compatibility of displacements at the junction between segments iandi+ 1 as

fLi+ • -- f2.1 = -- Z'g2.i + fo.i, gLi+ I = g2.i, (20) where fo.i is a line load produced by the reaction of an internal frame or the external pressure load from the acoustic medium and Z is a junction impedance from a ring or bulkhead. Here, f2., and g2.• can be eliminated in (20) using ( ! 9) leaving as unknowns only (f,g) at end 1.

II. FRAME IMPEDANCE

The frame is modeled by straight beams termed branches. The centerline of branches join at rigid or resilient corners. External concentrated forces and moments act on corners. The frame is linked to the shell at terminations

through termination mobilities y r. The segments joined at a corner are linked through linkage mobilities y, while a corner might be connected to the shell through a corner impedance z. The definitions above suggest that the frame can be broken into repetitive groups of corners, segments, and terminations as shown in Fig. 2(a).

Transfer mattices of each branch modeled as Timo-

shenko beam'ø are determined similar to the steps in Sec. I A for the shell. Each transfer matrix is of 12th order and may be written in the form

/-- branch

l T2 yT• cørner 1

/'• . . •'•YT3 T termination •.•, •T 3

Fy,My

z,Fz,Mz, lzz Fx. Mx

x,Fx. Mx. y, Fy,My, lyy

FIG. 2. (a) Branch, comer, termination group and (b) beam element.

g/i/ = tar where f/• = { Fx, F•., F z, Mx, M•, Mz} r, go = {u,v,w,O,, •, •,}r are the force and displacement vectors of the•h branch from the ith corner in a local principal coordinate system [s• Fig. 2(b)]. The vectors at the ith •gid corner {•,•)0' are related to {f,g}# by

(22)

where eo is + I at end 1 and -- I at end 2 of a branch, and is the concentrated excitation vector at the comer. If the

branch is terminated, the vectors at its corner end are relat•

to those at the termination by

q YT•

where g•, is a displacement excitation v•tor at the termination.

•t a 0 be the coordinate transformation matrix that transforms vectors from local to reference coordinates of

segmentj at corner L Then, equilibrium of• has the form

• a0•/e 0 = 0, (24) j--I

where N• is the number of branches from corner i. Also, let the first se•ent in a corner always be taken as a reference segment; then, compatibility of g at a rigid corner between the•h and first g gives

g0 = ao •a,.•,. (25) At each corner of the frame, enforcing (24) and (25) produces a set of simultan•us equations for all {f,g• in the frame.

A set ofdisplacemcnt excitations {g•} = - I is applied at •ch termination independently while all other terminations are held fixed. The solution matrix fr• is the impedance mat•x at terminations. It relat• forces •d moments at •ch

termination to displacements and rotations over all terminations as follows:

Fr = Zr'Gr + Fo, (26) T T T T T where = ..... = .....

and F o is a known force vector at terminations when Gr = 0. It represents reactions at terminations from extern• forces and moments applied to the frame with all te•inations clam•d.

III. ACOUSTIC LOADING

The Green's function surface element method begins by subdividing the. axisymmetric shell boundary into finite tings on which a source density that varies harmonically along the circumference is distributed. • Influence coefficients are determined relating pressure and normal gradient of pressure to source density:

p = A*-<r, ø•P = B*-<r, (27) where p is a vector formed from acoustic pressures on the

2455 J. Acoust. Sec. Am., Vol. 85, No. 6, June 1989 M. EI-Raheb and P. Wagner: Acoustic radiation from a shell 2455


finite rings, A* and B* are pressure and normal gradient influence coefficient matrices, tr is a vector formed from source density, and h is inward normal vector. Expanding p and O- into their harmonics

P = Po + • (P•½ cos nO + P•s sin nO), (28)

O- = O-o + •'• (o-•c cos nO + o M sin nO),

and, in turn,

l}n ---• AH 'O'H, -- BH'O' n , ( 29 )

T T T where pm T = (P0,P, ...... P, ..... ), etc. The coefficients in A M and B n are derived in Ref. 11.

To treat acoustic loading Pn, the pressure is assumed to act as a concentrated line load at each junction between segments. Its intensity equals the average integrated pressures over two segments meeting at that junction (see Ref. 9):

PH = EH'AH'O'H' (30a)

Let e m be a component OfEH.AH, and {3•. H be a row ofA m. If, in general, the segments undergo a change in radius a(s) with intrinsic position s along the generator, as in the case with cones and spheres, then for each Ctm, e H contains the two rows of loading

{ eH(Nxx)} I (sL){a(x)--a(x--Ax)} eH (Qxx) = -•' aH AX

+1 { a(x + Ax) --a(x)} 2 an (sR) Ax ' (30b)

where s L and sn are portions along the generator of in- fluenced points to the left and right of a junction while Ax is the length of a ring along the axis of revolution.

IV. INTERNAL AND EXTERNAL COUPLING

Figure 3 shows a segment of shell at a junction, acted upon by a concentrated force vector fr = { Fx, Fo, F, M•, M o, Mr} r at termination r where forces and moments refer to a local cylindrical coordinate system. Decompose fr in terms of line loads { Px, PO' Pr, rno} as follows:

{P•,Po,P. too} r= { F•, Fo, Fr, Mo} r

( ) 1 1+2 • cosn(0--0o) 2ira .= •

(31)

where fr acts at an angle 61 = 610 along the perimeter of the shell's cross section. Since M• and Mr are not natural moments for the shell, an approximation replaces M• with two radial forces identical in magnitude and opposite in direc- tion, separated by an infinitesimal distance along 61. Follow- ing a similar procedure, two axial forces replace M•. The resulting line loads are

N

{ &,, 0x} r= (M•, -- M•}T._-•I_2 Z n sin n(O -- 0o). •7'a n=l

(32)

x '- ; '-X)s {Fx, Mx; ,, •x)T (F r, Mr; w, •r)T

1 O•w v (Oxx;W,-1 •u+l {Nx: u,-• +•')S •-• 20xViS FIG. 3. Convention of forces and displacements of shell (S), and termination ( T ).

In shell variables, the termination force vector

fs = { N•, N•o, Q•, M• } r can be expressed in terms of the line loads in (31) and (32) as

{ N•,,,N•o, Q•,•,,M•) r

= {P•, + q3•,,po,p, + q3,, mo} r. (33) Expanding the left-hand side of (33) in terms of harmonics, for example

N•x = N • + •'• ( N }•, cos n 61 + N • sin n 0), (34)

then, equating coefficients of cos(n61) and sin(n61) in both sides of (33) determines the relations

f•s = •0 fTK' f•Ox = dO fTk'

ffs"=•c•. fr•, ffs"=d'. fr•, where

1 0 0 0 0 •5"o =•1 0 0 1 0 0 2ira 0 0 0 0 1

1 •o---- [0 1 0 0 0

2ira

s 0 0 0 d•= 1 0 c 0 0 Ira 0 0 s ( n/a ) c

0 0 0 0

s = sin(n0o), c = cos(n0o),

(35)

0 - (n/a)c' 0 0

0 0 '

s 0

'•n-- 1 d(.j•n) n dOo

fs • and f•ø s correspond to axisymmetric breathing and tor- sional motions, respectively. Combining all harmonics of the fs vector in the order

r = {f• , •T •IT •2T •IT•2T Fsjik) •T S • S • S •"'• S •tS •'"a• with use made of (35) produces

Fs•(•) = A•'fr•, (36)

2456 J. Acoust. Soc. Am., Vol. 85, No. 6, June 1989 M. EI-Rahob and P. Wagner: Acoustic radiation from a shell 2456


where

T T A; = o%5 oVT Index k denotes the k th termination andj(k) gives the junction number where the k th termination of the frame meets the shell.

For a frame with m terminations, the concentrated point forces and displacements at its various terminations are related by the impedance matrix given by Eq. (26):

F r = Zv.G r + F o, (26)

where grk = {u,v,w, tPx, •Py, •Pz}rrk are the elements of Gr. The components in gr• are related to shell displacements gs by the relati9ns

vT = Os, •bo = -Xs, (37)

1 (Or I Ou) I c•us toT = tos, •l,, = •- Ox a O0 s-- a OO Expanding gs in terms of its harmonics gn then substituting in (37) produces relations between gr• and

gr• = I'T•'gnjtk), (38) where

I'rk = { )',o, }'•o, )'st, )q2 ..... }'• 0'• .... }. Here, changing the sign of Otbo/O X while deleting transforms y.,.to rr(1 +6.,o) øq'mr and y•., to rr(1 + •5•o) •/m r-

The effect of a mass or moment of inertia at a point on the shell is to add a concentrated impedance at that point. Using l• from (36) and rr• from (38) in which 0 o denotes the circumferential angle of the mass or moment of inertia, and assuming a diagonal matrix M with components { M, M, M, Ix•, Ioo, I.}, where M is the mass and {I•, Ioo, I.} are the corresponding moments of inertia about a local (x,O,r) triad, then the effective impedance is

Z• = -- o2lk.M.rr•, (39)

wherej is the junction number nearest the concentration. The forces and displacements at each junction are relat-

ed to their values at other junctions by Eqs. (19) and (20). The major differences between these relations and those of the frame [ Eqs. (22)- (25) ] are that: ( 1 ) the transfer matrix is of the order 8 X N, where Nis the number of circumferential harmonics; and (2) the impedance matrix Zr introduces coupling among the junctions where it terminates. Denote Z?-' the submatrix coupling termination k• with termination k•. Then, a transformed submatrix can be formed by

Z• •'•'• = •,&,-Z?"rr•,. (40a) Assembling a larger matrix out of the Z½ © by adding overlapping submatrices and then adding contributions of concentrated mass, bulkhead, and ring which do not couple junctions, such as

Z•, (la-Zo-ra), (la,Za-ra), (40b)

leads to a matrix of global impedances of order 4 X N X n j, where nj is the number of junctions into which the shell was divided. Denote this matrix •r.

The forcing function may include Fo from (26), Pn from (30a), and whatever excitations are applied directly on the shell. If one of the latter is intermediate between junctions, it gives rise to both a force inhomogeneity 4• • as well as a displacement inhomogeneity u •. This is seen from a revision of Eq. (4)

S(x) = T(x)'S(O) + H(x-- x•)T(x-- x•).{ FoS} , (41)

where xv is the distance from the left junction where the force vector Fs is applied and H(x - xr) is the Heavyside function. Thus

Now, the force vectors • of the same order as • are assem-

bled out of •&• = A•.F•, and • while the displacement vector Y is resembled out ofv • alone becau• the frame ter-

minations are everywhere at junctions and hence do not con- tribute to displacement inhomogeneity. Also, matrices •a, Y•, •, • assembled out of parts of T(1) for the various segments permits writing the force and displacement relations as a set otglobal equations

Note that, in the a•ve equation, any coupling between different circumferential harmonies, or any coupling •tween more than two con•cutive junctions by means of a frame is contained in the matrix •. It is computationally e•cient, therefore, to reduce the order by partial solution that yields

•'• = -- • -- Pn + •'• •.T, (•)

To derive an equation in the source density v•tor ,, substi- tute PH = En'AH-* from (30a) into (•) to obtain the two p•s of the solution

-- ß + •a-• •-T}. (45) Then, continuity between shell and fluid motions at their interface as adapted from Ref. 9 leads to

{•, sin •, + fi•, cos •, + 1/( = {u• sin • + w• cos •}, (46)

d(a(x)) ta• •s • • ,

dx

where {•,•} and {u•,w•} are the restrictions of •a and g• to just those rows corresponding to axial and radial displacements u and w, andpv is the fiuid's m•s density. With the solution of (46) for ,, • and • are readily found.

A difficulty ads• out of coupling two systems with in- herently different natural coordinates. For this problem, the difficulty is manifested at low frequencies by the incompati- bility •tw•n linearized rigid body rotations of the shell and an internal frame. To see this, on a cylinder consider two points r• and r2 whose global Cartesian coordinat• are

2457 J. Acoust. Sec. Am., Vol. 85, No. 6, June 1989 M. EI-Raheb and P. Wagner: Acoustic radiabon from a shell 2457


r• r = (xk,a sin 0k, -- a cos 0k ), k = 1,2. (47)

The length l• ]r 2 -- r•] is found from

12= (x2--x•)2+2a211 -- cos(02-- 0•) ]. (48)

First, transform the coordinates by a linearized rotation about the global y axis (see Fig. 3), which results in new components

(•)r= {xk -- a cos Okq•,,a sin

-- (a + xk•)cos 0k}. (49) Under this transformation, length is not invariant and the linearized defect can be found from

AI =l•-•l-lr2-

= (1/2l) [ (in)2 _ 12 ]

• (a/I) (cos 02 -- cos 01) [x2(cos 02 -- 1)

-- x,(cos 0• -- 1) ]q•. (50)

107.2 dynes

0.3cm

The defect, as well as other motions, are proportional to The frame's impedance matrix reacts to this defect as it would to any other relative strain between two terminations, and responds elastically with inaccurate and possibly enor- mous forces at terminations. These, in turn, amplify the rigid body rotations. The selection of the globaly axis about which to rotate was arbitrary, and by symmetry the same defect results from a linearized rigid body rotation about any axis orthogonal to the shelfs axis of revolution.

A practical way out of this difficulty is to subtract from the displacements multiplying the frame's impedance matrix the exact rigid body rotations. To do this, a revised form of Eq. (43) is solved for {•,o•-•, • • } by replacing .ff with and co with a fiducial frequency co•. For accuracy in approxi- mating rigid body motion, co• must be necessarily small but finite for numerical reasons. In computing •, the frame is replaced by a set of concentrations of mass, avoiding the difficulty caused by the length defect when the frame's stiffness is included. Now, the solution •9, can be reduced to that part ,• n consisting of rigid body rotations only. Then, for each subsequent value of co, another revision of Eq. (43) is solved, namely

= - Y , (51) where •'v is a matrix like •e but assembled from the frame submatrices Z• © only [see Eq. (40a) ].

V. A SIMPLE EXAMPLE

Instead of proceeding with an application of the method to a complex configuration and examination of its results, a simple spherical shell, without internal structures, is tested here. This enables comparison with results of other investigators and it permits a close survey where difficulties might be anticipated, namely, near well defined characteristic frequencies of the internal acoustic Dirichlet problem.

FIG. 4. Geometry of sphedcal shell.

Putting aside then the complex structures treated above, comparison is sought with existing results for the spherical shell treated in Refs. 2 and 12. Its geometry and material properties are (see Fig. 4)

a z_ 10 cm, h = 0.3 era,

E = 2.0685 X 10 •2 dyn/cm •, p = 7.668 g/cm •, v = 0.3, p/= 1 g/cm 3, c/= 1.524• 10 • cm/s,

where a and h are mean radius and thickness of shell

(E, p,v) are modulus of elasticity, mass density and Pois- sows ratio of shell, andp/and c s are mass density and speed of sound of the acoustic medium. The shell is excited by a concentrated load Fatting radially with sinusoidal time dependence. The intensity ofFis 107.2 dyn. This is the value produced by a uniform pressure of 1 dyn/cm 2 over a solid angle of 72 ø as used in Ref. 12. The spherical shell is divided into ten elastic segments and each elastic segment includes nine curved acoustic ring elements producing 90 acoustic elements in total.

In Ref. 12, the solution was expanded in a series of Le- gendre polynomials and spherical Hankel functions. That this series converges poorly for large/ca is well known? In comparing results, then, differences may arise from differences in load distribution or from insufficient terms in the

series. Also, both problems become more important for large ka.

Sound-pressure levels along equatorial circles on the shell and at points remote from it were computed for the range 0.005•ka•3.2, where k = co/c/is the acoustic wavenumber. Figure 5 plots three directivity patterns for each ka. In these plots, the number at the top represents the largest pressure in dyn/cm • that can be graphed within the plot's limits. The first column plots po = IP(a)[; the second column plotspa = [ p(R)R/a[, where R = 20a is the radius of the equatorial circle in the farfield; and the third column which plots ph• = IRe(p(R)R/a[ is included so that by

2458 J. Acoust. Sec. Am., Vol. 85, No. 6, June 1989 M. EI-Raheb and P. Wagner: Acoustic radiation from a shell 2458


(a) ka = 0.005

Pa PR P•

0.12 0.006 0.006

(b) ka = 0.50 0.14 0.06 0.04

(c) ka -- 1.00

0.6 0.10 0.10

FIG. 5. Sound-pressure field.

2459 J. Acoust. Soc. Am., Vol. 85, No. 6, June 1989 M. EI-Raheb and P. Wagner: Acoustic radiation from a shall 2459


(d) ka = 2.00

Pa PR P• 8 3.2 2.4

(e) ka = 2.90

(f) ka = 3.00

16 0.5 0.4

FIG. 5. (Continue(l.)

2460 J. Acoust. Soc. Am., Vol. 85, No. 6, June 1989 M. EI-Raheb and P. Wagner: Acoustic radiation from a shell 2460


(g) ka = 3.10

. Pa PR P•

8 0.5 0.4

(h) ka = 3.11 10 0.5 0.4

(i) ka = 3.12

10 0.5 0.4

FIG. 5. (Co•tim,e(l.)



(j)ka-3.13

(k) ka - 3,14

(I) ka - 3.15

Pa PR P•

10 0.5 0.4

FIG. 5. (Continued.)



Pa PR P•

(m) ka = 3.20 40 0.5 0.4

FIG. 5. (Continued.)

comparing with column 2, the relative importance of radiation damping can be investigated.

Figure 5(a) shows directivity at ka = 0.005. At such low frequency, the acoustic pressure is caused by inertia added by the fluid. In fact, the shell translates as a rigid body along the line of action of the load. The translational ampli- tude w may be computed simply from the relation

F= r_O2W(m• q- ms), Po = (ap•./2)o2w, m• = 4rra2hp, m s = (2rr/3)a3 ps,

where ms and rn s are mass of shell and low-frequency approximation of mass added by the fluid. For F = 107.2 dyn, w--•3.7X 10 -6 cm and Pu = 0.107 dyn/cm 2, which agrees with the value computed. At this frequency, the oscillating shell radiates approximately as a dipole.

Figure 5(b) shows directivity at ka = 0.5. The dipole nature at the farfield is evident and the minimal difference

between concentrated and distributed loading is observed when compared with results of Reft 12. On the sphere, the pressure develops three complete lobes and a partial fourth at the point where the load is applied.

Figure 5(c) shows directivity at ka = 1.0. Although farfield pressure still is dipolar, the fourth lobe on Pu has become more intense due to increased elastic motions of the

shell. Farfield pressures still agree with Ref. 12. Figure 5 (d) shows directivity at ka = 2.0. This frequen-

cy is close to a bending resonance of the coupled shell. Here, Pa, PR, and P ,•e develop six lobes. Results in Ref. 12 agree in shape but differ in intensity. This is the first appearance of the difference whose two possible causes were given above.

Figure 5(e) shows directivity at ka = 2.9, which was the example in Ref. 2. In fact, when normalized by the factor (a2/F), Pa agrees closely with the results of Reft 2 in shape and magnitude while P h e appears rotated by 180*. An important feature of far field pressure for ka > 2 is the appearance of phase, revealing an increase in radiation damping. This is

evidenced by the growing difference between PR and P The results for ka>•3 cease to resemble those of Reft 12 [see Fig. 5(e) ].

For acoustic radiation as well as scattering problems in a domain exterior to a bounded volume, investigators have taken pains to treat those values of ka corresponding to acoustic resonances of the interior (see Refs. 14-16). This is because when formulating the Helmholtz equation as a Fredholm integral equation, the equation to solve the exteri- or Neumann problem is the transpose of that equation to solve the interior Dirichlet problem. 17 The preceding re- mains true if the words Dirichlet and Neumann are inter-

changed. This would lead to difficulties for rigid boundaries. However, other investigators dealt with compliant boundaries and encountered no difficulty at the critical frequencies. IS.10

TO examine the utility of the present method, a number of frequencies were tried near the first critical frequency, ka = rr. Directivities in the range 3.1 <ka<3.2 in intervals of Aka = 0.01 are summarized in Fig. 5(f)-(m). Note the smooth change from 14 lobes to 16 lobes in P•. Comparably smooth are the P r• directivities as a central lobe vanishes from one end while another appears at the opposite end. Specifically note the similarity among ka = 3.13, 3.14, and 3.15.

Vl. CONCLUSION

A method was devised for predicting the acoustic radiation from a complex shell that might contain various structures. In treating the structures as impedances, it was discov- ered that under rigid body rotations a defect arises between shell and frame displacements that leads to erroneous interface forces. These can be eliminated by directly subtracting the frame's rigid body rotations. An application of the Green's function surface element method is used to describe



the acoustic field. The acoustic fluid was coupled to the shell by concentrating the pressure at line loads and by imposing compatibility between shell and acoustic normal velocities. No difficulties were encountered near critical frequencies. Subsequent work will include application of the method to complex shell structures.

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