Acoustic radiation from fluid-loaded elastic plates I. Antisymmetric modes

Acoustic radiation from fluid-loaded elastic plates Antisymmetric modes

Barry Lee Woolley

Naval Ocean Systems Center, San Diego, California 92152 (Received 20 February 1981; accepted for publication 9 June 1981)

The Timoshenko-Mindlin plate equation of motion is modified in order to simultaneously specify the precise cutoff and asymptotic behavior of both antisymmetric modes of plate vibration described by it. A mathematical method for extending equations of motion to include higher order antisymmetric modes is presented. This method is illustrated by the development of an equation of motion for the first four antisymmetric modes of plate vibration.

PACS numbers: 43.40.Dx, 43.20.Rz, 43.20.Bi

II

INTRODUCTION

It is difficult to obtain solutions of problems of wave propagation in bounded elastic media using the three- dimensional theory of elasticity. In fact, solutions have only been obtained for infinite trains of harmonic waves traveling along infinitely long plates, semi- infinite media, and bodies bounded by circular or ellip- tical cylindrical surfaces. • If an additional boundary is introduced, such as an impedance discontinuity on an infinite plate (for example, a rib or change in plate density or thickness), a superposition of an infinite number of modes is generally required in order to satisfy the additional boundary conditions. The com- plexity of such problems as well as those associated with the treatment of transients has led to the establish-

ment of various approximate plate theories. These strength of material theories of plates involve a finite number of modes and result in simpler frequency equations. These theories have to be used to describe the

wave motion of plates if one wants to calculate the acoustic scattering from liquid-loaded plates with impedance discontinuities. But these approximate plate motion equations have their limitations.

The classical (Lagrange) plate equation of motion is only valid for thin plates at low frequencies: it does not describe the dispersion of the first antisymmetric mode of plate vibration and hence is only accurate up to approximately the classical coincidence frequency of the plate. It does not contain any symmetric modes. It only describes the first antisymmetric mode of plate vibration. A better plate equation for strictly antisymmetric modes of plate vibration is the Timoshenko- Mindlin plate equation. •' It is a two-dimensional theory of flexural motions of isotropic, elastic plates which was deduced from the three-dimensional equations of

elasticity by Mindlin. Mindlin's theory included the effects of rotatory inertia and shear in the same manner as in Timoshenko's one-dimensional theory of bars. 3-s The resulting Timoshenko-Mindlin plate equation was far superior to the classical (Lagrange) plate equation when it came to predicting the behavior of the straight- crested flexural waves in an infinite plate for frequencies above the classical coincidence frequency of the plate. However, the Timoshenko-Mindlin plate equation only gives the behavior of the first two antisymmetric or flexural modes of plate vibration. The cutoff frequencies for the first two antisymmetric modes ob-

tained from the' Timoshenko-Mindlin equation of motion can be made to be exact. But the Timoshenko-Mindlin

equation of motion can only exactly specify either the asymptotic behavior of the first antisymmetric mode or the exact cutoff frequencies for the first two antisymmetric modes. 6 It cannot simultaneously do both. Fur- thermore, the Timoshenko-Mindlin equation of motion has no adjustment parameter for the asymptotic behavior of the second antisymmetric mode of plate vibration. As a consequence of this last fact, the Timoshenko- Mindlin plate equation starts to give inaccurate modal angular positions for the second antisymmetric mode above a frequency that is approximately seven times the classical coincidence frequency. The Timoshenko- Mindlin equation also gives inaccurate results at grazing angles.

It is the purpose of this paper to improve the Timo- shenko-Mindlin equation of motion and to obtain a new equation of motion containing twice the number of antisymmetric modes now contained in the Timoshenko- Mindlin equation of motion.

I. IMPROVEMENT TO THE TIMOSHENKO- MINDLIN EQUATION

Consider an infinite elastic plate of thickness h, density p, Young's modulus E, and Poisson ratio (r (Fig. 1). Let the plate be fluid loaded with a liquid of density 9o. The applied load on the plate is denoted by q.

LIQUID 'X• LIQUID OR VACUUM

FIG. 1o An insonified infinite plate.

771 J. Acoust. Soc. Am. 70(3), Sept. 1981 0001-4966/81/090771-11500.80 ̧ 1981 Acoustical Society of America 771

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 132.174.255.116 On: Thu, 18 Dec 2014 12:30:21

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4oø

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PLATE PARAMETERS:

E = 2.168663 X 10•N/m 2 P = 7.8 X 10 3 kg/m 3 o = 0.283629

st ANTISYMMETRIC MODE (EXACT)

1 st ANTISYMMETRIC MODE (TIMOSHENKO-MINDLIN)

2nd ANTISYMMETRIC MODE (

•. •, .• •, m m

2nd ANTISYMMETRIC MODE. (TIMOSHENKO-MINDLIN)

3rd ANTISYMMETRIC MODE (E

[th ANTISYMMETRIC

MODE (EXACT)

O ø , 10 30 50 70 90 110 130 150 170 1 90 210 230 250 270

fh(kHz-in)

FIG. 2. Antisymmetric modal transmission peaks in an infinite steel plate.

Then the Timoshenko-Mindlin plate equation of motion for the transverse displacement of the plate w is

2 P 8 - 12 • - K-•- • W +ph at 2 ß

-(1 D V•.+ ph •' a •') - -,K•.Gh 12K•.• g• q, (1) where D =EhS/12(1-(y•') is the flexural rigidity of the plate, G =E/2(1 +•) is the shear modulus of the plate, and K•' =•/12 is a constant introduced by Mindlin to account for the fact that the transverse shear strains in

the plate are not truly independent of the thickness of the plate. The value of •' has been chosen in order to get the correct cutoff frequencies for the first two antisymmetric modes. Had •2 been set equal to 2 - 2/3 •/2, the ratio of the square of the Rayleigh wave speed to the

square of the shear wave speed, an incorrect cutoff frequency for the second antisymmetric mode would be obtained while the correct value for the asymptotic behavior of the first antisymmetric mode would be obtained. 6

In order to simultaneously specify the exact value of the asymptotic behavior of the first antisymmetric mode and the cutoff frequencies of the first two antisymmetric modes, more parameters in addition to g•' will have to be introduced into the Timoshenko-Mindlin equation of motion. Indeed, the correct asymptotic behavor of the second antisymmetric mode is also needed. This can be easily seen in Fig. 2. Figure 2 is a plot of the peak transmission in an infinite steel plate as a function of the angle of incidence, e, of an insonifying plane wave of speed c and circular frequency co versus the frequency-thickness product which is related to the dimen- sionless normalized frequency • =w/coo, where coc=c •' (ph/D) •/•'. coc is the classical coincidence frequency of the plate. The liquid loading the plate is water. • = •r2/ 12. It can be seen that the second antisymmetric mode of the Timoshenko-Mindlin plate equation deviates significantly in angie from what one obtains from the

772 J. Acoust. Soc. Am., Vol. 70, No. 3, September 1981

exact equation of elasticity. Two additional parameters besides •' are needed in order to specify the exact asymptotic and cutoff frequency behavior of the second antisymmetric mode. A total of three parameters are needed.

.

Now for harmonic time dependence the Timoshenko- Mindlin equation becomes (with q-0)

+ '2z) + z) w=o. A modified Timoshenko-Mindlin equation with five parameters can be written as

Ph3•2 V 2 + w - w =0 (3) V2+B 12D K2G ] D ' where •2 is defined to be •/12 for the rest of this paper. This equation reduces to the Timoshenko-Mindlin plate equation when A=B-C-F-H-1. Now we must determine the values of A, B, C, F, and H.

At the cutoff frequency for the second antisymmetric mode, we get from Eq. (3), with V-0,

- =0. (4)

The exact equation of elasticity gives the second cutoff frequency as co• =•G/ph •'. Putting this back into Eq. (4) one obtains the following one of the three equations relating the five parameters'

•F =/•. (5) Now let us consider how straight-crested harmonic

waves may propagate. It is sufficient to let w=cosk x (x- ct). Substituting this value for w into Eq. (3) and letting k-oo, we get the asymptotic value of c which is known to be the Rayleigh wave speed c• =[2- (2/3t/•')] t/•' X( c/p)z/2. We get

[-A + B(ph3/12D)c[][-C +(Fp/•'G)c•] =0, (6)

giving either

F=(•:•'G/OC[)C=[3'/•'•/24(3 '/•'- 1)]C (7a)

Barry Lee Woolley' Acoustic radiation I 772


or

B=(12D/phSc•)A=3'/2A/[(1-•)(3 '/2 - 1)]. (7b) A similar argument for the second antisymmetric mode whose asymptotic value of c is known to be the shear wave speed, c,=(G/p) •/•', gives

[-A +(Bph3/12D)c•][-C +(FP/K•G)•] =0, (8) whose solution is either

B=(12D/ph•)A=[2/(1 - c•)]A (9a) or

F = (K•'G/pc•)C: (• / 12 )C. (9b) Either of the two sets of equations relating the param-

eters A, B, C, and F may be used to obtain an equation of motion with precise cutoff and asymptotic behavior for both antisymmetric modes contained in the Timo- shenko-Mindlin theory. That is, either Eqs. (7a)and (9a) or Eqs. (7b) and (9b) may be used. Values for A and C have yet to be determined. They may be determined so as to obtain the best angular and magnitude fit to what one would obtain from the exact equatio•

of elasticity. As an example of one fit we present Fig. 3. In Fig. 3 we display the peak transmission in an infinite steel plate as a function of the angle of incidence, e, of an in•onifying plane wave of speed c and circular frequency w versus the frequency-thickness product. As in Fig. 2, the liquid loading the plate is water. Only the first two antisymmetric modes of plate vibration are shown as predicted by the exact equations of elasticity, by the Timoshenko-Mindlin plate equation, and by our appropriate modification to Eqs. (1) and (3):

12D •' •-• w+ D Ot 2

(H BFph•' 0:z) _ 1 AFD V•. +• q (10) -• - •c2Gh 12•2GD 0t

with A =0.955 and C = 1.0. The values of A and C were

chosen so that the angular separation (at the extreme right of the figure) between the modes as predicted by Eq. (10) would exactly equal the separation as predicted by the exact equatio• of elasticity. Since only the product of A and C is of importance, C was arbitrarily set equal to one. Now what have we gained?

,

ß

The precise cutoff and asymptotic behavior of the first two antisymmetric modes of plate vibration are given by Eq. (10). Hence for the first time we have excellent behavior at high frequency-thickness products for both modes. The Timoshenko-Mindlin plate equation not only gives very poor angular behavior for the second antisymmetric mode at higher frequency-thickness products, but it also gives very inaccurately low values for the magnitude and width of that mode at all frequency-thickness products. Equation (10) gives angular behavior for the second antisymmetric mode that may be tailored (by adjusting A), for a region of 50-kHz-in. or more, to be within a degree of the exact angular values above a frequency-thickness product of about 200 kHz-in. Furthermore, Eq. (10) gives good magnitude and width values for the second antisymmetric mode at all frequency-thickness products even where its angular separation from the position of the corresponding exact mode is significantly large--that is, even at low frequency-thickness products. These results are new and useful. Equation (10) gives good behavior for the magnitude and width of the first antisymmetric mode and asymptotically good behavior for the angular position of the first antisymmetric mode. The magnitude and width predictions of Eq. (10) for the first antisymmetric mode are good even down to a frequency-thickness product of 70 kHz-in., where the angular predictions of the equation are poor. The Timoshenko-Mindlin equation (when K •' is set equal to c•/• i•tead of •/12) gives slightly better predictions for the magnitude, width, and angular behavior of the first antisymmetric mode of plate vibration than those given by Eq. (10).

One would think that the method of this section could

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PLATE PARAMETERS E = 2.168663 X 10 TM N/m 2 o = 7.8 X 103kg/m 3 o = 0.283629

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1st ANTISYMMETRIC MODE (EQ (10) OF TEXT)

I st ANTISYMMETRIC MODE (TIMOSHENKO-MINDLIN)

''' ß ß ee....eeee..e "__"_•..._•..__...__....__....__...__....__...,,,,..,,_____.___•,

I st ANTISYMMETRIC MODE (EXACT)

2nd ANTISYMMETRIC MODE (EQ (10) OF TEXT)_...-•:•.. .......................................... "" ......ee.ee e..ee'

_ . ..................... ; ...... .-r,..--'-"-•SyMMETRIC ........... /• .... M_.O D__.E__( E X__.A.C T)

_ '" ODE

•- (TIMOSHENKO'MIND•IN) I I I I I I I I I I I I

30 50 70 90 11 O 130 150 170 190 21 O 230 250 270 fh (kHz-in)

FIG. 3, Modal transmis-

sion peaks as predicted by various theories for an in-

finite steel plate.

773 J. Acoust. Soc. Am., Vol. 70, No. 3, September 1981 Barry Lee Woolley: Acoustic radiation I 773


be used in a straightforward manner in devising differential equations of motion for three, four, or more antisymmetric modes. These equations of motion would be made to have precise cutoff and asymptotic values for the modes. The additional modes would be intro-

duced by adding additional factors to the left-hand term of the left-hand side of Eq. (1) while appropriately modifying the right-hand side of the equation and introducing undetermined coefficients as was done in Eq. (3). However, this cannot be successfully done. The reason for this is that all modes higher than the first antisymmetric mode are asymptotic to the same limiting value. This would make that set of equations which includes those equations of the type given by Eq. (8) an inconsistent set. Hence we are limited to using the present asymptotic methods for equations of motion which include no more than two antisymmetric modes of vibration. Any successful extension of the equations of motion to include the third or higher order modes of antisymmetric vibration will have to be a nonasymptotic extension. Such an extension will be made in the follow-

ing sections.

II. MATHEMATICAL FORMULATION

The methodology for extending the Timoshenko-Min- dlin plate equation heuristically follows from problems involving scattering from impedance discontinuities on infinite plates. ?-9 Nevertheless the resulting equations apply to plate problems with or without impedance discontinuities.

So without loss of generality consider an infinite plate with an impedance discontinuity (Fig. 4). The plate is liquid loaded and may be air or liquid backed. The origin of the coordinate system is taken to be at the upper plate-liquid interface above the impedance discontinuity of the plate. Let a plane sound wave arrive from the liquid and impinge upon the plate per- pendicularly to the line of attachment of the impedance discontinuity:

•o = e•k• ,i,, e+ •k• co, e, (11)

where 0 is the angle of incidence of the wave and k is

I I•PEDANCE DISCONTINUITY

FIG. 4. An infinite plate with an impedance discontinuity.

the wavenumber in the liquid. The total field potential may be represented as follows:

• =•o + V'etkX sin O-ik• cos 0+ (I•. (12)

Here • represents the field scattered by the impedance discontin•ty and V is the plate-wave reflection coefficient of the plate.

In scattering problems • is sought in a form that satisfies the reduced homogeneo• wave equation:

• f f•)e tx•+•(xz-•z) = dx, (13)

where f(•) is found to be a function of X whose denomina- tor • (X) is determined by the equation of motion used to character•e the plate. For the classical plate equation we have

•(X) =(X • - •)'/•(X 4 +F•) - F• (14)

and for the Timoshe•o-Mindtin plate equation we have 7

• m m 2 m DiX)=(X2-•)•/2(Z < k2F•+F;)-(F5 +XF]) (15) with ,

F•=(pw•/E)( 1 +•)[(2/• •) + 1 -•],

F•=(-ph•/D)[ 1 - (ph2•/12•G)] , (16) F• = WZpo/•ZCh ,

F• = ( w 2 Po/D )[ 1 - (oh 2 w' / 12 •' G) ],

where Po is the density of the liquid and art the other constants are as previously defined.

Now in an exact integral evaluation of Eq. (13) it becomes ap•rent that the poles of f(•) [which are the zeros of •(X)] determine the modes of plate vibration. But the modes of plate vibration are known from the Hayleigh-Lamb equation. s Hence ff we wish to create improved plate equatio• with additional modes or ff we simply wish to improve the behavior of the modes in existi• plate equations, we are presented with a mathematical problem of tailoring the behavior of the poles as a function of the frequency and the material parameters to the known behavior of the modes from the Ray- leigh-Lamb equation. Art the physics of the problem is already contained in the modal description derived from the Rayleigh-Lamb equation. The introduction of additional modes into the approximate plate equatio• is accomplished by the introduction of additional poles in f(•) or more conveniently additio•t zeros in •(k). Now let us turn to the problem of extendi• the Tim- oshenko-Mindlin plate equation to include the first four antisymmetric modes of plate vibration. In the process of introduci• these two additional modes, the behavior of the two antisymmetric modes described by the Timo- shenko-Mindtin plate equation wilt be made more realis- tic.

Now each new mode introduces four new zeros into

•(•) because of the physical symmetry of scattering denoted by $(0)=•-0) (Fig. 5). This symmetry is re-



IMPEDANCE IMPEDANCE DISCONTI N UITY DISCO NTI N U ITY

FIG. 5. Physically equivalent situations.

flected in the introduction of a pole's negative and the introduction of both of their complex conjugates as poles. Hence the most general equation we may write for which is an extension of the Timoshenko-Mindlin plate equation and which includes the first four antisymmetric modes is

•) (x) =(x* -/½)'/*(c'"'x" +,r,x ? +.•'"'x" +'r,x" + a'"'x '•

+),sXS w 2 -F•. X +74X +F•) -(MX 8 +75 xv +PX • +7•X 5

+ NX 4 +),vX 3 + F•'X 2 +),sX + F•'). (17)

However, the previously stated physical symmetry, $(0)=$(-8), requires all the odd terms to be identically zero. That is, all the y•'s are identically equal to 0. Hence we are left with

D(X) =(X 2 -/½)z/*'(GWXa +E•X e +c•X 4 Ft•X *' + F w -

- (MX 8 + PX e + N7t 4 + F•X •' + F•). (18)

We have ten unknown coefficients that have to be deter-

mined' namely, G •, E •, c• w, F•, F•, M, P, N, F•', and F•'.

In the next section we will use the following facts and procedures to determine the ten unknown coefficients. At low frequency-thickness products G •, E • M P, and N must be small compared to the other coefficients and a •, F•', F•, F•', and F•' must be very nearly 1, F•", F•', F•", and F•', respectively. Actually the values of a •, F•, F•', F•, and F• are determined by the exact solution and not by the Timoshenko-Mindlin approximation to the exact solution. Nevertheless, roughly it can still be stated that they are very nearly equal to the Timoshenko-Mindlin coefficients. Furthermore, at the frequency-thickness products at which the third and fourth antisymmetric modes are introduced, F•' and F•' must be identically equal to zero in order to obtain the correct cutoff frequencies for these modes. F• and F•' must still be zero at the cutoff frequency for the second antisymmetric mode. The Rayleigh-Lamb equation can be exploited to give the exact frequency-thickness product at which the antisymmetric modes are introduced. The relationship between the roots of Eq. (18) and its coefficients can be used to complete the determination of the ten unknown coefficients in •'(X). More particularly, by this method the value of G • can be unambiguously determined for any given frequency- thickness product above the point at which the fourth antisymmetric mode is introduced.

III. DETERMINATION OF CONSTANTS

First of all, we want to decide upon a suitable analytic form for the ten unknown coefficients introduced in the


last section. Since the coefficients in the Timoshenko-

Mindlin equation can be viewed as being expressed as the first term or the first couple of terms of an expansion in frequency-thickness product, it is reasonable to assume that the unknown coefficients can be written

by using the same sort of expansion including some higher order terms. Such an assumption would im- mediately give (•w, F•, F•, F•, and F•' the values needed to agree with the exact solution at low frequency- thickness products. In order to keep the mathematics tractable, it is reasonable to try to introduce as few as possible additional terms in such an expansion for each coefficient whenever a new mode is introduced. Further-

more, since G w, E w, M, P, andNmust be small compared to the other coefficients at low frequency-thickness products, it is necessary that their expressions be multiplied by some factor of either h or co or both. So we may tentatively write

G u, Gob 4 ph•'w •. G•h 4 ph•w•( ph•W•Co ) = 80 •G or 80 •G 1- •ZG

•- • P h2w • ph 2 w 2 E •(80)• • •G 1- •G C or

(80) "• ½G -½G C•+[•G] C

• • = 1 - ph2 w2 /Oh2 w• ) 2 •zC C4+•G Cs

F•=•(i+•) --+1- •G •o%•O

•G MoOr •2 G M• - •2 G

Pw•hS po (l Ph• w• ) P= •G -•zG Cm or

PoW•h• [ ph•w• •G Px 1- •C C•s+ C•

•0• ( •• •) N= •G No 1- •G C• or

poW% [ph• • • )•

for the coefficients in the expansion of the Timoshenko- Mindtin equation which includes the first four antisymmetric modes. The Ci's, El's, P•'s, Ni's, G•'s, and M•'s are dime•ionless constants which have to be determined. The factor of •4/80 muEiplying the Gi's was determined from the ratio of G • and F• which would be derived from an approx•ate plate equation including the f•st fo• ant•ymmetric modes of plate vibration using Mindlin's method of derivation of plate equations. •'s That •, we have refined our kinematical assumptions by using Mindlin's systematic method of ex•nding the displacements as power series in the

Barry Lee Woolley: Acoustic radiation I 775


thickness coordinate and thereby obtained the ratio between G w and F•'. Now that we have an analytic form for the ten unknown coefficients, we will proceed to find them.

The cutoff frequencies for the first four antisymmetric modes of plate vibration are given by the Rayleigh- Lamb equation. They are

h-•' l• E ) •'t2 3h-•' •2 E )•'t2 o, q- ' 2 ' and

1( E(1 - or) •/ h 0(l+c•)(1-

F• and F• must be zero at the nonzero cutoff frequencies of the modes contained in our approximate theory. From this it can be determined that

10 1-2(• 1 5(1-2(•) C8=C2ø:• -+8(1-cr)' C9=C2t=•436(1

and

Cl 0 --- C2 2 = (1 - 2c•)

72(1- (•) '

Now the constant term in Eq. (20) is the negative of the product of all nine roots in X •'. But these roots may be unambiguously determined from the Rayleigh-Lamb equation at frequencies which are greater than the cutoff frequency of the fourth antisymmetric mode. Hence one may use the constant term to determine the value of G w at any given point above the cutoff frequency of the fourth antisymmetric mode. Now we may exactly match G w to what it should be from the three-dimen-

sional equations of elasticity at any number of frequencies above the cutoff frequency of the fourth antisymmetric mode depending on the analtyic form we choose for G •. We have chosen G • in Eq. (19) to be such that it will exactly match the results from the three-dimensional equations at either one or two points. We could have chosen a more complicated form of G w which would be able to match the three-dimen-

sional equation results at more points, but that would result in a more complicated approximate equation of motion which is something we are trying to avoid. The form of G •', G •' =(Gok4/80)(ph2co•/•2G), which matches at only one point gives results at other points which diverge rather rapidly from the values given by the

We have determined F•' and F•'.

M, P, and N will only be significant in comparison withF•andF•ifM oorMt, PoørPt, and N O orN t are large. But if these latter constants are large they will violate the known condition that M, P, and N must be small compared to the other coefficients (i.e., a •, F•, F•', F•, and F•) at low frequency-thickness products. Hence M, P, and N are either insignificant or zero. We take them to be identically zero. We have determined M, P, and N.

The relationship between the 18 roots of Eq. (18) and its coefficients will now be used to complete the determination of the ten unknown coefficients in f)(X). First we must state a theorem we will use. tø In an integral rational equation of degree n, with the coefficient of X" unity, the coefficient of X "-• is the sum of the roots with the sign changed; the coefficient of .V '-e is the sum of all possible products of the roots, two at a time; the coefficient of X "-a is the sum of all possible products of the roots, three at a time, with the sign changed; and so on, the constant term being the product of all the roots multiplied by (-1)". Now if we set •)(X)=0 in Eq. (18) and perform a little algebra, we get the following integral rational equation in X=X•':

x + x + , , x + ,

+ (F•) 2 + 2F•a + 2Fra 2NF• - PF; X* +

+ (F•)2+gF•F• 2F• • X+ •)2 :0 ß

three-dime•ional equations of elasticity. Hence we choose the form

(20)

80 - c

and determine Gt and Co from two points chosen to be above the cutoff frequency of the fourth antisymmetric mode but below both the cutoff frequency of the fifth antisymmetric mode arid below the frequency at which the fourth antisymmetric mode shows disper- sive behavior attributable to the anticipation of the introduction of the fifth antisymmetric mode. After making our particular choices, we obtain

4.557467616 x 10-4h 4 ph2co 2 Gw_-. 80

x 0 +9.582533052 x 10-•' Ph•'•ø•' ) We have determined G •.

Having determined M and G w, we may now use the coefficient of the X 8 term in Eq. (20) and the three-dimensional equations of elasticity to determine the values of



E w in an analogous manner to the way the G w coefficient was determined. Having determined E w in this fashion, one may use the coefficient of the X ? term in Eq. (20) and the three-dimensional equations of elasticity to determine the values of a •. Likewise, having determined E • and a w in this fashion, the coefficient of the X 6 term in Eq. (20) and the three-dimensional equations of elasticity may be used to obtain the values of F•. Finally, after F• is determined, one may obtain F• using the three-dimensional equations of elasticity and the coefficients of the X 5 term in Eq. (20).

Although the procedure for determining E •, a •, F•', and F• outlined in the above paragraph is straightforward, there are several complications which must be noted before we give the results. Since an unam- biguous determination of the roots of the unloaded plate only exists where all the roots are real, one is restricted to exactly matching the values of E • • and F• at frequencies above the cutoff frequency of the fourth antisymmetric mode. Since the analytic forms of E • w, • , a and Fx are postulates, this admits of the possibility that their values at lower frequencies will differ from the optimum values that may be displayed by an ideal approximation theory. The determination of E •' •, • • , c• F•, and F• is of course quite dependent on the analytic form and values chosen to determine G •. Furthermore, each coefficient of the sequence E •, c• w, F•, and F• is quite dependent on the analytic form and values chosen for all the coefficients preceding it. As a result of this, the determination of G •, E •, and (• becomes somewhat of a trial and error art' the

coefficients have to be determined with reference to

each other. But they can be determined. It is found, understandably enough, that the determination of F•' and F• is not nearly as critical to the matching of the results from the three-dimensional equations as is the determination of G •, E w, and c• =. Furthermore, since the determination of F•' and F• involves products of the roots •aken three and four at a time, respec-

tively, it is more difficult to explicitly isolate the needed values of F• and F•. This is true because their contributions to the products may be confused with poorly determined contributions from the other factors.

Our determination of E w, a ', F•, and F•' by the above methods gives

,

E• _ 3.382904999 x 10-2h 2 ph 2C0 g (80) •/2

x (1-0.1505•95469 •: 1 - 0.2025644964 ph2 • •G - 5.22895•539x10 -•

x • •G +8.183934727x10 -s

•- 1.284815498 x 10-• (Ph•) 4 (21) F•: (1+•) • + 1- 1-0.2112432232

- 1.952415777X 1 k •G ]

+ 1.391694925x 10-s• •G ] -3 0•4935616x 10 '• (Phzw2• ••

and

•=W2po (l_0.1518806898Ph2W•) Fs •2Gh •G ' The added terms in a• and F• were found necessary to improve the angular behavior of the fo• antisymmetric modes contained in our theory. The final equation of motion for the first four antisymmetric modes of plate vibration is (with the constants G •', F•, F•, E F•', and F• as given above)

(G"'V s- E•V • +a'V 4 +F•V 2 *F•=(F•- F•V2)q. (22)

• 40 ø • •St A•MOOE (EQ (22) OF TEXT) O uJ 30 ø

I st ANTISYMMETRIC MODE (EXACT)

20 ø

10 ø

O o 10

PLATE PARAMETERS

E = 2.168663 X 10 • N/m 3 p-- 7.8X 103kg/m 3 O = 0.283629

2nd ANTISYMMETRIC MODE (EQ (22) OF TEXT•.-.•

- • • • •-'-'••••3rd ANTISYMMETRIC X / _1

2nd ANTISYMMETRIC •_...• J • • MODE (EXACT) • • • ........ c ....... c,,

i i

30 50 70 90 110 130 150 170 190 210 230 250 270

FIG. 6. Antisymmetric modal transmission peaks in an infinite steel plate.

777

fh (kHz-in)

J. Acoust. Soc. Am., Vol. 70, No. 3, September 1981 Barry Lee Woolley: Acoustic radiation I 777


7O ø

60 ø

w 40 ø

Z

U. 30 ø 0

Z 20 ø

10 ø

ldB

3dB -. - 6dB

2nd SYMM

MODE 3rd

I 30 50

I st ANTISYMMETRIC MODE

1st SYMMETRIC MODE

ETRIC 0 o

10 70 90 1 10

PLATE PARAMETERS E = 2.168663X 10" N/m 2 /3 _- 7.8 X 10 3 kg/m 3 o = 0.283629

2nd ANTISYMMETRIC MODE

3rd ANTISYMMETRIC MODE

4th SYMMETRIC

,thANTISYMM 'RIC I I

130 1 $0 170 190 210

fh (kHz-in)

230 250 270

FIG. 7. Transmission loss

as predicted by the exact equations of elasticity for an infinite steel plate.

Equation (22) is used for Fig. 6. In Fig. 6 we display the peak transmission in an infinite steel plate as a function of the angle of incidence, •, of an insonifying plane wave of speed ½ and circular frequency co versus the frequency-thickness product. As in Figs. 2 and 3, the liquid loading the plate is water. The first four antisymmetric modes of plate vibration are shown as predicted by the exact equations of elasticity and by Eq. (22). The angular behavior as predicted by Eq. (22) can be seen to be excellent for the range shown. Above f•--27 the angular behavior as predicted by Eq. (22) diverges from the behavior as given by the exact equations of elasticity. This is due to the anticipated introduction of the fifth antisymmetric mode: Eq. (22) is not an asymptotic theory. The magnitude behavior of the modes will now be discussed.

The T imoshenko-Mindlin plate equation gives good

magnitude behavior predictions for the first antisymmetric mode up to approximately f• =22. Thereafter, the canceling of the first antisymmetric mode by the first symmetric mode of plate vibration (as shown in Fig. 7 and seen in experiments •2) makes the magnitude predictions by the Timoshenko-Mindlin plate equation for the first antisymmetric mode irrelevant to physical observations. The Timoshenko-Mindlin plate equation does not give magnitudes for the second antisymmetric mode that are anywhere nearly correct where the magnitudes are of any significance. However, Eq. (22) gives good magnitude behavior predictions for the first three modes and very poor magnitude behavior predictions for the fourth antisymmetric mode. The poor behavior for the fourth mode is due to the anticipation of the introduction of the fifth antisymmetric mode which is not accounted for in F•'. A change in F• causes changes in the modal magnitudes without

70 ø

60 ø

50 ø

40ø

:30 ø

20 ø

10 ø

1

1st ANTISYMMETRIC MODE

(EQ. (22) OF TEXT)

'•' •'•. 1st SYMMETRIC ,,// MODE . •-

2nd

SYMMETRIC MODE

PLATE PARAMETERS

E = 1.22 x 10 TM N/m 2

p = 8.9 x 103 kg/m 3 o: 0.33

EQ. (22) OF TEXT ----- 6dB

12 dB

2nd ANTISYMMETRIC MODE (EQ. (22) OF TEXT)

10 30 50 70

2nd SYMMETRIC MODE

ß 3rd ANTISYMMETRIC (EQ. (22) OF TEXT)

3rd

SYMMETRIC

//

I

90 110 130

4th SYMMETRIC

MODE

4th ANTISYMMETRIC

MODE (EQ. (22) OF TEXT

150 170 190

fh (kHz-in.)

FIG. 8. Transmission loss

contours as predicted by exact equations of elasticity and modal peaks for antisymmetric modes as predicted by Eq. (22) of text for an infinite copper plate.



significantly affecting the angular positions of the modes.

Now since Eq. (22) was developed using the specific case of an infinite steel plate to guide the formulation, one may ask if the resulting equation is only good for steel. To answer that question we apply Eq. (22) to the case of an infinite water-loaded copper plate. The results are shown in Fig. 8. Figure 8 displays the peak transmission in an infinite water-loaded copper plate as a function of the angle of incidence, 0, of an insonifying plane wave versus the frequency-thickness product. As can be seen from the transmission loss contours calculated using the exact equations of elasticity, the angular behavior of the predicted modes shows that Eq. (22) is indeed not limited to any one given material. The magnitude behavior of Eq. (22) for the case of the infinite water-loaded plate used in Fig. 8 does not seem to be as good as was the case for the steel plate. (The canceling by symmetric modes of antisymmetric modes complicates comparisons.) Magnitude behavior can be easily modified by changing F• ø without significantly affecting the angular behavior.

The poor grazing angle behavior of a purely antisymmetric theory can be corrected only by the addition of a symmetric mode. As has been demonstrated, symmetric modes are needed to obtain the correct magnitude behavior of antisymmetric modes when the two start to coalesce.

IV..BOUNDARY CONDITIONS

We now wish to consider the uniqueness of any

boundary value problem we may encounter using the operator L from Eq. (22). Let u(x) and v(x) be functions with eight continuous derivatives. Let the boundary points or surfaces be denoted by a and b. Then there will be a unique solution if P(u, v), given by

b P(u, v): (uLv- vLu)dx , (23)

is zero. • That is, L must be self adjoint. We inte- grate by parts to obtain P(u, v). For sell-adjoint boundary conditions u must satisfy the same boundary conditions as v. If we let u or v =w (the vertical displacement of the plate) we obtain five distinct sets of the four independent boundary conditions that must be satisfied at both a and b in order for P(u, v) to be zero and the problem to have a unique solution. They are

Set I' w =0

w =0

w =0

w =0,

Set II: w=0

w =0

w =0

wtv=o,


Set III: w =0 (24)

H '• =0

w'" =0

w v =0,

Set IV: w =0

w"=0

Wiv =O

•0 vi =0 ,

Set V: w' =0

w m =0

wv=O

where the prime denotes differentiation with respect to x (without loss of generality). Alternately, each of the five sets may be made up of four linearly independent combinations of the four conditions presented in the set. Before going on, it is perhaps worthwhile to note that the combination w =0 and wt=O characterizes a fixed end, the combination w' = 0 and w t' = 0 characterizes a free

end, the combination w=O and w" =0 characterizes a hinged end, and the combination w • = 0 and w # = 0 characterizes a so-called free-fixed end.

Finally, if we consider the problem of a line impedance discontinuity at c

P(u, v) = (uLv - vLu)dx + (uLv - vLu)dx , (25)

where the boundary conditions are selfadjoint at a and b, we obtain the following five distinct sets of boundary conditions that may be satisfied by u =w (the vertical displacement of the plate) in order to make the solution of the problem unique.

Set I: w(c+)-w(c_)=c•

w'(cJ -w'(c_) =c..

w"(cJ -u,'(c_)

w"(cJ -w"(c_) =c, ,

Set II: w(c.)-w(c_)=c s

w'(cJ -w'(c_)

w"(c,)-w'(c_)

w'(cJ ,

Set III' w(c.)-w(c_)= c 9

w'(cJ -'w'(c_)

w'"(cJ -w"'(c_)

,

set

w"(cJ -,,#(c_)

w(cJ

w"'(cJ ,

Barry Lee Woolley: Acoustic radiation I 779


Set ¾: w'(c2-w'(c_)=c•7

w"(c,) - w" (c_) =

=c,,

= C=o .

The prime denotes differentiation with respect to x. The c•'s are constants or zero. The conditions physically represent statements on the continuity of the displacement, statements on the continuity of the slope, and restrictions on the higher order moments and forces which have to be derived for the particular problem under consideration. The c•'s are constants or zero.

V. DISCUSSION AND RESULTS

Mindtin took the approach of using both the asymptotic high-frequency behavior of plate modes of vibration derived from the three-dimensional equations of

elasticity and, more importantly, the cutoff frequencies derived from the three-dimensional equations of elasticity to fix the behavior of the plate modes of vibration considered in his strength of material plate approximate theories. He did this by appropriately choosing the parameters introduced in his theory so that the cutoff and/or asymptotic behavior matched what one would find from the three-dimensional equations of elasticity. However, that approach has its limitations because of its inability to simultaneously specify both the precise cutoff and asymptotic behavior of all modes with the limited number of parameters used by Mindlin. We have shown that this defect may be partially overcome by the introduction of more undetermined parameters. But one sacrifices agreement at the intermediate frequencies of interest to obtain the simultaneous precise cutoff and asymptotic behavior. One also sacrifices a certain measure of physical insight into the problem. The agreement problem in Mindlin's theory is an inherent problem since Mindlin's approach is one that uses asymptotic fits to a limited number of parameters. The problem is even more complicated by the inconsistency introduced by the higher modes all being asymptotic to the same limit. Hence some sort of modification of Mindlin's

approach is needed if one wishes to sensibly extend the theory to higher order modes than those already treated by Mindlin. One needs an approach which will match the results from the three-dimensional equations of elasticity at the intermediate frequencies of interest in addition to the cutoff frequencies. That is precisely the approach we take.

One does not want to introduce higher order modes of vibration into a theory in a manner that will unduly com- plicate the boundary conditions that have to be met in the solution of boundary value problems. Unduly compli- cating the b•andary conditions just shifts the difficulty of solving such problems from the differential equation onto the boundary conditions. It also may simply re- duce the set of possible boundary conditions to an un- physical set, to a set that has little or no correspon- dence to physical reality. When the foregoing con- siderations of simplicity are applied to the most general differential equation that can be written which incorporates a greater number of antisymmetric modes

780 J. AcoUst. Soc. Am., Vol. 70, No. 3, September 198•

and makes use of the physical symmetry displayed in Fig. 5, we have in effect restricted ourselves to undetermined coefficients which have a simple form. We take for our representation of these coefficients a logical expansion in frequency-thickness product that reduces to a theory which is a good approximation to the three-dimensional equations of elasticity at low frequency-thickness products. For four antisymmetric modes such simple coefficients for a logical expansion based on an extension of the Timoshenko-Mindlin

theory appear in Eq. (19).

The expansion that has been used for our theory for four antisymmetric modes has the benefits of a Mindlin plate approximate theory: the cutoff frequencies are precisely determined and the theory has a physical basis. But instead of using asymptotic slope information as Mindlin has done, we use modal information from the three-dimensional equations of elasticity at an intermediate frequency. Herein lies the advantage of our approach. By using modal information from the three-dimensional equations of elasticity at intermediate frequencies, we obtain greater ability and flexibility in tailoring the behavior of the modes contained in our theory. Hence we obtain much better behavior for these modes. Since higher order modes in general couple with ones of lower order, we have the same upper limit to the possible accuracy of our theory that exists in Mindlin's approach: namely, the applicability of the approximate theories are limited to the cutoff frequency of the lowest neglected mode or to a frequency below the cutoff frequency if the lowest neglected mode is strongly coupled to one of the lower modes.

Now our approach may be used to derive a needed approximate equation of motion if an exact frequency

equation is available to guide the work. Our •approach is not limited to plate theories. The existence of an approximate strength of material theory, such as the Timoshenko-Mindlin plate theory, is not essential to employing our method. However, the nonexistence of such theories may make the employment of our method exceedingly difficult. This would be due to the difficulty of mathematically deducing the needed material dependence of the approximate theory. Our method may be employed on a poor strength of material theory to extend the number of modes contained in it while improv- ing it or, alternately, to improve the theory without introducing any new modes. The method has the possibility of being used to guide the theoretical development of strength of material approximate theories since the mathematical matching of the exact modes at intermediate frequencies may reveal what the material dependence can or cannot be in such an approximate theory. It would be appropriate to mention here that our development of Eq. (22) for the' first four antisymmetric modes of plate vibration does not depend on the parochial heuristic development given in this paper but could have been made on a mathematical basis starting with the Timoshenko-Mindlin plate equation.

ACKNOWLEDGMENTS

The author acknowledges many helpful discussions during the progress of this work with Professor S. I.

Barry Lee Woolley: Acoustic radiation I '• 780


Hayek of the Applied Research Laboratory of the Penn- sylvania State University. The author also wishes to thank Dr. P. A. Barakos of the Naval Ocean Systems Center for his encouragement of this project and S. L. Speidel, also of the Naval Ocean Systems Center, for the transmission loss calculations shown in Figs. 7 and 8 and used throughout this work in calculating modal peaks using the exact equations of elasticity. This work was supported by the Independent Research program of the Technical Director of the Naval Ocean Systems Center.

1A. E. Armenakas, D.C. Gazis, and G. Herrmann, Free Vibrations of Circular Cylindrical Shells ( Pergamon, New York, 1969) and references cited therein.

2R. D. Mindlin, "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates," J. Appl. Mech. 18, 31-38 (March, 1951 ). ,

3Lord Rayleigh, Theory of Sound (Macmillian, New York, 1945), Vol. 1, p. 258.

aS. Timoshenko, "On the Correction for Shear of the.Differen- tial Equation for Transverse Vibrations of Prismatic Bars," Phil. Mag. Ser. 6, 41, 744-746 (1921).

5S. Timoshenko, Vibration Problems in Enggneering (Van Nos- trand, New York, 1937), p. 337.

6K. F. Graff, Wave Motion in Elastic Solids (Ohio State Univ., Columbus, 1973).

?B. L. Woolley, "Acoustic Scattering from a Submerged Plate I. One Reinforcing Rib," J. Acoust. Soc. Am. 67, 1642- 1653 ( 1980 ).

SB. L. Woolley, "Acoustic Scattering from a Submerged Plate II. Finite Number of Reinforcing Ribs," J. Acoust. Soc. Am. 67, 1654-1658 (1980).

9B. L. Woolley, "Backscattering from a Plate with a Thick- ness Discontinuity," J. Acoust. Soc. Am. 68, 345-349 (1980).

10j. V. Uspensky, Theory of Equations ( McGraw-Hill, New York, 1948).

lip. M. Morse and H. Feshbach, Methods of Theoretical Phy- sics (McGraw-Hill, New York, 1953), Vol. 1, p. 870.

12R. Richards, Ph.D. thesis, Pennsylvania S•te University ( 1980 ).



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