an approximate theory for the sound radiated from a periodic line-supported plate

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Journal of Sound and Vibration (1978) 61(3), 3 15-326

AN APPROXIMATE THEORY FOR THE SOUND RADIATED

FROM A PERIODIC LINE-SUPPORTED PLATE

D. J. MEAD

Department of Aeronautics and Astronauti cs,

Un iversity of Southampton, Southampton SO9 5NH ,

ngland

ND

A. K.

MALLIK~

Department of Mechanical Engineeri ng,

I ndian nstitute of Technology, Kanpur 208016, I ndia

(Received 13 March 1978)

An approxim ate metho d is presented for estimating the sound po wer radiated by an infinite

plate, supported elastically along parallel, equ i-spaced lines and subjected to a simple

pressure field convecting uniformly over the plate in a direction perpendicular to the supports.

Suitable complex mod es are assumed for the induced plate flexural wave motion, and an

energy method is used to estimate the plate response and the radiated sound pow er. The

effect on the plate response of the acoustic loading is taken into account. The influeqce of the

convection velocity (which may be subsonic) and of certain plate paramete rs is investigated.

1. INTRODUCTION

The sound power radiated by vibrating stiffened plate structures is important in aero-

nautical and naval applications. The structures which vibrate are frequently periodic, in

that they can be regarded as an assembly of many identical elements, identically inter-

connected Advantage may be taken of this spatially periodic property to simplify vibra-

tion and acoustic calculations.

It has already been shown [l) that plates which are stiffened periodically can radiate

sound when they are excited by a subsonic convected pressure field. The response of the

simpler periodic plates can be obtained in a closed form [2] provided the acoustic loading

on the plate does not significantly affect the plate response. In that case, the radiated sound

power can be reliably estimated. However, when the effect of acoustic loading on the

response is significant, the closed form solution is inapplicable. A special series form can

then be used, and Pujara [3,4] has introduced a series of “space harm onics” for this pur-

pose. However, he found that up to eleven terms were required in the series to provide

adequate convergence.

In a recent paper [S] the authors showed that the response of periodic beam s to convected

pressure fields can be predicted quite accurately by an energy method in which a suitably

chosen app roximate mode of wave motion is used. The space-averaged energy of vibration

was predicted surprisingly accurately. It is the purpose of the present paper to use the

same method to predict the sound power radiated by such a system. The energy method

used makes it particularly simple to include the effects of fluid (acoustic) loading in the

response and radiated power estimation.

tThh work was carriedout when the second

author was on

study leave as a Commonw ealth Scholar at the

Institute of Sound and Vibration Research , University of Southam pton, Southam pton,

England.

315

00224 0X/7 8/2303 15 + 12 02.00/O @ 1978 Academic Press Inc. (London) Limited

G



316

D. J. MEAD AND A. K. MALLIK

The system considered is an infinite uniform plate, with a single set of equi-spaced

elastic line-supports which make the plate into a periodic structure. A harmonic (and sub-

sequently a random) pressure field of plane w aves convects across the plate in a direction

perpendicular to the line-supports. The theoretical analysis of the radiated sound is

developed, and computed results are presented to show how the radiated sound pow er

varies with both frequency and convection velocity of the pressure field. Attention is con-

centrated only on the first free-wave propagation band of the periodic system. For the case

of the random pressure field (assum ed to have a frozen pattern) the average power radiated

from the whole of this propagation band is considered.

The results are compared with the sound power radiated by the plate as calculated in the

following ways

:

(4

W

(c)

by using the method of this paper, but estimating the plate response in the absence

of acoustic radiation dam ping; the response so calculated is then used to calculate a

radiated sound power

;

by using the closed form solution for the plate response [2], again w ith acoustic

radiation dam ping in the response ca lculation neglected; the sound power radiated

by this plate motion is then calculated;

by using the method of space harmo nics [3, 41 to estimate the plate respon se, full

account being taken of the acoustic radiation dam ping in the response calcula-

tion; the sound pow er radiated by this motion is then calculated and is the most

accurate value which can be determined-its accuracy is limited only by the numb er

of terms used in the space harmonics series.

2. SOUN D RADIATION FROM THE INFINITE PERIODIC PLATE

The plate to be considered is infinite in both x and y directions. The infinite set of identical,

uniform elastic line suppo rts lies in the y direction at the x-wise interval 1. The transverse

stiffness of these supports is

k

per unit length and the rotational stiffness is

k,

per unit

length (see Figure 1).

A harmon ic, plane pressure field convects across the plate in the x direction at frequency

o and convection velocity cP. The local pressure is independent of y and is expressed by

p(x, t) = pOeiCo r- pX),

(1)

where the wave-number, kp, is ojc,,.

,Support I lrveS

Rototlonal stiffness,

k per unit length

Tronslot~onal stiffness,

k per unit

length

Figure 1. Diagram of the periodically stiffened plate.



SOUND

R DI TION FROM PERIODIC PL TE

317

The plate displacem ent, w(x, t), generated by this pressure will also be independent of y,

and w ill be expressed by

w(x, t) = W(x) ei@,

(2)

where x is measured from the left-hand support of one of the plate bays. W (x) is in general

a complex function.

Since all the plate bays are identical, and are subjected to the same exciting pressure

field, the responses in all bays mu st be identical apart from a phase difference. In fact

w(x + 1, ) = w(x, t) eeikpr.

(3)

This latter property, which applies to all adjacent pairs of bays, makes it possible to

expand the whole displacemen t pattern w(x, t)

(- co < x < + co) in

this series of space-

harmonics:

w(x, t) = +f wi ei(wr-kjx),

(4)

where

j=-m _

kj = kp + 2jz/l, mj = f

f

’ W x)e”p dx.

(%6)

0

A full justification for this particular series is given in the Appendix. The non-sinusoidal

wave motion

W x)

has been decomposed into sinusoidal components so that the fluid

loading effects on the motion can be readily analysed. To do this, use is made of the simple

relationship between the acoustic rad iation pressure an d the amp litude of the sinusoidal

wave motion which is causing it:

&ad, x, r) =

iopc

J1 _ (kj,k)2

~je (m -k x ,

(7)

where

k

is the wave-num ber of free acoustic wave motion in the adjacent acoustic medium .

k = o/c where c is the speed of sound.) The total sound pressure acting at any point due to

all such components is

P&(XP , = ‘c”

iwpc

W ei ot -

kg

j=-a,

Jl

- k/k )’ ’

’

(8)

The total sound power radiated by the ‘plate c&r be shown to be equal to the sum of the

powers radiated by each component wave on its own. Each of these is given by

pbx, t w~Tx, 0 dx

per un it area of the plate, where wT(x, r) is the complex conjugate of the ith component of

the plate wave-displacemen t, and Re stands for “real part of” . Evaluating the integral, and

summ ing over all the wave compon ents, gives the total power radiated per unit area as

n(w) =

Re y -

1 pcwz 1wj 12

j=-a2

Jl - kJk)”

3.

THE EQUATIONS OF PLATE MOTION INCLUDING

SOUND RADIATION EFFE CTS

(10)

The plate displacement, W x), is governed by the usual fourth order differential equation

of motion When the external pressure fluctuation p(x, t) (equation (1)) and the acoustic



318

I). .1.MEAD AND A. K. MAl.I.lK

radiation pressure P,,~(x> t) are included in this. it becom es

where

D

is the flexural rigidity of the plate, Eh3/1 2(

I -~ v’), h

is the plate thickness, r is

Poisson’s ratio, 9 is the loss factor of the plate and m is the mass per unit area of the plate,

equal to P,,,~,

h. Wj is given by equation (6).

As in reference [5], an approxim ate solution is sought for this equation. and the Galerkin

metho d is used.

W x)

is express ed in terms of a series of suitable approxim ate comp lex

mod es each of which satisfies certain “wave-boundary” conditions approp riate to the

periodic nature of the system and its external loading. These m odes and boundary conditions

have been fully discussed in the previous paper [5]. The sth such modal displacement func-

tion is denoted by ,f’,(s) so that

W(x) =

1 5 4,.f,(-e tm

s=l

qs is the non-dimensional amplitude of the sth mode . It is convenient now to introduce the

non-dimensional co-ord inate, 5, given by

< = (2x/1) - 1)

(13)

which has its origin at the centre of the bay between adjacent support lines. Other non-

dimensional quantities are introduced as follows:

non-dimensional frequency, Sz = w12 m/D )’ l.

14a)

non-dimensional convection velocity, CV ’ =

cpl(m/D)', (14b)

non-dimensional sound velocity, SV = ~l(m,D) ~,

(14C)

non-dimensional pressure loading, P =

po13/D,

(14d)

non-dimensional density of the acoustic medium, p = /)air//pp,a,ch,

14e)

non-dimensional rotational supp ort stiffness, tiI =

kJD,

l4fl

non-dimensional translational supp ort stiffness, K = k,13/D.

14N

The virtual work metho d may now be used to set up the equations for the qr’s, by substi-

tuting equation (12) into equation (ll), m ultiplying throughou t by @* *j’:(c) (v = I to N

in turn) and integrating over one comp lete bay length, i.e., from 5 = - 1 to 5 = + 1. Tw o

additional terms must be added into the equations to allow for the virtual work of the trans-

verse and rotational constraints at one end of the bay (at 5 = + 1 say). After suitable non-

dimensionalization, the following set of simultaneous equations is obtained for the ql’s:

[K_ - Q2Mry + i&SV s)Cr,]jq~) = F’fQI). 15)

where

164

16b)



SOUND R DI TION FROM PERIODIC PL TE

Q, = f

s

+’ e-‘%/2mf: d,

-I

319

(164

These terms may be identified, respectively, as the non-dimensiona l stiffness mas s, acoustic

dam ping and exciting force corresponding to the rth generalized co-ordinate. The primes on

the j’s in & denote differentiation with respect to 5.

The total sound power radiated per unit area of the surface is

Re {(12/2)&:*J

[C,1{4,)).

074

In the special case when only one mode, qrfr, is allowed to participate in the motion, this

reduces to

Re {(12/2)~c(41 12C,,) = (021’/2)pc(q1 I2 Re(C,,).

(17b)

This may be expressed in non-dimen sional form as shown below. When evaluated for

unit value of the loading parameter,

P ,

it may be called the sound power admittance func-

tion, defined by

Y(8) =

22pSV/q, I2

Re(C,,).

(18)

Now suppose the external pressure field is varying randomly, instead of harmonically,

and is convecting as frozen free waves in the x-direction. Let the corresponding power

spectral density of the loading p arameter be S,(dl). The non-dimen sional time-averaged

sound power radiated is then given by

00 =

s

m S,(G)P(62) da.

(19)

0

4. FREQU ENCY BANDS OF RADIATION

If the external pressu re field convects over the infinite periodic plate at a subso nic velocity

sound will not be radiated at all frequencies. Those wave components of equation (4) whose

respective wave-nu mbers are greater in mag nitude than the acoustic wave numb er

k = o /c )

cannot radiate sou nd On the other hand from those compon ents for which

k , < o/c ,

sound crm be radiated. Now k j = k , + 2 j n / l = o / c ,, + 2 j n f l . Hence for sound to be radiated

131

-w/c < w/cP + 2jx/l c w/c.

(20)

When the convection velocity cP is less than c, this inequality can only be satisfied by certain

negative values ofj over certain frequency ranges. The upper and lower bounds of frequency

for a particular j are

In nondimensional form these are

@la, b)

It will be seen later that the sound power radiated in the first radiating band is much greater

than tha t in the subsequen t band s. Table 1 shows the frequency band s of radiation for a

periodically supported plate with several different convection velocities CV and with



320

I>. 1. MEAO ANT I A. h. MALLIK

Sli = 10.

It may be noted that for C‘V = 4 and 6, ,R,, __

Sdy,,so that two adjacent frequency

radiation bands are overlapping. Note also from equations (21a) and (21b) that L?,, = QU ,

at CV = Sk’/3, so that two adjacent radiation bands are just touching.

TABLE 1

Upper und lower firquenq hounds cfrudiution bands; SV = IO

Convection

velocity,

cv

___~ ~_~~~

2

4

6

20

+1

f4 Q”

No radiation

No radiation

No radiation

40n cc

-

Value ofj

0

-1

-2

.____

Ql Qu 0, QY

Q, Qu

No radiation

lOn/3 5n 20ni3 10n

No radiation 4&c/7 40n/3 8Oz/7 8&I/3

No radiation 1 h/2

30n

157I 60~

0 x 4on/3 x, 80x13 x

When the convection velocity is supersonic,

CV > SV

and the condition for sound to be

radiated is satisfied for all values of Q for j = 0. For other values of j (both + ve and - ve)

there will be frequency bands o f radiation starting at particular frequencies and extending

up to infinity. Table 1 show s one such example, with CV = 20.

5. CALCULATIONS PERFORMED

Com puter calculations have been carried out for the sound pow er radiated from a plate

on transversely rigid supports (k,

-+ m) but with rotational constraints of K = 4. The plate

loss factor has been taken as r~ = 0 .1, and the density ratio as p = 0.1. A single approxim ate

mod e has been a llowed and this has been chosen in the same w ay as in reference [S] for

substitution into equations (16a-d). N ote that with transversely rigid supports, f,(l) = 0

and the term involving K: in Krs (equation (16a)) becom es zero.

The sound pow er admittance function has been calculated for a range o f non-dimensional

frequencies, Q = 8 to 24. In the first place, three different values of the convection velocity

param eter have been considered: 0’ = 2. 4, 16. The sound velocity pa rameter, SI has

been taken to be 10. Nine terms in the series for C, 1

(j = -4 to + 4) were found to give

satisfactory convergence.

Altogether four different metho ds have been used to comp ute the sound radiation, with

varying degree s of approximation:

(4

(b)

(4

(4

the principal metho d of this paper, as embod ied in equations (15), (16) and (18);

acou stic radiation is fully taken into accou nt in determining the 4,‘s;

an approxim ate form of this metho d, with the acoustic radiation terms Cls in equation

(15)

for the response neglected ;

a metho d involving calculation of the exact, closed form solution for the plate response,

with acoustic radiation neglected; the sound po wer radiated by this motion has then

been comp uted by using equation (10);

the metho d of space harmonics to calculate both the plate response and radiated

sound po wer, with acoustic radiation fully included in the response calculation and

eleven terms used in the space harmonic series.

Calculations have also been made of the total sound pow er radiated by the plate when



SOUND

RADIATION

PROM PERIODIC

PLATE

321

excited by a random pressure field, convecting as frozen plane waves in the xdirection.

The pressure loading spectrum has been assigned a unit value over the frequency range

8 < 62 < 24, and zero outside this range. The first free-wave pro pagation zone of the periodic

plate lies well within the range P =

8 to 24, and so the range in which large plate responses

are generated is well covered. Accordingly, computer calculations of (L > given by equation

(19) have been carried out, with S,(0) = 10 and the lower and upper limits of integration

taken as 8 and 24 respectively.

The computer results are presented in Figures 2-4. Figures 2(a), (b) and (c) show the

sound power ad mittance function as calculated by the four different methods. Figure 3

shows the time-averaged random radiated power for the range of convection velocities

CV = 1 to 10 0 and for r) = 0.1. Figures 4(a), (b) and (c) show similar results for other values

of q (OXQO .25 and O-10)and with a different value of the density ratio p

(=

O-05).

6. DISCUSSION OF RESULTS

Figures 2(a), (b) and (c) compare the sound power radiated, as computed by the four

different methods. The method of space harmon ics is undoubtedly the most accurate, and

the values computed from it constitute the standard against which the other approxim ate

values sho uld be judged.

There is close agreement between the space harmonic values and the values obtained

from the principal method of this paper, even when the external pressure field convects at

subsonic velocities. The one exception occurs in the frequency range around 62 = 22 to 23

when CV = 2 (Figure 2(a)). In this range, the pressure field excites a wave with several

half-wave lengths between ad jacent supports. The simple polynomial mode chosen to

represent the plate motion cannot well represent such a mode. Notice, how ever, that the

scale of the radiated sound power for this frequency range is much smaller than for the

lower frequency range, so that the actual so und power radiated is of second or third order

proportions.

When the sound power is computed after neglecting acoustic radiation in the response

calculation, the values obtained are generally greater than those found by including

acoustic radiation. Indeed, at some frequencies (acoustic coincidence frequencies) the radia-

ted sound power calculated in this way approaches intinity, whereas it drops to xero if

acoustic radiation is included (see Figure 2(b)). Away from these coincidence frequencies the

difference is small, but this is due to the relatively large value of structural dam ping used in

the calculation (q = @l). With much smaller v alues of structural dam ping, of course, the

neglect of acoustic radiation in the response calculation leads to increasing inaccuracy in

the response and sound power calculation.

When a random pressure field convects over the plate at different convection velocities,

the mean sound power radiated is as shown in Figure 3. Notice the two distinct ranges

into which the figure divides. The sound velocity parameter is 10, so that pressure fields

with convection velocities of 10 and m ore are expected to cause large radiation. This is seen

to be the case, but there a lso exists a lower an d narrow convection velocity range in which

sound is radiated (centred on CV x 4). This is the range of maxim um plate response, but

the plate radiates very inefficiently in the particular frequency band co nsidered (8 < D < 24).

No sound power is radiated if CV lies between 6 and 10, since the frequency radiation

band s corresponding to these velocities do not begin until 0 > 24 (see Table 1) and the

calculated sound power shown on Figure 3 is for the frequency range from Q = 8 to 24.

Figure 3 does not display any results from the space harmonic method. Figures 2(a), @)

and (c) show such close agreement between the space harmonic method and the principal



322

I). I. MEAD AND A. K. MALLIK

/ / I

b)

Non-d~mensmal frequency, i-l

Figure 2. The sound power admittance function vs. non-dimensional frequency 0, for a plate with x, = 4,

K, + co, q = 0.1, jj = 0.1,

SV = 10. --o--

Principal method of this paper; A, present method, but with effect

of acoustic damping on response neglected;

-, closed form solution for response, with effect ofacoustic damping

on response neglected; A, space harmonic method. a) CV = 2; b) CV = 4; c) CV = 16.



SOUND RADIATION FROM A PERIODIC PLATE

323

I

I

5

IO

50

Convect ion ve loc i ty . CV

0

Figure 3. Average sound power radiated by the plate when subjected to a from& random pressure field moving at

different convection velocities, CK Plate characteristics and figure legends as for Figure 2.

I-

‘I _

.2 _

-5 _

.

I I

1

5 IO

50 1

I I

I

:b

P

I

5

I I

I

5

IO

50

Convect ion ve loc i ty . CY

r

I I

I

c)

I I

I

5

IO

50 lo

Fire 4. Avera& sound power radiated by the plate when subjected to a frozen, random pressure field moving

at di%kent convection velocities, CV.

K, =

4,

K,

+ co, p = O-05, SY = 10. Figure legends aa for Figure 2. (a) q =

O-02; b) q = 010; (c) q = 0.25.



324

1).

J. MEAD AND A. K. MALLIK

method of this paper that no difference would be perceived on Figure 3 between the mean

sound pow ers calculated by both m ethods. On the other han d, the sound power ca lculated

by neglecting acoustic radiation in the response calculation is seen in Figure 3 to be appre-

ciably greater than the power calculated by including acoustic radiation. This is seen more

clearly still in Figure 4(a) where the plate structural dam ping is much smaller (17= 0.02)

but the difference decreases as the structural dam ping increases (see Figures 4(b) and 4(c).

with q = 0.10 and 0.25 respectively). Figures 4(a), (b) and (c), relate to a less dense acoustic

medium (p = 0.05) and this feature also leads to a smaller difference between the sound

power calculated in these two ways (cf. Figure 3 with Figure 4(b), between which the only

difference is in the densitv of the acoustic medium ).

Mo st of the figures show close agreement between the results from the two approximate

methods (b) and (c), in which aco ustic radiation is neglected in the response calculations.

This shows that the approximate mode chosen for the energy analysis is a good approx i-

mation to the exact mode yielded by the closed form solution. The disadvantage of these

methods is that they lead to gross over-estimates of the sound power radiated close to

acoustic coincidence conditions, and to lesser over-estimates when the structural dam ping

is low. There is no advantage to be gained in using method (b) instead of the principal

method (a) of this paper, as the inclusion of acoustic radiation effects in method (a) consti-

tutes a negligible increase in comp utational effort. There could be some advantage in

using method (c) in the higher frequency propagation bands , when the mode given by the

closed form solution (but in the absence of acoustic radiation effects) may be more accurate

than the approximate modes w hich have to be chosen.

7. CONCLUSIONS

Simple, approximate complex modes can be used in an energy method to compute quite

accurate values of the sound pow er rad iated from a periodic supported plate. The method

allows the effects of acoustic radiation to be included in the response analysis. The sound

power radiated in the first propagation zone of the plate excited by a convected pressure

field can be computed surprisingly accurately by using a single approximate well-chosen

mode. The method has been demonstrated for a periodic plate with uniform plate bays,

but is applicable also to periodic plates with non-uniform bays.

It has been demo nstrated that the periodic plate can radiate sound even though the

existing pressure field convects over the plate at subsonic velocities. In such cases, sound is

only radiated in discrete frequency band s.

REFERENCES

1. D. J. MEAD1970 Journal ofSound and Vibration

11,

181-1 97. Free wave propagation in periodic-

ally supported , infinite beams.

2. D. J. MEAD 1971 Transactions of the American Society of Mechan ical Engineers Journal of

Engineering for Industry 93 183-192. Vibration response and wave propagation in periodic

structures.

3. K. K. F JJARA970 Ph.D. Thesis University of Southam pton England. Vibrations of and sound

radiation from some periodic structures under convected loadings.

4. D. J. MEAD and K. K. PUJARA 1971 Journal of Sound and Vibration 14 525-541. Space-harmonic

analysis of periodically supported beams: response to convected random loading.

5. D. J. MEAD nd A. K. MALIX 1976 Journal of Soun d and Vibration 47 457471 . An approximate

method of predicting the response of periodically supported beams subjected to random convected

loading.



SOUND RADI AT ION FROM A PERIODI C PLAT E

325

APPENDIX: FORM AL JUSTIFICATION FOR THE SPACE HARM ONIC SERIES

Let the x-wise leng th of the infinite pe riodic plate be written in the form L = lim nl,

n-ral

where

1 s

the distance between adjacent support-lines. The transverse d isplacement of the

plate under a convected harmonic pressure field may be written a s

w(x, r) = W(x) e”O’.

(Al)

The complex displacem ent function W(x) may be decomposed into a structural wave-

numb er spectrum given by

F(k,) =

s

L

W(x) eikxx dx,

W)

0

where k, is the wave-number. As usual, one can write

f

L

W(x) =

W(k,) ebikG

dk,.

0

(A3)

The spectral density of the displacement (in the wave-num ber domain) is

SJkJ = lim

(

W(kJI’/L.

L-m

(A4)

Now equation (A2) can be written as

s

(r+ 1)1

W(x)ei kr ” dx + . +

W (x) ei kxX x +

r l 1

(A3

Now the response of any bay of the periodic plate to a convected harmonic pressure field is

identical to that of the previous bay apart from a phase difference of s =

(3).) Hence

-

kJ. (See

equation

s

21

s

1

I, =

w(x) eikn’ = e-W

W(z) ikk+r) dz (where z = x - Z)

1 0

= eWz-kp)l

I

1

W(z) ik xz

z = e*‘Z,,

(A@

0

where 8 =

(kX

kJ1. Similarly, one can show that I,

= eiZeZ, and that Z

I

= ei(r-l)eZ

1’

Hence, from equation (A5),

FV(kJ = ,‘~KI (1/27r)Zi(l - e’“ B)/(l - e?}.

(A7)

Substituting this into equation (A4) one obtains

S,(kJ =

lim {(l/nZ)~(1/2a)Z,(l - e’“ B)/(l - ei”)12}.

n-00

(Ag)

It can easily be seen from this equation that

S,(kJ = 0

unless eie = 1; i.e., it only has non-

zero values for those values of k , for which eie = 1. This means that 0 = f2jlt (where j is

any integer), and

kX k, f 2j n/l.

(A



326

I>. .I. MEAD AND A. K. MALLIK

With the se discrete values o f kX I = k,, say), the integral of equation (A3) can be rewritten as

+ClI

where

W(x) = c TVj,Ck”

1A10)

.j= -a,

Wi

W(x) eikJ.r

x.

0

(Al

1)

Finally, equation (Al) can be expres sed in the form

W(X , ) = +i Wj e”“’ k,xt,

j= -41

(Al2)

wh ere Vj is given by equation (All). Equation (A12) is the series of “space harmonics”

used in this paper, and previously used by Me ad and Pujara [4].

an approximate theory for the sound radiated from a periodic line-supported plate

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