radiation from elastic plates by vibration inputs: I. Analysis

Journal

ofSound

ACTIVE

and Vibrafion (1990) 136(l), l-15

CONTROL

FROM

OF SOUND

ELASTIC

PLATES

TRANSMISSION/RADIATION BY

VIBRATION

INPUTS:

I. ANALYSIS-f’ C. R. FULLER Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A. (Received 14 November 1988, and in revised form 7 April 1989)

Active control of sound radiation from vibrating plates by oscillating forces applied directly to the structure is analytically studied. The model consists of a plane acoustic wave incident on a clamped elastic circular thin plate. Control is achieved by point forces, and quadratic optimization is used to calculate the optimal control gains necessary to minimize a cost function proportional to the radiated acoustic power (the transmitted field). The results show that global attenuation of broadband radiated sound levels for low to mid-range frequencies can be achieved with one or two control forces, irrespective of whether the system is on or off resonance. The efficiency of the control strategy is demonstrated to be related to the nature of the coupling between the plate modes of response and the radiated field.

1. INTRODUCTION In numerous industrial applications, structurally radiated noise is a persistent problem, the solution of which by passive methods is in many cases inadequate. The possibility of active noise control was suggested at least 30 years ago, but its implementation is a relatively recent development due to advances in fast microprocessors. A number of successful active noise control applications have already been demonstrated, particularly for low-frequency sound in one-dimensional fields, as summarized in the review articles of references [l] and [2]. Spatial control of three-dimensional acoustic fields is a far more difficult problem. Early work by Jesse1 [3] and later by Kempton [4] established the theory behind active control of free-field sound radiation using acoustic control sources. The review paper by Mangiante [5] also summarizes early attempts to actively control sound radiation in three dimensions. These works all demonstrated that even for a simple source, such as a monopole, many acoustic sources will be necessary to give global control (here global means throughout an extended volume or area) of sound when the control sources are located more than a quarter of wavelength away from the noise [6]. When the noise source is more complex or distributed, as for most structural noise sources, then the situation is exacerbated. Much work has been concerned with the active control of radiated noise from electrical transformers using acoustic sources [7-91 and again it was demonstrated that many acoustic sources will be needed for global attenuation. Recent analytical work by Mollo and Bernhard [IO] using a boundary element approach and Deffayet and Nelson [I I] using classical radiation theory have led to insight into control of more complex distributed sources. However, as shown in reference [ 111, when there are distinct phase changes across a source, as in a panel vibrating in a higher mode, t

Originally

presented

in brief form at Inter-Noise

0022-460X/90/010001+15$03.00/0

‘88, Avignon,

France,

August 1988.

@ 1990 Academic

Press Limited

2

C.

R.

FULLER

then an active acoustic source will have to be located near each antinodal point, especially for even-even modes, again implying that many control sources will be required by this method. Recent research work, however, suggests that for structurally radiated noise there is some advantage in applying the control action directly to the structure in the form of vibration inputs [12]. In experiments and theoretical investigation on controlling sound transmission into elastic cylindrical enclosures it was found that global reductions of the order of 15 dB could be achieved with one or two control forces applied directly to the cylinder [ 13-161. Here it was found that only one or two structural modes were coupling to the interior acoustic field; thus only a low number of control actuators were needed. By contrast, calculations show that the number of interior acoustic sources needed to achieve reasonable attenuation for the same situation is equal to at least twice the circumferential modal order of the interior response [17]. This result has also been somewhat validated by experiments [ 181. The present paper is concerned with the application of this technique to the more general problem of control of sound radiation from vibrating panels. To this end an analysis of an idealized system consisting of a baffled clamped circular elastic plate excited on one side by a plane acoustic wave as a noise input is presented. The sound field radiated from the other side of the plate (the transmitted field) is reduced by applying vibrating point control forces directly to the plate surface. In order to determine the complex amplitudes of the control forces, so as to minimize the radiated sound levels, a cost function is defined as the integral of the squared pressure amplitude over a hemisphere in the far field. The control force amplitudes necessary to minimize this quadratic cost function can be determined by matrix and tensor theory [17,19]. On determining the optimal control complex amplitudes, these can be resubstituted in the constitutive equations for the system and the minimized radiated field as well as other parameters can be evaluated. Of course, various other noise inputs could have been chosen, such as a point or moment excitation of the plate, but the acoustic plane wave was chosen here as a noise input as it is of different character to the control input and thus illustrates the potential of the technique. For active control of sound radiation from panels by forces, the author could find only two directly related works, both performed in the U.S.S.R. Knyasev and Tartakovskii [20] demonstrated in a brief experiment that it was possible to reduce sound radiation from a panel by using an active damping force. Veyalyshev et al. [21] analytically considered sound transmission through a one-dimensional plate and showed an increase in the plate transmission loss with one active control force applied to the plate. Both of these results are consistent with the results of the analysis presented in this paper.

2. ANALYSIS

The arrangement and co-ordinate system used in the investigation is shown in Figure 1. The analysis will proceed by first deriving the pressure radiated from the plate due to the incident plane acoustic wave (noise) and the point force excitation (control). The total radiated field is then obtained by superposition and used to derive a system cost function. The optimal control forces are derived and then resubstituted in the system equations in order to evaluate the performance characteristics of the technique. 2.1.

PLATE

RESPONSE

EQUATIONS

For the analysis presented here the acoustic propagating medium is considered to be air. Thus the fluid loading of the radiated (or scattered) field from the plate vibration is

ACTIVE

CONTROL

OF

SOUND

RADIATION

Rtgid baffle

lncldent

\ ’

‘,

\

\

\

\ \

\

\

l.o”Trol

// 1

---I / /

rxcular

I

’

/

/

/

/ / /

1. System

arrangement

; I

’ TransmItted I wove

/

/

/

Figure

p(R,8,:_’ ‘/

I

force(s)

Clamped plate

\

\

and

/

co-ordinate

system

small and the plate normal mode response and eigenvalues assumed here are for the in vacua case. For a thin circular plate with a clamped boundary condition at its edge, r = a, the out-of-plane displacement of the plate can be taken as [22] (a list of symbols is given in the Appendix)

where the radial

w(r, 0) = t

i

nz,

j=,

distribution

function

w,

i

‘OS Cne) ff sin (n6) ”

I

ejm,

’

is given by

H~j(klr)=J,(krr)-{J,(k:a)/Z,(k~a)}Z,(k,’r). Application of the clamped teristic equation

(1)

boundary

condition

at r = a allows derivation

Jl(kja)l.(kja)-J,(kra)I~(k:a)=O,

(2) of a charac-

(3)

from which eigenvalues kJ’a, for n = 0, 1,2, . . . and j = 1,2,3, . . . can be determined. Thus the indices (n, j) correspond to a particular plate mode of vibration, as discussed in reference [22]. For classical plate theory, the resonant frequencies of vibration for a mode (n, j) are then given as [22] w,, = (h/2a”)c,,(kja)‘, or in non-dimensional

wavenumber

(4)

form as

k,ja =(h/2a)c,,,(kJ~~)~/c,,. 2.2. RADIATION

FOR

AN

INCIDENT

PLANE

WAVE

(NOISE

(5) FIELD)

An acoustic plane wave of amplitude p0 incident to the normal axis of the plate at yi can be transformed to cylindrical co-ordinates as [23] oc pi,,< = p0 C E,,(-i)” cos (no) J,( k,r) eiw’, where k, = kOsin ( -yi). (67) “=O

4

C. R. FULLER

If one assumes that the plate vibration has little effect on the pressure forcing function, then the pressure field exciting the plate into motion will be very close to twice the incident pressure (blocked pressure). By applying the well known eigenfunction expansion theorem, which states that any forcing function may be expanded in terms of the system normal modes [24], and utilizing orthogonality of the plate modes of vibration one can derive the plate “forced” response to the incident acoustic wave as w,,,,.(r, 0) =2p,cc

e,,(--i)’ cos (nO)H,,A, An,(oJ2,p2)

n I

e

iw,

(8)

,

where A, is a coupling integral given by A, =

’H,,,(klr) J,(k,,r sin -yl) (r/a) d(r/a) I0

and A, is the mode normalization

(9)

constant given by [22]

A,, = m[(J,(kTa))2+(J~(kSa))21,

t 10)

where m = (p,h) is the mass per unit area of the plate. Now the far field radiation from a vibrating clamped circular plate expressed in terms of the plate displacement modal amplitude Wni has been derived using a Rayleigh integral approach by Hansen and Bies [25] as e iwrhik,,R

prad(R, 8, y)=-i”p,-w2a2W,

d2,

R,

(11)

where the coupling integral A, in this case is I AZ=

I0

Hq(k:r)

J,(k

sin Y) (r/a) d(r/a)

(12)

and the radiation co-ordinates are shown in Figure 2. Thus, from examination of equation (8), the pressure field radiated by the plate for an incident acoustic plane wave is given by

Figure 2. Radiation

co-ordinate

system

ACTIVE

2.3.

RADIATION

FOR

POINT

CONTROL

FORCE

OF

SOUND

EXCITATION

RADIATION

(CONTROL

5

FIELD)

By again using the eigenfunction expansion theorem, the response of the plate to a series of L point forces located at (r,, 19,)can be derived as by Cremer et al. [24] and is given by w,1.=;

/=I

H,j(k,‘r)H,i(kj’r,) cos (nt9 - nt3,) eiw’ 3 &%-A,j(w’,,- w2) n i

F;CC

(14)

where FI = F,/a’ has the units of pressure. By re-formulating equation (14) into equivalent modal amplitudes and plate characteristic functions and utilizing equation (ll), the pressure radiated from the plate for the series of point force excitations is found to be 2,2

pyyR, 2.4.

DERIVATION

8, y) =!y OF

OPTIMAL

L

C F’,CC I-1 n , CONTROL

Hq( kJ’r,) cos (~0 - nB,)A, eiwr-ikR &TA&+co2)

(15)

FORCES

2.4.1. Cost function dejnition In order to determine the complex amplitudes of the control forces, Fi, so as to minimize the radiated pressure field amplitudes, a cost function is defined as the integral of the squared pressure amplitude over a hemisphere in the far field at a particular radius R. Thus the cost function is given by (16) where s is the hemispherical surface area in the far field. By superposition the total radiated pressure, in the presence of control action, can be written as the sum of the noise and control fields, p:::‘(R, 0, a) =P& where the distribution B, =

+ ; F;A, /=I

(17)

i”( -i)“en cos (no) A, A, eiwrmikR Anj(&, -w’)

(18)

functions are given by

and A=P,

H,,,(kfr,) cos (nt9 - nOI) A, eiwrpikR Eml,i(W:, w’)

-

(19)

for the noise field and control field respectively. Note that for multi-plane wave input at different angles such as in a reverberant field, equation (17) could be expanded to include a series summation in plane wave amplitudes. Thus, substituting the expression for total pressure, given by equation (17), into equation (16) and performing the integration, where ds = R2 sin y dy d0 allows expression of the cost function in matrix form as [17] ,t3(F,)=~T[A]~*+~T[B]~T*+~T[B*]T~*+~T[C]~*,

(20)

where the control force vector is defined as FT=[F,,

F2 ,...,

F,]

@la)

6

C. R. FULLER

and the noise amplitude

input

vector is a scalar

for a single incident

wave,

ST= [Pal.

(21b)

Thus equation (20) expresses the cost function p as a real, homogeneous, quadratic function of the unknown complex amplitudes of the control forces, F,, I= 1, 2,. . , L. The matrices appearing in equation (20) have the forms [B] = 1 [ii~%*~] ds,

[A] =

[ L?,&*T]ds,

[C] = j- [I%*‘]

ds,

where the elements

of the vector

A are

AT= [A,, A?, . . . , AL] and the elements

(22a-c)

s

S

of the vector

(23)

6 are scalar for a single plane

wave input

BT= [B,].

(24)

Equation (24) can easily be extended to the multi-wave input case. By virtue of the orthogonality between circumferential modes on the interval (0,27r) it can be easily shown that a typical element in the sth row (s = 1,2, . . . , L) and mth column (m = 1,2, . . . , L) of the square [A] matrix has the form A,, = p:w4a4n

1 EX:,,, cos (ntl, - ne,,,), n

(25)

where 7112

JG=

I

c

0

Similarly,

a typical

-i”H,,j( kJr,)A,

-i”H,,,(kJr.Y)A, c

j &~A,j(Wfj-W2)

element

, eml,i(wf,,-w')

in the (row) matrix

sin (Y) dr.

(26)

[B] of row s, (s = 1,2, . . , L) is

B,, = 2p;w4a4 1 FE, cos (n0,) Yr,, n

(27)

where lr/2

VT =

I

0

Finally, a typical to a scalar is

element

-i” (-i)“A,A,

cj An,(C0tj-02) of matrix

-i”H,,(k/‘r,)A,

(28)

sin (y) dy.

c

1 &77A~,(WZ,,-WZ)

[C] which for the case of single wave input

reduces

(29) where n/2

z;, =

I

o

2a,(_in)(_i)n

As previously, it is straightforward multi-wave inputs. 2.4.2.

c

.i

44 Anj(0n.j-w2)

to extend

c2~.(-i”)~-i)“~~~~ .i

Anj(wnj-w

equations

sin

(?I

dy.

(30)

1

(29) and

(30) to the case of

Optimal solution

A solution for F is sought for equation (16) such that the cost function is a minimum, p,,( F,), the optimal solution. The amplitude(s) of the incident wave field(s) is/are known a priori, while the number and location of the control forces must be specified. Thus the only unknowns appearing in /3( F,) are the complex force amplitudes F,, I = 1, 2, . . , L. The vector F, such that the quadratic function p has its sole minimum value, can be determined by differentiating A with respect to F and setting the resulting equations to zero [ 17, 191. This is most easily achieved by re-writing equation (16) in an indicial form

ACTIVE

CONTROL

OF SOUND

7

RADIATION

and separating real and imaginary parts. Since FT[A]F* is greater than zero for all choices of F, then a unique minimum exists. The optimal solution for the control force vector is then found to be [17,19] I’= -[A*lm’[BITS.

(31)

For the following results, equation (31) will be used to determine the optimal control forces for various test configurations. After determining F, the minimized radiated field as well as other parameters can then be determined by re-substituting the optimal values of F into equation (20), etc. It also should be noted that the definition of p,J F,) allows evaluation of the transmitted or radiated power in the far field where spherical waves approach plane wave behavior in terms of the phase speed and impedance. Thus radiated power can be simply calculated by substituting all the specified and calculated optimal values into equation (20) and dividing the result by 2pl-c,,. 3. RESULTS

AND

DISCUSSION

Example results were calculated for a system with the physical properties summarized in Table 1. The plate was assumed to be undamped steel with a thickness to radius ratio of h/a = 0.005 and the acoustic medium was air. For the following calculations the acoustic plane wave was assumed to be incident to the plate at yi = 45”. One or two control forces were considered located at positions of (r,/a, I!?,) equal to (Oe5,O) and (0*75,0), respectively. TABLE

1

Material properties

Material

Phase speed (m/s)

Density (kg/m31

Thickness (h/a)

Incident wave angle (degrees)

Steel

2916 (shear)

7700

0.005

-

1.21

-

Air

343

45

In evaluating the response of equations (17) and (20), the first step is to solve the characteristic equation (3) for the system eigenvalues, kia. This was performed by using a Newton-Raphson technique, and in Table 2 are presented mode non-dimensional resonant frequencies. The results discussed here are for non-dimensional input frequencies, k,a = W/C,, close to the resonant frequencies of the (0,l) and (1, 1) modes. The coupling integrals A, and A2 of equations (9) and (12) have previously been solved exactly by Hansen and Bies [25], where the final expressions can be found. However, the integrals in equations (26), (28) and (30), expressions for the matrix elements, have to be evaluated numerically, and this was achieved by using a Simpson’s rule method. Finally, the infinite modal sums in these equations have to be truncated and for frequencies of the range 0 < k,,a < 1.5 a truncation value of n = 4,j = 5 (i.e., 25 modes were considered) was found to provide satisfactory convergence in the results. Higher frequencies would require inclusion of more modes, with an associated increase in computation time. All calculations were performed on a CYBER CY180-860 mainframe computer.

3.1. INPUT FREQUENCY OF kOa = 0.21 Sound pressure level directivity curves evaluated in the far field across the diagonal 13= 0, 180” are presented in Figure 3. The results are normalized to the largest amplitude

C. R. FULLER

8

TABLE

2

Plate eigenfrequencies, n

0

k,a 3

2

1

4

j\ 1 2 3 4 5

0.2171 0.8453 1.8938 3.3620 5.2498

0.4519 1.2928 2.5521 4.2306

0.7413 1.7978 3.2691 5.1587 7.4672

6.3285

1.0846 2.3596 4.0446 6.1461 8.6657

1.4807 2.9978 4.8781 7.1925 9.9239

y (degrees)

Plate Figure 3. Directivity control forces.

of radiated

pressure

field, k,,a = 0.21.-,

Noise; -.

-,

one control

force; - - -, two

calculated and are for the three cases of noise alone, one and two control forces positioned as specified. The noise input non-dimensional frequency is k,a = 0.21, which is very close to the non-dimensional resonant frequency of the (0, 1) plate mode, ko,a = 0.2171. The radiation pattern of the noise alone can be seen to be uniform due to the plate responding dominantly in the (0, 1) mode, being close to its resonance. Application of one control force reduces the radiated levels markedly by approximately 60 dB. The shape of the residual sound field, with a node at y = o”, indicates that a higher plate mode is now responsible for sound radiation. Application of two control forces simultaneously, has the effect of further increasing attenuation of the radiated levels by approximately 30 dB. In Table 3 are given the corresponding normalized radiated power distributions in each circumferential mode, n, of the radiation pattern, calculated as described previously. Use of one control force leads to large attenuation of the n = 0 component but results in control energy spillover into the n = 1 component, as evidenced by the directivity curves of Figure 3. Two control forces reduces this spillover effect, at the cost of a decrease in attenuation of radiated power due to the n = 0 mode. It is also interesting to study the plate out-of-plane displacement response, w(r, 0), under these control conditions. Plate response for the noise field alone is given by equation (8). The minimized plate field can be calculated by superposition of the response due to the noise field, equation (8), and that due to the control forces, equation (14), identical to the calculation of total radiated pressure. In Figure 4 the plate displacement amplitude is plotted across the diagonal of 6 = 0, 180” for the three previous conditions. The response is again presented normalized to the largest calculated value. As to be expected, the response for the noise alone is

ACTIVE

CONTROL

OF SOUND

RADIATION

3

TABLE

Circumferential power distribution (dB), k,,a = 0.21 n

Noise

0

0

1 2 3 4

I.0

0.6

0.4

02

0

-97.0 -123.4 -93.2 -126.3 -162.8

0.2

(8: 180”)

out-of-plane forces.

0.4

0.6

0.8

I.0

(8=0”) Radml poshon,

Figure 4. Normalized force; - - -, two control

Two forces

-112.9 -56.1 -96.52 -137.6 -200.3

-84.9 -155.8 -228.9 -304.5

0.8

force

One

plate displacement

r/a

amplitude,

k,a = 0.21. -,

Noise; -

-,

one control

predominantly in the (0, 1) mode. Use of the one control force markedly attenuates the plate vibrational amplitude, and now the plate is seen to be vibrating in a (1, 1) mode. Use of two forces changes the response distribution but does not markedly change its overall amplitude. However, Figure 3 demonstrates a further 30 dB increase in attenuation under this condition. Thus the second control force achieves radiation attenuation by generating a higher order mode distribution with a lower radiation efficiency (see later discussion) rather than purely suppressing plate out-of-plane vibration, w. Thus it is apparent that global attenuation of noise radiation can be achieved at this frequency with just one or two control forces applied directly to the plate. The efficiency of this control strategy can be understood by examining equations (13) and (15). The radiated pressure field for the noise and control have the same form of singularity at the plate resonant frequencies. Hence control energy is distributed where it is most needed, with reduced spillover. Of course, if the control field was from an acoustic monopole positioned over the plate (representative of an acoustic control source) then this would not be the case. From a transmission point of view, acoustic energy from two coupled 3-D fields, the incident and radiated, has to flow through a (effectively) 2-D structure, the plate. Applying the control action where the field dimensionality is reduced, usually results in more efficient control implementation in terms of numbers of control transducers. 3.2. INPUT FREQUENCY OF k,a =0.45 The second set of results are for a higher frequency of k,a = 0.45, which is close to the resonant frequency of the (1, 1) mode, k,,a = 0.4519.

10

C. R. FULLER

The radiation directivity patterns are given in Figure 5, for the same cases as previously. For the noise input alone the radiation pattern appears to be a somewhat distorted (1, 1) mode with large contributions from other modes. This result is surprising when one considers that the plate is being driven near the (1, 1) resonant frequency. Application of one control force alone does not achieve global reductions; in fact, the sound pressure levels in the sector of 8 = 0” are somewhat increased. Use of two control forces, however, is seen from Figure 5 to achieve an approximately 50 dB global attenuation in the radiated levels. An examination of the circumferential distribution of radiated acoustic power given in Table 4 shows that in fact for noise alone, even though the input frequency is near a (1,1) resonance, the radiated power is somewhat shared by the n = 0 and n = 1 modes. Use of one control force achieves only attenuation of the n = 1 mode and nearly all the power is now via n = 0 motion. When two forces are employed simultaneously both the n = 0 and n = 1 radiated powers are attenuated, and thus the net sound levels are reduced drastically. This explanation is also supported by Figure 6, in which is presented the plate normalized out-of-plane displacement amplitude, for the same conditions. For noise alone the plate is seen to be vibrating dominantly in the (1,1) mode as expected. Application of one control force leads to a large reduction in plate response. However, this was seen in Figure 5 to not correspond to a decrease in radiated sound levels. Use of two forces does not significantly change the plate response amplitude; rather, it increases its distribution complexity, resulting in a lower radiation efficiency and a corresponding decrease in radiated levels.

y (degrees) --0

Plate Figure 5. Directivity control forces.

of radiated

pressure

field, k,a = 0.45. -,

TABLE Circumferential

Noise; -

-,

one control

4

power distribution

(dB), kOa = 0.45

n

Noise

One force

Two forces

0 1 2 3 4

0.0 -2.6 -91.3 -187.0 -249.9

0.0 -25.3 -67.4 -103.8 -140.0

-48.5 -80.1 -43.4 -72,O -102.7

force; - - -, two

ACTIVE

IO

0,8

CONTROL

O-6 0.4 (8=180")

OF

O-2

SOUND

0

11

RADIATION

0.2

0.4 06 (8:O")

0.8

I.0

Radlal posh IO”, r/u

Figure 6. Normalized force; - - -, two control

out-of-plane forces.

plate displacement

amplitude,

k,a = 0.45. -,

Noise; -.

-, one control

This behavior can be further understood by studying the plate radiation efficiency for various modes, repeated here in Figure 7 from the work of Hansen and Bies [25]. At k,a = 0.45 the radiation efficiency of the (1, 1) is approximately 20 dB less than the (0, 1) mode. Hence, although the plate is vibrating dominantly in the (1, 1) mode, the radiation field is due to components in both modes. Thus, as shown by Mierovitch et al. [26], two independent controllers are needed to reduce markedly the radiated levels. Likewise, as the higher order modes have a decreasing radiation efficiency with modal order (at least for frequencies such that k,a d lo), control energy spilled into higher plate modes does not markedly affect the residual radiated modes. For the previous frequency of k,a = 0.21, only one force is needed by virtue of the fact that, at this frequency, the (0, 1) mode is the most efficient radiator, being of monopole characteristic. 3.3. PLATE TRANSMISSION LOSS For the incident acoustic plane wave the incident acoustic power is easily shown to be ITi = (IP01/2q~c0)~a2 cos

hi)

(32)

and, as discussed previously, the radiated acoustic power is rr = P(fi)/2PfC,. Thus the transmission

(33)

loss of the plate can be calculated from TL = 10 log,, [W/zY].

(34)

The plate transmission loss calculated by using equation (34) versus non-dimensional frequency is presented in Figure 8. For the case of noise alone the transmission is high at very low frequencies (ha i Oal), and approaches zero at a number of frequencies corresponding to plate resonant points, where the plate vibration theoretically goes to infinity. Of course, addition of slight damping into the analysis would most probably eliminate the very sharp dips in the noise transmission curves at the higher order mode resonant frequencies. Use of one control force significantly increases the plate transmission loss (or, correspondingly, the attenuation of the radiated field) up to koa = O-4. Above this frequency there are large frequency ranges where one single control force does little to increase the plate transmission loss.

12

C. R. FULLER IO’

I

I

I00 b

10-i

Figure

100 Non-dimenslonol

7. Plate modal

radiation

efficiency

0.5 Non-dImensIonal

Figure

8. Plate transmission

loss. -,

Noise; -

IO’ frequency,

-,

koo

(from reference

I.0 frequency,

one control

[25]).

koo

force; - - -, two control

forces.

Use of two control forces, on the other hand, is demonstrated to provide a large increase in plate transmission loss over virtually the complete frequency range of O< kOa < 1.5, irrespective of whether the plate system is on or off resonance. At higher frequencies the modal density of plate response increases, and the radiated field has contributions from an increasing number of modes, and thus more control actuators would be needed for good attenuation. The important result demonstrated here, however, is that large attenuations in radiated levels are achieved in the frequency range of Figure 8 with two control actuators, independent of source order or distribution, parameters which have previously led to difficulties in using acoustic control sources.

4.

CONCLUSIONS

By using an idealized analysis, the active control of sound radiation from plates by vibrating force inputs has been studied. The results show that for low to mid-range

ACTIVE CONTROL

OF SOUND

RADIATION

13

frequencies, large global reductions in radiated sound levels can be achieved with just one or two judiciously located point force actuators. High values of attenuation are achieved in this frequency range irrespective of whether the plate system is on or off resonance. Rather than just suppressing plate vibration completely, the high sound attenuation achieved was shown to be due to the control force modifying the plate source characteristics, effectively changing the response to a higher order distribution with an associated lower radiation efficiency. The technique would also be equally as effective for control of sound transmission through elastic plates or panels of various shapes and boundary conditions. ACKNOWLEDGMENT

The author is grateful for the support of this work by NASA Langley Research Center under grant NAG-1-390, Dr Harold C. Lester, Technical Monitor. REFERENCES 1. G. E. WARNAKA 1982 Noise Control Engineering 18(3), 100-110. Active attenuation of noisethe state of the art. 2. J. E. FFOWCS-WILLIAMS 1984 Proceedings of the Royal Society of London A395, 63-88.

Anti-sound. 3. M. J. M. JESSEL 1967 Breoet Fraqais No. 1,494,967. (ProcCdC Clectroacoustique d’absorption des sons et bruit genants dans des zones itendues). 4. A. J. KEMPTON 1976 Journal of Sound and Vibration 48, 475-483. The ambiguity of acoustic sources-a possibility of active control. 5. G. A. MANGIANTE 1977 Journal of the Acoustical Society ofAmerica 61(6), 1516-1523. Active sound absorption. 6. P. A. NELSON, A. R. D.CURTIS,S.J. E~~~orrand A.J. BULLMORE 1987 Journalqf‘Sound and Vibration 116, 397-414. The minimum output of free field point sources and the active control of sound. 7. 0. L. ANGEVINE 1981 Proceedings of Internoise 81, 303-306. Active acoustic attenuation of electric transformer noise. 8. S. ONODA and K. KIDO 1968 Proceedings of the 6th International Congress on Acoustics, Tokyo. Automatic control of stationary noise by means of directivity synthesis. 9. C. F. ROSS 1978 Journal ofSound and Vibration 61,473-476. Experiments on the active control of transformer noise. 10. C. G. MOLLO and R. J. BERNHARD 1987 AIAA Paper 87-2705. A generalized method for optimization of active noise controllers in three-dimensional spaces. 11. C. DEFFAYET and P. A. NELSON 1988 Journal of the Acoustical Society of America 84(6), 2192-2199. Active control of low-frequency harmonic sound radiated by a finite panel. 12. C. R. FULLER 1987 U.S. Patent No. 4,715,599. Apparatus and method for global noise reduction. 13. C. R. FULLER and J. D. JONES 1987 Journal ofSound and Vibration 112,389-395. Experiments on reduction of propeller induced interior noise by active control of cylinder vibration. 14. J. D. JONES and C. R. FULLER 1989 American Institute of Aeronautics and Astronautics Journal 27(7), 845-852. Active control of sound fields in elastic cylinder by multi control forces. 15. J. D. JONES and C. R. FULLER 1988 Proceedings of the 6th IMAC Conference, 350-521. Reduction of interior sound fields in flexible cylinders by active vibration control. of 16. J. D. JONES and C. R. FULLER 1987 Proceedings of Noise-Con 87, 371-376. Mechanisms active control for noise inside a vibrating cylinder. 17. H. C. LESTER and C. R. FULLER 1986 AIAA Paper 86-1957. Active control of propeller induced noise fields inside a flexible cylinder. 18. S. B. ABLER and R. J. SILCOX 1987 Proceedings of Noise-Con 87, 341-346. Experimental evaluation of active noise control in a thin cylindrical shell. 19. P. A. NELSON and S. J. ELLIOTT 1984 ISVR Technical Memorandum No. 647. A note on the active minimization of steady enclosed sound fields. 20. A. S. KNYASEV and B. D. TARTAKOVSKII 1967 Souiet Physics-Acoustics 13(l), 115-117. Abatement of radiation from flexurally vibrating plates by means of active local dampers.

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C. R. FULLER

21. A. I. VYALYSHEV, A. I. DUBININ and B. D. TARTAKOVSKII 1986 Soviet Physics-Acoustics

32(2), 96-98. Active acoustic reduction of a plate. 22. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill. 23. P. W. SMITH 1957 Journal of the Acoustical Society of America 29(2), 721-729. Sound transmission through thin cylindrical shells. 24. L. CREMER, M. HECKL and E. E. UNGAR 1973 Structure-borne Sound. Berlin: Springer-Verlag. 25. C. H. HANSEN and D. A. BIES 1976 Journal ofthe Acoustical Society ofAmerica 60(3), 543-555. Optical holography for the study of sound radiation from vibrating surfaces. 26. L. MEIROVITCH, H. BARUH and H. Oz 1983 Journal of Guidance, Control and Dynamics 6, 302-310. A comparison of control techniques for large flexible structures.

APPENDIX: : A-5111 B B, co C,, Cp P

h M,, i Jn, I, ko 4 k;

k“I L m

n PO $rod

r,

4

6 4

Y x

R 5 t W w,,

pPo E E,,

I;‘,,, A,,4 n’, PI

PS lo w,,, y;

(7

nR

LIST OF SYMBOLS

plate radius column vector of control distribution function elements of matrix [A] column vector of noise distribution function elements(s) of matrix [B] speed of sound in air element(s) of matrix [C] wave shear speed in plate material column vector of force amplitudes plate thickness plate radial characteristic function radial mode number Bessel functions of the first and second kind free wavenumber in air radial wavenumber in air plate eigenvalue resonant wavenumber total number of control forces mass per unit area of plate circumferential mode number incident pressure amplitude radiated pressure column vector of input pressure amplitudes polar co-ordinates cylindrical co-ordinates radial distance to observation point integral surface of cost function column vector of incident wave amplitude(s) time plate out-of-plane displacement plate displacement modal amplitude cost function optimal cost function =2,n=O;=l,n>O =l,n=0;=2,n>O incident wave angle plate mode normalization constant coupling integrals incident and radiated acoustic power fluid density plate density circular driving frequency plate mode resonant frequency conjugate conjugate differentiation with respect to argument

ACTIVE

CONTROL

Subwipts

cf inc

j I n

s tot

control field incident radial mode index control force number circumferential mode index matrix index number total field

Superscripts rad radiated

OF

SOUND

RADIATION

15