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Active control of sound radiation from a plateusing a polyvinylidene fluoride volumedisplacement sensor

F. Charettea) and A. BerryG.A.U.S., Mechanical Engineering Department, Universite´ de Sherbrooke, Sherbrooke, Que´bec J1K 2R1,Canada

C. GuigouV.A.L., Mechanical Engineering Department, Virginia Polytechnic Institute and State University,Blacksburg, Virginia 24061-0238

~Received 18 April 1997; accepted for publication 19 November 1997!

This paper presents a new volume displacement sensor~made of shaped strips of PVDF film! andthe experimental implementation of this sensor in an active control system. A design strategy for aPVDF sensor detecting the volume displacement induced by a vibrating 2D structure is presented.It is based on the modal representation of the plate response. It actually consists in designing aPVDF sensor, composed of several shaped PVDF strips bonded to the surface of the structure, insuch a way that the output signal of the sensor is directly proportional to the volume displacement.The design methodology is based on the experimental measurements of the plate mode shapes~eigenfunctions! and is valid for any type of boundary conditions. The experimental implementationof such a volumetric sensor in an active control system is then presented. The experimental resultsobtained validates this new type of volume displacement sensor. ©1998 Acoustical Society ofAmerica.@S0001-4966~98!00703-6#

PACS numbers: 43.40.Vn, 43.38.Fx@PJR#

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INTRODUCTION

The two main strategies for actively controlling soufields are active noise control~ANC! and active structuraacoustic control~ASAC!, first proposed by Fuller.1,2 TheANC approach is based on controlling the acoustic fieldusing loudspeakers as acoustic secondary sources, whilASAC approach directly modifies the response of the rading structure by applying structural inputs as secondsources, i.e., modal restructuring.2 In general, the ASAC ap-proach has been proved to require a smaller number ofondary sources for a global control of the acoustic field. Rcently, piezoceramic actuators embedded in or bonded tostructure have been successfully used as structural seconsources for ASAC applications.3,4 These integrateddistributed actuators overcome many of the disadvantageshakers.

In ASAC, the other component of prime importancethe error sensor, which defines the type of information tominimized by the controller. Polyvinylidene fluoride~PVDF!materials have been suggested as error sensors in the acontrol of structural vibration,5,6 and more recently in theactive control of sound radiation.7–10 In the latter case, thetype of information measured on the structure by such ssors has to be carefully defined so that the minimizationthis information effectively leads to a global attenuationthe sound field. In this instance, PVDF materials provdistributed sensors for which spatial filtering techniquesbe applied by tailoring the sensor shape, in order to se

a!F. Charette is currently working at the Vibration and Acoustics Laboraries ~VAL ! of Virginia Polytechnic Institute & State University.

1493 J. Acoust. Soc. Am. 103 (3), March 1998 0001-4966/98/103

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for example, the most efficient radiating modes in the strtural response.

Since active control is most efficient at low frequenciethe development of appropriate sensors to be used in ASshould be based on the low-frequency mechanisms ofsound radiation from vibrating structures. For planar radtors, it is well known that the sound radiation is directrelated to the velocity distribution over the surface of thstructure, for frequencies lower than the critical frequencywas shown11,12 that velocity distributions corresponding tcertain modes, i.e., odd–odd or symmetric modes for simsupported and clamped boundary conditions, are the mefficient radiators. Therefore, the sensor to be desigshould be able to mainly detect these well-radiating velocdistributions. A strategy originally investigated by Fulleet al.13 is to detect only the supersonic~radiating! compo-nents of the structural wave number spectrum. Anotherproach is based on the ‘‘radiation modes’’ concept; it wshown14,15 that the sound power can be written in termsradiation modes, which correspond to the eigenvectors ofradiation impedance operator of the structure and radsound independently. Recently, Snyderet al.10 have imple-mented the active control of sound radiation from a simsupported panel using PVDF film shaped in order to obsethe radiation modes of the panel. The first and most radiaof these radiation modes was found to be the piston-tmode. Such a mode actually represents the monopole beior of the structure and can be detected by measuring thevolume displacement over the surface of the structure. Baon this assumption, Rex and Elliott16 developed a ‘‘quadrati-cally weighted strain integrating sensor,’’ i.e., a quadratica

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Copyright by the Acoustical Society of America. Charette, F., Berry, A., & Guigou, C. (1998). Active control of sound radiation from a plate using a polyvinylidene fluoride volume displacement sensor. Journal of the Acoustical Society of America, 103(3), 1493-1503. doi: 10.1121/1.421301

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weighted sensor measuring the volume velocity of the strture. This sensor can be implemented in practice in the foof a piezoelectric cable or an optical fiber bonded on or ebedded in the structure. Using the same approach, Johand Elliott17 showed that the volume velocity of a twodimensional structure can be measured by quadraticweighting the sensitivity or the thickness of an extendPVDF sensor covering the whole surface of the structuThis solution being unrealistic in practice, it was appromated by shaping the sensor into quadratic or rectangstrips.18 They successfully implemented active controlvolume displacement using such sensor on a plate andserved significant reductions in the sound radiations. Hoever, the disadvantage of their sensor design is that it hacover the whole plate surface, which may not be availablemany applications. Recently, Guigouet al.8 have successfully implemented active control of volume displacementthe case of flexural beams, by using a single shaped PVstrip, and they observed significant attenuation of the raated sound field. The shape of the PVDF strip was basethe mode shapes~eigenfunctions! of the beam. This laswork provides the basis of the present investigation.

The objective of this paper is to extend the conceptsthe abovementioned paper in the case of 2D structures.implementation of an extended, shaped PVDF sensorvolume displacement sensor is discussed in the caserectangular panel with arbitrary boundary conditions;sensor shape is shown to be independent of the frequand type of excitation. Moreover, the strategy of minimizithe volume velocity keeps the control architecture vesimple~single input control system!. Experimental results othe volume displacement sensor and its use in an activetrol system are presented and discussed.

I. VOLUME DISPLACEMENT PVDF SENSOR

A. Description of the system considered

The mechanical system studied in this paper consista rectangular clamped plate on which two piezoelectrictuators are bonded; see Fig. 1. A piezoelectric actuatocomposed of two identical colocated piezoceramic patcon each side of the plate. The piezoceramic patches aresumed to be perfectly bonded to the plate surface. A smetric actuation~pure flexion! is induced in the plate whenthe two piezoceramic patches are driven 180° out of pha2

Here, one of the piezoelectric actuators is used to induceunwanted vibration~i.e., primary actuator! while the secondactuator is the control~i.e., secondary! actuator. The platedimensions and material properties are described in TabThe piezoceramic actuators characteristics and locationsgiven in Table II.

TABLE I. Properties of the clamped plate.

Lxpl 50.0 cm

Lypl 39.8 cm

Lzpl 3.15 mm

Young’s modulus (E) 6.531010 PaDensity ~r! 2800 kg/m3

1494 J. Acoust. Soc. Am., Vol. 103, No. 3, March 1998


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The total transverse displacement of the plate underaction of the primary and secondary actuators can be wrias a linear combination of modes,

w~x,y!5(i 51

l

WiC i~x,y!, ~1!

where Wi is the modal amplitude andC i(x,y) is the i theigenfunction. The modal sum in Eq.~1! is truncated to ‘‘I ’’modes.

B. Experimentally determined eigenfunctions

The design of the volume displacement sensor stwith the determination of the plate eigenfunctions. In thwork, these eigenfunctions are determined experimentaeven though theoretical approximations are available incase of clamped plate. The experimental determination ofeigenfunctions can lead to a better representation of aticular ‘‘real’’ ~i.e., experimental! plate displacement, sincetakes into account the imperfections in the boundary contions, plate material, external loading, etc. Furthermore,ing experimentally measured eigenfunctions would be naral for more complex plates for which no theoreticapproximations are easily available.

The plate was excited between 100 and 500 Hz, usthe primary PZT actuator, in order to obtain its modal chacteristics. A Bruel & Kjaer analyser was used to measthe transfer function between the velocity signal from a lavibrometer@at a total of 144 (12312) regularly spaced measurement points# and the excitation signal. The modal analsis STAR SYSTEMS V5.0 package was then used to procthe data. This software uses a frequency domain curve-fitanalysis to determine the modal characteristics. The expmental mode shapesC i(x,y) are then represented as a combination of polynomial trial functions,

C i~x,y!5 (p50

P21

(q50

Q21

Ai ,pqS x

LxplD pS y

LyplD q

. ~2!

Such trial functions are very general and may be usedplates with arbitrary boundary conditions. In order to detmine the unknown coefficientsAi ,pq of a mode shape, thefollowing system of equations has to be solved for eamode ‘‘i ,’’

@wmnpq#$Ai ,pq%5$wi ,mn%. ~3!

The vector$wi ,mn% contains thei th modal displacement othe mnth measurement point as provided by the STASYSTEMS package. The vector$wi ,mn% thus contains 144elements. The coefficients of the matrix@wmnpq# are givenby

wmnpq5S xm

LxplD pS yn

LyplD q

, ~4!

wherexm andyn are the positions of themnth measuremenpoint.

Reasonably small values ofP andQ in Eq. ~2! are suf-ficient to reconstruct the first few modes of the plate. Cosequently, the values ofP andQ are usually smaller than th

1494Charette et al.: Active control of sound radiation


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FIG. 1. Clamped plate studied.

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number of measurement points in each direction of the pand the linear system represented by Eq.~3! is overdeter-mined. Therefore, a least mean square technique is usedthe following system has to be solved,

@wmnpq#T@wmnpq#$Ai ,pq%5@wmnpq#

T$wi ,mn%. ~5!

The superscriptT indicates the transpose operator. A critrion must be used to evaluate the accuracy of the polynomapproximation obtained. The criterion suggested by Geand Wheatley19 is to increase the degree of the approximtion polynomials until a statistically significant decreasethe variance of the approximation error is obtained. The vance of the error is defined as



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M•N2P•Q21, ~6!

where ‘‘M ’’ and ‘‘ N’’ are the total number of measurinpoints in thex andy directions, respectively. Thus the orde‘‘ P’’ and ‘‘ Q’’ of the approximation polynomials are selected independently for each mode ‘‘i ’’ using Eq. ~6! ascriterion.

C. Structural volume displacement of a plate

In the present work, the cost function considered foractive control system is the structural volume displacem



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of the plate. The structural volume displacement is definethe integral of the transverse displacement over the surof the plate,

D5E0

LyplE

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w~x,y!dx dy. ~7!

Introducing Eqs.~1! and~2! into Eq. ~7! and performing thedouble integration yields the volume displacement ofplate, which can be written in matrix form as

D5LxplLy

pl^Wi&@Ai ,pq#$Dpq%. ~8!

The dimensions of the matrix@Ai ,pq# are ‘‘I ’’ rows by ‘‘ P3Q’’ columns. Each row of the matrix@Ai ,pq# contains thecoefficients of a given eigenfunctionsC i(x,y) over the poly-nomial trial functions. The line vectorWi& contains themodal amplitudes. The components of the column vec$Dpq% are simply given by 1/@(p11)(q11)#.

The volume velocity of the plate is simply Eq.~8! mul-tiplied by j v as harmonic excitation is assumed. Thuspure tone excitation, it is equivalent to minimize the volumvelocity or the net volume displacement. The strategy ofing the volume displacement as a cost function for an accontrol system have been shown previously8,15,18 to be veryefficient to reduce sound radiation in the low-frequenrange~i.e.,k0L!1, k0 is the wave number in air, andL is thecharacteristic dimension of the structure!.

D. Design of a volume displacement sensor usingshaped PVDF film

The charge response of an arbitrarily shaped stripPVDF sensor applied to a two-dimensional structurebeen derived by Leeet al.5 It was shown that the chargresponse of the sensor is a function of the integral ofstrain over the surface of application and is expressed a

q52Lz

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2 E0

LyplE

0

LxplFe31

]2w~x,y!

]x2

1e32

]2w~x,y!

]y2 12e36

]2w~x,y!

]x]y G•F~x,y!•dx dy,

~9!

whereLzpl and Lz

s are, respectively, the plate thickness aPVDF sensor thickness. Theei j are the PVDF sensor strescharge coefficients. It is assumed that no skew angle is aciated with the polarization of the PVDF film used in th

TABLE II. Ceramic piezo-characteristics.

Primary Secondary

Lxpz 3.81 cm 3.81 cm

Lypz 3.18 cm 3.18 cm

Lzpz 0.19 mm 0.19 mm

Pos. of the center alongthe x axis

16.0 cm 19.0 cm

Pos. of the center alongthe y axis

30.0 cm 15.0 cm

Young’s modulus (E11) 6.331010 Pa 6.331010 PaDensity ~r! 7750 kg/m3 7750 kg/m3



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study,e3650. The functionF(x,y) represents the variationof the film sensitivity with the position on the surfaceapplication. In practice, this variation of sensitivity is moeasily achieved by shaping the PVDF film as described infollowing.

First, consider a single PVDF strip parallel to thex di-rection of the plate, whose center line is located aty5yct , asshown in Fig. 2. The maximum width of the strip is note2mx , and the width at the positionx is 2mxF(x), whereF(x) is a function varying between21 and11. Negativevalues ofF(x) correspond to an inversion of the PVDF filmpolarity. It is assumed that the strip axis coincides with tdirection of maximum stress/charge coefficiente31. It is de-sired to find the output response of this strip when the fution F(x) is given by

F~x!5 (r 50

R21

a r S x

LxplD r

. ~10!

In this case, the charge response of the PVDF strip canwritten as

qx52Lz

pl1Lzs

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LxplE

yct2mxF~x!

yct1mxF~x!Fe31

]2w~x,y!

]x2

1e32

]2w~x,y!

]y2 Gdydx. ~11!

Introducing Eqs.~1! and ~2! into Eq. ~11! and integratingwith respect to the variabley leads to

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•F S 11mxF~x!

yctD q11

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yctD q11G1

e32qLypl

~Lypl!2 S x

LxplD p

•S yct

LyplD q21

•F S 11mxF~x!

yctD q21

2S 12mxF~x!

yctD q21G J dx. ~12!

FIG. 2. Top view of the finite plate with a typical shaped PVDF striconfiguration.



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At this point, it is necessary to introduce a simplifying asumption before integrating along the strip axis: If the widmx of the strip is small enough such thatumxF(x)/yctu!1then

S 11mxF~x!

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and

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Inserting these approximations into Eq.~12! and performingthe integration along thex axis, the charge of the PVDF striparallel to thex axis can be written in matrix form as

qx52Lz

pl1Lzs

2^Wi&@Ai ,pq#@T1pq,r #$a r%, ~14!

where $a r% is the vector of the shape coefficients of tPVDF strip. The coefficients of the matrix@T1pq,r # are givenby

T1pq,r52mxe31p~p21!

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LyplD q

12mxLx

ple32q~q21!

~Lypl!2~p1r 11! S yct

LyplD q22

. ~15!

Similarly, for a PVDF strip parallel to they direction atx5xct , with a maximum width 2my ~see Fig. 2! and a shapedescribed by

F~y!5 (s50

S21

bsS y

LyplD s

, ~16!

the charge response is given by

qy52Lz

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Ai ,pqE0

LyplH e31q~q21!Lx

pl

~p11!~Lypl!2

3S y

LyplD q22

•S xct

LxplD p11

•F S 11myF~y!

xctD p11

2S 12myF~y!

xctD p11G1

e32p

Lxpl S y

LyplD q

•S xct

LxplD p21

•F S 11myF~y!

xctD p21

2S 12myF~y!

xctD p21G J dy. ~17!

Using approximations similar to Eq.~13! for the terms (11@myF(y)/xct#)

p11, etc., and performing the integratioalong they axis, the charge of the PVDF strip parallel to thy axis can be written in matrix form as



-qy52

Lzpl1Lz

s

2^Wi&@Ai ,pq#@T2pq,s#$bs%, ~18!

where

T2pq,s52mye31q~q21!

Lypl~q1s21! S xct

LxplD p

12myLy

ple32p~p21!

~Lxpl!2~q1s11! S xct

LxplD p22

. ~19!

The total output chargeq5qx1qy written in matrix form is

q52Lz

pl1Lzs

2^Wi&$@Ai ,pq#@T1pq,r #$a r%

1@Ai ,pq#@T2pq,s#$bs%%. ~20!

It is required that the sum of the output charges be proptional to the net volume displacement of the plate. In tcase, the sensor formed by the 2 PVDF strips is a voludisplacement sensor. This requirement allows the unknocoefficientsa r and bs of the strip shapes to be determineEquating Eqs.~8! and ~20! row by row implies that themodal volume displacements of the plate are equal tomodal charge response of the sensor, which ensures thasensor is independent of the frequency; this yields

2Lz

pl1Lzs

2$@Ai ,pq#@T1pq,r #$a r%1@Ai ,pq#@T2pq,s#$bs%%

5LxplLy

pl@Ai ,pq#$Dpq%. ~21!

Equation ~21! forms a system ofI linear equations to besolved for theR1S unknown coefficientsa r and bs . Aunique solution is obtained ifR1S5I , I being the totalnumber of modes considered in the plate response. So,uniqueness condition relates the total number of modes toorder of the polynomials used to shape the strips of PVDNote that high order polynomials yield complicated shapof PVDF strips~with possibly many zeros, thus many polaity changes! implying difficulties in experimental implementation. Therefore, when the number of modesI considered islarge, it is worthwhile to consider more than one strip in tx and/ory directions. Adding strips on the plate allows thorder of the shape polynomials to be kept small for eastrip. Equation~21! can easily be extended to consider mothan one strip in thex or y direction. It is also interesting tonote that since matrix@Ai ,pq# is rectangular, it cannot beeasily dropped from Eq.~21!. Furthermore, the goal is toobtain a linear equation for each mode considered infrequency range of interest@eliminating matrix@Ai ,pq# wouldremove the modal representation in Eq.~21!#.

It is important to note that the volume displacement ssor given by Eq.~21! is independent of the excitation frequency and of the type, magnitude, and location of the etation. Also, the use of polynomial functions to reconstruthe plate response and to design the sensor shape, makemethodology valid for arbitrary boundary conditions thatnot allow rigid modes. It should be noted that since tPVDF film is a strain sensor, it is not sensitive to rig~whole body! modes.



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II. EXPERIMENTAL RESULTS

A. Experimental implementation of the volumedisplacement sensor

The procedure described in Sec. D of Part I was impmented experimentally on the clamped plate describedTable I. Two piezoceramic actuators~a primary ‘‘distur-bance’’ actuator and a secondary ‘‘control’’ actuator! werebonded to the plate. The characteristics and locations ofpiezoceramic actuators are described in Table II. The prerties of the PVDF film used in the experiments are listedTable III.

The shapes of the strips were determined usingI 57modes for the plate response and polynomial functionsorderR54, S53 for the shape functions of the strips in thx and y directions. Once the positionsxct and yct of thestrips have been chosen, the solution of Eq.~21! yields thecoefficients of the two PVDF strip shapes which form tvolume displacement sensor. The accuracy of the PVDFume displacement sensor turns out to be quite sensitive topositionsxct andyct . This is partly due to the approximatiointroduced in Eq.~13! to design the shapes of the PVDstrips. Therefore, after fixing the positions of the PVDstrips and solving Eq.~21! for the shape function@F(x) andF(y)# coefficients, Eqs.~12! and ~17! are numerically inte-grated and their sum is compared to the volume displacemmeasured directly. If the comparison shows that the tcurves corresponds well, then the chosen positionsxct andyct are valid ones. If not, other positions forxct andyct haveto be tested.

The velocity at a total of 144 (12312) regularly spacedpoints on the [email protected]., j vw(xix ,yiy) for ix and iy51 to12# was measured using a laser vibrometer. A Bruel & Kjaanalyzer was used to acquire the transfer functions betwthe excitation signal~to the PZT actuator! and the laser sig-nal. The measured volume displacement can then be evated as

Dmeasured5Lx

plLypl

144 (ix51

12

(iy51

12

w~xix ,yiy!, ~22!

which is a discrete version of Eq.~7!. Figure 3 presents theresults forxct50.259 m andyct50.176 cm. Figure 3~a! and~b! shows the shapes of the PVDF strips in thex and ydirection, respectively. Figure 3~c! presents the comparisoof the measured volume displacement~with a laser vibrome-ter! and the numerically calculated charge response@Eqs.~12! and ~17!# of the PVDF sensor when the excitationinduced by the primary actuator. Note, there is no ordinscale in Fig. 3~b! because the two curves were superpose

TABLE III. PVDF film characteristics.

e31 0.046 N/~V m!e32 0.006 N/~V m!Lz

s 0.028 mmmx 1.0 cmmy 1.0 cmYoung’s modulus (E11) 2.03109 PaDensity ~r! 1780 kg/m3



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facilitate comparison~i.e., the charge response of the PVDdoes not have the same unit/scale as the volume displment response!. Figure 3~c! shows an excellent agreemebetween the numerically calculated signal and the measvolume displacement, which confirms that the chosen ptions xct andyct are valid. Note that the shapes of the stri@Fig. 3~a!, ~b!# are almost symmetric. This is intuitively understandable since the experimental mode shapes of theto be observed by the PVDF strips are also symme~monopole modes!.

The sensor strips shown in Fig. 3~a! and~b! were imple-mented experimentally and taped to the clamped plate.total electrical signal from the PVDF sensor~the added sig-nal of the two PVDF strips! was first passed through a preamplifier ~with high input impedance and variable gain! andthen measured with a Bruel & Kjaer analyzer. Figure 4 a

FIG. 3. Results obtained for the positionsxct50.259 andyct50.176 m.

FIG. 4. Performance of sensor when the primary actuator induces the vtions.



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Fig. 5 present the PVDF sensor signal compared to the dmeasurement of the volume displacement when the vibraof the plate is induced by the primary PZT actuator andsecondary PZT actuator respectively. Note that the sigfrom the PVDF sensor was scaled to be equivalent tomeasured volume displacement at 170 Hz. It must becalled that the sensor is designed using the first seven exmental modes of the plates, so that the contribution of higorder modes is not considered in the shape of the senThus it is expected that the sensor signal deviates fromvolume displacement in higher frequencies. Also, the oresonance differences between the sensor signal and thesured volume displacement are presumably due to the hiorder modes contributions in the flexural strain of the plaPVDF materials measure strain and are thus more sensto higher order modes. In effect, this is more apparent incase of the primary actuator excitation, which is significanoff-center and thus excites more efficiently the higher ormodes of the plate. Note that in both cases the responsthe ~2,1! and ~1,2! modes is large, indicating that thesmodes have a nonzero volume displacement and are thuperfectly antisymmetric~this is mainly due to the presencethe PZT on the plate and the imperfections in the experimtal boundary conditions!. The fact that modes~2,1! and~1,2!have a volume displacement, shows that it is worthwhilebase the design of the sensor from experimentally measmode shapes~i.e., eigenfunctions!. It allows to take into ac-count all the experimental imperfections~i.e., boundary con-ditions, added mass, added stiffness, plate imperfectietc.! of the actual experimental setup. The sensor is desigfrom the measured modal characteristics. Therefore,modifications of the structure mechanical properties~addedmass, stiffness, boundary conditions, etc.! after the designstage will affect the response of the sensor.

B. Active control of volume displacement

This section describes the active control experimenting the volume displacement sensor developed previouThe plate was located in a semi-anechoic chamber forexperiments. Sound pressure levels were measured atdiscrete points equally distributed over a 1.2 m radius frthe center of the plate in the planex5L x

pl/2. Note, the mainpurpose of these sound pressure measurements was togood indication of the reduction that could be expected i

FIG. 5. Performance of sensor when the secondary actuator inducevibrations.



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realistic application. Therefore, no baffle was installaround the plate. Furthermore, in order to have a sinsound pressure value to quantify the reduction obtainedthe control, an average sound pressure level in the px5L x

pl/2 is defined as

^ p&5R2

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p/2

up~R,u!u2usin~u!udu, ~23!

whereR is the radius between the center of the plate andpositions of the measurements point,r0 is the density of air,andc0 is the speed of sound in air.u is the angle between thline perpendicular to the plate surface and the line that jothe considered point on the radiusR to the center of theplate. Since the sound pressure measurements were donine anglesu i , the discrete version of the last equation is

^ p&5pR2

9r0c0(i 51

9

up~R,u i !u2usin~u i !u. ~24!

The displacement of the plate before and after control wmeasured using a laser vibrometer at 144 regularly spapoints on the plate surface. The primary actuator was useinduce the unwanted vibration while the secondary actuawas used as the control actuator. The voltage appliedthese actuators was typically between 80 and 120 V peapeak. A feedforward filtered X-LMS controller was used.number of single frequency was tested~i.e., harmonic controlwas performed here!. The control results are presented fthree typical frequencies, two on-resonance and oneresonance frequency. For each of these frequencies, thesured displacement field as well as the sound pressure lein the planex5L x

pl/2 are displayed before and after contris applied.

Figure 6 presents the results obtained at 125 Hz, cosponding to an off-resonance frequency of the clamped p~below the first mode!. The structural displacement field before control clearly shows that the plate acts like a monoptype radiator with a maximum displacement close to the pmary ~i.e., disturbance! actuator location. After control, theplate is forced to act as a dipole type radiator by the secoary ~i.e., control! actuator. One can note that the vibratiolevel before and after control are globally the same in tcase~the absolute maximum displacement value is appromately 0.08mm before control, 0.06mm after control!. Theexperimental sound pressure levels in the planex5L x

pl/2 areshown in Fig. 6~b!. The reduction of the average sound presure level is around 16 dB. This reduction can be attribusolely to the fact that the plate deformation is changed fra monopole type radiator to a dipole type radiator~i.e.,modal restructuring2!. The control of the volume displacement has no appreciable effect on the vibration level atfrequency and only the radiation efficiency, which represethe capacity of the plate the generate sound, is decrease

Figure 7 presents the results obtained at 140 Hz, whcorresponds to the resonance frequency of the first moi.e., mode~1,1!. As seen in Fig. 7~a!, the plate displacemenbefore control corresponds to the~1,1! mode shape, withmaximum displacement at the center of the plate. This m

the



Redistri

FIG. 6. Experimental results at 125 Hz~off-resonance!.

etyc

thonueloonea

ee

nceterthecedgethe

tion

is a monopole-type radiator.2,11 Once again, after control thsecondary actuator forces the plate to act as a dipole-radiator. Note here the vibration levels after control are mulower than before control which indicates that in this casecontrol combines both vibration restructuring and vibratireduction. Figure 7~b! presents the measured sound presslevels. The reduction of the average sound pressure levaround 40 dB. The results of the volume displacement ctrol at 140 Hz~Fig. 7! shows that the sound attenuationresonance of a symmetric mode is associated with a decr



pehe

reis

n-

se

of the radiation efficiency~passage from a monopole typradiator to a dipole-type radiator!, combined with a decreasof the vibration amplitudes.

Finally, Fig. 8 presents the results for 320-Hz resonaof mode~1,2!. Note that the shape of the displacement afcontrol is globally the same as the one before control. Onother hand, the maximum displacement has been redusignificantly. Therefore, the 14-dB reduction of the averasound pressure level is associated with the reduction ofvibration levels. In this case, the sensor acts as a vibra



Redistri

FIG. 7. Experimental results at 140 Hz@resonance of mode~1,1!#.

th

onona

mentrol

inndnceThis

s/

sensor, since the plate displacement before control wasready a dipole-type radiator. It is interesting to note thatmeasured pressure directivity in thex5L x

pl/2 plane corre-spond to that of a monopole-type radiator~as expected!. Atthis frequency, it can be deduced from Fig. 8~a! that thepressure directivity in they5L y

pl/2 plane would be that of adipole-type radiator before and after control. Since the ctrol only reduces the vibrational levels, pressure reductiin they5L y

pl/2 plane are expected to be similar to that mesured in thex5L x

pl/2 plane.



al-e

-s-

III. CONCLUSIONS

This paper presents the development of a novel voludisplacement sensor for plates and its use in an active cosystem. The sensor is made of shaped strips of PVDF filmthe x and y directions and is independent of the type afrequency of excitation. The approach is very general, siit is based on experimentally determined mode shapes.allows consideration of any boundary conditions~that do notallow rigid modes!, and plates with added punctual mas



Redistri

FIG. 8. Experimental results at 320 Hz@resonance of mode~1,2!#.

ntrinetibnspa

t

ffi

su

e isent

vi-r-

theas

ol.els

res-trola

tural

stiffness and/or with internal defects. Results were presefor a clamped plate. The sensor was validated by compathe volume displacement measured with a laser vibromto the measured signal from the PVDF sensor. The contrtion of high order modes of the plate in the sensor respoas well as the location of the PVDF strips, is sensitiverameters in the sensor design.

A global reduction in the planex5L xpl/2 of about 16 dB

was obtained in the average sound pressure level foroff-resonance frequency case~125 Hz!. This reduction isprincipally associated with a decrease in the radiation eciency ~the plate goes from a monopole-type radiator todipole one!. For the on resonance case of mode~1,1!, at 140Hz, the obtained reduction in the average sound pres



edg

eru-e,-

he

-a

re

level is 40 dB. This large reduction for this resonance casattributed to the fact that the control of volume displacemdecreases simultaneously the radiation efficiency and thebration levels. Finally, for the results at 320 Hz, which corespond to the resonance of mode~1,2!, the reduction of 14dB of the average sound pressure level is associated withreduction of the vibration levels. In this case, the plate walready acting like a dipole-type radiator before contrTherefore, the controller mainly reduced the vibration levat this frequency.

The transversal displacement fields and the sound psure levels obtained experimentally before and after conclearly validates the use of PVDF film directly bonded toplate as a volume displacement sensors in an active struc



.xc

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or

tic

c-te

r-s

nyv

ia-ap

m

foron-

s forndra-

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eary

to

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ng

ia-n-

ve

ofPIE443,

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acoustic control~ASAC! system for harmonic excitationThis sensor could be used for ASAC under wide-band etation if an appropriate digital controller is implemented.

ACKNOWLEDGMENTS

The authors wish to thank Professor Bruno Paillard ahis student Martin Bouchard for the development and impmentation of the control code on the DSP card. This wwas funded in part by the F.C.A.R.~Formation de Cher-cheurs et l’Aide a` la Recherche!.

1C. R. Fuller, ‘‘Active control of sound transmission/radiation from elasplates by vibrational inputs: I. Analysis,’’ J. Sound Vib.136, 1–15~1990!.

2C. R. Fuller, S. J. Elliott, and P. A. Nelson,Active Control of Vibration~Academic, London, 1996!.

3C. R. Fuller, C. H. Hansen, and S. D. Snyder, ‘‘Active control of struturally radiated noise using piezoceramic actuator,’’ Proceedings of InNoise89, 509–511~1989!.

4R. L. Clark and C. R. Fuller, ‘‘Experiments on active control of structually radiated sound using multiple piezoelectric actuators,’’ J. AcouSoc. Am.91, 3313–3320~1992!.

5C. K. Lee and F. C. Moon, ‘‘Modal sensors/actuators,’’ J. Appl. Mech.57,434–441~1990!.

6Y. Gu, R. L. Clark, C. R. Fuller, and A. C. Zander, ‘‘Experiments oactive control of plate vibration using piezoelectric actuators and polnylidene fluoride~PVDF! modal sensors,’’ J. Vib. Acoust.116, 303–308~1994!.

7R. L. Clark and C. R. Fuller, ‘‘Modal sensing of efficient acoustic radtors with PVDF distributed sensors in active structural acousticproaches,’’ J. Acoust. Soc. Am.91, 3321–3329~1992!.

8C. Guigou, F. Charette, and A. Berry, ‘‘Active control of finite bea



i-

d-k

r-

t.

i-

-

volume velocity using shape PVDF sensor,’’ Acta Acust.~China! 82,772–783~1996!.

9R. L. Clark, R. A. Burdisso, and C. R. Fuller, ‘‘Design approachesshaping polyvinylidene fluoride sensors in active structural acoustic ctrol ~ASAC!,’’ J. Intell. Mater. Syst. Struct.4, 354–365~1993!.

10S. D. Snyder, C. H. Hansen, and N. Tanaka, ‘‘Shaped vibration sensorfeedforward control of structural radiation,’’ Proceedings of the SecoConference on Recent Advances in Active Control of Sound and Vibtion, April 28–30, 1993, Blacksburg, Virginia, pp. 177–188~1993!.

11C. E. Wallace, ‘‘Radiation resistance of a rectangular panel,’’ J. AcouSoc. Am.51, 946–952~1972!.

12A. Berry, J. L. Guyader, and J. Nicolas, ‘‘A general formulation for thsound radiation from rectangular, baffled plates with arbitrary boundconditions,’’ J. Acoust. Soc. Am.88, 2792–2802~1990!.

13C. R. Fuller and R. A. Burdisso, ‘‘A wave number domain approachactive control of structure-borne sound,’’ J. Sound Vib.148, 355–360~1991!.

14S. J. Elliott and M. E. Johnson, ‘‘A note on the minimization of thradiated sound power,’’ ISVR Technical Memorandum~1992!.

15M. E. Johnson and S. J. Elliot, ‘‘Active control of sound radiation usivolume velocity cancellation,’’ J. Acoust. Soc. Am.98, 2174–2186~1995!.

16J. Rex and S. J. Elliott, ‘‘The QWSIS—A new sensor for structural radtion control,’’ 1st International Conference on Motion and Vibration Cotrol Yokohama, pp. 339–343~1992!.

17M. E. Johnson and S. J. Elliott, ‘‘Volume velocity sensors for acticontrol,’’ Proc. Inst. Acoust.15, Part 3, 411–420~1993!.

18M. E. Johnson and S. J. Elliott, ‘‘Experiments on the active controlsound radiation using a volume velocity sensor,’’ Proceedings of S1995 conference on Smart Structures and Integrated Systems, Vol. 2658–669~1995!.

19C. F. Gerald and P. O. Wheatley,Applied Numerical Analysis~Addison-Wesley, New York, 1989!.



active control of sound radiation from a plate using a ... · pvdf sensor, composed of several...

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