Sensors

3a-\8 5

Copyrighl (t) 1996 IFAC 131h Triennial World Congress. San Francisco. US."

IDENTIFICATION OF A THIN PLATE WITH BONDED PIEZOELECTRlC ACTUATORS/SENSORS

Jin Kwon Hwang*.

Chong~Ho

Choi*. Chu] Ki Song':'*, Jang Moo

Lee *~':I:

* Department (~rCofltrol and In strumentation EIIKilleerillg. **'"

ACRe. ERC-ACI, Seoul Narional Universir.v. Seoul. Korea ** Kia Technical Cell1er, KIA Motors Co.. Sei}u/, Korea f)el'arlmellf of Mc(:hallical L>ej'ign and Production Engillel'l"ittR, Seoul Na rional Unil,etsily. Seeml. Korea

Ahstr.K:t: Thi s paper proposes an idcntiJ'k atio n method for a thin plate wlll're multiple piczoek(.;tric actuators and sensors are handed. Since a thin plate has small damping rat ios o f all modes. each modal signal ea n be ext racted with a handpass filt~r corresponding (Q th e IlHKic, With tbe hand pass fillers, the MIMO model can bc converted to several multi-input singh.'-output (MISO) Imxlels. Modal parameters uf the MISO models are oblain ed hy using ' .S lllcthod. The platl~ lIscd for an experi ment is an all-clamped plate with two pairs 0 1" pio.oc leClric actuator:-;
KcywonJ : Idl'ntiri cati o ll , MrMO model. Vibration, Leas.t-squares (I.S ) meth od

I. I NTRODlJCTlON Vibrati o n co ntrol~ have hee n studied fo r many years in nexihlc stru<.: turcs sueh <.L\ Ocams, plate:-., space vehicles and rnhnt JnllS ( Bailey and HlIbbard. 19};5: Daz and Poh. 1988; \Vil1iiJ11lS unu Juan g. 1992). Pict.oelectric marerial s play an impo rlunt ruk in tile application vihration control d ue to tlu.: ir ad\'i:lntages uf guml accuracy in sensing and act uation , li ght wl!ig ill. :-.J1lall sit.c. lllw cost and high force without reacliDI1 . Dimi tr iadi s. cl al. (11}Y I ) dcvclopcu i.I dynamic: model I'm till.' vihnltioll r csp(l ns~ or a s im ply-suprorted plale. C la rk . e t itl. i jl)I)3) found a d ynamic model o f il s illl~)l y - s UPIl()n cd plate by usin g pic/.oclcelric anualOrs experime nl a ll y. T /.Oll a nd Fu ( llJ94) sllldi ed piezoelectric maleri"ll s fo r stru<:tural monitoring alld l:onlrof o f e lastic n mtilHlUlI1 . l ;a l
pr

accelerome te rs hy rare fcedhat: 1-. control and HWcontrol. The models of these previou s researc hes were based on theoreti cal analyses and modal lc sti ngs. and most ur the control methods werc based on SISO system s, In tbe conventional modding of slructures. modal frequencies. arc obtained by modal testing or FEM . Damping ratios are es.timated hy modal tc stin g. Other parameters related 10 piezoelectric materials CMl he computed hy using c baracteristic con stants of r iCl.nclcctric materials and eigenfunetjol1s of ."arw:turl!s. T his analytical computation could he compli<.:atcd and hurd{:nsome depend in g o n s hapes and ho nding dirc<.:li o ns or pie/.(lc lcc(r il.'" maleri ~lls . This pa pel' proposes it simple ide ntificatio n me thod o f a Ihin pl
4521

designed to extra(.:( 1l1odal informatinn from sig nals of actuators and sensors. Using a bandpass filler for each mode transforms the identification prohlem or a MIMO systc m inln lhut uf several M1SO system s which are co mposed of linear modal equations of 1l1otion of a plate. Thc LS method is appl ied to MISO systems to find the corresponding modes. This idcntili c~llitlll method does not require c harm.:tc ri slic uuw thc piezoelectric materials nur the plidC. Th is me thod cu n he also applicd without c o nsidering the shapes and bonding dire<.:lions of actuators or sensors. Tht~ proposed method is applied to an allc,lampcd plate with lWO pairs of bonded piezoelectric actuators and sensors. The. experimental res ult s match well wilh the ide nt ified mode l.

or

2. lDENT IFICA TION OF A THIN PLATE WITH SONDED PIEZOELECTRIC ACTllA TORS/SENSORS The modal C4l1ati(lOS of a thin platc excited by multiple picl.ockclric actuator ... arc exprcssed i.l,' foll o ws (T7,ou and Fu. 1994)

~I

~

S:.' (x,,,.\·, )

~ 111" S.:' (xI'Yr)·

« "Ill'. <1I'S' = <'

11

r) '

"01/' . ",

a"
1, 2. \ .. ·. (4 )

=

Lt/", , q,, (f).

111 ==

1.2.

(S)

,,=1

D;'

i~

~

== Ill;,

fL;, V Cy,,.< x.y Jduly .

S,:'(x"y,)' 1/

= 1, 2, 3, ..

Ihl:" dnl11aill of the I- th picl.odeclri c aCluator

~ c luat n r.

displa",~c1l1 ...~nL S

with the

mea sured by pic/lldectric ~ensors are o r S0 1lSorS as follows.

lIlo...la! sensit ivi ty

(1)=='1' y ' (r L.. " ',, ' ",' ,r '" )(IU) ,,' . !!I

= 1.2 .· .. ,ns.

Since this system is a M [MO systcm and the displaccmcnts caused hy all the modes arc transformed into sensor outputs, it is difficult LO rind <:Ill the modal pantmders of ({". b". C,,/ .,nd tI"", dife~ tly , To obtain these paramelers easil y. Ihe followi ng /:act is used, h plays an imponanl role 111 developing lhe itlen tifiouion Illl'thod used in thi s paper.

Fact: A thin plate has small damping ralio and the equations of motion of a thin plate can be expressed by decollpled m(xial equations. So. the fre411ency respo nse funclioll of Cl thin platc ha.... TCSOllan<.:e peaks at all moues.

(2 )

pi
-

(.'",

'::", (r)

tlnd 111" is the 1ll01l1c lll -voltage con .;tanl whi<.:h is dett!rmined hy pic/.oeil'cLric pl\)pC'ni~s and clastic characteristics or a

~'",

@2':;"w,; , b" ~ W,~ ,

I

JJ. q,,, ( x.y) V : M,i.Ly)dxdy

cxprcs ~eu

Cl"

Without loss of gent!n.tlity, il is assumed that thert'! are two pie7.oelectric actualurs and two sensors in the plate. (Extension to an arhitrary higher form is straight forward_) Then. equations (3) can he 1'~\Vrilten as follows,

{fLcIJ,,(.\'.1') \7 ' M/I.,.,) d.\'d\'} " , (t),

rollowing form hy Grccn's lheorelll,

Till'

~lS

natural frequencies. damping nltio5 and modal se nsitivities are the objects to be found . For convenience. the unknown parameters to be identiried arc clefined (lS follow s:

( I)

(11;10.1101'. V : is the l.<.Iplucian opcralOr i.llld Mlx,y) is Ihe hcncl ing moment induced hy th c I-I h ac tuator. The integration term ill equation (I) call he <.:hunged into the

ih

In the syste m idcnlitical ion. the modal paramet.e rs such

/ =1

where f) is the rx, yJ domain or a plate. q,,(t) is the n-Lh n.alural coordinate. w" is rhc /I-th natural frequency. So, is lhe I/ -th damping rati o. 1'"CI .y) i ~ the 11-th c igcnfullction. Ita is the numhcr o r "'.:lIIaW ....... uP) is the ,'oltagc input of the

where

modal sensiti vity of Ih e m-Ih sens.or.

q" U}+

ii (J) + 2(,W"li .. (t ) + ll1,~ 1.1" (T ; " !

where Z1I/(1) is the output vo ltage of the m-th piezoelectric sensor. D;:. is the domain of the sensor, liS is the number of sensors and \'" is the voltage-displacement constant which is dClermineu hy the pie7.oe lectric properties and the clastic c haraClerislics or the plat e, S:(.~m 'Y"') denote~ the fI - lh

This Facl enabk-s each mode \,1' Ihe plate 10 be ide ntified individually. hy using Lht! bafldp~l sS filter des igned for Ihe corresponding [IllKle. By ~x<.: itin g the plate with random inputs, the frequency respon ses of sensor ourpu[ s can be computed. It is not difficult 10 design a filter whit:h has a profX:l' orde r and hundw'idth for each mode of the plate. Of eour~e , the cenler of the handwidth in the filter is sel to the peak freque ncy (0 be it.JcnliflcJ. Therefore. the s ignal s liltcrcd rrom (he inpuls and OUlputs have info nnmio n o n the mode of inlercsl .md little information 0 11 other modes. Ignori ng other modes in th~ lihered signals and deleting other natural coordinates, eq uations (4) and (5) for each mode can he approximated in thc following form . The approximated equation is v~llid only for the mod e selectcd by the bundpass fillel'.

4522

ij(f)+atiU)

t

/Jq(tJ=<'1 U/(f)+(.'~ U:(t).

:i = d, 1/(1), ,! III = d, qUI. (1)

where U{(/). lIi(t), s i g nal ~

b.

1"1'

of

cc'

l.I 1 It).

rl l

(13)

(7)

z' 1k + 21 + 0 zf l k + 11 + b / [k I = C. l',[k] + c: F, lkl.

(R)

v,lkl ~fI,'lkl+211,'lk -11+1I,'[k-21,

( 14)

u; [k - 2J

( 15)

(6)

v,1k 1~ 11; 1k I + 2 1/; 1k - 11 + =:,.'(1) and : ; (1) arc the fillered

" : (t),

: 1 (t)

and ~ (I l. The parameters

ll,

and cI~ for ~a('h modI.' in equati ons (6), (7) and

(S I arc Ihe ohjccts tLl hl' found.

First, the signal having more power is se lected between the

where represents Jisnctc-timc p(lrameters correspondin g to [he contillllous-time ones. For a suffi c iently small sampling interval. the dis cr~te · lime mndel deserihcs the continuous time system well. The re lari ons of parameters hctwct: n the continuous-time and the discrctl.! -lirne are computed )0 he as follows.

filtered outputs :. / (tl ;mll ::; ([I hy comparing the squared su m of the s
,

4(h-l)

a=

no-b-I)

loss of genen.IiLY. hen.: :.t fter. il i:-. as,",umed lhal Zj (I) ha~

more p()\vcr than :; (t). Then. d l is set t(1 onc. From equations (7) and (H), d c ca n he ohtained hy minimizing the tlll~ squared error sum following cost i'unction

T'(ii -" -I) (I + abs(p») 16 r~

,

I'

T(o-b-I)

16 ~I

c,

or

mill

4(o+b-l)

,b =

I I:; Ik 1- 1>=,' Ik 11'"

c. = -----,-,- - : - - " - - - r(ll - h -I ) ( I +abs(p»

(9)

(/1

( -I

whe rc : / Ik ) and :~ Ik, me the sampled signals of

z;' (1)

= I,

t1~

='

p.

where T is a sampling time interval. Equati on (13) is a

a nd ::.; (r). I' is the sC'-1li ng constant and K is the number of

linear equation with respect 10 ,:I 'kj.

S<:l l1lples. The algorithm ror minimizing equation (9) is well·

cost function is set to be the following squared error sum.

I-',rkl

and I';![k]. The

known as Ihe LS method. Then. d~ is set to p and the rcl
I

(10)

:: ( , ) = 1' ::1 It) .

To (:valuat(: olhl.'r parameters (/, h.

1" 1 and C 2 more
the filtered outputs

: . i (f I and

~:; (t).

., , @::I (I)+sgn(p).::~(t),

:: (I )

where

sgn(·)

the

is

~ign

function . Thi s

(I I)

makes the

magnitude uf :' (I) bigger than that of ~.t (t). With eq uations 0). (X) and (11). equation (6) 1.:3n he expressed only with

lIi (t l. If.; (f) ~md

:: I

l/ )

+ (I

:- I

(I)

:: I! t)

<:IS

By using the LS method 10 equation (10), the parameters Zi. 1), r l and Cl corresponding to eaeh mode cun he obt<:lined. The continuoll~-time parameters of each mode arc compulcd from the di screle- timc ones. The MIMO model of the continuous-Lime in ilK' state space form can he obtai ned from the conlinuous- limc parameters of e ach mod\.! by using equations (4) and (5). When two pairs of actuators and scnsors are hOlldcd on the plate ~lnd the numher of modes under consideration is n, the model in the state space is represented as

a MISO system. i.e_.

wl/I = A w (t) + B U(l l .

+ h : -. . (I)

= ", (I + ahsl/') l{ c, III It) + (", II{ (I))

z(1) =

( 12)

where ahs(p) is the ahsolutc valul' o/" p.l.n ord~r to apply the I.S method. the c.quation (12) is transrormed into the discrete-time equation . TlISliIl's rule is adopted because of" ils aCl: uracy and :-.impli(.' ity. Then I.!quallol1 (12) hccomes

where

w(t)

= [lJ1(r)

( IR)

ql(l) (/ ,1 f) £j)(t)

u(t ) = [11,(1) u,(llf.z(t)= [, A. B, C Cl rc

4523

( 17)

C w(l) .

(t)

,,(n)'.

£111 (1)

{ill(r)f.

The matrices

0 A=di agllnal(A 1• A I.··· . Ai" ., A il )' A i ~ [ -hi

B [0 =

()

c~

I'll

0

("

CI ~

IJ

( ' "

° ()

[ill,

[)

cl!

0

d ill

di •

D

d"

()

1/".:

'''J

-~J

cqutsit io n

i = 1, 2" ,

('. rJ

Actuator ,"-

) Actu"tcr

Amps

DSP

Sensor

LOIV-lew'l Voltage Amps.

80ard

( ,, '

O}

Higll-Ievel Voltage

Dat a

lamped Plate

Sensor

, 1/ •

IJ

3, e:XPERIMENTAl SYSTEM SET· UP The system for the expe .. ime nt is composed of
Fig. 2 Blod cii ngram of th e experimental sy stem . For data a..:quisiti o n o f the ac(uators and the se nsors, a D S P

board is implemented with " TMS 32UC 30 DSP of Texas Instrument rnc .. two ND con ve rle rs and o ne D/A c o n verle r a ll of w h ich ha v l.~

12-bir resolut io n. Two ho ards arc

supplemented. Onc has two low-level voltage amplifiers for .hc sensors and a l- hy-2 demuhiplexer. whi ch is connct: tcd to thc D/A convert!!;r. Tht!" other ha"> two high- level vohage ampliti ers (0 drivt! Ihe two piezoelcc lrit: actu ators. The bl ock diagram o f the experime ntal s.ystclTl is s ho w n in Fig .

2. -nle o u tputs ':': J(rl. z) t) o f ' he (wO !;cnsors are sampled Ihrough AID converters direc tly. The s ignals for the two al: tu.ato rs arc g enerated ind ependently. The in puts " 1(1) and thc

u;!(t) arc

flutpU(S of Ih ~

high-level

lwo

voltage

amplifiers.

4, EXPERIMENTAL RE SU LTS AND DISCU SSIONS

In thi s ex perime nt. the sa mpli ng freque ncy is se t 10 4 .R67

or

kHz and tile num hcr samples for th e experiment is te n tho usand. Bandlimitcd noise for each actuator is generated by filterin g
nm

,

A _- [

-1 .~7(IX}

11 - I.X6 19 x 11)'

- ~. () 4()R

A

Fig. I 1\11 cl amped pl .. uc ",·ilh l w p (;O- IOCi.lICd ac tuators and sensors .

-

~ -

[ "

- U261 x IO"

11 = [0 ()

4524

.

A~ A,

-..\ . ~on}

A

-:! .~ I4(J [-1.72 77 w -7.~:! I)} [-:1 .26J: x 10" -t).~997 J [

(J

= --6. I X40 x tU' ()

=

.=

x

- 288.66

(I

553.01

(,

- 2250.h

· 2 ~O.9~

(I

- 123lJ.!

(I

7lJ l.I6

r

o

-992.6:; \)

-2309.5 ()

27]4)\]'.

o

-610.57 ()

. 1.71.88

-46160

C . :;: [12:81 :: I

-.292S4 0

0 -.35961

o

.59l)07 0

()

[)

-.2108.'

- ....(...l.944

error ratio for each mode:

,

0

~I-I lkl-;I lkll' L..J

()

,,

0\

'"Itill

'"",,,,,

m

The natural frC4UL'IlCics and the damping ratios are found from A matrix by simple calculation and are summarized in Table I. In order to confirm the parameters obtained from the cX[1crimcnl. the FEM and the modal testing are also tried.

Nalural rn:quen<..:y (Hzl

6X.67

6H.91

1~5.16

125.2X 155.56

2 3

155.1) I

4

2()Y.20

207.75

5 6

lIS.OX

21 ? ..1X

2X7.51

2X-l.:;X

= 1.2,"',6,

K

error ratio for total modes: -".,."L'. . K, - - - - - -

FEM 6~.03

123.15 153.1.l 204.74 212.02 2H47

Damping ratio (%) Proposed method I).ll ~ 0.147 11.156 0.30 I D.lo3 D.268

I

where z;:,,,lk] is the signal ohtained from the m-lh output with the handpass filter for the l1-th mode and "A" represents an output from the Illodel. Table 2 shows the error ratios hetween the actual outputs and the model outputs. Although the parameters of eaeh mode are obtained from thc filtered signal, the error ratios for each mode arc not large. This indicates that the filtered signals have sufficienl information on the mode to he identified and hule information on other modes. Moreover. the ratios for total modes arc not much diJTerenl from those for each mode. This is not only hecause the resonancc peaks of modcs grc
Tahle :2 Error ratios betwyen the actual Qutputs and the modd outputs

Mode

Errnr ratios Error ratios for ;'l(f)

Error ratios for ':1(t)

3 4 5 6

9.999IxIO· 1.1537x1O' 2.0689x 10' 2.8123x I 0' 9.5559xIO' 2.5207x I ()'

1.2773x I 0' 1.4726xIO' 4.5471 x I ()' 1.0026x III ' 1.8539x 10' 9.0333xl()'

Tolal

I.I085xIO'

2.6020x I 0"

Modal testing 0.340 0.237 0.291 0.357 0.220 0.406

111=1.2.

~Iz"'[kll'

TahlL' 1 '\Jatural freQuencies andJlamping ratios

Modal tesulIg

n

~k",[kl- ':",[kJI'

(=

To apply the FEM, ANSYS is used and each element is considered as an eight-nodes structural shell. The results of FEM without the honding clleels of piezoelectric malerials arc sUIllmari7ed in Table 1. hut the damping ralios cannot he found 1'1'0111 lhe FEM. The results or modal testings by using lhc Half-pow(.'f approach arc also summarized in Table I. With the FEM and modal Lest, it is difficult and complicated to (;ompulc the Band C matrices which are related to piezoelectric l1latcri~lIs such as modal sensitivities and characLCristi( cnnstan(~ hccausc they depend on shapes and honding detections in this experiment. Tahle I shows tbat the !l:ltural frequencies ohtained hy the proposed mdhoJ are similar to those obtained hy other methods. The damping ratios ohlailll:u hy the proposed method arc slightly dillerent fronl those ohtained hy the modal testing. These diffcrcm.Ts could he GlUsc,d hy the following reason. Thc accelerometer is nol necessary in the proposed method while the mass and honding stillness nr the accelerometer and its cahle make the damping ratios increase in the modal testing f(n the plate. Though Band C matrices depend on the characteristics rclawd to the piezoelectric materials, it is easy to ohtain these Illatrices hy the pmposed method. The outputs of the idcnti ficd 11ltldd J.re <..:llJllputed with input signals 1J1(f) and 11 2(f). The error ralios hetween the actual outputs and the model output--. arc computed hy the following method:

Mode Proposed method

= 1,2.

[)

[)

I

2

4525

>3

- - - - _. .

.•

-

,.

lh ~

Fig. 3 Freque ncy respo nses o f la l <,(I). ( b) ~ ! (t)

..

,.

....

"

(h)

( a)

I

- _ . _. -

aClU al outpUL"i.

(Th e numocr:-; denote the l'orrcs po ndi ng modes)

•

handpass lilters arc designed according to the lower freq uency modes to be ident ifi ed . With thc filtl~rs a nd the charac teri st ics of the th in plate, the MIIv1 0 syste m is tra ns formed into severa l MI SO sys tems. whi ch are eas ier to solve. T he LS met hod is ilpplied to obtain the paramete rs of each mode. The ex perime nla l results. o f the all·cblll pcd plate show tha( the proposed method is prac tica l and powerful in findin g the paramc t~ rs:. The proposed method has severa l ad vantages. I ) 8 y us ing the bandpass fi hers . modes of interest (.;an be exp li l· itl y selected. 2) It does not need the l~ haractl.!risti c data o f pi ezoelectric materials ami c ige n· functinns nf the plate. ~) It can he uscd regardl ess of th e num ber of ac1uators a nd sr.: nsors, their locati uns, thei r bondi ng direc tio ns and the ir ~ hapes. 4 ) It can be easil y applied to di strihul cd struc ture, whe re damping ralios are sma ll. Acti ve vi hmti on cOnl rol o f the thin plate based o n this 111C1dcl will he prese nted l a h ~r.

I

I, :

I i

I:~~.

REFERE:-ICES

,

(h)

(a)

Fig. 4 Frequency responses o f lhl,." m odd outputs. ( a ) ~ , ( f) . (h) c., (I)

i

I '

::

a

,.

,· Ii

~

,

,. Ill·,

1101·'

(h i

( a)

Fig. 5 The ac tu all1u tpulS and the l11 0del out puts. Cal c. , (I). ( h I :, (1)

5. CONC LUSION

Bail"y, T. and lE Huhhard ( 1985). Distributed Piezoc let'tric- Po lymcr Ac ti \c Vibratio n Contro l of a Cantilevcred beam . AIAA j (lll mal oj GuidOlw e Gnd COl/tro!. VoL 8. No. 5. pp. 605-6 1 L Baz, A. and S . Poh ( 19RR). Perforlllilncc or a n Adive C ontrol System with Pici'.oe lcctric Actuators. JouI'I/al (~f Sound (11 /(1 Vibration, Vn!. 126(2), pp. 327-34 3. CI ark , RL. M. R. Flclll llling 'lIld C. R. Fuller ( 199:; ). Piezoe lectric AC lualOrs fo r J) jsl rihutcd Vi hrati on EX:('ita ti on o f Thin Plates: A Compari son between T heory and E xperime nt AS·WE. Journal (~rVib r01iol/ and Acoltstics. Vol. 115, pp. ::U2-.B9. Dimi triad is. EX. C.R. Fuller and C.A. Roger> ( 199 1). PicLoc lectric Actuators for Di strihutcd Vibratio n Excitation of Thin Plates. ASME. Journa l (~l Vibrlllioll mu! A ('ousf;CS, Vo l. 113, pp. 100- 107. FaIangas, ET, l .A. Dworak and S. Koshigoe ( 1'194). Conl ro lling Pla lc Vibr" ti on:-. Using Piezoe lec tric AC luators. lEE£. COlJ/ro! System, Vu!. 14(4 ). pp. 34-4 1. Tzou. H.S. a nd H.Q . Fu ( 1994 J. A Study o f Scgmc nl ation Distrihuted Piezoe le(; tril.: Sen sor and Ac tuators Pal1 I : T heoretical Ana lys is. jou rl/ol of Sound ond Vibratio ll. V(l!. 172, No. 2, pp. 247 -259. Willi ams. T. and IN . J ua ng <" 1 ~92). Scnsiti vil y the Tr.ms:miss ion Zeros o f FleX ibl e S pace Struc llIres. journal oj Gllidauce. COl/lrol alld n YllolJli('s, Vol. 15. No. 2. pp. 368-375.

or

or

Thi s paper propOSI.!S ,I simple and powe rful identification method for a thi ll pl;ltt: where multipl e ac tuators and scn~ors arc hon ded . Th e p ro po sed me1hod is ha'\ed on the j ~I L'1 that a th in pl.uc h a~ small d amping ratios. T he

4526