Cracks detection using active modal damping and piezoelectric components

(1)

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Cracks detection using active modal damping and

piezoelectric components

Baptiste Chomette, Amâncio Fernandes, Jean-Jacques Sinou

To cite this version:

(2)

ele tri omponents Authors: B.Chomette

a,b,∗

,A.Fernandes

a,b

,J-J. Sinou

c

a

CNRS,UMR 7190, Institutd'Alembert, F-75005Paris, Fran e

b

UPMC Université Paris 06,UMR7190, Institutd'Alembert, F-75005Paris,Fran e

c

Laboratoirede TribologieetDynamique desSystèmesUMR-CNRS5513,É ole Centrale de Lyon,

F-69134E ully,Fran e

Corresponding author: B.Chomette (baptiste. hometteupm .fr)

Abstra t

Thedynami sofasystemanditssafety anbe onsiderablyae tedbythepresen eof ra ks.

Healthmonitoringstrategiesattra tsoagreatdeal ofinterestfromindustry. Cra ksdete tion

methodsbasedonmodalparametersvariationareparti ularlye ientinthe aseoflarge ra ks

butare di ultto implementinthe aseof small ra ksdue tomeasurementdi ultiesin the

aseofsmallparametersvariation. Thereforethepresentstudyproposesanewmethodtodete t

small ra ks basedon a tive modal dampingand piezoele tri omponents. This method uses

thea tivedampingvariation identi ated withthe Rational Fra tion Polynomial algorithm as

anindi atorof ra ksdete tion. Thee ien yoftheproposedmethodisdemonstratedthrough

numeri alsimulations orrespondingtodierent ra kdepthandlo ationsinthe aseofanite

(3)

Cra ks an onsiderablymodifythedynami s ofa systemand ae t systemsafetyinseveral

indus-trialappli ations. Cra kmodeling an be therststep of ahealth monitoring strategy. Cra ks an

bemodelledusingseveral approa hes[14 ℄,forexample redu tioninelement stinessorpinnedjoint

at ra k lo ation [19 , 26℄. Rigid nite element method an be used to model ra ks with a set of

spring-damping elements of variablestiness as shown by Kulesza and Sawi ki [20℄. The approa h

anberstlinearifthe ra kissupposedtobeopened[12℄andse ondnonlinearifthe ra k

breath-ingistakenintoa ount[2 ,9,24℄. Christides andBarr[10℄shownthat ra ks anbemodelled using

exponential stinessredu tion inthe ase ofEuler Bernoulli beams. This modeltakesinto a ount

the se ond moment of area redu tion indu ed by the ra k. Sinha et al [31 ℄ proposed a simplied

model for the lo ation of ra ks based on a triangular variation of the stiness in beam stru tures

usingmeasuredvibrationdata. Thisapproa hbasedonalo alformulation oftheexibilitypermits

parti ularlytoin ludethe ra kmodelinaniteelementmodelassumingthatthestinessredu tion

allfall within asingleelement.

Onewaytodete t ra ksonstru turesistoemploymodaltestinginwhi h hangesinmodal

param-eters su hasvariations infrequen iesand mode shapesareusedto dete t damage. A reviewbased

onthe dete tion ofstru tural damage through hanges infrequen ieswasdis ussed bySalawu[28 ℄.

Moreover, Dilena and Morassi [11 ℄ proposed damage identi ation based on hanges in the nodes

of mode shapes. Theydemonstrated thatappropriate use of resonan es and antiresonan es an be

usedtoavoidthenon-uniquenessofdamagelo ationforsymmetri albeams. FaverjonandSinou[12 ℄

demonstrated that itis possibleto dete t the numberof ra ks in abeam and estimateboth ra k

positions and sizes despitethe presen e of noise levels. Cra ks an also be dete ted using observer

[30 ℄ or using non linearphenomena in the ase of ra ked beams submitted to for ed vibrations [1℄

or inthe aseof ra ked rotors[29 ℄.

Piezoele tri omponents are usually used in health monitoring systems oupling to a tive ontrol

strategies su h as a tive ollo ated ontrol [23 ℄, a tive modal ontrol [7, 15℄ or passive strategies

[4 , 17℄. These strategies an be used in omplex appli ations su h as urved panels in a ar body

asshownbyHurlebaus[18℄,using evaluated modalparameters [32℄ or inthe aseof 3Dme hani al

stru tures [16℄. There are more various nite element approa hes to model piezoele tri

(4)

aone-dimensionalbeamelement basedonEuler-Bernoulli theoryusing abilineardistributionofthe

ele tri eld potential [13℄. This model permits parti ularly to link one dimensional piezoele tri

and ra kstheoryinorder to develophealth monitoring strategies.

Inthe aseof ontrolledstru turesorsmart-stru turesin ludinga tuators, sensorsand ontrolloop,

ra ks an onsiderably modify the dynami s of the ontrolled system. Chomette et al. [6 ℄ shown

thata tive modal ontrol basedon Linear Quadrati Algorithm is highlysensitive to the variation

ofboundary onditionsand thus to stru ture modi ation. Theydemonstrated thatdieren es

be-tween the ra ked stru ture and the modal model used in the ontroller based on the un ra ked

stru turelead to ontrol performan e de rease. Chomette andSinou [8℄investigated thepossibility

ofdete ting transversal ra ksin ontrolled trussstru turesusing variationin theamplitude ofthe

ontrolledsystemfrequen yresponsefun tionandshownthatthe ontrol performan evariation an

be anindi ator of transversal ra k dete tion.

Thereforethepresentstudyproposesanewmethodtodete tsmall ra ksbasedonpiezoele tri

om-ponents anda tive modaldamping. Firstly the modelof a ra ked Euler-Bernoulli beam in luding

piezoele tri elements isproposed. The mainpurposeof the ontrol systemishereto dete t ra ks.

Consequently, the ontrol algorithm is se ondly designed inorder to see small ra ks. The novelty

of the proposedstrategy is to applied a tive ontrol to monitor thestru ture behavior. Finally the

e ien y of theproposed methodis demonstrated through numeri al simulations orrespondingto

dierent ra k depths and lo ations in the ase of the nite element model of a lamped- lamped

beam in ludingfour piezoele tri transdu ers.

2 Finite element modeling

Inthis se tionthe niteelement modeling of amulti- ra ked Euler Bernoullibeamin luding

piezo-ele tri omponents is proposed. Firstly the ra k modeling extended on several elements and the

piezoele tri modeling aredetailed. Se ondlythenite element modeling ofthe ompletestru ture

(5)

Christides and Barr [10 ℄ shown that the ee t of a ra k ina ontinuous re tangular beam an be

onsideredusinganexponential variationoftheexuralstiness

EI

where

E

istheYoung modulus

and

I

the se ond moment of area. This stiness redu tion an be approximated by a triangular

redu tionfor the ra ked element asexplained bySinhaetal. [31℄inorderto in lude theredu tion

inaniteelementmodel. Ifthestinessredu tionextendsovermorethanoneelementtheproposed

approa h have to beextended on several elementsas shownin Fig. 1. Thus thestiness redu tion

anbe written inlo al oordinates forthe

p

th

element oflength

ℓ

e

EI

±

(x

e

, p) = EI

0 ±

E (I

0 − I

c

)

ℓ

c

x

e

+ (p − 1) ℓ

e

− x

±

c

,

(1)

where

E

,

b

,

e

p

and

I

0

aretheYoungmodulus, theweigth,thedepthandthese ond moment ofarea

of theun ra ked beam respe tively.

e

c

and

I

c

arethe ra k depth and these ond moment of area

of the ra ked beam respe tively.

+

and

+

_c

− x

−

_c

= 2 ℓ

c

,

(2)

where

ℓ

c

= 1.5 e

p

isthe lengthoftheae tedstinessbasedonexperimentaldata[14 ℄. Thestiness

matri es are obtained using the standard integration based on the variation in exural rigidity

EI

±

(x

e

, p)

K

±

e,crack

=

Z

ℓ

e

0 EI

±

(x

e

, p)

N

′′

e

T

(x

e

)

N

′′

e

(x

e

) dx

e

(3)

where the shape fun tions N

e

are those for a standard Euler-Bernoulli element beam. Finally, the

element stinessmatrix in luding a ra k anbe writtenusing the

(6)

K

c

±

e

(p) = ±

E (I

0 − I

c

)

ℓ

c







6 (2X

c

+ℓ

e

)

ℓ

e

3 2 (3X

c

+2 ℓ

e

)

ℓ

e

2 −

6 (2X

c

+ℓ

e

)

ℓ

e

3 2 (3X

c

+ℓ

e

)

ℓ

e

2 2 (3X

c

+2 ℓ

e

)

ℓ

e

2 (4X

c

+3 ℓ

e

)

ℓ

e

−

2 (3X

c

+2 ℓ

e

)

ℓ

e

2 (2X

c

+ℓ

e

)

ℓ

e

−

6 (2X

c

+ℓ

e

)

ℓ

e

3 −

2 (3X

c

+2 ℓ

e

)

ℓ

e

2 6 (2X

c

+ℓ

e

)

ℓ

e

3 −

2 (3X

c

+ℓ

e

)

ℓ

e

2 2 (3X

c

+ℓ

e

)

ℓ

e

2 (2X

c

+ℓ

e

)

ℓ

e

−

2 (3X

c

+ℓ

e

)

ℓ

e

2 (4X

c

+ℓ

e

)

ℓ

e







(5) where

X

c

= x

±

c

− p ℓ

e

and

−

K

c

±

e

(p)

isthestinessredu tionindu edbythe ra konthe

p

th

element

andK

e

the element stiness matrixof anun ra ked element.

Usingthismodeling, ra klo ationisindependantofthemeshandthemeshsize anbemodiedto

takeintoa ountotherphysi al ouplinglikeele trome hani al ouplingusingpiezoele tri elements.

Due to the dete tion method based on linear ontrol, the ra k is supposed to be open and the

breathing ra kis not taken into a ount.

[Fig. 1 about here.℄

2.2 Piezoele tri nite element modeling

φu

andK

φφ

arethemass,theshort- ir uitstiness,theele trome hani al

ouplingandthediele tri matrixrespe tively. u,

φ

,FandQaretheme hani aldispla ementve tor

and ele tri al potential, the for e and the ele tri al harge ve tor respe tively. The generalized

ele trome hani al oupling oe ient

k

ℓ

is dened [17 ℄for the

ℓ

th

mode as

k

_ℓ

2 =

ω

2 co

ℓ

− ω

2 cc

ℓ

ω

2 co

ℓ

,

(7) where

ω

co

ℓ

and

ω

cc

ℓ

arethe

ℓ

th

pulsationswithopenandshort ir uitedele trodesrespe tively. The

generalized ele trome hani al oupling oe ient isusedto hara terize piezoele tri a tuators and

(7)

The nite element model of the omplete beam in luding multi- ra ks and piezoele tri elements

permits to take into a ount ra ksinthegeneralized ele trome hani al oupling oe ient. It an

be written usingequations (4)and (6)











M

uu

u

¨

+ (

K

cc

uu

−

K

crack

)

u

+ K

uφ

φ

=

F K

φu

u

+

K

φφ

φ

=

Q

,

(8)

where K

crack

is the ra k stiness matrix after assembling the dierent beam elements of stiness

K

±

e,crack

.

3 Modal Control

Modal ontrol is usually applied to redu e stru tural vibration intargeting the ontrol energy only

onthe modesof interestand permits to redu e thea tuators and sensorsnumber [15℄. The ontrol

algorithm is designed here inorder to dete t small ra ks. The Linear Quadrati Gaussian (LQG)

algorithm is hosen inthis appli ation to target the ontrol only on the mode of interest with the

weighting matrix inorderto seethe modalinuen e ofthe ra ks. The LQG algorithm isbased on

alinearmodelofthe stru ture.

3.1 Linear system

Thelinear ontrol [23 ℄ isbased ona linearmodelwhi h an be des ribed inthestate formby











˙

x

=

Ax

+

Bu

∗

+

W

w

y

=

Cx

,

u

∗

= −

Gx

ˆ

,

(9)

whereAisthedynami alsystemmatrix,BandCarethea tuator andsensormatri esrespe tively

and W is the ex itation matrix. x is the state ve tor in modal oordinates. y

=

t

_(y

1 y

2 )

is the

outputve tordetailedhereinthe aseoftwo sensors. u

∗

=

t

_(u

∗

1 u

∗

2 )

istheoptimal ontrol solution

oftheLinearQuadrati Regulatorproblem, detailedhereinthe aseoftwoa tuatorsand al ulated

using the estimated state x

ˆ

and the gain matrix G.

w

is the ex itation applied on the beam and

hoosentobeunitaryinthisstudy. Inpra ti alappli ations,awhitenoiseex itation anbeapplied

(8)

A

=







0

n,n

Id

n,n

−diag ω

ℓ

2 −diag (2ξ

ℓ

ω

ℓ

)







,

B

=







0

n,1

0

n,1

Π

a

1 Π

a

2 





,

W

=







0

n,1

Π

w







,

C

=







Π

c

1

0

1,n

Π

c

2

0

1,n







,

x

=







q

˙

q







,

(10)

with

ℓ = {1, n}

,whereqisthemodaldispla ement ve tor,

ω

ℓ

and

ξ

ℓ

arethenaturalfrequen iesand

damping respe tively.

Π

a

i

,

Π

ℓ

2 =

k

a

i

ℓ

2 C

a

i

p

(ω

cc

ℓ

)

2 1 − k

a

i

ℓ

2 ,

(11) where

a

indi ates a tuator.

k

is the ele trome hani al oupling oe ient and

C

p

thepiezoele tri

omponent apa ity. The piezoele tri omponent whose ele trome hani al oupling oe ient

k

is

al ulatedusingtheequation(7),ismodelledwithopenele trodesandalltheother omponentsare

modelledwithshort ir uited ele trodessothattheonlypiezoele tri ee tisdueto the onsidered

omponent.

3.2 Transfer fun tions in open and losed loop

Thetransferfun tion ofthe un ontrolled (open-loop)systemH

ol

an bewritten using equation(9)

for onesensorof response

y

1

as

H

ol

(jω) =

y

1 w

where

(

A

s

,

B

s

,

C

s

,

W

s

)

isthestatemodelthatrepresentsthestru tureinthenumeri alsimulations.

Frequen iesand damping variation indu ed by the ra ks are taken into a ount inthematrix A

s

.

Modeshapesandele trome hani al oupling oe ientvariationindu edbythe ra ksaretakeninto

a ountintheB

s

,C

s

andW

s

matri es. Thetransferfun tionofthesystem ontrolled( losed-loop)

H

cl

an be written for onesensor as

(9)

where

(

A

m

,

B

m

,

C

m

)

isthe state model usedbythe Luenberger observer [21℄ to evaluate thestate

ve tor. GandL arethe LinearQuadrati gain(LQ) andtheobserver gainof theLinearQuadrati

Gaussian algorithm (LQG)[3℄ respe tively.

3.3 Gain al ulation to dete t ra ks

In the ase of small ra ks the frequen y shift indu ed by the stiness redu tion annot be easily

measured. The proposedmethod onsistsinstudyingthedynami of the losed loopsystemandto

modifythe ontrol gainGinordertodete tthe ra ks. ThegainGisobtained usingthequadrati

riterion [23 ℄

J =

Z

∞

0

x

(t)

T

Qx

(t)dt

(14)

where Qis the lassi al weightingmatrix. It an bewritten for

n

modes

Q

=







Q

sup

0

n,n

0

n,n

Q

inf







.

(15)

The prin iple of the method is to design a little robust ontroller highly sensitive to thestru ture

modi ations. Therefore,the ontroller performan e must be high and the robustness low. Inthis

study, only the even modes are weighted using the Q matrix so that a frequen y shift and an

ele trome hani al oupling oe ient variation indu e a slip of the ontroller ee ts on the even

modes. In the ase of un ra ked stru tures only the weighted modes are highly a tively damped

using Q

sup

and Q

inf

and there is no ee t on theodd modes. Weighting oe ient are arbitrarily

hosen to target the ontrol only on the wanted modes. In this study, three ponderations (Q

1

, Q

2

and Q

3

) of the LQG ontroller are tested and the maximal weighting matrix Q

3

is dened so

that Q

sup

= diag (1 1e15 1 1e17 1)

and Q

inf

= diag (1 1 1 1 1)

. In the ase of ra ked stru tures,

thestru tural modi ations indu edbythe ra ksmodify the ontrollerdynami s and indu esome

a tive damping onthe oddmodeswhereasthedamping of theeven modesde reases.

=

20 mm

) presented in Fig. 2 in ludes two piezoele tri a tuators and two piezoele tri olo ated

(10)

foundintable 1.

[Table 1 about here.℄

Numeri alsimulationsofse tion4arebasedontheopen(without ontrol)and losed(with ontrol)

looptransfer fun tion between the ex itationand therst piezoele tri sensorofthe ra ked

stru -ture. Modalparameters of the un ra ked un ontrolled beamidentied using theRational Fra tion

Polynomial (RFP) algorithm [25℄ applied to the Frequen y Response Fun tion ofthenite element

modelaredetailed intable2.

[Table 2 about here.℄

3 = 17.6 mm

. In these ond

asetwo ra ksarelo atedat

x

c

1 = 6 mm

and

x

c

2 = 14 mm

. The ra kdepth

µ =

e

c

e

p

variesfrom

0

to

1

4

.

The ontrollooppresentedinFig. 3in ludesaLQG ontroller basedonaLuenbergerobserver.

Fre-quen yResponseFun tions(FRF)aremeasuredbetween theex itation

w

andtherstpiezoele tri

sensor. u

∗

is the optimal ontrol al ulated using the LQ gain G,

u

∗

1

and

u

∗

2

are the ontrol

om-posantsappliedonthea tuators1and2. yistheoutputve torwhere

y

1

and

y

2

arethe omposants

ofthesensors1and2. Themodel

ˆ

The study fo usses on the rst fourth modes. Three ponderations (Q

1

, Q

2

and Q

3

) of the LQG

ontroller aretested inthetwo ases inorder to showtheproposed method e ien y,withQ

3 >>

Q

2 >>

Q

1

.

4.1 First ase: three ra ks

In this rst ase three ra ks are lo ated on

x

c

1 = 2.4 mm

,

x

c

2 = 1 mm

and

x

c

3 = 17.6 mm

. A

(11)

is ae ted by the rst and fourth ra k. This rst study permits to know if the ele trome hani al

oupling oe ient of the a tuators and onsequently theB

s

and C

s

matri es of the ontroller are

ae tedornot. The oupling oe ient is al ulated forseveral lo ationsofone piezoele tri pat h

onthe beamfrom

x

as

= 0 mm

to

x

as

= 180 mm

,where

x

as

denes theleft orner oordinate ofthe

pat h, and for a ra k depth from

µ = 0

to

µ = 1/4

. The evolution of the oupling oe ient is

presented inFig. 4(a)to 4(d). In thesegures, the lo ation ofthepiezoele tri omponent hanges

along the beam to look into all the se tions of the stru ture where the stiness is ae ted by the

ra ks. The ex itation appli ation point is onstant. In the data shown in Fig. 5 to 8, the

piezo-ele tri omponents and the for e ve tor lo ation are onstants. There is a large variation of the

oupling oe ient for the rst and third mode when the pat h is above the se ond ra k. There

isa large variationof the oupling oe ient for thefourthmode when thepat h isabove the rst

or third ra k. There is a very small modi ation for the se ond mode. These modi ations will

be taken into a ount in the losed loop system dynami s al ulation parti ularly in the B

s

and

C

6=

C

m

). These dieren es an modify the losed loop

stru turedynami s and onsequently indu e some performan e variations.

Frequen y response fun tions of the un ontrolled and ontrolled beam for three ponderations Q

1

, Q

2

and Q

3

are shown in Fig. 5(a), 5(b) and 5( ) for the rst mode, in Fig. 5(d), 5(e) and 5(f)

for the se ond mode, in Fig. 5(g), 5(h) and 5(i) for the third mode and in Fig. 5(j), 5(k) and

5(l) for thefourthmode. It an beobserved thatthepresen e of ra ks de reasethefrequen ies of

ea h mode. Moreover, in reasing the ra k size de rease modal frequen ies. For the un ontrolled

beam, the frequen y variation indu ed by the ra k depth variation from

µ = 1/16

to

µ = 1/4

,

usually measured to dete t ra ks, is about -0.8 to -3.8% for the rst mode, 0 to -0.2% for the

se ond mode, -0.5 to -2.3% for the third mode and -0.4 to -1.7% for the fourth mode. Howeover,

due to thesmall depthofea h ra ks,dete tion ofthe presen eof ra ks basedon theevolution of

naturalfrequen iesappearstobenote ient. Identiedfrequen yanddampingforthreeweighting

matri es are presented in Fig. 6(a), 6(b), 6( ) and 6(d) for the rst fourth modes. The modal

(12)

ontrolled beamusingtheQ

1

weighting matrixis about 18.6to 33.8% for therstmode and7.6to

12.6% for thethird mode,72 to 85%for the rstmodeand 51.1 to 65.9% for thethird mode using

the Q

2

weighting matrix, 93.7 to 96.5% for the rst mode and 81.0 to 91.9% for the third mode

usingtheQ

3

weighting matrix.

4.2 Se ond ase: two ra ks

In this se ond ase two ra ks are lo ated at

x

c

1 = 6 mm

and

x

c

2 = 14 mm

. Frequen y response

fun tionsoftheun ontrolled and ontrolled beamfor threeponderationsQ

1

,Q

2

and Q

3

areshown

in Fig. 7(a), 7(b) and 7( ) for the rst mode, in Fig. 7(d), 7(e) and 7(f) for the se ond mode,

in the Fig. 7(g), 7(h) and 7(i) for the third mode and in Fig. 7(j), 7(k) and 7(l) for the fourth

mode. There is the same de rease of the frequen ies indu ed by the ra ks depend on the ra k

depth. For the un ontrolled beam, the frequen y variation indu ed by the ra k depth variation

from

µ = 1/16

to

µ = 1/4

, usually measured to dete t ra ks, is about -0.1 to -0.5% for the rst

mode, -0.8 to -3.6% for the se ond mode, -0.3 to -1.4% for the third mode and -0.1 to -0.4% for

the fourth mode. Consequently, the evolution of natural frequen ies appears to be not e ient to

dete t ra ks. Identied frequen y and damping for three weighting matri esare presentedin Fig.

8(a), 8(b),8( ) and8(d) for the rstfourthmodes. The modalstru tural damping variationinless

than2%forthefour modes. Thedamping variationfor the ontrolled beamusingtheQ

1

weighting

matrix isabout 5.0to 13.8% for therst mode and5.2 to 13.5% for thethird mode, 28.9to 64.7%

fortherstmodeand38.6to64.3%forthethirdmodeusingtheQ

2

weightingmatrix,66.1to90.9%

for the rstmode and 68.7to 90.7% for thethirdmode using theQ

3

weightingmatrix.

The index variation is summarized in Fig. 9. The dete tion index based on frequen y shift is

al ulated using the formula

100 ×

f

cracked

− f

uncracked

f

cracked

and based on a tive modal damping using

100×

ξ

cracked

− ξ

uncracked

ξ

cracked

(13)

indu easmallfrequen yvariationthat annotbeexperimentallymeasuredandaverylightstru tural

damping variation. Using a tive damping,thedamping variation ishighly in reased proportionally

tothe ra kdepthandindu esalargeamplitudevariationofthe ontrolledbeamfrequen yresponse

fun tionthat anbeeasilymeasured. The ontrolisinitiallytargetedonthese ondandthirdmodes.

Themodalparametersvariation,frequen yshift,modaldampingandmodeshapesvariation, indu e

somedieren esbetweenthestatemodelusedbythe ontrollerandtherealstru ture. These hanges

modifythe ontrollerbehaviourandindu esomea tivedampingontherstandthirdmodesinitially

very slightly ontrolled and are here used as an indi ator of ra ks presen e. These results an be

physi allyinterpreted. Thestinessof the ra ked stru tureis lowerthan theun ra ked ones. The

piezoele tri omponentsarethereforemoree ientandtheele trome hani al oupling oe ientis

in reased. Thisvariation istakeninto a ount inthe B

s

and C

s

matri es. Moreover, thefrequen y

shift in luded in the matrix A

s

leads to disturb the modal ontrol in the mode targeting. And as

the ontrol gains are not updated, the ontroller dynami s is hanged. The proposed method is

well adapted to dete t small ra ksin reasing theweighting matri esand onsequently the ontrol

voltage.

5 Con lusion

In this paper a new method to dete t small ra ks based on piezoele tri omponents and a tive

modal damping is proposed. The originality of the proposed strategy is to exploit the fa t that

modal ontrol is little robust. The main purpose of the ontrol system is here to dete t ra ks.

Consequently, the ontrol algorithm is designed in order to dete t ra ks and the weighting matrix

is hosen only to this task. Some weighting matri es an be tested by swit hing the ontroller

between several values up to the ra k dete tion. As the robustness of the ontroller is inversely

proportional to the ontroller performan es, the weighting oe ientsmust be in rease inthe ase

of small ra ks. This method does not ae t here the stru ture stability be ause the system is

more and more damped. In other appli ations, the ra k dete tion ould indu e some vibration

ampli ation instate of damping but iftheweighting matri esare tested in reasingly, ra ks must

be dete ted beforea too large ampli ation. Therefore, it an be applied to monitor thestru ture

dynami s and to dete t ra ks instead of redu e vibration. The model of the un ra ked stru ture,

ontained intheLuenberger observerpermitsto monitorthe ra kedstru turedynami s. Themain

(14)

method is validated withnumeri al simulation on a lamped- lamped ra ked beamin luding four

piezoele tri transdu erslinkedbytheLQG ontrol loop. Thisapproa h ouldbeextendedtoplates

andmore omplexindustrialsystems. Weightingmatrix oe ients ouldbeoptimizetodete tsmall

ra ksusing aminimumofenergy. Moreover, the proposedmethod anbeeasily implementedwith

a tuatorsandsensors traditionallyusedto dete t ra ks withfrequen yshiftbyaddinga ontroller

andan identi ation algorithm adapted tohighly damped ontrolled systems.

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hara teristi soffor edresponse,Me hani alSystemsandSignalPro essing 31(2012),382-404.

[2℄ M.A. Al-Shudeifat, E.A. But her, New breathingfun tions for thetransverse breathing ra k

ofthe ra kedrotorsystem: Approa handsub riti alharmoni analysis, JournalofSoundand

Vibration 31(2011),526-544.

[3℄ M. Athans, The role and use of the sto hasti linear-quadrati -gaussian problem in ontrol

systemdesign, Automati Control IEEE Transa tionon 16(1971), 529-552.

[4℄ A. Badel, D. Guyomar, E. Lefeuvre, and C. Ri hard. Piezoele tri energy harvesting using

a syn hronized swit h te hnique, Journal of Intelligent Material and Stru tures 17 (2006),

831-839.

[5℄ A.Benjeddou,Advan esinpiezoele tri niteelementmodelingofadaptivestru turalelements:

a survey, Computersand Stru tures 76(2000), 347-363.

[6℄ B. Chomette, D. Remond, S. Chesne, and L. Gaudiller, Semi-adaptive modal ontrol of

on-boardele troni boardsusingidenti ationmethod, SmartMaterialand Stru tures 17(2008),

1-8.

[7℄ B.Chomette,S.Chesne,D.Remond,andL.Gaudiller,Damageredu tionofon-boardstru tures

usingpiezoele tri omponentsanda tivemodal ontrol-appli ationtoaprinted ir uitboard,

Me hani al Systemand Signal Pro essing 24(2010), 390-397.

[8℄ B. Chomette and J-J. Sinou, Cra k dete tion based on optimal ontrol, Journal of Vibration

(15)

239 (2001),57-67.

[10℄ S. Christides and A.D.S. Barr, One-dimensional theoryof ra ked Bernoulli-Euler beams,

In-ternational Journal of Me hani al S ien es 26(1984), 639-648.

[11℄ M. Dilena and A.Morassi, Identi ation of ra k lo ation invibrating beams from hanges in

node positions, Journal of Sound andVibration 255(2002),915-930.

[12℄ B.FaverjonandJ-J. Sinou, Identi ationofanopen ra kinabeambyusingaposteriorierror

estimator on the frequen y responsefun tions withnoisy measurements, European Journal of

Me hani s A-Solid 28(2009), 75-85.

[13℄ A.Fernandes andJ.Pouget. Analyti alandnumeri almodellingoflaminated ompositeswith

piezoele tri elements, Journal of Intelligent Material Systems and Stru tures 15(2004),

753-761.

[14℄ M.I. Friswell and J.E.T. Penny, Cra k modeling for stru tural health monitoring, Stru tural

Health Monitoring 1(2002), 139-148.

[15℄ L.GaudillerandJ.DerHagopian,A tive ontrolofexiblestru turesusingaminimumnumber

of omponents, Journalof Sound andVibration 193(1996), 713-741.

[16℄ Z. Gosiewski, A.P. Koszewnik, Fast prototyping for the a tive vibration damping system of

me hani al stru tures, Me hani al Systems andSignal Pro essing (2012), inpress.

[17℄ N.W.HagoodandA.vonFlotow, Dampingofstru tural vibrationswithpiezoele tri materials

and passive ele tri al networks, Journal of Sound andVibration 146(1991), 243-268.

[18℄ S. Hurlebaus, U. Stöbener, L. Gaul, Vibration redu tion of urved panels by a tive modal

ontrol, Computer andStru tures 86(2008), 251-257.

[19℄ F. Ismail,A.Ibrahim,and H.R.Martin, Identi ation offatigue ra ks fromvibration testing,

Journal of Soundand Vibration 140 (1990), 305-317.

[20℄ Z. Kulesza, J. Sawi ki, Rigid nite element model of a ra ked rotor, Journal of Sound and

Vibration 331(2012), 4415-4169.

[21℄ D. Luenberger, An introdu tion to observers, Automati Control, IEEE Transa tions on 16

(16)

Universitélibre deBruxelles, 2001.

[23℄ A. Preumont, Vibration Control of A tive Stru tures: An introdu tion, se ond ed., Springer,

2002.

[24℄ N.PugnoandC.Sura e, Evaluationofthenon-linear dynami responsetoharmoni ex itation

of abeamwithseveralbreathing ra ks, Journalof Sound andVibration 235 (2000), 749-762.

[25℄ H.Ri hardson,MarkandD.L.Formenti, Global urvettingfrequen yresponsemeasurements

usingtherationalfra tionpolynomialmethod,inPro eedingoftheInternationalModalAnalysis

Conferen e and Exhibit,1985, pp.390-397.

[26℄ P.F.Rizos and N.Aspragathos, Identi ation of ra k lo ation and magnitude ina antilever

beamfrom thevibration modes, Journal of Soundand Vibration 183 (1990), 381-388.

[27℄ P.SadilekandR.Zem ik, Frequen y responseanalysisofhybrid piezoele tri antileverbeam,

EngineeringMe hani s 17(2010):73-82.

[28℄ O.S. Salawu, Dete tion of stru tural damage through hanges in frequen y: a review,

Engi-neering Stru tures 19(1997), 718-723.

[29℄ J.T. Sawi ki, M.I. Friswell, Z. Kulesza, A. Wroblewski, J.D. Lekki, Dete ting ra ked rotors

using auxiliary harmoni ex itation, Journal of Soundand Vibration 330(2011), 1365-1381.

[30℄ J.T. Sawi ki, Z. Kulesza, Anew observerbasedmethod for rotor ra kdete tion, Advan es in

Vibration Engineering 11(2012),131-141.

[31℄ J.K Sinha, M.I Friswell, and S.Edwards, Simplied models for thelo ation of ra ks inbeam

stru turesusing measuredvibration data, Journalof Sound andVibration 251 (2002), 13-38.

[32℄ U.Stöbener,L.Gaul, A tivevibration ontrol ofa arbodybasedonexperimentallyevaluated

(17)

1 Triangularstiness redu tionextendedon severalelements . . . 17

2 Multi- ra ked ompositebeamin ase1 and 2 . . . 18

3 Controlledmulti- ra ked ompositebeam . . . 19

4 Ele trome hani al oupling oe ient for one piezoele tri pat h on the beam from

x

as

= 0 mm

to

x

as

= 180 mm

andfor a ra kdepthfrom

). . . 20

5 Un ontrolled(--)and ontrolled()FRFofthemulti- ra kedbeamintherst ase

,Q

2

and Q

3

. . . 21

6 Frequen yanddampingintherst aseforthreeweightingmatri es(

:un ontrolled,

:Q

1

,

:Q

2

,

:Q

3

) . . . 22

7 Un ontrolled (--) and ontrolled () FRF of the multi- ra ked beam inthese ond

ase for the rst fourth mode (

:

µ = 0

,

:

µ = 1/16

,

:

µ = 1/8

,

:

µ = 3/16

,

:

µ = 1/4

) forthree ponderationsQ

1

,Q

2

andQ

3

. . . 23

8 Frequen yanddampinginthese ond aseforthreeweightingmatri es(

:un ontrolled,

:Q

1

,

:Q

2

,

:Q

3

) . . . 24

9 Index variation in the rst and se ond ase for the rst and third mode, al ulated

usingfrequen y (

) ora tivedamping for the three ponderations (

:

Q

1

,

:

Q

2

,

:

(18)

(19)

(20)

(21)

0

40

80

120

160

200

1

2

3

4

5

6

7

8 Patch location [mm]

Coupling coefficient [%]

(a)Firstmode

0

40

80

120

160

200

1

2

3

4

5

6

7 Patch location [mm]

Coupling coefficient [%]

(b)Se ondmode

0

40

80

120

160

200

1

2

3

4

5

6

7

8 Patch location [mm]

Coupling coefficient [%]

( )Thirdmode

0

40

80

120

160

200

2

3

4

5

6

7 Patch location (mm]

Coupling coefficient [%]

(d)Fourthmode

Fig. 4: Ele trome hani al oupling oe ient for one piezoele tri pat h on the beam from

x

as

=

0 mm

to

x

as

= 180 mm

and for a ra k depth from

µ = 0

to

µ = 1/4

(

:

µ = 0

,

:

µ = 1/16

,

−100

Amplitude [dB]

Frequency [Hz]

(d)Se ondmode,Q

1 1120

1130

1140

1150

1160

1170

−140

−130

−120

−110

−100

Amplitude [dB]

Frequency [Hz]

(e)Se ondmode,Q

2 1120

1130

1140

1150

1160

1170

−140

−130

−120

−110

−100

Amplitude [dB]

Frequency [Hz]

(f) Se ondmode,Q

3 2100

2150

2200

2250

2300

−160

−150

−140

−130

−120

−110

−100

Amplitude [dB]

Frequency [Hz]

(g) Thirdmode,Q

1 2100

2150

2200

2250

2300

−160

−150

−140

−130

−120

−110

−100

Amplitude [dB]

Frequency [Hz]

(h)Thirdmode,Q

2 2100

2150

2200

2250

2300

−160

−150

−140

−130

−120

−110

−100

Amplitude [dB]

Frequency [Hz]

(i)Thirdmode,Q

3 3450

3500

3550

3600

3650

3700

−160

−150

−140

−130

−120

Amplitude [dB]

Frequency [Hz]

(j) Fourthmode,Q

1 3450

3500

3550

3600

3650

3700

−160

−150

−140

−130

−120

Amplitude [dB]

Frequency [Hz]

(k)Fourthmode,Q

2 3450

3500

3550

3600

3650

3700

−160

−150

−140

−130

−120

Amplitude [dB]

Frequency [Hz]

(l)Fourthmode,Q

3

Fig. 5: Un ontrolled (- -) and ontrolled () FRF of the multi- ra ked beam inthe rst ase for

,Q

2

andQ

(23)

0 1/16

1/8

3/16

1/4

405

410

415

420

425

430 µ

Frequency [Hz]

0 1/16

1/8

3/16

1/4

0

0.005

0.01

0.015 µ

Damping

(a)Firstmode

0 1/16

1/8

3/16

1/4

1145

1150

1155

1160

µ

Frequency [Hz]

0 1/16

1/8

3/16

1/4

0

0.005

0.01

0.015

0.02

0.025 µ

Damping

(b)Se ondmode

0 1/16

1/8

3/16

1/4

2140

2150

2160

2170

2180

2190

2200

2210

2220

µ

Frequency [Hz]

0 1/16

1/8

3/16

1/4

0

0.002

0.004

0.006

0.008

0.01 µ

Damping

( )Thirdmode

0 1/16

1/8

3/16

1/4

3520

3540

3560

3580

3600

3620

µ

Frequency [Hz]

0 1/16

1/8

3/16

1/4

0

0.005

0.01

0.015

0.02 µ

Damping

(d)Fourthmode

Fig.6: Frequen yanddamping intherst aseforthreeweightingmatri es(

:un ontrolled,

:Q

1

,

Amplitude [dB]

Frequency [Hz]

(e)Se ondmode,Q

2 1100

1120

1140

1160

1180

−160

−150

−140

−130

−120

−110

−100

Amplitude [dB]

Frequency [Hz]

(f) Se ondmode,Q

3 2150

2200

2250

−150

−140

−130

−120

−110

Amplitude [dB]

Frequency [Hz]

(g) Thirdmode,Q

1 2150

2200

2250

−150

−140

−130

−120

−110

Amplitude [dB]

3

Fig. 7: Un ontrolled (- -) and ontrolled () FRF of the multi- ra ked beam in the se ond ase

,Q

2

andQ

(25)

0 1/16

1/8

3/16

1/4

425.5

426

426.5

427

427.5

428 µ

Frequency [Hz]

0 1/16

1/8

3/16

1/4

0

1

2

3

4

5

6 x 10

−3

µ

Damping

(a)Firstmode

0 1/16

1/8

3/16

1/4

1080

1090

1100

1110

1120

1130

1140

1150

1160

µ

Frequency [Hz]

0 1/16

1/8

3/16

1/4

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04 µ

Damping

(b)Se ondmode

0 1/16

1/8

3/16

1/4

2170

2175

2180

2185

2190

2195

2200

2205

2210

µ

Frequency [Hz]

0 1/16

1/8

3/16

1/4

0

0.002

0.004

0.006

0.008

0.01 µ

Damping

( )Thirdmode

0 1/16

1/8

3/16

1/4

3560

3570

3580

3590

3600

3610

µ

Frequency [Hz]

0 1/16

1/8

3/16

1/4

0

0.005

0.01

0.015 µ

Damping

(d)Fourthmode

Fig. 8: Frequen y and damping in the se ond ase for three weighting matri es (

:un ontrolled,

:Q

1

,

:Q

(26)

Mode 1

Mode 3

0

50

100 Index [%]

Case 1 : µ = 1/16

Mode 1

Mode 3

0

20

40

60 Index [%]

Case 2 : µ = 1/16

Mode 1

Mode 3

0

50

100 Index [%]

Case 1 : µ = 1/4

Mode 1

Mode 3

0

50

100 Index [%]

Case 2 : µ = 1/4

Fig. 9: Index variation in the rst and se ond ase for the rst and third mode, al ulated using

frequen y(

)or a tive damping for thethree ponderations (

:

Q

1

,

:

Q

2

,

:

Q

3

) and two ra ks

depth(

µ = 1/16

and

µ = 1/4

)

(27)

1 Elasti ,piezoele tri and diele tri onstants ofpiezoele tri materials . . . 27

(28)

c

E

11

(GPa)

c

E

22 c

E

33 c

E

12 c

E

13 e

31

(C/m

2

)

e

33 ǫ

S

33

(nF/m) 127.2 127.2 117.4 802.1 846.7 -6.6 23.2 8.85

₄

4 3590.4 1.18

×

10