Active vibration control of a fluid/plate system

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THÈSE

En vue de l'obtention du

DOCTORAT DE L’UNIVERSITÉ DE TOULOUSE

Délivré par l'Université Toulouse III - Paul Sabatier

Discipline ou spécialité : Systèmes Automatiques

JURY

M. Jean-Pierre RAYMOND, President

M. Miroslav KRSTIC, Rapporteur

M. Yann LE GORREC, Rapporteur

Mme. Lucie BAUDOUIN, Membre

M. Gildas BESANCON, Membre

Mme. Valérie BUDINGER, Membre

M. Edouard LAROCHE, Membre

M. Christophe PRIEUR, Membre

Ecole doctorale : EdSYS

Unité de recherche : LAAS - CNRS

Directeur(s) de Thèse : M. Christophe PRIEUR, Directeur de thèse

Mme. Lucie BAUDOUIN, Co-Directeur de thèse

Rapporteurs : M. KRSTIC and Y. LE GORREC

Présentée et soutenue par Bogdan ROBU

Le 03.12.2010

Titre :

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"R bdare, r bdare, r bdare, r bdare, r bdare ...

r bdare, r bdare, r bdare, r bdare ..."

Arhim. IlieCLEOPA

"That whi h does not kill you

makes you stronger."

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I wouldrst liketoexpressmygratitude towards myadvisors Mme. Lu ieBaudouin

etM.Christophe Prieurfortheir helpand onstanten ouragementthroughoutthese

years. They provided mewith freedomand supporttopursue my resear hideas and

I amgratefulI hadthe han etobenetfromtheirexperien e. Thank youespe ially

for the motivationnotes during the writing of the nal part of the thesis.

I also like to express my thanks to M. Miroslav Krsti and M. Yann Le Gorre

for making methe honorto review my thesis and for givingme useful ommentsfor

improving it's quality. I equally thank to M. Jean-Pierre Raymond for a epting to

be the president of my thesis ommittee. My thanks also go towards Mme. Valérie

Budinger for the great help with the experimental setup and for a epting to be a

memberofmythesis ommittee. Ialsothank toM.GildasBesançonandM.Edouard

Laro he fortheirparti ipationinmythesis ommitteeandfortheiruseful omments.

I also want to thank to Mme. Isabelle Queinne , head of the MAC group, for

wel omingmehereatLAAS-CNRS.MygratitudegoesalsotowardsM.DenisArzelier

for guiding me during the master proje t and for allowing me to benet from his

experien e during my thesis. I also wish to thank to all the members of the MAC

group, my edu ation has been enhan ed be ause of your advi es and presen e.

I would alsolike to thank to all my friends, to the ones that are stillnext to me

and to the ones that are far away. I'm glad that you entered my life, it's been a

wonderful growing experien e. I learnedmany things from ea h and any one of you

... I just hope that I learned allthe things I had to.

A spe ial thanks to all my high s hool friends. Thank you very mu h guys for

all your help and support in the di ultmoments, I ould not done it without you.

Thank youthat after allthese years we an still get together and have agreat time.

Finally, my deepest and sin ere thanks go to my parents and espe ially to my

brotherCos. Thebest brotherandthe bestparentsthat ouldeverexist. Thankyou

for always being here for me- I apologize for the times that I was not there for you.

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Introdu tion 1

1 Experimental devi e presentation 5

1.1 Chara teristi s of the experimentaldevi e . . . 6

1.2 Data a quisition hain . . . 8

1.3 A tuators and sensors . . . 11

1.3.1 Presentation of the piezoele tri phenomenon . . . 14

1.3.2 Optimalpla ement of a tuators and sensors . . . 15

1.3.3 Dynami of piezoele tri pat hes . . . 18

1.4 Con lusion of the hapter . . . 19

2 Mathemati al modeling of the system 21 2.1 Introdu tion . . . 21

2.2 Plate model . . . 23

2.2.1 Beam model . . . 23

2.2.2 Plate innite dimensionalmodel . . . 32

2.2.3 Plate nite dimensional approximation . . . 35

2.2.3.1 Computationof the dynami plate matrix

A

p

. . . . 36

2.2.3.2 Computationof the plateinput matrix

B

p

. . . 40

2.2.3.3 Computationof the plateoutput matrix

C

p

. . . 45

2.3 Tank model . . . 48

2.3.1 Sloshingof liquids -state of the art . . . 48

2.3.2 Tank approximation . . . 54

2.3.3 Tank innitedimensional model . . . 57

2.3.3.1 Generalequations . . . 57

2.3.3.2 Determination of for es and moments. . . 67

2.3.4 Tank nite dimensionalmodel . . . 71

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2.3.4.2 Determination of parameters for the mass-pendulum

model . . . 74

2.4 Complete modelrepresentation . . . 80

2.4.1 Innite dimensional oupling. . . 80

2.4.1.1 Inuen e of the liquidsloshing onthe plate movement 80 2.4.1.2 Inuen e of plate deformationon the liquidsloshing 81 2.4.2 Finitedimensional oupling . . . 81

2.4.2.1 Liquidsloshing inuen e onthe re tangularplate . . 82

2.4.2.2 Plate deformationinuen e on tankliquidsloshing . 84 2.4.2.3 Compa t writing of omplete model . . . 85

3 Controller synthesis - Theoreti al approa h 87 3.1 Energy omputation . . . 88

3.2 Pole pla ement and full state observer . . . 90

3.3

H

∞

ontroller . . . 97

4 Experimental results 105 4.1 Inuen e ofthe a tuator dynami s . . . 105

4.2 Choi eof the suitableamount of modes . . . 107

4.3 Model adjustments . . . 110

4.3.1 Computationof the naturalfrequen y . . . 111

4.3.1.1 Computationof plate natural frequen ies. . . 111

4.3.1.2 Computation of the natural frequen ies of sloshing modes . . . 120

4.3.1.3 Naturalfrequen iesofthe ompletesystem: plateand tank . . . 122

4.3.2 Computationof modaldamping . . . 125

4.3.3 Modelmat hing problem . . . 125

4.4 Pole pla ement ontroller . . . 127

4.5

H

∞

robust ontroller . . . 133

4.5.1 Synthesis of a

H

∞

ontrollerwithout lters . . . 134

4.5.2 Synthesis of a

H

∞

ontrollerwith lters . . . 140

4.5.2.1 Matlab

Robust ControlToolbox ontroller . . . . 141

4.5.2.2 HIFOO ontroller . . . 143

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4.5.2.4 Simultaneous redu ed-order HIFOO ontroller . . . . 150

4.6 Comparison of the ontrol methods . . . 153

General on lusion 157

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1.1 Experimentaldevi e ISAE-ENSICA . . . 5

1.2 Experimentaldevi e, detailedpresentation of main omponents . . . 6

1.3 Deformation ofthe re tangularplate (

1 st

mode) . . . 6

1.4 Re tangular plate without ylindri altank . . . 7

1.5 Equipped experimentalsetup . . . 8

1.6 Detailview of harge amplier . . . 9

1.7 Detailview of a quisitionsystem and xPCTarget . . . 10

1.8 Detailview of high voltage amplier . . . 11

1.9 A tuators onne ted tothe plate . . . 12

1.10 Sensors onne ted tothe plate . . . 12

2.1 Beamwith a exion movement . . . 24

2.2 Plate bending along

x

axis . . . 32

2.3 Theplateand thetwobeamssele tedforthe hoi eofthe Ritzfun tions 34 2.4 Qualityfa tor

Q

. . . 38

2.5 Neutral berof the re tangular plate . . . 43

2.6 Cylindri altank onne ted tothe plate . . . 50

2.7 Horizontal ylindri altank . . . 51

2.8 Natural angular frequen y

ω

n

of the rst transverse sloshing modes (extra ted from [48℄) . . . 52

2.9 Natural angular frequen y

ω

n

of the rst longitudinal sloshing modes (extra ted from [48℄) . . . 53

2.10 Implementing the rst method(only one re tangular tankis shown) . 54 2.11 Equivalenttanks . . . 56

2.12 Coordinate system for a partially lled re tangular ontainer under external a eleration . . . 58

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2.14 The mode shape of the rst three antisymmetri waves (from left to

right). . . 66

2.15 Masspendulum and mass spring me hani almodels . . . 71

2.16 Me hani al model with one xed mass and 3 sloshing masses, repre-senting fuel sloshingunder longitudinalex itation . . . 73

3.1 State feedba k ontrol . . . 91

3.2 Feedba k ontrollaw and observer. . . 96

3.3 Standard

H

∞

problem . . . 99

3.4 Standard

H

∞

problem . . . 103

4.1 Bode plot of the system with and without onsidering the a tuator dynami s,tank lllevelof

0.9

. . . 106

4.2 Experimental Bode plot for a tank ll level of

0.9

in the frequen y range

[0, 200]

Hz.

#1

is the rst exion mode of the plate,

#2

is the rst sloshing mode of the liquid,

#3

and

#4

are the se ondand third sloshingmode ofthe liquid(they are almostinvisibledue totheir very smallamplitude),

#5

is the rst torsion mode of the plate,

#6

is the se ond exion mode of the plate,

#7

is the third exion mode of the plate,

#8

is the forthexion mode ofthe plate,

#9

isthe fthexion mode of the plate,

#10

is the sixth exion mode of the plate,

#11

is the se ond torsionmode of the plate,

#12

isthe seventh exion mode of the plate,

#13

is the eightexion mode of the plate . . . 108

4.3 First threemodaldispla ements ofthe free-free beam . . . 112

4.4 First ve modaldispla ements of the lamped-freebeam . . . 113

4.5 Plate rst exion mode at

2.301

Hz,

η

1 (y, z) = Y

1 (y)Z

1 (z)

. . . 114

4.6 Plate se ond exion mode at

14.413

Hz,

η

2 (y, z) = Y

2 (y)Z

1 (z)

. . . 115

4.7 Plate third exion mode at

40.3583

Hz,

η

3 (y, z) = Y

3 (y)Z

1 (z)

. . . 115

4.8 Plate rst torsion mode at

49.2027

Hz,

η

4 (y, z) = Y

1 (y)Z

2 (z)

. . . 115

4.9 Plate se ond torsion mode,

η

8 (y, z) = Y

3 (y)Z

2 (z)

(not taken into a - ount during the modeling phase) . . . 116

4.10 First

4

modal displa ementsof the plate . . . 118

4.11 ExperimentalBode plot for the plate and a tankll level of

0.7

. . . 122

4.12 ExperimentalBode plot the plate and a tank lllevel of

0.9

. . . 123

4.13 Frequen y mat hing forthe tankllinglevel

e = 0.7

(numeri almodel - plainline and experimental set-up- dotted line) . . . 126

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4.14 Frequen y mat hing forthe tankllinglevel

e = 0.9

(numeri almodel - plainline and experimental set-up-dotted line). . . 127

4.15 Feedba k ontrollaw and observer. . . 128

4.16 Pole/zeromap of the open-loopsystem (

×

for the poles,

◦

forthe zeros)129

4.17 Experimentaloutputof the ofopen-loop(dottedline) and losed-loop

(plain line) systems using a pole pla ement ontroller with a tank ll

level of

0.9

. . . 130

4.18 Voltagedelivered bythepolepla ement ontrollerduringexperiments,

tankll level of

0.9

. . . 131

4.19 Frequen y response of the polepla ement ontroller,tanklllevelof

0.9

131

4.20 Standard

H

∞

problem . . . 133

4.21 Temporal response for robust ontrollers using Robust Control

Tool-box,withoutlters; simulationsonasystem with the same amountof

modes; tank ll level equal 0.9. Thin line is obtained with

d

12 = 0.1

, plain linewith

d

12 = 0.25

and boldlinewith

d

12 = 1

. . . 135

4.22 Voltagedeliveredbytherobust ontrollers;tanklllevelequal0.9with

d

12 = 0.1

(thin line),

d

12 = 0.25

(plain line) and

d

12 = 1

(boldline) . . 136

4.23 Bodeplotoftherobust ontrollerssimulatedonanaugmentedsystem;

tank ll level equal 0.9. The thin line is for

d

12 = 0.1

, plain line for

d

12 = 0.25

and boldlinefor

d

12 = 1

. . . 137

4.24 Temporal response for robust ontrollers using Robust Control

Tool-box,withoutltersandwith

d

12 = 0.1

; testsonanaugmentedsystem; tankll level 0.9 . . . 137

4.25 Pole/ zero map for the open-loop system augmented with one mode;

tankll level equal0.9 . . . 138

4.26 Pole/ zero map for the losed-loop system augmented with one mode

and withthe ontroller omputed with

d

12 = 0.1

; tanklllevel equal0.9138

4.27 Pole/zero map of the previously omputed ontroller,

d

12 = 0.1

; tank lllevelequal 0.9 . . . 139

4.28 Standard

H

∞

problemwith lters . . . 141

4.29 ExperimentalBodeplotoftheopen-loopsystem(plainline)andofthe

losed-loop system (bold line) omputed with the Robust ontroller

fromMatlab for 2 modes and a xed tank llingof

0.7

. . . 142

4.30 ExperimentalBode plot ofthe open-loopsystem (thinline) and ofthe

losed-loop system using a HIFOO ontroller and a Robust ontroller

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4.31 Experimental Bode plots of the open-loop system (thin line) and the

losed-loop system (bold line) with a HIFOO ontroller omputed for

2 modes and a xed tank llingof

0.9

. . . 145

4.32 Comparison between a

1 st

and a

4 th

order HIFOO ontroller;

experi-mental results for a xed tank lling

e = 0.7

and omparison to the open-loopsystem (thin line) . . . 146

4.34 ExperimentalBode plots for the open-loopsystem (plain line) and of

the losed-loopsystem (boldline) using HIFOO ontroller -

e = 0.7

. 147

4.33 ExperimentalBode plots for the open-loopsystem (plain line) and of

the losed-loopsystem (boldline) using HIFOO ontroller -

e = 0.9

. 147

4.35 Experimental output of the losed-loop ontroller using HIFOO

on-troller(boldline)andoftheopen-loop(dottedline); platedeformation

of

10

m,

e = 0.9

. . . 148

4.36 VoltagedeliveredbytheHIFOO ontroller;platedeformationof

10

m,

e = 0.9

. . . 149

4.37 ExperimentalBodeplot, omparisonbetweentheopen-loop(plainline)

and the losed-loopsystem with HIFOO omputed onsidering

2

or

8

modes of the system; xed tanklling of

0.9

. . . 149

4.38 HIFOO ontroller al ulated for the tank ll level

0.9

and tested on the tank ll

0.7

. . . 150

4.39 Experimental Bode plot of the open-loop system (dotted line) and

of the losed-loop system (bold line) using simultaneous HIFOO

on-troller -

e = 0.9

. . . 151

on-troller -

e = 0.7

. . . 152

on-troller -

e = 0.5

. . . 152

4.42 Temporalevolutionofthe experimentaloutputforthe losed-loop

sys-temswith polepla ement ontroller(plainline)and HIFOO ontroller

(boldline); plate free end deformationof

10

m,

e = 0.9

. . . 153

4.43 Experimental Bode plots for the losed-loop system with pole

pla e-ment ontroller (plain line) and HIFOO ontroller (bold line);

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1.1 Plate hara teristi s . . . 7

1.2 Chara teristi s of the ylindri al tank . . . 8

1.3 Chara teristi s of the piezoele tri pat hes . . . 13

2.1 Plate Ritz fun tions (

Z

1

means the mode is a exion mode while

Z

2

means isa torsion mode) . . . 36

4.1 Modal energeti ontribution rate of ea h mode . . . 109

4.2 Natural frequen ies of the beams asso iated tothe plate . . . 112

4.3 Natural frequen ies of plate modes -analyti al al ulus . . . 114

4.4 Natural frequen y of plate modes when the tank hole is taken into a ount -numeri al al ulus . . . 117

4.5 Comparison between the naturalfrequen ies in the ase where one or two a tuators are used . . . 119

4.6 Natural frequen y and the mode des ription for the rst

4

platemodes 120 4.7 Comparisonofthesloshingfrequen iesobtainedfromtheexperimental urvesand with dierentapproximationmethods. Tank lllevel

h

s

2R

=

0.7

. Forother tank lllevels, the results respe t the same pattern. . . 121

4.8 The measured naturalfrequen ies for the omplete system (plate and liquid)when the tank islled up to some arbitrarydepths . . . 123

4.9 Chara teristi s of the mass-pendulum systems for tankll level

0.7

. 124 4.10 Chara teristi s of the mass-pendulum systems for tankll level

0.9

. 124 4.11 Measurement of the dampingof ea h vibration mode . . . 125

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General ontext

The new generation of airplanes and spa e shuttles need to y further and further

away. Thus, the problem of fuel apa ity has ome to the attention of the s ienti

world. In a ordan e, the tanks used to sto k the fuel need to be enlarged. The

drawba k is that larger quantities of fuel imply that a potentially larger quantity of

liquid an be subje t to movements if ne essary preventive measures are not taken.

Therefore,thequestionof ontrollingtheliquidbehaviorhasarisenandNASAstarted

from the early '60s to on entrate on this issue. The rst omplete study was done

by Abramson [2℄, based on many other studies dealing with this issue as [22℄, [34℄,

[92℄just to ite fewof many.

The ore problem with large quantities of liquid in large tanks is that, a

phe-nomenon of sloshing o urs at low frequen y. As the sloshing frequen ies get lower,

aninterferen e withthe ontrolfrequen iesgeneratedby pilotsmay o ur. Thismay

lead to a ontinuous ex itation of the liquidwhi h, in return, will ae t the vehi le

stability. Besides, this an even leadtothe non- ontrollabilityand destru tionof the

vehi le [48℄. Even if su h extreme ases are not willing to o ur, the liquid strange

behavior an still pose serious problems [46℄. As an example, [3℄ and [129℄ give a

listsofairplanesthat were onfronted tothis issue duringthetestingphase: Douglas

A4D, Lo khead P-80, Boeing KC-135, CessnaT-37, North Ameri anYF-100.

More-over, liquid unpredi table movement also ae ted the NEAR spa e raft whi h had

to interrupt hisinsertion burn due to large fuel rea tions. Even though the fuel was

nally ontrolled, the mission wasstilldelayed for almost a year [138℄.

In order to minimize the sloshing, various methods an be used. Firstly, the

ontainers with liquid an be divided, using baes, in several smaller ontainers so

that the eigenfrequen ies of the sloshing modes are in reased [124℄, [130℄. Se ondly,

sin e the lo ation of the ontainers also ae ts the damping of the stru ture [20℄,

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theliquidfreesurfa e analsoin reasethenaturalsloshingfrequen ies[20℄. Fourthly,

a ontrol system an also be arefully hosen so that sloshing modes are attenuated

or atleast not ex ited toomu h. We will on entrate our work on this lastmethod.

In order to ontrol the sloshing,one needs to ompute for ea h mode the natural

frequen y, the mode shape and then the total for es and moment that it generates.

Exa t solutions though, are possible only for very few spe ial ases, su h as verti al

ylindri al tankor a re tangulartank [67℄. Furthermore,in the ase when the exa t

solutionsexist,the ouplingbetweenthesesolutionsandtheequationsofuidmotion

is too omputationally demanding even with super omputers [48℄. Based on these

remarks, some approximationsof the liquidsloshing have to be found. As presented

in [18℄, a good approximation is obtained by onsidering ea h sloshing mode as a

system withasingledegreeoffreedomandrepresenting iteitherasamass-pendulum

system ora spring-mass system. Even thoughboth methods are equivalent [67℄, the

mass-pendulumsystemisusuallypreferredduetosomesmalladvantages(his natural

frequen y varies withthe hangesinaxiala elerationasthe sloshingfrequen y does

[48℄). Finally,the os illatinguid an berepresented asa simpleme hani alsystem,

in whi h the lo ation and the magnitude of the model variables are determined to

givethe same for es and moments asthe liquid does.

Another hara teristi of airplanesand spa eshuttles ofthe future isthe in rease

of their size. As they be omelarger, inorder to redu ethe overall weight, the wings

andtaildenitelyneedtobelighter,thusmoreexible. See[13℄forthe AirbusA-380

ase or [137℄ for the NASA A tive Aeroelasti Wing (AAW) on ept. The study of

exible stru tures has aptured the attention of resear hers for many years and is

well overed in the literature. Asanexample, one an he k the works of [30℄ or[56℄

where the theory is presented and experimental resultsare given.

It is well known that, espe ially in the ase of large airplanes, a great part of

the fuel is on entrated in the wings. Thus, for some airplanes, the quantity of fuel

arried in the wing tanks be omes a large per entage of the total wing mass [92℄.

Thus, the wing willbe onsiderably inuen ed by the liquid movements.

On the other hand, due to their hara teristi s, smart materials have been used

for many years now, espe ially in the eld of ivil engineering, for measuring and

attenuating the deformations of stru tures [29℄. Therefore, the question of how it

an be useful to use them for ontrolling the exible devi es arose. Sin e the rst

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suppress the vibration of stru tures [11℄, [23℄, [45℄, [65℄, [66℄, [144℄. However, up to

our knowledge, only few works have addressed the oupling between liquid sloshing

and exible stru ture [79℄, [132℄. Moreover, even fewer onsider this oupling in the

ase of airplanes [108℄,[109℄.

The devi e we are working on follows these lines, the purpose being to ontrol,

using piezoele tri pat hes, a exible plate onne ted to a tank lled with liquid.

Furthermore,this devi ewas onstru tedtohave, inlowfrequen y domain,thesame

behavior asa real plane wing [110℄.

Thesis outline

The manus ript is onstru ted as follows.

The rst haptergivesadetailedpresentation ofthe experimentaldevi ewewant

to modeland ontrol: a re tangularplate lamped atone of itsends, onne ted toa

ylindri altankat itsother end. Afterageometri hara terization of the stru ture,

the a quisition system is detailed and analyzed. The nal part of the hapter

on- entrates on the presentation of a tuator and sensor pat hes. Sin e they are made

from piezoele tri materials, a brief des ription of the piezoele tri phenomenon is

rst given. Then, some detailsare given on the optimal pla ement of these pat hes.

Finally, the a tuator speed and his inuen e on the total dynami of the system is

analyzed.

Chapter 2 gathers the steps of the mathemati almodeling of the devi e and

de-tailsthe omputationofthestru turemodel. Eventhoughnumeri almethodsarethe

most employed forthe model omputationof omplexstru tures likeours, we hoose

towork with ananalyti alpro edure. Itwilllead toa more tediousmodelingphase,

but, takingintoa ountmanyme hani al onsiderationswillshowitsinterestduring

the ontroller omputation phase. The main idea we follow for the omputation of

the modelisrst toget two separate partialdierentialequation (PDE)models, one

for the plate and one for the tank with liquid, and se ond to put them together by

studyingthe mutualinuen e. Thus, the modelisrst writtenusing PDEsand then

is approximated using the Ritzmethod for the plate and using me hani al analogue

systems for the sloshing. Finally, the nite dimensional system is written under the

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In Chapter 3 the theoreti al bases of the Chapter 4are set. Sin e the ontrollers

we ompute are basedonthenite dimensionalmodel,the issueof hoosingthe

suit-able amount of modes for the model approximation needs to be ta kled. A method

based on the energeti ontribution of ea h stru ture mode solves this issue. Then,

the theory to ompute a polepla ement ontroller oupled witha full state observer

is briey reminded. Finally,the frameof robust

H

∞

ontrol is briey presented and more attention is given to the parti ularities of the method implementation in the

ase of innite dimension systems.

The ore problem of ontrolling the experimental devi e is treated in Chapter 4.

After testing the inuen e of a tuator dynami s, the issue of hoosing the right

amount of modes for the model approximation is onsidered. Based on te hni al

onsiderations of airplanes and on the energeti ontributionof ea h mode, a hoi e

of thenumberofmodes tobe onsidered ismade. Numeri alsimulationsand

experi-mentaltestsare ondu ted afterward. First, apolepla ement ontrolleris omputed

and tested. Se ond, a

H

∞

ontroller, robust toexternal perturbations,is omputed. Using the HIFOO pa kage, redu ed order ontrollers an also be found. Moreover,

the simultaneous ontrol problem with redu ed order ontrollers is also onsidered.

Simulationsand tests are shown and analyzed.

The manus riptends withalast hapterdealingwiththe on lusions ofthiswork

and with perspe tives for further resear h.

Ea h hapter (ex ept for the last hapter whi h presents the general on lusions

of the manus ript) ends with ashort on lusion dealing with the ontributionof the

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Experimental devi e presentation

This hapter isdevoted tothe des ription of the experimentaldevi e we are working

on. It is lo ated at I'Institut Supérieur de l'Aéronautique et de l'Espa e - É ole

NationaleSupérieured'IngénieursdeConstru tionsAéronautiques(ISAE-ENSICA)

inToulouse,Fran e. Thedevi e ispi turedinFigure1.1and ithas been onstru ted

tohavethesamebehavior,inlowfrequen ies,asareal planewingwithfuel(see[110℄

or [114℄).

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1.1 Chara teristi s of the experimental devi e

The experimentaldevi e is omposed of analuminum plate and aplexiglas tip-tank

lled withliquid. The plateisre tangular, lamped atone side andfree ontheother

threesides. Atthefreeendoftheplate,oppositetothe lampedend,is onne tedthe

ylindri altank, asit anbeseen onFigures1.2and1.3. The tankisina horizontal

position and it an be lled with water ori e up to anarbitrary level.

Figure1.2: Experimentaldevi e, detailed presentation of main omponents

PSfrag repla ements

x

y

z

O

w(y, z, t)

Figure 1.3: Deformationof the re tangularplate (

1 st

mode)

The length of the plate is along the horizontal axis and its width is along the

verti al one (see Figure 1.3). At the lamped end, there are two a tuators glued

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aluminium and has the hara teristi s depi ted in Table 1.1 below. A view of the

plate withoutthe ylindri altank an be seen inFigure1.4.

Plate length L

1.36

m

Plate width l

0.16

m

Plate thi kness h

0.005

m

Plate density

ρ

2970

kg m

−3

Plate Young modulus Y

75

GPa

Plate Poisson oe ient

ν

0.33

Table 1.1: Plate hara teristi s

Figure1.4: Re tangular plate without ylindri al tank

The tankis entered at

1.28

m fromthe plate lamped side and is symmetri ally spread along the horizontalaxis. Due to the ongurationof the whole system, the

tankundergoesalongitudinal movementwhen theplate has aexionmovementand

a pit hmovement if the plate has atorsion movement.

The geometri al hara teristi s of the horizontal ylindri al tank are given in

Table 1.2. It an beremoved orlled with i eorwater. Ifthe tank islled with i e,

it an be easily modeled by a steady mass [123℄ equal to the empty tank mass plus

the mass of the i e.

The ratio between the liquid height and the total height of the tank gives the

tank ll level, whi h is a good indi ator of the tank behavior. When the tank ll

level is lose to

0

or lose to

1

(the tank is almost empty or almost full), there is no sloshing behavior, and the modeling pro ess is similar to the ase of frozen water.

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Tankexterior diameter

0.11

m Tank interior diameter

0.105

m

Tank length

0.5

m

Tank density

1180

kg m

−3

Tank Young modulus

4.5

GPa

Table 1.2: Chara teristi s of the ylindri altank

The interesting ases are when the tank ll level is between these values. In this

ase a sloshing phenomenon o urs, whi h is hara terized by a periodi motion of

the liquidfree surfa e. This motion reates periodi for es and moments of for e. It

is in this situation that this work is pla ed, therefore, we will further onsider only

the ases for whi h the sloshing motion o urs. A more omplete des ription of this

phenomenon willbe given laterin Se tion2.3.1 of Chapter 2.

The movementof the plate isgenerated by some piezoele tri a tuators while

in-formationaboutplatedeformationaregatheredusingpiezoele tri sensors. Moreover,

the a tuators anbeusedasa ontrolinputorasa perturbationinput. Moredetails

about the a tuators/sensors geometryand behavior are given below inSe tion 1.3.

Let usrst des ribe the data a quisition hain.

1.2 Data a quisition hain

PSfrag repla ements

Equipped Structure

P iezoelectric

Actuator

P iezoelectric

Sensor

Charge

Amplif ier

High V oltage

Amplif ier

DSpace Card

+

Computer

P late

+

T ank

Figure 1.5: Equipped experimentalsetup

Inorder tore ordthe informationtransmittedtothe a tuatorsand given bythe

sen-sors, some a quisition hain is used. A s hemati representation of the experimental

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present the dierent omponentsof the hainthat makepossiblethe implementation

of numeri al ontrollers. They are listed below starting from the signal delivered by

the sensor until the voltagedelivered tothe a tuator.

Thedatadeliveredbythepiezoele tri sensorisrst olle tedbya hargeamplier

before being delivered to the DSpa e

ard. The harge ampliers, one for ea h sensor, are of type 2635 and are made by Brüel & Kjaer [33℄. Their pi ture along

with the onne tions to the experimental devi e are presented in Figure 1.6. The

prin ipleofthe hargeamplieristoset, usinganoperationalamplier,anullvoltage

between the sensor ele trodes sothat the eventual parasite apa itan e vanishes. In

this way, all the harges on the sensor ele trodes are send towards a apa itan e

where a voltage, orresponding to the harge dieren e, is measured. For further

details about the ele tri s heme of the devi e one an read referen e[81℄.

Figure 1.6: Detailview of harge amplier

The signaldeliveredby the hargeamplierissenttoa omputerusing aDSpa e

ard. Using the same ard, the signal delivered by the omputer is send tothe high

voltage amplier. The ontrol laws are implemented on the omputer and exe uted

in real time with asele ted samplingtime of

0.004

s.

Inordertomanipulatethedierentsignals,deliveredtothea tuatorsandre eived

from the sensors, the software xPC Target from Matlab

(26)

omputer. Itallowsthe real timeexe utionof aSimulinkmodelonthe omputervia

anoptimized real-time kernel.

The xPC Target reates a real-timetesting environment for Simulink models by

onne ting a host omputer, a target omputer and the experimental devi e under

test. Visual details of the a quisition hain are presented in Figure 1.7 where the

master( omputer ontheleft side)and slave( omputeronthe rightside) omputers,

along with the DSpa e

ard an be seen. The master omputer, on whi h are runningxPCTarget,SimulinkandanC- ompiler,is onne tedtotheslave omputer

via a single TCP/IP ommuni ations link. The slave omputer is onne ted to the

experimental setup. Based on the Simulink model, a ode is generated by

Real-Time Workshop and downloaded to the target omputer via the ommuni ations

link. During thea quisitionpro ess,the resultsare storedonthe slave omputerand

then an beuploaded to the master using Matlab

and xPCTarget software.

Figure 1.7: Detailview of a quisitionsystem and xPC Target

Finally, the voltage delivered tothe plate, by the DSpa e

ard, is ampliedby a highvoltage amplier. Ithas anamplifyinggain of

13

and an deliveramaximum voltage of

±100

V. In order to be fun tional, it has to be powered at

±15

V and

±100

V.One voltage amplier onne ted toa sour e delivering

±15

V an be seen in Figure 1.8. Althoughthe devi e is home-made at ISAE-ENSICA, his hara teristi s

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Figure1.8: Detailviewof highvoltage amplier

1.3 A tuators and sensors

As presented earlier, there are two a tuators and two sensors whi h are glued onthe

plate towards the lamped side (see Figure 1.9for the a tuators and Figure 1.10 for

the sensors). The a tuators are glued on one side of the plate while the sensors are

glued onthe otherside, thus there are two pairs of ollo ateda tuators and sensors.

Sin etheyareallmadefrompiezoele tri materialssomedetailedinformationisgiven

in this se tion on erningtheir behavior.

The piezoele tri erami s belong to the larger group of ferroele tri materials,

that isto say, materialswhi h are spontaneously polarized(without anele tri eld

being applied).

The piezoele tri a tuators are made from PZT (Lead zir onate titanate), model

PIC 151. The material model used (PIC 151 is onsidered a "soft" PZT) it is the

standard material used for a tuators. In order to reate a moment, both a tuators

lengthen when a voltage is applied to their ele trodes. The two sensors (made from

PVDF -Polyvinylideneuoride, arelatively new lass of piezoele tri materialsused

as sensor devi es) are lo ated on the opposite side of the plate with respe t to the

a tuators. They deliver a voltage proportional to their deformation. The

hara ter-isti s of the ollo ated sensors and a tuators are given in Table 1.3. Both a tuators

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Figure1.9: A tuators onne ted to the plate

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Instrumente (PI) ompany [1℄. A tuator length/width/thi kness

0.14

/

0.075

/

5e

−4

m Sensor length/width/thi kness

0.015

/

0.025

/

5e

−4

m A tuator/Sensor density

7800

kg m

−3

A tuator/Sensor Youngmodulus

67

GPa

A tuatorpiezoele tri oe ient (

d

31

)

−210e

−12

mV

−1

Sensor piezoele tri oe ient(

e

31

)

−9.6

C(m)

−2

A tuator/Sensor Poisson oe ient

0.3

Table 1.3: Chara teristi s of the piezoele tri pat hes

The piezoele tri materials are generally used to attenuate the vibrations and

measure the deformationof stru tures (see [23℄, [29℄, [53℄ among other referen es for

some examples). In the ase of exible stru tures, many studies alsoinvestigate the

use of piezoele tri pat hes to ee tively suppress the vibrations (see for instan e

[11℄, [45℄, [66℄, [140℄, [144℄). Indeed, piezoele tri pat hes oer a fast response and

have a large bandwidth, they are light and low ost, and have good sensing and

a tuating apabilities. Moreover, they are self-sensing a tuators, thus they an be

simultaneouslyusedasa tuatorsandsensors. However, onlyafewresultsarealready

availablein the literature foruid-stru ture systems (see [108℄or [109℄)for the same

stru ture as ours. For other stru tures, one an he k referen e [79℄ whi h gives a

re enttheoreti alresultand[132℄whi hvalidatesthea tive ontrolmethodbymeans

of experimentalresults.

Despite these advantages, some pre autions need to be taken. First of all, the

voltagelimitationsof thematerialsshouldbe onsidered. Inorder toavoidthe

depo-larizationofthematerial,the voltageappliedintheoppositedire tionofthematerial

polarizationneedstobe arefully ontrolled(maximumallowan eforPZTmaterialis

around

500

Vmm

−1

). Se ond, are should alsobetaken whenthe materialis exposed

to very high temperatures. The limittemperaturefor a piezoele tri material is

de-nedastheCurietemperatureand,againinthe aseofaPZTmaterial,isaround

250

degrees Celsius (ex eeding this limit the material is not being ferroele tri anymore

thus loosingall piezoele tri properties). In our ase thoughthese onsiderationsare

respe tedsin e the ambienttemperaturearoundthe experimentalsetup doesnot

ex- eed

30

degrees Celsius, while the voltage delivered to the piezoele tri a tuators is rst limited by the voltage amplier(see Se tion 1.2).

The stru tures that integrate piezoele tri a tuators and sensors on a exible

system are oftenknown asa tivestru tures orsmartstru tures, whilethe ontrolon

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additional materials are glued to in rease the stru tural damping of the stru ture

and redu e the vibrations [71℄, [136℄). The ontrol is a tive due to the fa t that

the equipped devi e is self-sensing and self- ompensating, due to the piezoele tri

pat hes.

1.3.1 Presentation of the piezoele tri phenomenon

Both a tuators and sensors use the piezoele tri ee t. Letus shortly des ribe it.

The existen e of the ee t was dis overed in the 1880 by the Curie brothers on

quartz rystals. When a stress is applied, these rystals have the property to

de-velop a proportional ele tri moment. Our purpose here is not to give a omplete

hara terization of the phenomenon but just some details that will help the reader

to better understand the behavior of the a tuators/sensors. The modeling will be

given in Se tion 2.2.3.2 of Chapter 2. For a detailed des ription of the piezoele tri

phenomenon [100℄,among others, gives a omplete hara terization.

The piezoele tri ee t is twofold: the dire t piezoele tri ee t (also known in

the literature asthe generator ee t) presented above and the onverse piezoele tri

ee t. The latter is dened as the shape hange of a piezoele tri rystal when an

ele tri eld isapplied. Moreover, it an beseen asathermodynami onsequen e of

the dire t ee t.

As it an be seen from the above statements, piezoele tri materials experien e

both ele tri andme hani alphenomena. Therefore,the ompletepiezoele tri

equa-tion is dened as a ombination between:

•

a me hani alphenomenon, des ribed, for an elasti materialexperien ing only small perturbations, by the tensor expression of the lassi al Hook law

on-ne ting the strain

ǫ

to the stress

σ

by the means of the omplian e tensor

s

[100℄:

ǫ = sσ;

(1.1)

•

anele tri phenomenon, des ribed bythe ele tri behavior ofthe material on-ne ting the ele tri displa ement

D

to the ele tri eld intensity

E

and the ele tri permittivity

κ

[100℄:

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Moreover, inthe ase ofthepolarizationofa rystalprodu edby anele tri eld,

(apiezoele tri rystalforinstan e),thelastequationdes ribingtheele tri behavior

be omes:

D = κ

0 E + P

(1.3)

where

P

isthepolarization harge perunit areataken perpendi ulartothe dire tion of polarization (or short polarization) and

κ

0 = 8.854 × 10

−12

Fm

−1

is the va uum

permittivity .

Atthesametime,ea htypeof piezoele tri ee t(dire tor onverse) isdes ribed

by his own spe i relations.

•

Ontheone hand,the dire tpiezoele tri ee tisdes ribed by arelationlinking up the polarization harge

P

of the stress

σ

applied tothe rystal sides:

P = dσ

(1.4)

where

d

is a onstant value alled piezoele tri modulus [100, Chapter 7℄;

•

Ontheotherhand,the onversepiezoele tri ee tisdes ribedalsobyarelation between the strain

ǫ

, responsible for the hange of shape of the material, and the intensity of the ele tri eld

E

[100, Chapter 7℄:

ǫ = dE

(1.5)

where the oe ient

d

is the same asin (1.4).

By ombining the relations (1.1) and (1.3) with (1.4) and (1.5) we obtain the

omplete piezoele tri equations [75, Chapter 13℄:

ǫ = sσ + dE,

(1.6)

D = dσ + κ

0

E.

These equations willbe later used inSe tions 2.2.3.2and 2.2.3.3of Chapter 2 to

ompute the analyti almodelof a tuators and sensors.

1.3.2 Optimal pla ement of a tuators and sensors

The optimal pla ement of a tuators and sensors is a key problem in the ontrol of

exible stru tures. Due to the nature of exible stru tures, spatially distributed

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the study of the optimal pla ement is natural when some performan e riteria need

to be obtained. There are many referen es whi h suggest dierent methods for a

better positioning of the a tuators as [44℄, [52℄, [63℄, [66℄ or [86℄ by analyzing the

ontrollabilityand observabilitymatri esforaxed amount ofvibrationmodes oras

[8℄ by studyingthe energy spa e of the stru ture. Even the thi kness of the a tuator

an be al ulated in order to have optimum values for the bending moment of the

a tuator. Forthislastissueone an he ktheworkof[81℄wheretheauthor omputes

the suitablethi kness ofapiezoele tri pat hinordertohavemaximumvaluesofthe

bending moment for aspe i plate stru ture.

In the experimental devi e of this thesis, the position of a tuators and sensors

was already xed and ould not be hanged. Thus, we do not onsider the optimal

positionproblem. Wegivenevertheless, inthe followinglines,somedetailsaboutthis

interesting issue. In the literature, two main types of approa hes an befound:

•

The losed-loopapproa htype onsistsrstat hoosingthe ontrollawto imple-menton the stru ture and then todetermine, for this spe i law, the optimal

pla ement of a tuators and sensors. In this ase, the lo ation of a tuators and

sensors is treated as some extra design parameters in the ontrol law

ompu-tation. For more details one an read referen e [141℄. The greatest advantage

of this method is the optimization for a spe i ontrol law but the greatest

drawba k of the method is also the fa t that the position of sensor/a tuator

pat hes depends onthis ontrol law;

•

The open-loop approa h type onsists in treating this problem independently from the ontrolled design problem. This ase has the main advantage that

several ontrollaws anbetestedforthesame a tuator/sensorpositioning. For

more details one an onsider [61℄, [66℄, [69℄, [81℄, [95℄ or [96℄ among many

others. In the following lines, we give some details on erningthis method.

There are several open-loopapproa hes in the literature on erning the optimal

pla ementofa tuatorsandsensors. Forexampleone an he k[95℄wheretheideasof

ontrollabilityandobservabilityofa tuators/sensorsareemployed. Anotherapproa h

an be read in [66℄, where the a tuators/sensors are ollo ated and pla ed at the

lo ationwhere the highestpositionsensitivity of ea h mode is experien ed.

We will now explain briey the method detailed in [95℄ sin e it is very easy to

implement.

This methodis basedonthenotionsof ontrollabilityfora tuator pla ementand

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will also be briey detailed, for the general ase of a linear system, in Se tion 3.2

of Chapter 3. This approa h seems natural if we think that, usually, a tuators need

to be pla ed where they have the highest authority to ontrol the system while the

sensorsshouldbepla edwherethey havethe higheststrength toobservethe system.

The methodis omputed separatelyfor the piezoele tri a tuators and sensors.

On the one hand, for the a tuators, adieren e is made between the modal

on-trollability and the spatial ontrollability. The modal ontrollability measures the

ontrollerauthorityoverea hmodeof theexiblestru ture whilethespatial

ontrol-labilitymeasures the a tuatorauthority onlyoverthe presele tedmodes(usually the

rstvibrationmodessin e thelowfrequen ymodes tendto ontributemorethanthe

high frequen ymodes tothe stru turevibrations). This dieren eisnaturalsin ewe

want the a tuator tohave ahigh authority overthe sele ted modes but, atthe same

time, to have a low authority over the non sele ted ones. This is espe ially true in

order toprevent the spilloveree t (ex itation ofhigh frequen y modes). Therefore,

in the ase of the a tuators, the optimization problem proposed by [95℄ is to

maxi-mize the spatial ontrollability measure while keeping some a tuator ontrolover all

modes, thus keepingsome levelof modal ontrollability.

On the other hand, for the sensors, the optimization problem in ndingtheir

lo- ation is formulated in a similar way in referen e [95℄ by dierentiating the modal

observability (observability of the sensor over all the modes) from the spatial

ob-servability (observability of the sensor over some sele ted modes). Finally the

opti-mization problemis formulated in order to maximize the spatial observability while

maintaininga minimumlevelof modalobservability.

After nding the optimal position of a tuator lo ation and of sensor lo ation

separately, the inherent question is wether or not this method an be implemented

forthe position omputationofboth piezoele tri a tuatorsand sensors. It isproven

in [95℄ that it iseasier to nd the optimalpla ement of a ollo ateda tuator/sensor

pairbystudyingonlythe ontrollabilityortheobservabilityand notboth (whi h an

be time onsuming).

Forour experimentalsetup, assaid earlier, the positionof the a tuators and

sen-sors was xed inadvan e. Thus, we did not study the problemof optimalpla ement

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1.3.3 Dynami of piezoele tri pat hes

Another thing that should be onsidered is the inherent dynami s of a tuators and

sensors. This is animportant issue during the modeling of the piezoele tri pat hes

sin e their dynami s may modify the total dynami of the modeled system.

As detailedearlierinSe tion1.2,some highvoltageampliersareused beforethe

piezoele tri a tuatorsforthe ontroloftheexiblestru ture. Arstorderdynami al

modelof this type of a tuator, similar tothe one in[131℄, is omputed below:

τ ˙v + v = ku

(1.7)

where

u

is the input voltage and

v

is the output delivered voltage. Moreover, the onstants have the values

τ = 4.85e

−7

s and

k = 1

, determined from the te hni al

spe i ations in order tox the ut-o frequen y of the modelat the same level as

the amplier bandwidth. Based on these issues, the minimal period of the output

voltage delivered by the amplieris

3.25e

−5

s.

At thesame time, weneedto omputethe maximal response speed for the

piezo-ele tri a tuator. Weremarkthat,ifthespeedofthea tuatorislargerthanthespeed

of the voltage amplier,then we donot need to take into onsideration the a tuator

dynami s. In this ase, the speed of the piezoele tri a tuator response saturates

after the voltage amplier does.

A ording to the te hni al spe i ations from PI Cerami atalog [1℄, the PZT

rea hes his nominal displa ement in

1/3

of its resonant period, provided that the ne essary urrentisdelivered. Besidesthis, theresonantperiodisdenedas

T

0 =

L

N

1

,

where

L

is the length of the piezoele tri a tuator and

N

1

is the frequen y onstant for the transverse os illationof a slimrod polarizedin the longitudinal dire tion. In

our ase, the length in taken from Table 1.3 while the frequen y onstant for the

PIC 151 material is

N

1 = 1500

. Therefore, the resonant period of the piezoele tri a tuator is

3.11e

−5

s.

As it an beseen, the maximal speed for the a tuator islarger thanthe maximal

speed for the voltage amplier. Thus, for a given ex itation, the a tuator response

time is mu h smaller than the one of the voltage amplier. Therefore, his dynami

an benegle ted sin e is not interfering in the response time of the total stru ture.

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1.4 Con lusion of the hapter

In this hapter we gave a general presentation of the experimental devi e we are

working on. The a quisition hain that will help us implement the ontroller for

vibration attenuation is also shown. Moreover, the hara teristi s of the plate/tank

system along with those of the piezoele tri a tuators and sensors are presented.

These hara teristi s will allow us to ompute the analyti al modelof the devi e in

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Mathemati al modeling of the system

2.1 Introdu tion

In this hapter we detail the dierent steps to build the mathemati al modelof the

uid/stru ture system depi ted earlier. We an nd in the literature two dierent

approa hes on erningthe modelingof su h devi es:

•

A numeri al approa h based on nite element method (FEM). The method approximates the distributed parameter system with an unlimited number of

degrees of freedom and modes by a nite dimensional dis rete system. To

do this, the whole stru ture body is divided in several subdivisions or nite

elements. Finally, the nite element des ription of the stru ture is a sum of

beam and lumped mass elements. Further on, the mass and stiness matri es

arefoundfromtheexpressionofthekineti andpotentialenergiesforthesystem

with nite degrees of freedom. As a result the nite element method provides

a quite good approximation for the frequen ies and mode shapes. For further

details about the des ription of the method one an he k for example [83℄

or [147℄. The ases where FEM is employed during the modeling phase are

numerous, asanexample one an he k [86℄, [133℄for aexibleplate system or

[108℄, [109℄ for auid plate system, among many others;

•

An analyti al approa h whi h allows to nd an analyti al solution, of innite dimension, for the ele trome hani alinnite dimension problem. Forthis ase

also, the referen es in the literature are numerous. Among many others, for a

exible stru ture system we an ite [63℄, [81℄, [107℄.

Usuallyin theliterature, for"simpler "a ademi alstru tures likebeamsoreven

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onewhi hallowsthe omputationofasimplemodel. Whilethinkingofmore omplex

stru tures, likethe one inour ase, the approa hmostlyemployed inthe literatureis

usingthenumeri almodelingbasedonFEMmethod. Eventhoughthismethodoers

thepossibilitytomodelitemswitha ompli atedshape,theirstru turegeometry an

not hangein time. Tothe best of our knowledge,only stru tures that are in asolid

form(oilpipelines,plates,beams,ringsofdierentshapesandsizes,fulltanks) anbe

modeled,but we an not modelthe liquidsloshing. Nevertheless, re entadvan es (

∼

year2006-2007)intheANSYS

software(niteelementmethodsimulatorsoftware), showthatare enttoolboxon omputationaluiddynami s alledFLUENT

might

be able to solve this typeof issue.

In our ase though, this method is di ult to use. Using nite element method,

the liquid, an only be modeled as a "frozen liquid" whi h a ts as a steady mass

with no sloshing phenomenon. Moreover, in our ase, the sloshing behavior is of

great importan esin eitsigni antly hangesthe system dynami sespe iallyinlow

frequen ies. Forastudy that onsiders the ouplingbetween aexible stru ture and

a uid one an he k [98℄ or again [25℄. In the latter, the ee t of the uid is taken

into a ount in the FEM modeling phase by means of an added mass formulation

detailedin [97℄.

Foranother exampleone an he k the work[114℄ forthe same stru ture asours.

InthisworktheauthorusestheFEMto omputethenumeri almodelofthestru ture

without liquid (therefore without any sloshing behavior). Even though the

experi-en es in [114℄ are done for three ases: empty tank, full tankand half full tank, the

ontrollers are omputed by always onsidering the tank tobeempty.

Therefore, we hoose togoonwith the analyti alapproa heven thoughwe think

that it leads toa more omplexmodelingphase.

In this hapter we are going to detail the dierent steps that will lead us to a

omplete model of the disposal. Sin e the plate and the tank an be viewed as two

separate entities oupled together, the main idea wehave inmind is to ompute two

separate models and then tounite them. Therefore, we willrst ompute one model

fortheplateand anothermodelforthetankwithliquid. Finally,inordertohavethe

omplete stru ture modelwe study the intera tions between the two models, that is

to say the way the behavior of one modelae ts the behaviorof the other.

More pre iselywewillrst writeamodelfortheexibleplateandanotherforthe

ylindri altankwithliquidusingpartialdierentialequations. The ouplingbetween

the twoinnitedimensionalmodels isobtainedby studyingthe inuen eof theplate

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innite dimensional modelby taking intoa ount only a nite numberof modes for

the plateandliquid. Basedonthis, the ouplingofthetwonitedimensionalmodels

is alsoexpressed innite dimension.

2.2 Plate model

In this se tion wedetail the onstru tionof the modelfor the re tangularplate with

piezoele tri a tuators and sensors. The partial derivative equation (PDE) plate

model is well known in the literature. For a more detailed presentation one an see

for example [30℄ or[56℄.

We start from the beam equation (whi h is a

1

-dimensional plate), for the sake of simpli ity during the modeling phase. We then study the plate and ompute an

innite dimensionalmodelusing partial derivativeequations (wewill see inthe next

se tions that the plate model is onstru ted on the basis of the beam model). The

obje tive istogivea lassi al state-spa eapproximation(nite dimension)using the

Ritz methodto approximate the PDE model. We willget:

(

˙

X

p

(t) = A

p

X

p

(t) + B

p

u(t)

y(t)

= C

p

X

p

(t)

(2.1)

where

X

p

is the state-spa e ve tor of the plate and

A

p

,

B

p

,

C

p

are respe tively the dynami , ontrol and output matri es.

u(t)

will be the ontrol (input) variable (the voltageappliedtothepiezoele tri a tuator)and

y(t)

theoutputvariable(thevoltage delivered by the piezoele tri sensor).

2.2.1 Beam model

The beam represents the transposition of a plate ina

1

-dimensionalspa e. Sin e we are dealingwith a beam, whi h is des ribed by onlyone dimension asit an be seen

in Figure2.1, we dedu e thatonly the exion movementis possible.

ThebeamPDEmodelingiswellknownintheliterature,one an he kforexample

[30℄where models for dierent types of beams are presented.

We onsider an homogeneous beam, lamped at one end and free at the other,

of onstant se tion, whi h has the length

L

and the mass

m

. By denition, the dimensionsofthebeam rossse tionare mu hsmaller(intheoryarenulldimensions)

than the length of the beam.

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O

PSfrag repla ements

y

x

w(y, t)

beam

Figure2.1: Beam with aexion movement

the main inertia axes. We start the study of transverse beam vibrations supposing

that the beam has only exing movements.

We make the lassi al inemati hypotheses asin [56℄:

•

thebeamisuniformand omposedofahomogeneous,isotropi elasti material;

•

the beam isredu ed toitsneutral ber, whi hby denition willbe the part of the beam that does not feel any onstraint, thus the axis where the elements

are neither lengthened orshortened;

•

Bernoulli hypothesis: plane se tionsremain plane,thus onlydeformations nor-maltothe undeformedbeamaxis are onsidered. This isequivalentto thefa t

that shear deformations are negle ted;

•

the beam deformationisonly alongthe

x

axis. Thisdeformation

w

istherefore written asa fun tionof the oordinate

y

dened along the beam length and of time

t

:

w = w(y, t);

•

the hypothesisof geometri allinearityis veried. This isequivalentto thefa t that the deformationshavea innitelysmallamplitude. The normal

longitudi-nalstrain tensor

ǫ

y

is therefore alinear fun tion of displa ement and rotation:

ǫ

y

= −x

∂

2 _w

∂y

2 .

Under these hypotheses and assuming that a exion moment

m

y

is a ting on the beam, the al ulus of potential and kineti energies lead to the following movement

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∂

2 _w

∂t

2 +

Y I

m

l

∂

4 _w

∂y

4 = 0

(2.2) where

m

l

=

m

L

isthelineardensityofthe beam,

I

theareamomentof inertia(se ond moment ofinertia) ofthe beam ross se tion about the beam neutralaxisand

Y

the Youngmodulusof the beammaterial. Moreover, for abeam ofre tangularse tionof

height

h

and width

l

, we write the area inertialmomentum as(see [30℄)

I =

lh

3

12

and

the linear density as

m

l

= ρlh = ρS

, where

ρ

isthe density of the beam material.

Con erning the initial onditions, they are dened as:

w(y, 0) = w

0 (y)

and

∂w

∂y

(y, 0) = w

1 (y)

(2.3)

where

w

0

and

w

1

stand for the initialdeformationand velo ity respe tively.

Clamped-free beam

As one an read in referen e [30, Chapter 8℄, the boundary onditions of the beam

are written for the lamped side by onstraining the transverse deformationand his

derivative tobe null:

w(0, t) =

∂w

∂y

(0, t) = 0

(2.4)

and for the free side by onstraining that the bending moment and Kelvin-Kir ho

edge rea tion (whi h depends on the transverse shearing for e and the derivative of

the bendingmoment) are alsoequal tozero:

∂

2 _w

∂y

2 (L, t) =

∂

3 _w

∂y

3 (L, t) = 0.

(2.5)

Of ourse, other boundary onditionsare possible(see [30, Chapter 8℄)and some

of them willbeused latter inthis work (forthe "free-free"beam forinstan e).

First, the beam vibrationresponse is obtainedby solving the homogeneous

equa-tion (2.2) with the initial onditions (2.3) and the boundary onditions (2.4) and

(2.5). In our ase, homogeneousbeam with onstant se tion,it ispossibleto nd an

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variable separation method orFourier de omposition method[30, Chapter 8℄:

w(y, t) =

∞

X

i=1

Y

i

(y)q

i

(t).

(2.6)

To ensure the onvergen e of the series, we hoose the fun tions

{Y

i

}

i

as a set forming aHilbert orthogonalbasis

(L

2 ₎

ofthe eigenfun tions ofthe spa edierential

operator

∂

4 ∂y

4 = ∆

2

. Theexisten eofthis basisisduetothefa tthat

∆

2

isa ompa t

and symmetri operator [32℄. Therefore, the fun tions

{Y

i

}

i

have to be a solutionof the eigenvalues problem:

d

4 _Y

i

(y)

dy

4 = λ

i

Y

i

(y), y ∈ [0, L]

(2.7)

Y

i

(0)

=

dY

i

dy

(0) = 0,

d

2 _Y

i

dy

2 (L) =

d

3 _Y

i

dy

3 (L) = 0.

whi hhas aninnity of solutions

(λ

i

, Y

i

)

detailedbelow.

Sin e

{Y

i

}

i

isan orthogonalbasis, one an use the s alarprodu t to ompute the beam displa ement

w

:

w(y, t) =

∞

X

i=1

< w(y, t), Y

i

(y) > Y

i

(y) =

∞

X

i=1

q

i

(t)Y

i

(y)

where

< Y

i

, Y

k

>= δ

ik

, the Krone ker delta symbol, equal to

1

when

i = k

and

0

otherwise.

Combining the previous equation with (2.2), we an rewrite the homogeneous

equation as:

∞

X

i=1

q

i

d

4 _Y

i

dy

4 +

ρS

Y I

∞

X

i=1

d

2 _q

i

dt

2 Y

i

= 0.

Using (2.7)weget:

∞

X

i=1

q

i

λ

i

Y

i

+

ρS

Y I

q

¨

i

Y

i

= 0.

The s alar produ t with

Y

k

, for

k ∈ N

∗

gives:

∞

X

i=1

q

i

λ

i

< Y

i

, Y

k

> +

ρS

Y I

q

¨

i

< Y

i

, Y

k

>

= 0

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and using the orthogonality of the hilbertianbasis, weget:

q

i

λ

i

+

ρS

Y I

q

¨

i

= 0.

Therefore, the inemati parameters

q

i

verify the dierentialequations, for

i ∈ N

∗

:

¨

q

i

(t) +

Y Iλ

i

ρS

q

i

(t) = 0,

(2.8)

q

i

(0) =< w

0 (y), Y

i

>

L

2 ,

˙q

i

(0) =< w

1 (y), Y

i

>

L

2 .

and the modaldispla ements

Y

i

verify the dierentialequations (2.7).

Therefore, the solutionsof the ordinarydierentialequation (2.8) are given by:

q

i

(t) = E

i

cos ω

i

t + F

i

sin ω

i

t

where

ω

i

=

s

λ

i

Y I

ρS

(2.9)

and

E

i

,

F

i

are omputed fromthe boundary onditions.

Wethenndthemodaldispla ements

Y

i

bysolvingthedierentialequation(2.7). From(2.9),weinferthatthereareonlytwopossible asesfor

λ

i

forthe " lamped-free"beam:

λ

i

= 0

and

λ

i

> 0

. The third ase

λ

i

< 0

is not valid, sin e it willimply that, as the other plate oe ients are positive, there are vibration modes with a

omplex naturalangular frequen y.

Let usrst onsider the simpler ase when

λ

i

= 0

. From (2.7)we have

d

4 _Y

i

dy

4 (y) = 0

whi hhas apossible solutionofthe followingshape:

Y

i

(y) = A

i

y

3 _{+ B}

i

y

2 + C

i

y + D

i

.

Solving this equation using the boundary onditions we nd the oe ients

A

i

=

B

i

= C

i

= D

i

= 0

, thus

Y

i

(y) = 0

. This solution is again not valid sin e, as detailed

earlier, the

Y

i

(y)

are forming anorthogonal basis thus they an't beequal tozero. Let us now onsider the ase

λ

i

> 0

. Again we need to solve (2.7) with the boundary onditions. There are several approa hes in the literature for writing the

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total expression of the modal displa ement as a sum of sine, osine, hyperboli sine

and hyperboli osine fun tions, ea h fun tion multiplied by an unknown onstant

whi hneeds to be determined.

Another more elegant and faster approa h is the one proposed by [56℄. We write

the solution of the equation as:

Y

i

(y) = A

i

s

1 (Ω

i

y) + B

i

c

1 (Ω

i

y) + C

i

s

2 (Ω

i

y) + D

i

c

2 (Ω

i

y)

(2.10) where

(Ω

i

)

4 = λ

i

=

ρS

Y I

(ω

i

)

2

(2.11)

was used to simplify the writing. The fun tions

s

1

,

c

1

,

s

2

,

c

2

are independent and dened as:

s

1 (Ω

i

y) = sin(Ω

i

y) + sinh(Ω

i

y),

c

1 (Ω

i

y) = cos(Ω

i

y) + cosh(Ω

i

y),

s

2 (Ω

i

y) = − sin(Ω

i

y) + sinh(Ω

i

y),

c

2 (Ω

i

y) = − cos(Ω

i

y) + cosh(Ω

i

y).

Asusual,the onstantsfromthedispla ementequation(2.10)arefoundbywriting

the boundary onditionsof the beam. Asit an be seen,the fun tions:

s

1

,

c

1

,

s

2

and

c

2

an be easily obtained one from another by a simple derivative operation. Thus, the boundary onditions, that use the derivative of the oordinate up to the third

order,areveryeasytoexpress. After he kingtheboundary onditionswenoti ethat

we haveonly four equationsbut ve unknown elements: