1
1
WITH FULLY NONLINEAR COUPLING CONDITIONS
ABSTRACT
Theresearch ofH.T.B. wassupp orted inpartbytheAirForceOceof ScienticResearch undergrants
AFOSR-90-0091andAFOSR-F49620-93-1-0198. Thisresearchwasalsosupp ortedbytheNationalAeronautics
and Space Administrationunder NASA Contract Numb ersNAS1-18605 andNAS1-19480 whileH.T.B. was
a visiting scientist and R.C.S. was in residence at the Institute for Computer Applications in Science and
Engineering(ICASE),NASALangleyResearchCenter,Hampton,VA 23681. H.T. Banks
Centerfor ResearchinScienticComputation
North Carolina State University
Raleigh, NC27695
R.C.Smith
Departmentof Mathematics
IowaState University
Ames,IA 50011
Ametho dologyforestimatingphysicalparametersinaclassofstructuralacousticsystems
is presented. The general mo del under consideration consists of an interior cavity which is
separatedfromanexteriordisturbancebyanenclosingelasticstructure. Piezo ceramicpatches
are b onded to or emb edded in the structure; these can b e used b oth as actuators and
sen-sors in applications ranging from the control of interior noise levels to the determination of
structural aws through nondestructive evaluation techniques. The presence and excitation
of the patches, however, changes the geometry and material prop erties of the structure as
wellas involvesunknown patch parameters,thus necessitatingthe developmentof parameter
estimationtechniqueswhichareapplicableinthiscoupledsetting. Indevelopingaframework
for approximation, parameter estimation and implementation, strong consideration is given
to the fact that the input op erator is unb onded due to the discrete nature of the patches.
Moreover,the mo del is weakly nonlinear as a result of the coupling mechanismb etweenthe
structural vibrations and the interioracoustic dynamics. Within this context,an illustrating
mo delisgiven,well-p osedness and approximation resultsare discussedandan applicable
pa-rameterestimationmetho dologyis presented. The schemeisthen illustrated through several
numericalexampleswithsimulationsmo delingavarietyofcommonlyusedstructuralacoustic
The recent success of piezo ceramicmaterialsas sensors and actuators inapplications
involv-ing structural vibrations has spawned intense study into questions regarding the mo deling
of piezo ceramicactuator/sensor interactions with underlying structures, the optimal design,
placementand numb erof actuators/sensors to b e used,and the developmentofeective
con-trol strategies ina varietyof environments. When b ondedto or emb eddedinathinstructure
(b eam, plate or shell), piezo ceramic patches derive their actuating capacity from the
piez-electric prop erty that an induced voltage pro duces a strain in the material thus leading to
the p otentialfor pro ducing in-planeforcesand/or momentswhenthe patches are mountedin
pairs. Conversely,their sensingcapabilities are due to the dual piezo electriceect;namely,a
mechanical deformationleads to the generation of a prop ortional voltage across the element
whichcan then b e used to measureaccumulatedstrain.
However, the b onding or emb edding of patches in the underlying structure changes not
only the geometry of the structure but also physicalprop erties such as the density, stiness,
Poissonratio and dampingco ecients,when indeed,many of these parameters are unknown
evenforthehomogeneous,uniformhoststructurematerial. Thisnecessitatesthedevelopment
ofeectiveparameteridenticationmetho dstob eusedwhenestimatingsystemparametersin
applications such as exp erimentalmo del validation, the determination of optimal placement
and numb er of patches, the use of piezo ceramicpatches in nondestructive evaluation (NDE)
techniques, and the implementation of mo del-based control schemes. We p oint out that the
estimationof physical parameters in this setting diers from that considered in muchof the
previous literature (see [6] and the references therein) in that here the patch contributions
to the systemlead to unb ounded (discontinuous) inputand output op erators due to the fact
that the patches coveronlydiscrete p ortionsof the structure.
Parameter estimation metho ds for distributed parameter systems involving unb ounded
op erators have b eendevelop ed and testedin the case inwhichpiezo ceramicpatchesare used
as sensors and actuators when b onded to a transversely vibrating b eam[15, 16]. There,
t-to-data techniques involving PDE mo dels were develop ed which could b e used to estimate
unknown b eamparameters given various data forms. Moreover, in that setting, results p
er-tainingto convergenceand continuousdep endenceon datawereobtainedinavarietyof cases
involving physically tractable metho ds forexciting the system and measuring data.
In this work, we develop an analogous metho dology which can b e used for estimating
physical parameters in structural acoustic systems. In the systems of interest, an exterior
disturbance (e.g. noise source, mechanical or electromagnetic force) is separated from an
interior cavity by a thin elastic structure (a b eam, plate or shell). As energy is transferred
from the exterior eld to the structure, vibrations develop which then lead to unwanted
interior noisethrough acoustic/structureinteractions. Control ofthis unwantedinterior noise
is accomplished through sensing and actuating via piezo ceramic patches which are b onded
to the structure. Before mo del-based control schemes can b e implemented, however, the
physicalparametersofthestructure(whichnowincludesthepatches)mustb eestimatedfrom
data which iscollected b oth on the structure and from the acoustic resp onse in the enclosed
cavity. Although similarities exist b etween the problem of estimating physical parameters
f to warrant in-depth study of techniques for the latter coupled system. Finally, we note that
although the initial imp etus for developing distributed parameter estimation techniques for
structural acoustic systems was motivated by mo del-based noise control considerations, the
same techniques can b e used when p erforming vibration analysis or using NDE metho ds to
determinestructural aws inthese coupledsystems.
The structural acousticproblem used here to motivate and illustratethe development of
an appropriate parameter estimationmetho dology consistsof a 2-D enclosed cavity which is
separated from the p erturbing exterior disturbance by a thin b eam. This mo del represents
a 2-D slice from a 3-D mo del for several exp erimentalapparata b eing used in the Acoustics
Division,NASALangleyResearchCenter,totestmo deling,parameterestimationand control
strategies. We add, however, that the metho dology b eing presented is equally valid for
es-timatingparameters in many 3-D mo dels representing various exp erimentalsetups currently
in use (see the mo dels in [9, 10 ]). This 2-D mo del was chosen simply b ecause it simplies
the discussion and more clearly illustratesthe pro cess involved in developing the parameter
identication techniques.
Moreover, the mo deling, mathematical, and computational considerations presented here
are valid whether the external sourceor disturbance of the elasticstructure (denoted by in
the mo del b elow) isacoustic, mechanicalor electromagneticexcitationof the structure. The
principleideas inthis pap er are also p ertinentwhetherwe assumethat the piezo ceramicsare
b onded to the structure or are emb edded within the structure. For the sake of deniteness,
we assume that the piezo ceramicpatches are b onded to the structure (the structural mo dels
are somewhat dierent if the patches are emb edded).
The mo delb eingusedtoillustratethe metho dologyisweaklynonlineardue tothemanner
inwhichthe structural vibrationscouplewith the interior acousticelds. Whilelinearization
provides a very go o d approximation to the system dynamics (see [1]), we retain the
non-linearity here so as to illustrate some of the general analytic assumptions which are made
when extending well-p osedness and parameter convergenceresults for the corresp onding
lin-earproblemtoaweaklynonlinearproblemofthistyp e. Thisalsofacilitatesthedemonstration
of numerical techniques which can b e used when simulating, testing and implementing the
parameter estimation scheme in the nonlinear problem. In discussing parameter estimation
metho ds for structuralacoustics problemsofthis typ e, our emphasisis on the formulationof
theprobleminamannerwhichisconducivetoapproximationandimplementationb othinthe
linear and nonlinear formsas wellas undera variety ofinteriorcavity dampingassumptions.
In the second section of this presentation, a mo del for the 2-D system b eing used to
illustrate the parameter estimation metho d is presented. Details regarding the mo deling of
the acoustic and structural comp onents as well as coupling conditions are given, and care is
taken to motivate the assumptions which lead to various damping conditions in the system
mo del. In formulating the strong form of the system mo del, details are also givenregarding
the interactions b etween the piezo ceramic patches and the underlying structure (b eam) as
wellas theweaklynonlinearinteractionsb etweentheb eamand theinterioracousticeld. To
provideaformulationwhichisconduciveto approximationinthecontextofunb oundedinput
op erators as well as facilitates the discussion of well-p osedness results, the weak form of the
systemequationsisthendevelop edandp osed intermsofsesquilinearformsandthe b ounded
R S
\ 6
0
=1
2 System Model
b b D
b
pe pe Dpe
b` b` b` D
b`
N
i i
t t
a b h
E ; c
w t;x
T b
E c
T E c
@ t
underlyingthe well-p osedness results forthe linear and nonlinear problemsare discussed.
A discretizationmetho d suitableforsimulationsand theimplementationofthe parameter
estimationmetho disoutlinedinSection3. Thisdiscussioniskeptrelativelybriefsincedetails
regardingthecorresp ondingnitedimensionalsystemforthelinearizedproblemcan b efound
in[1]. However,thesectiondo esprovideadditionaldetailsconcerningthediscretizationofthe
nonlinear comp onentofthe op erator and abriefalgorithmforcarrying outthis discretizetion
is included.
A parameter estimation scheme suitable for data consisting of displacement, velo city or
accelerationmeasurementson the b eam,voltage measurementsfromthe patches,or pressure
measurement inside the cavity is presented in Section 4. Assumptions on the form of the
unknown parameters are discussed and conditions leading to convergence and continuous
dep endenceon data results for the linear and nonlinear problems are outlined.
In Section5,numericalexamplesdemonstratingthe parameterestimationtechniqueswith
a variety of data typ es and metho ds for exciting the system are presented. Sp ecically,
examplesare given in which the force to the system is provided by a numerically simulated
acoustic source, simulated voltage inputs to the patches, and a simulated hammer impact
to the b eam. The simulated data under comparison inthe examplesconsists of acceleration
values of the b eam,voltage (accumulatedstrain)values from the patches,and voltagevalues
in conjunction with interior acoustic pressure values. The conclusions from this study and
physical considerations concerning the implementationof the metho dare summarizedinthe
nal section of the pap er.
The mo del of interest consists of an exterior disturbance which is separated from an interior
cavity( ) by a commonelasticb oundary 0 ( )that is mo deledby an Euler-Bernoullib eam
as depicted in Figure 1. The b eam is assumed to have length , width and thickness .
The Young's mo dulus, mass density (in mass p er unit volume) and damping co ecient for
the homogeneous b eamare denoted by and ,resp ectively. Due to the nature ofthe
exterior forces and the manner in whichthe patches are excited,we will b e considering only
transversevibrations ( ).
Bonded to the b eam are piezo ceramic patches which are mounted in pairs as depicted
in Figure 1. In this discussion, it is assumed that the patches have thickness , width ,
Young's mo dulus , density , and damping co ecient . Moreover, it is assumed
that the b onding layersfor eachpatch haveequalthicknesses,Young's mo duli,densitiesand
dampingco ecients,and theseparameters are denotedby , , and , resp ectively.
Weemphasizethat these assumptions are made solelyforclarity of presentation,and similar
resultscanb eobtainedinananalogousmannerforthemoregeneralcaseinwhichthe patches
and b onding layershave dieringthicknesses and materialprop erties(see,forexample,[12]).
Finally, we assume that inside the cavity, there is a region ~
=
~
, p ossibly all of
, with ~
( ) = , which provides a rst approximation to the interior material (eg.,
j
f
n
^
Ω
y
(t)
Γ
^
Γ
0
(t)
0
a
w(t,x)
Γ
0
1
~
Ω
Ω
~
2
Ω
~
3
Ω
~
4
n
R
f 0
0
0
Fig. 1. The2-D domain.
2.1 Acoustic Component
t N
t
P
~
U
P t;~x P ~x p t;~x
t;~x ~x t;~x
~
U t;~x ~
U ~x ~u t;~x
~
x x;y
<
t
E materialthatabsorbsenergyinotherexamples)whichdisruptsanddampstheinterioracoustic
elds. This region is assumed to have p ositive measure and may or may not b e small as
comparedto ( ) (seeFigure1 where =4).
Weconsiderrst the acousticwavedynamicsinside the cavity( ). The variablesof interest
to us are the pressure , density , and the velo city , each of which can b e represented in
termsof a meanand uctuatingcomp onent
( )= ( )+ ( )
( )= ( )+^( )
( )= ( )+ ( )
(here we have taken = ( ) and are assuming that the rate of sound travel is suciently
rapid so that littleheat transfer takes place). For the range of magnitudes involvedin these
problems( 140dB), itisusualto assumelinearrelationswhenconsideringconstitutivelaws
andforcebalancing[20],andwemakethatassumptionthroughout theanalysiswhichfollows.
In theregion( ) ~
,airdampingisomittedduetothe relativelysmalldimensionsofthe
typ e of exp erimentalcavities of interest to us. Hence in that region, an increase in pressure
( 0 2 2 0 2 2 2 2 0 0 0 0 f f t t f f f t t @ @t 0 r1
0 r1
0 r1 0 r1
0r 2 2 2 2 r2 r2 r2 0r E p ~s
p E ~s
~s t;~x
~s ~u
p E ~s d s~
E d
<
@ ~s
@t
p
; ~x t =
; ~x
@ p
@t E
p ; ~x t =
@ p @t E p d
p ; ~x :
~ u ~s
~u ~! ~u t
~
! ~u
t t
~u :
t
= or = (2.1)
( ( ) denotes the displacement of the center of gravity of an innitesimal element of the
mediumandsatises = ). Onthe otherhand,the materiallump edin ~
willprovidesome
mediumdampingand here we assumethat a changeinpressure yields
= ~ ~ (2.2) where ~ and ~
denote the bulk mo dulus of elasticityand damping co ecientof the medium
in ~
. Wep ointout thatthis useofageneralizedHo oke'slawinwhichstressisprop ortionalto
alinearcombinationofstrainandstrainrateisdoneundertheassumptionthatrelativelylow
acousticfrequenciesare excited( 1000 Hz),and issimilarto the constitutivelawleadingto
Kelvin-Voigtdampingofvibrationsinelasticmaterials. Wealsoemphasizethatthis damping
mo delshouldb econsideredasarstapproximationtotheactualacousticdampingmechanism
in the medium ~
, and dep ending on the sp ecic materials involved,the manner of acoustic
excitation, and the geometry of the physical system, more comprehensive mo dels may b e
required to accurately describ e the mediumdamping.
Force balancing in the acoustic cavity,weobtain the relation (the Euler equation)
= (2.3) where = ( ) ~ ~ ~
denotestheequilibriumdensityofthemedium((2.3)isequivalenttothelinearizedmomentum
equationinuid dynamics). Bytaking two timederivativesof(2.1) and (2.2),the divergence
of (2.3), and eliminatingcross terms,one arrivesat
= 1 ( )
~ = ~ ~ 1 + ~ ~ 1 ~ (2.4)
Taking the curlofthe momentumequation(2.3) andnotingthat = we obtain (
)=0. Hencethevorticity = isconstantintimein( ). Underthe assumption
that the initial vorticity (0) is zero, we may conclude that =0 for all timeor that
the ow is irrotational in ( ). Thus in ( ), there exists a scalar velo city p otential such
that
= (2.5)
The ow is more complex near and in the region ~
as a result of the viscous eects and
medium damping. This can p otentially lead to rotational comp onents in the acoustic eld
whichingeneral necessitatestheuse of avectorp otential. Asarst approximation, however,
we are assuming that the rotational comp onents near and in ~
are negligibleand a relation
( ) 8 < : ( 2 2 2 2 2 2 2 2 2 0 2 2 1 2
2.2 Beam Component
r 0 2 2 2 2 2 2 2
r 1 2
2 [ M 0 M t tt f f t f f t tt t t
b b` b` pe pe
@ @t p ; p @ @t t p @ @t E
; ~x t =
@ @t E d
; ~x :
n
c ~x d ~x
c ~x
E = ; ~x t =
E= ; ~x
; d ~x
; ~x t =
d= ; ~x ;
@
@t
c d ; x;y t ;t> ;
n ; x;y ;t> :
d ~x
~
x t = @ t t
t
w
w @
@x
t;x t;x;w t;x f t;x <x<a; t > ;
w t;
@w
@x
t; w t;a
@w
@x
t;a t> ;
f
t;x;w t;x
x x
x hb b T T x
=0
whichimpliesthat = since no acoustic sources or sinks are present in( ). Use of this
pressure expression (actually = ) after dierentiation in time once in equations (2.1),
(2.2) then yields
= 1 ( )
~ = ~ ~ 1 + ~ ~ 1 ~
Finally,hardwall b oundary conditions are assumedsince one of the exp erimental
congu-rations(acylinderwithendplates)thatwehavemo deledhasconcretewalls. With^ denoting
the outward unit normal to the cavity and ( ) and ( ) denoting the sp eed of sound and
dampingco ecientsgivenby
( )= ( ) ~ ~ ~ ~ ( )=
0 ( )
~ ~ ~ ~
the acoustic resp onsecan b e mo deledas
= 1 + 1 ( ) ( ) 0
^=0 ( ) 0 0
(2.6)
We emphasize that this mo del was derived under the assumption that the only acoustic
damping in the cavity o ccurs in the region ~
and hence ( )= 0 in the rest of the chamb er
and b oundary, i.e. for ( ) ~
and ( ) =0 0 ( ). Moreover, we have assumed that
the ow is irrotationalin the region( ).
Throughforceandmomentbalancing,theequationofmotionforthetransversedisplacements
of the b eamare found to b e
+ ( )= ( ( ))+ ( ) 0 0
( 0) = ( 0) = ( )= ( )=0 0
where is the total b eam moment, is the force due to the exterior disturbance eld and
( ( )) is the backpressure due to the ensuing acoustic waves inside the cavity (this
lattertermisingeneralnonlinearsinceitseecto ccurson thesurfaceofthe vibratingb eam).
For pairs ofpatches having edges and , the density of the structure is
(
h i
M 0
K
K
0 0
1 2
1 2
2
2
3
2
3
3 3
3
3 3
31
3 3 3
3 3
3
3
3
31
pe
pe
pe
D
pe
B
pe
b b` b` pe pe pe
D Db D
b`
b` Dpe pe pe
B
pe b`
b` pe b` b` pe b`
b`
D
pe
D
pe
x
; x x x
;
x x
t;x M t;x M t;x
M
M
M t;x EI x @ w
@x
c I x @ w
@x @t
M t;x V x
V
EI x E
h b b
E a E a x
c I x c
h b b
c a c a x
E bd h T T :
a a a h= T h= a h= T T
h= T
d
c
c ( )=
1
0 otherwise
lo calizes the patcheects b etween the endp oints and (see[11 ,12 ]for details).
The general b eammoment
( )= ( ) ( )
consists of an internal comp onent ,dep ending on material and geometricprop erties ofthe
b eamand patches,and an external comp onent (the control term)whichresultsfromthe
activationof the patches through an applied voltage (seeFigure 2(a)).
For a b eamundergoing pure b ending motion with out-of-phase excitation of the patches,
the internaland external momentsare givenby
( )= ( ) + ( )
( )= ( )
where isthe voltage into the patches. As shown in[11 ],the stiness, damping and control
constants forthe combinedstructure are
( )=
12 +
2
3
[ + ] ( )
( )=
12 +
2
3
+ ( )
= ( +2 + )
(2.8)
Theconstants and givenby =( 2+ ) ( 2) and =( 2+ + )
( 2+ ) result fromthe integrationof stressesthrough the b onding layerand patch,and
is a piezo ceramic constant which relates the amount of strain pro duced in the patch to
the levelof voltage b eing applied.
At this p oint itis worth commenting further on the damping term whichistakento
b eacombinationoftheKelvin-Voigtdampingco ecientforthepatchandthedampingwhich
resultsfromthepro ductionofcurrentwhenthestructurevibrates. Thislattercontributionto
the damping resultsfrom the piezo electriceect ofthe patches whichdictates that avoltage
is pro duced when the patch is subjected to in-plane strains. Under the assumption that the
Kelvin-Voigt (material) and electrical damping have approximately the same typ es of eect
inthe patch,we havecombinedthe twointo the co ecient whichmust b e consideredto
b e unknown and like the other parameters, must ultimately b e estimated using data tting
techniques with exp erimentaldata when considering actual applications. We also p oint out
thattheexpression(2.8)caneasilyb egeneralizedtoincludethep ossibilityofdieringmaterial
prop ertiesin the twopatches or b onding layers(again, see[11, 12 ]).
The fact that the patches generate avoltage when strainedimpliesthat they can b e used
+
+
Voltage Input
Generated
Moment
Actuator
Strain
Accumulated
Generated
Voltage
Sensor
(a)
(b)
2
1
0
0 0
Z
" #
S Fig. 2.
2
2
2 1
0
0
1
K K 0
K
0 r 1
0 r 1
s
S x
x
S
S
t
t
t
t
2.3 Coupling Conditions
V t
@ w
@x
t;x dx
@w
@x t;x
@w
@x t;x
t;x;w t;x
w t;x
t;x;w t;x | ; <x<a ;t>
w
t
t;x;w t;x t;x;w t;x |
Acoustic cavity with piezoceramic patches; (a) patch generation of pure bending moments,
(b) patch sensingof accumulated strain, and (c) xed domain and boundary .
( )= ( ) = ( ) ( ) (2.9)
where the sensor constant dep ends up on piezo electric material prop erties as well as the
geometryandsizeofthepatch(see[18]). Hencethevoltageprovidesameasureofaccumulated
strain inthe b eam (seeFigure 2(b))thusenabling the patch to serveas asensor in a variety
of applications involving the measurementof b eamvibrations.
In the mo del discussed thus far, the structural and internal acoustic resp onses are coupled
through the backpressure ( ( )) on the surface of the b eam. A second coupling
equation isthe continuity ofvelo city(or momentum)condition
( )= 1
( ( )) ^ 0 0
whichresultsfromtheassumptionthatthe b eamisimp enetrableto air(orwhateverthe uid
in the interior cavity). We p oint out that the velo citycondition provides aform of damping
to theb eamwhichissimilarto thatobtained withthe incorp orationof viscous(air) damping
eects(mo deledby atermofthe form inthe b eamequation). As notedinthe examples,
theinternalKelvin-Voigtdampinginconcertwiththe couplingeectsdueto thecontinuityof
velo cityand backpressurecausesab eamresp onsewhichdiers somewhatfromthatobserved
with an uncoupled, undamp edb eam havingthe samedimensions.
The mo delwehave develop ed has nonlinearities inthe (i)variable domain( ),(ii) back
i
i !
X 2
r 1 2
0 r 1
0 K
K K
111
K 2
0
2
2
2
2
3
2
2
2
=1
0 0
1 1
tt t
t
tt D
t
s
i B
i
i pe
t t
i
th
pe
th
S
i
B
i
D
B
i
D s
c d x;y t ;t> ;
n x;y ;t> ;
t;x;w t;x | w t;x <x<a ;t> ;
w @
@x EI
@ w
@x
c I @ w
@x @t
t;x;w t;x f t;x @
@x
u t x
<x<a ;
t > ;
w t;
@w
@x
t; w t;a
@w
@x
t;a t> ;
;x;y x;y ; w ;x w x
;x;y x;y ; w ;x w x :
u t i
i
; i
; ;s ;EI c I
;;EI c I
For the coupledsystem in which pairs of patches are b onded to the b eamand excited
out-of-phase,the acoustic,structuraland couplingcomp onentsjust discussedcan b ecombinedto
yieldthe nonlinear mo del
= 1 + 1 ( ) ( ) 0
^ =0 ( ) 0 0
1
( ( )) ^= ( ) 0 0
+ +
= ( ( ))+ ( )+ ( ) ( )
0
0
( 0) = ( 0)= ( )= ( )=0 0
(0 )= ( ) (0 )= ( )
(0 )= ( ) (0 )= ( )
(2.10)
Here ( ) is the voltage b eing applied to the patch and denotes the characteristic
function overthe patch.
We p oint out that the piezo ceramic material parameters (see (2.9)) and =
1 as well as the b eam parameters and are considered to b e unknown and
are estimated using inverse problem techniques as discussed in later sections. While the
expressionsgivenin(2.7)and (2.8)canb eusedas startingvaluesinthe parameterestimation
routines, exp erimental evidence (see [15, 16 ]) has indicated that the nal parameter values
canvaryquitesignicantlyfromtheanalyticvaluesduetothecontributionsfromtheb onding
layer,variation inthe measurementof physicalconstants, and nonuniformities inthe various
materials. This combined with the lack of analytic expressions for the damping constant
necessitates the estimationof these parameters b efore mo del-based control strategies can b e
implemented.
We also emphasize that the parameters and are piecewiseconstant in
na-ture due to the presence and diering material prop erties of the b onding layer and patches
(see(2.8) as wellas the resultsin [15]). This leads to dicultieswith the strong form of the
systemequationssince itnecessitatesthe secondderivativesof the Heavisidefunction
(equiv-alently, derivatives of the Dirac delta) thus yielding an unb ounded control input op erator.
Thedierentiationofthe discontinuousmaterialparametersalsoleadsto dicultieswhen
ap-proximatingthedynamicsof thecoupledsystem. Toavoid theseproblems,itisadvantageous
to formulate the problem in weak or variational form (the use of the variational form also
p ermitsthe use of basis functions having less smo othness than required for those used when
0 f f i i 0 0 0 0 0 0 0 0 0 0 0 0 0 Z ( ) Z Z n o Z X Z R R R R Z Z Z Z Z Z
n h io
Z
X
Z
2.5 Weak Form of the System Equations
f
f t t
t t t t D s i B i i pe t t t t D s i B i i pe
0r 1 r 1
0r 0 r
Z
r 1 r
0
0 r 1
r 1
r 1r r 1r
0
K
2
r 1r
r 1r 0
K 0 1 1 0 1 1 () 0 2 0 0 2
0 () 0
2 2 2 2
0
=1
2
0
()
0 () 0
0 0 2 0 0 0 2 2 0 2 0 2 2 0 =1 2 0
| n
~v ~v
;w
;w ;;w
p
t;x;w t;x | n
t | n
t;x;w t;x |
t;x;w t;x n
x x
x x a
c d c dw
w d w EID w D c ID w D w d
u t D d fd:
;a ;` <x<a; y
c
d! wd
d! EID w D d
c
d d! c ID wD w w d
u t D d fd
mo del (2.10) is completely equivalent to the nonlinear mo dels whose linearizations are the
basis of the investigations in [1, 8 , 10 , 11] if one makes the approximation ^ ^
and replaces the coupling terms and = inthose mo dels by and = ,
resp ectively. That is, the p otentialused in those references diers fromthe one used here by
a multiplicativefactor .
An appropriate choice for the state of the second-order problem (2.10) is the pair ( )
consisting of the acousticp otentialand b eamdisplacement. Itfollows immediatelythat with
thischoice,the statefortheprobleminrst-orderformis =( _
_)whichcontainsthe
pressure(since = _
) as wellas the b eamdisplacementand velo city.
To facilitate the writing of (2.10) in weak form we make several assumptions. First we
assume that ( ( )) ( ^+^) is small (this is reasonable if either is small near
0 ( ) or if the b eam has small slop e so that ^ ^). Note that with this assumption,
the velo city term ( ( )) ^in (2.10) can b e approximated by the normal term
( ( )) ^whichariseswhendevelopingtheweakformoftheequation. Inaddition
to makingthis approximation, we multiply the cavity and b eam equations in(2.10) by
su-ciently smo othtest functions and , resp ectively,and integrateby parts (Green's theorem)
in the usualmanner (see[1]). Assumingthat satises the b oundary conditions ( )= ( )
at =0 and = ,wethen obtain
1 ( _ ) + 1 + _ _ + _ + _ + _ ( ) = ( ) + (2.11)
In the integrals ab ovewemakethe further approximations that = where isthe
xeddomain [0 ] [0 ],and = where 0 is theb oundary 0 =0as
depictedin Figure2(c). Wethen obtain the nonlinear rst order variationalform
0
0
0
0
H
V
3 3
H
V
V H
V V
t
H 2
V 2 2
2
f 2 g
r 1r
h i h i
h i h i
0 r 1
! '
!
* ! !+
Z Z
* ! !+
Z Z
* ! !+
* ! !+
2 2
0
1 2
0 0
2 1 2 1
2
0 0
2
0 0
2
0
0 2
0
0
0
2 2
1
2.6 Abstract First Order Formulati on
V H
V V H V H L L V
H H L H L H
H H H x x x ;a
H V
w ;
c
d! w d
w ;
d! EID w D d
; ; ;
; ; ; :
| w
H V V , H
H , V H
;w ; V
The statespaceand spaceoftest functionsare takento b ethe pro ductspaces =
and = wherethe Hilb ertspaces and are givenby =
() (0 )and =
() (0 ). Here
()and
()denotethequotientspacesof and overthe
con-stantfunctionsand (0 )isgivenby (0 )= (0 ): ( )= ( )=0 at =0
(the use of the quotient space results from the fact that the p otentials are determined only
up to a constant). Fromenergy considerations, the and inner pro ducts are takento b e
= 1
+
= 1
+
while the pro duct space inner pro ductsare givenby
8
9
7
3
= 8 7 + 9 3
8
9
7
3
= 8 7 + 9 3
We p oint out that in the variational form the derivatives have b een transferred fromthe
plate and patch moments onto the test functions. This eliminates the problem of having to
approximate the derivativesof the characteristic function and the Dirac delta as is the case
with the strongform ofthe equations. Moreover,we note thatthe continuityof velo city
con-dition ^= of the strong form b ecomes a natural b oundary condition inthe weak
or variational form (2.11). Hence it is not directly imp osed on any numerical approximate
solutions. The coupling condition willtherefore b e satised only approximately in
computa-tional solutions as well as in the conceptual formulation given in making the approximation
to pass fromthe strong form(2.10) to the variational form(2.12).
The system (2.12)can b e formallyapproximatedby replacing the state variablesby their
nite dimensional approximations and constructingthe resulting matrix system,and it is in
this form that we willconsiderapproximation strategies inSection 4. In order to discussthe
well-p osedness ofthe problem,however,itisadvantageous to p ose theprobleminanabstract
Cauchy format as discussed inthe nextsubsection.
Following the theoreticalworkin [2,3 ],it isadvantageous to formulatethe problemin terms
ofsesquilinearformsand the b oundedop erators whichtheydene(seealso[1, 10 ]for further
examples detailing the abstract formulation of structural acoustic systems in this manner).
We b egin by p ointing out that the Hilb ert spaces and form a Gelfand triple
withpivotspace (furtherdetailsconcerningthebasicdenitionsandfundamental
0 0 0 0 i D V V V V V V V D V H Z Z Z Z n o Z Z Z Z R R
r 1r
r 1r 0
j j j j j j 2
j j
2
h i
n
j j j j j j 2
j j
jr j
jr j
j j
jr j jr j
2
n
j j 0
n 1 0 0 2 2 2 0 2 0 2 2 1 2 1
1 1 1
1 2 2 2 1 1 1 1 2 2
2 3 3
2 4 2 4 2 ~ 0 2 2 0 2 2 1 ~ 2 2 0 2 2 4 2 ~ 2 2 2 4 2 4
2 4 2
2
;
d! EID w D d ;
;
c
d d! c ID w D w d :
; c ; c
; c ; c > V
; ; ; V V ; ; meas
; c ; c
; c ; c > V :
V
;
d
c
d! c I D w d
k d! k EI D w d
c
d! d!
V
meas >
H
; c ; c H
c ;
meas V
(8 9)= 1
+
(8 9)= 1
+ +[ ]
(2.13)
With these denitions, it is straightforward to show that and satisfy various
conti-nuity,symmetryand co ercivity conditions. Namely, satises
(8 9) 8 9 for some lR (b ounded)
Re (8 8) 8 for some 0 ( -elliptic)
(8 9)= (9 8) (symmetric)
(2.14)
for all 8 9 (the b oundedness results from Schwarz's inequality for inner pro ducts in
conjunction with equivalence results for various Sob olev norms, while the -ellipticity and
symmetryof follow directlyfromthe fact that (8 9)= 8 9 ).
The prop erties of dep end up on the form of damping in the cavity . If damping is
includedin aregion ~
as discussedinSection2.1with ( ~
)=0,then satises
(8 9) 8 9 for some lR (b ounded)
Re (8 8) 8 forsome 0 ( -elliptic)
(2.15)
The -ellipticityfollows fromthe factthat
Re (8 8) = +
+
8
with the nal inequality resulting from the observation that = for
all 8 . On the other hand, if the cavity is taken with medium damping in ~
with
( ~
) 0(thecaseof \weakcavitydamping"),thenone can onlyestablishthe weaker
-semiellipticitycondition
Re (8 8) 8 0 ( semielliptic)
for (in this case =0 since (8 8) contains no full acoustic comp onentwhen damping
is weak in the cavity). As discussed in the next section, the inclusion of medium damping
withinthe region ~
where (
~
) =0and the resulting -ellipticb ehaviorof yields
i 0 3 3 3 3 3 Z X 2 4 3 5 3 3 3 3 3 3 3 formally 0 =1 2 2 1 1 2
1 2 1 2
1 2 1 2 2 1 V ;V s i B i i pe V ;V
b t t
t t t t
tt V ;V t t V ;V t i V ;V i t t t t t tt V ;V t V ;V 2L
h i K
2 h 1 1i
0
0
h i h i
1 1 1 !
B
F G Z V 2
2L
h i
Z H 2
Z AZ C Z
V
C Z B F G Z
A f 2H 2 2 g
A
0 0
h i h i
B U;V
Bu; u D d
V ;
F ;f= G z;z ; w
w t;x;w t;x t;x;w t;x t;x;
z t ; z t ; z t ; Bu t F G z;z ;
V
z t t; ; ;w t; V , H
u t ;Bu t
t ;F t t ;G z t ;z t V V
A ;A V;V
A ; ;
i ; A A
t z t ;z t ;w ;;w V H
t t t; t
t; t u t t t
; V;A A H
I
A A
:
t;x;w
t;x; G z;z
z t ; z t ; z t ; Bu t F ;
Hilb ertspace containing the voltageinputs and wedene the op erator ( ) by
9 =
for 9 , where is the usual duality pairing. Finally, the external forcing and
nonlinear p erturbation terms are given by = (0 ) and ( ) = (0 ~
( )) where
~
( ) = ~
( ( )) ( ( )) ( 0) denotes the nonlinear p erturbation to
the linear coupling term.
With thesedenitions, wecan write the system(2.12)in the abstract weakor variational
form
( ) 9 + ( ( ) 9)+ ( ( ) 9)= ( )+ + ( ) 9 (2.16)
for 9 in . We reiterate that the state for the problem in second-order form is given by
( )=( ( ) ( ))in .
Top ose thesysteminrst-orderform,weformthepro ductspaceterms ( )=(0 ( )),
( ) = (0 ( )) and ( ( )) = (0 ( ( ) ( ))) in = and dene the op erators
( ) by
8 9 = (8 9)
for = 1 2 (the existence of and is guaranteed by the b oundedness of and ).
Then, for the state ( )= ( ( ) ( ))=( _
_) in = , the weak form(2.16) is
equivalent to the system
( )= ( )+ ( ( )) (2.17)
in where
( ( ))= ( )+ ( )+ ( ( )) (2.18)
and
dom = 2=(7 3) :3 7+ 3
= 0
(2.19)
(further details concerning the formulation of the rst-order system in the linear case can
b e found in [1]). The representation (2.17) is formal in the sense that the manner in which
dierentiationand the resultingsolution existshas yetto b esp ecied. This willb e discussed
next inthe context of provingwell-p osedness results for the systemmo del.
Finally, in discussing well-p osedness results and parameter convergence in the sections
whichfollow, itproves usefulto comparethe nonlinear system with that which results when
b oth coupling termsare linearized. The latter is found by replacing the term ( ) by
( 0) in the various expressions for the coupled system(equivalently, take ( ) =0).
This yieldsthe second-order variational form
! t
! t
! t
1
2
0
2
1 2
1 2
0 0
0 0
2.7 Model Well-Posedness 3
3 3
3 3 3
3 3
3
3 3 3
3
3 3
3 3
W
3 3
H
3 3 3 3 3
t t u t t
V
meas H
C t
t e t !
V t
t
t; t
u t t t B U;V
u t ;Bu t V V V
t t
V
; V;A A H
A A
:
; ;
> !
t e
V
Z AZ B F
V
A H
n
A
T H
jT j A
T H
H T
A A H!H H C Z
B F G Z V H 2L
B 2 f g 2 2 V
T H T
W f g2
A
A f 2Hj 2 0 2 g
A
0
0
W A A
h i h 0A 0A i
jT j W
A f g2 W A
( )= ( )+ ( )+ ( ) (2.21)
in .
Whilethemainpurp oseof thepap er isthe careful mo delingpresented ab oveandthe
demon-stration (by examples in Section 5 b elow) of a feasible numerical approach to estimation of
parameters, we give here a brief discussion of p ossible theoretical considerations. The ideas
we discuss are admittedly incompletebut theydo make a start on well-p osedness of the full
nonlinear structural acoustics mo del.
The rst step in determining the well-p osedness of the semilinear system mo del is to
argue that generates a semigroup on . As noted earlier, the sesquilinear form is
V-elliptic, continuous and symmetricwhile iscontinuous and -ellipticif strong damping
( (
~
)=0)isincludedviatheregion ~
or -semiellipticif dampingisweakoromitted
inthecavity. Inb othcases,theLumer-Philipstheorem(withargumentssimilartothosegiven
in pages 82-84 of [2]) can b e used to prove that the op erator dened in (2.19) generates
a -semigroup ( ) on the state space . The semigroup satises the exp onential b ound
( ) for 0 (where in fact, =0 due to the fact that isdissipative as shown in
[2]). Moreover,ifstrongmediumdampingisincludedviathe region ~
(whichimpliesthat
is -elliptic),the semigroup ( )is analytic on .
For the problem thus p osed, the state lies in which implies that the semigroup ( )
generated by :dom is dened on . The nonlinear forcing term ( ( ))=
( )+ ( )+ ( ( )),however,liesin ratherthan sincethecontrol term ( )
denes the pro duct space control term ( ) = (0 ( )) 0 = .
To remedy this, \extrap olation space" ideas and arguments similar to those presented in
[3,4 , 19] are used to extend the semigroup ( ) on to a semigroup ~
( )on a larger space
0 so as to b e compatiblewith the forcing term.
As detailedin [10 ],the space of interestis denedin termsof dom where
dom = =(8 9) 9 8 9
=
9
8 9
Sp ecically,the space =[dom ]is takento b e dom with the inner pro duct
8 9 = ( )8 ( )9
for some arbitrary but xed with (recall that the original solution semigroup
satises the b ound ( ) ). As proven in [4], the resulting norm is equivalent to
the graph norm corresp onding to . Moreover, we have that 0 = [dom ]
t t t t Z ! a priori 0 0 0 2 0 0 2 2 3 3 3 H 3 3 3 3 3
Lemma 1. (Well-Posedness of the Nonlinear System)
A W
A A 2W
A h A i
2H 2W
A A AH H
A T W
T H W T
A
7!
7!
T
W
C 2 !
Z T Z T 0
Z
Z 2 Z Z
Z
C 2 !
W
0
C Z B F G Z Z
Z 2 H
T ; C t t t F B
t u t
t F t
t
X
;T X X t ;T
X
t t t s
Bu s F s G s
ds
Bu F G L ;T ;V
t
;T X X
Bu F
G z;z ; w
F Bu
G z;z
F Bu
t; t u t t t t
H
V t
Fromthe denition of and the equivalenceof the norm with the graphnorm
corre-sp onding to ,we can dene ~
2 by
~
2 ( )= 2
for all2 , . Withthis denition and theRieszrepresentationtheorem,itisshown
in[10]that ~
isan extensionofthe originalop erator fromdom to allof . Finally,
as proven in [4],the op erator ~
is the innitesimalgenerator of a -semigroup ~
( ) on
whichisan extensionof ( )from to (notethat ~
( )isalso analyticif strongmedium
cavity dampingisincluded).
Having extended the op erator and hence the generated semigroup to a space which is
compatiblewith theforcing function,weare nowinap ositionto discusscriteriaon the input
terms and which guarantee the existence of a unique solution to the system mo del. In
the corresp onding linear problem, under reasonable regularity conditions on ( ) and
( ),one can immediately arguethe existenceof aunique strong solution to the system
intermsofthe extendedsemigroup ~
( ). For the semilinearproblemof interest,however,the
nonlinear nonhomogeneous terms must satisfy certain continuity criteria in order to obtain
similarresults. For example, if we let denote the reexive Banach space and assume
that :[0 ] dened in(2.18) iscontinuousin on [0 ] and uniformlyLipschitz
continuous on , then the integralequation
( )= ~ ( ) + ~ ( ) 0 ( )+ ( )+ ( ( )) (2.22)
is well-dened for + + ( ) ((0 ) ). Moreover, for (0) = , the solution
( ) of (2.22) is a unique mild solution to (2.17) (see Theorem 1.2, page 184 of [21]). In
addition, if : [0 ] is Lipschitzcontinuous inb oth variables,then it follows from
Theorem1.6, page189 of[21] that (2.22)providesthe strongsolution to (2.17)interpretedin
the sense.
The required continuity of the nonhomogeneous terms and is demonstrated in[10]
and hencethe remainingquestion concernsthe Lipschitzcontinuityof the nonlinearcoupling
term ( ) = (0 ~
( )). This is a dicult condition to assure . However, if we
assume that the input terms and are suciently smo oth so as to assure the necessary
continuity in ( ),then the nonlinearsystemis well-p osed. Theseresults are summarized
in the following result.
Consider the nonlinear system
representedby(2.16),(2.17)or(2.22)andassumethat and aresucientlysmo oth
sothat ( ( ))= ( )+ ( )+ ( ( ))isLipschitzcontinuousinb oth and . Then
(2.22) with is the unique strong solution to (2.16) b oth when strong acoustic
cavity damping is assumed and when weak or no damping is present in the cavity (in
whichcase, is only -semielliptic).
These results can b e further extended when strong acoustic damping is assumed via the
region ~
and hence is -elliptic and ~
=0 y
x
i 3
0
6
X
X X
1
0 0
1
=1
=0
+ =0
7! 7!
n
Z 2H A Z Z
G Z
3 State Approximation
3.1 System Approximation
N
n
i N
i
n
i
n
i
th
N
m
j m
i j N
ij a
i `
j
t u t t F t
;T V
meas
w t;x w t B x
B i
t;x;y t P x P y
Theorem 1. (Linearized System: Equivalence of Solutions)
to (2.20) for suciently smo oth nonhomogeneous terms. Moreover, the following theorem
summarizesthe corresp onding result for the linearizedproblem.
Considerthesystem
rep-resentedby (2.20)or (2.21)andsupp ose thatthe mappings ( )and ( )from
[0 ] to lR and , resp ectively, are Lipschitz continuous. Furthermore, assume that
mediumdamping of the form (2.2) is present in the region ~
with (
~
) = 0.
Thenforeach =dom ~
,wehavethat(2.21)takenwith (0) = has aunique
strong solution givenby (2.22) with ( )= 0. Moreover,this strong solution is equal
to the weaksolution of (2.20).
The pro of of the equivalency b etween strong and weak solutions follows that given in [3]
for general second-order systems with unb ounded input terms. We p oint out that numerous
numerical results have suggested similar results for the nonlinear problem and the case in
which damping is omitted in the cavity even though we have not b een able to extend the
results equatingthe strongand weaksolutions to coverthose cases.
Themo delingandwell-p osednessdiscussionthusfarhasb eenfortheinnitedimensional
non-linearstructuralacousticsystem. Inthis sectionwediscussaGalerkinschemefordiscretizing
theproblemwhichcanb eusedwhensimulatingthesystemdynamics,estimatingthephysical
parameters, and calculating control gains (see [1]). This is accomplished by approximating
the b eamdisplacementandacousticp otentialbyspline andsp ectral expansions, resp ectively.
As detailed in [1] where the corresp onding linear problem was considered, cubic splines are
suitablefordiscretizingtheb eamdisplacementsincetheysatisfythesmo othness requirement
as well as b eing easilyimplementedwhen adaptingto the xed-endb oundary conditionsand
patch discretizations. Sp ecically, the approximate b eam displacement is taken to b e the
linear combination
( )= ( ) ( )
where is the cubic splinewhich has b eenmo died to satisfythe b oundaryconditions.
The acoustic p otentialis approximatedby the Galerkin expansion
0 0 x y 6 0 0 N M M # t # t M
A A w t
# t
# t B
u t F t M M # # g g 6
f g f g
0
0
f g f g
0
2
H 2
H 2H
111 111 2
0 2 111
A 0 0 _ ( ) ( ) = 0 ( ) ( ) _ ( ) + 0 ~ ( )+ 0 ~ ( ) 0 0 (0) _ (0) = " # 2 4 3 5 2 4 3 5 2 4 3 5 2 4 3 5 2 4 3 5 " # 2 4 3 5 " # 2 4 3 5
h i h i
=1
=1 + =0
=1 1 =1 1 2 1 1 2 1 2 1 2
1 11 12
2 21 22 2
31
32 22
1 11 12
2 2
2
1 2 1 2 1
1 21 11 22 12 22 11 12 31 32
2 2 0
0 i i k m k a i ` j m ;m
i;j ;i j
x y m c m i m i n b n i n i m i n i th N m c n b
N N N
N N N N N N N
N N N
N
N N N
N N N N N N
N N N
N N N N N
N
N N N
N N N
N N
T
N N
N N N N
m
N N N
n T s T N N N N N N N N N
N N N
N N N N N
N N
P x P y
;a ;` i j
B x; P x P y m
m m
m n
H B H B B B i
N m n
H H H
H H
M M ;M ;
M M ;M ; A w t
A
A w t A
A A ;A
B ;B ; F t ;F t :
# t t ; t ; ; t ;w t ;w t ; ;w t N
m n u t u t ; u t s
M A
M ;A
A
M M
V
A A w t
B F t y
y t y t B u t F t
y y
where ( ) and ( ) denote the standard Legendre p olynomials that have b een scaled
by transformation to the intervals [0 ] and [0 ], resp ectively. The condition + = 0
eliminates the constant function thus guaranteeing that the set of functions is suitable as a
basis for the quotient space. We take ( ) = ( ) ( ) where
( +1)( +1) 1.
The and 1dimensional approximating cavity and b eam subspaces are taken to b e
= span and = span , resp ectively, where and are the
cavity and b eam bases describ ed ab ove. Dening = + 1, the approximating state
space isthen takento b e = and the pro duct spacefor the rst order systemis
= .
Byrestrictingtheinnitedimensionalsystem(2.12)to ,oneobtainsthenonlinear
nite dimensionalsystem
with
=diag [ ]
=diag [ ] ( ) =
0
( )
=diag [ ]
and
~
= 0
~ ~
( )= 0 ~
( )
Thevector ( )=( ( ) ( ) ( ) ( ) ( ) ( )) containsthe 1=
( + 1) 1 approximate state co ecientswhile ( ) = ( ( ) ( )) contains the
voltagevalues. Thematrices and arethemassandstinessmatriceswhicharisewhen
solving the uncoupledwaveequation withNeumannb oundary conditions while and
are the mass, stinessand dampingmatriceswhicharise whensolving the damp edb eam
equation with xed b oundary conditions. The matrices and result from the choice
of inner pro duct (see[1]). The contributions fromthe coupling termsare contained inthe
matrix and op erator ( ( ))whilethecontrol, forcingand initialtermsarecontained
in ~
, ~
( )and ~ ,resp ectively. Amoredetaileddescriptionofhowthevariouscomp onent
matricesare constructedcan b e found in[1].
The equivalentnite dimensionalCauchy equation is then
_ ( )= ( ) + ( )+ ( )
(0) =~
0
q
only once A
0 2
111 0 2
2 0 0 2 2
h i
Z
X
Z
X
h i h i h i
0
32
32
32
0
1
=1
0
=1
32
32 32 32 1
32 32 32 1
32 32 32 1
3.2 Algorithm for Constructing the Nonlinear Component
Numerical Algorithm for Creating the Nonlinear Component :
y t A y t B u t F t
y y
A
A w t
y t
n m A w t
i;j ; ;m ;m n m
A w t B x P x P w t B x dx
f x dx c f x :
n c
A w t
A ;A A
A B x ; A c B x ; A P x
A n n ; A n n A n m
N N N N N
N N
N
N N
N N
N N
x y
N N
p;ij a
n
p a
i `
j n
J N
J n
J
a
n
k
k k
q k
N N
N
a N
aw
N
bj
N
a
ik n
k i
N
aw
k i k
n
k i
N
bj
ij a
j i
N
a q
N
aw
q
N
bj
q x
dimensional problem,it willalso b e usefulto consider the corresp onding problem
_ ( )= ( )+ ( )+ ( )
(0) = ~
(3.2)
when considering issues such as parameter convergence. The linear op erator is obtained
by considering the linearizationof ( ( ))as discussedin [1].
The determination of state trajectories involves the rep eated construction of the nonlinear
op erator ( ( )). Althoughmostofthecomp onentsofthisop erator needtob econstructed
onlyonce,the ( 1) matrix ( ( ))mustb edeterminedat eachstepinthesolution
of the ordinary dierential equation (ODE) (3.1) and hence its formation must b e made as
ecientas p ossible.
We rst p oint out that for =1 ,this ( 1) matrixhas the entries
( ) = ( ) ( ) ( ( ) ( ))
where the integrals are evaluated viaa Gaussian quadrature ruleof the form
( ) ( )
Here isthenumb erofquadraturep ointsand isthequadratureweight. Thesequadratures
andhence the formationofthe matrixcan b eecientlyaccomplishedinthe manneroutlined
in the following algorithm.
( )
1. Createthe matrices and whichhave the comp onents
= ( ) = ( ) = ( )
(hencedim = ( 1) dim =( 1) anddim = ( +1)).
Thesematricesneedto b e formed and canb e createdb eforesolvingthe ODE
system.
2 2 3 2 2 0 1 1 2
1 0 2
4 Parameter Estimation Problem
K K 111
K 111 K
K 111 K 2
2
C C fZ g0
Z C C 0 h i X 32
32 2 32 2
32 32 1 32 2 32
32 32 32
32 32 32 32
32 32 32 1
1 1 2 1 2 2 2 2
A ;A A
;EI;c I ;i ; ;s; z
Z q ;EI;c I; ; ; ;
; ; q Q Q
q Q
J q t ;q z
z z t
;x;;w
x
J q
@ w
@t
t;x q z
x A w t ; x n
j m
b P x ; b n
A w t b ones ;m ; A w t n m
A w t A A w t ; A w t n m
ind j m
ind j m
A w t ;ind ind A w t ; A w t n m
A w t A A w t ; A w t n m :
N a N q y ` j q N bj N x N bj N q x N bj N N bj N bj N N bj N q x x x N b N N bj N N b N q N N f N aw N b
N N N
N a N aw N bj b D S i B i b D B B s S S s i i i i i i i
^= ( ) dim ^= 1
for =1: +1
^
= (^) dim
^
= 1
( ) = ^
(1 +1) dim ( ) = ( +1)
( ) = ( ) dim ( ) = ( +1)
1=( 1) ( +1)+1
2= ( +1)
( ) (: 1: 2)= ( ) dim ( ) =
end
( ) = ( ) dim ( ) =( 1)
Bycreatingthe matrices and oine, thetimeneededto solvethe ODE
system is reduced thus improvingthe eciencyof the schemefor parameter estimationand
control applications.
Thegoal ofthe parameterestimationproblemisto determinethe\true"materialparameters
and , =1 givendata measurements ^from someobservable
sub-space ofthe statespace. To p ose this mathematically,welet =(
) and assume that where denotes an appropriately chosen admissible
parameter space. The parameter estimationproblem isto then seek whichminimizes
( )= ( ) ^ (4.3)
given p ointwise temp oral measurements ^ = ^( ) at given p oints on the b eam and in the
cavity. Note that this minimization is p erformed subject to = ( _
_) satisfying the
coupled system equations (2.12) or (2.16). Dep ending on the exp erimental apparatus, the
data observations may consist of p osition, velo city,acceleration, or accumulatedstrain
mea-surementsat p ointson theb eamas wellas pressuremeasurementsinsidethecavity. Theform
of the op erator dep ends up on whetherone isp erforming the estimationpro cedures inthe
timedomainor inthe frequency domain. In the timedomain, isthe identitywhereas itis
the Fourier transformfor estimationin the frequencydomain.
For time domain estimation with data consisting of p osition, velo city, or acceleration
measurementsat p oints on the b eam, the tcriterion functionalto b e minimizedis
111 0 X ! X 8 < : ! 9 = ; S X X X
K 0 0
K 0 0 j 0 j
C
f g f g f g f g
111
111
111
0 0 0
111
K 111 K K 111 K i
S
i i i
i
i
S
i i i t i i
D
k ij
i ; ;s
j ;
ij
s
k
k k s
s
k
k k s
D
s
k
k k s
k k k
s B B s S S s 1 2 2 1 2 2 1 2 2 2 =1 =12 2 +1 =1
1 3 2 +1
2 +1
=1
1 3 2 +1
2 +1
=1
1 3 2 +1
1
1 3 2 +1
1 1
J q
@w
@x
t;x q
@w
@x
t;x q z
p x;y
J q
@w
@x
t;x q
@w
@x
t;x q z t ;x;y p
;EI c I
x ;a s s
x c B x ; c c c
EI x c B x ; c c c
c I x c B x ; c c c
B x H x x H x x
c c c
; ; ; ; ;
Q
accumulatedstrain measurements,an appropriate functionalis
( )= ( ; ) ( ; ) ^ (4.5)
(see(2.9)). Herethedataconsistsofthevoltagemeasuredacrossthepatches. Thepatchdata
can also b e combined with pressuremeasurements ^ at p oints ( )in the cavity to provide
a tcriterionfunctional
( )= ( ; ) ( ; ) ^ + ( ) ^ (4.6)
whichin somecases has moresensitivity than that in (4.5) which considers only strain
mea-surements.
Theab ovetcriteriacanalsob eusedwithdatathathasb eentransformedtothefrequency
domain (in which case is the Fourier transform), and this is indeed a common pro cedure
for starting the optimization pro cess with data in which several frequencies are excited(see
the commentsin the examplesas well as [13, 14]). In this case, optimization is qualitatively
p erformedinthefrequencydomainuntilfrequenciesmatchsucientlysothattheoptimization
routineswillconverge with the timedomaindata.
To facilitatethe estimationof thematerial parameters and ,wenowmakesome
assumptions regarding their spatial b ehavior. Becausethe b eam and patches are considered
to b e homogeneousas wellas uniforminwidth and thickness,itisreasonable to assume that
thedensity,stinessanddampingparametersofthecombinedb eam/piezo ceramicpatchesare
piecewiseconstant innature (see for example,[15]). A suitablepartition is then taken to b e
= 0 where the 2 p oints are the endp ointsof the piezo ceramic
patches. Finally,we assume thatthese parameters have the form
( ) = ~ ( ) = = = ( ) = ~ ~ ( ) ~ =~ = =~ ( ) = ^ ~ ( ) ^ =^ = =^ (4.7)
wherethepiecewiseconstant basisfunctionsaredenedby ~
( ) ( ) ( ).
The co ecientconstraints = = = ,and so on, resultfrom the uniformity ofthe
b eam in areas not covered by patches. Finally,we recall from the denitions (2.8) and (2.9)
that the patch parameters and are simply constants which dep end
on piezo electric prop erties,the geometryand sizeof the patch,and patch and b onding layer
prop erties.
Although isnitedimensional withthe ab oveassumptions on the parameters,the
3 2 2 1 2 2 1 2 2 1 2 2 Theorem 2. (Linearized System)
0
K 0 0
K 0 0 0
2
2
0 ! !1
j 0 j j j j j 2
2 7! ! ! ! X X ! X 8 < : ! 9 = ; N N t N N i N i i N i S N i N i i N i S N i N i i N t i i N f N t N N N N V
i i i V V
N N N N N t N t t N N t N
w H V
J q
@ w
@t
t;x q z ; ; ; ;
J q
@w
@x
t;x q
@w
@x
t;x q z
J q
@w
@x
t;x q
@w
@x
t ;x q z t;x;y p ;
w
Q Q
d H V V V
H
" N N :
q q V
q ; q ; d q;q ; ; V
i ; q;q Q
q Bu F t q Q L ;T ;V :
q q q Q
z t q z t q V
z t q z t q V
t > z z z z
H
however,corresp ondingminimizationproblemsinvolvingthe stateapproximationscan b e
de-velop edandusedwhenestimatingphysicalparameterswiththeset-to-datatechniques. With
, and denedinthe lastsection, nitedimensional functionalscorresp onding
to thosein (4.4), (4.5) and (4.6) are
( )= ( ; ) ^ =0 1 2 (4.8)
( )= ( ; ) ( ; ) ^ (4.9)
and
( )= ( ; ) ( ; ) ^ + ( ) ^ (4.10)
resp ectively. The approximate b eam displacement and acoustic pressure at the
various p ointsare foundbysolving eitherthe nonlinear nitedimensionalsystem(3.1)or the
system(3.2) if one is considering the linearizedproblem.
The following theorem taken from [16 ] sp ecies conditions under which convergenceand
continuousdep endence(ondata)ofthesolutionstothelinearizednitedimensionalparameter
estimationproblemsinvolvingthe functionals (4.8), (4.9) and (4.10) can b e exp ected.
Let b e a compact subset of a metric space ~
with
metric and assume that approximates inthe sense that for each 8 ,
there exists8 suchthat
8 8 ( ) 0 as (4.11)
Furthermore, assume that ( ) and ( ) dened in (2.13) are -elliptic, continuous,
and satisfy the continuity with resp ect to parametercondition
( )(8 9) (~)(8 9) ( ~)8 9 for 8 9 (4.12)
for =1 2 and ~ . Finally,assume that
( + )( ; )is continuous from to ((0 ) ) (4.13)
For arbitrary suchthat in , one then has the convergence
( ; ) ( ; ) in norm
( ; ) ( ; ) in norm
(4.14)
for 0. Here and are solutions to the linearized system (2.20) and and
1
3 2
s
D
N
2
n
Z
Z
Z Z
Z
1
Lemma 2. (Nonlinear DampedSystem)
Q L ;a
> ; EI > c I > ;a
meas F
Bu q
q
; q
G
Bu t F t G t
Bu F Bu F G G t
t
G the metricspace
~
= [ (0 )] [lR ] with elements satisfying the co ecient constraints
sp ecied in(4.7) as wellas the physical constraints 0 0 and 0on (0 ). In
consideringtheremaininghyp otheses,wenotethatthesesquilinearformssatisfytheellipticity,
continuity and parameter continuity conditions as long as dampingis assumedin the region
~
with (
~
) = 0 (see (2.14) and (2.15)). Moreover, the input term ( +
)( )satises the condition (4.13). Finally, the convergencecondition sp ecied in (4.11) is
satised as aconsequence of the approximating prop erties of the cubic splines and Legendre
p olynomials ina Galerkinsetting (see[6] for further details).
Hence, for the linearized problem with strong acoustic damping via ~
, a subsequence of
solutions to the problems involving the minimization of the functionals (4.8) with =
0 1, (4.9) or (4.10) subject to (3.2) will converge to a solution of the original problem of
minimizingthe functionals(4.4), (4.5)or (4.6) subjectto (2.20). The convergencein thecase
involving the minimization of (4.8) with acceleration data do es not follow directlyfrom this
theorembut canb eobtainedusingresultsfromthetheoryofanalyticsemigroupsinamanner
analogous to thatused in[7].
With b oundedness and Lipschitz continuity assumptions on the nonlinear coupling term
( ),similarresultscanb eobtainedforthenonlinearproblemassummarizedinthefollowing
remark.
Consider the systemwith the nonlinear input
term ( )+ ( )+ ( ( )). If,in addition to assuminga continuity condition of the
form(4.13)(with + replacedby + + ( )), wealso assumethat ( ( ))is
continuousin ,uniformlyLipschitzcontinuousin ,anddisplaysatmostane growth
at , then convergence results analogous to those summarized in Theorem 2 can b e
obtained for the nonlinear system.
Wep ointoutthatthesecontinuityassumptionson werealsomadewhendiscussingthe
well-p osedness of the nonlinear system. Details concerning these conditions as wellas arguments
leadingto the pro of for the nonlinear case can b e found in[5].
As indicated previously, the assumption of medium damping inside the cavity is often
inappropriate in applications of interest. Whilewe have not extended Theorem 2 to include
thecaseinwhichacousticdampingisomittedorisweak,extensivenumericaltestssuggestthat
parameterconvergenceandcontinuousdep endenceoftheparametersondataisb eingobtained
in the same manner exhibited by the system having b oth strong acoustic and structural
damping. Thisisdemonstratedbythe resultsinthe followingexamplesfor thenonlinear 2-D
i
0
!
X 2
r 1 2
r 1
0 K
2
2
K
K
2
K
K
2
2
2
2
2
3
2
2
2
=1
0
3 10 2 2
3 3
10 2 3
31
31
10 tt
f t
tt D
t
s
i B
i
i pe
t t
f
b
D f
pe pe
B
pe b`
B
b`
S
S
c x;y ;t> ;
n x;y ;t> ;
t;x;w t;x n w t;x <x<: ;t> ;
w @
@x EI
@ w
@x
c I @ w
@x @t
t;x;w t;x f t;x @
@x
u t x
<x<: ;
t> ;
w t;
@w
@x
t; w t;:
@w
@x
t;: t> ;
;x;y ;x;y w ;x w ;x
d ;
x y a : m ` m
: m : m : m
kg=m E : N =m : kg=m EI : Nm
c I : kgm =sec : kg=m c m=sec
=
T : m b : m
mil
E : N =m kg=m
E bd h T T
: Nm=V
T : m
d : m=V
V
To test the parameter estimationmetho dology, the general problem
= 1 ( ) 0
^ =0 ( ) 0 0
( ( )) ^ = ( ) 0 6 0
+ +
= ( ( ))+ ( )+ ( ) ( )
0 6
0
( 0)= ( 0)= ( 6)= ( 6)=0 0
(0 )= (0 )= (0 )= (0 )=0
was considered. For these examplesweassumedno cavitydampingand hence 0 =
throughout. The cavity was assumed to have and dimensions = 6 and = 1
with a b eam at one end having length 6 , width 1 and thickness 005 (see Figure3).
The density and Young's mo dulusfor regionsof theb eam devoid ofpatches weretakento b e
=2700 and =71 10 whichyields =135 and =7396
for the linear mass density and stiness parameter (see Table 1 for a compilation of the
structural parameters for the system). The Kelvin-Voigt damping parameter was chosen to
b e = 001 . Finally, the values =121 and = 343 were used
for the atmospheric density and sp eed of sound.
In the examples, we consider a system in which the b ounding end b eam has b onded
to it a centered piezo ceramic patch covering 1 3 of its length as shown in Figure 3. The
patch is assumed to have thickness = 000508 and width = 1 (we p oint out
that the chosen thickness value corresp onds to 20 which is a commercially available
thickness for piezo ceramic patches). The Young's mo dulus and density were taken to b e
= 63 10 and = 7650 which are reasonable for a patch made from
G-1195 piezo ceramic material.
From (2.7) and (2.8), wesee that the density and stiness co ecientin the region of the
combined b eam and patch (Region 2) will b e greater than that of the b eam(Region 1) (see
Figure 3). We also assume that the damping co ecient will b e slightly larger in Region 2
than Region 1.
Fortestingpurp oses,thestructuralparametervaluesinregionscoveredbythepatcheswere
chosenas sp eciedinTable 1. Asseen there,the constant = ( +2 + ),which
arises when mo deling the actuation due to the patch, was taken to b e = 0067
(this latter value was obtained by assuming a b onding layer of thickness = 0001 and
taking =19 10 whichisthe value sp ecied forG-1195 piezo ceramicmaterial).
The constant whicharises when usingthe patches as sensors was takento havethe value
1
0
Region 1
Region 2
0.2
0.4
0.6
Fig. 3.
K K
K K
K
K
K D
B S
D
B S
D
B
D B
S ;EI;c I;
;EI c I
;EI;c I
;EI;c I; Acousticchamber with onecentered 1/3 length patch.
The followingexamplesdemonstratethe numericalestimationofthe materialparameters
and using various techniques for exciting the system and observing the
resp onse. In the rst example,the natural frequenciesfor the fully coupled systemwere
de-terminedbysimulatinganimpulsehammerimpactto the centerofthe b eam. The knowledge
ofthesenaturalfrequencieswasthenusedwhencho osingthefrequenciesoftheexcitingforces
in the other examples. In the second example, a p erio dic uniform forcing function (mo
del-ing a uniform exterior acoustic pressure eld) was applied to the b eam for a short interval
of time and then set to zero. This forcing function was chosen so that three system mo des
wereinitiallyexcitedand then allowedto b egin decayingdue to the dampingthe b eam. The
accelerationofthe centerofthe b eamwasused asdataforestimatingthematerialparameters
and (since no voltage was applied in this exampleand the patch was not used for
sensing, and were not estimated). In the third example, the system was excited by
the applicationofap erio dicvoltageintothe patch. Again,the systemwas excitedforashort
time interval and then allowed to freely decay in energy. Acceleration data at the center of
the b eamwas used to estimatethe four parameters and . The patchwas used
b oth for actuating and sensing in the fourth and fth examples. In Example 4, a p erio dic
voltage was applied for an initial time intervalafter whichthe systemenergy was allowedto
decay. Duringthedecayinterval,twosets ofdatawerecalculatedandacomparisonwas made
b etweentheresultsobtainedwheneachwas usedforrecoveringtheparameters
and . The rst data setconsisted solely of the voltagepro duced by the patches during
vi-bration(and hencecontainedstrainmeasurements)whilethe secondcontainedacombination
of voltagemeasurementsfrom the patch and acoustic pressure values frominside the cavity.
Finally,a simulatedvoltage spike to the patches was used to excitethe system inExample5
(withaneectsimilarto thatobservedwhenanimpulsehammerisusedtoexcitethe system)
with the patches again b eing used as a sensor throughout the remainderofthe timeinterval.
Thusin the last two examples,the \smart material"asp ects of the structure wereutilizedin
