subjected

6thWorldCongressofStructuralandMultidisciplinaryOptimization

RiodeJaneiro,30May-03June2005,Brazil

Topologicaldesignofcontinuumstructuressubjectedtoforcedvibration

NielsOlhoff,JianbinDu

InstituteofMechanicalEngineering,AalborgUniversity,Aalborg,Denmark,no@,jd@

ct

Thispaperdealswiththeproblemoftopologicaldesignoptimizationofelastic,continuumstructureswithoutdampingthatare

subjectedtotime-harmonic,design-independent(or–dependent)

importantobjectiveofsuchadesignproblemisoftentodrivetheeigenfrequenciesofthestructureasfarawayaspossiblefromthe

prescribedloadingfrequencyjectiveisimplemented

inthispaperbyminimizingthedynamiccomplianceofthestructuresubjecttothegivenfrequencyoftheloading,usingthevolumetric

densitiesofmaterialinthefiniteelementsintheadmissibledesigndomainasdesignvariables,andapplyingtheSIMPmodelto

alstructuralvolume,theboundaryconditions,andthematerialaregiven.

Onlyloadingfrequenciesthataresmallerthantheoptimumvalueofthecondeigenfrequencyofthecorrespondingproblemforfree

vibrations,arediscusdinthispaper.

ds:Topologyoptimization,dynamiccompliance,vibrationlevel,loadingfrequency

uction

Themethodoftopologyoptimizationfirstappearedintheliteraturein1988,andwasoriginallydevelopedfordeterminingthe

distributionofanelasticmaterialwithinanadmissibledesigndomainthatsubjecttoagivenstaticloadingyieldsthestiffestpossible

structureforaprescribedamountofmaterial,eBendsøeandKikuchi[1]andBendsøe[2].Incomparisonwithusualsizingandshape

optimization,topologyoptimizationimpliesmoredesignfreedomsinceitallowsnewholesandconnectionstobegeneratedinthe

structureandhergyoptimizationisthereforean

importantpreprocessingtoolforsizingandshapeoptimization(eOlhoffetal.[3]).Duringthelastdecade,themethodhasbeen

gyoptimizationhasthereforebecomeastandardtoolforsynthesis

ofpartsorwholestructuresintheautomotiveandaerospaceindustries,anditisrapidlyspreadingintootherengineeringdesigndisciplines.

ThereaderisreferredtotheexhaustivetextbookbyBendsøeandSigmund[4]andthereviewarticlebyEschenauerandOlhoff[5]forrecent

developmentsandpublications.

Liketheconceptofcomplianceinstaticstructuraldesign,dynamiccomplianceisaproperglobalmeasureofthedynamic

msoftopologyoptimizationwiththeobjectiveofminimizingthedynamiccompliance(maximizingthe

dynamicstiffness)ofstructuressubjectedtotime-harmonicexternalloadingofgivenfrequencyandamplitudewere,e.g.,studiedby

Maetal.[6]andJog[7].TopologydesignsubjecttotransientexternalloadingwasstudiedbyMinetal.[8],wherethedynamic

complianceisdefinedrelativetoaspecifiedtimeinterval.

Aproblemthatmayariinstructuraltopologyoptimizationundertime-harmonicdynamicloadingisthatthestaticcompliance

(correspondingtothesameloadingamplitude,butzerofrequency)mayincreatoaveryhighlevel(iak[9])duringthe

emecasthestaticcomplianceactuallytendstoinfinity,

whichreflectsthatadr,wehavefoundthatthedesign

objectiveofthedynamicproblemcanbeimplementedalongdifferentoptimizationpaths,andthatitispossibletoavoidtheproblem

,intheprentpaper,anapproachispropodinwhichthestaticcomplianceof

thestructureisconstrarithmdevelopedforthis

handlestheoptimumdesignproblembyacontinuationtechniquewheretheloadingfrequencyisquentiallyincreadfroma

calexamplesareprentedtodemonstratethevalidityoftheapproach.

Asanextensionrelativetotraditionaltopologyoptimization,ourstudyalsoincludescaswherethedynamicloadingis

design-dependent,i.e.,caswhereboththelocatio

methodsdevelopedbyHammerandOlhoff([10],[11];ealsoDuandOlhoff[12],[13])areemployedtohandlethedesignproblems

ndofthepapersomeillustrativeoptimumtopologyresultsareprented,bothforcas

ofdesign-independentanddesign-dependentloading.

ationoftopologydesignproblemforminimizationofdynamiccompliance

Theproblemofoptimizingthetopologyofacontinuumstructure(withoutdamping)forminimumvalueoftheintegraldynamic

structuralcompliancecanbeformulatedinadiscreteformasfollows:

2

.),,1(,10

,)(,0

,)(

:

|}|{min

0

**

1

2

ee

N

e

ee

p

d

e

Ne

VVVV

toSubject

C

E

L=≤≤<

=≤−

=−

=

∑

=

ρρ

αρ

ω

ρ

PUMK

UPT

(1)

Thedynamiccomplianceiqualtotheworkdonebytheexternaldynamicforcesagainstcorrespondingdisplacements,andis

proportionaltotheexpressiongivenforC

d

in(1),whereUdenotesthedisplacementamplitudevectorofthesteady-state,time-

harmonicvibration,monicexternalloadingvectorp(t)withthe

givenexcitationfrequency

p

ωcanthenbeexpresdasti

petωPp=)(andthedisplacementresponvectorasti

petωUa=)(.The

symbolsKandMreprenttheNdimensionalstructuralstiffnessandmassmatrices,bolρ

e

is

thevolumebolαdenotesthevolumefractionof

availablematerialandisgivenby

0

*/VV,whereV

0

isthevolumeoftheadmissibledesigndomainandV*istheprescribedtotal

bolN

E

reprentsthetotalnumberoffiniteelements.

Itisnotedthatthedynamicstiffness

d

KdefinedasMKK2

pd

ω−=maybenegativedefinitewhentheloadingfrequency

p

ω

hasahighvalue,ca,theproductoftheamplitudesofthe

externalloadingvectorandthedisplacementvectorofthestructuralresponmayattainanegativevalue,andinordertoincludethis

possibilityinourproblemformulation,weutheabsolutevalueoftheaboveproductasthedynamiccompliancedefinedinour

problem,eEq.(1).Thefirstequationlistedintheconstraintsofproblem(1)isthedynamicstateequationandcanbesolvedina

directwaybyGausliminationorbyusingatechniqueofmodalsuperposition.

PMaterialmodel

TheSIMP(SolidIsotropicMicrostructurewithPenalty)interpolationmodel(e,e.g.,Bendsøe[2];Rozvany,etal.[14];Bendsøeand

Sigmund[15])isudheretogetherwithafilteringtechnique(e,e.g.,Sigmund[16]),inordertoavoidcheckerboardformationand

dep,thefiniteelementelasticitymatrixE

e

is

expresdintermsoftheelementvolumetricmaterialdensityρ

e,

0≤ρ

e

≤1,andthepenalizationpowerp,p≥1,as

*)(

e

p

eee

EEρρ=,(2)

where*

e

Eistheelasticitymatrixofacorresponding

analogywith(2),foravibratingstructurethefiniteelementmassmatrixmaybeexpresdas

*)(

e

q

eee

MMρρ=,(3)

where*

e

Mreprentstheelementmassmatrixcorrespondingtofullysolidmaterial,andthepowerq≥romexceptionsbriefly

discusdinthefollowingction,normallyq=balstiffnessmatrixKandmassmatrixMforthefiniteelement

badstructuralresponanalysbehindtheoptimizationcannowbecalculatedby

∑∑

==

==EE

N

e

e

q

e

N

e

e

p

e

1

*

1

*,MMKKρρ,(4)

where*

e

Kisthestiffnessmatrixofafiniteelementfullysolidmaterial,andN

E

denotesthetotalnumberoffiniteelementsinthe

admissibledesigndomain.

zedvibrationmodes

Withvaluesassignedtopandqasstatedabove,applicationoftheSIMPmodelforproblemsoftopologyoptimizationwithrespectto

dynamicstructuralresponforprescribed,lowexcitationfrequenciesmayleadtotheoccurrenceofspurious,localizedvibration

alizedmodesmayoccurinsub-regionsofthedesigndomainwithlowvaluesofthematerialdensity(e.g.0≤ρ

e

≤0.1),

wheretheratiobetweenthestiffness(with,say,p=3intheinterpolationformula)andthemass(withq=1)inate

thespuriousmodeswehavetthemassverylowviaahighvalueofthepenalizationpowerinsub-regionswithlowmaterialdensity

(Pedern[17];Tcherniak[9]).Thus,theinterpolationformula(3)forthefiniteelementmassmatrixwasmodifiedas

⎩

⎨

⎧

≤

>

=

)1.0(,

)1.0(,

)(

*

*

ee

r

e

eee

eeρρ

ρρ

ρ

M

M

M,(5)

wherethepenalizationpowerrischontobeaboutr=6,i.e.,muchlargerthanthepenalizationpowerpforthestiffness,whichiskept

unchangedatavalueaboutp=3.

ivityanalysis

ThensitivityoftheobjectivefunctionC

d

inproblem(1)withrespecttothedesignvariablesρ

e

isgivenby

,)()(||UPUPUPUPTTTT′

+

′

=

′

=

′

signC

d

(6)

whereprimedenotespartialderivativewithrespecttoρ

e

,andsign()sitivityP

′

oftheloadvectorwillbe

3

zeroifitisdesign-independent,otherwiitcanbehandledusingthemethoddescribedbyHammerandOlhoff([10],[11]),andalsoby

DuandOlhoff[12],[13].ThensitivityU

′ofthedisplacementvectorisgivenby

,)()(22UMKPfUMK

′

−

′

−

′

≡=

′

−

pp

ωω(7)

wherethensitivitiesofthestiffnessandmassmatricescanbedirectlyobtainedfromtheSIMPmaterialmodel,.(4).Thevectorfis

knownasthepudoloadandisdefinedbythetermontheright-handsideofEq.(7).InsteadofsolvingEq.(7),theadjointmethod(ee.g.

TortorelliandMichaleris[18])maybeudtocalculatethensitivityoftheobjectivefunctioninamoreefficientmanner,whichgivesthe

followingresult

.))(2)((2UMKUPUUPTTT′

−

′

−

′

=

′

pd

signCω(8)

Accordingly,theoptimalityconditionforproblem(1)canbeexpresdinthefollowingformbymeansofthemethodofLagrange

multipliers,

,0))(2)((2=Λ+

′

−

′

−

′

ep

VsignUMKUPUUPTTTω(9)

whereΛistheLagrangemultipliercorrespondingtothematerialvolumeconstraint,andthesideconstraintsforρ

e

optimizationproblem(1)canbesolvedbyusingthewell-knownMMAmethod(Svanberg[19])oranoptimalitycriterionmethod,

fixedpointmethod,asdevidinChengandOlhoff[20].

calexamples

5.1Topologydesignofa2Dinletsubjectedtohydrodynamicpressureloading

Thixampleilluidflowinthechanneloftheinitial

inletisasshowninFig.2(a),andisassumedtoexertauniformhydrodynamicpressureloadingofgivenfrequencyandamplitudeon

rodynamicpressureisadesign-dependentloadinganditistreatedasdescribedinthepaper(Duand

Olhoff[12]).ThematerialoftheinletisisotropicwithYoung’smodulusE=107,Poisson’sratioυ=0.3andthespecificmassγ

m

=1(SI

unitsareudthroughout).Theavailablematerialvolumefractionistas40%.Thedesignobjectiveistominimizethedynamic

.2(a).2(b-d)showoptimized

topologiesandtheassociatedloadingboundariesforthreegivenloadingfrequencies

0=

p

ω

(staticloading),

800=

p

ω

and

1000=

p

ω

.TheloadingfrequenciesarealllowerthantheoptimumobtainablevalueΩ

opt

=1328ofthefundamentaleigenfrequency

sthatwhentheloadingfrequencyisincreadfrom0to800,theshapeoftheinletisslightlychangedwhilethe

optimumtopologyoftheinletremainsthesame(Fig.2(b),(c)).Whentheloadingfrequencyisincreadupto1000,boththetopology

andtheshapeoftheinletarechanged(Fig.2(d)).

Asacomparison,Fig.3showstheoptimumtopologyoftheinlet(forthesamematerialvolumefractionasabove)whenit

performsfreevibrationsatthemaximumvalueΩ

opt

=damentaleigenfrequencyofthe

initialdesignoftheinlet(withuniformlydistributedmaterialovertheadmissibledesigndomain,eFig.2(a))is455.

zedtopologies(40%volumefraction)andloadingboundariesof2Dinletforthreedifferentloadingfrequencies.(a)

Admissibledesigndomain,loadingandsupportconditions.(b)ω

p

=0.(c)ω

p

=800.(d)ω

p

=1000.

4

mtopologyofthe2Dinlet(for40%volumefraction)obtainedbymaximizingthefundamentaleigenfrequencyoffree

imumfundamentaleigenfrequencyisΩ

opt

=1328.

,whentheloadingfrequencyis

muchlowerthantheoptimumfundamentaleigenfrequencyofthestructure,theresultingtopology(eFig.2(c))obtainedbythe

dynamicdesignoftheprentpaperissimilartothestaticdesignthatsustainstheamplitudeoftheloadingatzerofrequency,eFig.

2(b),whichimpliesthatthedynamicdesignisdominatedbythespatialdistributionoftheamplitudeoftheexternalloadingvector.

However,iftheloadingfrequencyisclortothevalueoftheoptimumfundamentaleigenfrequencyofthestructure,thedesignis

dominatedbythedynamicrequirement,anddrivesthefundamentaleigenfrequencyofthestructureasfarawayaspossiblefromthe

ntermediatevalueoftheloadingfrequency,theoptimumtopologyoftheinletisakindof

compromibetweentheloadingamplitudedominateddesignandtheeigenfrequencydominateddesign(.2(b-d)andFig.3).

5.2Minimumdynamiccompliancedesignofaplate-likestructure

Thixampleconcernsoptimumtopologydesignofaplate-likestructurewithsupportconditionsasshowninFig.4(a).Thefinite

elementmodelofthestructureconsistsof600(30×20×1)8-node3DbrickelementswithWilsonincompatibledisplacementmodelsto

-harmonic,concentratedtransverexternalloadp(t)=Pcosω

designobjectiveistominimizethedynamiccomplianceoftheplateforaprescribedloadingfrequencyω=ω

p

=80andavolume

fractionof50%forthegivensolidmaterial,whichhastheYoung’smodulusE=1011,Poisson’sratioυ=0.3andthespecificmassγ

m

=damentaleigenfrequencyoftheplateintheinitialdesign(eFig.4(a))isΩ

1

=61.6,i.e.,lessthanthegivenloading

zationofthedynamiccompliancedrivesthedesignawayfromtheresonancepointwhichimpliesacontinual

decreaofΩ

1

asshowninFig.4(b).Asaresult,thestaticcomplianceofthestructureincreasveryquickly(Fig.4(c)).Fig.4(d)

showsthatatiterationstep30,dicatescreationofarigidbody

vibrationmodeinassociationwiththefirsteigenfrequency,andthatthestructurecannoteffectivelysustainthestaticloadassociated

withω=0.

0

20

40

60

80

100

Iterationnumber

F

r

e

q

u

e

n

ci

e

s

ω=80

Prescribedloadingfrequency

Ω1

p

Firsteigenfrequency

(a)(b)

0

0.5

1

1.5

2

x10-6

Iterationnumber

D

y

n

a

mi

c

a

n

d

s

t

a

t

i

c

c

o

m

pl

i

a

n

c

e

DynamiccomplianceCfor

StaticcomplianceC

ω=ω=80

(ω=0)

p

s

d

(c)(d)

Figure4.(a)Admissibledesigndomain(a=3,b=2andc=0.1)withloadingandsupportconditions.(b)Iterationhistoryforthefirst

eigenfrequencyΩ

1

oftheplate(Ω

1

<ω

p

=80).(c)Iterationhistoriesforthedynamicandstaticstructuralcompliance(thelatter

correspondstothesameloadingamplitudebutfrequencyω=0).(d)Materialdistributionatiterationstep30.

5

50

60

70

80

90

100

110

120

130

Iterationnumber

Firsteigenfrequency

ω=ω=80

Prescribedloadingfrequency

F

r

e

q

u

e

n

ci

e

s

Initialloadingfrequencyω0

p

Ω1

51015202530

0

1

2

3

4

5

x10-7

Iterationnumber

DynamiccomplianceCfor

loadingfrequency

ω=ω=80

StaticcomplianceC(ω=0)

St

r

u

c

t

u

r

al

c

o

m

pl

i

a

n

c

e

p

d

s

(a)(b)

51015202530

0

1

2

3

4

5

6

x10

-7

Iterationnumber

D

y

n

a

mi

c

c

o

m

pl

i

a

n

c

e

ω=100

ω=80

ω=60

ω=0

p

p

p

p

ω

p

--Prescribedloadingfrequency

(c)ω

p

=80(d)

Figure5.(a),(b)Iterationhistoriesforthefirsteigenfrequencyoftheplate,theloadingfrequency,andthedynamicandstatic

compliances.(c)Optimumtopologies(50%volumefraction)forω

p

=80.(d)Iterationhistories(steps5to30)ofthedynamic

compliancesoftheplatesubjecttofourdifferentloadingfrequencies.

(a)ω

p

=0(staticloading)(b)ω

p

=60(c)ω

p

=80(d)ω

p

=100

zedtopologiesoftheplate-likestructureforfourdifferentloadingfrequencies.

Inordertoavoidsuchastaticallyweakdesign,,we

startoutthedesignproblemwithavalueω=ω0oftheloadingfrequencythatislowerthanthefirsteigenfrequencyΩ

1

oftheinitial

design,andwethenquentiallyincreaωuptoitsoriginallyprescribedvalueω=ω

p

=80(Fig.5(a)).Intheconvergedresult,a

structurewithminimizeddynamiccomplianceandimprovedstaticstiffnessisnowobtained(eFig.5(b,c)).Fig.5(d)showsthe

iterationhistoriesofthedynamiccomplianceofthestructureunderthesameloadingconditionsasFig.4(a)butwithfourdifferent

prescribedloadingfrequenciesω

p

=0(staticloading),ω

p

=60,ω

p

=80andω

p

=imumdynamiccomplianceincreasas

.6(a-d)givetheoptimumtopologiesoftheplate-likestructurecorrespondingtothe

oadingfrequencyincreas,theoptimumtopologyofthestructureisgraduallychanged

fromthedesignthatisdominatedbythestaticbehaviourofthestructure(ignmainlydependsonthespatialdistributionof

theloadingamplitude)totheonethatisdominatedbythedynamicbehaviourofthestructure(ncyrespon).Toillustratethis

point,wemayperformthetopologyoptimizationtaskofmaximizingthefundamentaleigenfrequencyoffreevibrationsoftheplate.

Hereby,wefindthattheoptimumvalueofthefundamentaleigenfrequencyisΩ

opt

=127.6,andthecorrespondingtopologyaswellas

theiterationhistoryofthefundamentaleigenfrequencyareshowninFigs.7(a)and(b).

6

60

70

80

90

100

110

120

130

Iterationnumber

F

u

n

d

a

m

e

n

t

al

ei

g

e

n

f

r

e

q

u

e

n

c

y

(a)Ω

opt

=127.6(b)

Figure7.(a)Optimumtopology(50%volumefraction)associatedwiththemaximumvalueΩ

opt

=127.6ofthefundamental

eigenfrequency.(b)Iterationhistoryofthefundamentaleigenfrequencyoftheplate.

Finallyletusconsideracawithaprescribedvalueoftheloadingfrequency,e.g.ω=ω

p

=150,whichishigherthanthe

optimumvalueofthefundamentaleigenfrequencyΩ

opt

=,asdiscusdearlier,toensureareasonablestatic

stiffnessofthedesign,weintroduceanupperbound8105−×=≤

ss

CC

forthestaticcomplianceC

s

intheformulationoftheproblem.

TheoptimumtopologyresultforthisproblemisshowninFig.8.

mtopology(50%volumefraction)forω

p

=150withanupperboundonthestaticcompliance,i.e.

ss

CC≤).

Fig.9(a)showstheiterationhistoriesofthedynamiccomplianceoftheplatesubjecttothehigherloadingfrequency(ω=ω

p

=

150)andfourdifferentupperboundconstraintsonthestaticcomplianceC

s

(associatedwiththesameloadingamplitudebutzero

frequency).ThegraphsshowthattheoptimumdynamiccompliancedecreasastheupperboundconstraintonC

s

.

9(b),iterationhistoriesareshownforminimumcompliancetopologydesignoftheplatesubjecttoagivenupperboundconstrainton

thestaticcompliance(7105.0−×=≤

ss

CC

)raphsshowthatforthehighloading

frequencydesigns,thedynamiccomplianceofatureis

oppositetothatobtainedbyminimumcompliancetopologydesignsubjecttoprescribedlowerormediumloadingfrequenciesshown

inFig.5(d).Asaconclusion,variationsoftheminimumdynamiccompliancewithrespecttodifferentloadingfrequenciesaredepicted

inFig.10(a),andFig.10(b)prentsthestaticcompliancesassociatedwiththeminimumdynamiccompliancedesignssubjectto

differentprescribedloadingfrequencies.

0

0.5

1

1.5

2

2.5

3

3.5

4

x10-7

Iterationnumber

D

y

n

a

mi

c

c

o

m

pl

i

a

n

c

e

o

f

s

t

r

u

c

t

u

r

e

C=4x10-7

s

C=3x10-7

sC=1x10-7

s

C=0.5x10-7

s

DynamiccompliancesCfor

ω=ω=150

anddifferentupperboundsonC

p

d

s

0

1

2

3

4

5

6

x10-7

Iterationnumber

D

y

n

a

mi

c

c

o

m

pl

i

a

n

c

e

o

f

s

t

r

u

c

t

u

r

e

DynamiccompliancesCfor

ω=ω=130

p

ω=ω=150

ω=ω=180

ω=ω=200

p

p

p

withconstraint

C<=C=0.5x10

s

s

-7

d

(a)(b)

Figure9.(a)Iterationhistoriesofthedynamiccompliancesoftheplatesubjecttoahighloadingfrequency(ω=ω

p

=150>Ω

opt

=

127.6)andfourdifferentupperboundconstraintsonthestaticcomplianceC

s

,i.e.

ss

CC≤

.(b)Iterationhistoriesofthedynamic

compliancesoftheplatesubjecttoagivenupperboundconstraintonC

s

(7105.0−×=≤

ss

CC

),forfourdifferentloadingfrequencies

ω

p

=130,ω

p

=150,ω

p

=180andω

p

=200,allofwhicharehigherthantheoptimumvalueofthefundamentaleigenfrequencyΩ

opt

.

7

0

0.5

1

1.5

x10-7

Loadingfrequencyω

Ωopt

Mi

ni

m

u

m

d

y

n

a

mi

c

c

o

m

pl

i

a

n

c

e

C

C=0.5x10s

-7

d

C<=C

s

s

Dynamicdesignwithout

constraintonC

s

Dynamic

Designwith

constraint

p

0

0.5

1

1.5

x10-7

Loadingfrequencyω

ΩoptSt

at

i

c

c

o

m

pl

i

a

n

c

e

C

of

mi

ni

m

u

m

d

y

n

a

mi

c

c

o

m

pl

i

a

n

c

e

d

e

si

g

n

s

C=0.5x10

s

-7s

C<=C

s

s

Dynamicdesignwithout

constraintonC

s

Dynamic

designwith

constraint

p

(a)(b)

Figure10.(a)MinimumdynamiccompliancesC

d

entloadingfrequencies.(b)StaticcompliancesC

s

(correspondtothesame

loadingamplitudebutzerofrequency)associatedwiththedesignsinFig.10(a)atifthe

prescribedloadingfrequencyisclotoorhigherthantheoptimumvalueΩ

opt

=127.6ofthefundamentaleigenfrequencyforthe

correspondingproblemoffreevibrationsoftheplate,anupperboundconstraint

ss

CC≤

isprescribedforthestaticcompliancein

ordertoavoidobtainingastaticallytooweakstructurefromthedynamicdesign.

sions

Problemsofstructuraltopologyoptimizationwiththeobjectiveofminimizingthedynamiccompliance(maximizingtheintegral

dynamicstiffness)ofcontinuumstructuressubjectedtotime-harmonicforcedvibrationwithprescribedfrequencyandamplitudeofthe

quencyω

p

oftheloadingisassumedtobesmallerthantheoptimumvalueofthe

ultsshowthatthedesignobjectiveofminimizingthe

dynamiccomplianceyieldsastructurewhoeigenfrequenciesoffreevibrationsaregenerallyfarfromthegivenexcitationfrequency

ω

p

ofthedynamicloading,whichimpliefficientavoidanceofresonancephenomenaandreductionofthevibrationlevelofthe

structure.

Itisfoundthatthedesignobjectiveoftheforcedvibrationproblemmaybeimplementedalongdifferentoptimizationpaths

accordingtodifferentlevelsoftheexternalexicitationfrequencyω

p

.Forcaswheretheloadinghasalowerormediumvalueofω

p

,

theminimumdynamiccompliancedesignprocessmaybedrivenbyacontinuationtechniquewheretheloadingfrequencyis

quentiallyincreadfromasufficientlylowinitialvalueuptoitsprescribedvalue,ω

p

.Thisproceduredeliversthedesiredresultthat

theoptimumstructureisassociatedwithminimumdynamiccompliancesubjecttotheprescribedloadingfrequency,andalsoimplies

aneffectiveimprovement(decrea)peralsorevealsthatiftheloadingfrequencyω

p

is

increadfromalowervalueuptotheoptimumvalueΩ

opt

ofthefundamentaleigenfrequencyofthecorrespondingfreevibration

problem,thentheoptimumtopologyofthestructureisgraduallychangedfromadesignthatisdominatedbythestaticbehaviourofthe

structure(i.e.,thedesignmainlydependsonthespatialdistributionoftheloadingamplitude)toadesignthatisdominatedbythe

dynamicbehaviourofthestructure(i.e.,itsfrequencyrespon).Finally,iftheloadinghasahighvalueoftheofexcitationfrequency

(i.e.,somewhatbeloworabovetheoptimumvalueΩ

opt

ofthefundamentaleigenfrequencyofthecorrespondingfreevibration

problem),wehavefounditexpedienttointroduceanupperboundconstraintonthestaticcomplianceinordertomaintainareasonable

staticstiffnessofthedesign.

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