Multi Degree of Freedom

7/29/2019 Multi Degree of Freedom

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Twodegreeoffreedomsystems

Equationsofmotionforforcedvibration

Freevibrationanalysisofanundampedsystem


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Systemst atrequiretwoin epen entcoor inatesto escri et eir

motionarecalledtwodegreeoffreedomsystems.

masseachofmotionofs stemin thes stemtheof

typespossibleofnumbermassesofNumberfreedomofdegrees

oum

er


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T erearetwoequations oratwo egreeo ree omsystem,one oreac

mass(preciselyoneforeachdegreeoffreedom).

Theyaregenerallyintheformofcoupleddifferentialequationsthatis,

eachequationinvolvesallthecoordinates.

Ifaharmonicsolutionisassumedforeachcoordinate,theequationsof

motionleadtoafre uenc e uation that ivestwonaturalfre uenciesof

thesystem.


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weg vesu a e n a exc a on, esys emv ra esa oneo esenaturalfrequencies.Duringfreevibrationatoneofthenatural

frequencies,the

amplitudes

of

the

two

degrees

of

freedom

(coordinates)

mode,principlemode,ornaturalmodeofvibration.

Thus

a

two

degree

of

freedom

system

has

two

normal

modes

of

vibration

correspondingtotwonaturalfrequencies.

Ifwegiveanarbitraryinitialexcitationtothesystem,theresultingfreevibrationwillbeasuperpositionofthetwonormalmodesofvibration.

However,if

the

system

vibrates

under

the

action

of

an

external

harmonic

force,theresultingforcedharmonicvibrationtakesplaceatthefrequencyoftheappliedforce.


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Asisevi ent romt esystemss ownint e igures,t econ igurationo a

systemcanbespecifiedbyasetofindependentcoordinatessuchas

length,angle

or

some

other

physical

parameters.

Any

such

set

of

coordinatesiscalledgeneralizedcoordinates.

generallycoupled

so

that

each

equation

involves

all

coordinates,

it

is

alwayspossibletofindaparticularsetofcoordinatessuchthateach

equat ono mot onconta nson yonecoor nate. eequat onso mot on

arethenuncoupled andcanbesolvedindependentlyofeachother.Such

aset

of

coordinates,

which

leads

to

an

uncoupled

system

of

equations,

is

calledprinciplecopordinates.


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Equationsofmotionforforced

vibration Consi eraviscous y ampe two egreeo ree omspringmasssystem

showninthefigure.

Themotionofthesystemiscompletelydescribedbythecoordinatesx1(t)

andx2(t),whichdefinethepositionsofthemassesm1 andm2 atanytimet

.


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vibration T eexterna orcesF1 an F2 actont emassesm1 an m2,respective y.

Thefreebodydiagramsofthemassesareshowninthefigure.

Thea lication ofNewtonssecondlawofmotiontoeachofthemasses

givestheequationofmotion:


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vibration Itcan eseent att e irstequationcontainstermsinvo vingx2,w ereas

thesecondequationcontainstermsinvolvingx1.Hence,theyrepresenta

systemof

two

coupled

second

order

differential

equations.

We

can

thereforeexpectthatthemotionofthem1 willinfluencethemotionof

m2,andvicaversa.


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vibration T eequationscan ewritteninmatrix ormas:

, , ,

respectively andx(t)andF(t)arecalledthedisplacementandforce

vectors,respectively.whicharegivenby:


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vibration Itcan eseent att ematrices m , c an area 2x2matricesw ose

elementsaretheknownmasses,dampingcoefficienst,andstiffnessofthe

system,respectively.

Further,thesematricescanbeseentobesymmetric,sothat:

Freevibrationanalysisofanundampedsystem

,

F1(t)=F2(t)=0.Further,ifthedampingisdisregarded,c1=c2=c3=0,andthe

equationsofmotionreduceto:


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Freevibrationanalysisofan

undampedsystem

Weareintereste in nowingw et erm1 an m2 canosci ate

harmonicallywiththesamefrequencyandphaseanglebutwithdifferent

amplitudes.Assuming

that

it

is

possible

to

have

harmonic

motion

of

m1

andm2 atthesamefrequency andthesamephaseangle,wetakethesolutionstotheequations

as:

w ere 1 an 2 areconstantst at enotet emax mumamp tu eso

x1(t)andx2(t)and isthephaseangle.Substitutingtheabovetwosolutionsintothefirsttwoequations,wehave:


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undampedsystem

Sincetheaboveequationsmustbesatisfiedforallvaluesoftimet,the

termsbetweenbracketsmustbezero.Thisyields,

theunknownsX1 andX2.Itcanbeseenthattheaboveequationcanbe

satisfied

by

the

trivial

soution

X1=X2=0,

which

implies

that

there

is

no

v rat on. oranontr v a so ut ono 1 an 2,t e eterm nanto

coefficientsofX1 andX2 mustbezero.


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undampedsystem

Theaboveequationiscalledthefrequency orcharacteristicequation

0}))({(})({)( 2232212

1322214

21 kkkkkmkkmkkmm

because

solution

of

this

equation

yields

the

frequencies

of

the

characteristicvaluesofthesystem.Therootsoftheaboveequationare

ivenb :


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undampedsystem

ss ows a sposs e or esys em o aveanon r v a armon csolutionoftheform

when=1 and=2 givenby:

WeshalldenotethevaluesofX1 andX2 correspondingto1 asandthosecorrespondingto2 as .


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undampedsystem

ur er,s nce

theaboveequationishomogeneous,onlytheratios andr2= canbefound.For ,theequations

give:

Noticethatthetworatiosareidentical.


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undampedsystem

enorma mo eso v ra oncorrespon ng o can eexpressed,respectively,as:

Thevectors ,whichdenotethenormalmodesofvibrationareknownasthemodalvectorsofthes stem.Thefreevibrationsolutionor

themotion

in

time

can

be

expressed

using

wheretheconstants aredeterminedbytheinitialconditions.


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undampedsystem

n a con ons:

Eachofthetwoequationsofmotion,

involvessecondordertimederivatives;henceweneedtospecifytwo.

Thesystem

can

be

made

to

vibrate

in

its

ith

normal

mode

(i=1,2)

by

subjectingittothespecificinitialconditions.

However,for

any

other

general

initial

conditions,

both

modes

will

be

excited.Theresultingmotion,whichisgivenbythegeneralsolutionoftheequations

can

be

obtained

by

a

linear

superposition

of

two

normal

modes.


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undampedsystem

n a con ons:

Since

and

alreadyinvolve

the

unknown

constants

and

constants.arecandcwhere

)()()(

21

2211 txctxctx

wecanchoosec1=c2=1withnolossofgenerality.Thus,thecomponentsofthevector canbeexpressed as:)(tx

wheretheunknown canbedeterminedfromtheinitialconditions


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undampedsystem


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undampedsystem


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undampedsystem


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Examp e:Fin t enatura requenciesan

modeshapesofaspringmasssystem,which

isconstrained

to

move

in

the

vertical

direction.

Solution:Theequationsofmotionaregiven

Byassumingharmonicsolutionas:

thefrequencyequationcanbeobtainedby:


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Thesolutiontotheaboveequationgivesthenaturalfrequencies:


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From

theamplitude

ratios

are

given

by:


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From

Thenaturalmodesaregivenby


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T enatura mo esare

givenby:


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Itcan eseent atw ent esystemvi ratesinits irstmo e,t e

amplitudesofthetwomassesremainthesame.Thisimpliesthatthe

lengthof

the

middle

spring

remains

constant.

Thus

the

motions

of

the

mass1andmass2areinphase.


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W ent esystemvi ratesinitssecon mo e,t eequations e ows ow

thatthedisplacementsofthetwomasseshavethesamemagnitudewith

oppositesigns.

Thus

the

motions

of

the

mass

1and

mass

2are

out

of

phase.Inthiscase,themidpointofthemiddlespringremainsstationary

foralltime.Suchapointiscalledanode.


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Usingequations

themotion(generalsolution)ofthesystemcanbeexpressedas:


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eequa ono mo ono agenera wo egreeo ree omsys emun erexternalforcescanbewrittenas:

Weshall

consider

the

external

forces

to

be

harmonic:

where istheforcingfrequency.Wecanwritethesteadystatesolutionas:

whereX1 andX2 are,ingeneral,complexquantitiesthatdependon andthesystemparameters.Substitutingtheabovetwoequationsintothefirstone:


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Weo tain:

IfwedefineatermcalledmechanicalimpedanceZrs(i) as:

an wr te t e rst equat on as:

where


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Multidegreeoffreedomsystems

Modelingofcontinuoussystemsasmultidegreeoffreedomsystems

Eigenvalueproblem


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ss a e e ore,mos eng neer ngsys emsarecon nuous an haveaninfinitenumberofdegreesoffreedom.Thevibration

analysis

of

continuous

systems

requires

the

solution

of

partial

,

.

Infact,analyticalsolutionsdonotexistformanypartialdifferentialequat ons. eana ys so amu t egreeo ree omsystemont e

otherhand,

requires

the

solution

of

aset

of

ordinary

differential

equations,whichisrelativelysimple.Hence,forsimplicityof,

multidegreeoffreedomsystems.

Forasystemhavingndegreesoffreedom,therearenassociatednaturalfrequencies,eachassociatedwithitsownmodeshape.


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multidegreeoffreedomsystem.Asimplemethodinvolvesreplacingthedistributedmass orinertiaofthesystembyafinitenumberoflumpedmassesor

rigidbodies.

Thelumpedmassesareassumedtobeconnectedbymasslesselasticanddampingmembers.

Linear

coordinates

are

used

to

describe

the

motion

of

the

lumped

masses.

Such

modelsarecalledlumpedparameteroflumpedmassordiscretemasssystems.

Theminimumnumberofcoordinatesnecessarytodescribethemotionofthelumpedmassesandrigidbodiesdefinesthenumberofdegreesoffreedomofthesystem.Naturally,thelargerthenumberoflumpedmassesusedinthemodel,the

.


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indicatethetypeoflumpedparametermodeltobeused.

Forexample,thethreestoreybuildingshowninthefigureautomaticall su estsusin athreelumpedmassmodelasindicated

in

the

figure.

Inthismodel,theinertiaofthesystemisassumedtobeconcentratedasthreepoint

levels,andtheelasticitiesofthecolumnsarereplacedbythes rin s.


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Anot erpopu armet o o approximatingacontinuoussystemasa

multidegreeoffreedomsysteminvolvesreplacingthegeometryofthe

systemby

alarge

number

of

small

elements.

Byassumingasimplesolutionwithineachelement,theprinciplesof

compatibility andequilibrium areusedtofindanapproximatesolutionto

. .


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UsingNewtonssecondlawtoderive

equationsof

motion

amultidegreeoffreedomsystemusingNewtonssecondlawofmotion.

.

masses andrigidbodiesinthesystem.Assumesuitablepositivedirectionsforthedisplacements,velocitiesandaccelerationsofthemassesandrigidbodies.

2. Determinethe

static

equilibrium

configuration

of

the

system

and

measure

thedisplacementsofthemassesandrigidbodiesfromtheirrespectivestatic.

3. Drawthefreebodydiagramofeachmassorrigidbodyinthesystem.

,bodywhenpositivedisplacementorvelocityaregiventothatmassorrigid

body.


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equationsof

motion

4.App yNewton ssecon awo motiontoeac massorrigi o ys own y

thefreebodydiagramas:

Example:Derivetheequationsofmotionofthespringmassdampersystem

s ownint e igure.

d l d


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equationsof

motion

Draw ree o y iagramso massesan app yNewton ssecon awo

motion.Thecoordinatesdescribingthepositionsofthemasses,xi(t),are

measuredfrom

their

respective

static

equilibrium

positions,

as

indicated

inthefigure.TheapplicationoftheNewtonssecondlawofmotionto

massmi gives:

or

Theequationsofmotionofthemassesm1 andm2 canbederivedfromthe

aboveequationsbysettingi=1alongwithxo=0andi=nalongwithxn+1=0,

.


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T eequationso motioninmatrix ormint ea oveexamp e can e

expressedas:


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43/67

oranun ampe sys em, eequa onso mo onre uce o:

Thedifferential

equations

of

the

spring

mass

system

considered

in

the

example,canbeseentobecoupled.Eachequationinvolvesmorethanonecoordinate.Thismeansthattheequationscannotbesolvedindividuallyoneatatime;theycanonlybesolvedsimultaneously.

Inaddition,thesystemcanbeseentobestaticallycoupledsincestiffnesses are cou led that is the stiffness matrix has at least onenonzerooffdiagonalterm.Ontheotherhand,ifthemassmatrixhasatleastoneoffdiagonaltermnonzero,thesystemissaidtobedynamically

coupled.Further,

if

both

the

stiffness

and

the

mass

matrices

have

nonzero

offdiagonalterms,thesystemissaidtobecoupledbothstaticallyanddynamically.


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T eequationso motion ora ree yvi ratingun ampe systemcan e

obtainedbyomittingthedampingmatrixandappliedloadvectorfrom:

0kxxcxm

inwhich0isazerovector.Theproblemofvibrationanalysisconsistsof

determiningtheconditionsunderwhichtheequilibriumconditionexpressed

b the above e uation will be satisfied.

Byanalogy

with

the

behavour

of

SDOF

systems,

it

will

be

assumed

that

the

freevibrationmotionissimpleharmonic(thefirstequationbelow),which

xxx

xx

22 )sin(

)sin()(

t

tt

Intheaboveexpressions, representstheshapeofthesystem(whichdoes

notchangewithtime;onlytheamplitudevaries)and isaphaseangle.Thethirde uationabovere resentstheaccelerationsinthefreevibration.

x


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u s u ng

xxx

xx

22

)sin(

)sin()(

t

tt

intheequation0kxxcxm

2

which(since

the

sine

term

is

arbitrary

and

may

be

omitted)

may

be

written:

xxm snsn

0xmk 2

Theaboveequationisonewayofexpressingwhatiscalledaneigenvalueorcharacteristicvalueproblem.Thequantities aretheeigenvaluesor

characteristic

values

indicating

the

square

of

the

free

vibration

2

frequencies,whilethecorrespondingdisplacementvectors expressthecorrespondingshapesofthevibratingsystem knownastheeigenvectorsormodeshapes.

x


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can es own y ramer sru e a eso u ono sse osimultaneousequationsisoftheform:

0

Henceanontrivialsolutionispossibleonlywhenthedenominatordeterminantvanishes.Inotherwords finiteam litudefreevibrationsare

mk2

possibleonlywhen

02

mk

.ExpandingthedeterminantwillgiveanalgebraicequationoftheNthdegreeinthefrequencyparameter forasystemhavingNdegreesof2

.

TheNrootsofthisequation representthefrequenciesoftheNmodesofvibrationwhicharepossibleinthesystem.

2232221 ,....,,, N


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emo e av ng e owes requency sca e e rs mo e, enexhigherfrequencyisthesecondmode,etc.

Thevector

made

up

of

the

entire

set

of

modal

frequencies,

arranged

in

sequence,wi eca e t e requencyvector.

2

1

Normalization:

N

3

wasno e ear er a ev ra onmo eamp u eso a ne rom eeigenproblemsolutionarearbitrary;anyamplitudewillsatisfythebasicfrequencyequation

2andonlytheresultingshapesareuniquelydefined.


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Int eana ysisprocess escri e a ove,t eamp itu eo one egreeo

freedom(thefirstactually)hasbeensettounity,andtheother

displacementshave

been

determined

relative

to

this

reference

value.

This

iscallednormalizingthemodeshapeswithrespecttothespecified

referencecoordinate.

Other

normalizing

procedures

also

are

frequently

used;

e.g.,

in

many

computerprograms,theshapesarenormalizedrelativetothemaximum

sp acementva ue neac mo erat ert anw t respecttoany

particularcoordinate.Thus,themaximumvalueineachmodalvectoris

unity,whichprovidesconvenientnumbersforuseinsubsequent

calculations.


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enorma z ngproce uremos o enuse ncompu erprograms orstructuralvibrationanalysis,however,involvesadjustingeachmodalamplitudetotheamplitude ,whichsatisfiesthecondition

n

Thiscanbeaccomplishedbycomputingthescalarfactor1nn m

nM mT

n vmv

where representsanarbitrarilydeterminedmodalamplitude,andthen

computingthe

normalized

mode

shapes

as

follows:

nv

2/1 MvBysimplesubstitution,itiseasytoshowthatthisgivsthedesiredresult.Aconsequenceofthistypeofnormalizingtogetherwiththemodalorthogonalityrelationshipsrelativetothemassmatrixisthat

where isthecompletesetofNnormalizedmodeshapesandIisanNxNidentitymatrix.Themodeshapesnormalizedinthisfashionaresaidtobe

Imn

T

n

orthonormalrelativetothemassmatrix.


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Amodelofafourstorythreebayframecanbeevaluatedto

determinethemodeshapes. This2Dmodelisfromatypical

.

,

interest. Thefirst

mode

usually

has

the

largest

contribution

to

thestructure'smotion. The eriodofthismodeisthelon est

andthenaturalfrequencyisthelowest.

Pleaseclickonthemovietostart!


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Firstmodesha e


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Secondmodesha e


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Thirdmodesha e


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Examp e:


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56/67

So ution:

Whenthecharacteristicequationpossessesrepeatedroots,the

corres ondin

modesha es

are

not

uni ue.


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So ution:


58/67

Anunrestraine systemisonet at asnorestraintsorsupportsan t at

canmoveasarigidbody.Itisnotuncommontoseeinpracticesystems

that

are

not

attached

to

any

stationary

frame.

Suchsystemsarecapableofmovingasrigidbodies,whichcanbe

consideredasmodesofoscillationwithzerofrequency.

, .

systemsthat

are

not

properly

restrained,

rigid

body

displacements

can

takeplacewithouttheapplicationofanyforce.Thus,denotingapossible

r g o y sp acement yur,we ave

0Kuf rr

,issingular.Inthiscase,thebelowequationcanonlybesatisfiedwhen

=0. 0uMK r 2


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T erigi o y isp acementsaret ose isp acementmo est att e

elementmustbeabletoundergoasarigidbodywithoutstressesbeing

developed

in

it.

Rigidbodydisplacementshapesarealsoreferredtoasrigidbodymodes.

Asystemcan,ofcourse,havemorethanonerigidbodymode.Inthemost

, . ,

spacecraftor

an

aeroplane

in

flight

has

all

six

possible

rigid

body

modes,

threetranslationsandthreerotations,onealongeachofthethreeaxis.

Rigidbodymodesofaplanestresselement


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T enatura mo escorrespon ingto i erentnatura requenciescan e

showntosatisfythefollowingorthogonalityconditions.When :rn

Proof: Thenthnaturalfrequencyandmodesatisfy

rnrn m

nnn mk

2

T

Similarly

the

rth

natural

frequency

and

mode

shape

satisfy

r

nTrnn

Tr mk

2

rrr mk2


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Premu tip ying y gives:rrr m n

r

T

nrr

T

n mk 2

Thetransposeofthematrixontheleftsideof will

equalthe transposeofthematrixontherightsideoftheequation:n

Trnn

Tr mk

2

Subtractingthe

first

equation

from

the

second

equation:

rTnnr

Tn mk

2

Theequation istruewhen whichforsystemswith

022 rTnrn m

0

T

mrn

positivenaturalfrequenciesimpliesthat rn

Modalequationsforundamped


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q p

systems T eequationso motion ora inearMDOFsystemwit out ampingis:

p x

haveillustratedbefore fora2dofsystemsubjectedtoharmonic

excitationisnotefficientforsystemswithmoreDOF,norisitfeasiblefor

. ,

to

transform

these

equations

to

modal

coordinates. Thedisplacementvectorx ofaMDOF systemcanbeexpandedinterms

ofmodalcontributions.Thus,thedynamicresponseofasystemcanbe

expressedas:N

)()()(1

qt rr

r qx



63/67

systemsN

Usingt eequation ,t ecoup e equationsinxj t

givenbelow

p x

)()()(1

qrr

r qx

canbetransformed toasetofuncoupledequationswithmodal

coordinatesqn(t)astheunknowns.Substitutingthefirstequationintothe

p

p N

rrrrr

N

r

(t )q(t )q

p T

NT

NT

Tn :givesbyequationin thiseach termyingPremultipl

p nr rrnrrr n 11



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systems ecauseo eor ogona yre a ons ,a

termsineachofthesummationsvanishexceptther=nterm,reducingtheequationto:

rnrn m

or

(t )q(t )q nnnnnnn

p

wherennnnn

)()( ttPKM TnnnTnnnTnn pkm

ea oveequa onmay e n erpre e as eequa ongovern ng eresponseqn(t)oftheSDOF systemwithmassMn,stiffnessKn,andexcitingforcePn(t).

ere ore

n sca e

t e

genera ze

mass

or

t e

nt

natura

mo e,

n

thegeneralizedstiffnessforthenthmode,andPn(t)thegeneralizedforceforthenthmode.Theseparametersonlydependonthenthmode.


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W en ampingisinc u e ,t eequationso motion oraMDOFsystem

are:

p x Usingthetransformation

p )()()( ttqt r

N

r qx

wherer arethe natural modesofthesystemwithoutdamping,these

equationscan

be

written

in

terms

of

the

modal

coordinates.

Unlike

the

caseofundam eds stems,thesemodale uationsma becou led

throughthedampingterms. However,forcertainformsofdampingthat

arereasonableidealizationsformanystructures,theequationsbecome

,

.equationintothefirst,weobtain:

p NNN

ttt p rrr 111


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Premu tip yingeac termint isequation y gives:n

T

NT

NT

NT

ttt

whichcanberewrittenas:

rrr 111

(t )P(t )qK(t )qC(t )qM nnn

N

rrnrnn 1

w erer

TnnrC c

HereCisanondiagonalmatrixofcoefficientsCnr.

(t)PKqqCqM


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Themodalequationswillbeuncoupledifthesystemhasclassical

damping.ForsuchsystemsCnr=0ifnrandCn canbeexpressedas:

Forsuchsystems:

nnnn

)(tPqKqCqM nnnnnnn n:

)(2 2 nnnnnnn

M

tPqqq

mode.nthfor theratiodampingtheiswhere n

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Multi Degree of Freedom