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Multi Degree of Freedom
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Transcript of 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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Equationsofmotionforforced
vibration T eexterna orcesF1 an F2 actont emassesm1 an m2,respective y.
Thefreebodydiagramsofthemassesareshowninthefigure.
Thea lication ofNewtonssecondlawofmotiontoeachofthemasses
givestheequationofmotion:
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Equationsofmotionforforced
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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Equationsofmotionforforced
vibration T eequationscan ewritteninmatrix ormas:
, , ,
respectively andx(t)andF(t)arecalledthedisplacementandforce
vectors,respectively.whicharegivenby:
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Equationsofmotionforforced
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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Freevibrationanalysisofan
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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Freevibrationanalysisofan
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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Freevibrationanalysisofan
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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Freevibrationanalysisofan
undampedsystem
ur er,s nce
theaboveequationishomogeneous,onlytheratios andr2= canbefound.For ,theequations
give:
Noticethatthetworatiosareidentical.
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Freevibrationanalysisofan
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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Freevibrationanalysisofan
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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Freevibrationanalysisofan
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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Freevibrationanalysisofan
undampedsystem
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Freevibrationanalysisofan
undampedsystem
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Freevibrationanalysisofan
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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UsingNewtonssecondlawtoderive
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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UsingNewtonssecondlawtoderive
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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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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So ution:
Whenthecharacteristicequationpossessesrepeatedroots,the
corres ondin
modesha es
are
not
uni ue.
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So ution:
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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
Modalequationsforundamped
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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
Modalequationsforundamped
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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