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8/10/2019 Assumed Modes Methods
1/17
G.Le
ng,MEDept,NUS
7Assumed
ModesSumm
ation
Whenexactnaturalmod
esandfrequenciesaremodes
aredifficu
lttofind...
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G.Le
ng,MEDept,NUS
7.1
Whatar
eassumedmodes?
Asetoflinearlyindependentfunctionsi
thatsatisfythe
g
eometricboundaryconditio
nsbutnotnec
essarilythena
tural
boundarycond
itionsofthesystem.
Notes
1.
linearlyind
ependentie
2.geometricie
3.naturalie
8/10/2019 Assumed Modes Methods
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ng,MEDept,NUS
E
xample:Assumedmodesforabar
L
u(x,t)
x
B
oundarycond
itions:u|x=0
=0andu/
x
|x=L=0
Q
uestion:Can
yousuggesta
function(x)
thatsatisfiesthese
boundaryconditions?
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ng,MEDept,NUS
Firstguesstry(x)=x.
W
illthiswork?
W
hatsthenext
possibleguess?
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G.Le
ng,MEDept,NUS
Q
uestion:Can
youfindahig
herorderassu
medmode?
8/10/2019 Assumed Modes Methods
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ng,MEDept,NUS
Qu
estion:Cany
ouguesstheg
eneralformfortheassumedmodes?
A
possibleseto
fassumedmodeswouldbe
Homework:Provethisandco
mparewithth
eexactmodes!
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G.Le
ng,MEDept,NUS
Example:A
ssumedmod
esforasimplysupported
beam
x
x=L
x=
-L
y
Geometricboundaryconditions:V(L)=
0andV(-L)=0
Whatisapos
sibleassumed
mode?
8/10/2019 Assumed Modes Methods
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G.Le
ng,MEDept,NUS
C
anyoufinda
higherordermode?
H
owdothesea
ssumedmodescomparewiththefirsttwo
modes
fromtheexactsolution?
8/10/2019 Assumed Modes Methods
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G.Le
ng,MEDept,NUS
7.2
Approxim
ationofcontinuoussystem
sviaassumedmodes
A
ssumedmode
scanbeused
toformaMDOFapproxima
tefor
thecontinuous
systemeg:
F
orabar,theelasticstrainenergyandkin
eticenergyare
givenby:
L
V
=
1/2
EA(u/
x
)2dx
0L
T
=
1/2
A(u/
t)2dx
0
8/10/2019 Assumed Modes Methods
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G.Le
ng,MEDept,NUS
i=N
U
singNassum
edmodes,we
letu(x,t)=
i(x)qi(t)
i=1
T
heelasticstrainenergythen
:
L
V
=
1/2
0
w
hereqT
=
{q1,...,qN}
L
Kij
=
EAi(x)j(x)dx
0
8/10/2019 Assumed Modes Methods
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G.Le
ng,MEDept,NUS
S
imilarlythekineticenergyis:
L
T
=
1/2
0
w
hereqT
=
{q1,...,qN}
L
Mij
=
Ai(x)j(x
)dx
0
8/10/2019 Assumed Modes Methods
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G.Le
ng,MEDept,NUS
F
inallyiftheexternalaxialforceperunitlengthisp(x,t),thenthe
v
irtualworkdonebythisloa
dingis:
L
W
=
0
UsingNassumedm
odes,u(x,t)
=
SothegeneralizedforceQT
={Q1,...,QN}is:
L
Qi(t)
=
p(x,t)
(x)dx
0
8/10/2019 Assumed Modes Methods
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G.Le
ng,MEDept,NUS
ThereforebyLagrangesformulation,the
NDOFapproximation
forthebaris: M
q
+
Kq
=
Q
Question:Howgoodisthisapproximatio
n?
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ng,MEDept,NUS
Example
:MDOFapproximationfo
rauniform
bar
L
u(x,t)
x
1.Boundaryco
nditions:u|x=0=0andu/x|x=L=0
2.Assumedmodes,2DOFa
pproximation
1(x)=x/L(x/L2)
2(x)=(x/L)2(x/L2)2
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ng,MEDept,NUS
Th
emassmatrix
isgivenby:
L
Mij
=
Ai(x)j(x
)dx
0
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ng,MEDept,NUS
Th
estiffnessmatrixisgivenby:
L
Kij
=
EAi(x)j(x)dx
0
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G.Le
ng,MEDept,NUS
H
encetheapproximatedfirsttwofrequenc
iesare
3.7/(2L)(E/)1/2an
d11.8/(2L)(E/)1/2
Comparewith
theexactfirst
twonaturalfr
equencies: