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7Assumed

ModesSumm

ation

Whenexactnaturalmod

esandfrequenciesaremodes

aredifficu

lttofind...

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7.1

Whatar

eassumedmodes?

Asetoflinearlyindependentfunctionsi

thatsatisfythe

g

eometricboundaryconditio

nsbutnotnec

essarilythena

tural

boundarycond

itionsofthesystem.

Notes

1.

linearlyind

ependentie

2.geometricie

3.naturalie

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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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Firstguesstry(x)=x.

W

illthiswork?

W

hatsthenext

possibleguess?

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Q

uestion:Can

youfindahig

herorderassu

medmode?

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Qu

estion:Cany

ouguesstheg

eneralformfortheassumedmodes?

A

possibleseto

fassumedmodeswouldbe

Homework:Provethisandco

mparewithth

eexactmodes!

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Example:A

ssumedmod

esforasimplysupported

beam

x

x=L

x=

-L

y

Geometricboundaryconditions:V(L)=

0andV(-L)=0

Whatisapos

sibleassumed

mode?

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C

anyoufinda

higherordermode?

H

owdothesea

ssumedmodescomparewiththefirsttwo

modes

fromtheexactsolution?

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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

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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

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S

imilarlythekineticenergyis:

L

T

=

1/2

0

w

hereqT

=

{q1,...,qN}

L

Mij

=

Ai(x)j(x

)dx

0

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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

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ThereforebyLagrangesformulation,the

NDOFapproximation

forthebaris: M

q

+

Kq

=

Q

Question:Howgoodisthisapproximatio

n?

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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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Th

emassmatrix

isgivenby:

L

Mij

=

Ai(x)j(x

)dx

0

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Th

estiffnessmatrixisgivenby:

L

Kij

=

EAi(x)j(x)dx

0

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H

encetheapproximatedfirsttwofrequenc

iesare

3.7/(2L)(E/)1/2an

d11.8/(2L)(E/)1/2

Comparewith

theexactfirst

twonaturalfr

equencies: