8/6/2019 Undamped Free Vibraion Derivation

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Single Degree of Freedom SystemThere are four cases

1.Undamped free vibration.

2.Damped free vibration3.Undamped force vibration

4.Damped force vibration

1.Undamped free vibration

Using hooks lawFa u

or F=-kx k is conxtant of proportionality

or F=-ku___________(1)

Aslo we know that

F=maF=mu___________(2) u=double derivative of

displacement (u)

8/6/2019 Undamped Free Vibraion Derivation

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Compare eq (1) and (2)

mu=-ku

mu+ku=0

This is linear equation of order 2 (i.e 2nd order)

divided by m we get

m

k

u

um

ku

Auxiliary

umku

s!

!

!

2

"

8/6/2019 Undamped Free Vibraion Derivation

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or

tm

kBt

m

kAtu

tm

kBt

m

kAtetu

m

kb

anda

btBbtAetu

iseq

im

kandi

m

k

i

im

ku

at

sincos)(

)sincos()0()(

,

0

)sincos()(

)2(

1

!

!

!

!

!

!

s!

8/6/2019 Undamped Free Vibraion Derivation

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8/6/2019 Undamped Free Vibraion Derivation

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m

k

m

kwT

Timeperiod

wntutuwntutu

AineqBandAvaluesofPuttingthe

BB

BinequattuNowputting

uAAu

n

TTT

222

cos)(0cos)(

)(""""

00

)(0)0(

0

!

!

!

!!

!!

!!