Single Degree of Freedom – Free Vibration

fixed / ground

c k

m

FBD of single mass

x

damper FspringF

m staticx

0

kxxcxm

maFFF xscx

Solution A: Critically damped

occurs noscillatio no , thereforenote

)()(

ratio damping the, 2

c let

(undamped)frequency natural g/x /let

1damped Critically A. Case

00021

n

static

txxxeetAAtx

m

mk

)(ζ

ntt

n

nn

Case A – Critically damped

Case B - Overdamped

frequency natural damped1

2

1,

2

1

1exp(1exp()(

2)(

noscillatio no -1 Overdamped B.Case

2

00000

22

121

nd

d

n

n

n

nnt

tsts

xxB

xxxA

tBtAetx

eAeAtx

)(ζ

n

Case B - Overdamped

Underdamped

andfrequency natural damped theis ,)1(

sincos)(

nOscillatio -10 ed UnderdampC. Case

1/22

000

nd

dd

nd

t txx

txetx

)ζ(

n

Case C - Underdamped

Example D - less damping

SDOF – Forced Vibration

tutu sin)( 0

)( tx

c k

m What force does the mass “feel?”

road

Summing the forces in the x direction:

c

k

tFtckukxxcxm

tkutcukuuckxxcxm

u(t)

tuututu

kuuckxxcxm

uxkuxcxm

1

02/1222

0

00

00

tanfor

)sin()sin()(

obtain we RHS, thesimplfyingby and

sincos

find we,for ngsubstituti therefore,

cos then,sin)( sincehowever

obtain weg,rearrangin

0)()(

ratio damping the, 2

c

andfrequency natural undamped the,)/(

for

])/2()/1[(

)sin()/(F)(

n

2/1

2/122220

m

mk

wtktx

n

nnss

From D. Eqns we obtain steady state solution

What force does “Mass” feel?

2/1

2222

2

0

t

)/2()/1(

)/2(1

isibility transmiss theof magnitude thewhere

)sin( show that can weAnd,

)(F

find wedamper, and spring thefrom results

mass the toed transmitt force theSince

nn

n

t

ssss

t

T

T

tTF

F

kxxckxxct

F

Example ¼ car model

symbol value unitsweight W 1000 lbf mass m 2.588 lbf/(in/sec^2)

spring k 100 lbs/indamper c lb/(in/sec)

damping ratio zeta 0.500 n/anatural frequency 6.216 rad/sec

0.989 Hzdamped nat. freq. 0.857 Hz

n

d

http://www.me.metu.edu.tr/courses/ME432/soft07.html

Effect of damping on forced vibration

Transmissibility

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.0 5.0 10.0 15.0

Forcing Frequency (Hz)

Tra

nsm

issib

ilit

y