Review Basic Concepts, Tutorial 3 (Vibration) 

 

Harmonic Forced Vibrations 

Undamped  Damped 

EOM: tFkxxm o sin  

 Steady‐state response: 

)sin( tX  

 

EOM: tFkxxcxm o sin  

 Steady‐state response: 

)sin( tX  

 

20

mkF

X  n

n

,0

  222

0

cmk

FX   tan 21

mkc  

or  or  or  or 

20 /1

1/ nkFX

 

1/ 1/,0

n

n   22220 /2/1

1/

nnkF

X

 

21

/1/2tan

n

n

 

 

m

tFo sin

x

k

m

tFo sin

x

ck

0 1 2 3 40

1

2

3

kFX p

/0

n / / n 1 2 3 4

2/

0

n /

kFX/0

n /

Review Basic Concepts, Tutorial 3 (Vibration) 

 

Harmonic Forced Vibrations (Caused by Rotating Unbalance) 

EOM: 

 

Steady‐state response: 

)sin( tX  

 

 

                

222

2

)()(

cMkmeX   2

1tan

Mkc

 

or  or 

222

2

)/2())/(1(

)/(

nn

n

meMX

  2

1

)/(1/2tan

n

n

 

 

 

M

x

meω2

k/2 k/2

e ωtm

c

tex sin

0 1 2 3 4 50

1

2

3

ζ=1

0.5

0.30.15

ζ=0

MXme

n /

n /

2 sinMx cx kx me t

Static equilibrium position