Transcript FORCED VIBRATIONS

FORCED VIBRATIONS
Mechanics of Vibrations
k
m
F0cos(wt)
x
mx  kx  F0 cos(wt )
x p (t )  X cos(wt )
Eqn. of motion
Response (particular soln of the diff. eqn.)
F0
 st
Amplitude
X

2
2
k  m w 1 r
F
Static deflection (w=0)
 st  0
k
w
r
Frequency ratio
wn
x(t )  C1 cos(wnt )  C2 sin(wnt )  X cos(wt )
Mechanics of Vibrations
Total response
k
m
F0cos(wt)
c
x
mx  cx  kx  F0 cos(wt )
x p (t )  X cos(wt   )
X
F0
(k  m w2 ) 2  c 2 w 2

 st
(1  r 2 ) 2  ( 2r ) 2
cw
1 2 r
)

t
an
(
)
2
2
k  mw
1 r
F0
w
c
 st  ,...r  ,... 
k
wn
2m wn
  t an1 (
x(t )  C1e
(    2 1 ) wn t
 C2 e
Phase angle
Damping ratio
(    2 1 ) wn t
Mechanics of Vibrations
 X cos(wt   )
Response Under F0eiwt
mx  cx  kx  F0 eiwt  F0 [cos(wt )  i sin(wt )]
x p (t )  Xei ( wt  )  X [cos(wt   )  i sin(wt   )]
X
F0
(k  m w2 ) 2  c 2 w2
  tan1 (
H (iw) 
H (iw) 
 st

(1  r 2 ) 2  (2r ) 2
F0
cw
w
1 2 r
)

tan
(
)...,


,...
r

st
k  m w2
1 r 2
k
wn
X  i
X
e ,... H (iw) 
F0 / k
F0 / k
1
(1  r )  (2r )
2 2
2
e
 i
1

1  r 2  i 2r
Mechanics of Vibrations
Frequency response
k
F0cos(wt)
m
F0sin(wt)
c
x
1
H (iw) 
1  r 2  i 2r
mx  cx  kx  F0 cos(wt )
mx  cx  kx  F0 sin(wt )
x p (t )  Re( st H (iw)eiwt )
x p (t )  Im( st H (iw)eiwt )
Mechanics of Vibrations
k
Response Under
the Motion of the
Base m
c
y  Y sin(wt )
y
x
mx  cx  kx  ky  cy  kY sin(wt )  cwY cos(wt )
 cw
 Y k  c w sin(wt   ),...  tan (
)
k
x p (t )  X [cos(wt   ' )]
2
1
Y k 2  c 2 w2
Y [1  (2r ) 2 ]
2
X
2
(k  m w )  c w
 '  tan1 (
2 2
2
2

(1  r 2 ) 2  (2r ) 2
cw
1 2 r
)

tan
(
)
2
2
k  mw
1 r
Mechanics of Vibrations
Displacement
Transmissibility =
Td = |X/Y|
Response Under Rotating Unbalance
Mx  cx  kx  m0ew2 sin(wt )
M
m0 
Mass of the system
Rotating unbalanced mass
e
Distance of the unbalanced mass from the center of rotation
w
Rotation speed
m0 e 2
x p (t )  X sin(wt   )  Im[
r H (iw)eiwt ]
M
MX
1
2
 r H (iw) ,...H (iw) 
m0 e
1  r 2  i 2r
Mechanics of Vibrations