Download SELF-EXCITED VIBRATION

SELF-EXCITED VIBRATION
The force acting on a vibrating object is usually external to the
system and independent of the motion. However, there are
systems in which the exciting force is a function of the motion
variables (displacement, velocity or acceleration) and thus varies
with the motion it produces (called coupling). Friction-induced
vibration (in vehicle clutches and brakes, vehicle-bridge
interaction) and flow-induced vibration (circular wood saws, CDs,
DVDs, in machining, fluid-conveying pipelines) are examples of
self-excited vibration
Example 1: a mass supported by a spring is carried by a moving
belt through friction.
x
m
v0
The friction coefficient at the mass and belt interface is a function
of the relative velocity between the mass and the belt as

gradiant – (>0)
v
The equation of motion of the mass is
mx  kx  mg   0 [1   ( x  v0 )]mg   0 (1  v0 )mg   0mgx
or
mx  0mgx  kx   0 (1  v0 )mg
negative damping causing (initially) divergent vibration
As vibration grows, velocity x and hence relative velocity x  v0 .
This causes the friction coefficient to decrease (see m – v curve)
and then vibration decreases. This cycle of increasing and
decreasing vibration repeats itself forever (unless there is
structural damping). The moving belt can sustain vibration — selfexcited vibration.
Example 2: a solid oscillating in a fluid can interact with the fluid
and produce interesting behaviour. The relative airflow against the
oscillating solid modifies the velocity vector and thus the lift and
drag force acting on the solid.
Consider a long cylinder of length l
supported by a spring k and a damper c.
V
k
d
x
The equation of vertical motion of the
cylinder in the flow is
mx  cx  kx  f (t )
The lift force f is
f (t ) 
1
V 2dlCL
2
Which term varies with time on the right ?
Karman Vortices
shedding occurs in Re=[60, 5000]
http://www.algor.com/products/analysis_replays/turbulent_flow/d
efault.asp?sQuery=vortex%20shedding
Due to vibration of the cylinder, the lift coefficient is no longer a
constant because of shedding of vortices. It may be expressed as
CL  CL0 sin  t
(C
L0
 1 for cylinder)
and

2πksV
d
Vd
 1000, ks  0.21.
where ks is the Strouhal number. For Re 

So when
V
d
2πks
k
d
 n
m 2πks
the cylinder vibrates violently (in resonance) in the flow. When
flow velocity becomes high enough, the flow becomes turbulent,
and the lift force becomes random.
Flutter is a phenomenon of self-excited vibration.
L

x
V
G
O
V
M
D
A rigid wing attached to a rigid support through a spring
Vertical motion:
mx  kx   L cos  D sin 
where
L

2
(V 2  x 2 )lcCL 

2
(V 2  x 2 )lc
CL


D

2
(V 2  x 2 )lc
CD


and
x
V
  0  
  arctan( )
Assume small displacement so that
  tan 
then
x
V

cos  1
V 2  x 2  V 2
dCL
dC
( 0   ) cos  D ( 0   ) sin  ]
2
d
d

dC
dC
dCD
 V 2lc[ L  0  ( L 
 0 ) ]
2
d
d
d
mx  k x 
finally
mx 

2
Vlc(
V 2 lc[
dCL dCD

dC

0 ) x  kx  V 2lc L 0
d
d
2
d
V
O
Torsional vibration (assuming centre of gravity and aerodynamic centre
coincide):
( me 2  I )  K  Le  M
where the pitching moment


dCL
( 0   )
2
2
d


dC M
M  V 2 lc 2CM  V 2lc 2
( 0   )
2
2
d
L
V 2lc 2CM 
V 2 lc
The above equation becomes

dC
dC

dC
dC
( me 2  I )  [ K  V 2lc( L e  M c)]  V 2lc( L e  M c)]0
2
d
d
2
d
d
The static divergence speed is
Vdiv 
2K
lc(
dC L
dC M
e
c)
d
d
for a symmetric aerofoil
Vdiv 
2K
dC
lc L e
d
The high the Vdiv , the greater speed capacity the aircraft has.