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SELF-EXCITED VIBRATION

The force acting on a vibrating object is usually external to the

system and independent of the motion. However, there aresystems in which the exciting force is a function of the motionvariables (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.

The friction coefficient at the mass and belt interface is a function

of the relative velocity between the mass and the belt as

The equation of motion of the mass is

m

x

v0

v

gradiant (>0)

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xmgmgvmgvxmgkxxm 00000 )1()](1[ ++===+

ormgvkxxmgxm )1( 000 +=+

negative damping causing (initially) divergent vibration

As vibration grows, velocity x and hence relative velocity 0vx .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 self-excited vibration.

Example 2: a solid oscillating in a fluid can interact with the fluidand produce interesting behaviour. The relative airflow against the

oscillating solid modifies the velocity vectorand thus the lift and

drag force acting on the solid.

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k

x

V d

Consider a long cylinder of length l

supported by a spring kand a damperc.

The equation of vertical motion of the

c linder in the flow is

)(tfkxxcxm =++

The lift forcefis

L

2

2

1)( dlCVtf =

Which term varies with time on the right ?

http://www.algor.com/products/analysis_replays/turbulent_flow/default.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

Karman Vortices

shedding occurs in Re=[60, 5000]

http://www.algor.com/products/analysis_replays/turbulent_flow/default.asp?sQuery=vortex%20sheddinghttp://www.algor.com/products/analysis_replays/turbulent_flow/default.asp?sQuery=vortex%20sheddinghttp://www.algor.com/products/analysis_replays/turbulent_flow/default.asp?sQuery=vortex%20sheddinghttp://www.algor.com/products/analysis_replays/turbulent_flow/default.asp?sQuery=vortex%20sheddinghttp://www.algor.com/products/analysis_replays/turbulent_flow/default.asp?sQuery=vortex%20shedding

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tCC sinL0L

= ( 1L0 C for cylinder)

and

d

Vks2=

where sk is the Strouhal number. For 1000Re >= Vd

, 21.0s k .

So when

s

n

s 22 k

d

m

k

k

dV

==

the cylinder vibrates violently (in resonance) in the flow. Whenflow velocity becomes high enough, the flow becomes turbulent,

and the lift force becomes random.

Flutter is a phenomenon of self-excited vibration.

OV

G

A rigid wing attached to a rigid support through a spring

M

x

D

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V

L

Vertical motion: sincos DLkxxm =+

where

+=

+=+=

D22L22

L

22)(

2)(

2)(

2

ClcxVD

ClcxVlcCxVL

and

)arctan(0

V

x==

Assume small displacement so that222

1costan VxVV

x+=

then

])d

d

d

d(

d

d[

2

]sin)(d

dcos)(

d

d[

2

0

DL

0

L2

0

D

0

L2

CCClcV

CClcVkxxm

+=

=+

finally

0

L2

0

DL

d

d

2)

d

d

d

d(

2

ClcVkxx

CCVlcxm =+++

Torsional vibration (assuming centre of gravity and aerodynamic centre

coincide):

MLeKIme +=++ )(2

where the pitching moment

O

V

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)(d

d

22

)(d

d

22

0

M22

M

22

0

L2

M

22

+==

+==

ClcVClcVM

ClcVClcVL

The above equation becomes

0

ML2ML22 )]d

d

d

d(

2)]

d

d

d

d(

2[)(

c

Ce

ClcVc

Ce

ClcVKIme +=+++

The static divergence speed is

)d

d

d

d(

2

ML

div

cC

eC

lc

KV

+

=

for a symmetric aerofoil

eC

lc

KV

d

d

2

L

div=

The high the divV , the greater speed capacity the aircraft has.