whirling of shafts

7/26/2019 Whirling of Shafts

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Introductio

n

Whirling of shafts

In engineering, we have seen many

applications of shaft and a rotor system.

Power transmitting shafts always have

either gear, pulley, sprocket, rotor or a

disc attached to a shaft as shown in theFigure



Introductio

n

Whirling of shafts

Shaft

disc

bearings

Problems in shaft and a rotor

systems:

(i !nbalance in rotor"disc

(ii Improper assembly

(iii #eaker bearings



Introductio

n

Whirling of shafts

Unbalance in rotor / disc

$op view of a rotor

rotor

%eometric

centre

&ass

centree

m

For perfect balancing

(i &ass centre (centre

of gravity has to co'

inside with the

geometric centre

(ii m.e unbalance )



Introductio

n

Whirling of shafts

Unbalance in rotor / disc

$op view of a disc

e

m

ω

*entrifugal force



Whirling of shafts

Static dynamic



Whirling of shafts

$op view of the disc

P' %eometric center

%' centre of gravity

+' center of rotation

+ P %

d e

otating shafts tend

to bend out at certain

speed and whirl in an

undesired manner,which affects the

working of machine

and the shaft may

also fail due to largedeflection at the

center



Whirling is defined as the rotation of planemade by the bent shaft and line of centers

of bearings as shown in Figure

Whirling of shafts



Whirling of shafts

neglecting damping

Assumptions

(i the disc at the mid'span has an

unbalance

(ii the shaft inertia is negligible

and the shaft stiffness is same

in all directions

(iii any internal damping is

neglected

-

-"



Whirling of shafts

neglecting damping

+ P %

d e

P' %eometric center

%' centre of gravity

+' center of rotation

e' eccentricity

d' deflection of shaft

*entrifugal force


estoring force(spring force) /.d ω

e(dm0+



Whirling of shafts

neglecting damping

12uating both the forces

3ivide numerator and

denominator by /

/de(dm0=+

m0/

em0d

−

=

r 4

er d

−

=



Whirling of shafts

neglecting damping

It is observed from above e2uation that theoretically,

the deflection of the shaft tends to infinity when r 4,

i.e ωωn.

$he speed of the shaft under this condition is referred

as critical speed of shaft .

r 4

er d

−

=



Whirling of shafts

neglecting damping

*ritical speed

r 4

r

e

d

−

=

) 4 5 6

)

4

5

6

d " e

ω"ωn (r



Whirling of shafts

neglecting damping

If r 74 8elow critical speed

d is 9ve

which indicates that disc

rotates about + ( centre of

rotation and + and % (*entre of gravity are

opposite each other

%P

+


r 4

er d

−

=



Whirling of shafts

neglecting damping

If r 4 ;bove critical speed

d is <ve

d → 'e, which indicates +,

approaches % and discrotates about center of

gravity.

%P

+


r 4

er d

−

=



Important

It is desired to run the shaft at speed much higher

than the natural fre2uency of the shaft rotor system,

which has reduced whirling of shaft.

Whirling of shafts

neglecting damping



Whirling of shafts with damping

Damping is the resistance to motion

;ir or

+il

For the analysis of the systems

with damping an additional

assumption is made, i.e the

e=ternal damping force isproportional to the velocity of the

disc at geometric center.




Force diagram

+

P

%

/d cωd

b

d

e

$hree forces acting on the shaft under e2uilibrium:

(i centrifugal fore at % acts racially outwards

(ii restoring force at point P acts radialy inwards and

(iii damping force at P acts radialy outwards.

mb0




$op view of the disc at time t

=

y

+

P

% (=g,yg

y

x

d

e

φ

ψ ωt

t e.cos0==g +=t e.sin0yyg +=




$he e2uation of motion for the system in > ' direction is:

$he e2uation of motion for the system ? ' direction is:

F

)/==c=m g =++

)/==ce.cos00=m(

=++− t

t e.cos0m0/==c=m =++

t e.sin0m0/yycym =++



$he governing e2uation of motion of the system is:

Solution of governing differential e2uation

-et, =(t, the steady state solution of e2uation of motion is:

;bove 12n has to satisfy governing 12n.

$ransient solution Steady state solution


(t=(t==(t pc +=

>cos(0=(t @−= t

t e.cos0m0/==c=m =++



Aectorial representation of forces

eference a=is

KX-m 2 X

O

A

B F

t

Impressed force

KX

Spring force

c X

Damping

force m 2 X

Inertia force

X

Displacementvector





$he steady state response of the system in x, horiBontal direction is :

From triangle +;8

( ) ( ) em0c0>m0/> =+− >

( ) em0c0

m0/> =+−

( ) ( )

c0m0/

em0>

+−=




3ividing by /

(Cr (4

er

> r +−=

/c0

/m04

/

em0

>

+

−

=




$he steady state response of the system in x,horiBontal direction is :

Similarly, the steady state response of the system in y,

Aertical direction is :

( ) ( )

@cos(0

Cr 4

=(t

−

+−

= t

r

er

( ) ( )@sin(0

Cr 4y(t

−+−

= t

r

er




$he deflection of shaft is :

=

y

+

P

% (=g,yg

y

x

d

e

φ

ψ ωt

y=d +=

( ) ( )

Cr 4

er d

r +−

=




$he deflection of shaft is :

*ritical speed

( ) ( )

Cr 4

r

e

d

r +−

=

) 4 5 6

)

4

5

6ξ).)

ξ).4

ξ).

ξ).5

ξ).6

ξ).D

ξ).E)E

ξ4

d " e

ω"ωn (r




$he phase angle is :

−= −

4

r 4

Ctan@

r

) 4 5 6 D)

)

6)

)

G)

4))

4)

46)

4)

4G)

ξ4.)

ξ).E)Eξ).D

ξ).ξ).4ξ)

P h a s e

a n g l e , ψ

ω"ωr (r



Summary

3ue unbalance in a shaft'rotor system, rotating shafts tend

to bend out at certain speed and whirl in an undesired

manner

#hirling is defined as the rotation of plane made by thebent shaft and line of centers of bearings

$heoretically, the deflection of the shaft tends to infinity

when r 4, i.e ωωn.$he speed of the shaft under this condition is referred as

critical speed of shaft.



Summary

It is desired to run the shaft at speed much higher than the

natural fre2uency of the shaft rotor system

$heory indicates that at higher speeds the shaft tries to

rotate at centre of gravity, and deflection of the shaft is

negligible



Theory questions

#hat do you understand by critical speed of shafts.

3erive the necessary relations and thus, e=plain what ishappening in the system carrying a shaft having an

unbalanced disc at its centre is operated above and

below critical speed.



Theory questions

$he speed of the

shaft under the

condition when

r 4, i.e ωωn isreferred as

critical speed of

shaft .

3erive the relation

*ritical speed

r 4

r

e

d

−

=) 4 5 6

)

4

5

6

d " e

ω"ωn (r

whirling of shafts

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