4 Chapter 4 Interference Back Lash

7/25/2019 4 Chapter 4 Interference Back Lash

1/47

1

M

L

K

N

R

RA

rar

Addendum

Circles

Pitch

Circle

Base Circle

P

O1

O2

Pinion

Pitch

Circle

H

FE

G

Gear

Proile

Base Circle!heel

Figure shows a pinion

and a gear in mesh

with their center as

O1andO2 respectively.

MN is the common

tangent to the basic

circles and KL is the

path of contactbetween the two

mating teeth.


2/47

2

M

L

K

N

R

RA

ra

r

Addendum

Circles

Pitch

Circle

Base Circle

P

O1

O2

Pinion

Pitch

Circle

H

FE

G

!heel

Consider, the radius of the

addendum circle of pinion

is increased to O1

N, the

point of contact L will

moves from L to N. If this

radius is further increased,

the point of contact L will

be on the inside of basecircle of wheel and not on

the involute profile of the

wheel.


3/47

M

L

K

N

R

RA

ra

r

Addendum

Circles

Pitch

Circle

Base Circle

P

O1

O2

Pinion

Pitch

Circle

H

FE

G

!heel

!he tooth tip of the pinion

will then undercut the

tooth on the wheel at theroot and damages part of

the involute profile. !his

effect is "nown as

interference, and occurs

when the teeth are being

cut and wea"ens the

tooth at its root.


4/47

#

M

L

K

N

R

RA

ra

r

Addendum

Circles

Pitch

Circle

Base Circle

P

O1

O2

Pinion

Pitch

Circle

H

FE

G

!heel

In general, the

phenomenon,

when the tip oftooth undercuts the

root on its mating

gear is "nown as

interference.


5/47

$


6/47

%


7/47&

M

L

K

N

R

RA

ra

r

Addendum

Circles

Pitch

Circle

Base Circle

P

O1

O2

Pinion

Pitch

Circle

H

FE

G

!heel

'imilarly, if the radius of

the addendum circles of

the wheel increasesbeyond O2(, then the tip

of tooth on wheel will

cause interference with

the tooth on pinion. !he

points ( and N are called

interference points.


8/47)

M

L

K

N

R

RA

ra

r

Addendum

Circles

Pitch

Circle

Base Circle

P

O1

O2

Pinion

Pitch

Circle

H

FE

G

!heel

Interference may be

avoided if the path of

the contact does note*tend beyond

interference points. !he

limiting value of the

radius of the addendum

circle of the pinion is

O1N and of the wheel is

O2(.


9/47+

M

L

K

N

R

RA

ra

r

Addendum

Circles

Pitch

Circle

Base Circle

P

O1

O2

Pinion

Pitch

Circle

H

FE

G

!heel

!he interference may only

be prevented, if the point

of contact between the twoteeth is always on the

involute profiles and if the

addendum circles of the

two mating gears cut the

common tangent to the

base circles at the points

of tangency.


10/471

-hen interference is ust prevented, the ma*imum length

of path of contact is (N.

sinrMPapproachofpathMaximum ==

sinRPNrecessofpathMaximum ==

( ) sinRrPNMPMN

MNcontactofpathoflengthMaximum

+=+==

( )( )

tan

cos

sin"

RrRr

contactofarcoflengthMaximum

+=+

=


11/47

11

1. /eight of the teeth may be reduced.

2. 0nder cut of the radial flan" of the pinion.

. Centre distance may be increased. It leads to increase inpressure angle.

#. y these tooth correction, the pressure angle, centre

distance and base circles remain unchanged, but tooththic"ness of gear will be greater than the pinion tooth

thic"ness.


12/47

12

Gear

Proile G

E F

H

Pitch

Circle

Pinion

O2

O1

P

Base Circle

Base Circle

Pitch

Circle

Ma#$

Addendum

Circles

r

ra

RA

R

N

K

L

M

!heel

!he pinion turns

cloc"wise and drives

the gear as shown in

Figure.oints ( and N are

called interference

points. i.e., if the

contact ta"es placebeyond ( and N,

interference will occur.


13/47

1

Gear

Proile G

E F

H

Pitch

Circle

Pinion

O2

O1

P

Base Circle

Base Circle

Pitch

Circle

Ma#$

Addendum

Circles

r

ra

RA

R

N

K

L

M

!heel!he limiting value of

addendum circle radius

of pinion is O1N and the

limiting value ofaddendum circle radius

of gear is O2(.

Considering the critical

addendum circle radius

of gear, the limiting

number of teeth on gear

can be calculated.


14/47

1#

Gear

Proile G

E F

H

Pitch

Circle

Pinion

O2

O1

P

Base Circle

Base Circle

Pitch

Circle

Ma#$

Addendum

Circles

r

ra

RA

R

N

K

L

M

!heel3et

4 5 pressure angle

6 5 pitch circle radius

of gear5 7mT

r 5 pitch circle radius

of pinion

5 7mt

! 8 t 5 number of teeth

on gear 8 pinion

m 5 module


15/47

1$

Gear

Proile G

E F

H

Pitch

Circle

Pinion

O2

O1

P

Base Circle

Base Circle

Pitch

Circle

Ma#$

Addendum

Circles

r

ra

RA

R

N

K

L

M

!heel

aw5 9ddendum constant

of gear :or; wheel

ap5 9ddendum constant

of pinion

aw. m 5 9ddendum of

gear

ap. m 5 9ddendum of

pinion

< 5


16/47

1%

From triangle O1N, 9pplying cosine rule

( )

( )

sinsin

sin21sin2sin

1

2sin

%&cos2sin

cos2

2

2222

2

222

2222

222

11

22

1

2

1

RPOPN

r

R

r

Rr

r

R

r

Rr

RinrRr

RrRr

NPOPNPONPPONO

==

++=

++=

++=

++=

+=


17/47

1&

3imiting radius of the pinion addendum circle=

2

1

22

1

21 sin21

2sin21

++=

++=

tT

tTmt

rR

rRrNO


18/47

1)

Addendum of the pinion = O1N - O1P

++=

++=

1sin21

2

2sin21

2

2

1

2

2

1

2

t

T

t

Tmt

mt

t

T

t

Tmtmap


19/47

1+

Addendum of the pinion = O1N - O1P

( )( )

++

=

++=

1sin21

2

1sin212

2

1

2

2

1

2

GG

at

t

T

t

Tt

a

p

p

The equation gives minimum numer of teeth required

on the pinion to avoid interference!


20/47

2

"f the numer of teeth on pinion and gear is same#

$=1

( )

+

=1sin'1

2

2

1

2

pat


21/47

21

1. 1#7O Composite system 5 12

2. 1# 7O Full depth involute system 5 2

. 2O Full depth involute system 5 1)

#. 2O'tub involute system 5 1#


22/47

22

From triangle O2(, applying cosine rule and simplifying,

!he limiting radius of wheel addendum circle=

2

1

22

1

22 sin21

2sin21

++=

++=

Tt

TtmT

Rr

RrRMO

Addendum of the pinion = O%M- O%P

++= 1sin21

2

21

2T

t

T

tmTmap


23/47

2

++

=

++=

1sin211

1

2

1sin212

2

1

2

2

1

2

GG

aT

T

t

T

tTa

p

w

The equation gives minimum numer of teeth required

on the &hee' to avoid interference!


24/47

2#

P()CH L(NE

Pc

)h

a

*RACK

c

!he rac" is part of toothed wheel of infinite diameter. !he

base circle diameter and profile of the involute teeth are

straight lines.


25/47

2$

Base Circle

c

RACK

P(N(ON

P()CH L(NE


26/47

2%

P()CH L(NE

P(N(ON

RACK

c

M

L

HP

K


27/47

2&

P()CH L(NE

P(N(ON

RACK

c

M

L

HP

K

r

3et

t5 (inimum number of

teeth on the pinion

r5 itch circle radius of

the pinion

5 7 mt

= Pressure an+le

AR$m , Addendum o rac-


28/47

2)

P()CH L(NE

c

M

L

P

K

P(N(ON

RACK

H

!he straight profiles of

the rac" are tangential

to the pinion profilesat the point of contact

and perpendicular to

the tangent PM! oint

is the limit ofinterference.


29/47

2+

P()CH L(NE

RACK

c

M

L

HP

K

r

P(N(ON9ddendum of the rac"=

( )

2

2

2

2

sin

2

sin2

sin

sin

sinsin

sin

R

R

At

mt

r

OP

OP

PLLHmA

=

=

=

=

===


30/47

Gear

P

Pinion

!he gap

between thenon>drive face

of the pinion

tooth and the

adacent wheeltooth is "nown

as backlash!

If the rotationalsense of the pinion

were to reverse,

then a period of

unrestrained pinionmotion would ta"e

place until the

bac"lash gap closed

and contact with the

wheel tooth re>established

impulsively.


31/47

1

ac"lash is the error in motion thatoccurs when gears change direction.

!he term ?bac"lash? can also be used

to refer to the si@e of the gap, not ust

the phenomenon it causesA thus, one

could spea" of a pair of gears as

having, for e*ample, ?.1 mm of

bac"lash.?

9 pair of gears could be designed to have @ero bac"lash,

but this would presuppose perfection in manufacturing,

uniform thermal e*pansion characteristics throughout the

system, and no lubricant.


32/47

2

In the case of a large gear and a small pinion, however,the bac"lash is usually ta"en entirely off the gear and the

pinion is given full si@ed teeth.

ac"lash can also be provided by moving the gears

farther apart. For situations, such as instrumentation andcontrol, where precision is important, bac"lash can be

minimised through one of several techniBues.

!herefore, gear pairs are designed to have some

bac"lash. It is usually provided by reducing the tooth

thic"ness of each gear by half the desired gap distance.


33/47

For instance, the gear can be split along aplane perpendicular to the a*is, one half

fi*ed to the shaft in the usual manner, the

other half placed alongside it, free to rotate

about the shaft, but with springs betweenthe two halves providing relative torBue

between them, so that one achieves, in

effect, a single gear with e*panding teeth.

9nother method involves tapering the

teeth in the a*ial direction and providingfor the gear to be slid in the a*ial direction

to ta"e up slac".


34/47

#

M.

N.

RR.

r.r

Base Circle

Base Circle

P

O1

O2

M

N

R

RA

rar

P

O1

O2

!heel

Pinion

/tandard

0cuttin+

Pitch Circle

/tandard0cuttin+Pitch Circle

c

/tandard

0cuttin+

Pitch Circle

/tandard0cuttin+Pitch Circle

c

c. Oeratin+Pitch Circle

.

.

Figure a

Pinion

Figure b

!heel

Backlash


35/47

$

3et r5 standard pitch circle radius of pinion

(5 standard pitch circle radius of wheel

c5 standard centre distance 5 r )(

r*5 operating pitch circle radius of pinion (*5 operating pitch circle radius of wheel

c* 5 operating centre distance 5 r* ) (*

+5 'tandard pressure angle

+*5 operating pressure angle

h 5 tooth thic"ness of pinion on standard pitch circle5 P/%

h*5 tooth thic"ness of pinion on operating pitch circle


36/47

%

3et , 5 tooth thic"ness of gear on standard pitch circle

,1 5 tooth thic"ness of gear on operating pitch circle

p 5 standard circular pitch 5 2 rD t 5 26D!

p* 5 operating circular pitch 5 2 r1Dt 5 261D! .5 change in centre distance

5 ac"lash

t 5 number of teeth on pinion

T 5 number of teeth on gear.

Involute gears have the invaluable ability of providing conugate action

when the gearsE centre distance is varied either deliberately or

involuntarily due to manufacturing andDor mounting errors.


37/47

&

===

=

=

==

1

.cos

cos

.cos

cos.

.cos

cos.

cos.cos.

...

ccccccNow

cc

cc

c

c

R

R

r

r


38/47

)

On the operating pitch circle=

10... Hhp

ac!lashthic!nesstoothofsumpitchOperating

++=+=

+=

+=

Rhin"in"RH

r

hin"in"rh

tr#in"olutome#

2.$$.2.

2.$$.2.

"


39/47

+

'ubstituting h and , in the eBuation :1;=

( ) ( )

in"cin"cc

c

c

chp

Rrin"Rrin"R

R

r

rhp

R

hin"in"R

r

hin"in"rp

++

+=

++++

+=

+

++

+=

.$.2$.2..

.

...$2..$2..

.

2.$$.2

2.$$.2.


40/47

#

[ ]

[ ]

[ ]

$.$.2.

.2

$.$.2.

2

22

.2

$.$.2.2

in"in"cc

crr

t

in"in"cc

c

t

r

t

r

in"in"ccchp

=

+=

+=

[ ][ ]

$.$.2

$.$.2

.

.

2

in"in"cac!lash

in"in"cr

r

rrt

==

=

!here is an

infinite number of

possible centre

distances for a

given pair of

profile shifted

gears, however

we consider only

the particular

case "nown asthe extended

centre distance.


41/47

#1

!he important reason for using non standard gears

are to eliminate undercutting, to prevent interference

and to maintain a reasonable contact ratio.

!he two main non> standard gear systems=

:1; 3ong and short 9ddendum system and

:2; G*tended centre distance system.


42/47

#2

!he addendum of the wheel

and the addendum of the

pinion are generally made

of eBual lengths.

/ere the profileDrac" cutter

is advanced to a certain

increment towards the gear

blan" and the same Buantity

of increment will be

withdrawn from the pinion

blan".


43/47

#

!herefore an increased

addendum for the pinion

and a decreased

addendum for the gear

is obtained. !he amountof increase in the

addendum of the pinion

should be e*actly eBual

to the addendum of thewheel is reduced.


44/47

##

!he effect is to move the

contact region from the

pinion centre towards

the gear centre, thus

reducing approachlength and increasing

the recess length. In this

method there is no

change in pressureangle and the centre

distance remains

standard.


45/47

#$

6eduction in interference with constant contact ratio

can be obtained by increasing the centre distance.!he effect of changing the centre distance is simply

in increasing the pressure angle.


46/47

#%

In this method when the pinion is being cut, the profile

cutter is withdrawn a certain amount from the centre of

the pinion so the addendum line of the cutter passes

through the interference point of pinion. !he result is

increase in tooth thic"ness and decrease in tooth space.


47/47

Now If the pinion is meshed with the gear, it will be

found that the centre distance has been increased

because of the decreased tooth space. Increased

centre distance will have two undesirable effects.

4 Chapter 4 Interference Back Lash

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