chapter 4 spur gears gear profiles

7/25/2019 Chapter 4 Spur Gears Gear Profiles

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It is essential for correctly

meshing gears, the size of the

teeth ( the module ) must be

the same for both the gears.

Another requirement - the

shape of teeth necessary for

the speed ratio to remain

constant during an increment

of rotation; this behaior of thecontacting surfaces (ie. the

teeth flan!s) is !no"n as

conjugate action.


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Although many tooth

shapes are possible

for "hich a mating

tooth could bedesigned to satisfy the

fundamental la", only

t"o are in general use%

the cycloidal andinvoluteprofiles.


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*he portion of the

Inolute +ure that"ould be used to

design a gear tooth


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lue arro"s sho"s

the contact forces.

*he force line runs

along commontangent to base

circles.


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*he inolute profile of gears has important

adantages;

0 It is easy to manufacture and the centerdistance bet"een a pair of inolute gears

can be aried "ithout changing the elocity

ratio. *hus close tolerances bet"een shaft

locations are not required. *he most

commonly used conjugate tooth cure is

the involute curve.(rdman 2 3andor).


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#. In inolute gears, the pressure angle, remains

constant bet"een the point of tooth engagement

and disengagement. It is necessary for smooth

running and less "ear of gears.ut in cycloidal gears, the pressure angle is

ma4imum at the beginning of engagement,

reduces to zero at pitch point, starts increasing

and again becomes ma4imum at the end of

engagement. *his results in less smooth running

of gears.


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$. *he face and flan! of inolute teeth are

generated by a single cure "here as in cycloidal

gears, double cures (i.e. epi-cycloid and hypo-

cycloid) are required for the face and flan!respectiely. *hus the inolute teeth are easy to

manufacture than cycloidal teeth.

In inolute system, the basic rac! has straight

teeth and the same can be cut "ith simple tools.


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1. 3ince the cycloidal teeth hae "ider flan!s,

therefore the cycloidal gears are stronger than

the inolute gears, for the same pitch. 5ue to this

reason, the cycloidal teeth are preferred speciallyfor cast teeth.


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#. In cycloidal gears, the contact ta!es place

bet"een a cone4 flan! and a concae surface,

"here as in inolute gears the cone4 surfaces

are in contact. *his condition results in less "earin cycloidal gears as compared to inolute gears.

6o"eer the difference in "ear is negligible.


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$. In cycloidal gears, the interference does not

occur at all. *hough there are adantages of

cycloidal gears but they are out"eighed by the

greater simplicity and fle4ibility of the inolutegears.


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#'

1.A normal dra"n to an inolute at pitch point is

a tangent to the base circle.

#. 7ressure angle remains constant during themesh of an inolute gears.

$. *he inolute tooth form of gears is insensitie

to the centre distance and depends only onthe dimensions of the base circle.


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&. *he radius of curature of an inolute is equal

to the length of tangent to the base circle.

'. asic rac! for inolute tooth profile has straightline form.

. *he common tangent dra"n from the pitch

point to the base circle of the t"o inolutes is

the line of action and also the path of contact of

the inolutes.


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. 8hen t"o inolutes gears are in mesh and

rotating, they e4hibit constant angular elocity

ratio and is inersely proportional to the size of

base circles. (9a" of :earing or conugateaction)

.


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*he follo"ing four systems of gear teeth are commonly used

in practice%

1. 1&=> +omposite system

#. 1& => ?ull depth inolute system

$. #/> ?ull depth inolute system

&. #/>3tub inolute system


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*he 1&=> composite system is used for generalpurpose gears.

It is stronger but has no interchangeability. *he tooth

profile of this system has cycloidal cures at the top

and bottom and inolute cure at the middle portion.

*he teeth are produced by formed milling cutters or

hobs.

*he tooth profile of the 1&=> full depth involutesystem"as deeloped using gear hobs for spur and

helical gears.


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*he tooth profile of the #/ofull depth involute system

may be cut by hobs.

*he increase of the pressure angle from 1&=oto #/o

results in a stronger tooth, because the tooth actingas a beam is "ider at the base.

*he 20o stub involute system has a strong tooth to

ta!e heay loads.


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r

A

O

Gear

CB

F

E

Base Circle

Addendum Circle

Pitch Circle

ra

*he study of the geometry ofthe inolute profile for a gear

teeth is called inolumetry.

+onsider an inolute of base

circle radius ra and t"o points and + on the inolute as

sho"n in figure. 5ra"

normal to the inolute from

the points and +. *henormal and +? are

tangents to the ase circle.


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D

ra

r

A

OGear

C

B

rbrc

F

E

Base Circle

Addendum Circle

Pitch Circle

cb

Let

ra= base circle radius

of gear

rb= radius of point B

on the involute

rc= radius of point Con the involute

bt

ct


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$$

bt

ct

D

ra

r

A

OGear

C

B

rbrc

F

E

Base Circle

Addendum Circle

Pitch Circle

cb

bt

ct

and

!b= pressure angle for

the point B

!c= pressure angle for

the point C

tb= tooth thic"nessalong the arc at B

tc= tooth thic"ness

along the arc at C


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bt

ct

D

ra

r

A

OGear

C

B

rbrc

F

Ecb

Base Circle

Addendum Circle

Pitch Circle

( )cos

#$%cos

cca

bba

rr

rr

OCFandOBEFrom

=

=

ccbb rr

Therefore

coscos =

bt

ct


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bt

ct

r

O

Base Circle

ra

Gear

Ecb F

B

rc

rbA

Addendum Circle

Pitch CircleC

D

?rom the properties of the Inolute%

Arc AE = Length BE and

Arc AF = Length F

( )

=

==

===

functioninvolutecalled

isExpression

Inv

AOEAOB

OE

BE

OE

ArcAEAOE

bb

bbb

bbb

b

tan

tan&

tan

tan

bt

ct


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bt

ct

r

O

Base CircleraGear

E

bc

F

B

rbrc A

Addendum Circle

Pitch Circle

CD3imilarly%

ccc

CCc

c

Inv

AOFAOC

OF

BE

OF

ArcAFAOF

=

==

===

tan&

tan

tan

b

bbb

b

b

r

t

r

tAOBAOD

BpotheAt

tan

int

+=

+=

bt

ct


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r

O

Base Circlera

Gear

E

bc

F

B

rbrc A

Addendum Circle

Pitch Circle

C D

c

ccc

b

c

rt

r

tAOCAOD

CpotheAt

tan

int

+=

+=

bt

ct


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Catthicknesstooth

rr

t

invinvt

r

tinvr

tinv

r

t

r

tequationsabovetheEquating

c

b

b

cbc

c

cc

b

bb

c

ccc

b

bbb

=

+=

+=+

+=+

&&

&

&

tan

tan

'


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Base Circle

Addendum Circle

r

O

ra

Gear

E

F

B

rc

rbA

CD

cb

Pitch Circle

bt

ct

@sing this equation

and !no"ing tooth

thic!ness at any point

on the tooth, it is

possible to calculate

the thic!ness of the

tooth at any point

chapter 4 spur gears gear profiles

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