gr cal

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Profile contact ratio, Active profile diameter, Lowest and Highestpoint of single tooth contact

In a pare of meshed and rotating gears, distance between the starting andterminating point of contact of a tooth with the mating tooth on line of

action is called length of line of action

Ratio of the degree of rotation from starting to termination of contact of onetooth to, the degree of rotation completed by next mating tooth at the sametime is called profilecontact ratio. That is equal to, length of line of actiondivided by base pitch. More than one teeth will share the load if contact ratiois more than one. This factor is very important while calculating loadtransmitting capacity of gears. Long addendum gears will have high contact

ratio when compared to short addendum gears. Long addendum gears mayhave a contact ratio of more than two. High contact ratio will ensure smoothpower transmission and load sharing by more number of teeth, but pressureangle has to be less than standard addendum gear.

Diameter at the point of termination / start of contact by the mating geartooth addendum is called active profile diameter. There will not be anycontact with mating teeth after this radius. Hence profile accuracy after this

radius is not important. But true involute form is a must from addendum toactive profile point.

When gears are meshed and rotating together, a pare of teeth will make andterminate the contact. Even though the contact ratio is more than one, onlyone tooth can be in contact with mating tooth for a short degree of rotation.Lowest point on profile where single tooth contact exists is lowest point ofsingle tooth contactand highest point on profile where single tooth contact

exists is highest point of single tooth contact.

Usually gear stresses are calculated by applying the load at the highestpoint of single tooth contact ( lowest point of single tooth contact for internalgears). If gears are of lower accuracy, more than one pare of teeth may notshare the load even though the contact ratio is more than one. Hence toothstresses has to be calculated by applying load at highest point tooth ofcontact (lowest point of contact for internal gears). That is at the tip of the

addendum.

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rog2 - rbg

2 + rop2 - rbp

2 - ( Sin ox Co ) = Ll a

( rog

2

- rbg2

- Lla)

2

+ rbg2

= rag

( rop2 - rbp2 - Lla)2+ rbp

2 = rap

( rop2 - rbp2 - Pb)2+ rbp

2 = rlp

( rog2 - rbg2 - Pb)2+ rbg

2 = rlg

( rop2 - rbp2 - Lla + Pb)2+ rbp

2 = rhp

( rog2 - rbg2 - Lla + Pb)2

+ rbg2 = rhg

External gears

Gear

Pinion

Outer radius-

Radius at highest pointof single tooth contact

Base circle

Radius at start of active profile-

Radius at lowest pointof single tooth contact

Operating

Pitch circle

Operating

Pitch circle

Base circle

Line of action

Outer radius-

Radius at highest point

of single tooth contact

Radius at lowest point

of single tooth contact

Radius at start

of active profile

Operatingce

nterdistance

rog

rhg

rbg

rag

rlg

rhp

roprap

rlp

rbpRadius of base circle-

Radius of base circle-

L la

C

oOperating

Pressure angle

Operating

Pressure angle

Length-

o

o

Pb (Pbt)-For helical gearsCrp =

Ll a

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( rop2 - rbp2 - Pb )2+ rbp2 = rl p

( rop2 - rbp2 + ( Sin o x Co) -Pb

)

2+ rbi

2 = rli

( rop2 - rbp

2 - Ll a + Pb )2+ rbp2 = rh p

( rop2 - rbp2 + ( Sin o x Co) -Ll a + Pb

)

2+ rbi

2 = rhi

Internal gearing

Ring gear

Operating

Pitch circle

Operating

Pitch circle

Line of action

Length-

Base circle

Base circle

L la

Active profile radius-

Pinion

Active profile radius-

Outer radius- ropInner radius- rii

rap

rai

Base circle radius- rbi

Base circle radius- rbp


of single tooth contact lir

Radiusathighestpo

int

ofsingletoothconta

ct

rhi

Operatingcenterdistance

oC


of single tooth contactrlp

Radius at highest point

of single tooth contact rhp

rop2 - rbp

2 + (Sin o x Co ) - rii2 - rbi

2 = Ll a

( rop2 - rbp2 - Ll a )2+ rbp2 = ra p

( rop2 - rbp2 + ( Sin o x Co) )2 + rbi2 = rai

Crp=Ll aPb (Pbt)-For helical gears

o

oo- Operating pressure angle

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Span measurement and measurement between pins/balls

Circular tooth thickness and space width of gear tooth can be controlled bycontrolling, span measurements (base tangent length) or measurementover/between pins/balls.

Usually tooth thickness of external spur / helical gears are established bycontrolling the base tangent length (measured by gear tooth micrometer).Tooth space width of internal spur/helical gears are established bycontrolling measurement between pins/balls. Span measurement in helicalgears must be normal to the helix. In helical gears, measurement over Pinswill not give a accurate reading. Hence, instead of pins, Ballsare to be usedfor helical gears and balls are to be aligned with one of the gear faces whiletaking measurements. Pins are to be used for measurement in spur gears.

For span measurement, number of teeth to be spanned has to beestablished in such a way that the gear tooth micrometer flange will makecontact as tangent near the mid point on the gear profile. Similarly formeasurement over pins/balls, selected pin/ball has to make a contact nearthe mid point of gear profile, and should not touch the root. If pin/balltouching at the root, small relief has to provided on measuring pin/ball.

It is really not very important to make measurement at mid point of profile.Measurement can be taken at any point on the profile, but contact point(tangent point) must be on involute profile.

Span measurement is not possible in helical gears, if F Sin b x M n

N =Z x

+ 0.5Non corrected spur gears

N =

d b

D + 2k( )Cos-1

x Z

+ 0.5 Corrected spur gears

N =

Znx n

+ 0.5Non corrected helical gears

N =

Zn1x n1

+ 0.5corrected helical gears

Zn1 =Z

Cos3

1

1= Tan-1

( )(D + 2k) x TanD

n1= Tan -1(Tan t1 x Cos 1)

d bD + 2k( )t1= Cos

-1

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b = ( ) For spur GearstD +

aorb

+ -Z

Mop= 2 (q + ao) For even no. of teethq=rb

Cos b

Mop= Cos x 2 q + 2ao For odd no. of teeth2 xZ

b = ( ) For Helical GearsttD +

aorbx Cos b

+ - Zt

All angles are in radians

( )( )

External teeth

Odd no.of teethEven no.of teeth

b

Pitch circle

Base circle

b

Measuring pin/ball

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b= ( )sD +ao

rbFor spur Gears

b=( )stD t+ -ao

rb x Cos bFor Helical Gears

q=rb

Cos bMbp = 2 (q + ao) For even no. of teeth

Mbp= (Cos ( ) )+ 2ao For odd no. of teeth2 xZx 2 q

Internal teeth


Odd teethEven teeth

Measuringpin/ball

bb

Pitchcircle

Basecircle

b

Mbp

q

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M =

Mn=(Mt x Cos b )

Mt=

2 tt

(N-1) 2rb

DZ

Spur gears2 t 2

+ +

Helical gearsrb2 t+D

+Z

(N-1) 2

M

Pitch circle

Base circle

Helicalgear

Mt

Mn


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Tip interference - Internal Gearing

In internal gearing, if difference in number of teeth between ring gear andpinion is very small, Pinion teeth will interfere with ring gear teeth and thisis called tip interference.Hence proper care to be taken while deciding

number of teeth of ring gear and pinion to avoid Tip interference. Thisproblem can also be avoided by giving required profile correction to ringgear / pinion.

Difference in number of teeth between ring gear and pinion is also adeciding factor for assembling pinion and ring gear. Avoiding the tipinterference will only ensure the axial assy of pinion in to the ring gear andmay not ensure the radial assembly of pinion. If radial assembly is

required to be ensured, sufficient profile correction to be given to ring gear/ pinion, or difference in number of teeth between ring gear and pinion tobe increased.

Tip interference may also pose a problem while generating ring gear teethon machines. It can be avoided by selecting proper gear cutting tool,otherwise cutter will cut part of ring gear teeth near the tip of theaddendum.

Center distance

Ring gear

Pinion

Tip interference

AA

A

A

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= Cos - 1 ( )Co 2+ rii2 - rop

2

2 x Cox rii

1=

Sin xCorop

Sin - 1 ( )+ - pi + g

p= ( )tp

Dp+ - dop

dop = Tan dop - dop

dop= Cos-1( )

rbp

rop

g= ( )s

Di+ - dii

dii= Tan dii - dii

dii= Cos-1 ( )

rbi

rii


Tip clearance - Internal gearing

A

A

A

A

Inner radius

Ring gear

Outer radius

Pinion

Ring gear

Pinion

Operating

centerdistance

rop

iir

Clt= (1 - ) x rii

Cltmust be positive to avoid tip interference

Co

g

p

1

Clt

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Interference of base circle - Internal gearing

If above condition is met, base circle interferenceproblem will be avoided.

(rop2

- rbp2) Lla

This problem can be avoided by providingpositive correction to pinion or by increasinginside diameter of the ring gear.

This problem can also be avoided bydecreasing the difference in number of teethbetween ring gear and pinion.

Pitch circle-Ring gear

Base circle-Ring gear

Pitch circle-Pinion

Base circlePinion

We encounter this problem In internalgearing, if difference in number ofteeth between ring gear and pinion istoo large. Ring gear teeth inner edgewill try to dig in to the pinion profilenear base circle.

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Under cut

Pitchcircle

radius

A=m

Base circle

Pitch circle

Material removed from involute profile near

the root diameter is called undercut. Usuallythis will occur while hobbing small pinions,when hob tries to generate profile belowthe base circle. Hence profile generationbelow the base circle is not possible, hobwill remove material from involute profilenear base circle. Under cut will increase thebending stress because of reduction in

tooth thickness near root.Magnitude of under cut depends on pressure angle, number of teeth,and helix angle. Increase in pressure angle, number of teeth and helixangle will reduce the under cut. Under cut can be avoided by providingpositive correction to pinions. Helical gears will have less under cut whencompared to a spur gear of same number of teeth and normal pressureangle.

Pitchcircle

radius-m

Under cut

Hob

To avoid under cut this condition must be satisfied

Minimum number of teeth

to avoid under cut Std Add and Ded

2Cos x db

D2

- A

Cos2 xZ xm (Z xm) - 2A

If addendum = module

Zminimum=2

Sin2

Zminimum=2 xA

Sin2 xm

For helical gears substitute with t

Pressure angle Min. No. of teeth

( spur gears)

14.50

32

200

18

22.50

14

250

12

gr cal

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