The document discusses center distance variation, minimum number of teeth, and contact ratio in gear systems. It explains that the center distance between gears can be varied within limits without affecting the velocity ratio. Varying the center distance can help eliminate interference between gears. It also discusses how to calculate the minimum number of teeth needed to avoid interference, and defines contact ratio as the ratio of the length of the arc of contact to the circular pitch, representing the number of pairs of teeth in contact at any given time.

1. PREPARED BY …….,
Mechanical-3B1_Group_03
KINEMATICS OF MACHINES
CENTER DISTANCE VARIATION , MINIMUM NUMBER OF TEETH , CONTACT RATIO .

2. • Centre distance is the distance between any two components in gear
terminology.
• The center distance can be manually varied within limits without
affecting the law of gearing.
• Therefore by modifying center distance, we can eliminate the
interference.
CENTRE DISTANCE VARIATION

3. • Centre distance between
pinion and wheel is as
described in given figure.
• If the center distance is
increase over its standard
center distance, then it’s
pressure angle Φ increase as
shown in figure .

4. • Modified center distance
between pinion and wheel is
as shown in given figure.
• With this effect the length
corresponding to interference
point M and N change to point
M’ and Ń , it’s allows the path
of contact KL not to extend
beyond the new interference
points M’ and Ń , there by
avoiding interference .

5. • Now, consider two involute gears
meshing as shown in given figure.
• According to law of gearing,
𝜔1
𝜔2
=
𝑂2𝑃
𝑂1𝑃
=
𝑂2𝑁
𝑂1𝑀
…………(1)
• By varying the centre distance we
have,
(𝜔1)′
𝜔2 ′ =
𝑂2𝑃′
𝑂1𝑃′
=
𝑂2𝑁′
𝑂1𝑀′
………….(2)

6. • From figure we can say, O2N’ = O2N = 𝑅 𝑏 = Constant
& O1M’ = O2M = 𝑅 𝑏 = Constant…………..(3)
• From equation (1), (2) and (3),
𝜔1
𝜔2
=
(𝜔1)′
𝜔2 ′ =
𝑂2𝑃
𝑂1𝑃
=
𝑂2𝑃′
𝑂1𝑃′
.
• Hence, it is proved that if we change the centre distance within limits of law
of gearing , it will not affect velocity ratio. i.e. the centre distance needs to be
constant.
• Therefore by modifying center distance, we can eliminate the interference.

7. • In order to avoid interference, the minimum number of teeth on gear
should be such that it results the base circle to lie below the dedendum
circle.
• This minimum number of teeth is called as critical or minimum number
of teeth to avoid interference.
• The relative position of the base circle and the addendum circles below
or above each other depends upon the number of teeth , pressure
angle and module.
MINIMUM NUMBER OF TEETH

9. • As explain the maximum value of the addendum radius of the
wheel to avoid interference can be up to BE from the figure .
• (BE)2 = (BF)2 + (FE)2
= (BF)2 + (FP+PE)
= (R cosθ)2 + (R sinθ+ R sinθ)2
= R2 cos2θ + R2sin2θ+r2 sin2θ+ 2 r R sin2θ
= R2 (cos2θ+ sin2θ) + Sin2Φ(r2+2rR)
= R2 + ( r2+2rR) sin2θ

10. • (𝐵𝐸)2 = 𝑅2 1 +
1
𝑅2 𝑟2 + 2𝑟𝑅 sin2 𝜃
• BE = R 1 +
𝑟
𝑅
sin2 𝜃
𝑟
𝑅
+ 2 .
• Therefor, the maximum value of the addendum of the wheel can be
equal to, [ BE – pitch circle radius (R) ].
• Let t = number of teeth on pinion
T = number of teeth on wheel

11. • Now, R =
𝑚𝑇
2
, r =
𝑚𝑡
2
and G =
𝑇
𝑡
= gear ratio
• Hence, 𝑎 𝑤 𝑚𝑎𝑥=
𝑚𝑇
2
1 +
𝑡
𝑇
𝑡
𝑇
+ 2 sin2 𝜃 − 1
=
𝑚𝑇
2
1 +
1
𝐺
1
𝐺
+ 2 sin2 𝜃 − 1 .
• Let the adopted value of number of the addendum in some case be 𝑎 𝑤
times the module of teeth then this adopted value of the addendum
must be less then the maximum value of addendum to avoid
interference.

12. • i.e. =
𝑚𝑇
2
1 +
1
𝐺
1
𝐺
+ 2 sin2 𝜃 − 1 ≥ 𝑎 𝑤𝑚𝑎𝑥 OR
T ≥
2𝑎 𝑤
1+
1
𝐺
1
𝐺
+2 sin2 𝜃 −1
.
• In the limit,
T =
2𝑎 𝑤
1+
1
𝐺
1
𝐺
+2 sin2 𝜃 −1
This gives minimum number of teeth on the wheel for given values of
gear ratio.

13. • The minimum number of teeth on the pinion is given by ,
t =
𝑇
𝑔
.

14. • The contact ratio or number of pairs of teeth in contact is defined as
ratio of length of arc of contact to the circular pitch.
• Therefor, number of teeth in contact,
n =
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑎𝑟𝑐 𝑜𝑓 𝑐𝑜𝑛𝑡𝑎𝑐𝑡
𝐶𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑝𝑖𝑡𝑐ℎ
n =
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑎𝑟𝑐 𝑜𝑓 𝑐𝑜𝑛𝑡𝑎𝑐𝑡
𝜋.𝑚
CONTACT RATIO

15. • The arc of contact is length of the pitch circle traversed by a point on it
during the mating of a pair of teeth thus , all the teeth lying between
the arc of contact will be meshing with the teeth on the other wheel .
• As the ratio at the arc of contact to the circular pitch is also the contact
ratio.
• For continuous transmission of the motion , at least one tooth of one
wheel must be in contact with in another tooth of the second wheel ,
therefor , n must be greater than unity .

16. • If n lies between 1 & 2 , the number of teeth in contact at any time will
not be less than 1 and never more than 2 .
• If n is between 2 & 3 , it is never less tan two pairs of teeth and not
more than three pairs and so on .
• If n is 1.6 , one pair of teeth are always in contact , whereas two pairs of
teeth are in contact for 60% of the time .