PREPARED BY ……., Mechanical-3B1_Group_03

KINEMATICS OF MACHINESCENTER DISTANCE VARIATION , MINIMUM NUMBER OF TEETH , CONTACT RATIO .

• 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

• 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 .

•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 .

•Now, consider two involute gears meshing as shown in given figure.

•According to law of gearing, = = …………(1)

•By varying the centre distance we have,

= = ………….(2)

• From figure we can say, O2N’ = O2N = = Constant & O1M’ = O2M = = Constant…………..(3)

• From equation (1), (2) and (3), = =

• 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.

• 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

•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

• =

•BE = R .

•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

•Now, R = , r = and G = = gear ratio

•Hence, = =

• Let the adopted value of number of the addendum in some case be times the module of teeth then this adopted value of the addendum must be less then the maximum value of addendum to avoid interference.

• i.e. = ≥ OR T .• In the limit,

T This gives minimum number of teeth on the wheel for given values of gear ratio.

•The minimum number of teeth on the pinion is given by ,

t = .

•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 =

n =

CONTACT RATIO

•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 .

• 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 .