DO YOU WONDER WHAT OUR PLANETARY GEAR SETS & HOW ARE THE GEAR RATIOS DERIVED

Planetary gear set
                      PLANETARY GEAR SETS 

The planetary gear set mostly consists of Four main parts as shown in Figure Below.

Planetary gear set

  • Sun gear
  • Planet gear
  • Ring gear
  • Carrier

Each of these components can be the input, the output, or can be held stationary. Depending upon input, output, and stationery element the following gear ratios can be calculated.

S.No.InputOutputStationary

Ratio

ASun (S)Planet Carrier (C)Ring (R)1 + R/S
BPlanet Carrier (C)Ring (R)Sun (S)1 / (1 + S/R)
CSun (S)Ring (R)Planet Carrier (C)-R/S

Holding any two components together will deliver 1:1 gear reduction. The first gear ratio listed above is a reduction where the output speed is slower than the input speed. The second is an overdrive, where the output speed is faster than the input speed. The last is also reduction but with opposite output direction (hence represented by negative sign).

As most of the industrial applications demand speed reduction, our concern is with only the first type of reduction where the ring gear is held stationary. Input is given through the sun gear and output is the planet carrier.

Let’s look in detail how the  Gear ratio is derived for this kind of reduction

  1. From the basic law of planetary gear set we have,

    Planetary Gear Set
      Basic Law of Planetary Gear Set

rr= rs + dp,

rr=rs+ 2rp

2rp= rr-rs        – (Eqn. 1)

Where, rr= Radius of Ring Gear,

rs = Radius of Sun Gear,

dp = Diameter of Pinion Gear

rp = Radius of Planetary Gear

Now, we know Linear velocity ‘V’= Angular velocity x Radius = N x r  (as ω = 2πN)

Therefore,  For an object moving in circle its Linear velocity is V= N x r

  1. In planetary gear set, as the input is connected to sun gear and output is given to carrier

Gear ratio ‘i’= Ns/Nc

Where, Ns= Speed of Sun Gear, Nc= Speed of Carrier

  1. Linear Velocity of the carrier is angular velocity x distance from the centre

Vc= Nc x (rs + rp)      – (Eqn. 2)

  1. The instantaneous linear velocity of the point of contact between sun and planet is

Vs= Ns x rs        – (Eqn. 3)

Also, instantaneous velocity equals to twice the velocity of carrier

Vs= 2Vc= 2Nc x (rs + rp)        – (Eqn. 4)

Equating Eqn. 3 and Eqn.4 we get,

Ns x rs = 2 x Nc x (rs + rp)

i = Ns/Nc = 2 x (rs + rp) / rs

= 2rs + 2rp / rs

From Equation 1 we have,

2rp= rr – rs,

=(2 rs + rr – rs)/ rs

= (rs + rr) / rs

=1 + rr / rs

= 1 + Zr/Zs

Here, Zr is No. Of teeth on Ring Gear, Zs is No. Of teeth on sun gear. Hence Gear ratio for planetary gear set is, Ratio of Number of teeth of ring gear to number of teeth of Sun gear plus One. In this way the gear ratio for planetary gear set is derived…!!

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