9.9 Calculation for Planetary gear mechanism
1. E ngagement between Internal gear and pinion(External gear)

Pinion(External gear)au=

No. of teeth of Pinion(speed reduction)
No. of teeth of Internal gearb) When Internal gear is driveru=

No. of teeth of Internal gear(increase speed)
No. of teeth of Pinion        2. Planetary gear mechanism
Planet carrier

Most mechanism of Planetary gear
comes with compact design and highreductive gear ratio consisting of Sun,
Planet, Internal gears and Planet carrier.

Sun gearInternal gear

CCentre distance ‘a’ for the Internalgear trains are shorter than the External gear trains. Internal gear train
operates in the same gear direction.
Calculation formulas for transfer ratio‘u’ are as follows.
a) When pinion is driverInternal gear

Basic gear axis for Planetary gear train mechanism(2K-H)

Planet gear

CB

CB

B

SS

S

A

AA

(a)
(Types of planetary)(b)(Types of solar)(c)(Types of star)

Types ofmechanism

Fixedmember

Input

Output(a) Types ofplanetary

Internalgear

Sun gear

Planetcarrier(b) Types ofsolar

Sun gearInternalgear

Planetcarrier(c) Types ofstar

Planetcarrier

Sun gearInternalgear

A : Sun gear
B : Planet gear
C : Internal gear
S : Planet carrier

Formula ofgear ratio

Ratio range

1zC + 1zA
1zA + 1zC
1− zCzA

1/3 - 1/12

1/1.2 - 1/1.7

1/2 - 1/11

・z: No. of teeth. A & C: Sun and Internal gear
・’-‘ symbol indicates output revolving direction

Interference of Internal gear

Interference will occur when design provides insufficient Number of teeth between Internal and Planet gears (Externalgear) during assembly. Please refer to the causes and types of interference as follows.
Interference

Phenomenon

Cause

Interference

Phenomenon

Cause

Involute interference

Unworkable conditions when a Tooth
Insufficient No. of teeth for Trimming
tip of Internal gear cuts into Dedenpinion
interference
dum of pinion during operations.

During assembling, pinion canSame as trochoid
be assembled to axial direction
interference
but not to radius direction.

Trochoid interference

Tip of pinion after engaging with Sun Difference in No. of teeth
Filletgear interferes to Tooth tip of Internal b e t w e e n I n t e r n a l a n dinterference
gear causing unworkable conditions. Planet is insufficient.

Insufficient No.
Tooth tip of pinion touchedof teeth for pinion.
Dedendum fillet of Internal gear(insufficient Tooth
causing unworkable condition.
depth of pinion).

Relationship among the gears in a Planetary gear mechanism① No. of teeth of Internal gears = (No. of teeth of Sun gear + 2 ) × No. of teeth of Planet gear.
No. of teeth of Internal gear + No. of teeth of Sun gear
= Should be integer number
The number of planet gears②③ Prevent the Tip interference among Planetry gears.
m( ZB + 2) < m( ZA + ZB ) sinʌ(n : The number of Planet gears)
nNo. of teeth of Internal gear (z2)100908070605040z2=∞, z1=17.1
ginmedrv

CuelinofrimTeedrvz2

30 z2=33.2
20=z1

Cu

linofidhooc

Tre

nc

re

rfe

tein

(z1=26.8 , z2=35.2)

100

Range of Number of teeth for pinion and KG-Internal gears

A = 20°
Involute interference curved line

When designing Planet gear, please achieve following conditions.

(b)
10 20 30 40 50 60 70 80 90 100
The number of Planet gears(z1)Fig. 6  Interference of Internal gearNo. of teeth of Internal gear

Range of No. of teeth for pinion

No. of teeth of Internal gear

Range of No. of teeth for pinion

60

21 - 4496

19 - 8072

20 - 56100

19 - 8480

20 - 64108

19 - 9284

20 - 68120

19 - 10490

19 - 74

117