---
title: "Assembly condition of planetary gear trains"
subject: "Theory of Mechanism and Machine I"
chapter: 6
key: "semester_6/theory_of_mechanism_and_machine_i:planetary-gear-assembly-condition-applications"
source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/planetary-gear-assembly-condition-applications
site: BCT Engineering
---

# Assembly condition of planetary gear trains

Theory of Mechanism and Machine I · Chapter 6: Simple and Planetary gear trains

- Asked in: year not given, 6 marks
- Video: [Epicyclic Gear Train: The 3 Assembly Conditions](https://www.youtube.com/watch?v=2QLmYs4dY7o) — Frederic Bruyere
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/planetary-gear-assembly-condition-applications
- Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-6

## Question

Derive the condition for assembling a planetary gear train having a sun gear, a ring gear and N equally spaced planets. A planetary train has a sun gear of 24 teeth, planets of 24 teeth and a ring gear of 72 teeth. Check whether 3, 4, 5 and 6 equally spaced planets can be assembled. List four applications of planetary gear trains.

## Short answer

Assembly condition: (z_sun + z_ring)/N must be integer. With z_s=24, z_r=72 (sum 96), 3 and 4 planets are feasible; 5 fails the tooth condition; 6 meets mesh but tips interfere. Applications: automatic transmissions, differentials, reduction gearboxes, watch mechanisms.

## Answer

#### Condition for equal spacing
Let the sun have $z_s$ teeth and the ring $z_r$ teeth. Fix the ring and place the first planet in mesh with both. Now rotate the arm to bring the first planet to the next planet position, a fraction $1/N$ of a revolution away (angle $360^\circ/N$). The sun turns through

$$
\theta_s = \frac{360^\circ}{N}\left(1 + \frac{z_r}{z_s}\right)
$$

(the arm turns $1/N$ revolution, and with the ring fixed the sun turns $1 + z_r/z_s$ times as fast as the arm).

A second planet can be placed in the vacated position only if the sun has turned through a whole number of tooth pitches, that is, $\theta_s = k \times \dfrac{360^\circ}{z_s}$:

$$
\frac{z_s + z_r}{N} = k \quad \text{(integer)}
$$

So the sum of the teeth on the sun and ring must be divisible by the number of planets. For the geometry (equal module) we also need $z_r = z_s + 2z_p$.

#### Check: $z_s = 24$, $z_r = 72$
Geometry: $z_p = (72 - 24)/2 = 24$ teeth, which agrees with the given planets.

$$
z_s + z_r = 24 + 72 = 96
$$

| Planets $N$ | $96/N$ | Assembly condition met? |
|---|---|---|
| 3 | 32 | Yes |
| 4 | 24 | Yes |
| 5 | 19.2 | No |
| 6 | 16 | Yes |

The planets must also not touch each other. The planet centre radius is $m(24+24)/2 = 24m$, and the planet outside diameter is $m(24 + 2) = 26m$. For $N = 6$, the distance between neighbouring planet centres is $2 \times 24m\sin 30^\circ = 24m$. This only equals the pitch diameter, so the tooth tips (26m) would overlap: **6 planets cannot be fitted**. For $N = 4$ the distance is $2 \times 24m\sin 45^\circ = 33.9m > 26m$, and for $N = 3$ it is $41.6m$, so both are fine.

**Result: 3 and 4 planets can be assembled; 5 fails the tooth condition; 6 meets the tooth condition but the planets would interfere.**

#### Applications
1. Automatic transmissions of vehicles (speed change by locking elements).
2. Differentials of automobiles (bevel epicyclic train).
3. Reduction gearboxes of wind turbines, hoists and cranes.
4. Turbine speed reducers; watch mechanisms and epicyclic screwdrivers/hand drills.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For N equally spaced planets to assemble, which condition must hold?
   A. z_r - z_s divisible by N
   B. z_p divisible by N
   C. (z_s + z_r)/N is an integer
   D. z_s divisible by z_r
   Answer: C. The sun must turn through a whole number of tooth pitches, which requires (z_s + z_r)/N to be an integer. (Key point: Assembly condition)

2. For equal modules, the ring teeth are related to sun and planet teeth by:
   A. z_r = z_s + 2 z_p
   B. z_r = z_s + z_p
   C. z_r = 2 z_s + z_p
   D. z_r = z_s - 2 z_p
   Answer: A. The ring pitch radius equals sun radius plus planet diameter, so z_r = z_s + 2z_p. (Key point: Geometry relation)

3. For z_s = 24 and z_r = 72, which number of planets fails the assembly condition?
   A. 3
   B. 4
   C. 6
   D. 5
   Answer: D. 96/5 = 19.2 is not an integer, while 96/3, 96/4 and 96/6 are integers. (Key point: Which planet counts assemble)

4. Why is N = 6 not practical here even though 96/6 is an integer?
   A. The sun would be too small
   B. The planets just touch, since the chord between centres equals the planet pitch diameter
   C. The ring would not mesh
   D. 96 is not divisible by 6
   Answer: B. The chord between neighbouring planet centres is 24m, equal to the planet pitch diameter, so adjacent planets just touch. (Key point: Planet clearance)

## Related in Theory of Mechanism and Machine I

- [Epicyclic gear train tabular method](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aepicyclic-gear-train-tabular-method) — Theory of Mechanism and Machine I, chapter 6
- [Compound gear train speed numerical](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Acompound-gear-train-speed-numerical) — Theory of Mechanism and Machine I, chapter 6
- [Compound planet epicyclic train numerical](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Acompound-planet-gear-train-ring-speed) — Theory of Mechanism and Machine I, chapter 6
- [Straight bevel gear geometry and formative teeth](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Astraight-bevel-gear-geometry-formative-teeth) — Theory of Mechanism and Machine I, chapter 5
- [Interference and minimum number of teeth](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Ainterference-minimum-teeth-pinion) — Theory of Mechanism and Machine I, chapter 4
- [Spur gear pair dimensions numerical](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aspur-gear-pair-geometry-numerical) — Theory of Mechanism and Machine I, chapter 4

## Related in other subjects

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