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How to check planetary gear results: three worked cases

Barbudenge · 2026-09-21 · Worked example

Revised on 2026-10-01: added speed checker and tooth condition check.

An animation may look plausible even when its input, output or fixed member differs from the assumptions used on paper. This worked example analyses the same tooth counts in three configurations. The goal is to obtain results you can check by substitution before using a simulation tool.

1. Inputs and assumptions

Consider a simple planetary train with a 24-tooth sun S, a 72-tooth internal ring R and 24-tooth planets P supported by carrier C. Assume spur gears of equal module with reference geometry and no profile shift. This is a kinematic analysis: it does not calculate stress, backlash, friction or load capacity.

Start with the geometry. The pitch diameters in this arrangement require \(N_R=N_S+2N_P\). Here, \(72=24+2\times24\). With a module of 2 mm, the sun, ring and planet pitch diameters are 48 mm, 144 mm and 48 mm. The distance from the central axis to a planet axis is \(\frac{48+48}{2}=48\,\mathrm{mm}\). The internal mesh gives the same distance: \(\frac{144-48}{2}=48\,\mathrm{mm}\).

This equality is necessary but does not complete the design. Multiple planets require checks for spacing, tooth phasing, interference and manufacturing conditions. Here we only need a coherent input set for studying rotational speeds.

2. One equation for three cases

Choose the same positive direction for every axis, viewed from the same side. Speeds n are in rpm. To relate sun and ring motion, subtract the carrier speed from each:

The negative sign comes from the train observed with the carrier stationary: the external sun–planet mesh reverses direction, whereas the internal planet–ring mesh does not. To avoid division by zero when members rotate together, use the equivalent form without a denominator:

There are three speeds and one independent relation. Two conditions must be specified to determine the third speed: for example, an input speed and a stationary member. Saying only “input at 1,200 rpm” does not define the operating condition.

3. Calculated results

Fixed ring: a 4:1 reduction

Set \(n_R=0\) and \(n_S=1\,200\,\mathrm{rpm}\). The carrier speed is \(\frac{24\times1\,200}{96}=300\,\mathrm{rpm}\). Input and output rotate in the same direction. Defining the ratio as \(i=\frac{n_{\mathrm{input}}}{n_{\mathrm{output}}}\) gives \(i=\frac{1\,200}{300}=4\).

Substitute into the original equation: \(\frac{1\,200-300}{0-300}=\frac{900}{-300}=-3\). The sun speed relative to the carrier is 900 rpm, but its absolute speed remains 1,200 rpm. Mixing these two measurements is a common source of errors.

Fixed sun: a 4:3 reduction

Now set \(n_S=0\) and use the ring as the input at \(n_R=1\,200\,\mathrm{rpm}\). The carrier rotates at \(\frac{72\times1\,200}{96}=900\,\mathrm{rpm}\). The input/output ratio is 4/3, approximately 1.333. No tooth counts changed; choosing a different fixed member and input changed the reduction.

Fixed carrier: reversed rotation

Set \(n_C=0\) and \(n_S=1\,200\,\mathrm{rpm}\). The equation becomes \(24\times1\,200+72n_R=0\), giving \(n_R=-400\,\mathrm{rpm}\). The ring rotates opposite to the sun. The signed ratio is −3; the reduction magnitude is 3:1.

Checker: two known speeds

Use the same sign convention for sun and ring. This checker solves the equation for a simple planetary assembly and checks whether the planet would have a positive integer tooth count. It does not analyse load, interference or assembly phasing.

4. Checking a simulation

Open Engrenarium Web and use an equivalent arrangement with the same teeth and constraints. Compare the calculated speeds rather than just the appearance of the animation. If labels or viewing directions differ, first identify which members correspond to this example's sun, ring and carrier.

1. Record the driven member and the fixed member.
2. Check the sign convention and displayed units. To convert rpm to rad/s, multiply by \(\frac{2\pi}{60}\).
3. Substitute all three speeds into \(24n_S+72n_R-96n_C\). The result should be zero, apart from rounding.
4. Repeat the analysis with different constraints while keeping the tooth counts unchanged.

This is an independent checking procedure, not a report of a physical experiment or validation of a particular software release. Compare any small residual with the precision of the displayed values. A large discrepancy calls for checking inputs and constraints.

5. What the calculation does not establish

Reduction is not efficiency. A 4:1 ratio does not specify efficiency. In the ideal fixed-ring case, neglecting losses, carrier torque magnitude can be four times the sun input torque. Real hardware requires loss and component-strength calculations; the housing also carries the reaction torque of the fixed member.

Planet rotation is not carrier rotation. A planet can spin about its own axis while its centre travels with the carrier. With the carrier fixed, planet centres stop moving, but the planets themselves can still spin.

Reversing the ratio definition inverts the number. If output speed is 300 rpm and input speed is 1,200 rpm, output/input is 0.25 and input/output is 4. Both descriptions agree when the definition is stated.

6. Exercise: two moving inputs

Keep the tooth counts and impose \(n_S=600\,\mathrm{rpm}\) and \(n_R=-200\,\mathrm{rpm}\). What is the carrier speed? Before opening the solution, note that sun and ring contributions have different weights in the equation.

Conceptual basis and further reading

The numbers in this guide form an illustrative example calculated for this page. See the general derivation in Planetary gears: general case and train value and shaft constraints in Simple and compound gear trains. To go beyond a simple planetary set, continue with Compound planetary trains and differentials.