A compound epicyclic train uses gears rigidly joined on an orbiting planet shaft to couple different gear meshes. This calculator models two internal rings: the first is fixed, the carrier is driven, and the second ring is the output. Both planet tooth counts matter. The two exposed gear planes make their common shaft motion visible.
Compound Epicyclic Train Interactive Calculator
Drive the blue carrier and watch a rigid pair of planet gears mesh with a fixed ring and an output ring. The two views expose separate axial planes of the same mechanism. All four tooth counts affect the kinematics or geometry.
Equation Used
- Fixed first internal ring, rotating carrier input, second ring output, one rigid compound planet with no slip.
- Internal mesh center distance a = m(Nring − Nplanet)/2 in each axial plane. The two planes may need different modules.
- One planet pair is shown; a multi-planet design needs additional phasing and clearance checks.
- Illustrative tooth outlines use 20-degree involutes without manufacturing root details. Conservative small/close-count cases show pitch circles only.
- No sun gear is specified and no sun-to-carrier conversion is included.
Carrier-driven two-ring compound train. The original ring-difference shortcut has been replaced because it ignored the planet counts. Both meshes must share a carrier radius; the second module is calculated to satisfy that geometry. This is a kinematic model, not a manufacturable gear selection.
One planet shaft, two internal meshes
The blue carrier turns around the main axis and carries the compound planet shaft. In the first plane, a planet rolls inside the fixed ring. In the second plane, the other planet half turns at exactly the same angular speed because both halves are rigidly joined. Its internal mesh determines the motion of the output ring.
The end views use separate zoom levels to show each mesh clearly. Matching dark marks on the planet halves reveal their common spin. The gold ring mark shows accumulated output rotation. Playback continues through successive carrier revolutions without resetting the gear angles at the end of each turn.
Unlike the previous drawing, this model does not show a fictitious sun input without a sun tooth count. The input here is explicitly the carrier. The former shortcut based only on ring-tooth difference has also been removed: it was not the ratio of the entered compound train.
Internal-mesh center distance is half the difference of pitch diameters. A rigid compound planet needs the same center distance in both planes. The calculator reports the relative module required to satisfy that condition; it does not silently assume every independent tooth-count choice can use one common module.
Use the model to compare kinematic arrangements
Compound planetary arrangements occur in compact gear transmissions. The example here is intended to explain a carrier-driven two-ring arrangement and its speed relationships. It is not a model of a particular commercial gearbox.
Use it to compare tooth-count changes, direction reversals and stationary-output cases. Gear material, torque, efficiency, tooth geometry, bearing loads and assembly details require separate design work.
Derive the ratio in the carrier frame
Let ωc be carrier speed, ωp compound-planet speed and ωo output-ring speed. The first ring is stationary. For its internal mesh, (ωp − ωc)Pl = (0 − ωc)Nf. For the second internal mesh, (ωp − ωc)Ps = (ωo − ωc)No.
Eliminating planet speed gives ωo/ωc = 1 − Nf·Ps/(No·Pl). The reported carrier/output ratio is the reciprocal. A negative value means opposite directions; zero output speed is displayed as stationary.
The geometry also requires m1(Nf − Pl) = m2(No − Ps). Therefore m2/m1 = (Nf − Pl)/(No − Ps). Equal modules are possible only when the two tooth-count differences are equal in this unshifted pitch model. Matching center distances is necessary but is not a complete interference or assembly check.
Example: the original four tooth counts
Enter the counts
Fixed ring Nf = 100, output ring No = 99, fixed-side planet Pl = 24, and output-side planet Ps = 22.
Calculate the speed relationship
Output/carrier speed = 1 − (100 × 22)/(99 × 24) = 0.074074. The carrier/output ratio is 13.5:1, and 100 carrier turns produce approximately 7.407 output turns in the same direction. The earlier 100:1 answer did not account for the planet counts.
Check the carrier radius
The two tooth-count differences are 76 and 77. Their required module ratio is 76/77 = 0.987013. These counts therefore do not give the same carrier radius with a common module in this model. Selecting actual modules and tooth profiles remains a separate design problem.
Kinematics are only one design check
A speed result does not establish a usable gear set. Each internal gear and mating planet must have compatible tooth geometry. Low planet tooth counts or a small ring-to-planet difference can cause interference. Root shape, profile shift, backlash and the assembly path also matter.
The drawing uses pitch circles for conservative small or close-count cases instead of showing apparently valid teeth. If a ring has no more teeth than its planet, the parts are shown separately and an assembled ratio is not reported. A multiple-planet arrangement needs additional phasing and planet-to-planet clearance checks.
Questions about the compound train
Why do both planet sliders change the result?
The two rigidly joined planet halves couple the internal meshes. Their tooth-count ratio appears directly in the output/carrier speed equation.
Does a one-tooth difference between the rings guarantee 100:1?
No. The planet counts and the driven member must also be specified. With the default four counts and carrier input, the ratio is 13.5:1.
What does a negative ratio mean?
The output ring turns opposite the carrier. It is a direction result, not an error.
Why can the output be stationary?
When Nf·Ps equals No·Pl, the two relative mesh relationships cancel the carrier motion at the output ring.
Are different modules allowed between the two planes?
The two separate meshes can use different modules, provided each planet matches its own ring and both meshes share the same carrier radius. The reported module ratio is a geometric requirement, not a stock gear recommendation.
References
- Morini Gears Design: simple and compound planetary gears, including a two-ring compound arrangement in Example D.
- KHK: internal gear technical information, internal mesh center distance, rotation relationships and tooth-interference considerations.
The equations above are derived for the explicitly defined carrier-driven arrangement. They are not a claimed ratio for the manufacturer’s pictured gearbox.
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