Educational Model of Planetary Differential Gear

My Educational Mechanical Examples Series

This model is one of my educational mechanical mechanism examples on 80mm x 80mm base plates.
You can find all models in the series in this collection => [Mechanical Mechanism Examples]

The present model

This is an educational model of a planetary differential gear system, also known as Mechanical Paradox Planetary Gear System, which can realize much higher reduction ratio than a planetary gear system alone.

FOLLOW THIS LINK FOR THE INTRODUCTION VIDEO → https://youtu.be/QcAEG79P_sY 
*Somehow, the editor of printables is currently broken and does not allow me to embed the video.

Brief Description

In this system, a differential gear stage is added after a planetary gear stage.

In the planetary gear stage, the sun gear serves as the input, and the revolution of the three planetary gears— which orbit inside a fixed internal ring gear around the sune—forms the output. The planet gears have 12 teeth, the sun gear has 6 teeth, and the fixed internal gear has 33 teeth. If we look at the mechanism from a coordinate system that revolves together with the planet gears around the sun gear, so that each planet appears fixed in position while still spinning, then for each one rotation of a planet gear, the sun gear rotates twice, and the internal gear rotates in the opposite direction by 12/33 turns. Observing the motion with the internal gear held fixed, the sun gear effectively rotates 2+12/33 turns while the planet gears revolve 12/33 turns around the sun. Thus, the reduction ratio of the planetary stage is:

12/33 : 2+12/33 = 12 : 78 = 1 : 6.5 .

In the differential gear stage, overlapping the fixed internal gear, a second internal gear with exactly three fewer teeth is placed, and this free internal gear is also engaged by the planet gears. Because the two internal gears differ by three teeth, their phases coincide at only three points around the circumference, while everywhere else their phases gradually shift. At the point of engagement with the planet gears, the phases must match, so the free internal gear is forced to rotate by the amount corresponding to the tooth-count difference as the planet gear revolves around the sun. Consequently, for each full revolution of the planet gears around the sun, the free internal gear advances by three teeth. Since the free internal gear has 30 teeth, this differential stage provides a reduction ratio of:

3 : 30 = 1 : 10 .

Combining the planetary reduction (1 : 6.5) and the differential reduction (1 : 10), this compact coaxial gear system achieves an overall reduction ratio as high as 1 : 65.

This system is also called as a mechanical paradox planetary gear system because two internal gears with different tooth counts mesh simultaneously with the same planet gears—seemingly violating the ordinary geometric constraints of involute gears. How the paradox can be solved? At first, as is visually apparent, the diameter of an internal gear in a planetary system must equal the diameter of a sun gear plus two planet gears. Usually, for gears of the same module, the pitch diameter is proportional to the tooth count, so one expects the following relationship:

(internal gear teeth) = (sun teeth) + 2 × (planet teeth).

In this model, with a 6-tooth sun and 12-tooth planets, the expected internal gear tooth count is:

6 + 12 × 2 = 30.

However, this system uses both a 30-tooth internal gear and a 33-tooth internal gear—with identical diameters.

This is made possible by applying positive and negative profile shift to the two internal gears: the 30-tooth gear is enlarged, while the 33-tooth gear is reduced, so that both acquire the same diameter. Because the 30-tooth gear is enlarged, the sun gear must also be enlarged accordingly. Due to it, its teeth become thicker and the usual undercut seen in unmodified small-pinion involute gears is eliminated.

The differential stage operates on a principle similar to that of a harmonic drive. In a harmonic drive, two gears with different tooth counts are made to mesh by elastically deforming one of them so that their diameters coincide. In contrast, in the planetary differential system, none of the gears undergo any deformation. Instead, the two internal gears with different tooth counts mesh indirectly through the rotating planet gears, which mediate the phase difference between them.

Understanding gear profile shift resolves the paradox and clarifies how this well-designed gear system operates.

Related Model

Case

This model is compatible with the case included in my first set.

Printing

• Use the models named ???-printable.stl for printing.
  The models named ???-assembled.stl are provided just to show how they should be assembled.
   

• Use well-dried PETG to have better dimensional accuracy.

• Use 0.1 mm or 0.08 mm layer height to have smoother surfaces.

• Use slow printing speed for overhangs.

• Select “Random” seam position to have smoother rotation.
  Randomly distributed seam should be easily worn out after some wearing.Printing

Sanding and Filing

Sometimes, the gears suffer from the stringing effect and/or elephant foot effect, resulting in a too tight fit to the shafts (they are designed with a 0.15 mm radial clearance). 

If you see rough surface on the shafts due to stringing, sand off the roughness with a small piece of sand paper.

If you feel the gears do not rotate smoothly due to an elephant effect, widen the hole slightly by using a thin round bar file.

Without those issues, the parts should rotate very smoothly with minimal friction.

Assembly

Just snap assemble the parts.

Other examples

You may also be interested in the models in my educational mechanical mechanism examples.

Find them in this collection:
https://www.printables.com/@osamutake_3341417/collections/2728214

Happy printing!

Acknowledgement

I got into gears thanks to K.$uzuki's amazing articles and YouTube videos. Many of the mechanisms shown in this series came from the introductions on his website. He also makes excellent gear models himself. This series wouldn’t have existed without his inspiration.

I learned a lot about technical detail of designing gear tooth profiles from Haguruma-No-Hanashi website. I’m truly grateful for that.