This was some early work that I did on planetary gearboxes.

It’s no longer relevant to the current project, but it might be interesting for some.

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The gearbox

This is the 2nd article in a series on mechanics. It is an extension on A few notes on planetary gear sets. Nothing new or novel is being shown here. This is an educational post and a building block for future work.

This is a basic forward/reverse gearbox. It's about as simple as it gets.

One can do much more than this with just two planetary gear sets.

This set is simpler than other designs as it has no clutches or meshing requirements. Of course, those parts do exist for good reasons in many applications.

The design decisions here are all for demonstrating the principles of how it works, rather for any practical application. Hopefully it is easy enough to understand.

It looks like this.

Fig2: the gearbox from the motor end.

The operation modes are:

Forward at 1/3rd input speed.
Reverse at 1/3rd input speed.
Stop while the input may still rotate.
Neutral: where input and output is free to move independently.
Locked: where nothing is free to move

There is one motor (in red) driving an input shaft (in blue)

There are 2 planetary gear sets that connect the input shaft to the output. Finally, the output shaft (in green) is at the opposite end.

The 3 red arrows show where one would apply a brake to the system in order to apply one of the modes.

Fig 3: the gearbox from the output end

The input (blue) is passed all the way through the output (green) so that we can see most the functionality from just one camera angle.

Here is a schematic for the design.

Fig 4 (CC0): a base schematic for the gearbox. Blue for input, green for output, red for brakes, orange for neutral, and gray for planet gears.

You can see that the schematic is based on a cross-sectional look at the 3D model.

Fig 5: CC0 The cross section

Forward mode

In forward gear we apply the forward brake. This locks the ring on the 2nd gear set.

Fig 6 (CC0): forward mode

Fig 7 (CC0): forward schematic

The 2nd gear set is now in a Fixed Ring mode.

This gear set has a 1+2=3 ratio. (as we explored in the last post)

It means that the carrier (output) will follow the sun (input) at 1/3rd of the speed.

The 1st gearset has no effect on the system. It is in Neutral mode. The sun is moving at the input speed and the ring is moving at the output speed. From the perspective of this gear set, the sun and ring both become inputs which drive the carrier. The carrier will move at 1/2 of the input speed, but it's not connected to anything, so it's speed is not relevant to this mode.

Reverse mode

To achieve reverse, we select the reverse brake instead of the forward brake.

Fig 8 (CC0): reverse mode

This puts the first gear set into Fixed carrier mode. The ratios we use here are 1+3=4 (rather than 1+2=3 as in previous examples). Now the output moves at -1/3rd of the input speed

Fig9 (CC0): reverse mode schematic

The 2nd gear set is now in Neutral mode mode and will have no effect. As before, the ring will find a speed that is driven by both the sun and carrier. Again, this speed is not very important because it doesn't go anywhere.

Stop mode

For stop mode, we simply hold the output fixed using it's brake.

Fig 10 (CC0): stop mode

In this case, both gearsets are now in Neutral mode.

The input may still turn, but now the motion is only transferred to the free spinning parts: the 1st carrier and 2nd ring. Their speeds become 1/4 and -1/2 respectively.

Fig 11 (CC0): stop mode schematic

The result is that the input can rotate freely while the output is fixed.

Neutral mode

Neutral mode works the same as stopped mode. Except no brakes are engaged and the output is free to move.

There are an infinite number of speeds one could demonstrate here. Let's just look at output is 1/3 and input is 0.

Fig 13 (CC0): neutral mode

Again, we see that all the motion goes to the freely moving parts.

Fig 14 (CC0): neutral mode schematic

Of course, this is not a perfect neutral. Any friction in the gears will transmit some of the power between input and output. This would make it a bad choice for anything with a very low idle load on the output. Like a propeller on a boat for example.

This is where clutches inevitably become a good idea. It depends on your application.

Locked mode

Activating any 2 brakes will result in all parts, including the input to be stopped. This is because any combination of 2 brakes will cause on of the gear sets to become locked to zero speed.

fig 15 (CC0): It should also be pretty intuitive that you can't drive and stop something at the same time; or go forward and reverse at the same time. Nonetheless, this mode needs a name. That and we get to use magenta for something.

Changing gears (modes)

Exactly what happens when you change between gears is beyond this basic intro. Things like momentum and friction need to be considered based on your application.

Like most gearboxes, one typically needs to go via neutral in order to change gears. This includes going to and from the stop mode.

This is fine if your project plays nicely with momentum: like a car.

But it’s not great if you work with precision: like a robot arm or a crane.

One might imagine changing from forward to stop as quickly as possible. In reality momentum of all the parts will be challenging. This will also be some effects back on the input. The simple way to keep precision is to bring the input to a stop. Then go via Locked mode to change gears. Otherwise, you need to add more complex parts like clutches, torque converters and/or control systems.

What is it good for?

On its own. Probably not much. It's just a conceptual model. Especially if you have an electric motor can reverse itself.

But, I do have an application in mind where this is a small part of a larger solution. So I put this post together to reference back to.

Making the animations.

This section is about CADing up and animating the parts in Fusion 360.

The end goal was to create a looped animated GIF where all the important parts are visible. It should be easy to understand.

Fig 16 (CC0): the individual parts

There is no “physics” in this software, so all the rotations needed to be calculated. (rather than to have a physics engine let the gears push each other around). While fusion 360 has joints motion links, these are (frustratingly) not available in the animation tool so all of the movement needs to be done manually.

This is the full table.

	Gear	Teeth	Ratio	Forward	Reverse	Stop	Neutral
Input	Sun 1	24	1	1	1	1	0
Output	Ring 1	72	3	1/3	-1/3	0	1/3
	Planets 1	24	1	0	-1	-1/2	1/2
	Carrier 1	96(4)*	4	1/2	Brake	1/4	1/4
Input	Sun 2	36	1	1	1	1	0
	Ring 2	72	2	Brake	-1	-1/2	2/3
	Planets 2	18	0.5	-1	-3	-2	1
Output	Carrier 2	108(3)*	3	1/3	-1/3	Brake	1/3

Table 1: the relative speeds of all parts for each operational mode.

It would take lot of extra work to go into detail on the math. You either love math and can do it yourself, or you’re not interested at all and won’t read it.

Overall, the process was:

Fill out all of the input and output speeds. Including the brake settings for each mode.
Apply the equations below from Wikipedia to complete the rest of the table
Find a set of gear teeth that will both work and animate nicely.

Here are the equations.

This simulator is a good way to find some nice teeth counts to satisfy the gear ratios. One thing to keep in mind is the goal to make a short looping GIF animation. Each gear needed a number of teeth that would work with the fractions in the table.

i.e.: if ring 1 had 64 teeth. After a 1/3rd rotation, it would have moved 21.333 teeth. The animation would loop here and would glitch. With 72 teeth, it moves 24 whole teeth and there will be no jump in the animation.

*. For Carriers, this is the “effective” number of teeth. In brackets are the number of planets on the carrier. This also needed to be selected to achieve a loop-able animation and is the only reason why set 1 has 4 planets and set 2 has 3 planets.

Earlier works.

The design we looked at today has some flexibility in terms of its gear ratios. Here, forward and reverse are matched. But you can also tweak these to get some unmatched values.

Before this design, I had a model that did 1/9 forward and -1/3 reverse. It also featured some nice disk brakes as well. I later dropped these in favour of the red arrows. Disk brakes added too much of a production look and made it more difficult to focus on what is happening.

Fig 17: an alternative design with a greater difference between the forward and reverse speeds.

Fig 18: another angle

Fig 19: the cross section for the alternate design

Fig 20: original concept sketches of the schematics

License

All the images on this page with CC0 are in the public domain and can be found on Wikimedia Commons.

I didn’t bother with all of the images because some of them are probably not as useful. Feel free to contact me if you want me to upload a CC0 version of something. It probably won’t be an issue.

Next

Next will likely be a look at worms.

This article ends a bit abruptly.

CUT

From here its easy to add the gear ratios. The sun1 and ring1 needs to to be 1:3. We know from before that the effective ratio of the pla

	Gear	Teeth	Ratio	Forward	Reverse	Stop
Input	Sun 1		1	1	1	1
Output	Ring 1		3	1/3	-1/3	0 (brake)
	Planets 1		1
	Carrier 1		4		Brake
Input	Sun 2		1	1	1	1
	Ring 2		2	Brake
	Planets 2		0.5
Output	Carrier 2		3	1/3	-1/3	Brake

I then used the equations to fill out the directly affected parts.

Here the ratios are worked out. The inputs are connected together and always 1. The output is connected to the first ring and 2nd carrier. We

	Gear	Teeth	Ratio	Forward	Reverse	Stop
Input	Sun 1		1	1	1	1
Output	Ring 1		3	1/3	-1/3	0
	Planets 1		1
	Carrier 1		4		Brake
Input	Sun 2		1	1	1	1
	Ring 2		2	Brake
	Planets 2		0.5
Output	Carrier 2		3	1/3	-1/3	Brake

We then use this equation to get the missing speeds carrier 1 and ring 2. For the N values we can use the ratios (since we haven’t picked any teeth yet)

Carrier1:

Ring 2:

	Gear	Teeth	Ratio	Forward	Reverse	Stop
Input	Sun 1		1	1	1	1
Output	Ring 1		3	1/3	-1/3	0
	Planets 1		1
	Carrier 1		4	1/2	Brake	1/4
Input	Sun 2		1	1	1	1
	Ring 2		2	Brake	-1	-1/2
	Planets 2		0.5
Output	Carrier 2		3	1/3	-1/3	Brake

To get the planet rotations we can pull these equations from Wikipedia. Either works.

Planets 1:

Planets 2:

	Gear	Teeth	Ratio	Forward	Reverse	Stop
Input	Sun 1		1	1	1	1
Output	Ring 1		3	1/3	-1/3	0
	Planets 1		1	0	-1	-1/2
	Carrier 1		4	1/2	Brake	1/4
Input	Sun 2		1	1	1	1
	Ring 2		2	Brake	-1	-1/2
	Planets 2		0.5	-1	-3	-2
Output	Carrier 2		3	1/3	-1/3	Brake

	Gear	Teeth	Ratio	Forward	Reverse	Stop
Input	Sun 1	24	1	1.00	1.00	1.00
Output	Ring 1	72	3	0.33	-0.33	0.00
	Planets 1	24	1	0**	-1.00	-0.50
	Carrier 1	96(4)	4	0.50	Brake	0.25
Input	Sun 2	36	1	1.00	1.00	1.00
	Ring 2	72	2	Brake	-1.00	-0.50
	Planets 2	18	0.5	-1.00	-3.00**	-2.00
Output	Carrier 2	108(3)*	3	0.33	-0.33	Brake

*Effective number of teeth of the carrier. And (number of planets)

**Planet speeds a calculated from these equation:

CUT

Forward rotation. 1st planets

From the logic table we can see that for a rotation of the input, some parts will turn by 1/3 and others by other fractions

With all the parts moving at different speeds, the conse

I chose 72 teeth for both rings because it’s a number that is divisible by both 3 and 4. This made it easier to create the GIF animations.

	Ring 1	Sun 1	Planets 1	Ring 2	Sun 2	Planets 2
Ratio	3	1	1	2	1	0.5
Teeth	72	24	24	72	36	18
Fractions	1/3 = 24	1 = 24			1 = 36

This is also the only reason why there are 4 planets on the first set and 3 on the second set.

The animated gifs in this video are looped.

When I made the gifs in the last post, I was able to simply loop a 1/4 turn of the carrier. In this case, all the various gears would rotate by a full integer number of teeth, and hence I could make a seamless animation.

For these animations, it was a bit trickier. Looking at the logic table. All teeth counts should be divisible by 3 and 4. The planets in the 2nd set should be 3 and the planets in the first should be 4.

the 2nd set is a 1+2=3 ratio, previously we had 32 teeth in the sun. This is not divided by 3, so a 1/3rd ru

Since this is not divisible by 3 we will select 36. this gives 72 for the ring and 18 for the planets. All are divestible by 3.

For the

To get the files to be as efficient as possible, I had to calculate the frames

to get the planet rotations we need to apply a different equation:

Fig 16: an alternate design where the forward gear is 3 times slower than reverse

https://upload.wikimedia.org/wikipedia/commons/5/5f/Simple_reversing_gearbox_in_neutral_mode.gif

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