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# Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

## Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

(OP)
Hi Guys, Hopefully you can help me find answers to my questions.

It sounds like a simple problem and i know I will drive some people away by saying that i am trying to
calculate the gear ratio out of a planetary-style gearbox, but that is NOT the whole story! this is very similar to a harmonic drive.

The whole story is that a while back i came up with the idea of taking two planetary gearboxes and interconnected the planets,
i can do this in Fusion 360 by adjusting the module of one planetary gearsets until the center to center distance of the planets is equal between the two planetary's, and the reason i thought to do this is to get some very high gear ratios(100:1+) out of 3D printed plastic parts without using much of any bearings or hardware.

heres one that i have made in the past http://www.thingiverse.com/thing:2054378 , i was not able to calculate the speed of anything at that time, and designed it without knowing the output gear ratios at all, now i have gotten a bit more educated on the subject, but still can't figure this out completely.

# of Planets 4
Sun #1= 26
Planet #1=38
Ring #1=102
Sun to Carrier Ratio: 4.9261 rotations of sun per revolution of carrier
Planet Gear Rotations per revolution around ring gear 2.6845

Sun #2 = 29
Planet #2 = 37
Ring #2 = 103
Planet Gear Rotations per revolution around Ring Gear = 2.7838

if you interconnect the planet gears between these two planetary gearboxes, drive Sun gear #1, and hold Ring Gear #1 stationary, What will the ratio between Sun Gear #1 and Ring Gear #2 be?

its too easy to calculate these when i keep the planet gears with the same number of teeth in both gear sets, but I think i could achieve much much higher gear ratios if i don't constrain my design like that. Thanks in advance!

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

hi, if you have the 2 x planet gear sets on the same carrier, you would need each planet gear from one set to rotate independently from the other set if they are different sizes.

Can I ask why you cant calculate this yourself? Do you know about planetary gears and ratios etc? Its quite simple really, even wikipedia provides the calculations...

Basically you have 2 x separate planetary gear sets, and the common link is that the carrier gear rotational speed (W(C)) will be the same for both sets.

i come to the conclusion (using your constraints) that: W(R2)/W(S1) = N(S1)*(N(S2)+N(R2)) / (N(R2)*(N(S1)+N(R2)))

where W(R2) is your output (ring 2) and W(S1) is your input (sun 1).

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

I just noticed that the PCD of your planet gears is different; 64 (26+38) for set 1 and 66( 29+37) for set 2.

Still, the calculation should hold.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

(OP)

"hi, if you have the 2 x planet gear sets on the same carrier, you would need each planet gear from one set to rotate independently from the other set if they are different sizes."
- this is not true if the ring gears and sun gears can spin independently.

I'm sorry you have misunderstood me. The planets are not interconnected by the carrier, in fact there isn't a carrier, the planets are connected to one another directly. I.E. the planet gears are one single component with 37 teeth on one end and 38 teeth on the other end. And held In place solely by the double helix teeth, the picture in the link should help clarify.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

(OP)
Hey thanks for your help, i thought about this all day and i think i figured out the solution, i even tested it and it worked out.
Seriously though, i didn't think i would solve this one myself...

TS1 = Teeth Sun #1
TP1 = Teeth Planet #1
TR1 = Teeth Ring #1

TS2 = Teeth Sun #2
TP2 = Teeth Planet #2
TR2 = Teeth Ring #2

For the final gear ratio : 1 / ((TS1/(TR1+TS1) * ((TR2-((TR1/TP1)* TP2))/ TR2))

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

The term 'carrier' applies to the item that carries torque from the planets to the next stage. In this case the carrier is integrated into each planetary gear and would be the 'carrier' term in any calculations.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

(OP)
Thank you 3DDave I was not aware.

I'm very excited that I got this figured out, the ability to calculate the outcome of the gearing has allowed me to write a program to auto calculate the highest achevieveable gear ratios and some of them are totally insane. I just completed a prototype that is 1,400:1 and it's half the size of a hockey puck... pretty impressive for 3D printed parts!

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

OP, I think you need to be careful here "this is not true if the ring gears and sun gears can spin independently."

If you have the two planet sets connected directly (common shaft), and the planets are different sizes, then the gearbox WILL lock up because you have a fixed ring gear for set 1 and a fixed sun gear for set 2.

I cant understand how you got any models to work like this?

The reason I see it would lock up is as follows:

Say set 1 has a very large planet gear and a small sun gear. it would take only a few rotations of the planet gear for the 'carrier' to do ONE revolution. (As the ring gear is fixed)

Now say set 2 had a very large sun gear and a very small planet gear. The planet gears will have the same centre axis (PCD) as that of set 1 as they are on the same shafts, this also means they must have the same rotational speed as the planets on set 1.

Now, if you want to do ONE revolution of the carrier (same as set 1), the planet gears on set 2 will have to rotate many times as they are so small. But this will not work, as the planet gears must rotate the same as set 1 as they are connected.

Therefore I dont believe this will work, the only way you can share a common carrier is to have the planet gears from different sets rotate independently.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

(OP)
The sun gear on set #2 is not fixed. And does not even need to be used, it's only to hold the planet gears against the ring gear.

I have actually made this, if you fast forward to the end of this video you can see how it works
There's other videos on my channel of a Robotic Arm I made with this concept.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

hi, i just noticed that you are not holding sun gear 2 and its free to rotate, so ignore my comments :)

i just began to work out the ratio and it is a very long calculation, which will take too long for me to reduce so I wont bother. will trust yours is correct :P

(OP)
Thanks!

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

To wrap my mind around it, I thought about it this way:

The second stage planets "roll" around the inside of the second stage ring, but they are not actually just rolling. Their orbit "ground" speed and their rotational "rim" speed are independent of each other, and don't agree. The difference is the rim speed of the stage 2 ring, kind of like a differential pulley arrangement.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

(OP)
Would you guys believe me if I told you that I thought of this myself, but I'm actually not a engineer at all... I'm an electrical apprentice.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

You could make this whole arrangement a little easier to assemble by just using a smooth ring instead of the second ring gear, without a slit and sized so that it will either just slip over the second stage planets or is a very light press fit over them.

Because your first and second stage planets are connected and your first stage ring gear is fixed, you don't need a second ring gear at all. Since you're not using the second stage ring to provide output torque, you don't really need torque transfer from the teeth. The second stage ring will still rotate, driven by friction with the planet teeth.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

jgKRI,

the 2nd stage ring gear is the output, this would have the MAXIMUM torque! you would definitely require gears ti mesh.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

#### Quote (kiwitom235)

the 2nd stage ring gear is the output, this would have the MAXIMUM torque! you would definitely require gears ti mesh.

#### Quote (Rossrohrer)

The sun gear on set #2 is not fixed. And does not even need to be used, it's only to hold the planet gears against the ring gear.

.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

(OP)
The #2 ring gear needs

to have teeth because it is the output. The #2 sun might not need teeth because it doesn't even need to be there, but I would rather not have it rounding off all of the planets.

### RE: Gearbox is VERY complicated to Calculate Differential Planetary, Harmonic

yes Rossrohrer you are correct.

I am not sure why jgKRI is quoting two other posts and then not typing anything to explain?

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