Warning! some math included

As I was asked by some and as I am a CD kinda guy I use this platform to talk a bit of math on the gearbox
first some basics
R = Number of teeth in ring gear|
S = Number of teeth in sun (middle) gear|
P = Number of teeth in planet gears|

I will hang a 1 or 2 at the end of it as its a compound planetary with 2 planetary used

The first constraint for a planetary gear to work out is that all teeth have the same pitch, or tooth spacing. This ensures that the teeth mesh. In our case we decided on 2.25mm modulus with 14.5 deg pressure angle and a 30 deg double helical setup

If you don’t use any tooth shifts which we did not except the one for going helical the formula is
R = 2 × P + S

To relate the turns the complete formula is

Tr =Turns of the ring gear
Ts = Turns of the sun gear
Ty = Turns of the planetary gear carrier

( R + S ) ×Ty = R × Tr + Ts × S

On the first stage the input is the sun so Ts= 1 and R1 is held steady so its 0 so it becomes wiht
S1 = 21
P1 = 21
R1 = 63 ; (S+2P)
(21+63) * Ty = 63 * 0 + 1* 21 => 84Ty = 21 => Ty=1/4
So for each turn of the S1 gear the planet carrier makes a 1/4 turn

Furthermore for each Ty each planet turns 3 times as it has 21 teeth and the ring has 63 and 63/21 = 3

Now for the planets to be equally spaced (R + S) which is 84 must be divisible by of planets with no remainder - so resulting in an integer. The max planets we can fit is 4 that is also resulting in an integer division with 84. That also means that the 2nd planetary will also have 4 planets and R2+S2 will have to be also an integer multiple of 4

So the first reduction is from sun to planet carrier which is 4:1

So if we pick P2 = 22 teeth than as it is connected to P1 to maintain the same center distance S2 has to be 20 teeth which means based on R= S + 2P => R2 = 64 (20 + 22*2)

As P2 is connected to P2 as a solid stack that means for every revolution P1 makes P2 makes one and for 3 revolutions of P1 to make 1 Ty one complete circle around P2 also makes 3 turns and as it has 22 teeth it moves a distance of 66 teeth (3*22) so as the ring one stays static with the 3 revolutions of P1 anmd R2 has 64 teeth but gets moved 66 that means it advances 2 teeth which each revolution of Ty. And with 64 teeth it needs 64/2= 32 revolutions of TY to make 1 Revolution or a reduction of 32:1

So now we have a reduction of 32:1 on a top of a 4:1 from S1 to Ty giving us a total reduction of 32*4= 128 to 1. Meaning with every turn of S1 R2 the output will do a 1/128 of a turn.

With a bevel gear input of 19:41 the overall reduction is close to 276:1

All that is left now is to examine the positional relation of P1 to P2

If we look at P1 and where it is at

degrees 0 90 180 270
tooth 0 15.75 10.5 5.25 And P2 is at (based on P1)
" 0 16.50 11.00 5.50 And on R1 at
" 0 15.75 31.50 47.25 And the R2 tooth is at position
0 16.00 32.00 48.00 for it to mesh P2 would have to be on
Tooth 0 16.76 11.52 6.29

This is based on the fact of the motion between R1 and R2 where R1/P1 = 3/1 (63/21) and R2/P2 = 64/22

So each P2 except the one in the 0 position needs to be turned .2619 teeth or (360/22) * .2619 =4.2857 degrees Hence making the need of designing 4 individual stacks and making sure the gears are in the right position.

To assemble the stacks they were broken in 3 parts

And then bolted together In cad a single stack was designed first and then split into the 3 pieces to make sure all holes line up Furthermore the bolts help give the plastic more rigidity .

S2 in theory is not necessary for the system to work but at R2 is the output and will take some load S2 helps P2 to stay aligned and transfers that load through the bearing to the central shaft. Some more "Unloading was done by tying the planet carrier into the ring holder via “bearing wheels” on the input side

And tying the output side planet holder to the shaft so to not only guarantee planet to planet position but also planet to center axle position

And then tying the ring gear R2 Also back to the center axle to transfer most of the load of the belt drive there (except of course the radial one from being a compound planetary gear system) and allowing fot the mount of a gear output in line with the central shaft

The 3 CIM Motors will mount to this

Also allowing to skin it and fully enclose the input side. On the output side I will most likely skin the Wheel that connects the ring to the shaft to keep dirt out and grease in leaving a small gap between R1 and R2 that probably will get a TPU gasket (not designed yet)


Really neat post! Just a heads up though, but trigger warnings are usually used when the post is discussing something really sensitive that could trigger flashbacks to traumatic events/topics (i.e. assault), hence the term trigger warning – it isn’t really meant to be used in this lighthearted context. Maybe just shortening it to “Warning! Some math included” might be a better title? (@mods?)

Right now it feels like the post is making fun of TWs, though I’m sure that wasn’t the intent


Ill change it It was directed at some who bust me that way with all my math


Instead of playing up the “scariness” of the math, it goes a long way towards helping students if you try to make it approachable and treat it like you treat any other thing.


I try and somehow it does not let me change the title never changed a title before


I think the mods got it :slight_smile:

All over sudden the edit took or the mods got it lol

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Why do people fear math?

I mean, it’s OK to hate on it, but fear?

to be blunt: because the experience of math class for many students is more “elaborate societal hazing ritual” than it is an exploration of useful/important concepts


I fear math. I fear it largely because I struggle with it and it’s hard to admit I struggle with it. So it’s less the math I fear and more the social stigma from being ashamed that I struggle with it.

Also, discourse supports mathjax, by wrapping your equations in $'s you can do inline equations A^2 + B^2 = C^2. If you do double $$ on a new line with the equations between them you can do longer equations.

( R + S ) ×Ty = R × Tr + Ts × S

It may help make the math easier to read and/or more approachable.


Thanks for telling me I didnt know that

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@Oblarg I dunno, I think that’s more of a hate thing. Mt daughter doesn’t fear math, but those timed drills in second grade made her hate it. Well, maybe dislike intensely is a better phrase.

Very confused, I’ve noticed that the ones good at math are the ones who get picked on in school.

Hazing comes from multiple sources. Some hazing comes from fellow students, some from the educational institution itself.

Even if the teachers have the best intentions, they have a script to follow.

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I hate factoring. I also hate having the calculate the Reynolds number from scratch. Gearing at least follows nice geometric logic.

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