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)
