Optimisation of the Original MPRT
So okay, I decided to see if I could optimise the original MPRT.
Not a trivial task, but doable with brute force:
- For each Pressure Angle (PA) from 15 degrees to 25 degrees in steps of 1 degree.
- For each Diametral Pitch (PD) from 0.75 to 1.25 in 0.05 steps.
- Test for interference for 1/10 rotations of the tooth.
Test for interference?
Basically test if any points from the external gear are inside the planet gear.
Okay, the results suggest any PA between 20 and 25 degrees is okay, and for the original MPRT configuration (18/21/-60 & 21/-57) the optimal DP for the output ring is 94.0% of the PD of the drive gear set.
But like most code you do have check the results, so for each 1/10 tooth rotation of the optimal result, DXFs were export and checked.
Now, at least to me, the gears are meshed (no backlash) but the contact is between the outer teeth, not the centre tooth (see image below):
All the other 1/10 tooth rotations are similar.
The next rabbit hole?
The contact area is mostly on the low area of the planet and high area of the ring. These are not important areas (I think?) for normally meshed gears. The meshing may be improved by trimming the high area of the ring gear.
Ideally the pitch circle of the planet and the ring gear should touch (the red and green circles):
So the plan would be to set the ring gear pitch circle to touch the planet pitch circle, and then "trim" the ring gear top of tooth area until there is no interference (for 1/10 tooth rotations).
Need to work out how to code "trim". Hopefully an easy fix for the Make Gear code pops out of the experimentation.
Results or Any Easter Eggs?
After a long day hacking my code, here is a solution:
Note the white line. It is to show how straight the tooth profile.
Now, is it optimal, can I do better? What are the results telling me?
And the answer is not "42" as it has 57 teeth!
Like usual I have to sleep on it.
The answer is that the gear pitch circles are not valid!
- As the gear is locked it has to rotate 57 to 21, and the gear pitch circles are not in this ratio (as their Diametric Pitch (DP) are different).
- The effective pitch circles have to be 57 to 21.
- So the teeth have to contact either side of the centre, where the effective pitch circles must overlap or cross.
For tooth clearance purposes, using the gear pitch circles for gear design is appropriate.
So in conclusion, the design is now fully constrained and zero backlash, optimal by our definition.
Note, as I trim gear teeth for interference, the pressure angle is no longer important.
Universal Epicycle Gear Formula
I found the universal epicyclic gear formula:
Okay, I rearranged it a little to suit my tastes.
- Ts = Turns in the Sun
- Tc = Turns in the Carrier
- Tr = Turns in the Ring
Note: The Planets play not role
- Ns = Number of teeth on the Sun
- Nr = Number of teeth on the Ring
Basically, you set Ns and Nr, and two of the three Ts, Tc and Tr, and solve for the unknown.
Let's try Daren's design:
- Ts=1 (one turn)
- Tr=0 (fixed)
- Tc=18/78 (=1/4.333)
- Ts=1 (one turn)
- Tr=-0.01214575 =(-1/82.333)
A reduction ratio of -82.333 was the previous calculation.