Compound Planetary gears

[wfrancis]

Hello - I've been trying to calculate some compound planetary gears but I've not been successful in that I can successfully calculate the ratios but they don't work with the ring gear results.  In my case I'm trying to make something very similar to what is described here:

http://www.mathworks.com/help/physmod/s ... ygear.html

I want to drive the sun, have the stepped planets stationary and take power off the ring gear. In my application I need weird ratios less than 2 so this is necessary.

What would be most useful are all the simple rules necessary that given a desired ratio I could programmatically find valid combinations. This thread was very useful for single ratio planetary gear sets and I added a little script I wrote find valid combinations. I'd love to be able to do the same for compound gear sets.

http://gearotic.com/ESW/FavIcons/index.php?topic=1074.0

Thank you!

[Nate]

If I understand correctly, there are four pitch radii here, r_sun, r_solarplanet, r_ringplanet, and r_ring, and the gear ratio should be

r_sun/r_solarplanet * r_ringplanet/r_ring

And, from geometry, it's clear that:

r_sun + r_solarplanet + r_ringplanet = r_ring

These are periodic gears, so you must have:
r_sun/r_solarplanet is rational
r_ring/r_ringplanet is rational

You probably want to be able to have all of the planetary gear stacks be the same, to ensure that, add the following conditions:
#sun_teeth / #planets is an integer
#ring_teeth / #planets is an integer

I think that's it.  (Besides obvious stuff like r_sun and r_solarplanet having the same pitch.)

The fact that you can have different pitches on the ring and sun gears gives you a lot of freedom about how to set things up.

[wfrancis]

Nate,

Thank you so much for your reply. Given the dearth of non-kinematic equation driven examples (of which I poorly understand) I appreciate your detailed reply. I regret that I'm having a little difficulty digesting your explanation vs my understandings, though.

For example, when you gave your equation for doing the ratio calculation (r_sun/r_solarplanet * r_ringplanet/r_ring) this is very different (and doesn't appear to be equivalent) to the one I've been using. Following your nomenclature, I've been using (r_sun * r_ringplanet) / (r_solarplanet * r_ring), as taken from the T1*T3 / T2*T4 example on this page:

http://www.schsm.org/html/gear_ratio_calculations.html

I'm assuming that my two planet gears share a shaft and that it's immaterial for this portion of my calculations that the final gear is a ring gear. For example, for a desired ratio of 2.5 (ignoring for a moment that 2.5 can be achieved without a compound arrangement) my program suggested (as one of many given a range of teeth to try for all gears)  r_sun=28, r_solarplanet=14, r_ringplanet=10, r_ring=50. At each stage of the arrangement I also checked the two rules for my number of planets looking for integers as mentioned in the first thread and it all looked good. I exported the gear profiles into SolidWorks (pro tip: 'fit spline' on the teeth for a much lighter weight part) and went about making a test model I could 3d print. It all looked good until I tried to fit the ring gear and it clearly was a bit too small. bummer. (I kept all my diametral pitches the same in this case, but yes the option to vary them has appeal). Clearly I need to figure something else out ...

The arrangement above fails another one of your tests, too - r_sun + r_solarplanet + r_ringplanet = r_ring. In fact I put that conditional into my program and none of my otherwise "valid" ratios worked. Looking at the example I linked to (mathworks) in my initial message I'm trying to get my head around why that would be true given that the ring can be smaller. If I do that equation with my example from above it does suggest that my ring gear should be 52 teeth and maybe that would work given that 50 is too small? I'll explore that some more in my searching program.

Anyway, you've given me a lot to consider and I'll think and code some more and see what happens.

Will

[Nate]

... Following your nomenclature, I've been using (r_sun * r_ringplanet) / (r_solarplanet * r_ring), as taken from the T1*T3 / T2*T4 example on this page:

http://www.schsm.org/html/gear_ratio_calculations.html

Those are the same - they just look different.  In other words:
(T1 * T3)  / ( T2 * T4) =  T1 / T2 * T3 / T4

 r_sun=28, r_solarplanet=14, r_ringplanet=10, r_ring=50.

... If I do that equation with my example from above it does suggest that my ring gear should be 52 teeth and maybe that would work given that 50 is too small? I'll explore that some more in my searching program.

That should work, but you'll get a different ratio.

Edit:

r_sun=32 r_solarplanet=16 r_ringplanet=12 r_ring=60

Is a bit bigger, but should give you the same 2:7 ratio.

[wfrancis]

Nate, thanks for your reply.

After sending my last note I did realize that 1:.04 is the same as 1:2.5, so, yes, you are correct about the equations being the same. I just needed to adjust my thinking :)

So far your rules have worked very well. I'm surprised at how few combinations work (a handful out of a few hundred thousand possibilities) for a given ratio.

My next challenge is to do a similar calculation but to turn the ring gear into a spur gear - sometimes I need the 'ring' to spin in the same direction as the sun but I'd like to maintain the planets. I think I got my head around why the geometry works in your example so I'll see if I can figure it out for a concentric spur gear. I think the other rules still hold so already I'm most of the way there.

Edit:

I think it's sun + (r_solarplanet - ringplanet) = internal spur gear


Will

[Nate]

Edit:

I think it's sun + (r_solarplanet - ringplanet) = internal spur gear

That looks right to me.

So far your rules have worked very well. I'm surprised at how few combinati ons work (a handful out of a few hundred thousand possibilities) for a given ratio.

(Ignore this section if you don't like math.)

It's not that strange.  

Let's say the drive ratio is written as a fraction a/b in lowest terms.  (With a fixed diametral pitch, the ratio has to be rational.)

Then for some constant c
(r_sunplanet)*(r_ring) = b * c
(r_ringplanet)*(r_sun)=a * c

So for each prime factor of b, you get to pick r_sunplanet, or r_ring, and similarly for each prime factor of a, you have to pick r_ringplanet or r_sun.  There's a similar sort of restriction on the prime factors in c, and, an additional limitation because of the radius addition rule.