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# Planetary Gear Confusion

11/02/2017 3:01 PM

I have a working planetary gear set (part of an existing mechanism), with three planets, that has gears with the following numbers of teeth:
Number of teeth on the sun (Ns) = 6
Number of teeth on the ring (Nr) = 48
Number of teeth on each planet gear (Np) = 19
From geometry, since they fit together, the Pitch Diameters of each should sum according to:
PDr = PDs + 2PDp
Since they mesh, they should all have the same Diametral Pitch, then:
Nr = Ns + 2Np (this equation is referenced in many sources).
However in this case, Ns + 2Np = 44, not 48.

Can anyone help me understand why the equation relating the number of teeth in each gear doesn't hold in this case?
Thanks!

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#1

### Re: Planetary Gear Confusion

11/02/2017 3:16 PM

planetary gear calculator/sim

If I input the sun and ring as stated, I get planetary gears of 21 teeth.

If I input sun and planetary gears as stated, I get a ring gear with 44 teeth.

This is odd!

Are you sure someone did not miss the tooth count?

10/19/48 works

6/19/44 works

7/20/45 works

You can set the inputs and get the answers for various configurations.

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#2

### Re: Planetary Gear Confusion

11/02/2017 3:29 PM

Agree with J.S. Tooth count must be amiss in one or more members.

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#3

### Re: Planetary Gear Confusion

11/02/2017 4:25 PM

I have attached below photos of the sun, planets and ring with numbers on each gear tooth. I believe my counts are correct.

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#4

### Re: Planetary Gear Confusion

11/02/2017 4:46 PM

Then someone needs to go back over the theory, and make sure we are talking apples and apples in the variable names.

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#9

### Re: Planetary Gear Confusion

11/02/2017 7:51 PM

Indeed, counting was crazy. I superimposed numbers on the teeth to avoid confusion. Here's a bigger photo of the ring gear with numbers. If that isn't 48 teeth I'm retiring.

EDIT: Sorry, this was supposed to be reply to #7.

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#17

### Re: Planetary Gear Confusion

11/03/2017 8:54 AM

That is obviously 48 teeth.

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#7

### Re: Planetary Gear Confusion

11/02/2017 7:17 PM

I know what it's like counting teeth on gears. I found a bunch of gears at a junkyard to build a clock drive for a telescope. I almost went nuts counting teeth to get the ratios I needed to step a 1RPM timer motor down to 24 hours. It helps to mark the tooth you start on with a pencil.

I counted your ring gear in the picture several times and I get 42, and I can see the planet obviously is 19. That would leave 4 for the sun gear. I don't think I've ever seen a 4 tooth gear!

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#8

### Re: Planetary Gear Confusion

11/02/2017 7:42 PM

(Editing #7 after time out!!!!)

I see the sun gear is 6. So the ring gear should be 44. I count 42 so that's 5 percent low. If the ratio of tooth size is off by this much, it might work, with more friction. I have no idea why it would be designed that way, perhaps to achieve a slightly different gear ratio.

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#13

### Re: Planetary Gear Confusion

11/02/2017 9:30 PM

I'm not sure that I am actually seeing the Sun gear in those pics.

I can see what looks to be a drive gear with 6 teeth on the outside of the carrier, and that is most likely linked to the Sun.

This may just be my interpretation of the photo, but is it possible that the Sun actually has maybe 10 teeth on it as that would work mathematically for the train.

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#14

### Re: Planetary Gear Confusion

11/02/2017 9:37 PM

Good eye -- this gearbox is actually a multi-stage, of two identical planetary sets, so the photo, I'm showing the sun for the second set (which is connected to the carrier from the first); the video link shows the sun for the first, but in any case, both suns have 6 teeth -- that's the easy one to count :)

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#5

### Re: Planetary Gear Confusion

11/02/2017 5:19 PM

Thank you for trying to help -- I agree, it seems that the theory is missing something. Please let me know if there's something I can provide to help clarify; I'm stumped.

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#6

### Re: Planetary Gear Confusion

11/02/2017 6:56 PM

The theory is pretty straightforward. If the gears mesh, the teeth are the same size, the same number of teeth per inch. So it's a matter of circumference length. The number of teeth on a gear is proportional to its diameter (circumference / π). The diameter of the ring gear is the diameter of the sun gear + 2 x the diameter of the planetary gear.

https://woodgears.ca/gear/planetary.html

The assumption is that the number of teeth per inch is the same on the sun, planets, and ring gear.

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#10

### Re: Planetary Gear Confusion

11/02/2017 8:00 PM

Appreciate the help!
Per the image you included, and using the shorthand I included in my original post, from the geometry the pitch diameters (PD) should be constrained by PDr = PDs + 2PDp.
The gears mesh, this is a consumer product that works very well, and so I think the assumption that the number of teeth per inch (diametral pitch, DP) is the same on all three (sun, ring, planets) seems correct.
However, since PD = N / DP, where N is the number of teeth on the gears, if the DP is the same for each gear, then Nr should be equal to Ns + 2Np. This same relationship also appears on the woodgears page (R = 2 x P + S), but mysteriously isn't true in this case, as the gearbox works.

What are we missing?

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#16

### Re: Planetary Gear Confusion

11/03/2017 8:44 AM

That is also true if there are four planets, but in this case we are talking about a 3 planet system, so does that mess up the formula? It does not appear to, since your link clearly states that a four planet system can convert to a three planet system by removing one gear, and re-arranging to a symmetric pattern.

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#11

### Re: Planetary Gear Confusion

11/02/2017 8:11 PM

OK, this is a little far out, but there is a patent on a planetary gear system where there are two ring gears engaging the planetaries. One of the ring gears is mismatched with a different number of teeth. If one ring gear is held stationary, the other will rotate slowly. Thus the system acts as a speed reducer.

Planetary speed reducer
US 4429594 A

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#20

### Re: Planetary Gear Confusion

11/03/2017 9:24 AM

Not far out at all, since that is the action clearly demonstrated in the video. There must be some numerical accounting for this, but my noggin is backlogged with other issues.

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#22

### Re: Planetary Gear Confusion

11/03/2017 2:48 PM

These systems are pretty common in low budget winches used for ATVs or off road vehicles.

For example, a 1/2" wide ring gear on the case with 100 teeth, a ring 1/2" wide ring gear on the drum with 102 teeth, 1 inch wide planetary gears meshing with both ring gears. Every time the carrier makes one complete revolution the drum moves two teeth in relation to the case.

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#23

### Re: Planetary Gear Confusion

11/03/2017 3:06 PM

hence the difference in tooth count from the formula?

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#24

### Re: Planetary Gear Confusion

11/03/2017 3:20 PM

I don't believe this explanation works here, there is only one ring gear in the case I'm asking about.

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#25

### Re: Planetary Gear Confusion

11/03/2017 3:27 PM

I was just referring to the patent that Rixter posted. Not the OP.

I should have been more clear.

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#12

### Re: Planetary Gear Confusion

11/02/2017 8:26 PM

Here's a video of the gearbox in motion. The sun gear (center) is normally driving, but motor and drive shaft have been removed, so for the video I'm back-driving it.

https://www.dropbox.com/s/ql1rzrifok1c6rx/IMG_7427.MOV?dl=0

I wonder if one or more of the gears profiles has been "shifted" (http://khkgears.net/gear-knowledge/introduction-to-gears/profile-shifted-gears/) -- could that be why the standard relationship doesn't apply?

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#15

### Re: Planetary Gear Confusion

11/02/2017 9:49 PM

Yeah, I think you might have it. It changes the effective gear size. That makes sense.

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#18

### Re: Planetary Gear Confusion

11/03/2017 9:04 AM

So what matters more? Effective pitch diameter or tooth count?

I am not a gear head (obviously), but this is a curious thing to me.

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#19

### Re: Planetary Gear Confusion

11/03/2017 9:14 AM

Tooth count determines the gear ratio. The teeth can't slide past one another. The tooth profile allows you to put more or fewer teeth per inch on two engaged gears and still mesh smoothly. With a different tooth profile, there can be some deviation from the planetary gear tooth count formula.

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#21

### Re: Planetary Gear Confusion

11/03/2017 10:48 AM

It's very possible that the gears are operating slightly out of their pitch circles. A ring gear oversized by two teeth and planet gear carriers also with center distances just slightly larger than PD would call for. If so, you will feel it as extra backlash in the gear set. However, the teeth themselves may be slightly tweaked and you have a totally custom pitch. Depending on the intended life and intended duty cycle, the clearances could be intentionally loose.

The product doesn't look like they intend to be able to swap in commodity gears and it's not unusual for a manufacturer that intends to be sole source for replacement parts to make non-standard components. It is entirely possible that the sun, planet and ring gears may be individually tweaked for form, pressure angle and PD to mesh at the specific ratio of the box, depending on whether the gears are primarily for timing, for power transmission or both.

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#26

### Re: Planetary Gear Confusion

11/04/2017 7:46 AM

Surely it's diameters that count not how many teeth, once you have the diameters then you cut the teeth to suit, the 6 teeth on the sun gear are relevant only to the final drive and has no significance to the planet gear at all.

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#27

### Re: Planetary Gear Confusion

11/04/2017 5:42 PM

Not exactly... Relative diameters do count, but for all teeth to mesh correctly they each must be the same size and profile, this means that there will be an identical number of teeth per inch on each gear. So if you have a particular number of teeth on the Sun and Ring, then that will determine the number on the planets.

Put differently, a specific number of teeth on any two will determine the number on the third. Desired outcome determines the ratios, power determines the size of teeth, and you change the diameter of the whole to accommodate that.

In a simple gear train, the idler has no effect on final ratio so, although their size and profile must still match, its tooth count is unimportant, it simply transfers motion from the driver to the driven although, depending on the number of idlers, rotational direction can be reversed. In an Epicyclic however, any of the gears can become the driver or driven, and any the idler, so all can influence final ratio.

If, as has been suggested, these gears have been shifted, it may be that due to the need for a specific ratio or combination of ratios, the manufacturer went with this process as the best method to achieve it. Whilst not ideal, it was obviously considered acceptable, and the alternative may have been a more expensive meshed or stepped compound setup.

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#28

### Re: Planetary Gear Confusion

11/06/2017 4:05 PM

Thanks all for the help with this.

If it’s true that the profile has been shifted, it’s still not totally clear why the basic algebraic relationships don’t hold.

The consensus seems to be that even in the case of a shift, the diametral pitch (number of teeth per inch) of all gears still need to be equal.

It also seems that even in the case of the shift, geometrically, the ring pitch diameter should equal the sun pitch diameter plus twice the planet pitch diameter (PDr = PDs + 2PDp).

If the pitch diameter is equal to number of teeth divided by the diametral pitch (PD = N /DP), then it still follows that Nr = Ns + 2Np, which isn’t true here.

So, which assumption is incorrect? It seems possible that:

a) The diametral pitches are different between the different gears, but the shifted tooth profiles still allow them to mesh smoothly.

b) The pitch diameter is no longer equal to the number of teeth over diametral pitch for the gears with shifted profiles.

c) The ring pitch diameter no longer equals the sun pitch diameter plus twice the planet pitch diameter even though they still fit together.

My guess is that either a) or b) is likely no longer true, but I’m not sure how to confirm this. Does anyone know where I can find more information about shifted profile gears?

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#29

### Re: Planetary Gear Confusion

11/06/2017 4:53 PM

I am wondering if there is a back-lash (schlock) specification we missed here somewhere. What is the effect of too many teeth on ring gear? Does this make it run tight? loose? neutral??

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#30

### Re: Planetary Gear Confusion

11/06/2017 7:03 PM

It runs loose.

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