Gearing Puzzle #2

If your gear tooth spacing or your diameter do not match you're going to have a problem....the bigger the difference the bigger the problem....

1/3 RPM

I don't see how the carriers (or "spiders") can be exactly alike; carrier B has to accommodate ring spacings of 100/100/99, so must be a bit off of 120/120/120 degree angles.

Sloppy machining tolerances and a hammer are the key to assembly.

Shhh, don't tell, and I won't.

Maybe it's to even out the wear by slowly rotating one gear in relation to the other gear....

Yes. Lots of devices use relatively prime numbers so that mating elements rotate among the elements they engage with, thereby equalizing wear. 5/7 versus 4/6 screw compressor lobe configurations are a case in point. They can also achieve large gear ratios with few stages.

Can I be excused this one?

I have a note from Mrs Cat

Del

It won't turn at all.

For a planetary gearset to work, there is a definite correlation between the numbers of teeth on each set of gears.

The ring gear will have a number of teeth equal to the number of teeth on the sun gear plus twice the number of teeth on one planetary gear.

This cannot change. If you understand that the pitch diameters of the sun gear plus two planets is equal to the inside diameter of the ring gear, you will see that this is so. Picture it without teeth for an easier understanding of this. Obviously all teeth will be of the same pitch otherwise it would not mesh correctly.

In your example, with a sungear of 100 teeth and planetaries of 50 teeth, then the ring gear would have just 200 teeth, not 300 or 299 as you surmise. To get more teeth on the ring gear would require making their pitch smaller than those on the sun and planets, and then it would not work at all.

Good point:

OK so make it:

100, 50, 200 and 199

I'm not trying to be a smart ass but did you count the teeth correctly?

I'm just saying that the gear teeth are fairly small and easily missed?

Actually it's from a set of bonus point questions my professor gave us on a test once from a mechanical drives class I had back about 20 years ago.

We had to figure out if it would work and if so what was the input to output ratio plus what direction would the output turn in relation to the input.

The second part is what happens if ring gear B has one more tooth than ring gear A.

Formulas for calculating gear ratios of planetary systems:

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

I think there's something wrong here. All the teeth must be nominally the same size for it to work. So the diameter of the planets is 1/2 the sun. So the diameter of the outer ring = 2*sun (= sun dia + 2*planet dia). So the outer has 2*sun's teeth = 200 teeth. Unless I'm missing something.

Both outers must have same number of teeth, unless you're going to change some of the other diameters etc. If both have 200 teeth I make it ring gear B goes at 150 RPM. Though it made my brain ache working out the formulas and I could be wrong

Okay fine. The sun gears have 100 teeth and the planetary gears have 50 and the ring gears have either 199 and 200 teeth or 200 and 201 teeth.

Happy now?

Do I detect a hint of sarcasm? A word of apology for an elementary mistake would be more appropriate.

This reminds me of a joke: What does a drunken shortstop have in common with the Ancient Mariner?

We'll let that marinate (pun intended) for a while.

He stoppeth one of three.

Okay? Apology accepted?

I don't know what elementary you went to that was covering planetary gearsets at that level but mine barely had covered all 23 letters of the alphabet by the time I was in 6th grade.

It seems fairly elementary that the planet gears have to fit the gap between the sun and the outer ring.

Seems to me if they both had 300 teeth they would both be stationery. Therefore, 1 tooth difference means 1 tooth movement of ring per 1 input shaft rotation.

Do you realize that "stationery" is home or office paper supplies and has no connection with "unmoving"?

No matter how hard you push the envelope, it is still stationery!

Responding to being off topic, the system is not impossible if some simple minor adjustments are made - the size of one of the gear sets would be 1/300th different and one might have to limit the planets in the 299 tooth ring set to just one. In this scenario, the gear ratio from one ring to the other is 300:299. Therefore, zero movement of the one ring equals one tooth movement of the other ring. As the input does one revolution it covers 300 teeth. Since the other ring has only 299 teeth, it will move 1/300th for this revolution.

Ring gear 300/100 sun = 3revs of shaft for the 299t ring gear to retard 1 tooth.

Good question.

Jim

Good point; it is actually one revolution of the planet carrier that would move the movable ring 1 tooth, not one shaft rotation. However, I don't think this is direct 3 to 1 (ring to sun) because the planet carrier is moving, which alters that relationship by 1 rotation, I believe? - 4 to 1?

Having a chance to think through this now, I do believe it would be 4 input rotations per 1 planet carrier revolution which would change the movable ring 1 tooth since the rings are 1 tooth different. For simplicity let's make the sun and the planets both 100T and put the rings back to 300T. If we rotate sun, planet carrier, and ring all CW together to 4 o'clock, the input shaft rotation would be 1/3rd CW. If we then fix the planet carrier and rotate the ring gear CCW 1/3rd back to 12 o'clock, we have put the ring back to its fixed position, the planet would rotate CCW one rotation, and the sun would rotate CW one rotation making its total rotation 1 and 1/3rd. Repeating these two steps two more times would put the planet carrier all the way around back to 12 o'clock and would result in the total of 1.333 x 3, or 4 total rotations for the input. In short then, 4 input rotations would equal one planet carrier rotation, and that would result in the second ring advancing one tooth - a gear ratio of 1196 (299 x 4) to 1.

Sorry, I can't agree with some of that, and unless I'm mistaken some of the calculations in Rixter's link #12 are wrong.

Using the terminology in the link. As number of teeth is clearly proportional to gear radius (and diameter) and only ratios are relevant, take radii as S = 100, P = 50, R = S + 2*P = 200.

My calculation goes -

For any rotation, the arc travelled by each point of contact between gears must be the same.

On the first gearset, for sun rotation angle T_s, arc = T_s*S. The arc moved by the point of contact of planets and ring also = T_s*S. This point is at radius R, therefore the angle of the contact point = T_s*S/R. A straight line can be drawn through the point of contact of planet and ring, centre of planet, point of contact of planet and sun, and centre of sun. So the planet carrier also goes through angle T_s*S/R and hence T_y = T_s*S/R.

The link says T_y = T_s*S/(R + S). This uses ( R + S ) ×T_y = R × T_r + T_s × S but I don't see where this comes from, no derivation is given.

As a check, taking an extreme case S = R, my formula gives T_y = T_s, which intuitively seems right to me, but the formula in the link gives T_y = T_s/2.

My formula gives T_y = 100rpm*100/200 = 50rpm.

If anybody can find something wrong with my argument I'll stand (or sit) corrected.

Quite possibly nobody is interested so I won't run on about the 2^nd gearset just now, but I make speed of ring 167rpm (had a rethink from my #14)

Also I don't see the significance of ring gear tooth numbers differing by 1. If the teeth and the other gears are all the same there can't be different numbers. Some posters seem to think it gives a differential effect hence high overall ratio, but if so I don't see how.

Firstly, rereading the initial scenario, I don't even think there are any binding issues with two full sets because it does not say the planet axes are coaxial; it only says the carriers are locked together. This allows for the possibility of changing pitch diameters, as I first suggested, without needing a parallel offset coupling, and in which case the second sun would be welcome. Secondly, I wish I wasn't on vacation. This scenario could be set up easily enough in SolidWorks and proven out. Thirdly, limiting the number of teeth to simple divisors, is like saying we can't make a 100t spur gear mate with a 299t ring gear. Ring or spur, it is entirely possible to create the setup, even though we could also just evaluate the scenario theoretically. Fourthly, if both sets were perfectly equal and one 300 ring is stationary, the other 300 ring would also be stationary. Loosing or gaining one tooth on the ring would have to force the second ring to advance or lag one tooth per one carrier rotation. I think the scenario should be set up in CAD with the second ring at 297t and the sun and planets at 99t to make things easier by keeping teeth equal to pitch diameters. In this scenario the second ring would advance 3t per carrier rotation. (Even though it is entirely possible to have 100t with a 99.6666 pitch diameter). Willing to "eat crow" if I am not correct on this.

Couple of quick points

The ring isn't 300, it's 200 (on the 1st gearset at least) see spades #9 and my #14. OP accepted this (grudgingly ) in #16.

I can't agree both outer rings are stationary if = diameters. The planet carrier of set 1 is rotating (though slower than the 100rpm sun). This drives planet carrier 2 at same rpm. If sun 2 were stationary outer ring 2 goes slightly faster than planet carrier. But sun 2 is doing 100rpm also, which must be added. That's how I get 167rpm.

Have a good vacation, be good to discuss more if you want when you're back.

It says the carriers of both sets are locked together and the Suns share the input shaft, both turning at 100 RPMs. If set a is identical to set b and both the Suns and the carriers are rotating the same, it seems to me that everything is identical if the rings were the same. Therefore, in the "all the same" scenario if the one ring is fixed the other ring is going to stay in place as well. Also, the only thing that changes by changing the planets from 50 to 100 is the number of times the sun has to rotate in order for the carrier to go around 1 time. If the one ring is fixed, one planet carrier will move the second ring the distance that is equal to the difference in the number of teeth between the two rings; whether the sun has to go around 2, 4, or 24 times. At least, that's my view of the situation.

From my understanding of the OP you do have it correct. It is not a real world scenario it is purely conceptual. In this question it is stated that the largest ring gear is stationary. In the real world this can happen in an auto gearbox. The question then is what happens to the second, free, ring gear. No more, no less.

Jim

Exactly!

It's conceptual puzzle so as far as I am concerned anyone can change any gear to have any number of teeth to suit their meshing and synchronisation scenarios just so long as the rotatable second ring gear has a tooth difference of at least one plus or minus in comparison to the fixed first ring gear and the sun gears are of the same tooth count in reference to each other and the planetary gears are of the same tooth count in reference to each other as well.

As for the planetary gears I never said that there were 3 on each carrier or that they had an equal spacing either. I just said that their carriers were locked together.

Please (also jlucas) ignore most of my earlier posts (if you hadn't already!). I still think it's elementary that the ring gear teeth must = sun + 2*planet, but I found it hard going to come up with the formula. I'm not as young as I used to be! Re-doing the calcs I agree with Rixter's link #12. Humble pie consumed .

Putting some figures in, I make it that if both gearsets are identical, the ring gear of set 2 is stationary. Keeping the sun gear of set 2 = set 1 = 100 teeth, but increasing planet (each) to 51 teeth, and the ring gear to 202 teeth, ring gear of set 2 does 0.221rpm, same direction as the sun gears. With planets 55 teeth, carrier 210, ring gear of set 2 does 1.075rpm. So there is a differential effect, as you may have guessed all along.

I think I've got it right this time, but any comments…..?

I will concede that we are going to need sun T *2 planet T = ring T in a conventional planetary system wherein you have more than 1 planet and the ring and sun are coaxial. However, I don't know that we are in agreement on the 2nd ring rotation. Analyzing the setup visually, I have to look at what would happen with a tangible model. Perhaps an easier way to visualize it would be to work with complete rotations, instead of the 1/3rd rotations I made earlier. In the first set (100t sun, 50t planets, 200t ring), completing one carrier rotation would look like this: rotate everything as one locked unit 1 rotation CW. Next keep the carrier in place and rotate the ring CCW 1 rotation. With the carrier fixed the sun would turn an additional 2 rotations for a total of 3 rotations. This translates into ring 0, sun 3, and carrier 1. If we do the same with 51t planets and a 202t ring, we get ring 0, sun 3.02, and carrier 1. In other words, if the sun moves 3 and the carrier stays at 1 (which they have to do since they are both locked to set A), the ring moves 2/100ths or 2 teeth. With 55t planets and a 210t ring, we get 3.10 for the sun, which converts to 10 teeth movement for the ring if the sun is constrained to 3 rotations per 1 carrier rotation. Backing the sun up 10 teeth with the carrier fixed will move the ring 10 teeth. I think if you draw the gears out and run through this simple 2 step process, you will start to see what the movement has to be. Still sold on the difference in ring teeth determining the second ring's movement per carrier rotation.

I follow some of that, but as far as I can see it doesn't give the speed of ring 2 as a function of numbers of teeth, which was the question. Can you elaborate (if you can be bothered )

I had trouble visualising and calculating the speed of the ring, when sun is fixed and planet carrier rotates, from first principles.

So I looked at the general case, giving following

1. Carrier fixed, rotate sun CW by T_s1. Ring rotates T_r1 = -T_s1*S/R (arc of rotation same for both, so angle varies as 1/diameter.)

2. Rotate the whole thing CW by T_c (no internal movement). Then sun rotation T_s = T_s1 + T_c , so T_s1 = T_s - T_c. And ring rotation T_r = T_r1 + T_c = T_c - T_s1*S/R.

Substituting for T_s1 and jiggling the algebra gives

(R + S)*T_c = R*T_r + S*T_s as in the link.

Can use this for any combination of inputs and gear sizes ring (provided of course ring gear teeth = sun + 2*planet!)

I just finished rendering out an animation for the 100t suns, 50t and 51t planets, and 200t and 202t rings. Everything worked out as predicted with the ring moving 2 teeth for 1 carrier rotation. I will try to upload the animation in a few days and provide a link to it. This should bring clarity to how the gearing configuration operates.

Here is an animation that shows how things work - https://www.youtube.com/watch?v=InLprFDZv-E (If you have any viewing troubles try using Google Chrome, Explorer would not play it for me.) You can also find it by searching "jeff lucas gearing" at YouTube.

As predicted - 3 sun rotations = 1 carrier rotation = 2 teeth rotation of ring B. Sun A and sun B at 100t, planets A at 50t, planets B at 51t, ring A at 200t and ring B at 202t. 2 teeth*101 = 202 teeth or 1 ring B rotation, which means one needs 303 (3*101) input shaft rotations to get 1 ring B rotation. In short, the gear ratio is 303:1 (input sun to output ring).

Ring B rotation is the same as the difference in number of teeth between ring A and ring B for every one carrier rotation. I don't believe you can get it to work (in reality) any other way.

We have a winner!

You get one of my rare GA points.

Not quite yet. Ring gear B in his answer is two teeth LARGER than ring gear A. Your original question had the second ring gear one tooth SMALLER. Also the direction of rotation has not been answered. As the original question is conceptual only and it doesn't require it to work in the real world, i still believe my answer at #24 correctly answers the original question as put.

Edit. Answer as at #23

Jim

What do you mean? You were simply responding to my answer 21, which was already on target with 1 carrier to 1 tooth. Plus, your response (not answer) at 23 is incorrect because in a 100t sun, 100t planet, 300/299t ring scenario the shaft rotates 4 times for 1 tooth, which I also said. The direction is determined by which ring is larger - fixed or moving. If moving is larger it goes forward and visa versa. The animation shows the direction for the larger moving ring scenario. If you look at my answers throughout you will see that I have had the answer very early on, but could not convince everyone without actually animating it. I am offended that you would try to take credit for my work.

Re-reading your comment at 23, perhaps I was misinterpreting it as a response to my earlier answer. My apologies if I misinterpreted. Nevertheless, your answer is incorrect at 3 input rotations with a 300t fixed ring and the animation shows the direction - forward for a larger moving ring. With a 299t moving ring, you would have to make the second sun 99t, which was also not as the initial question was stated. I think I covered the odd sizes (binding) issue as well with my comments about velocity and slipping along teeth as opposed to rolling across tooth surfaces. The initial scenario as stated had issues that needed to be addressed. I think we agreed that we were after the principle. If I had to animate something with 299t / 300t ring I would have had to either animated it with a 99t sun B or allowed for slipping, which would have been impossible to see and would have obscured the results.

Refer to my comment 41 regarding tooth rolling vs. slipping and resulting velocity fluctuations.

Refer to my comments 24 and 25.

I do sincerely apologize for being so emotional about this, but after all the work that I put into addressing and clarifying the issues as well as answering the question for any given scenario, I just don't think you can give a short incorrect answer that, additionally, doesn't address any of the issues that needed to be considered and steal my "time in the spotlight".

To summarize:

I correctly said 1 tooth, but incorrectly said 1 shaft rotation.

You incorrectly said 3 shaft rotations.

I correctly said 4 shaft rotations (for 100/100/300-299).

I correctly generalized that 1 carrier rotation determines ring movement according to the difference in ring teeth no matter what the other gears might produce for the input shaft rotation.

This gives us the answer for the general set up, no matter how many teeth the various gears have - one carrier rotation produces a rotation of ring B that is equal to the difference in teeth between ring A and ring B.

You disregarded tooth slipping (and velocity) issues, which in a sense produces a bit of a camming situation as opposed to conventional gearing.

I clarified and addressed the tooth slipping issues and built a non-tooth slipping animation to depict accurate operation according one of the conventional configurations, which was suggested by Codemaster. The animation shows the direction of movement (even though direction of movement was not part of the initial question).

I could have done a conventional configuration animation with 300t and 299t rings, but it would have resulted in either sliding tooth meshing or making sun B 99t.

This gives a clear and accurate picture of how this particular planetary gear configuration works. You can change gear sizes, which will change input shaft rotations but will not change the carrier rotation relationship - one carrier rotation produces a rotation of ring B that is equal to the difference in teeth between ring A and ring B. Additionally, if you don't want slipping you need to have ring teeth equal to sun teeth plus planet teeth *2. Having 100t/99t suns, 100t planets, and 300t/299t rings produces a 1 tooth rotation for 4 input shaft rotations (1 carrier rotation). Having 100t suns, 50t/51t planets, and 200t/202t rings produces a 2 tooth rotation for 3 input shaft rotations (1 carrier rotation), as depicted in the animation. If the moveable ring is larger it will rotate in the same direction as the input shaft; if it is smaller it will rotate in the opposite direction.

Actually, JIMRAT, I am not positive, but I think you may also be wrong on the retarding of the ring gear in this non-conventional tooth slipping scenario. In a conventional scenario the sun B and planets B would have fewer teeth going through engagement (sun being 99t), but in your tooth-slipping scenario the number of teeth for sun and planets are the same for both the A set and the B set. Seems to me that this might actually CAM the 299t ring forward, as opposed to moving in reverse as it would be for a conventional set up.

I really don't have time to do another animation, but as I think it through, I am pretty confident that if set A is 100, 100, 300 and set B is 100, 100, 299 the B ring will be forced forward in the same direction as the input shaft, which would be opposite the direction that it would go if set B was 99 sun, 100 planets, and 299 ring. The planet tooth positions for 100/100/299 would be in the precise position as for the 300 ring as they go around the system. Because the tooth span for a 299 ring is shorter it would force the B ring forward so that the planets could still cover a total of 300 tooth increments for one time around. In short, unmatched sliding teeth will not produce the same result as matched-distance teeth.

Ok, disregard my "pretty confident" thoughts in 55 and 56, I think it will actually pull the extra tooth from the other end and make the B ring go backward, the same as conventional gearing. The extra tooth in the sun gear is thus accomodated by the slight sliding from tooth to tooth. I think you are correct on this point.

Yes you are right and i was wrong in my haste at #23. The 299t ring will ADVANCE one tooth.

No, I was saying that you were right about the ring direction, you were only wrong about the number of sun rotations - it is 4 not 3.

Since I realize that it is quite difficult to visualize the camming effect of unmatched gear tooth spacing, I went ahead and created another animation: https://youtu.be/dMzqrvMfxPk - again use your google chrome browser. You can also find it by searching "Unmatched Tooth Spacing".

The animation has 10 times fewer teeth so that it is easier to see what is going on, but it closely parallels the 100/100/299-300 version. Everything being in multiples of 10 (in general) the input shaft rotation is the same - 4 times around.

See my comment 57 - I wound up agreeing with you on direction as I thought more about it yesterday. The new animation I did today confirms the correct direction.

Love the animation. I wasn't trying to steal your thunder i just disagreed with you. Here is some new thunder for you;- https://www.youtube.com/watch?v=uT3SBzmDxGk

I am glad you used a single arm carrier in your animation as it was with a single arm carrier that you can get the smaller ring gear to advance instead of retard. To do this the planet has to be fixed to the 300t ring gear's planet. Or one planet that is 2w where w is the width of the ring gear. In this scenario the 299t will be half a tooth out at 6 o'clock and from here it is easy to visualise that as the planet moves around the 300T ring gear it has to push the 299T forward to keep the grooves aligned. In this case there can only be a single planet. In summary; with free planets and a multiple spoked carrier the 299T ring does indeed retard. If the suns and planets are fixed the 299T ring will advance.

I learned something in this exercise. Thank you.

Jim

I think we both learned something in all this exercise; for me that includes not getting too uptight, in addition to better understanding gearing. Just to clarify, you will see small translucent gray dots around the perimeter in the camming animation. Those dots represent the fixed 30 tooth ring. At first, I also thought that there could only be one planet, but I don't believe that that is the case with the double planetary set scenario. I believe with two separate planetary systems, like the initial design, one can have many planets but they will be at odd angles because the ratio of the sun to 299 ring is so odd. If the planets from one set to the other set were coaxial we would be stuck with one arm, but in the initial design the planets of the two different sets only share the carrier, so the planets for the 300 tooth ring can be equally spaced, while the planets for the 299 tooth ring can be unequally spaced. I only put in a single arm because the same planet is meshing with both the 30t ring and the 29t ring. Separating out the planets would change the single arm limitation. This, of course, is assuming that the cramming teeth can be shaped such that the movement of the 299t ring is uniform not changing in velocity from tooth to tooth. Don't you agree?

BTW, incredible cello duet!!! Actually have a music composition degree, and that's the best cello duet I've ever heard. Not entirely unlike using a 299 tooth ring with a 300 tooth ring while keeping the planets and sun gears the same. (Trying to stay on topic.)

I fail to see why this answer got any GA votes as it does not comply with the request of tcmtech - post#38 in that the underlined section of his post below has not been met.

"It's conceptual puzzle so as far as I am concerned anyone can change any gear to have any number of teeth to suit their meshing and synchronisation scenarios just so long as the rotatable second ring gear has a tooth difference of at least one plus or minus in comparison to the fixed first ring gear and the sun gears are of the same tooth count in reference to each other and the planetary gears are of the same tooth count in reference to each other as well".

A look at the proffered answer will show that the planets have a different tooth count - planet A=50, planet B=51 - whereas they were required to be identical to each other.

I reiterate my earlier prediction #9 that the configuration as required by the question will not work.

He got a GA vote from me for putting forth the effort to try and prove his point of how it might work.

Anyone else who does the same to prove their point if they disagree with or can expand on his answer to show something different might get one too.

As for the gearing setup as long as the sun gears are locked to the same shaft and the carriers of the planetary gears (be it one set or a dozen sets) are locked together and the two ring gears are such that one is in a fixed position and the other can rotate but has a slightly different tooth count +- 1 - 9 or so teeth (Preferably and odd number if possible) I will consider anyones arguments for the systems workings to be valid from that.

As for answers gear ratio and or output RPM of the movable ring gear for or against the input shaft turning the sun gears at 100 RPM is fine with me.

Also everyone is open to peer review as well so my GA can easily be canceled out by anyone else's counter vote.

Fair enough?

If not go take some Ritalin and calm down a bit. It's just a frigging logic puzzle for crying out loud.

Gee...I'm glad I don't work on your product development team..This is at least the third set of different rules that you have formulated for the same project.

Unless you opt for a stepped compound set, the moment that you deviate from the rule that the ring gear tooth count must equal the sum of the tooth counts on the sun gear plus twice that of one planet, then the arrangement won't work no matter how many iterations of the rules you wish to produce.

Returning to the original question - but with a slight modification - sun gear A only is driving, sun gear B is free to rotate on the shaft (this will be explained later at * & **).

Sun gear A is driving at 100 rpm, and carriers A and B are coupled together with 50 tooth planets, ring gear A is locked in position (let's assume that it has the correct number of teeth = 200 rather than the posited 300), then both carriers will rotate at 1/3rd the speed of sun gear A, ie 33.333 rpm and in the same direction.

*Sun gear B will rotate in the opposite direction to both sun gear A and the carriers.

If you could then somehow force ring gear B (let's accept that it has one more tooth than ring A. ie B = 201) over planetary set B , which will be difficult as it has an incorrect number of teeth, then ring gear B will also rotate at 33.333 rpm as it will be tooth bound to planet B and will be unable to rotate freely around carrier B.

** Sun gear B will also be locked up by the incorrect tooth count of ring B and will be forced to rotate in concert with carrier B and ring B at 33.333 rpm. If it were to be fixed to the drive shaft then the entire assembly would be frozen with sun A attempting to rotate at 100 rpm but sun B both restricting that to 33.333 rpm and attempting to rotate in the opposite direction, something would surely break.

For your latest design requirement - allowing differing tooth counts on the sun gears or planets to accommodate rings with different tooth counts to produce a working arrangement - say sun A = 100t, planets A = 50t, ring A = 200t, sun B = 100t, planets B = 51t, ring B = 202t - then with ring A fixed and the both suns driving at 100 rpm, carrier A would try to rotate in the same direction at 33.333 rpm and carrier B will try to rotate in the same direction at only 33.112 rpm - as both carriers are tied together, binding will result.

Maybe that's why it's called "Park"?

I can't agree with your last paragraph. If ring B were fixed, carrier B would try to rotate at 33.113rpm (I or rather Mathcad, makes it) resulting in binding. But ring B is not fixed, and the set-up rotates it at 0.221rpm.

That's why ring gear B is not fixed and can freely rotate.

Ring gear A is fixed and has X amount of teeth. Ring gear B is free to move if it needs but has X (+-) 1 to 9 teeth.

Codemaster - Upon revisiting the problem, I am convinced that you are indeed correct. In my last statement I was forgetting that ring B was not fixed.

tcmtech - you cannot have any more or less teeth on ring gear B without also having more or less teeth on either sun gear B or planets B. All teeth in the set must be of the same size and pitch or binding will result whether or not the ring gear is free to rotate. Therefore the number of teeth on the ring gear is absolutely controlled by the number of teeth on the sun and planet gears, this cannot be changed, you cannot add or subtract a ring gear tooth at will.

See my comment 71. Conventional gearing limits tooth numbers to equal tooth spacing, but it is possible to have unmatched tooth numbers through less conventional methods. See my animation link at comment 59 and open it in google chrome, not explorer.

Your 2^nd para - agree entirely, some posters seem to have missed that point. Also, can't have ring 301 or 299 teeth without changing the sun to an odd number of teeth. If you just change planet teeth, ring changes by an even number.

Unless I've missed it, nobody has answered the OP's question directly - what is the rpm of ring B? Saying things like it turns 1 extra tooth doesn't give a figure I can easily understand! The nearest thing to an rpm for ring B I can see is in #45 by jlucas, for sun A 100 teeth, planets A 50, ring A 200, planets B 51, ring B 202. Says 303 sun A turns for 1 ring B turn, so for sun 100rpm, means ring B 100/303 = 0.33rpm, same as my figure below. That's different from my earlier figure, cock-up on the algebraic front I'm afraid !

Using the general formula (R + S)*T_c = R*T_r + S*T_s, I calculated ring B rpm for a number of scenarios.

For 1^st set, T_r = 0 so T_c = S*T_s/(R₁ + S).

For 2^nd set,

T_r = ((R₂ + S)*T_c - S*T_s)/R₂

= ((R₂ + S)*S*T_s/(R₁ + S) - S*T_s)/R₂

Taking sun 100rpm clockwise, and sun A 100 teeth, planets A 50, ring A 200, I get

planet carrier 33.33rpm

Planets B 51, ring B 202 - ring B 0.33rpm CW

Planets B 55, ring B 210 - ring B 1.587rpm CW

Planets B 49, ring B 198 - ring B 0.337rpm ACW

Planets B 45, ring B 190 - ring B 1.754rpm ACW

Some posters have worked on set A having sun A 100 teeth, planets A 100, ring A 300.

That gives planet carrier 25rpm and

Planets B 101, ring B 302 - ring B 0.166rpm CW

Planets B 105, ring B 310 - ring B 0.806rpm CW

Planets B 99, ring B 298 - ring B 0.168rpm ACW

Planets B 95, ring B 290 - ring B 0.862rpm ACW

I agree with your math and subsequent summaries, as well as what the tooth numbers need to be with conventional equally matched gear tooth spacing; that's why I did the first animation with the tooth numbers that I had. However, the point that some posters seem to be missing is that, in addition to conventional means, it is possible to adjust tooth widths and gaps in such a way that teeth do some sliding against each other and the 100/100/299 scenario can be accomplished (as is done in lobe pumps). Sloppy or not, my unmatched tooth spacing animation does not show any tooth intersecting/interferance, which was the point of the simple example. Therefore, allowing for sliding allows for this unconventional approach, in addition to a more conventional approach. If one is determined not to think outside of the box, I don't know that I am going to change that. Nevertheless, it is my view that 100/100/299 can be made to function by using non-standard teeth and letting things slide, as is done with many lobe pumps.

OK thanks, I was looking at it from a purely theoretical viewpoint, you no doubt know a lot more than me about gearboxes in practice (and animations).

Just a comment about lobe pumps, and eg screw compressors which have unequal numbers on the male and female rotors - they don't have a ring gear to contend with!

Actually, a fairly complete summary of the operation of the system with the same number of teeth OR different numbers of teeth was given in my comment 54. My comment 59 provides a link to how one can have unmatched gear tooth spacing without binding. And, my comment 70 provides a bit more clarity on how the camming animation relates to the original scenario.

I saw that animation. It's very easy to draw a sloppy model that looks like it will work, but actually manufacturing one to correct gear tolerances would not happen.

I already said it was non-conventional and the teeth would slide against each other. All you have to do is search images for lobe pumps and you will see many examples of unmatched gear tooth spacing that are being manufactured, sold, and used. The concept is entirely possible, whether or not you agree with it.

"I think I've got it right this time, but any comments…..?"

No useful comments from me. As I recall I got the questions wrong and didn't show up for class the day they reviewed them.

This was just the remnants of a odd old question that came floating up in my brain while thinking about gears that one day and just figured people of an engineering mindset might find it worthy of a few minutes of puzzlement.

BTW, in a real world scenario we would, of course, loose the second unnecessary sun gear and just drive a second set of planets with the first set of planets which would eliminate all the binding problems.

What makes you think the second set of planets will bind? The second ring gear is free to rotate. The OP question is how far and in which direction. I think you could work it out very quickly if you ignored the "real world" and thought of the question as a conceptual question only.

Jim

I don't think it will bind; but my initial misinterpretation was that the planets of set a were coaxially locked with the planets of set b, which would create some sizing issues with slightly different sized rings. But, this was a misinterpretation - only the carriers are locked, not the actual planet gears. So, I'm with you - no binding.

The OP did say that the sun gears were mounted on the same shaft, but he didn't say they were concentric. It's deuces wild now! You need to see Gillett Burgess's "Elliptical Wheels on a Cart" cartoon and poem to appreciate the possibilities.

The "binding" is more a question of how uniform the movement is from tooth to tooth. Spur gears are typically set up to keep the teeth rolling across each other in such a way that there is neither a velocity change nor slipping. (This of course gets a bit more obscure with helical/spiral gears.) This constant velocity rolling cannot happen if tooth to tooth distance is not the same for each gear. Therefore, the 100t sun, 50t planets, 199t ring scenario would require a slight bit of slipping and/or velocity change from tooth to tooth.

This also addresses the author's Gearing Puzzle #1.