Calculation of gear positions 2

[GFinCA]

I have a problem that has so far stumped our small project engineering staff. I have a gear train that comprises a pinion (pinion 1) that meshes with two idlers (idlers 1 & 2), that in turn mesh with an internal gear. Another pinion (pinion 2) also meshes with idler 1. Both idlers mesh with the same internal gear. The idlers, pinions, and internal gear have fixed centers. Pinion 1 drives idler 2, and Pinion 2 drives idler 1. Both idlers drive the internal gear. The internal gear is an arc segment. Due to space issues, idler 1 can become disengaged from the internal gear at extreme travel, so Pinion 1 is meshed with Idler 1 to provide timing of idler1 so idler 1 can mesh with the internal gear on return. We would like to split drive torque from the pinions to internal gear through the idlers, so don't want pinion 1 to transmit torque to idler 1. Therefore, our desire is to have greater backlash between pinion 1 and idler 1 than between pinion 1 and idler 2. Backlash between the idlers and internal gear are the same. We are pretty sure this can be done by altering the positions of pinion 1 and idler 2. We can mathematically determine the gear centers with equal backlash, but are scratching our heads about how to do it with unequal backlash. Is there a mathematical way, either closed or iterative, to determine the theoretical positions of the idlers and pinion for this configuration, or can it only be accomplished through trial and error? I have a sketch of the gear configuration that I can email, if desired. Thanks for any help you can give.

G

 

[diamondjim]

G,
I am quite curious why you are using the
system that you have shown. What is the
advantage? Are you using pinion number
2 to try to take out the backlash in the
system by having is run in reverse?
English Muffin,
You seem to be caught in detail, I think
G is asking for principles only. I do not
think gearguru is attacking you but is
focused on the problem. We all learn on
this forum. We all have opinions based
on our experience or interpretations of
that experience which may or may not be
correct. We all are in a learning mode.
I do admire your tenacity. You would
probably make or are a good researcher.

 

[EnglishMuffin]

diamondjim
Funny - I hadn't thought about what this thing was actually supposed to do. I'm as curious as you are.
And attacking me? Gearguru? Why, only the other day he gave me a star! You mustn't take my remarks too seriously -I have a rather warped sense of humor.
As far as learning goes - you got that right, I don't think a day goes by that I don't learn something, either on this site or somewhere else.
I think my last post is simply my best shot at answering the original question as I now understand it. I thought the problem was that I didn't get into enough detail ! It probably can't be answered to GFinCA's satisfaction without getting into the details!
Its not tenacity, by the way, - I type fast and I just don't have anything better to do right now!

 

[EnglishMuffin]

diamondjim :
Oh - by the way, on your "researcher" remark, I think "might have made" would be more like it. At my age, its rather bitter-sweet to be told that you "might make" anything! I sound young though, don't I? (I wouldn't want it any other way!).

 

[gearguru]

DiamondJim, E.Muffin:
It is probably my non-Oxford English, what creates problems. Sorry, in another language I could express it better, but you would not be able to read it... Anyway, here is my point again:
Let's keep the internal gear and both idlers stationary and in their theoretical tight-mesh position.
Now if we swing the pinion1 around the idler2, the pinion1 will also rotate around it's own center, because it remains in tight mesh with the pinion2. But idler1 can not move with pinion1 - it is held in position by the internal gear. Therefore if we want the clearance between pinion1 and idler1, especially on both sides of the teeth, WE ALSO HAVE TO MOVE IDLER1 (or idler 2 in the opposite direction).
This way (grounding the gears as mentioned above) it is the easiest to visualize what happens in this gear train when relocating the pinion.
I'll look at E.M.'s calculations, will post my finding later.
Have a nice weekend, gentlemen!
gearguru

 

[EnglishMuffin]

GearGuru :
I didn't post my derivation of that equation (too much DETAIL!)- its just a solution of the scalene triangle formed by the lines joining the centers of idlers 1 and 2 and pinion 1, taking into account the fact that the gears have to roll, as you say. Pinion 1 rolls like a planetary, and idler 1 just rolls in a linear direction. The roll angles are coupled by the tooth ratios. The pitch radii are operating radii, in other words the solution starts by assuming that GFinCA's existing equal backlash solution is good and takes it from there. It also assumes that the ring gear is actually a straight rack - which looks like a reasonable assumption for such a small increase in spread between idler 1 and 2. I'm sure that this solution, even if correct, could be simplified with further approximations - that delta^2 at the end could probably be dispensed with for example. Don't see any way of avoiding the trancendental nature of the equation (whatever it turns out to be), and an iterative solution - when you've got pure angles and trigonometric ratios all in one equation it would seem inevitable.

 

[EnglishMuffin]

GearGuru - one other thing - I considered idler 2 fixed and allowed idler 1 to move - I just found it easier to visualize that way. I think you are visualizing idler1 fixed and letting idler 2 move. But the solution should come out the same either way.

 

[gearguru]

E. M.
The equations are OK. I was ready to criticize that you did not take into account the internal gear, then I found your last posts. I took your word "EXACT" too exactly. But I agree that the solution is "good enough".
If I had to work on similar problem I would probably use the vectors locating the centers of gears and solve/approximate it in Mathcad.
I think we did enough for GFinCA, I hope that it was not his college project.
This is my last post in this thread.
gearguru

 

[EnglishMuffin]

gearguru :
Mine too I expect (is GFinCA still reading this I wonder?)- sorry about the "exact" comment - thought you might pick up on that after I wrote it - have to be on your toes all the time with you guys! Just possible that the equations might actually be simpler or more elegant if you used a true curved sector gear. Glad the equations weren't screwed up - hope solution converges for CFinCA - probably requires double precision. Unless it's come up before, I think I'm going to post a question about the pros and cons of mathcad, matlab, mathematica etc - don't know much about any of them.

 

[GFinCA]

Wow! I get back to work today and find my mailbox full of new replies to my post! Thanks for all the interest. I think EnglishMuffin has given me what I was looking for, a way to determine the locations through mathematics. We would prefer that the backlash be equal, but didn't want to complicate the problem too much, so thanks for bringing that assumption in. As to the application, the geartrain will be a high-reliability redundant drive system for an aircraft door. Both pinions are required to be capable of driving the internal gear independent of the other, hence the need not to have pinion 1 drive idler 1. As mentioned previously, the gears are intermeshed to keep idler 1 in synch with idler 2 when it leaves the internal gear. Of course pinion 1 can drive idler 1 when idler 1 leaves the internal gear, but this only happens during maintenance.

Thanks for all the help!

G

 

[Pekelder]

EnglishMuffin,
That's right, the elastic deflection was caused by dynamics, but the drawing helped to get a grip of the rotation recuired to get the CLUNCK in other words, to get an idea when things would go wrong, and what we could do to prevent is. Changing the first few teeth of the rack, giving it more 'backlash' could help (we ended up not trying sadly), because the pinion would then not collide head-on with the rack, but at an angle to the flank of a tooth from the pinion. To get this, we would have to grind the head of the teeth further backwards, reducing the amount further op the rack. Because of the elasticity in the system, the rack will cause both pinions to get back in mesh. (We hoped, but never proved!).

Regards,

Pekelder

 

[EnglishMuffin]

Pekelder:
So hopefully this won't be a problem for CFinCA, since his system is relatively stiff torsionally. - don't want to be responsible for any aircraft crashes, do we?

 

[GFinCA]

pekelder, EnglishMuffin,

EM is right that our torsional stiffness is high, not to mention our speed is extremely low (18 rpm at the idlers), so I can't see there will be any issue.

Thanks for the reply,

G