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Author Topic: Irregular Gears for Keplerian Motion  (Read 416 times)
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Confused
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« on: December 07, 2018, 01:48:38 PM »

Hi there!  I'm new to the forums, but by way of a quick introduct ion, I'm a biologist turned digital artist doing some 3D printing projects so my interest in maths, engineeri ng and design is kind of as an enthusias t and hobbyist rather than in any kind of professio nal capacity.  As a quick heads up, I don't have a copy of gearotic motion (yet - if I do end up making things that require a lot of gears, it's on my wishlist) so my experienc e is mostly using Blender (which is both free and has an extensive API for scripting); hopefully you won't hold that against me. Smiley  I'm not terribly coordinat ed in person and don't have access to much in the way of tools and machining, so this project is going to be 3D printed, mostly using stereolit hography or laser sintering (so the shapes will be more precise than filament depositio n prints, but less precise than machined items e.g. cut using a lathe or router)

This is the third time I'm trying to write a post to this forum, because previousl y the process of writing out the issue in detail highlight ed the mistakes I was making - so thanks for inadverta ntly helping me along with this project already!

I'm trying to build an orrery that demonstra tes correct (or at least a close approxima tion of) Kepler's laws of planetary motion, roughly summarise d (for bodies orbiting the sun) as 1) bodies orbit as ellipses with the sun at one focus , and 2) the line from the body to the sun sweeps out a constant area per unit time - the implicati on being that a body on a noticably elliptica l orbit will move noticably faster at perihelio n (the closest approach to the sun) as it does at perihelio n (the point where it's furthest from the sun).  Anyone who's played Kerbal Space Program will know exactly what all this means, but if you're having trouble visualisi ng it, here's an animated gif of the orbit of Halley's comet; note the increase in speed as it passes the sun.



Most orreries assume circular orbits, which isn't actually so bad an assumptio n for the major planets since they all have fairly low eccentric ity.  Hence to show the effect properly, I'm trying to model a comet.  My initial thought was to model Halley's comet, but the eccentric ity is so extreme (0.967) that I'm worried it would jam up rather than turning smoothly, so I chose Tempel 2, with a much more moderate eccentric ity of 0.537.  I've worked out a fairly simple way of tracing out the elliptica l path using a modificat ion of a Tusi couple , but creating a system for turning the mechanism at the correctly varying speed has proven to be quite a head-scratcher; the relations hip between the input rotation (the Mean Anomaly in astronomi cal terminolo gy) and the output rotation (the Eccentric Anomaly, which can be used to drive the Tusi couple) is given by Kepler's Equation, which is non-trivial to solve to find the Eccentric Anomaly.  I won't bore you too much with the details, but after several months of tearing my hair out, poring over spreadshe ets and writing increasin gly complicat ed python scripts, I've finally managed to put together a rough approxima tion of a gear pair that should give the correct speed ratio:



Unfortuna tely, I anticipat e this gear having (at least) one major problem - as the egg-shaped input gear goes from about 12 o'clock to 2 o'clock, it seems like it's in great danger of coming unmeshed from the apple-shaped output gear and leaving it behind.  So I'm casting around looking for a way of keeping the gears meshed and turning throughou t the cycle, and preferabl y in a way which will also allow the mechanism to run backwards .  My first (and enduring) thought was to have pins strategic ally placed on the gears, and having an elastic band between them, so that there's tension in the output gear pulling it into that part of the cycle and keeping the teeth meshed.  However, in order to be reversibl e, I think it'd need it to be at apex of the gear, which might result in it getting pulled back in the wrong direction .  Plus, you'd have to drive the band to it's maximum tension at the point where the gear ratio is at it's worst, and I'm already a little concerned about the uneven input torque.  An alternati ve might be to mount one pin on the output gear, and another pin on another gear driven from the input shaft, but rotating in the same direction as the output gear, so that it's always going to be pulling the gear in roughly the right direction .



I've also considere d some other ideas, like having a set of circular gears driven by the same input, connected to the output by a pre-tensioned drive shaft that would be able to cope with the varying position of the output gear but would keep enough tension to have it chase the input gear when it's at risk of coming unmeshed; the big problem with that is that it's pre-tensioned, so will only drive the gear in one direction, and will tend to pull the gears free if it's run back.

I also feel like all of these solutions will necessari ly mean that it will be hard to "pause" the orrery in a set position; the tension in the system will try to pull it (forwards or back) to a particula r resting point, but I guess that's just something I might have to live with.  Although another idea might be to have some kind of structure along the rolling line of the gears (i.e. the shape without the teeth where the two gears meet) and then have a set of rollers under tension to try and hold them against each other.  I'm not sure how easy that would be to make, and whether it would risk catching or jamming the mechanism



Again, if it's relevant, the technolog y I'll probably be using will be some form of stereolit hography or laser sintering to make the pieces, so not as rough as FDM prints but also not as precise as hand-tooled metalwork .

Any other thoughts or suggestio ns?  Problems with the solutions suggested?  Obvious solutions I've missed?  Other things that would make this kind of gear untenable?
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ArtF
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« Reply #1 on: December 07, 2018, 11:23:18 PM »

Hi:

   So these are two gears that are centrodes, functiona l gears with that curve describin g them? If so, I
dont think youd have a problem with them falling out of mesh, if the elliptica l is too high it can happen,
but it looks like your pressure angle never gets too acute, nowhere near the fallout of mesh point anyway..

  Gears fall out of mesh when the pressure angle gets too high, it doesnt look like these do.Am I missing
something? Should run backwards as well since the shafts dont move during rotation.

Art
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Confused
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« Reply #2 on: December 08, 2018, 10:49:21 AM »

I'm not 100% sure what "centrode" means, and googling has left me more confused - I think probably "yes", but there may be some subtlety to what you're asking that I'm missing!  The idea is that with a constant input over time, the rotation of the output gear should vary following a curve like this; I calculate d an instantan eous gradient (based on small deviation s in input, since there isn't an algebraic solution for this curve) at points along the curve, which allowed me to work out the required gear ratio at each point.  I generated a set of reference points at equal angular intervals around the input, and then used trigonome try to work out firstly the curve for the second gear, and then points based on distance around the circumfer ence.  The gears should have effective ly equal circumfer ences (there's a little error because of the approxima tions used, but they can be made arbitrari ly small by using more reference points and smaller fractions - the limits of float precision notwithst anding).

The involutes were generated by taking a "flat" gear with a 20? pressure angle, and sort of simulatin g it being on a tangent line rolling along the surface of each gear.  Strangely, that gave unexpecte d results for the flattened part of the output gear for reasons I couldn't quite figure out, so a few of the teeth were generated by using the input gear as a template.

At any rate, thanks for the reassuran ce that falling out of mesh isn't as much of a danger as I was worried about.  I'm still a little concerned because I'm not sure how close the tolerance s are going to be when I print it out, so if there ends up being slack in the gears that might still be a problem, but I think that might end up being a depthing issue rather than anything that needs a full Heath Robinson solution.
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ArtF
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« Reply #3 on: December 08, 2018, 03:43:04 PM »

Hi:

>>I'm not 100% sure what "centrode" means, and googling has left me more confused - I think probably "yes", but there may be some subtlety to what you're asking that I'm missing!

  A cedntrode to a gear is a mating gear calculate d so the two gears shafts stay the same distance apart during rotation. You can have gears whewre the shafts have to move apoart and get closer together during rotation. These are non centrod gears. You can enter your numbers in Gearotic under functiona l gears to see a gear/centrode pair of the angualr relations hips your showing. Fucntiona l gears are designed for that. I dont think mesh is an issue in this case so long as the shafts dont move as part of the design.

Art
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kit
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« Reply #4 on: December 21, 2018, 07:22:41 AM »

Confused,

Basically you need to buy a copy of Gearotic and be amazed at how much you get for so little. If it doesn't do exactly what you want Art might just modify it until it does. All at no extra cost.

Kit

PS  Merry Christmas Art. I hope I haven't just dropped you in it  Grin
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ArtF
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« Reply #5 on: December 21, 2018, 07:43:46 AM »

Hi Kitt:

  Mery XMas back at you. I live my life deep in it, so dropping me there isnt an issue. Smiley

Only a few days to Christmas, where does the time go? Im almost finished a rewrite of Vexx's database
because it was driving me crazy on large files, and the new Guiloche wizard is about ready for release.
Heres a couple of laser burns I did a few days ago in testing..

Art
 


* 20181209_160415.jpg (3012.33 KB, 4128x2322 - viewed 56 times.)

* guiloche3.jpg (367.04 KB, 856x909 - viewed 44 times.)
« Last Edit: December 21, 2018, 05:06:13 PM by ArtF » Logged
kit
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« Reply #6 on: December 22, 2018, 05:11:00 AM »

You never cease to amaze me! I can see I need to upgrade my software and have a more serious play with Vexx.

Building a CNC router that actually works as well as I want it to and various other projects (plus having to earn a living) have kept me busy over much of 2018 but 2019 might actually see me complete a clock! I've spent plenty of time with Gearotic and CamBam, just need to get the machine finished to cut out the parts.


Kit
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ArtF
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« Reply #7 on: December 22, 2018, 08:51:05 AM »

Kit:

  In truth I didnt know what a guilloche was until a Gearotic user requested I take a look
at it. I knew the word but hadnt really consider their history. Im in his debt, they are
very interesti ng as a subject.
   This is my 12th year of a retiremen t where I promised myself Id learn something new
every day.( Did you know a spring gets physicall y heavier when stretched? I didnt.)

  Glad to hear your cnc is coming along. Its always fun getting that first cut! Try a ticker
first, youd be surprised just how much a clock will stretch your own springs.. (I wonder
if that makes one heavier as well? ..nah..)

Art


Art

   
 
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Nate
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« Reply #8 on: January 15, 2019, 10:09:16 AM »

...
I've also considere d some other ideas, like having a set of circular gears driven by the same input, connected to the output by a pre-tensioned drive shaft that would be able to cope with the varying position of the output gear but would keep enough tension to have it chase the input gear when it's at risk of coming unmeshed; the big problem with that is that it's pre-tensioned, so will only drive the gear in one direction, and will tend to pull the gears free if it's run back.
...

Any other thoughts or suggestio ns?  Problems with the solutions suggested?  Obvious solutions I've missed?  Other things that would make this kind of gear untenable?

You could look into a toothed belt-and-pulley system instead of gears.

Using helical or herring-bone gears may help with meshing issues and is pretty easy to do when you're 3D printing the gears.  You can also do more sophistic ated things to mess with the tooth form, though I'm not sure how much that can help in practice.

It seems like making a linkage that handles the orbits crossing each other is going to be a challenge .

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Mooselake
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« Reply #9 on: January 23, 2019, 03:46:25 PM »

Did you know a spring gets physicall y heavier when stretched?
...

 youd be surprised just how much a clock will stretch your own springs.. (I wonder
if that makes one heavier as well? ..nah.)

That's my new thing for the day Smiley

If I understan d right any increase in potential energy will cause an increase in mass so a clock will get heavier when wound.  It might only be a gazillion th of a gram, but it will be heavier.

Thought I'd have a CNC router down here by now, but never underesti mate how much work lurks in a new (to us) house and just how much time can disappear .  At least there's no snow in the forecast, unlike Mooselake North where it's 6 to 8 inches for tonight

Kirk
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ArtF
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« Reply #10 on: January 23, 2019, 04:55:20 PM »

Kirk:

   Exactly right. Kinetic energy is included when they calculate the energy of
a body and mass is proportio nal to that. As you said, probably a gazzillio nth
of a gram.  The kinetic energy added divided by Speed of light squared
is the increased mass. Extra weight would depend on where you are..

  >>Thought I'd have a CNC router down here by now, but never
underesti mate how much work lurks in a new (to us) house

  Having renovated three over years I feel your pain. Never again. Smiley

Art
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