Hyperbolic Worm Gears

by jsteuben, published

Hyperbolic Worm Gears by jsteuben May 26, 2014


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50754Views 11095Downloads Found in Engineering


Worm gears find many uses in engineering applications due to their compact size and high gear reduction ratios. A disadvantage of this type of gear set is that only one tooth on the spur gear is engaged (or "in mesh") with the worm gear at any time. This causes high contact stresses which can accelerate wear.

This thing is a demonstration model of a hyperbolic (also known as an enveloping) worm gear. Both the worm gear and the spur gear teeth are cut from hyperbolic surfaces. This allows multiple teeth on the spur gear to engage the worm simultaneously. This reduces the contact stress between the gears, and allows higher torques to be transmitted through the gearset. A video showing the motion of the hyperbolic worm gears is shown here: http://youtu.be/O-MFFjBNCb8.

In this design 12 teeth on the spur are theoretically in constant mesh with the worm gear, although manufacturing tolerances result in less-than-perfect contact. The gear reduction is 72:1 (72 revolutions of the worm gear produce one revolution of the spur gear). Both the worm and spur are constructed using true involute gear tooth profiles. The worm gear is horizontally mounted on a pair of knife-edge bearings, and the spur is fastened to a vertical shaft using a c-clip.

This model makes an excellent hands-on demonstration of worm gears for a wide range of students. It can also be used as a challenge: computing and modeling the worm gear geometry is non-obvious. The solidworks files for the worm and spur have been attached for anyone interested.


Print one copy of all parts except bearing_cap; you'll need two of those. All parts should be printed in the default orientation with no support. A layer thickness of 0.2 mm is best for the involute and spur gears; the layer thicknesses on the other parts is not critical.

Four M3 (or similar) screws are needed to affix the bearing caps to the base.

The worm gear requires a bit of cleanup - a large pad is included to improve print bed attachment which should be trimmed off. Some of the gear teeth are printed with very steep overhangs, a little work with a sharp razor will be needed to get smooth gear rotation.

A 10mm socket or drill driver can be used to spin the gearset more rapidly. It is advisable to lubricate the knife-edge bearings holding the worm gear with graphite to avoid excessive friction.

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Is this a correct 72: 1 gear ratio?
I made a star tracker using your worm gear
However, when I used it, the rotation speed of the spur gear was doubled

The worm rotates normally, but the speed of the spur gear is fast
Worm = 1 turn in 20 minutes -> Spur gear = 1 tooth in 20 minutes

But I moved the spur gear one tooth about 10 minutes

you are sure about the gear ratio? In my opinion there 1:24. worm with a triple thread

can you please post the whole assembly as one STL? I would like to try and print it all at once!

Made one out of PLA, and to make things even more exciting, I scaled it down: 0.8.
Had to adjust speed, and cooling fan, but the wormgear came out pretty neat at second print. Of course there were overhangs, but I removed them with a knife.
At first it was a bit reeeeeally hard to twist, because of parts fitting too tight, but now I rotated it with a standard Nema17 stepper, and ran very smooth!
This is an extremely clever idea, best ever 3D printable wormgear possible. I love that the main gear has no backlash at all. It's steady as hell :)

This is amazing! Thank you for posting the Solidwork files! I LOVE how the worm gear was done! Rotating a gear around a circle with a twist, brilliant! I can't help but think something similar could have been done for the spur gear, rather that just cutting the teeth off.

I see now that this is not really possible because the diameter of the worm gear varies.

finished today printing it. Doesn't run as smooth as I expected :( it was very hard to turn. I connected a drill machine to it and it run nicely (although some PLA was flying from the worm gear). I then tried reverse, and for some reason the wheel tried to lift and the worm miss-aligned, and that was the end of it. :(
Any suggestions as to why this might of happened?

A little bit late answer to this. The worm gears are usually self-locking. You cannot use the the wheel to move the worm. Only the worm can turn the wheel.

That's actually a feature rather than a problem. Quite useful for locking the angle or position of the driven gear and no additonal mechanism needed to hold its place. Think of an electric motor adjusting a car seat recliner, or perhaps driving wheels on a rover that will stay parked when stopped on an incline.

I'm also trying to devise a simple swing-wing mechanism based on that locking aspect. Never thought of the hyperbolic gear with it's better engagement, not sure how I'd approach making that in Blender. Pretty cool though.

But yeah, only the worm does the driving.

could you also put all the original files. for example i don't like printing with support structure. but the bearing cap has 2 pins, while the base has holes for those. This means that the cap needs support to be printed correctly. if you switch the pins with the holes, (pins on the base) you can print everything without support :)
Support, even if done perfectly, unless it's the water soluble kind, will never leave a smooth surface.


This prints perfectly with no supports.

Hyperbolic Worm Gears
by MJHeed

Could you maybe post some of the calculations? I used to know how to calculate gears but I finished my education long time ago and I can't remember anything :)))
It would be nice to see how to actually calculate, at least the basic stuff. I need a worm gear as well and yours seems the best around, but it would be hard to modify :(

I Made one with ABS, works like a charm and I will be using it.
Could you remake it to use three small steel bearings instead of the plastic ones ? . :) Please.

Im working on designing one in SOlidworks, ill post here when its done(it will use 605zz's/623zzs)

Hey thanks, umm 605zz / 623zzs is that the bearing sizes in mm or model ??
I am using the first in a large marine fish tank to move the wave maker pumps back and forth slowly to simulate a more natural current. Over time the bearing situation would be better.

I could not print that. There are lot of issues.. But thank you anyway.

the upper part of the work printed badly, do you recommend printing with supports or is there anything else i can go for it to print better?

Thanks got it now:-)

Please upload the locking ring to fix the gear. Can't see it in the files!

This is probably a stupid question, but what is hybridbearing.stl for? Do I need that? I don't see it in any of the images or in the video. I'm assuming socket_driver.stl is for when you want to drive this thing with a drill?

I was wondering the same thing? Anyone know what this is for?

Hi Jsteuben,

i cannot open your solidworks files because i have 2010 version.. could you be so kind to send me parasolid files?

Many thanks

since the main gear has 72 teeth and the worm gear has 3 teeth (or whatever you would call having 3 threads on the worm gear) wouldn't the reduction be 24:1 ?

any idea how much torque these can handle before they break?

i am having trouble with slicing the worm gear all I get is the individual teeth but the rest of the gear doesn't print
any idea what I mitt be doing wrong?

Nice design. I adore the locking ring on the brown gear.

Brilliant design! Thanks for sharing and thanks for including the solidworks files. To someone like me being able to look back at the modelling stages is priceless. :)

Searched but couldn't find that article about threads but here's one showing force distribution.
This person claims: "...the first thread takes a third of the load, the first three threads take
three-quarters of the load, and the first six threads take essentially
the whole load..."
You no doubt already know all this stuff while I don't even know that it applies to gearing.
I apologize for hijacking the conversation and will serve my penance by printing this thing.

"... 12 teeth on the spur are in constant mesh with the worm gear..."
Are you sure all 12 teeth are making contact?
I was reading about drilling and tapping holes in metal. One crusty old machinist said something I'd never thought about.. something like "Regarding strength, a nut (or a hole) doesn't actually use more than 2 or 3 threads since regardless how many, only 2 or 3 make contact with the bolt's thread."
I suspect it's the same situation with gears, sliding mechanisms, and similar. One or a few points of contact carry the whole load at any particular moment in time.

Ah yes, this is a good question. In the mathematically perfect world of CAD, all the teeth are in harmonious simultaneous mesh. However, as you point out, due to manufacturing tolerances (not to mention the effects of layered manufacturing) only a few points are in actual physical contact. Here is where the argument made by the old machinist breaks down: Local deformation occurs at the highly-stressed initial contact points when torque is applied to the worm. This serves to redistribute the stress more widely across contact patches on the same, and on other, gear teeth. Even with this redistribution, the contact is not as complete and uniform as the CAD model would naively imply.
Basically, the argument made by the crusty old machinist neglects the elastic and plastic deformations present in bolted joints. I've actually worked with fasteners in critical applications where we torqued bolts beyond the point of permanent plastic deformation in order to ensure full-length thread engagement.
Regardless, the point about the importance of manufacturing tolerances is a good one, and I've edited my description a bit to make this more clear.

Deformation wasn't exactly neglected. In fact it is the reason only 2 or 3 threads (at the end of the bolt) make contact. The rest move away from contact with the bolt's thread as the bolt stretches under normal torque.
A bolt might be torqued so high as to stretch it even further to get more threads in "contact", but those new contacts would be on the opposing surfaces of the threads. I don't see where that exercise provides any mechanical advantage, not to mention the bolt being much closer to its breaking point.
This gear "neglects" a few things as well. For instance, (assuming it is perfectly formed) there's roughly 12X the friction losses a conventional worm suffers, and it certainly ignores the cost and difficulty of manufacture as compared to a properly sized standard gear.
.. but I really hate to cap on this design... it's beautiful. You know your stuff. I'm jealous..

Comments deleted.

Would there be 12x the frictional losses? If I remember my schooling friction force is proportional to normal force. So, if the load is distributed across more than one tooth up to the theoretical 12 teeth, I would think the friction per tooth would ultimately reduced.

Since the load is distributed among the teeth, it makes sense the
friction is also distributed. Would you say the cumulative frictional
losses of 12 teeth making contact is more, less or equal to one tooth
making contact?

Would there be 12x friction losses? Intuitively, I would think that the friction torque would be roughly equivalent, maybe even worse in the spur gear design...I happen to be working on a steering box now and was wondering if there was any advantage to cut the sector/rack gears for a hyperbolic mesh.
Anyway, great model jsteuben!

I wasn't speaking of torque. Shaft bearings take care of that. I was thinking tooth contact. Friction causes wear, and gears wear at the teeth because that's where the friction is. All 12 teeth in contact would seem to mean some amount of increased friction.
If the objective is to transmit more power without using bigger gears, then more teeth in contact will distribute the load. If there is more friction it is an acceptable price to pay.
If bolt/nut thread force distribution and gearing are related, and if 33% of a load is carried by one tooth while virtually 100% is carried by 6 teeth, then perhaps a 6 tooth spur is enough. It won't look quite as pretty but should get the job done.

Edits never appear on this main page so let me say my statement:
"A bolt might be torqued so high as to stretch it even further to get more threads in "contact", but those new contacts would be on the opposing surfaces of the threads. I don't see where that exercise provides any mechanical advantage, not to mention the bolt being much closer to its breaking point..."
is wrong ... except for the breaking point part.
I have no excuse for that.. just watching TV... otherwise sober and paying attention.

This is a clever idea, is this not used more due to manufacturing difficulty's?

Yes, that's correct. These things are quite expensive to produce, even using CNC machinery.