# Zero Backlash Cylcoidal Roller Rack & Pinion



## strantor (Feb 20, 2020)

Good morning,
I would like to discuss an idea for a (hopefully)-simpler, -more capable, -more accurate, and -more easy-to-DIY linear motion drive that I have conceived (albeit not the first person to think of it). The idea is, for example in the case of a CNC router table or plasma table, to simplify the linear motion strategy by integrating the travelling rollers/wheels and the drive components into a singular assembly, such that the gantry pinion could ride on the fixed rack, and be supported by the rack. Where the typical involute rack & pinion would bind up if the weight of the gantry were placed on the pinion gear and forced down into the rack, this one would (hopefully) not bind up, but would roll smoothly as if it were rolling across a flat surface. I think it would be easy to CNC mill the rack out of flat stock, and likewise easy to cut the pinion; no special tooling or gear hobs required.

I've been tossing this idea around in one form or another for years. I've brainstormed, researched, and discussed it to such lengths that I can't reliably remember the order in which my thoughts were formed, but here's a brief timeline of how I _think_ I got to where I am now.

1. years ago I saw this video or one like it, of an academic physics exercise involving a tricycle with square wheels, rolling on a specifically lumpy surface, whose lumps canceled out the angles of the wheels








It seemed to me that a superior traction could be achieved this way, and could possibly be modified to have _absolute_ traction (zero backlash, zero slip) if modified to meet that need.


2. Sometime last year or the year prior, I became fascinated with the idea of using roller chain sprockets as gears, by wrapping a length of chain 360deg around a round piece of plate (welded in place) and then having a sprocket mesh with it. Then broadened the concept to include the idea of welding a roller chain down to a flat surface (rack) and having a sprocket drive against it (pinion). Then it seemed more economical (chain isn't cheap) to wrap a short section of chain around a plate (pinion) and have a chain-tooth profiled stationary rack. I discussed all of this on an internet forum, can't remember which one, possibly this one. I probably owe some of these thoughts to other people. It was either presented to me in that discussion, or around that time I discovered of my own accord: Nexen Roller Pinion System.







My idea is either the same as Nexen's, or very similar; I can't tell. Their rack seems to have flat tooth profiles, in much the same way as an involute rack has flat (trapezoidal) profiles. Maybe this is a knock-off of Nexen, but I'm hoping it's an improvement upon it: *cycloidal tooth profile*. A cycloid is a special kind of curve; not parabolic, not sinusoidal, not Cartesian, not catenary, not circular... none of those. It's its own kind of curve and it has no fixed radius.



			Cycloid
		




> The cycloid is the curve described by a point on a circle with radius _R_ rolling without slipping on a line (_D_)








Here is the cycloidal trace that would be left by a single roller in an application like the Nexen RPS:




Multiple roller traces:




Multiple roller traces with convergence:




From that last image, it should start to become apparent where the "cycloidal tooth profile" comes from. Here is a picture of the resultant rack:




Here is animation of how it would look if a pinion like the Nexen RPS were to travel over a rack with a cycloidal tooth profile (maybe it already does. but I think/suspect they're using flat tooth profiles):







I'm hopeful that the cycloidal tooth profile, being a perfect match for the cycloidal traces of the rollers, will enable the pinion to not only drive a gantry, but support the gantry as well, in the same way as the square-wheel tricycle. And that's part of the reason I suspect Nexen is using a flat tooth profile; if they were using a cycloidal profile, they should have a demonstration somewhere, of the pinion rolling atop a rack, bearing a load. Either that or my assumptions are incorrect and even with a cycloidal profile there would still be binding... But look at that trike! Its wheels aren't binding!

In any case, even if it can't bear a load, I still think it would be a superior system especially for us DIY types. Do we agree that it could be made in the home (CNC) shop more easily than an involute rack & pinion? Do we agree that it would be zero backlash and more accurate?

I badly want to build and try this, but my CNC is in pieces currently with a long road ahead. If anyone else would like to give it a shot, I would await the results with heavy anticipation. If there are any takers, I could provide a CAD file of the cycloidal rack per the desired dimensions.

EDIT: I forgot to mention. The pinion rollers would be on bearings; probably needle bearings. Maybe the endcap bearings of automotive universal joints would suffice.


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## homebrewed (Feb 20, 2020)

For a load bearing application you definitely want the pinions to be free to turn.  Otherwise you will have a large frictional component that will reduce the efficiency and life of the system.  Complications like that (requiring bearings and an axle for each pinion) probably explain why the concept hasn't been used.  And you likely will need to harden the rack and pinions.  But for lighter loads a plastic rack might suffice.

Neat concept!  And nice animation.


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## JRaut (Feb 20, 2020)

I'm certainly no expert, but I've done a bit of clock making/repairing in the past. Unlike the common involute gear tooth profile that we all know and love, the clock- and watch-making world has been using a cycloidal gear geometry for a very, very long time. And, interestingly enough, very small pinions are difficult to manufacture, so clockmakers have traditionally made "Lantern" style pinions using pins instead of teeth, very much like your rolling animation.

Even in those cases I describe, they are never made without backlash. So perhaps you're on to something, but I'm not sure using a cycloidal gear tooth geometry is the whole story.


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## strantor (Feb 20, 2020)

homebrewed said:


> For a load bearing application you definitely want the pinions to be free to turn.  Otherwise you will have a large frictional component that will reduce the efficiency and life of the system.  Complications like that (requiring bearings and an axle for each pinion) probably explain why the concept hasn't been used.  And you likely will need to harden the rack and pinions.  But for lighter loads a plastic rack might suffice.
> 
> Neat concept!  And nice animation.



The Nexen roller pinions do have bearings, and that is part of what I intended to preserve in my design.










strantor said:


> The pinion rollers would be on bearings; probably needle bearings. Maybe the endcap bearings of automotive universal joints would suffice.








you may be correct about hardened rollers and pinions. It seems to me that since the rollers would _roll_, there wouldn't be much friction to wear them or the rack down, but I could be wrong. 

Your point about complexity is noted and not argued, however with the commonly available parts (like U-joint bearings) the actual manufacture and implementation of the system should not be overly complicated I wouldn't think. The pinion would just consist of a couple of plates with boles bored and a central mandrel.


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## strantor (Feb 20, 2020)

JRaut said:


> I'm certainly no expert, but I've done a bit of clock making/repairing in the past. Unlike the common involute gear tooth profile that we all know and love, the clock- and watch-making world has been using a cycloidal gear geometry for a very, very long time. And, interestingly enough, very small pinions are difficult to manufacture, so clockmakers have traditionally made "Lantern" style pinions using pins instead of teeth, very much like your rolling animation.
> 
> Even in those cases I describe, they are never made without backlash. So perhaps you're on to something, but I'm not sure using a cycloidal gear tooth geometry is the whole story.


I forgot to mention the horology lantern pinion as another source of inspiration, but that is among them.









I have not seen one that is designed to mesh with anything close to a cycloidal profile. I suspect that isn't warranted in the application. In that application it would (I suspect, also no expert) only matter that the gear ratio and mesh is consistent; it would not matter if there is a few arc-seconds of play in it.


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## FOMOGO (Feb 20, 2020)

It could be designed to run in an oil bath. Should run for a long time with little to no maintenance, with very high accuracy I would think. Mike


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## strantor (Feb 23, 2020)

FYI I'm working on a python script that spits out cycloidal rack profiles parametrically for CAD/CAM. Details here. I will post the code here when it's complete. Here's how it looks so far:


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## Winegrower (Feb 23, 2020)

Maybe add in a meshing tolerance variable and track the input/output transfer function...there could be funnies at singularity points like the bottom of the tooth.


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## strantor (Feb 23, 2020)

Winegrower said:


> Maybe add in a meshing tolerance variable and track the input/output transfer function...there could be funnies at singularity points like the bottom of the tooth.


What would that look like? 
Could the same be achieved by specifying the roller diameter as .376 instead of .375? 
That would effectively trim 1/2 thou off of every dimension of the tooth profile. Or is there something specific that needs to happen at the crown and the root of the profile only?


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## stupoty (Feb 23, 2020)

Ave had A video on a zero backlash cycloydial gearbox thingy






Sort of seemed relevant


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