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Ken I

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Oct 31, 2010
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Cape Town, South Africa
Since this is a fairly long post, I'm going post the first part and the remainder as an attached PDF file.

Gear Cutting – Home Engineering Style.

It is possible using AutoCad or similar drafting package to develop you own custom (as well as standard) gear profiles.
From this you can also make your own cutters and make the necessary compromises to suit.
Involute Gear Profiles.
I am only going to deal with metric (Module) involute gears and I’ll leave it up to you to figure out that diameteral and circular pitch are merely variations of the same theme (metric or imperial).
A basic gear profile represents the circular tooth profile that would be generated from a “Rack” vis :-

A MAAG gear shaper in fact planes gears by using a rack shaped cutter – with each cut the “rack” is advanced and the gear rotated – in so doing it generates a true involute curve.
In my diagram above the green and red lines indicate the position of the cutter flanks as viewed from the perspective of the rotating gear in 2° rotation increments.
(A real MAAG cutter would have slightly taller teeth and corner radii to cut the clearance at the tooth root.)

The involute curvature is a constantly changing radius and is therefore difficult to make accurately (other than by generating using a MAAG or FELLOWS gear shaper – or using appropriately profiled cutters or hobs.
However in most cases a simple radius does very well – but there are limits.
The greater the number of teeth – the better the fit of a simple radius.
Typically for 16 teeth and over you will have no problem, 12 and lower tends to be problematical.
For smaller numbers of teeth you will find increasing incidence of interference and the need to use multiple radii which is somewhat outside the realm of home machining.
That said, for this tutorial I am going to make a pair of Module 1.5mm with the pinion being only 8 teeth and spur wheel 16 teeth. So the 8T is going to be problematical.
This was to repair a ride-on toy for my grandson who was delighted with the result (regardless of the slightly “grumbly” fit up of the gear. (Just to prove Grandad can fix anything.)
Basic Metric Gear Nomenclature.

For a standard Module gear the module is the height of the tooth above and below the centerline or pitch circle diameter – in this case 1.5mm so a tooth is approximately 2xM or 3mm in this case – there is also clearance which in almost all gear calculations is 0.138M or in this case ±0.2mm.

The pitch circle diameter is the module times the number of teeth (in this case Ø24.0) to which must be added a further two modules for the outside diameter (in this case Ø27.0)

The pressure angle for most gears is 20° (prior to WWII 14½° was commonly used – the greater the angle the greater the thrust between axles as the gears try to drive themselves apart – so in an attempt to limit this they tried to stay below the friction angle to eliminate this thrust – but with ballbearings and better lubrication and bearing materials this became less of an issue and 20° was adopted for its greater tooth strength and lesser interference problems with small tooth counts.)
What I Intend To Show.
I am going to show how to develop the tooth profiles using AutoCad (or similar).
Firstly I am going to approximate the spur wheel profile to the nearest cutter radius we might have to hand in order to make a cutter.
Then I am going to use the resultant compromised tooth profile of the spur wheel to generate the pinion gear profile and once again choose an approximate and convenient radius to make the cutter for the pinion.
Whilst I have done this to demonstrate home engineering capability, the method can be used to accurately determine wire cutting paths for wire cut EDM and the method can also be used for generating non-standard gears – for example for matching centerlines that do not equate to the normal pitch or making a gear with a deliberately oversized PCD vis :-

The above small pinion gear used in Slotcar racing – leaves far too little material to the Ø2.0mm driveshaft bore so the pitch circle is made larger – results in a non-standard tooth profile – but it does work.

Note: True involute gears “roll” over each other – there is no “sliding” of the gear faces.
When you deviate as above you do introduce sliding and therefore friction and wear problems – but it does work (see later).

First Step – Generate Spur Wheel Profile.
The illustration below demonstrates how to develop the tooth profiles using any CAD package.

Firstly draw the circular outlines of your gear and draw the rack adjacent to it vis:-
Copy a single flank (without clearance) from the middle of the rack crest to the centerline of your circular gear (In blue above).
Since this is a 16T gear, each tooth is going to occupy 22½°
We need to array rectangularly left and right at a pitch commensurate with a convenient angle.
I chose 2° of gear rotation which equates to 0.41887902mm of rack movement.

Next array a number of these lines to left and right (at this pitch) – I change the colour to keep track.

You could of course choose to array the flanks in say 0.5mm increments but then the rotation of each will be 2.387324° - you don’t want to have to type that in at each rotate command – so stick to single digit degrees and let the array command take care of the horrible divisions.

Next rotate each of these lines individually about the gear center - the first green line 2°, the second 4° the third 6° and so on- Repeat with the red lines -2° and so on.
Repeat until you have sufficient lines defining the profile.

Sooner or later the lines start to disappear away from the profile indicating where the tooth starts to disengage.

So now you have this :-


Next draw a three point arc – typically starting at the outer diameter intersect, the pitch line intersect and ending at the nearest inside line termination (black line above) vis:-
On measuring this radius I get R4.33mm – note gap – radius does not perfectly emulate involute – don’t worry at this stage we will (attempt to) fix that with the pinion.

Since I don’t have a R4.33 cutter (although I could spin one down in a tool and cutter grinder) I go for R4.5 or Ø9.0 milling cutter that I will use to form my hob (see later).
Next I redraw the arc using a SER (start, end, radius) method to draw the curve starting at the OD intersect plus any point you fancy and add the radius – this sounds a bit hit and miss – but play with it until you like the look of the fit – don’t worry if at this stage that it crosses some of our projected rack lines (interference) we will deal with that problem with the pinion.

Next add root and tip radii (if you want to) and complete the spur gear drawing.

For the rest, open the PDF.........


  • GearCutting.pdf
    1.2 MB · Views: 662
I use Gearotic Motion to generate the CAD files and then cut the gears on my laser. Finish and precision is OK for low load applications. There are several online gear generators available.

Busted Bricks - I have also had gears laser cut and it works fine for thin plate gears like you show. Waterjet cutting also works but is limited to M2 or larger tooth forms because of the 1.2mm diameter kerf. Machine must have a "dynamic head" to cut at an angle to eliminate "flare" on the cut.
Wire cut is best.
This for one-off stuff like we need - for production you're into hobbing, grinding etc. etc.
Regards - Ken
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Thanks for posting that, very clear format, that even I understood it.

Much appreciated.
there's a number of places where a stack of thin plate gears are used to increase load capacity there is also a method for spring loading the gears in opposite directions on a common shaft to make low backlash gear set ups
Just to show you can do really dumb stuff with a waterjet.....

Module 5 gear in 25mm thick black granite - started out with the intention of making a base out of it - but now I'm trying to figure out how to incorporate it into a working version of my improbability drive.
My brother suggested its for Fred Flintstone's car.
Regards, Ken
"Note: True involute gears “roll” over each other – there is no “sliding” of the gear faces."

This is a common misconception. This is only true when the contact point is passing through the pitch circle. Gear teeth faces must slide at points above and below the pitch circle.

See here: Classification of Types of Gear Tooth Wear - Part I : Gear Technology November/December 1992
"As indicated in Fig, the wear is almost nil in the pitch zone where the sliding speeds are low or nil, and becomes more and more pronounced a we move away from this zone. This zone is maximum at the tip circle and at the active dedendum circle, where the sliding speeds are the highest."

This uneven wear is partly why gear teeth lose their involute form and become noisy as the wear out.
rklopp - I had honestly never given that much thought - but its what I was taught (a long time ago) but logically since the volute contact point is always tangential to the pitch circle - then the thrust velocity can only match the pitch centreline velocity at the pitch circle, therefore at any other point it must be faster/slower therefore it must be at a vector to the pitch circle tangent - therefore there must be a lateral component vector at any contact point above or below the pitch circle.
There ! I proved it to myself logically - thanks for correcting my decades of ignorance.

Ah well I'm probably in good company - I can think of a few commonly held misconceptions that are as a result of teachers ignorance.

"even if you prove it to me, I still won't believe you !" Doug Adams - Salmon Of Doubt.

Regards, Ken
I have look making gear hobs.
Basically you need to a simple make a hob on the lathe.
A simple milling cutter would make first cut's
The hob would used for finishing the form of the teeth.

To further elaborate on rklopp's comments about gears not rolling and to further illuminate my own ignorance - I took to drawing the following vector diagram (for the gear example quoted earlier).

I have taken the point of contact at the plus/minus 1/2 a tooth rotation - which is as bad as it gets before the next tooth takes up the load and although this is a bad example (small tooth count), I was surprised at how large the sliding velocity actually is.
The pressure contact point is above the pitch circle for the spur gear and hence it's velocity is larger than the pitch circle velocity (1.1075 V) and the velocity of the pinion obviously slower (0.8725 v).
Their vectors are obviously tangential to the pressure point - so the difference between the vectors represents the difference in velocity between the teeth - that is the velocity at which they are sliding across each other.
Wow 57.5% of the pitch circle velocity.
Obviously the problem is more severe for small tooth counts (where the number of teeth in engagement starts to approach 1.

Dammit - now I have to re-evaluate all I have ever considered critical about gears to prevent relative motion between them.

Pretty much means you can do as you please as long as you have considered the resultant generated profiles will produce constant angular velocities.

(expletive deleted) Regards, Ken
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Pete, Obviously you are correct, but you got me to thinking is it a linear or other relationship ? - so I repeated the same ratio with half the modulus gear - so twice the number of teeth and the velocity came down by 41.4% - so not linear - looks suspiciously like square root of 2 minus one.

So its a geometric relationship (shocker).

Conclusion: Using the smallest tooth profile that will support the load might be the "rule" to consider - it will almost certainly run smoother and quieter.

Also based on what I posted earlier (in the PDF), its involute will more closely approximate a radius and thus be more suited to a home-made cutter.

Regards, Ken
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I have thought about making hobs for non standard pitches, if you look at Ivan Laws book on gears, there is a description of the Eureka a tool for making your own involute cutters for ‘one tooth at a time’ milling, I think the attachment could be adapted for hob making by feeding the cutter along the hob using the lathe screwcutting set up to the pitch needed, the tool would be a simple ‘v’ shape.
The problem with just screwcutting the blank is that there is no relief to the teeth, the Eureka deals with this.
Frankly, I have found that it is usually possible to find the right hob on the used market or to use the nearest standard and fiddle the blank size and depth of cut. I then test them by making a test rig consisting of a plate with two shafts at centre distance and trying the pair of gears altering the larger one until they run sweetly. For low low load applications this works well after all that is how traditional clock makers did it.
Going back to the Eureka, I would be most interested in hearing from anyone who has used one to make hobs. The other problem is choice of material, using HSS would be best but getting that heat treated here in the UK is prohibitively expensive so it’s back to silver steel (drill rod).

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