Technical Question - Pressure Angle

I would like to experiment with making my own gears using essentially a flycutter with straight sides to form the curved shape.

For those who think I'm nuts, see

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The articles above describe doing this with a shaper, but there are many options along the same theme.

My interest is in CNCing the process, and to that end, developing a spreadsheet listing all the variables for all the possibilities.

From that spreadsheet, I can (manually) plunk the values into a CNC program and start making gears.

So, I've been reading (and reading and reading) all about gears.

I think I understand most all of the numbers, formulas, constants and variables involved, save one: Pressure Angle.

I have not seen, in all my reading, an explanation of Pressure Angle that makes any sense to me.

I've seen all kinds of mechanical drawings showing an angle, calling it "Pressure Angle", and labelling the angle (usually 14.5 degrees), with absolutely no indication what that angle represents or how it's derived.

Often, there is an IDENTICAL drawing beside it with the angle now indicated to be 20 degrees, the "other common pressure angle".

Now, my reading of gears tells me that the cycloidal shape of gear teeth means that a (section of a) circle contacts another (section of a) circle.

And my simple remembrance of geometry tells me that a circle contacts another circle at a POINT, whereas an angle is defined by two intersecting LINES.

A pair of points don't define any sort of angle.

So what's a "Pressure Angle" ?

Any clarification greatly appreciated.


Reply to
Alan Rothenbush
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"PRESSURE ANGLE (f), for involute teeth, is the angle between the line-of-action and a line tangent to the pitch circle at the pitch point. Standard pressure angles are established in connection with standard gear-tooth proportions. (Figure 1.1)", From

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found by plugging "pressure angle" into Altavista. Came with lots of other interesting-looking pages. If I read it correctly it's the angle at which the teeth meet, so a 0 degree pressure angle would imply a tooth that's square to the radius, and a 90 degree pressure angle would imply a smooth cylinder.

Reply to
Tim Wescott

you know, thats a really good question. though i'm sure there's a 'formula' for it, its probably wont tell you much more than the adopted standards (for p.a.) do.. that is, i dont think you have much choice given standard gear geometry.

if you make it any bigger, you'll get binding between the teeth (unless you drop every other tooth); smaller and it would be an ineffiecient gear design (poor performance, [in]ability to transmit power, etc).

from what i understand, the smaller angle (14 to 20deg) is traditional and used where backlash might be problematic. larger.. up to 25deg allows for higher loading.

for spur gears that is.

again, i think it the angles are a nature of the geometric requirements of having two gears intermesh. try drawing a few ... better yet, have your favorite cad program draw a few based on a short list in a spreadsheet.. and then just have a qualitative look at the results.

if you're fixed on finding an equation, let me know, as you've got my interest up. couldn't be too hard as there'll soon be a tradeoff between available contact area and powertransmission.

let us know what you find out,


Reply to

Keep reading, Alan. At the simple level, you can think of the two "common" pressure angles as analogous to coarse or fine threads. Beyond that, I bone up by reading Machinery's Handbook. As I recall, if you look at a picture of two correctly shaped gears meshing, they will meet at one point, but the line which is mutually tangent at that point forms an angle with the line between the gear centers, and this angle is the pressure angle.

I once bit the bullet and ordered a set of gear tooth gages from McMaster. You can get a pretty good intuitive feel by just looking at the different gages what's going on. Next time you're down near Seattle stop by and I'll show 'em to you.

Grant Erw> I would like to experiment with making my own gears using essentially

Reply to
Grant Erwin

Hmmm. It's an involute but I'll get to that later.

It's easier to understand if you have two meshing gears handy you can play with. Preferably not helical ones or it gets even more complex.

Place the two gears flat on a table so that they are in mesh. Lay the edge of a ruler across the centres of both gears. Now rotate the gears a bit until two teeth are touching each other exactly on the line of the ruler.

Where the two teeth touch each other is at the pitch radius of both gears. If you look at the contact point and place another ruler such that is tangential to both teeth you will find this ruler is at an angle to the first one. That angle is the pressure angle.

Now go to this site

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and scroll down to the diagram about backlash at the foot of the page. It doesn't cover it but it's a perfect diagram to understand pressure angle. It shows two teeth meshing under the line joining the gear centres. You can see the tangent to the contact point would point off about 20 degrees to the North North West. That's your pressure angle again.


The faces of gear teeth are shapes called involutes. You generate an involute by rolling a straight line around a circle. That's again almost impossible to understand without a diagram. So go here.

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Let it load and scroll down. You'll find a diagram showing how an involute is made and also stuff about Unwin's Construction which simplifies the process of generating involutes.

I think there's a better way to describe the generation of an involute. You can form one if you unwrap a string that's wound around a circle. If you keep the string pulled tight such that it's always tangential to the circle as it unwraps then the end of the string will describe an involute shape. Get a bar and a string and play with this in practice and hopefully you'll see how it works.

Now go back to the first site. You'll see a diagram of two involutes on base circles touching each other. The nice thing about touching involutes is they roll around each other rather than sliding against each other. Involute shapes mean gear teeth don't wear out so fast or generate high frictional losses. If gear teeth were sections of circles as you state above they would slide rather than roll.

Once you understand pressure angle and involutes for straight cut gears it's easier to understand their extension to helical gears which are basically just twisted straight cut gears.

Hope this helps you.

Dave Baker - Puma Race Engines

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I'm not at all sure why women like men. We're argumentative, childish, unsociable and extremely unappealing naked. I'm quite grateful they do though.

Reply to
Dave Baker

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I'll have a try, since the quoted definition found and offered by another poster is perhaps not the clearest of possible ones (comes from trying to be too technically correct for all cases, I suspect.

Let's start with a pair of gears in mesh. Draw a line between the centers of the two axles. Now, rotate the gears until a contact point at the pitch diameter (roughly half way from root to crest) is on that same line.

The Pressure angle is measured between that line and a line drawn tangent to the contact points of the two tooth faces.

That angle is not too self-obvious in the drawings, and they probably *did* use the same drawing, except perhaps the line itself. :-)

The greater the pressure angle, the more of the torque is translated into a force trying to separate the two axles (and gears). I'm not sure what the precise benefit of the higher pressure angle is, but at least one disadvantage is that higher stress on the bearing mounting points.

Perhaps the higher pressure angle results in less rubbing between the two teeth?

Just my thoughts on the subject.

Enjoy, DoN.

Reply to
DoN. Nichols

"DoN. Nichols" wrote: (clip) The greater the pressure angle, the more of the torque is translated into a force trying to separate the two axles (and gears). I'm not sure what the precise benefit of the higher pressure angle is, but at least one disadvantage is that higher stress on the bearing mounting points. Perhaps the higher pressure angle results in less rubbing between the two teeth?(clip) ^^^^^^^^^^^^ You are right about the advantage of lower pressure angles (generally 14 1/2 degrees.) The force vector is resolved into two components--the one tangent to the pitch circles does useful work. The one parallel to the line of centers is wasted, and, as you mention, creates useless bearing loads. The question is, why would anyone ever increase the pressure angle? Because a high pressure angle results in stubbier teeth, which can carry heavier loads. The contact between gear teeth usually alternates between double-tooth and triple-tooth contact. As the tip of a tooth rotates out of contact with its mating tooth, there is an interval where the load is carried by fewer teeth (usually two), before the next pair comes in and picks up the load. There is slight bending of the teeth under load, which can result in vibration. Also, as the teeth deflect under load, the meshing is slightly imperfect, so wear is increased.

The disadvangtage of too low a pressure angle is most evident on gear having only a few teeth, where the teeth become fat in the middle and narrow at the root.

Reply to
Leo Lichtman

OK, I think I finally got it. Enough different explanations to the same question and blink, the light finally went on.

MANY thanks to all.

But as always, one question answered results in another two questions asked !

I'm now thinking about this involute shape. I guess I always knew it was not a section of a circle, but preferred to think of it as such because the math was easier.

Question 1.

Is there a general formula to describe the involute shape ?

Question 2.

Referring to the shaper gear cutting article, using a drum that rotates and a blank that moves, with small cuts made by a flat sided tool, can this method produce the correct shape of tooth for pitches coarser than a rack ?

I'm willing to forget about the "undercut" of coarse teeth, as the undercut isn't really a working surface. It just seems too easy that a fixed "rotation to movement" ratio would produce this funny shape.

Without thinking about his TOO hard, it seems as if the shape cut using this method would be somehow related to a sinusoidal curve .. or is that what an involute actually is ?

Again, all the reading I've done describes the shape using a "ruler and compass" mentality as opposed to some formula to be plugged into something electronic.

( Could be because most of my books are at least 20 years older than I am and so looking for the Fortran code is kinda a waste of time. Heck, in one of the them, the involute is described as "new-fangled" and the author questions whether it will catch on !)

Thanks again, for all the previous help, and whatever future thoughts the group might have.


Reply to
Alan Rothenbush

Alan Rothenbush wrote in news:


The involute spline curves you are discussing are generally cut on a gear hobber. A multi-flute, straight-flanked helical cutter of the proper pitch and pressure angle is used to produce the involute form of the gear. Both cutter and gear need to turn at precise speeds for this to occur.


Reply to
Jim Pond

I think the pressure angle bit has been answered..

I've studied and collected books on the subject of gear making. The best I've seen is "Gears and Gear cutting" by Ivan Law. There's also a good book on gear forms with all the equations by Barber-Coleman. Ebay is about your only source.

Don Foreman is building a device for me that will allow gear hobbing on any bridgy type mill with a dividing head. The "brainbox" will drive the indexer to turn exactly one gear tooth for every 1 revolution of the spindle. A fellow named John Stevenson has a prototype running.


Reply to
Karl Townsend

It results in a tooth that is thicker at the root, capable of handling more torque at the cost of slightly less smooth running.

Reply to
Lennie the Lurker

If you're handy with math it isn't too hard to derive. A buddy of mine who used to follow this NG even wrote a program to generate points to plot on a 10X transparency which he used on an optical comparator to see where his shaper bit wasn't an involute, then he'd lap some more. In the end, he made shaper bits which he then used to make gears for some nonstandard lathe. I derived the formula - I'm sure you can too if you think a little. Of course, that only gets you a formula for one side - the "gully" in between and the flats on top aren't involute at all as I recall.

Surely you don't mean "pitches coarser than a rack". A rack is like a gear which has infinite radius. Anyway, there are machines which generate gears using a method sort of like the one you describe (gear hobbers) but I have not heard of anyone using a shaper to do this.

The Brits are big on making gears in home shops. I subscribe to "Model Engineering Workshop" and it has had many articles over the years. Of course, these are written in Brit-speak and also suffer from terrible editing as is the norm in those magazines. But there is some method of using circular approximations which lots of those guys refer to all the time which is supposed to be simple both to understand and to implement (on a shaper) but I'm damned if I ever figured it out. Ahh .. it's on the Web, written up by our own long-departed John Stevenson whom some will remember:

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Good luck, if you ever figure it out please explain it!


Reply to
Grant Erwin

"Alan Rothenbush" wrote: (clip) Without thinking about his TOO hard, it seems as if the shape cut using this method would be somehow related to a sinusoidal curve .. or is that what an involute actually is?(clip) ^^^^^^^^^^^^^^^ An involute is not a circle, or a sinusoidal curve, or any other curve but an involoute. An involute is a special kind of spiral. Here is how to visualize it: Put a circular object, like a tin can on a piece of paper. Wrap some string around it, with a pencil at the free end. Starting at the surface of the can, keeping the string tight, draw a curve on the paper. That curve is an involute. In a gear, only the very first part of the curve is used, so you don't see a spiral.

Now, this is how to understand the relation of the involute to gearing. A pair of gears in mesh can be thought of as two rollers, one driving the other by friction. Those are the "pitch circles" of the gears. They can also be thought of as two smaller rollers, having the same size ratio, but not actually in contact. These are connected by a crossed belt, the

*string.* As this string/belt unwraps from one of the rollers, a point on the string (think of a knot) traces an involute, and as it approaches the other roller and wraps on, it traces another involute. Since these two involutes are traced by the same knot, they are always in contact, and they are the two tooth surfaces! The angle of the string is the pressure angle, and the two circles are called the base circles.
Reply to
Leo Lichtman

Reply to
Jerry Wass

As a parametric equation: Let T = angle Theta. x = cosT + T sinT and y = sinT + T cosT

You were not entirely wrong in your assumptions about circular arcs being involved in gear profiles. Involute is only one of many tooth forms in use. Go here for a good description and animation of involute and cycloidal gears in operation.

Richard Coke

Reply to
Richard Coke

Interesting technical discussion. It may well be the only one going on here right now!

My intrusion was to ask yet another question along these lines, specifically how did gearing get standardized on one or two pressure angles? It would seem to me that manufacturers would each have set up their own processes with a different pressure angle, and no other ones would mate.


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Reply to
jim rozen

As I was saying....

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Richard Coke

Reply to
Richard Coke

Last December, several of us were using the priciple in those articles,

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generate gears on a metal shaper. There was a lengthy discussion on the Yahoo shaper group:
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Basically, the gear blank translates and rotates under a reciprocating cutter which has the shape of a rack tooth. This is a similar process to that used in a gear shaper, except that instead of using a rotating cutter with multiple teeth, the single cutter is used. One tooth space is cut at a time, so the process is agonizingly slow. But the process is fascinating to watch and very educational, and allows you to generate a perfect involute form with basic tooling.

Here's a picture of the setup:

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Good Luck,

-Bill Fill Olympia, WA

Reply to
Bill Fill

I may be missing something, but I believe that the reason this is often done on a shaper is that the shaper will cut the entire width of the tooth with each stroke. So that after each stroke you make the table and dividing head movements.

With a mill, it's going to take forever, because after each table/dividing head movement, you are going to take a pass on the other table feed axis.

It's a slow process on a shaper. It's going to be many times slower on a mill.

Unless I'm missing something.

John Martin

Reply to

The way I think of it is to think of an Acme thread first. The Acme thread has a 29 degree tooth angle. Use a tool bit for a Acme thread and instead of cutting a spiral, cut parallel grooves. These will mate perfectly with a 14.5 pressure angle rack.

I have used a method similar to what you want to do to make gears. Instead of using a shaper, I turned steel as above and then milled it similar to the way a tap is. Then I used it in a mill to cut the gear very much as you describe. It will produce the correct shape of gear teeth.

Reply to
Dan Caster

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