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
http://www.sfu.ca/~alan/shaper/gears1.pdf
and
http://www.sfu.ca/~alan/shaper/gears2.pdf
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.
Alan
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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 http://www.sdp-si.com/D190/HTML/D190T34.htm , 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.
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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, -tony
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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 Erwin Kirkland, Washington
Alan Rothenbush wrote:

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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
http://www.robotgames.net/Resources/Gears/gears.htm
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.
Involutes.
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.
http://www.ul.ie/~nolk/gears.htm#Rotating%20spur%20gears%20in%20mesh
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 (www.pumaracing.co.uk) 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.
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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.
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"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.
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Lichtman says...

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.
Jim
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snipped-for-privacy@d-and-d.com (DoN. Nichols) wrote in message

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

Alan,
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.
Jim
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In rec.crafts.metalworking, you wrote:

I understand, but that requires a separate (and expensive) hob for each DP of gear.
By using a straight sided cutting tool of appropriate shape (not too far removed in shape from a lathe tool used to cut ACME threads), the theory is that with appropriate rotation and linear motion, ANY DP gear can be cut, and all of it without the need to syncronize the cutter rotation to the blank rotation.
Alan
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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.
Karl
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Alan Rothenbush wrote:

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: http://www.metalwebnews.com/howto/gear/gear1.html
Good luck, if you ever figure it out please explain it!
Grant
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Olympia, WA
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I'm NOT handy with math ( wasn't in my long lost youth and have forgotten what little I know back then ) and so my question.
I'm plodding along though, and should get it eventually.

Yes I do.
Again, the theory goes something like this.
A rack is a gear of infinite radius.
The "spiral" that is the involute for the tooth of a gear of infinite radius is in fact a straight line.
So a gear can mesh correctly with a rack of straight sides.
Therefore, a tool of straight sides SHOULD be able to create a tooth of the correct shape.

As documented in a followup article, yes, guys are doing it on a shaper.
My plan is to take the mechanics out of it .. remove the need to create drums of just the right size (for each DP) and simply do it under CNC.
And yes, I do plan to CNC my shaper to do this (the job's half done right now anyway), although it's just as easily on a CNC HMill or VMill with appropriate shaped flycutters.

Here's one VERY simplistic method of doing this.
Let's begin with a mechanical shaper - ram motion by arm power alone.
Grind a toolbit a lot like an ACME threading toolbit, whose angles match the pressure angle of the gear you want make.
Put the gear blank in a rotary device. See the pic in the articles I quoted previously.
Line the toolbit up with dead center of blank.
Lower the cutter so it just touches the blank.
Move the cutter (table) left so it's out of the way.
Lower the cutter to the full depth of the tooth you want to cut, (See why you have to move the cutter?)
Move the cutter (table) back right until it JUST touches the blank.
Take a shaper stroke to remove .. darn near nothing.
Rotate the the blank 1/1000 of a degree while moving the table 1/1000 of an inch to right.
Take another shaper stroke. Now a TINY bit of metal is removed.
Repeat the above steps for a LONG time and a tooth with apparently curved sides appears. The curve is in fact composed of zillions of straight line segments, but by making the steps small enough, it is, for all practical purposes, a curve.
And people CLAIM that the tooth so formed IS an involute.
This I can't prove, and I'm not sure I'm smart enough to ever prove it, but I am going to build it and see.
Alan
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I've just started to subscribe, after buying it locally for a few years.
And yes, I love the English !
I also particularly like the different approaches some people take to their projects. There was one recent article with both metric and imperial dimensions on the same _piece_, and used several different kinds of fasteners; Whitworth, BSW (which might be the same as Whitworth), BSF, metric, UNC and UNF.
Man, that's a lot of taps to keep track of !
Alan
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"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.
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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
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As I was saying....
http://www.pathcom.com/~u1068740/gears.html
Richard Coke
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