Miter Gear Cutting

[Ray C]

Part 2:

Just like the discussion about spur gears, part II will define all the terms of the formulas to define the critical aspects of the gear. We’ll also plug-in the numbers to get values.

One important value is the Pitch Cone Radius (abbreviated PCR). From the picture in part 1, there was a pitch line. The PCR is the length of pitch line from the end of the thick part of the tooth to the imaginary point where it intersects the shaft axis. This has no real physical measurement but is needed to solve many other aspects of bevel gears.

Pitch Cone Radius = Pitch Diameter / 2x Sin Pitch Cone Angle. (hereon: Pitch cone radius abbreviated PCR)

Addendum Angle = Addendum Amount / PCR and similarly, Dedendum Angle = Dedendum Amount / PCR (Note that these values are expressed as a pure tangent; thus, arctan is needed to get the actual value of the angle). These are the angles above and below the pitch line that outlines the addendum and dedendum.



There is also “virtual teeth” but, we’re going to leave that until we talk about the setup before making the gear.

For a 30 tooth gear with a DP (Diametral Pitch) of 25.4 (aka. Module 1 gear) here are the bare minimum values needed. There are other values that can be calculated but, they are more heavily needed to verify the gear once it is made.

Some plug & chug…

Pitch Diameter PD = No of Teeth / DP = 30/25.4 = 1.1811
Addendum = 1/PD = 0.0394
Dedendum = 1.157/DP = 0.0456
PCR = PD/2xsin PCA = 1.1811/2x0.7071 = 0.8352
Addendum Angle = arctan 0.0394/0.8352 = 2.7010 (degrees)
Dedendum Angle = arctan 0.0456/0.8352 = 3.1251 (degrees)
Angular Addendum = Addendum x cos Pitch Cone Angle = 0.0394 x cos 45 = 0.0279


Now, for the physical characteristics that you can relate to:

Cutting Angle: This is the angle to setup the mill and corresponds to the line in part 1. Cutting angle = Dedendum Angle - Pitch Angle
Cutting Angle = 3.1251 – 45 = -41.8749 (degrees) [the minus means it’s a line that cuts below the pitch angle].

Surface Angle. When you cut the blank, this is the angle of the tapered surface of the stock. Surface Angle = Pitch Angle + Addendum Angle
Surface Angle = 45 + 2.7010 = 47.7010 (degrees)

Outside Diameter. This is the diameter of the blank at the wide part. Outside Diameter = PD + 2 x angular addendum
Outside Diameter = 1.1811 + 2 x 0.0279 = 1.2368

Tooth Thickness (this is the thickness of the tooth at the pitch line at the wide end of the OD): Tooth Thickness = 1.571 / DP
Tooth Thickness = 1.571/25.4 = 0.0619 BTW: Tooth thickness is good to know now but really needed when determining how to cut the tooth.

And finally…

Tooth Depth: Depth of tooth at the wide-end (base) of the gear. Tooth Depth = 2.157 / DP
Tooth Depth = 2.157/25.4 = 0.0849


So, let all this soak in. Also, keep in-mind this represents the techniques used for manual generation of bevel/miter gears when standard involute cutters are used. The end result will produce an approximation of an ideal gear (just like any other gear that is made with cutters that cut ranges of teeth). If you want a perfect gear, use a CNC machine or a special-purpose gear machine.

I have not decided yet if I'll include another part that talks about setup or, if I'll skip right to making the part and include comments. FWIW, I'm on a special project at work and need to get-up at obscene hours in the morning to speak with people on the other side of the earth. Sleep is going to be at a premium for a while. It's going to be a couple days before I have enough time to write-up more about this.

Ray

 

[Ray C]

Alright then... I had a crazy week at work due to a crazy schedule and to make matters worse, several folks were away and all their work got funneled my way. Last night, I double-check all the calculations, calculated all the compound angles and trimmed the parts to size. Both of the blanks were heat treated to Rc 32. The mill table angles were also set.

Here's a quick pick of one of the blanks. I'm going to write-up another post on how to set the compound and mill angles because, it's very easy to make a mistake.



I'm going to make the actual parts now, then come back for the write-up and finish this off. See you in a couple hours or so....

Ray

 

[Ray C]

Home stretch. -No spoiler alert till we get to the end.


The gear blank has to be the proper size before cutting the teeth. The critical areas are the face width and the angle of the surface.

Earlier, the surface angle was defined to be the pitch angle plus the addendum angle which was 45 + 2.7010 = 47.7010 (degrees). That angle is with respect to the blue and green lines in the first post. This is NOT what you set your compound to because the compound angle is normal to the surface of what is being cut. The compound angle that’s needed is 45 – 2.7010 which is 42.299 degrees. Be really careful about this because it’s very hard to eye-ball the difference between 42 and 47 degrees and it’s an easy mistake to make.

The compound is nowhere near accurate but, after going through the triangle method of setting it up, it shows roughly 42.3 degrees. If someone needs to see the triangle method (again) let know. It's my cheap and dirty way of setting the compound. I'm sure I posted somewhere here at least a couple times over the years.



Double check! When measuring the part this way, you should get twice the value of the “textbook” calculated value (2 x 47.7010 = 95.4). Believe it or not, that protractor is amazingly accurate for the $20 it cost.



In order for a miter gear to function properly, the face has to fall within a certain range. The range is the smaller of:

Pitch Cone Radius / 3 OR 8 / Diametral Pitch. In this case the optimal face was 0.278. Here's the catch. The face length is not with respect to the surface of the part, its in relation to the pitch line. In this case, the actual linear distance was solved as a trig problem and because the angle is just a few degrees, it's only a few thou different. Anyhow, I cut the Z length down to preserve the OD and get the desired face length. This was done while it was in the lathe.




Part dimension are good. Now for the mill…

The cutting angle defined earlier was 3.1251 – 45 = 41.8749 degrees.



If you look at the top picture of the diagram, this time, the axis of measurement is the same as the axis of cutting so, you don’t need to “flip” the angle around as was done with the lathe. This is a top view of the mill table. The table moves back/forth in the Y direction to make the cut.


There are lots of ways to set angles on a mill. Here’s how I when a critical setting is needed. I’m all ears if anyone has a easier/faster and more accurate way of doing it.

Look at the second picture in the image above. To setup the angle, put a straight shaft in the rotary table and eye-ball it to 41.x degrees. Touch an indicator to the shaft at point A then, zero your DRO (or dials) then, move back to point B exactly 0.5000 inches. Then, move to point C just until the indicator touches. Since 0.5/tan 41.8749 is 0.5578, if the angle is perfect, the Y axis will read 0.5578 just when the indicator touches at point C. Adjust the angle and repeat as necessary. Last night, I had to do this about 10 or 12 times and then check it a couple more times after that. Things are going good if the process is repeated and the DRO is showing values within a thou of calculated.





For this, the gear the bore ID’s are 3/8” and I made a special “arbor” to hold them. Just a piece of junk out of the junk pile but, it was a very precise fit. Last thing I do is indicate and make sure it spins true. In this case, TIR was surprisingly good and within 1 thou (you get lucky sometimes).



In this case, with 30 teeth, it was just 3 cranks w/no extra holes. -How nice!


Here is something absolutely critical: The "Virtual" tooth gear cutter must be determined. This is a 30 tooth gear but, you cannot use the cutter for a 30 tooth spur gear. Virtual Gear Cutter = #Teeth / cos (pitch angle) in this case: 30/cos 45 = 42.4. Round-up the number properly then you must use the gear cutter for a 42 tooth spur gear even though this gear has only 30 teeth.

The centerline of the cutter was found using the same technique shown in how to cut spur gears. Before really cutting, I make a light test cut to make sure the line gets wider. It's hard to see in this picture, but it does get wider. This verifies the angle is set in the right orientation.


The table is cranked in until the cut depth is reached which was calculated in the first part as 0.0849.

If you do all the setup properly you only need to make one pass. When using the proper cutter and when setting the addendum and dedendum angle properly, you only make one pass. If the gear does not fit when done, you did something wrong.

So, buzz, buzz, buzz... It took about 55 minutes to actually cut both these gears.



When done, they needed just a little deburring. As expected, the HSS cutters had no problem at all cutting Rc32 metal.



Even though the outer faces of the gears are NOT 45 degrees, they press together perfectly and make a perfect 90 degree angle. That's because the outer edges (defined by the cut line) are resting on the axis defined by the addendum and dedendum line. In this picture, if I were to remove the square, the gears would stay locked together in a perfect 90 degree angle.



So, the only drawback to using generalized cutters for miter gears, is the very tops/edges of the gears endure some friction. If the gears were cut with cutters dedicated for that number of teeth, friction would be at it's theoretical lowest value.

So, that was fun... Hope you enjoyed the show. I wanted to get this over with and may have forgotten something. Let me know if I did.

Ray

 

[Ray C]

Oops... I just realized that a small detail was left out regarding cutting depth. As mentioned, the calculated tooth depth is 0.0849". That is measured at the rear of the tooth but, you have no real way to measure it because the physical geometry is somewhat abstract. Once setup, the tooth depth is generated by motion in the Y direction, influenced by a position set in the X direction. Solution: Math. This was not mentioned in the Machinery's Handbook because physical implementation is left up to operator. After a few minutes with pen and paper, it's obvious, the amount of X Feed once the part is touched off at the back end = Tooth Depth x Cos Pitch Angle. In this case: 0.0849 x cos 45 = 0.0600".

This, BTW, is just as critical as all the other precise measurements and setups in doing this. If you start messing around with manually tweaking the depth, it will be game-over and the gear won't hold the 90 degree angle when simply rested and balanced like this.

Those 2 gears have a happy spot where one just perfectly interlocks and rests on the other. When in-use of course, a couple thou of backlash space is needed.


Earlier, there was a discussion about off-setting the angle and making 2 or 3 passes. Upon digging into this and reaching back into the memory bank, those techniques are used mainly when hand-ground "fly-cutter" like shaping tools are used. Because the tools are usually cut at 60 or 75 degree angles (similar to a HSS threading tool), the offsets and multiple passes are needed for cutting gears.

You don't need to do that when using rotary DP or Module-type cutters.

Ray

 

[benmychree]

Read Brown Sharpe's book "Practical Treatise on Gearing" You will find that there is no mention of using fly cutters to cut bevel gears, and that special bevel gear cutters of the multi tooth type are used, and have a narrower point than ordinary involute gear cutters due to the geometry of a bevel gear as opposed to a spur gear, and that the only method set out is the rolling method to arrive at a shape of tooth that in usable, even then, filing the teeth must sometimes be resorted to for a smooth running gear. What you state in the last paragraphs is simply not true, do some reading by the people and companies who were the basis of the industry that we have today, not one's "memory bank".

 

[macardoso]

Ray,

Thanks for the amazing write up. I may have to venture into gear cutting yet!

Benmychree, I'd be interested in the 3 pass method you desribed, can you explain how to do this? Also is filing required for commercially made bevel gears? That seems expensive and labor intensive.

-Mike

 

[benmychree]

The best thing to do in reading about the three pass method is to resort to Brown & Sharpe's book that I referred to above, I think you could access it on line. As to filing regarding commercially available gears, it is not necessary, as they are made by a generating method that generates a correct tooth shape, something that a milling cutter can only approximate. Try searching "bevel gear generator".

 

[Ray C]

Read Brown Sharpe's book "Practical Treatise on Gearing" You will find that there is no mention of using fly cutters to cut bevel gears, and that special bevel gear cutters of the multi tooth type are used, and have a narrower point than ordinary involute gear cutters due to the geometry of a bevel gear as opposed to a spur gear, and that the only method set out is the rolling method to arrive at a shape of tooth that in usable, even then, filing the teeth must sometimes be resorted to for a smooth running gear. What you state in the last paragraphs is simply not true, do some reading by the people and companies who were the basis of the industry that we have today, not one's "memory bank".

Please read more carefully, I did not say fly cutters. -Read carefully!. Also, you are simply wrong about the only method set out is the rolling method -Period. Also, the technique I outlined is customized for the case of miter gears -not the generalized case of bevel gears. Once again, please think before commenting. If you are concerned this is coming from my memory bank, you'll be happy to know that absolutely everything outlined in these techniques comes from the 28th edition of Machinery's Handbook particularly the pages between 2086 thru 2090 and 2211 thru 2144. Rather than taking unsubstantiated reckless pot-shot statements about gear making, I took the time to take credible resources and distill the information down to a format understandable by someone interested in the topic. -And I urge you... please read the very first paragraph of the first post of this thread.

The implication that these gears are all wrong is amusing. The proof is in the pudding, the techniques outlined in my thread produced gears that work -and work very well. They were used in a test setup last week which is the beginning of another long term thread I'm working on that will outline how to make spirals and other forms. When I finally produce those spirals, show how they're made -and show them in operation, I guess I'll have to put-up with another post with wild pot-shots about how it's all wrong.

Ray

 

[benmychree]

I stand by all that I have stated, and quote from authority, you might do well to read their book.

 

[Ray C]

I stand by all that I have stated, and quote from authority, you might do well to read their book.

Likewise -and rather than just making farting noises in the back of the classroom followed by stating the names of some books, please take 10-15 hours of your personal time, to execute, photograph and write-up a live demonstration. And maybe it will be my turn to embarrass myself with some pot-shot statements at someone else's expense.

Ray