Miter Gear Cutting

Ray C

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Ok... Planning to do this in a few parts and am writing this as I go. The goal is to show the nitty-gritty basics about a miter gear and how to make one with a standard involute tooth surface. The intention is for folks to understand enough so when they open Machinery's Handbook and open the section on gears, you can follow along and then advance to more complicated gear types.

The tutorial on basic spur gears has most of the basic definitions and it's best to read that first before digging into this. https://www.hobby-machinist.com/threads/cutting-gears-diametral.69653/#post-584129 The format of this will be the about the same... A few "classroom lessons" with definitions and terms, followed by actually making the gear. I won't be able to start making the gear for a few days due to my FT job getting in the way of fun.

Part 1 ....

A miter gear is a special case of bevel gear. A miter gear connects shafts that are at right angles. A bevel gear can connect shafts at arbitrary angles (but there are some practical limitations).

FYI: For this example, we're talking about straight-tooth gears. The configuration of the teeth (curved etc) does not change the fact that the gear is a bevel-type gear.

A pair of mating miter gears can be different sizes but must be the same diametral pitch (for US sized gears) or modulus (for metric gears). This rule is true for all types of gears.

Diametral Pitch has the same meaning as for spur gears. If the gear is defined to be a 16 DP gear then, for every inch of diameter there will be 16 teeth. As diameter increases tooth count increases according to that ratio.

See the basic picture of the major features.

The yellow line is the Pitch Line. Since miter gears connect intersecting shafts, the Pitch Line of a miter gear is 45 degrees. The Pitch Line is with respect to the blue line which is the centerline of the gear and/or axis of the shaft. The Pitch Line does not correspond to any physical feature of the gear. Even though this could be called a 45 degree gear, there is no physical component that is 45 degrees.

The green line (surface line) is on top of the surface of the Addendum. The angle with respect to the pitch line is the "Addendum Angle" and everything above the yellow line up to the green line is the "Addendum" part of the tooth. The Addendum angle is usually a few degrees (we will calculate it later). The metal gear blank must have a "Surface Angle" that equals the pitch angle (45 degrees) plus the Addendum angle.

The red line is the "Cut Line" and rides at the bottom of the tooth. The angle between the pitch line and the cut line is called the Dedendum angle. This angle is not numerically the same as the Addendum angle. The milling cut angle will be "45 degrees minus the Dedendum angle".

Notice that all the lines intersect at an imaginary point at the top of the gear. The physical characteristics (defined by the surface and cut lines) have different angles. As a result of this, the tooth depth of a miter gear is not constant along its length. Look a the red circles and you can see tooth depth is different from top to bottom. The term tooth depth and "Whole Depth" will come-up later so be apprised, when measured, it is the depth at the deepest end of the tooth.
Pic1.JPG

Some definitions for Part 1:

Just like with the spur gear, the Addendum = 1 / Diametral Pitch. This is physically measured from the pitch line to the top of the tooth at the tallest part of the tooth (at the OD of the gear).

The Dedendum for bevel gears depends on which standards the gear is being made to. There are multiple variations and usually, gears made by different standards should still mesh but, the clearance at the root and addendum might not be suitable for a given application. In this example the dedendum is being assumed to be the classic model shown in Machinery Handbook. Dedendum = 1.157 / Diametral Pitch. This is measured from the pitch line at the tallest part of the tooth (at the OD of the gear).

If you stick to this standard (which is recommended when milling gears manually) the Whole Depth = 2.157/ Diametric Pitch or Whole Depth = Addendum + Dedendum. The values will come out the same if you stick with this convention.


Almost all of the physical features are now defined. Next, we will define a few intermediate terms that help make the math easier. After that, we'll show formulas to determine everything you need to make a gear. That is: the "Cutting Angle", "Stock Surface Angle", "Whole Depth", "Outside Diameter" and something called "Virtual Teeth" (which is used to determine which cutting gear to use).

Until we meet again...

Ray
 

benmychree

John York
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I have cut bevel gears, it takes a special cutter made for bevel gears, and it takes a minimum of three cuts to form each teeth, a central cut is taken through all the teeth, then the blank is offset to one side and the cutter is realigned with the small end of the tooth and a cut taken through all the teeth in turn, then the blank is offset to the other side of the central cut, and the process repeated until the tooth is the correct dimension at both ends. When the central cut is made, the tooth is way too thick at the big end. A standard involute cutter of the correct pitch will make a tooth space that is too wide at the small end, hence the need for a special cutter, made more narrow than the standard cutters. The amount of offset is a matter of experimentation , but a formula approximates it; even then, Brown & Sharpe's book on gearing says that some filing of the teeth may be necessary for satisfactory work. See Brown & Sharpe's book "Practical Treatise on Gearing".
 

Ray C

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I have cut bevel gears, it takes a special cutter made for bevel gears, and it takes a minimum of three cuts to form each teeth, a central cut is taken through all the teeth, then the blank is offset to one side and the cutter is realigned with the small end of the tooth and a cut taken through all the teeth in turn, then the blank is offset to the other side of the central cut, and the process repeated until the tooth is the correct dimension at both ends. When the central cut is made, the tooth is way too thick at the big end. A standard involute cutter of the correct pitch will make a tooth space that is too wide at the small end, hence the need for a special cutter, made more narrow than the standard cutters. The amount of offset is a matter of experimentation , but a formula approximates it; even then, Brown & Sharpe's book on gearing says that some filing of the teeth may be necessary for satisfactory work. See Brown & Sharpe's book "Practical Treatise on Gearing".

For manual cutting on a mill, here's a couple methods I'm aware of. One is a 2-pass method where the first pass is made with the cutter raised 1/2 of the chordal distance and the second pass with the indexer advanced to a half-angle followed by lowering the cutter 1/2 chordal distance.

There is also the 3 pass method where you take a center cut then 2 more passes (similar to the 2 pass method) moving the cutter 1/3 chordal distance up then down and 1/3 the angle offset. 2 Pass vs 3 pass is done depending on having indexing values that can accommodate all the angle settings.

Ray

EDIT: Fixed misspelling.
 
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benmychree

John York
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Yes, the two pass method is mentioned in B&S, but it seems that until the amount of offset that is necessary is established, the 3 pass is used as a starting point to establish that dimension. And then, the milling method can only create an approximate gear tooth shape, as opposed to generating methods, such as a bevel gear generator.
 
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tertiaryjim

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Have you looked at the use of a shaper setup to "Generate" the gears?
I think with some effort it could be done on a vertical mill but probably wouldn't be a good use of time. Too much effort to determine and connect the gearing.
I'm sure someone has done it on a horizontal mill.
Nice wright-up.
 

benmychree

John York
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I am sure that it is not possible to "generate" a bevel or miter gear by any "common" process by any common milling machine, whether horizontal or vertical orientation.
 

Ray C

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Have you looked at the use of a shaper setup to "Generate" the gears?
I think with some effort it could be done on a vertical mill but probably wouldn't be a good use of time. Too much effort to determine and connect the gearing.
I'm sure someone has done it on a horizontal mill.
Nice wright-up.

Hi there Tertiaryjim. Thanks...

Well, I don't have a shaper so that settles that... These gears are for a table-top project and I wanted to tackle making them myself.

Ray

PS: I did have a misspelling and will go back and correct it.
 

Ray C

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All,

Be apprised that many gears commercially available are approximations of the ideal tooth configuration. For example, if someone makes spur gears using cutters like the ones shown, you are not producing an ideal gear. Since each of those cutters can make a range of sizes, compromises are being made. The fact remains, such gears are commonplace yet, world keeps spinning. Same is true for bevel/miter gears.

The techniques I'm outlining and some of the terms I'm using come from very old books dating back to the late 30's and 40's. Regarding days-gone-by... The overwhelming majority of the lathes and mills people talk about on this website have "imperfect" gears in them.

Gear cutters.jpg

Ray
 

benmychree

John York
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Yes, that is correct for the usual type of "range type" cutters; although I have never seen one, the old books say that for closer work, "half number" cutters were available to more closely approximate "perfection".
I don't see how machining a bevel gear on a shaper could be any different than milling, the cutter moves in a straight path with either machine, so unless some convoluted accessory device was made to roll the blank, there would be no difference.
Later, I may post how I made a foundry pattern for a cast tooth miter gear, it is about 12" in diameter and about 1 DP, I made it for a local historic grist mill that dates back to the mid 1840s.
 

tertiaryjim

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benmychree
Your right about rolling the blank. Some people have posted information about it in the past.
 
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