Basic Gear Information

[KellyJones]

I get the impression that many folks on this forum find gears and splines a bit of a mystery. I am a gear designer by trade and weekend (relative newbie) home shop machinist. I thought I might be able to repay the many machinists on this forum for their sage advice and wisdom by posting small, digestible bits about how gears and splines are designed, specified and manufactured. If the members find this useful I will continue to post on a semi-regular basis, and try to answer questions as they come up.

Some of this may seem pretty basic, but bear with me.

Gears are usually specified by the number of teeth, diametral pitch (or module), and pressure angle. The number of teeth is pretty obvious, but what are those other two things?

Diametral pitch (or module) is simply a measurement of the size of the tooth. Diametral pitch is defined as the number of teeth divided by the pitch diameter in inches. (More about the pitch diameter later.) The module is the pitch diameter in millimeters divided by the number of teeth. As one member is won't to say, standards are so important everyone must have their own. Since I am most comfortable with imperial units, I will refer to diametral pitch, or simple "pitch" from here on.

The pressure angle is a characteristic angle that shows up in many gear calculations. The simplest way to think of it is as the angle of the flank of a tooth of a rack (as in rack and pinion) with a vertical line.

OK, so what is this pitch diameter of which you speak? Glad you asked. Imagine that we have two parallel shafts, and we want to turn one in the opposite direction from the other. The simplest way is to make two pulleys and connect them with a rope or belt. We draw a line between the two pulley center lines and observe that to get the driven pulley to go opposite the driver pulley, the belt had to cross the line of centers. (get yourself a piece of paper and pencil, I'll wait.)

The intersection of the belt and the line of centers is called the pitch point. A circle centered on either pulley and drawn through the pitch point is called the pitch circle. Notice that the pitch circle for each pulley is tangent to the pitch circle of the other pulley. The line representing the belt is called the line of action. Now draw a line through the pitch point and normal (perpendicular) to the line of centers. The angle between the last line and the line of centers is the pressure angle.

(The circles representing the pulleys are called the base circles, but as a home shop machinist, you probably won't ever deal with them.)

You have now sketched out the basics of a gear pair. The length of the line of centers is your gear mounted center distance. The number of teeth divided by the diametral pitch is your pitch diameter.

In my next post we'll discuss the geometry of the gears themselves.

[KellyJones]

One of the problems facing HSMs is, given a gear sitting on a bench, how do I order a replacement (or make a replacement)?

One of the first things you need to figure out is the diametral pitch. If you have removed the gear from an operating machine, it's pretty simple. The center distance is equal to the sum of the teeth on both gear divided by twice the pitch. Almost. In a well designed gear set, particularly with a ratio over 2:1, the designer may increase the center distance by about .002" or so the improve the performance of the mesh.

Measure your center distance, count the teeth, and calculate the pitch. For almost every machine you are likely to encounter, the pitch will be an integer number. Most pitches will be 10, 12, 14, 16, 18, 20, 24, 48, or 64. Even with increased center distance, your calculated pitch should be very close to one of these numbers. If it's not, get advice.

You can double check your work by measuring the outside diameter and root diameter of the gear. The OD should be equal to (the number of teeth +2) / pitch. The root diameter should be equal to (the number of teeth - 2.5) / pitch. There are exceptions to this, but I doubt you will run into them on most lathes or mills.

Ok, what about the pressure angle? Without specialized tools from a gear shop, it's difficult to measure. The vast majority of all the gears you are likely come across will have either a 14.5 or 20 degree pressure angle. If you are lucky, the gear will have "14.5 PA" marked on it somewhere. Most likely not, though. Look at the tooth. For a 14.5 degree tooth, the base of the tooth is nearly the same thickness as the tip of the tooth. For 20 degree tooth, the base is a little thicker. Machinery's Handbook has some silhouettes for comparison.

Next post we'll talk about the involute and how it's generated.

[NP317]

Interesting and useful information. Please continue.
Thanks.
~RN

[Harold_V]

Thanks, Kelly. While I can't speak for others who have worked in the trade, I know that my exposure to gears has been non-existent, so I welcome anything you can provide that would be useful in widening my horizons. Please do continue!

Harold

[tornitore45]

We can always use a gear tutorial.
The real purpose of this obvious replay is to get e-mail pings when the subject is updated.

[ken572]

Kelly

Thank You, for taking the time to tutor us on this subject.

Ken.

[KellyJones]

Thanks for the words of encouragement.

Last time we said we would discuss the involute and how it's generated. We're going to get a little theoretical here, so bear with me.

If we have two arbitrary cams, mounted on different centers, and use one to drive the other, the two cams will (hopefully) contact at a single point. We can draw a line through the point of contact that is normal (perpendicular) to both cams and that line will intersect the line of centers at a point. (If you don't recall high school geometry, you will have to trust me on this one.) The intersection of the line of action and the line of centers is called the pitch point (just like last time). If we want the output cam to rotate at a constant speed (with a constant input speed), the line of action at every point of contact must intersect the line of centers at a constant point. (You might want to draw this one.) This characteristic is called conjugate action. If the pitch point moves back and forth along the line of centers as the two cams rotate, the output cam will keep changing speed. Obviously, we want all gears to have conjugate action.

So for any arbitrary shaped cam, we can draw a mating cam such that the line of action (at every point of contact as the input rotates) intersects the line of action at a constant point. There are literally an infinite number of such combinations. Three of the better known shapes that satisfy this requirement are the cycloid, the Novokav, and the involute curves. Indeed, Swiss clocks were made for centuries with cycloidal gears.

The act of picking a shape for the first cam and calculating the shape of second cam is "generating" the curve. It is very common to pick a gear of infinite radius (e.g. a rack) for the first gear and using it to generate the mating gear tooth. We start by laying the basic rack on a circle representing the pitch circle of a gear such that the pitch line of the rack is tangent to the pitch circle of the gear. Then cut away all parts of the circle under the rack. Then move the rack a small distance along its pitch line, and rotate the gear in the same direction and by the same amount as measured along the pitch circle. Cut away the parts of the gear covered by the rack again. Keep going, and eventually the mating gear tooth profile emerges.

Here is a graphic:
http://www.uniontool.co.kr/products/mitsubishi.html

The drawing is from the perspective of sitting on the work piece and watching the rack go by.

Next post, we will talk further about the involute and why we use it.

[KellyJones]

So what does an involute curve look like? It's not a circular arc, or an ellipsoid, or anything else. Perhaps you've hear this one before. Take a can and wrap a string around it. Keeping the string taut, unwind the string from the can. The path traced by the end of the string is an involute curve.

http://search.yahoo.com/search?p=drawin ... ype=293224

Now of all the possible curves that provide conjugate motion, why was the involute finally picked? Well, there are three main reasons we like involute curves. First, all involute gears of the same pressure angle and pitch are interchangeable. (More about this in a moment.) Second, The basic rack has straight sided teeth. Third, involute gears are relatively immune to changes in center distance.

When we say involute gears are interchangeable, we don't mean it in the same way that parts of a machine are interchangeable. What we really mean is that any gear, with any number of teeth, can be cut with the same cutter as long as it is the same pressure angle and diametral pitch. (It will become obvious later that I am talking about cutting with a generating method here, not form cutting.) This is an important consideration for a gear manufacturer. Notice that it's called the American Gear Manufacturer's Association (AGMA), not the American Gear Designer's Association.

The fact that the basic rack has straight sides is also an advantage in building tooling.

When we say that involute gears are relatively immune to center distance changes, we are talking +/-.002" or so. Recall the reputation the Swiss had for precision in building clocks? They had to because they were using cycloidal gears, not involutes. Tiny changes in center distances messed up their conjugate action. Bad news for a clock.

Next time we will cover how most industrial gear are made, and then start talking about cutting tools most HSMs will need to make spur gears.

[tornitore45]

Beside knowledge, Kelly has a talent for explaining the real fundamentals.
Thanks

[TomB]

Kelly, your explanations are excellent and very much appreciated. Thank you. But now for a question.

I have seen the figure with the rack teeth forming the involute ever since my ME classes in the 60's but nobody ever explains how the figure extends to the thickness of the gear. Just based on analysis I can understand why the gear's axis and the hob's axis must be at the angle which squares the hob's pitch to the desired gear face. But then does the process require the hob to make a full pass across the face of the gear which would then be followed by a tiny rotation of the gear or does the hob make a full pass around the gear which is then followed a tiny move across the gear? It seems both tiny moves would have to be synchronized with the hob rotation position but what is the required synchronization?

Thanks Tom

[KellyJones]

Thanks for the comments folks.

Tom - I couldn't ask for a better straight man. If I understand your question correctly, that is today's topic.

We've talked about the involute, and the basic rack, but how does that allow us to cut parts (on an industrial scale)? Good question.

Hobbing is an example of a generating process, in that the tool mimics the basic rack, or another mating gear, and the tooth on the workpiece is generated from the kinematic action between the tool and the work. (Bear with me here.) Generating is a continuous process, meaning all the teeth on the work are created at the same time. (We'll talk about form cutting in another post.)

The hob contains several copies of the basic rack:

http://en.wikipedia.org/wiki/Hobbing#/m ... nsions.png

If you look carefully at the sketch of the hob on the right you can see the teeth look like a rack in cross section. In fact, each row of teeth looks like a rack. These teeth are offset to form a helix around the hob. If you spin the hob at speed under a stroboscope, it will look like the rack is moving from one end of the hob to the other. The hobbing machine spins the hob at the precise speed to cause the apparent left to right speed of the hob teeth to match the rotation of the work through dedicated gearing in the machine. Although the hob does not move left to right, it does plunge into the work piece, causing the depth of the tooth on the workpiece to get deeper. The hob also will move along the axis of the workpiece to machine any arbitrary face width.

Here is an excellent video of the process:

http://www.ask.com/youtube?q=gear+hobbi ... o&qsrc=472

The fact that the operator clamped the workpiece between two finished gears is a bit misleading, so ignore those. In essence, you will see that hobbing is much like running a worm on a worm wheel, in which the worm is cutting its mating teeth on the workpiece.

You should now be able to visualize how the rack (hob) removes a little bit from the workpiece, then the rack advances a small amount, and the work rotate the same amount (as measured on the pitch line), and the rack (hob) takes another bite, just like the pictures in your old text books. Do this continuously, and you have the hobbing process.

It should also be clear now why manufacturers like the interchangeability of the involute form. We can cut any number of teeth on the workpiece with the same tool by simple changing out the change gears.

Ok, so how do we cut internal gears, or external gears up against a shoulder (where I can't get a hob)? Well, recall that we can cut any conjugate tooth by arbitrarily selecting the tooth shape of the first gear. I said we usually select a rack, but we don't have to. If I cut Gear A with a rack, and sharpen its teeth, I can then use Gear A to cut Gear B, and Gear B will mate with any gear cut with the rack. This process is called shaping. Here is an excellent video of the shaping process:

http://www.ask.com/youtube?qsrc=1&o=283 ... ar+shaping

In this video the process is slow enough for you to see the generating action as the shaper and workpiece roll together. This is the exact same generating process as the hob. (On the hob, the teeth are wrapped around a cylinder and the movement of the tool is simulated by the virtue of the lead of the hob.) The shaper is actually rotating as if it were meshing with the work. As with the hobbing machine, movement of the work on the shaper is controlled by change gears.

Next post we'll look at how to cut gears in a home shop environment.

[tornitore45]

OK there is a rigid synchronization between hob and gear to be cut. As the hob moves axially a thick spur gear is generated.

How helical gear with "slanted" teeth are generated?
Is it possible to interpose a mechanism into the change gear train that advances the phase of the meshing relationship proportionally to the axial movement? Like generating a stack of identical thin gear but stacked with relative rotation one from the next.