Bill, you typed your last post while I was replying to John -

You are getting confused with pitch diameter, and all the other measurments there are for cogs. They are very simple. The number of teeth. The other measurements are included for us so that, for example, I make gearboxes for my trains. I need to know the ptich diameters of gears so that I can drill the centres for positioning the gears the correct distance apart. I need to know the overall diameter of the gears so I can machine clearances. However, the working part of the gear is the number of teeth.

If you have  two gears, one with 10 teeth and one with 20, and mesh them, the small one has to turn twice to drive the large one once. If you convert the large one to a flat rack, the small gear, if it turns twice, will move along the rack by 20 teeth - it doesn't matter what physical size the gear is. If you then measure the distance on the rack between 20 teeth, that is the distance travelled. If we assume the teeth are uniform, which most are, then the centre of one tooth to the centre of the other will be the same along the rack, and through all the teeth, so wear or shape does not matter.

Yes, that is diametral pitch - or modular pitch which are derived the same way. The distance between the teeth or pitch for nonstandard gears is still a factor to contend with. Open out the pitch curve, pitch circle or 'rolling circle' into a rack and the pitch circle becomes a straight line but the distance between the teeth - pitch - remains the same. So as I was saying; that 16 toothed pinion at 16 diametral pitch has a pitch circle of one inch. It's the conversion to or opening out to a rack that becomes an indefinite figure - rounded to 3.1416 in the diametral or inch method. I brought this up because John mentioned a 16 DP, 16 toothed gear....

Do you recall the gear hobbing machine that a modeler in the UK designed years ago? The first one was built of pieces of barstock and structural shapes - angle and channel then castings were made of the machine and it was offered as a rough casting set. I still use the one I built back in the early 70's. I like gears, and the more angle of contact - the better I like them. I've made cutters for helical gears and cut them on this machine.

Yes - I agree with you, the ptich circle is a crucial measurement - but, if you open it out into a straight rack. if the pitch is 16 teeth per inch, then the distance between the tips of 16 teeth is 1 inch irrespective of where the pitch circle is (usually about 1/2 way down the tooth). This is because the tooth is roughly trianglar and therefore the centre point of the tooth will be in the same position for the whole of its depth.

The distance between the tip of the teeth when it is  in a circle or pinion will be in excess of this, but as you "straighten it out" into a rack, all what were the spokes of the wheel become vertical and parallel, and therefore all the same distance apart.

In the diagram you have posted this shows the distance between the teeth is regular - and is equal to the pitch circle of your cog. What is crucial in this debate is - what is the distance between the 16 teeth (on the rack)

In the diagram you have posted this shows the distance between the teeth is regular - and is equal to the pitch circle of your cog. What is crucial in this debate is - what is the distance between the 16 teeth (on the rack)

The distance would be for the 16 toothed 16 DP gear: 1 inch pitch diameter X Pi = total distance for one revolution of the pinion on the rack = 3.1416; so Pi /16 = .1963 (16DP) between teeth (same as it is in the pitch circle and pitch diameter).

If the pinion were 20 teeth with the same diametral pitch (16), the pitch dia would be 1.25" so the distance for one revolution of the pinion on the rack will be Pi X 1.25" = 3.927" and the same .1963 (16DP).

It is in very rare cases that these figures among DP gears will equal an even number.

The tooth is divided on the pitch line or pitch circle with the addendum =1/DP and the dedendum = 1.157/DP with a working depth of 2 X 1/DP and the clearance is .2/DP    ---     that's all I can remember.

Bill C.

".... then the distance between the tips of 16 teeth is 1 inch irrespective of where the pitch circle is (usually about 1/2 way down the tooth)."

No, that is incorrect. A 16dp gear with 16 teeth is 1" in DIAMETER. The CIRCUMFERENCE occupied by 16 teeth is pi*D, so 16 teeth occupy a distance of 3.14159.... inches.

Read BClelmens' post.

A 16dp gear with 16 teeth is 1" in DIAMETER

.................has a pitch diameter of 1".....not th OD of the gear.
 RC

John,

Now with all this gear stuff straight in your mind, go back to Jim's first response post to you and it will all fit. Use what you know about these gears and racks and do the math that Jim explains.

Bill C.