#### Re: Metric Transposing

john kling

Do machines of the electronic lead screw variety work well with those impossible to represent  precisely by gear threads? I take it these to have some limit of accuracy based on the motor steps.

From: Paul Alciatore
To: SOUTHBENDLATHE@...
Sent: Wednesday, December 11, 2013 4:19 PM
Subject: [SOUTHBENDLATHE] Re: Metric Transposing

Nelson,

>I guess what I'm saying is the 127 gear will work for only one pitch
>in metric. All the other threads would need there own set up of
>gears. Changing gears on a lathe is a pain to say the least. I would
>also mention the cost of all those gears.
>Nelson Collar

If you do not understand the math and haven't done it, then do not
make such statements. I have made and sold the 127/100 transposing,
compound gear and do understand and have done the math. I have an
Excel spreadsheet that details all (well a lot of) the possible
thread pitches, both using the transposing compound and without it.
It lists all common and some uncommon metric threads and the gear
train used to cut them PRECISELY. It is for use with manual change
gears, not a QC gear box.

It is all about ratios. Gears have a integral or whole number of
teeth: 10, 11, 12, 13, etc. They can not have a fractional number of
teeth (like 12.5) and they particularly can not have an irrational
(like pi) number of teeth. So the ratios that you can get with gears
can ALWAYS be expressed by fractions with whole numbers for the
numerator and denominator. ALWAYS! That is mathematically certain.
The international groups that are responsible for such things have
changed the DEFINATION of the inch to make it precisely 254/10 mm or
25.4mm. This is now an exact definition and all further decimal
places in this number are zeros. PERIOD. No rounding. No fudging. No
NOTHING. It is exactly precise BY DEFINITION.

The number 254 is not a prime number. It is equal to 2 x 127 and both
of these numbers ARE prime. So, when we are converting a thread pitch
(like the lead screw's) we must include a gear with 127 teeth or a
higher multiple of that number (254, 381, 508, etc.) But 127 is high
enough so it is generally the only one of these that is used.

gears can be in a QC gear box or mounted behind the head stock on the
banjo. And those additional gears will be different for each
additional pitch in order to get the correct, the precisely correct
metric pitch. But, for any metric pitch that is a rational number (a
number that can be expressed as a fraction with whole numbers in both
the numerator and the denominator) you CAN cut that precise thread
with an English lead screw and a 127 tooth gear in the gear train.
Yes, you do need a set of gears. And yes, they can cost some money.
But I have had good luck on E-Bay.

Inch while metric threads are set up in MilliMeters of lead. These
are reciprocal units: one is a length and the other is 1/a length.
When you use whole numbers or simple fractional steps in one of these
systems you get a different kind of sequence of steps than in the
other system. What this means is, it should be NO SURPRISE that a
quick change gear box that is set up for one system will not make a
neat sequence of common threads in the other. With a QC gear box you
have a hit and miss combination of threads of the other system that
are possible with a given set of gears leading into that box. So, if
you are using a QC gear box, you may have to change other gears in
the string before the box in order to get a complete set of metric
threads. More work, but it IS POSSIBLE for any standard and most non

Second, There are thread pitches that can not be precisely cut with
any set up on a lathe and these pitches are perhaps the origin of
your statement that some are not possible. They are NOT metric
pitches. They are the pitches needed for some worms. A worm gear in
English measure is generally cut with a diametric pitch (DP) which is
defined as a certain number of teeth per inch of diameter. But,
because the circumference of a circle (the pitch circle) is equal to
the diameter x pi and because pi is an irrational number, the pitch
of the worm screw is irrational weather expressed in TPI or inches of
lead. Therefore, since gears can only be set up for rational ratios,
it is not mathematically possible to set up any gear ration to cut a
precise worm. There are ways to calculate a gear train that is close
to the exact value, but it can not ever be mathematically precise.
This may be the source of your statement.

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