25-08-2017, 09:32 PM
Worm Gears and Worms
Worm Gears and Worms.doc (Size: 33 KB / Downloads: 27)
At the February meeting I discussed my 'hobbyist method' for making miniature
worm gears and worms. There seemed to be some genuine interest in what I said
so I decided to write a bit about it for those who may have missed some of what
I presented rather quickly.
BACKGROUND
First we need to discuss a bit of (spur) gear terminology and mathematics. In
the system used in America (metric gears are very similar but the nomenclature
is somewhat different), gears are characterized by:
N = number of teeth on gear (self-explanatory)
PD = pitch diameter (see below)
P = diametral pitch (see below)
OD = outside diameter of gear (self-explanatory)
Picture two cylinders which are rolling against each other without slipping.
This is the model for all gears. The diameters of these cylinders correspond
to the pitch diameter of the gears. To form gears, we imagine adding teeth to
the cylinders. These teeth extend above the pitch diameter by an amount
called the ADDENDUM and below the pitch diameter by an amount called the
DEDENDUM.
The meshing of the teeth during the rolling of the cylinders does two things.
First it guarantees that the rolling will occur without slipping - this is why
gears are used in applications where the angular relationship of the axes must
be maintained, as in the leadscrew drive on a lathe for threading. Second, the
meshing of the gears allows the transmission of far more torque than would be
possible in two rolling cylinders that depend on only friction to transmit
motion.
To reduce wear on the teeth when they mesh and unmesh, we would like the
contact line between two meshing teeth to remain fixed during contact - no
relative motion => no wear. Although I don't intend to prove it here, this can
be accomplished by forming the tooth face in the form of an epicycle. Imagine
a small circle rolling around the large circle formed from the pitch diameter.
The curve traced out by a point on this small circle will be an epicycle.
The diametral pitch of a gear is analogous to the pitch of a screw thread.
In mathematical terms,
P = N/PD
It is the ratio of the number of teeth to the pitch diameter of the gear.
Theoretically, P can take on any value (just as any screw thread pitch is
possible) but, in production, gears are made to a few standard pitches (just
as screws come in standardized values of threads/inch). *Some* example
standards are:
20, 24, 32, 48, 64, 72, 80, 96, 120, 200
If making your own gears, you can, of course, use any pitch you wish, though,
in the interest of preserving the ability to incorporate commercially
available gears, you may want to consider using standard values.
For two gears to mesh, the pitches must be the same. This fact is what allows
us to calculate gear ratios in terms of the number of teeth on each gear.
The speed ratio of two gears is the ratio of the two pitch diameters that
describe the theoretical cylinders rolling against each other, e.g.:
For two cylinders, diameters PD1 and PD2 inches, to roll without slipping, the
surface speed of the two cylinders must be the same.