## Cycloidal Gear Form and Clock Wheels

Hello all, this is my first post. My name is Chris, I teach Chemistry 11-18 for a profession but I am an hobby engineer and have a huge interest in clocks and watches - horology.

The gears in clocks are called wheels to a horologist and the profile of the tooth is of a cycloidal form. My understanding of this is that the gear ratio is based on two circles linking together, this is the pitch circle diameter, PCD. To allow the wheels to turn, (because friction against two smooth sides would not be enough) you add the teeth, above the PCD to form the addendum of the tooth and to allow them to mesh with enough depth, you also take something away from the PCD to form the dedendum. The addendum and dedendum form the tooth but it is actually the PCD of both gears which give you the correct gear ratio.

Using a table like the one below, or using an online calculator such as the ones in the links, I can then calculate all the tooth profile details.

LINKs to Calculator: http://www.ryerson.ca/~v7chan/cncproject/gear/Cycloid%20Gear%20Calculator.html

From this info, I can then use CAD or solidworks and draw my wheel:

The example below is of a: 0.8Module, 60 teeth, 48mm pitch circle diameter, 50.15484 mm tip diameter, 2.51968mm tooth pitch, 1.259mm tooth width and gap between teeth, 1.0769mm addendum, 1.2598mm dedendum.

My question is that I want to know more about the cycloidal form and actually be able to draw it from points plotted rather than the innacurate form from approximated calculations.

If you looks at the diagram below, the tooth profile is being formed from the drawing on many circles based on the PCD, I believe these are hypocycloidal and epicycloidal curves which generate intersects to plot the tooth:

If the link doesnt work - google "Drawing Cycloidal Curves Chest of Books" : Cycloidal Gears

Basically I want to really understand how this cycloidal curve is formed and using pen and pencil, how I would go about drawing a cyloidal gear like the one I have done in solidworks (using the same dimensions) but drawing the epicycloidal and hypocycloidal curves to "form" the teeth.

There is info here and discussion of the maths, but it loses me very quite quickly and I just cant seem to then transfer this info to actually go and use it to draw a wheel....

Designing Cycloidal Gears

I hope someone could help me out and offer any advice. If this is in the wrong forum, please feel free to move it!

Chris