Having bored the socks off people with helical gears I'm now going to bore the pants off the same people by discussing internal gears.
All the books say that internal spur gears are not the same as external spur gears, they're inside out, but then move on without giving details. Chocolate teapot territory! Out of interest I decided to look into the design of internal gears. It turns out to be quite simple, in principle. In a similar fashion to external gears an involute curve is scribed from the base circle. There are two differences; the shape produced is the 'space' rather than the tooth, and the addendum and dedendum are swapped. I produced a spreadsheet to calculate the points on the involute of a circle, for a given base circle diameter, from the cartesian parametric equations for an involute curve. I then imported the points into my 3D CAD system and with a bit of fiddling produced a solid model of an internal gear. When importing the comma separated values for the involute I could specify the points be joined by straight lines or a spline. I chose lines as that meant I could trim the involute curve at a later stage. The gears were designed for 1DP and scaled to 10DP for the gears I was going to make.
One surprise was that the base circle (PCD times the cosine of the pressure angle) was larger than the PCD minus twice the addendum. So what are you supposed to do with the involute curve inside the base circle? I just extended it as a tangential line. It was simple to derive a mathematical inequality to determine how many teeth a gear needs before the base circle is smaller than the root diameter of the gear; 62.8 for a pressure angle of 14½º. I downloaded a pinion from the net and 3D printed both to look at form and fit:
These gears suffer from trim interference, where the gears will not assemble radially, but once assembled axially they run together smoothly. In practice the plastic gears can be forced together radially, but there is an audible click when they engage. I re-designed the gears using a 20º pressure angle, and this time I designed the pinion from first principles too. I chose 45 teeth for the internal gear (makes the indexing simple) and 21 teeth for the pinion. Here are the revised 3D printed gears, and steel blanks:
I made the form tool to cut the internal gear by milling the shape on the end of a ¼" square piece of HSS, using the CNC mill:
I only cut the profile 2mm deep, as the remainder would be ground away to provide relief. Some top rake and the side and front reliefs were ground by hand. Cutting the internal gear is standard using the slotting head on the Bridgeport and a rotary table. Once set up the X and Z axes were locked and the depth of cut set in Y using the DRO. To the left of the picture, as well as the 3D printed gears, is a crib sheet with the indexing angles, every 8º, as a sanity check:
The pinion was made on the CNC mill, using the 4th axis. I had a lot of trouble with the CAM software not doing what I wanted, and generating toolpaths that didn't match the toolpath it was showing. I always check the G-code using NCPLOT to draw the toolpath from the actual code; it doesn't always match what the CAM program thinks it is doing.
Since this was a one off gear I ended up with a 2mm ballnose cutter doing everything even though it wasn't the quickest way. I left the CNC mill to get on with it, while I did a days paid for work:
I had to do a little fettling of the pinion with needle files to get the gears to run smoothly. That is no real surprise as the clearances in the 3D model, at 1DP were only a few thou. The two gears now run smoothly together in either direction:
The next step is to look at the mathematics of the various types of interference problems that can occur with internal gears.
Andrew
