I had a problem that I could only solve with two gears that met at a 90 degree angle. Normal gears don't work that way - the teeth would get stuck or break off. I needed a gear that tapered inward at a 45 degree angle. Angled gears like this are called straight bevelled gears. Here's how I did it.
image courtesy of http://en.wikipedia.org/wiki/File:Beve_gear_schematic.png. Drawing by Ralf Pfeifer
Step 1: Gear Template Generator
I downloaded and installed a copy of http://woodgears.ca/gear/.
When I wrote this instructable the price was $26 and totally worth it. Have you seen my open source robot arm? It uses the same gear templates. By the time you read this I'll have 3 or 4 robots that use gears generated with this program.
I set the "shaft spacing" to the distance between the center of the gears. I set the same number of teeth on both gears. I picked the size of the hole in the center that I wanted. Feel free to change these options to suit your tastes.
Remember: The diameter of the gear is different than the shaft spacing because the teeth have to overlap.
I turned off most of the options on the right and turned on "draw unmeshed".
I saved my project and then exported as a DXF file.
Step 2: Lofting the model
In a 3D modelling program (I used Solidworks) I imported the DXF file.
In my head I pretended the drawing was lying flat on a table. I copied the design and raised it 5mm, the thickness of my gear.
The top of the gear has to be smaller than the bottom of the gear. But how much smaller? I knew I wanted a 45 degree angle, so I did the math. I know that at a 45 degree angle if I go up (gear pitch radius) the pitch will be zero. It scales linearly. So (gear pitch radius - gear thickness)/(gear pitch radius) will give me the amount to scale.
As you can see in the drawing, my scale factor was 0.77272...
Once I had the large gear on the bottom and the small gear on the top, I used a "loft" to build the outside edge of my gear.
Finally I exported to an STL file and 3D printed my gear in about 10 minutes.
Step 3: Final thoughts
Anything other than a 45 degree gets more interesting. I think it's tan(angle) * (radius - thickness) / radius.
If you are using a 3D modelling program you can add a twist between the top and bottom to make spiral bevel gears. As the gears turn the teeth touch in a different way, which can have advantages. Ask your favorite engineer about the specifics.
If you liked this project, please tell your friends and check out my website.
Thanks for reading!