If you aren't familiar with how Stirling engines work, there are plenty of resources online- the Wikipedia page is a good place to start.
Step 1: Design
I found that an aluminium drinks can would fit neatly inside a steel soup can, so decided to use that for the displacer on the grounds of being cheap, light and easy to work with. The steel soup can provides a sturdy body for the engine and also meant I could ake the bottom removable to allow maintenance.
I found some decking to use as the base, some 1" x 1/2" pine to use as supports for the moving parts, and a piece of 3/4" aluminium strip to use as the beam. The engine is a messy combination of metric and imperial units- that's what happens when you use scrap materials!
Step 2: Bill of materials
Two empty steel food cans, at least one with a tapering bottom
An aluminium drink can that fits closely inside the soup can without touching the sides
Two thin metal discs (eg lids from food cans)
Copper or brass pipe, as round as possible, around 40 - 50mm long
Power piston *
A low-friction ball bearing
A CD or other disc to use as the wheel
Pennies for weights
About 12" of metal or wood thick enough not to bend too much
Frame of engine
Thin wood (1" x 1/2" or larger) for supports, at least 24" (600mm) long
A piece of plank at least 12" long and 4" wide (300 x 100mm) for the base
Thin brass tube 1/8" inside diameter and steel rod 1/8" diameter- about 12" of each (the precise measurement of these isn't as important as the rod being a snug sliding fit inside the tube)
Two terminal blocks
JB Weld or other high temperature epoxy
Fast-setting epoxy (eg Araldite)
Rotary tool with grinding and cutoff bits
Step 3: Power piston
Others have made pistons by casting JB Weld inside their chosen power cylinder. You can read more about that here- I didn't do this because I had the machined piston/cylinder combo, but it seems to be quite a popular method.
The piston is a 20mm x 20mm solid cylinder with a 1/8" hole bored in one end. A short section of steel rod was glued into the hole, and a section of brass tube glued over that. The end of the tube was flattened and drilled to 2.5mm to accept a conrod made of coathanger wire. In engine terms, this is the small-end bearing.
Step 4: Flywheel
I initially tried gluing a short piece of brass tubing to the surface of the wheel to act as a crank, but the glue bond between the metal surfaces was weak and kept breaking. Drilling a small hole into the wheel and gluing a piece of coathanger wire into that made a much stronger crank. A piece of wood with a short section of brass tubing fixed into a hole drilled through it provided a support for the flywheel.
Step 5: Displacer
The length of the displacer is important- it should take up about two thirds of the space inside the cylinder. I worked out how far the crank on my engine was going to move (the "throw", about 20mm), and subtracted that from the length of the displacer cylinder (90mm) to find the largest size the displacer could be (70mm).
I actually glued the rod into the displacer with it inside the engine once I had assembled the displacer cylinder to make sure the two lined up- if everything was measured with sufficient precision this would be unnecessary. I used JB Weld for this join as fast-setting epoxies tend to soften at high temperatures.
Step 6: Displacer cylinder
The top of the displacer cylinder was drilled in the centre and a section of brass tubing glued in with JB Weld to form the displacer rod seal. A hole large enough to accommodate the power cylinder was cut into the side using a rotary tool and the power cylinder glued in.
When assembled, a short section of brass tube was glued onto the end of the displacer rod, flattened and drilled like the power piston to provide a joint for the beam conrod.
I also cut out the bottom of a tuna can and glued that around the top of the displacer cylinder to use as a water jacket for cooling, but this was actually less effective than plain air cooling, and developed a leak that was letting water into the cylinder so I removed it.
Step 7: Connecting rods
These are a very convenient size and shape to join the end of a piece of wire perpendicularly to another piece. One of the screws was removed and a hole drilled sideways all the way through the brass, and the other screw used to clamp the end of the conrod in the connector. I got this idea from reukpower's Coke Can Stirling Instructable.
The conrods all have a Z-shaped bend in the middle which allows for length adjustment by tweaking the bends.
Step 8: Framework of the engine
Despite working out all the dimensions beforehand, my apparent inability to drill a hole straight through a piece of wood coupled with my cheap wobbly power drill resulted in a certain amount of trial and error being applied to the placement of the engine parts.
Step 10: Design tweaks
After all these mods, the engine runs more reliably than any fuel source I have to test it with :) It will run at approximately 150-200rpm on a decent candle flame, and has run at nearly 500rpm when I accidentally overheated it slightly with a large meths burner. The engine is mechanically sound at that speed- my concern is the displacer overheating and the epoxy that keeps it attached to the rod failing.