The beta stirling engine is a fun and simple engine to make, but it requires some minor complexity to move the displacer differently than the piston moves. It requires you to put a hole through your piston in order to drive the displacer from a different point on the flywheel. The necessity of moving the displacer differently derives from the idea that there is on part of the engine that is hot and one part that is cold. If you are willing to have two hot spots or two cold spots then the movement of the displacer can be simplified to the following animation:


When the displacer is in the center it displaces all the hot air into the two cold ends of the engine. When the displacer and piston are all the way to the left then they combine to force air to the hot spot of the engine. I have called this the double cooled stirling, but two hot areas and one cold area would also work. It seems mechanically simpler to double cool it, and double cooling keeps your piston from being heated, so for simplicity sake I am just naming this the double cooled beta stirling engine.

Online discussion of this design can be found HERE

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