See the animation below. Consider that as the air in the hot end heats up, the high pressure air forces in on the sealed displacement piston and the flexible diaphragm will be forced to collapse, bringing the piston to the hot end of the cylinder, forcing the air to the cold end. As the air cools and contracts the pressure drops, and the lower pressure causes the diaphragm on the sealed displacement piston to distend, forcing the displacer to the cold end and the cold air to the hot end. The linear oscillation would of course be used to drive a linear alternator, simplified in this diagram as magnets moving outside of a fixed coil.
There can be no question at all that this design can forcefully shove the piston one way or the other as the temperature changes, the one and only thing that can be in doubt is the question of oscillation. It seems to me that the mass of the piston would have the same effect as a flywheel, and that a clear and forceful oscillation could be achievable. The specifics of the oscillation would certainly be determined by factors that include the mass of the piston and the density and composition of the working fluid, but certainly there exists a sweet spot where the most forcefull oscillations would be achievable.
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 HEREReturn to Dave's Planet