Hi Nick, there is a "Ringbom" Stirling, which has a piston which moves just the displacer, and has a separate power piston which moves according to the pressure differential, and there is a "resonance" or "acoustic resonance" type where it has a single piston, a heat chamber with good insulation from the cylinder and piston with the cylinder well cooled, and relies on a "resonating pressure pulse" established by the heat raising the pressure, and the power piston setting up a pulse, which will settle into driving the piston at the resonant frequency of the pressure wave in the closed cycle, as it alternates between pressure and temperature change, as per the first law of thermodynamics, I believe, which says in a closed system, the pressure of a gas will be directly proportional to the relative temperature, so the piston changing the volume allows the temperature to turn into pressure when the volume is reduced, and pushes the piston out, which increases the volume, allowing the air to cool, with the excess heat carried off by the mass of the cylinder, piston and the rest of the "cool" end of the engine, which takes it below its original pressure, so it expands faster when all the air is pushed into the hot end, and raising the pressure again. The phase shift takes place in the transfer of the heat to pressure, and the change of pressure with the "loss" of heat when the piston has been moved, and has lowered the pressure by increasing the volume, and at the same time, losing some of the energy of the heat also through transfer, causing the resonant effect which acts in place of a displacer. This engine seems to work by applying the same principle of a "hydraulic ram", pumping system which uses the inertia of running water to establish a resonance, and pump by just changing the pressure in a closed chamber with valves to bleed off excess water, and keep a resonating pressure change continuing and the inertia of the water actually doing the pumping. There is a video of this form of single piston Stirling among the videos available at the beginning of this post. It has taken a lot of mind bending to get around this idea and make it make sense, but having seen it work, it must make sense.

mad jack