Well, Stirling engines run off of the expansion and contraction of air gases. Inside the engine, there is a cylinder. A displacer runs inside the cylinder, made of steel wool. The displacer is used to move the hot and cold gases around the cylinder. When the displacer moves down, it forces the air from the bottom of the cylinder to the top. When it moves back up, it pushes the air from the top of the cylinder back down to the bottom. The bottom of the cylinder is heated, the top is cooled. The heat causes the gases inside the cylinder to expand, and the cold water causes the gases to contract. The cylinder is sealed off, and the gases inside never exit the engine itself.
The balloon is used as a "power piston", which is used to power the engine itself.
The displacer and power piston do not "move up and down" at the same time. The displacer travels first, and when it reaches half way to the top of its "stroke", the power piston is at the bottom of its "stroke". Therefore, when the displacer is completely at the top of its stroke, the power piston has traveled half way up of its stroke. When the displacer is at the bottom of its stroke, the power piston is half way down its stroke.
The power piston is 90 degrees out of phase from the displacer. When the displacer is at the top of its stroke, the air is cooled. When its cooled, the gases contract, which causes the balloon to "suck down", which pulls the crankshaft back down into the next rotation. The power piston is supplying the power, the displacer is only used to move the hot and cold air.
The flywheel is used to "keep it going". The flywheel momentum keeps the crankshaft rotating. My flywheel is just a paint can, with a terminal block epoxied in the center of it.
I tried to describe it the best I could.
