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Maybe I'm misunderstanding flywheels, but wouldn't a primative one just add a certain amount of 'inertia' to the power system?
All flywheels, primitive or modern, exist for the sole purpose of storing and releasing inertia, though in gyroscopes the inertial transactions are secondary to the purpose of resisting angular torque.
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If you hook up a flywheel that spins at, say, 100RPM, then it seems like the whole system would slow down until the flywheel is up to speed.
The system would "spin up" based on its total resistance versus the torque of the power source from 0 to its operational RPM. If the system were already turning and you placed an at-rest flywheel in communication with that system, i.e. "popped the clutch", the system would slow down and speed back up just as you described. Or something would break.
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Once the power source is taken off, the flywheel will keep going at the same rate but very quickly start slowing down
The flywheel will immediately start slowing down, but very slowly, at least if it's sized appropriately to the task.
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I think the belief here is that a flywheel will keep everything running at full power until it gets drained
Well, yes, if you define "full power" as any RPM within operating tolerances and "drained" that point at which flywheel RPM drops below system-nominal RPM.
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unless you have like 3x as much flywheel capacity as you have power requirements, so even once it starts spinning down, you have more power than you need for a while.
That's exactly how flywheels work, though 3x is a little steep. Your flywheel basically needs to be big enough in a system that spins fast enough to smooth out fluctuations on the input side to within whatever your tolerances are on the output side. RPM tolerances for millers are pretty relaxed.
RPM tolerances for record players are so tight the early flywheel-based systems had to have governors and clutches to keep the record spinning at a constant rate regardless of flywheel RPM above nominal. But we aren't playing records on our millstones, just grinding stuff up with them.
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You'd need a ton of these to be useful, and they wouldn't last long.
You only need 1 if it's axial mass is sufficient, and they last virtually forever.
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Plus the strain would mean you'd need a lot more metal components (though maybe you should anyway).
For water milling rock works just fine. Remember that you don't have to guess at what a flywheel would need to be: they exist in the real world, in the case of water mills ever since the 11th century. In some mills the millstones themselves had enough axial mass to act as flywheels. But not our DF mills, hence our need for flywheels.