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An electric motor doesn't have to have a "stator", so...
This Idea was inspired by some annotations to another Idea (link).
The typical electric motor has two major interacting components, usually known as the "rotor", which rotates, and the "stator", which is stationary.
It is known that if the stator is mounted such that it is also allowed to rotate,
then it will rotate in the direction opposite to the direction in which the rotor rotates. Both parts will simply rotate at half the normal rotation speed, relative to the stationary ground. Relative to each other, though, each part of the motor works as if the motor is rotating at "normal" speed.
So let us construct an electric motor with two "interactors" that are disk-shaped and basically very similar. The two disks have axial bearings, and are placed on the same (stationary!) axle, and are located with a quite-small gap between them. When electricity is applied, electromagnetic forces between the two disks cause them to rotate in opposite directions.
If the disks are identical, then we will want each disk to incorporate both electromagnets and permanent magnets, so that the overall device can act as either a motor or as a generator. The disks could instead be constructed so that all the electromagnets are in one disk and all the permanent magnets are in the other (reduces the need for "brushes" that carry electric power to a rotating object).
Now we can consider constructing this type of motor with two somewhat massive disks, such that when the motor is running at maximum speed, we have two energy-storing flywheels, which are rotating in opposite directions. A primary advantage of this design is that we don't need a separate motor/generator connected to the flywheel axle, as is typical for most energy-storing flywheel setups.
The secondary advantage, of course, is that two spinning disks lets us store twice as much energy as one (if, as pointed out in an annotation, the overall flat-disks electric motor runs at twice the normal speed, relative to each other).
And a third advantage, if the first annotations in the linked Idea are correct, is that this type of energy-storage flywheel will be more suitable for installing into vehicles, than ordinary flywheels that merely rotate in one direction, because certain gyroscopic effects will be reduced. (I have my doubts about the truth of that, but the first two advantages are sufficient, by themselves.)
Inspiration for this Idea
As mentioned in the main text (see the first two annotations of this linked Idea). [Vernon, Apr 09 2012]
Powering something that is turning
While not applicable in all situations, there are times when it is possible to have a solid electrical connection between turning and stationary objects. Brushes are not always needed! [Vernon, Apr 09 2012]
Regarding an oppositely-rotating "stator"
As mentioned in an annotation [Vernon, Apr 09 2012]
I'm linking this not just because it was mentioned in an annotation, but because a TwinFlywheel would be good for that kind of car (so I'll be linking this Idea on that page, also). [Vernon, Apr 09 2012, last modified Apr 10 2012]
||You've codeified a couple decent ideas... I read a while back about an electric (not electronic, electric) motor "differential" in which both sides of the motor spun in opposite directions (and there was a gear on one side so both wheels spun in the same direction).
||[FlyingToaster], yes, I linked that in my "Steam-Electric Hybrid" Idea. I can put that link here, too.
||//we get to store twice as much energy//
||I'm not seeing this. Are you saying that just by allowing the stator to rotate freely you can double the energy storage? I would have thought that, all else being equal, the stator and rotor move at the same relative speed thus store the same amount of energy.
||[xaviergisz], we can run this motor such that, from the perspective of either "interactor", it goes twice as fast as an ordinary electric motor. Then both parts, relative to the stationary ground, are moving at the normal speed of an ordinary electric motor --which in turn might be connected to an ordinary flywheel. And that's why the total energy stored here can be doubled.
||Thanks for pointing that out, though (main text edited to reflect it).