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In a split-flap display there is a barrel which rotates by a small angle N times to get to some next display element. However I fail to understand, how it is possible that the barrel rotates exactly by some desired angle? How is the rotation error after each flip is not accumulating? The only explanation I can think of is that it has sensors that sends signal back to the the motor to stop it.

Or is it possible that a motor can make exact rotations without error accumulation? I don't think it is possible unless the motor itself has built-in controller systems which resets the position each time to eliminate the error.

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The barrel is usually rotated by a ratchet mechanism, not directly by a rotary motor. The ratchet might be driven by such a motor (e.g., a clock) or by a solenoid (e.g., railroad station status board).

In the former case, the display is initialized manually. In the latter, there is usually a microswitch that indicates the index position of the barrel.

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