Digikey have (despite my prior suggestion au contraire) a large number of motors.
As an example only something like this MAY fit inside your spec - 24mm dia x about 12mm deep. Some what large as a "wristwatch".
Or you could lay this down with a right angle drive and hub drive of he "hand(s)".
About 16mm long and about 5mm across so you could fit inside an eg 20mm x 7 or so mm "watch" with suitable magic.
BUT
You could almost certainly build your own motor with less effort than you might expect.
A stepper motor type arrangement with a multi tooth disk and only two windings will allow unidirectional drive. Three windings and a toothed disk will allow you to make a basic stepper motor. You can add zero current hold in place with a permanent magnet if desired.
As Olin notes - it would be easier to implement something that LOOKED like what you have described but used a "sold state" display of some sort. Doing it mechanically is liable to be worthwhile only if you value the result for its artistic merits.
I haven't even bothered watching after "only DC motors can be used as a generator".
As far as I am aware, a motor can be of the following families:
- Permanent magnet DC brushed. DC back emf.
- Coiled stator DC brushed (as a separate winding, or internally wound as series or parallel). DC back emf IF the stator is powered. Single phase universal motors are a subset of series connection types for which, regardless of the polarity of the voltage, torque is always generated (needs moveable brushes or a different wiring to change direction though).
- Permanent magnet AC synchronous (three phases). Three phase AC back emf.
- Coiled rotor AC synchronous. I think those generally are not brushed but rather rectify the current induced by the stator. If brushed, no back emf unless the rotor is powered.
- DC brushless. This one is basically a permanent magnet AC synchronous with hall sensors built in, to be able to electronically switch the phases. The back emf is however square or trapezoidal to maximise flux linkage.
- Stepper motor (2, 3, 5 phases). Close to the PM AC synchronous in its construction, except that the motor is made to maximise the number of stable equilibrium positions of the rotor (many alternating magnetic poles at the rotor or variable reluctance). Back emf depends on how it's driven.
- AC asynchronous (3 phases). The rotor is a closed loop (a coil, or a squirrel cage made of bars) which creates its field from currents induced by the stator. Can only be used as a generator beyond the synchronous rpm (+voltage at stator). AC back emf (TBC).
- AC asynchronous (single phase). The motor cannot be self-started unless an out-of-phase auxiliary supply is created via a reacting capacitor, and fed to windings 90° from the main windings. Can only be used as a generator beyond the synchronous rpm (+voltage at stator). AC back emf (TBC).
There are many more (e.g. hybrids), but I think those represent 95% of the production. I'm sure I've missed a few important ones, please feel free to comment and I'll update the list.
The biggest clue to the type of a motor is the number of wires, but as you can see this is not enough. Some motors cannot generate power without an excitation, some not at all, and even if they do, the back emf is funny sometimes (trapezoidal for example) depending on its construction.
You could plan to try the various types of supplies on the motor, ramping up the voltage, and see if it does anything, but what's your "OK that's not it, better cut the power before I smoke it" point? If you don't know what type of motor it is, I assume you don't know anything about it. Including the voltage and current ratings, Max rpm. You could get that from eyeballing it, but there is no guarantee then.
For your specific problem though, if you are certain your motor is a DC bruhless but you don't know if the inverter+control circuit are integrated, look at the number of wires. Generally the motor does not have a circuit built in, and an ESC must be connected to it. You will have to identify which wires are the hall sensors.
ESC might or might not be used for current generation, it depends on how they are made. I don't think there can be any harm in hooking up a resistive load compatible with its current range at the input and test it.
Best Answer
Everything really depends on how the sign is mounted to the motor. If the sign is centered, then barring a requirement for top speed or how quickly it will spin up, just about any motor will do. If the sign is not centered then at torque calculation is needed. To calculate the torque multiply the weight of the object by the distance away from the center of the shaft. For example one pound two feet away would be two foot-pounds of torque.