My budget is about five hundred dollars. I have next two-three month to spare on this project.
"When you're spinning these flywheels you're going to get a tremendous gyroscopic effect that would make the quadcopter impossible to maneuver."Of course, but this is also true for each of the rotors on your quadcopter One possible solution is to use two flywheels with the same speed and shape but with different directions. (i.e. exactly the same solution you use for your quadcopter)
I never said it would be a good idea to use a flywheel, I just said that you can compensate the gyroscopic effect with two flywheels of different direction. "Why would you use power to spin a flywheel to make power??"I think the most important argument was the short time needed to speed up the wheel. If you could fly some minutes and recharge within some seconds, you would be flying much longer than with a battery with which you could fly for 20minutes but need 3hours to recharge.