What is the effect of torque transmitted through a gear system to the angular acceleration of a flywheel?

Question

For a school project, I am trying to design a system that will spin a flywheel at a high rpm to store energy. My initial idea was to accelerate the flywheel via a compound gear train with an approximately 1:200 gear ratio. The radius of the flywheel is 8cm and it weighs 2kg. My driving force is 250N through an arm of 20cm.

I wrote a simple Matlab code that first calculates the applied torque on the flywheel by multiplying my initial torque with 1/200(the gear ratio) and then uses the formula T=I*alpha to get the angular acceleration. At each iteration the code calculates the corresponding air resistance(then the net torque) and finds the time required to reach a limiting value where the resistance equals my driving torque(alpha = 0).

My problem is that when I divide the initial torque by the gear ratio I end up with a mathematical equation suggesting that increasing the gear ratio makes the flywheel spin slower. The resulting rpm vs time graphs seem appropriate but I know there is a logical fault in that statement, however I can't find a way to mathematically implement the fact that the gear ratio actually increases the speed while decreasing the torque. Any help on this subject is much appreciated.

Thanks

Answer 1

You are quite correct that the torque on the far side of your 1:200 gearbox will be smaller - 200 times smaller in fact. However remember that torque is related to angular acceleration not (at least not directly) to angular velocity.

So reducing the torque with your gearbox just means the flywheel will accelerate more slowly than it would without the gearbox. The only thing opposing the acceleration is the air resistance, and you ignored the air resistance (maybe put everything in a vacuum chamber) the flywheel would keep accelerating and given enough time would spin arbitrarily fast.