Amount of energy that can be stored from Faraday flashlight / linear alternator?

You’ll have to forgive me if this question is either extremely elementary or does not make sense — I am a civil engineer with little E/M physics background beyond high school education.

I am working on a project that involves the harvesting of energy from what is effectively a huge Faraday flashlight or linear alternator. (A Faraday flashlight is an application of the linear alternator, correct?) When the apparatus is shaken (for a finite period of time), a magnet moves through a coil, inducing an e.m.f. in the coil. The magnitude of the induced e.m.f. in the coil, as a function of coil geometry, magnet strength, and magnet position, can be found in this paper (equation 20).

I’d like to quantize the maximum amount of electrical energy produced in this process that can be stored in a capacitor or battery bank. As far as capacitors are concerned, using E=0.5CV^2 would be nice, but two issues present themselves:

1. There is no maximum value of C, implying that an infinite amount of energy could be stored in the capacitors from this finite process, something which is a clear violation of the Law of Conservation of Energy.
2. As the shaking is arbitrary, so too is the voltage created in the coil through Faraday’s Law of Induction. Since V changes with time, I have no idea what value to use for it.

Considering this, how would one go about measuring the amount of energy stored in a capacitor fed by an arbitrary voltage source?

I have no education with regard to batteries, but I understand that they operate in a fundamentally different way than capacitors and do not have an “equivalent capacitance” or anything like that. I also understand that they are generally more appropriate for large-scale energy storage. What would be the best way to measure the amount of energy stored in a battery bank fed by an arbitrary voltage source?

EDIT

I understand the solution to capacitor issue #1. Since capacitors have a charge time equal to about 5*tau=5*R*C (that gets it to about 98% charge), the amount energy that builds up in the capacitor is dependent on the amount of time the voltage source supplies a voltage. I haven’t done the math, I would guess that the amount of energy that has built up in the capacitor over a given time will never exceed the amount of energy expended by the voltage source over that same time.

I’m still not sure what to do about issue #2, though. Do the same rules surrounding time constants apply when the voltage is rapidly changing? That is to say, if the voltage supplied by the voltage source momentarily exceeds the voltage built up across the capacitor, do the same rules tau=R*C and Vc=Vs(1-exp(t/tau)) apply? Could I use a diode to avoid capacitor discharge if the voltage supplied by the voltage source is lower than the voltage built up across the capacitor?