Flywheel Energy Fluctuation

Many machines produce or consume torque in gusts. A single-cylinder engine delivers a hard push once every two revolutions and drags the rest of the time. A punch press does all of its work in a few degrees of crank rotation. A reciprocating compressor loads the shaft heavily on one stroke and lightly on the other. A flywheel is the buffer that carries the shaft across the gaps.

It works by trading speed for energy. When the driving torque exceeds the load, the surplus goes into speeding the flywheel up; when the load exceeds the drive, the flywheel gives energy back and slows. The size of that speed swing is the coefficient of speed fluctuation, $C_s = (\omega_{max} - \omega_{min})/\omega_{mean}$, and the energy traded across the cycle is the difference of two kinetic energies:

\[E = \tfrac{1}{2}I(\omega_{max}^2 - \omega_{min}^2) = \tfrac{1}{2}I(\omega_{max}+\omega_{min})(\omega_{max}-\omega_{min}) = I\,\omega_{mean}^2\,C_s\]

using $\omega_{max} + \omega_{min} \approx 2\omega_{mean}$, which is accurate to well within the precision of anything else in the calculation.

Notice what E is NOT. It is not the flywheel's stored energy, $\tfrac{1}{2}I\omega^2$; it is the sliver traded across one cycle, and the ratio between them is $2C_s$. For a typical $C_s$ of 0.02, the fluctuation is four per cent of what the wheel is holding. Confusing the two is the standard error on this page, and it goes the expensive way — it makes the required flywheel come out about fifty times too large.

The square on the speed is the design lever. Doubling the running speed cuts the required inertia by four, so the cheapest flywheel is nearly always the fastest one the bearings and the burst margin will allow, and that is the entire reason modern flywheel energy storage runs at tens of thousands of rpm in a vacuum. Against it, rim stress also goes as the square of the speed, so speed is bought against burst risk rather than for free. Once the required I is known, put the mass at the RIM — inertia goes as radius squared, so a rim-heavy wheel does the job at a fraction of the mass of a solid disc.

What $C_s$ is acceptable is a question about the machine and not about the flywheel. Punch presses and rock crushers live happily at 0.10 or worse, which is exactly why a press can be driven by a modest motor: the flywheel lends its energy to the stroke and the motor spends the rest of the cycle winding it back up. An engine driving an alternator needs an order of magnitude tighter, because the speed fluctuation appears directly in the output frequency. The hard part of the calculation is not this equation at all — it is finding E, which means plotting driving and resisting torque against crank angle and integrating the largest accumulated area between them, from one crossing to the next.