Flywheel Moment of Inertia for a Speed Fluctuation
Flywheel Moment of Inertia for a Speed Fluctuation
Machines with a cyclically varying torque demand — reciprocating engines, punch presses, crushers — see their shaft speed fluctuate over each cycle as energy input and output momentarily fall out of step; a flywheel is added specifically to store and release that energy fluctuation, smoothing the speed variation to within an acceptable tolerance. Sizing the flywheel's moment of inertia is a direct calculation from how much energy fluctuates per cycle, how fast the machine runs, and how much speed variation is tolerable.An undersized flywheel lets speed swing more than the coefficient of fluctuation allows, which for many applications (generators needing steady frequency, precision machinery, punch presses needing consistent stroke force) is directly unacceptable — but an oversized flywheel adds unnecessary weight, cost, and rotational inertia that slows the machine's ability to start, stop, or change speed, so this is a genuine sizing optimization, not a "bigger is always better" problem.
The required flywheel moment of inertia is I_fw = dE/(omega_avg^2·C_s), the cyclic energy fluctuation divided by the square of the average angular speed and the coefficient of fluctuation. where dE is the energy fluctuation over one cycle, f_avg is the average rotational speed, C_s is the coefficient of fluctuation (the allowable fractional speed variation), omega_avg is the average angular speed, and I_fw is the resulting required flywheel moment of inertia.
Inputs
dE := 500 J=500 Nm
f_avg := 16 Hz=16 Hz
C_s := 0.02=0.02
Converting the average speed to angular units puts it into the form the flywheel sizing formula requires.
omega_avg := 2·pi·f_avg=100.53 Hz
Dividing the cyclic energy fluctuation by the average speed squared and the allowable fluctuation coefficient gives the moment of inertia the flywheel needs to absorb that energy swing within the specified speed tolerance.
I_fw := dE/(omega_avg^2·C_s)=2.47 kg·m^2
Results
A required moment of inertia of about 2.47 kg·m^2 here would then guide the flywheel's physical dimensions — since I scales with mass times radius squared, a flywheel achieves a given inertia more efficiently by concentrating mass at a larger radius (a rim-heavy design) than by simply adding overall mass near the hub. Because the required inertia is inversely proportional to the square of the average speed, a higher-speed machine needs a dramatically smaller flywheel for the same energy fluctuation and speed tolerance — this is one reason high-speed machines are often lighter overall than low-speed ones handling comparable power. Tightening the allowable coefficient of fluctuation C_s (for a smoother-running application) directly and proportionally increases the required flywheel inertia, so this tolerance should be set based on the actual downstream sensitivity to speed variation, not chosen arbitrarily tight.
Assumptions & limitations
- Cyclic energy fluctuation dE is taken as a known input; in practice it comes from integrating the torque-speed curve over one cycle.
- Coefficient of fluctuation C_s is the allowable fractional deviation from average speed, a design choice based on application sensitivity.
- Flywheel is treated as a simple lumped rotational inertia; its own detailed shape/mass distribution is a separate sizing step.
- Steady cyclic operation is assumed — transient startup speed variation is not covered by this formula.