Turning Moment Diagrams • Coefficient of Fluctuation • Energy Storage — Simulate • Explore • Practice • Quiz
Flywheel energy storage is one of the oldest and most elegant methods of smoothing power delivery in mechanical systems. A flywheel is a heavy rotating disc mounted on a shaft that stores kinetic energy during periods of excess torque and releases it during periods of deficit. This energy buffering action is critical in internal combustion engines, punch presses, wind turbines, and braking energy recovery systems. The fundamental equation governing flywheel energy storage is E = ½Iω², where I is the moment of inertia and ω is the angular velocity in rad/s.
A turning moment diagram (also called a crank effort diagram) plots the instantaneous torque produced by an engine or machine against the crank angle over one complete cycle. For a four-stroke single-cylinder engine, the cycle spans 720° (two full revolutions). The torque varies significantly — it peaks during the power stroke and drops during compression, exhaust, and intake strokes. The mean torque is a horizontal line representing the average resisting torque. Areas above this line represent energy surplus (the flywheel absorbs energy and speeds up), while areas below represent energy deficit (the flywheel releases stored energy and slows down). The maximum energy fluctuation — the difference between maximum and minimum energy levels — determines the required flywheel size.
The coefficient of fluctuation (Cs) measures the variation in flywheel speed and is defined as Cs = (Nmax − Nmin) / Nmean, or equivalently (ωmax − ωmin) / ωmean. For most industrial applications, Cs ranges from 0.002 for spinning machinery to 0.2 for crushing machines. The energy stored in a flywheel that compensates for fluctuation is given by ΔE = Iω2Cs. Rearranging this equation allows engineers to calculate the required moment of inertia: I = ΔE / (ω²Cs). For a solid disc flywheel, I = ½mr², while for a rim-type flywheel, I = mr². These equations form the basis of flywheel sizing for any mechanical application.
In reciprocating engines, the flywheel smooths out the pulsating torque from individual cylinders, ensuring steady output speed. Punch press operations require enormous torque for very brief periods — the flywheel accumulates energy over the idle stroke and delivers it instantly during the punching stroke. In wind energy systems, flywheels buffer the variable input from wind against a steady generator load. Modern regenerative braking systems use high-speed flywheels to capture kinetic energy during deceleration and return it during acceleration, improving energy efficiency by up to 30%. Advanced flywheel materials such as carbon fibre composites allow extremely high rotational speeds, enabling compact designs with energy densities exceeding 100 Wh/kg.
In Simulate mode, select an application (Engine, Punch Press, Wind Energy, or Braking) and adjust the mass, radius, RPM, and load variation sliders. The left canvas shows an animated spinning flywheel with energy level indication. The right canvas displays the turning moment diagram with surplus (green) and deficit (red) shaded areas, mean torque line, and a live tracking marker. Switch to Explore mode to study 12 concepts across Energy Basics, Fluctuation, and Design categories with formulas and worked examples. Practice mode generates random numerical problems, and Quiz tests your understanding with 5 randomised questions drawn from a pool of 15.