A flywheel is fitted to the crank shaft of an engine having W amount of indicated work per revolution. Permissible limits of coefficient of fluctuation of energy and speed are C_E and C_S respectively. The kinetic energy of the flywheel is given by

Concept:

Coefficient of fluctuation of energy:

\({C_E} = \frac{{Maximum\;fluctuation\;of\;energy}}{{Work\;done\;per\;cycle}}\)

\({C_E} = \frac{\Delta E}{W} \Rightarrow \Delta E=C_E.W\)

Coefficient of fluctuation of speed:

\(\begin{array}{l} {C_S} = \frac{{Maximum\;fluctuation\;of\;speed}}{{Mean\;speed}}\\ {C_S} = \frac{{{ω _1} - {ω _2}}}{ω } = \frac{{2\left( {{ω _1} - {ω _2}} \right)}}{{\left( {{ω _1} + {ω _2}} \right)}} \end{array}\)

ω = mean angular speed during the cycle (rad/s)

\(ω = \frac{{{ω _1} + {ω _2}}}{2}\)

Maximum fluctuation of energy: ΔE = CE. W

\(\begin{array}{l} {\rm{\Delta }}E = \frac{1}{2}Iω _1^2 - \frac{1}{2}Iω _2^2 = \frac{1}{2}I\left( {{ω _1} + {ω _2}} \right)\left( {{ω _1} - {ω _2}} \right)\\ {\rm{\Delta }}E = Iω \left( {{ω _1} - {ω _2}} \right) = I{ω ^2}\frac{{\left( {{ω _1} - {ω _2}} \right)}}{ω } = I{ω ^2}{C_S}\\ K.E = \frac{1}{2}I{ω ^2} = \frac{1}{2}.\frac{{{\rm{\Delta }}E}}{{{C_S}}} = \frac{{{C_E}W}}{{2{C_S}}} \end{array}\)