Turning moment diagram and fluctuation terms

A turning moment diagram (crank-effort diagram) is a plot of the turning moment (torque) on the crankshaft against the crank angle.

Four-stroke engine diagram

A cycle takes two revolutions ($720^\circ$): suction, compression, expansion (power) and exhaust.

 T
 ^                              __
 |                             /  \
 |                            /    \
 |- - - - - - - - - - - - - -/- - - \- - - - - - - - -  T_mean
 +-.________.---.           /        \--.________.----> theta
 |               \         /
 |                '._____.'
 0             180           360           540           720
    suction      compression    expansion      exhaust

• Suction stroke ($0^\circ$ to $180^\circ$): small negative torque (the crank must pull the piston).
• Compression stroke ($180^\circ$ to $360^\circ$): large negative torque (work is absorbed in compressing the charge).
• Expansion stroke ($360^\circ$ to $540^\circ$): large positive torque (gas pressure drives the piston).
• Exhaust stroke ($540^\circ$ to $720^\circ$): small negative torque (the crank must push the piston).
• The mean torque line is drawn so that the area above it equals the area below it; its height is $T_{mean} = \dfrac{\text{work per cycle}}{4\pi}$.

Torque above the mean line accelerates the shaft; below it, the shaft slows down. The flywheel absorbs and returns this energy.

Definitions

• Fluctuation of energy ($\Delta E$): the difference between the maximum and minimum kinetic energy of the flywheel in one cycle, i.e. the largest swing of the cumulative area between the torque curve and the mean torque line.
• Coefficient of fluctuation of energy ($C_E$):

$$C_E = \frac{\Delta E}{\text{work done per cycle}}$$

• Coefficient of fluctuation of speed ($C_s$):

$$C_s = \frac{\omega_{max}-\omega_{min}}{\omega_{mean}} = \frac{N_{max}-N_{min}}{N_{mean}}, \qquad \omega_{mean} = \frac{\omega_{max}+\omega_{min}}{2}$$

Its reciprocal is the coefficient of steadiness. Typical $C_s$: about 0.002 to 0.005 for electric generators, about 0.03 for machine tools, and 0.2 for crushers.