--- title: "Turning moment diagram and fluctuation terms" subject: "Theory of Mechanism and Machine II" chapter: 2 key: "semester_7/theory_of_mechanism_and_machine_ii:turning-moment-diagram-four-stroke" source: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/turning-moment-diagram-four-stroke site: BCT Engineering --- # Turning moment diagram and fluctuation terms Theory of Mechanism and Machine II · Chapter 2: Turning Moment Diagram and Flywheel - Asked in: year not given, 6 marks - Video: [TOM 02 Turning Moment Diagrams and Flywheel | Sizing the Flywheel](https://www.youtube.com/watch?v=h1mFotC4Vxc) — Sunil Mohod - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/turning-moment-diagram-four-stroke - Whole chapter: https://www.bctengineer.com/md/semester_7/theory_of_mechanism_and_machine_ii/chapter-2 ## Question Explain the turning moment diagram of a single-cylinder four-stroke engine. Define fluctuation of energy, coefficient of fluctuation of energy and coefficient of fluctuation of speed. ## Short answer The turning moment diagram for a single-cylinder four-stroke engine has negative torque in suction, compression and exhaust strokes, and a large positive torque in expansion stroke. Fluctuation of energy ΔE is maximum minus minimum kinetic energy; C_E = ΔE / work per cycle; C_s = (ω_max - ω_min) / ω_mean. ## Answer 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. ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A turning moment diagram is a plot of: A. Torque on the crankshaft against crank angle B. Piston force against piston displacement C. Speed of the flywheel against time D. Gas pressure against cylinder volume Answer: A. It is also called the crank-effort diagram: turning moment versus crank angle. (Key point: Turning moment diagram) 2. In a four-stroke engine cycle, which stroke gives the large positive torque and which gives the large negative torque? A. Suction positive; exhaust negative B. Compression positive; expansion negative C. Expansion positive; compression negative D. Exhaust positive; suction negative Answer: C. Gas pressure drives the piston in expansion, while compression absorbs work. Suction and exhaust give small negative torque. (Key point: Torque during strokes) 3. The mean torque of a four-stroke engine is found from: A. Work per cycle / 2 pi B. Work per cycle / 4 pi C. Work per cycle x 4 pi D. Maximum torque / 2 Answer: B. A four-stroke cycle takes 720 deg = 4 pi rad, so T_mean = work per cycle / 4 pi. (Key point: Mean torque) 4. Fluctuation of energy is: A. The work done per cycle B. Max minus min kinetic energy of the flywheel in a cycle C. The area of the expansion loop only D. The mean kinetic energy of the flywheel Answer: B. Delta E = E_max - E_min, the largest swing of cumulative area between the torque curve and the mean torque line. (Key point: Fluctuation of energy) 5. The coefficient of fluctuation of speed C_s is: A. (w_max - w_min) / w_mean B. (w_max + w_min) / w_mean C. w_mean / (w_max - w_min) D. Delta E / work per cycle Answer: A. C_s = (w_max - w_min)/w_mean with w_mean = (w_max + w_min)/2; its reciprocal is the coefficient of steadiness. 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