Flywheels and Turning-Moment Analysis

Introduction and Context

Machines driven by engines, motors, and reciprocating mechanisms rarely receive or deliver torque uniformly throughout a cycle. In a single-cylinder engine, for example, useful work is produced mainly during the power stroke, whereas the other strokes consume energy. A punching press absorbs energy rapidly during a short working interval and may require energy storage between successive operations. These variations produce fluctuations in shaft speed unless an energy-storage device is provided.

A flywheel is a rotating energy-storage device. It absorbs excess energy when the driving torque exceeds the resisting torque and returns energy when the resisting torque exceeds the driving torque. A governor performs a different function: it regulates the mean speed over several cycles by controlling the energy supplied to the engine. Confusing these two functions is a common design error.

Turning-moment analysis provides the basis for determining instantaneous torque, work per cycle, fluctuation of energy, coefficient of fluctuation of speed, and flywheel size.

Learning Objectives

After studying this chapter, the reader should be able to:

• construct and interpret turning-moment diagrams;
• distinguish driving torque, resisting torque, mean torque, and fluctuating torque;
• calculate work and fluctuation of energy from torque–angle diagrams;
• determine flywheel mass or moment of inertia;
• explain the difference between flywheel action and governor action;
• apply these principles to engines, presses, pumps, and cyclic machines;
• identify assumptions, limitations, and practical design considerations.

1. Fundamental Concepts

1.1 Turning moment and angular displacement

The turning moment, or torque, about a shaft is

where T is torque in N·m, F is the tangential force in N, and r is the perpendicular distance from the shaft axis in m. More generally, if a force acts at an angle, only its tangential component contributes to torque.

For an incremental angular displacement dθ, the work done by torque T is

where θ is measured in radians. Thus, the work over a complete cycle is

for a single-revolution cycle. For a cycle occupying θ_c radians, the upper limit is θ_c. Since radians are dimensionless, N·m multiplied by radians gives joules.

The instantaneous power transmitted by a shaft is

where P is power in watts, T is torque in N·m, and ω is angular speed in rad/s. If rotational speed is N revolutions per minute,

These equations assume that torque and angular velocity are measured about the same shaft and that losses are either negligible or included in the torque balance.

1.2 Driving and resisting torques

Let T_d(θ) be the driving torque and T_r(θ) the resisting torque. The net accelerating torque is

If T_a is positive, the shaft accelerates; if it is negative, the shaft decelerates. For steady operation over a complete cycle, the net work must be zero:

Therefore, the work supplied per cycle equals the work absorbed per cycle. This condition does not mean that torque is constant at every instant.

The mean resisting torque is defined by

where W_r is the resisting work per cycle. For a machine operating at constant mean speed, the mean driving torque equals the mean resisting torque, provided losses are included consistently.

2. Turning-Moment Diagrams

A turning-moment diagram plots torque against crank angle or shaft angle. The horizontal axis represents angular position, usually in degrees, and the vertical axis represents torque. For an engine, the diagram may be obtained from gas pressure, piston geometry, connecting-rod motion, inertia forces, and friction. For a machine tool or press, it may be obtained from the required work profile of the operation.

The area under a torque–angle curve represents work. If the resisting torque is constant, it appears as a horizontal line. The area between the driving-torque curve and this line represents the change in kinetic energy of the rotating system.

When driving torque lies above resisting torque, the flywheel gains energy. When it lies below resisting torque, the flywheel loses energy. The cumulative area must return to its initial value at the end of a complete steady cycle.

2.1 Construction procedure

A practical construction procedure is as follows:

1. Divide the cycle into angular intervals.
2. Determine the torque at each angular position.
3. Plot driving torque against angle.
4. Calculate the mean torque from total work divided by cycle angle.
5. Draw the mean resisting-torque line, or plot the actual resisting torque if it varies.
6. Determine cumulative work by integrating the torque difference over successive intervals.
7. Identify the maximum and minimum cumulative values.

For numerical data, the trapezoidal rule is commonly used:

where W_i is the work over one interval, T_i and T_{i+1} are endpoint torques, and Δθ is the angular interval in radians. The approximation improves as the interval becomes smaller or as the curve is represented more accurately.

2.2 Turning-moment and energy diagrams

A turning-moment diagram shows instantaneous torque. An energy diagram, sometimes called a cumulative work diagram, shows the accumulated integral of torque with respect to angle. If the cumulative work curve is denoted E(θ), then

The fluctuation of energy is the difference between the maximum and minimum values of E(θ):

This quantity is the maximum energy that the flywheel must release and absorb during the cycle. It is not the total work supplied per cycle.

3. Flywheel Theory

3.1 Energy stored in a rotating flywheel

The rotational kinetic energy of a flywheel is

where I is the mass moment of inertia in kg·m² and ω is angular speed in rad/s. If the speed changes from ω₁ to ω₂, the change in stored energy is

For a flywheel operating near a mean speed ω, with a relatively small speed fluctuation Δω,

This follows by neglecting the small second-order term ½I(Δω)². The approximation is appropriate when Δω is much smaller than ω.

The coefficient of fluctuation of speed is

where ω̄ is the mean angular speed. Since angular speed is proportional to rpm, the equivalent expression is

For small fluctuations, the required flywheel inertia is therefore

This formula assumes that ΔE is the energy fluctuation over the relevant cycle, speed fluctuation is small, and the flywheel is the dominant energy-storage element. If the shaft, pulley, coupling, and other rotating parts have significant inertia, their equivalent inertia must be included.

3.2 Flywheel sizing from allowable speed fluctuation

The exact relation may be written using the limiting speeds:

Hence,

The approximate formula is usually convenient for preliminary design; the exact formula is preferred when speed variation is not small.

If a rim is treated as a thin ring of mass m and mean radius R,

Thus,

For a disk, the inertia is approximately

if the mass is distributed uniformly to radius R. Real flywheels often include a hub, arms, and rim; their total inertia should be calculated by summing component inertias or by using a manufacturer’s certified value. Concentrating mass near the rim stores more energy for a given total mass, but structural stresses and rim speed impose limits.

3.3 Stress and practical limits

The approximate circumferential stress in a thin rotating ring is

where σ_t is tensile stress in Pa, ρ is material density in kg/m³, and v is rim speed in m/s. Since

rim stress increases with the square of speed. The expression is an idealized thin-ring relation; actual design must account for geometry, keyways, shrink fits, stress concentrations, fatigue, thermal effects, overspeed margins, and applicable safety standards.

A flywheel should be balanced statically and dynamically. It must also be guarded because failure at high speed can release substantial kinetic energy. Inspection for cracks, corrosion, looseness, and fatigue damage is essential.

4. Worked Example: Flywheel Inertia

A machine has a fluctuation of energy of 4.8 kJ per cycle. Its mean speed is 300 rpm, and the permissible coefficient of fluctuation of speed is 0.02. Determine the required flywheel moment of inertia, neglecting other rotating inertia.

First convert speed:

Using the small-fluctuation relation,

Substitution gives

Therefore, the required flywheel inertia is approximately 243 kg·m². This is a preliminary value. A final design would include the inertia of all rotating components, the actual speed limits, transient conditions, shaft strength, balancing, and a suitable safety factor.

5. Worked Example: Fluctuation of Energy from Discrete Torque Data

Suppose a machine completes one cycle through six equal angular intervals of 60°. The net accelerating torques at the successive stations are 0, 80, −40, −100, 20, and 40 N·m. Assume the listed values represent the torque at the boundaries and that the final station coincides with the initial station, so the final torque is 0 N·m for closure. Using the trapezoidal rule, determine the approximate fluctuation of energy.

The interval angle is

The interval works are:

The cumulative works are approximately 41.9, 62.8, −10.5, −52.4, −20.9, and 0 J. Hence,

The positive and negative areas balance, as required for steady operation. The result is an approximation because the torque between stations has been represented by straight lines.

6. Flywheel versus Governor

A flywheel and a governor both influence speed, but they operate on different time scales and by different mechanisms.

A flywheel cannot maintain constant mean speed when the load remains permanently higher than the available driving power. It can only supply stored energy temporarily. A governor cannot eliminate rapid cyclic speed fluctuations if the engine torque itself is highly uneven; that is the flywheel’s role.

7. Applications

7.1 Reciprocating engines

In a single-cylinder engine, gas pressure produces a high positive torque during the power stroke. Compression and the remaining strokes require torque. The flywheel absorbs energy during the power stroke and returns it during the other strokes. Multi-cylinder engines reduce the magnitude of fluctuation by phasing their power strokes, but a flywheel is still commonly required.

The torque diagram of an engine depends on piston position, gas pressure, connecting-rod angle, inertia forces, friction, and pumping work. A correct analysis must distinguish indicated work from brake work and include mechanical losses when sizing a flywheel for the actual crankshaft.

7.2 Punching and shearing machines

A press may require a large amount of energy during a short part of its stroke. The motor supplies average power, while the flywheel supplies peak power during cutting or forming. After the operation, the motor restores the flywheel’s speed. The design must consider stroke frequency, material thickness, shear geometry, clutch engagement, and repeated thermal loading.

7.3 Machine tools and pumps

Planers, reciprocating pumps, compressors, and positive-displacement machines generate periodic torque demand. A flywheel reduces pulsation and helps prevent stalling. In pumps, it may also reduce pressure pulsation and mechanical vibration, although hydraulic accumulators or variable-speed drives may be more suitable in some systems.

7.4 Energy-recovery systems

Modern regenerative systems use rotating or electrical storage to absorb transient energy. A high-speed motor-generator and flywheel can recover braking energy and return it during acceleration. Such systems require containment, vacuum or low-friction bearings, power electronics, monitoring, and fault protection. Their control objective remains distinct from basic cyclic smoothing.

8. Applied Case: Illustrative Punch-Press Design

Background

Consider an illustrative mechanical punching press operating at 120 strokes per minute. The motor provides nearly constant average power, but punching requires 2.0 kJ during a short interval. Between punches, the motor restores energy to the flywheel.

Problem

The press designer permits a coefficient of speed fluctuation of 0.04 at a mean speed of 120 rpm. The required fluctuation of energy, determined from the press torque–angle diagram, is 2.0 kJ. Determine the ideal flywheel inertia.

Analysis and solution

The mean angular speed is

Using the preliminary sizing formula,

Thus, an idealized design requires approximately 317 kg·m² of total effective rotational inertia. If the motor rotor, shaft, clutch, and gears contribute 55 kg·m², the flywheel itself must provide approximately 262 kg·m², subject to the exact speed-limit calculation and dynamic analysis.

Outcome and engineering interpretation

The result shows why presses often use substantial flywheels even when their average motor power is moderate. The flywheel supplies peak instantaneous energy, while the motor supplies the average energy over the full cycle. The final design must verify clutch torque, acceleration time, shaft torsional stress, rim stress, bearing loads, braking behavior, guarding, and emergency-stop performance.

Learning points

1. Flywheel size is governed by energy fluctuation and permissible speed fluctuation, not by average power alone.
2. Energy supplied by the flywheel must be restored before the next demanding operation.
3. The flywheel does not replace a governor or an appropriate motor controller.
4. Safety assessment is essential because stored rotational energy can be very large.

9. Limitations, Design Practice, and Contemporary Issues

The ideal equations assume a rigid shaft, known torque variation, negligible backlash, small speed fluctuation for the approximate formula, and constant or suitably averaged losses. Real machines may exhibit torsional elasticity, resonance, variable friction, clutch slip, control delays, and nonperiodic loading. A measured torque signal may also contain noise and harmonics that require filtering without removing physically important peaks.

Good practice includes:

• obtain torque data from validated analysis, test, or simulation;
• use actual crank-angle resolution adequate for sharp torque changes;
• distinguish cyclic fluctuation from load transients;
• include the inertia of every rotating component reflected to the shaft;
• check exact maximum and minimum speeds when variation is not small;
• perform stress, fatigue, balancing, overspeed, and containment assessments;
• validate the design through instrumentation and controlled testing.

Computer-aided multibody dynamics and finite-element analysis can calculate torque histories, stresses, and vibration modes. Data-driven and AI-based monitoring can help detect imbalance, bearing degradation, or abnormal torque patterns, but such tools should support—not replace—validated physical models, engineering judgement, and safety certification. Any control system that changes fuel, electrical input, or clutch engagement should be assessed for fail-safe behavior and cybersecurity where it is connected to industrial networks.

Chapter Summary

Turning-moment analysis describes how torque varies with shaft angle and how that variation produces cyclic changes in stored kinetic energy. Work is the area under a torque–angle curve, while fluctuation of energy is the difference between the maximum and minimum cumulative work values. A flywheel absorbs energy when driving torque exceeds resisting torque and returns it during deficient-torque intervals.

The rotational energy relation E_k = ½Iω² leads, for small speed variation, to ΔE ≈ Iω̄²C_s. Consequently, flywheel inertia is determined by the required energy fluctuation, mean speed, and allowable coefficient of speed fluctuation. A governor regulates mean speed by changing energy input; it does not perform the same function as a flywheel. Applications include engines, presses, pumps, compressors, machine tools, and regenerative energy systems. Practical design must also address stress, fatigue, balancing, containment, transient loads, and control-system safety.

Key Terms

Turning moment: Torque about a rotating shaft.

Driving torque: Torque supplied by the engine or motor.

Resisting torque: Torque required by the driven machine and losses.

Accelerating torque: Driving torque minus resisting torque.

Turning-moment diagram: Plot of torque against shaft angle.

Torque–angle area: Work transmitted over an angular interval.

Mean torque: Work per cycle divided by cycle angle.

Cumulative work diagram: Plot of accumulated net work against angle.

Fluctuation of energy: Difference between maximum and minimum cumulative energy.

Flywheel: Rotating member that stores and releases kinetic energy.

Moment of inertia: Measure of resistance to angular acceleration.

Coefficient of fluctuation of speed: Speed range divided by mean speed.

Mean speed: Average operating rotational speed.

Rim speed: Tangential speed at the flywheel radius.

Governor: Device that regulates mean engine speed by controlling energy input.

Cyclic load: Load that repeats with shaft position or time.

Dynamic balancing: Correction of rotating mass distribution to reduce vibration.

Regenerative storage: System that captures and later returns transient energy.

Review Questions

Short Answer Questions

1. What physical quantity is represented by the area under a turning-moment diagram?
2. Define fluctuation of energy in a cyclic machine.
3. State the principal difference between a flywheel and a governor.
4. Why is mass placed near the rim of a flywheel?
5. What assumption permits use of the approximate flywheel-inertia formula?

Long Answer Questions

1. Explain how a turning-moment diagram is constructed and converted into a cumulative energy diagram.
2. Derive the approximate relation between fluctuation of energy, flywheel inertia, mean speed, and coefficient of fluctuation of speed.
3. Discuss flywheel action in a single-cylinder reciprocating engine.
4. Explain the design and safety considerations for a punching-press flywheel.
5. Compare flywheel regulation, governor regulation, and modern regenerative energy storage.

Analytical / Application Questions

1. A machine has an energy fluctuation of 3.6 kJ, mean speed of 240 rpm, and allowable coefficient of speed fluctuation of 0.025. Calculate the required ideal flywheel inertia using the small-fluctuation approximation.
2. A turning-moment diagram is available at equal angular intervals. Describe a numerical procedure for finding the maximum and minimum cumulative energies and explain how the reference datum affects the result.
3. A press operates correctly at steady production speed but stalls when material thickness increases. Determine whether the primary problem is likely flywheel sizing, governor action, motor power, or a combination, and justify the diagnostic measurements required.

Multiple Choice Questions with Answers

Question 1

The area under a torque–angle diagram represents:

A. Angular acceleration

B. Work

C. Rotational speed

D. Power factor

Correct Answer: B. Work

Question 2

A flywheel primarily reduces:

A. Cyclic fluctuation of speed

B. Mean load permanently

C. Fuel consumption directly

D. Bearing friction to zero

Correct Answer: A. Cyclic fluctuation of speed

Question 3

For a rotating shaft, instantaneous power is given by:

A. P = T/ω

B. P = Tω

C. P = Iω

D. P = F/ω

Correct Answer: B. P = Tω

Question 4

If the driving torque exceeds the resisting torque, the flywheel generally:

A. Loses kinetic energy

B. Gains kinetic energy

C. Stops immediately

D. Changes the mean radius

Correct Answer: B. Gains kinetic energy

Question 5

The exact change in rotational kinetic energy between two speeds is:

A. ½I(ω₂² − ω₁²)

B. I(ω₂ − ω₁)²

C. ½m(ω₂ − ω₁)

D. T(ω₂ − ω₁)

Correct Answer: A. ½I(ω₂² − ω₁²)

Question 6

A governor differs from a flywheel because a governor:

A. Stores energy only in its rim

B. Regulates energy input and mean speed

C. Operates only during one crank angle

D. Cannot respond to load changes

Correct Answer: B. Regulates energy input and mean speed

Question 7

A flywheel has ΔE = 1000 J, mean speed ω̄ = 20 rad/s, and C_s = 0.05. Using the approximate formula, its required inertia is:

A. 2 kg·m²

B. 25 kg·m²

C. 50 kg·m²

D. 500 kg·m²

Correct Answer: B. 25 kg·m²

Question 8

For a thin-ring flywheel, the approximate moment of inertia is:

A. ½mR²

B. mR²

C. m/R²

D. 2mR

Correct Answer: B. mR²

Question 9

The coefficient of fluctuation of speed is:

A. Mean speed divided by speed range

B. Speed range divided by mean speed

C. Maximum speed divided by minimum speed

D. Angular acceleration divided by torque

Correct Answer: B. Speed range divided by mean speed

Question 10

The approximate tensile stress relation for an ideal thin rotating ring is:

A. σ_t ≈ ρv²

B. σ_t ≈ ρ/v²

C. σ_t ≈ mv

D. σ_t ≈ Iω

Correct Answer: A. σ_t ≈ ρv²