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Inertia and Flywheels by the Numbers
Moments of inertia, reflected inertia, servo ratios and duty cycles, heavy starts, flywheels and spinning safely, on one timber line
How to work with rotating inertia: moments of inertia, reflection through drives, ratio choice, move profiles and RMS torque, heavy starts, flywheel energy and design, and safe rim speeds.
Start readingAbout this book
Inertia is the property that makes rotating machinery hard to start, hard to stop and useful for storing energy. It decides how big a servo motor must be to move a carriage in time, how long a large fan or chipper takes to reach speed and how hot its motor gets doing so, and how much a flywheel smooths a machine whose load comes in bursts. It also decides how much energy a rotor carries, and so how much a guard must contain if something comes apart.
This book shows how to work with inertia by the numbers. It starts with the moments of inertia of machine parts and built-up rotors and the units they are quoted in, then the torque to change speed, the reflection of inertia through drives, the choice of a gear ratio, and the move profiles and RMS torque that size a servo. It then covers heavy starts, flywheels and their design, and finishes with the stresses and energies of spinning parts.
It is for mechanical, machine design, automation and maintenance engineers. It assumes engineering training and comfort with torque, speed and energy; the formulas are explained as they are used. Each chapter works the running case with real numbers, includes one plausible mistake and how it was found, links the KEVOS® tools that do the chapter’s calculations, and ends with an exercise and its worked answer.
The running case
Every chapter follows one production line. This is an illustration. A sawmill near Mount Gambier, South Australia, is commissioning DL-3, a docking and chipping line that cuts sawn boards to length and chips the offcuts.
- Infeed: eight lagged steel rollers, 180 mm over the lagging, feeding boards at 1.5 m/s.
- Length pusher: a servo-driven carriage on a toothed belt that pushes each board to length against the saw line, with moves of up to 1.8 m at 2 m/s.
- Docking saw: a 900 mm circular saw blade on a direct-coupled 15 kW motor at 1,480 rpm.
- Chipper: a 1,400 mm steel disc chipper at 600 rpm, belt-driven by a 75 kW motor at 1,480 rpm, started direct on line.
Masses, dimensions, speeds, motor data, limits and loads are the case mill’s working figures or are calculated from stated assumptions. They do not describe KEVOS® work or any real business, machine or supplier’s data.
How to use the book
Chapters 1 and 2 build the inertias the rest of the book uses. Chapters 3 to 6 cover accelerating loads and sizing drives, ending with servo duty cycles. Chapter 7 covers heavy starts. Chapters 8 and 9 cover flywheels, and Chapter 10 covers the stresses, speeds and energies of spinning parts. Readers interested only in servo sizing can read Chapters 1 to 6; those interested in flywheels and heavy rotors can read Chapters 1 to 3 and 7 to 10.
Every calculation in the book was checked by running it, and the KEVOS® tools linked from each chapter were checked against the calculations they perform.
Source limits and professional responsibility
This book is educational. Motor starting limits, servo ratings, rim speed limits and material strengths come from the makers of the machines and materials concerned, and the figures used here are illustrations. Rotating machinery stores energy that can injure or kill; its guarding and stopping are governed by the model WHS laws and by Australian machinery safety standards, and a real design must be done or checked by a competent engineer. Work on such machinery is done with it stopped, isolated and locked out. Decisions about real machines remain with the competent people responsible for them.
Who this book is for
Mechanical and machine design engineers, automation and controls engineers, and maintenance engineers who size motors, servos, gearboxes and flywheels or check the rotating parts of production machinery.
What you will learn
- Calculate the moment of inertia of machine parts and built-up rotors, and convert datasheet inertias between units without the common factor-of-four error.
- Find the torque to accelerate and stop a rotating load, and reflect inertia and load torque through gears, belts and pulleys to the motor.
- Choose a gear ratio for a servo axis, build a move profile, and check RMS and peak torque against a motor's ratings.
- Estimate the run-up time and rotor heating of a heavy direct-on-line start, and judge when a start is acceptable.
- Size a flywheel for an energy fluctuation and speed variation, and check rotating disc stress, rim speed and the energy that guarding must contain.
Inside this book
Chapters
- 01Moments of inertia of machine partsHow to calculate the moment of inertia and radius of gyration of discs, rings, tubes, blocks and offset parts, and why the mass far from the axis is the mass that counts.
- 02Built-up rotors and datasheet unitsHow to add the parts of a built-up rotor into one inertia with its stored energy, and how to convert the inertias on motor and servo datasheets, including WR² and GD², without a factor-of-four error.
- 03Torque to speed up and slow downHow much torque it takes to bring a rotor to speed or stop it in a given time, how many turns and how much energy that involves, and how a motor's rating compares with the torque the change needs.
- 04Inertia through gears and beltsHow inertia and load torque look from the motor through a gearbox, belt or screw: the ratio squared, linear masses as m r², efficiency, inertia ratio and accelerating torque.
- 05Choosing the ratioHow the gear ratio trades motor torque against speed, why accelerating torque is least when reflected and rotor inertia are equal, and how top speed and real gearboxes set the ratio.
- 06Move profiles and RMS torqueHow to turn a repeated move into acceleration, run, deceleration and dwell torques, check RMS torque against a motor's continuous rating and peak torque against its peak, and choose the profile shape.
- 07Starting a heavy rotorHow long a motor takes to bring a heavy rotor to speed, how much energy it dissipates in its own rotor while doing so, and how start time and starts per hour are checked against the motor's limits.
- 08Flywheels and speed fluctuationHow a flywheel carries a machine through peak loads by slowing: energy released between two speeds, the coefficient of speed fluctuation, rim speed, and the inertia a duty needs.
- 09Designing a flywheelHow to size a flywheel rim from an energy fluctuation and speed variation: inertia, bore, mass, rim speed and rotating stress, and the effect of putting it on the fast or slow shaft.
- 10Spinning safelyHow to check a rotating disc's stress, rim speed and speed margin, why a central hole doubles the stress, what an overspeed does, and how much energy a guard must contain if a rotor fails.
Apply This in Practice
Use these KEVOS® tools to try the method from this reading with your own numbers.
Applies this method
Composite Rotor Inertia and Energy (Up to Three Shapes, Offsets, Stored and Released Energy)
Adds up to three bodies about a common axis into a rotor's inertia and radius of gyration, with its stored energy and the energy released for an allowed speed drop.
Applies this method
Flywheel Energy and Speed Fluctuation
Gives a disc or ring flywheel's inertia, the energy released between two speeds, the stored energy, the coefficient of speed fluctuation and the rim speed against a limit.
Used in Flywheels and speed fluctuation
Applies this method
Flywheel Rim Design (Inertia for Speed Fluctuation, Size and Stress)
Sizes an annular flywheel from an energy fluctuation and speed variation: inertia needed, bore, mass, speed range, rim speed and rotating stress against a safety factor.
Used in Designing a flywheel
Applies this method
Gear Ratio for Inertia Matching (Optimum Ratio and Motor Torque)
Finds the gear ratio that minimises accelerating torque, the largest ratio within the motor's top speed, and checks a chosen ratio against the motor's speed and peak torque.
Used in Choosing the ratio
Applies this method
Mass Moment of Inertia
Gives the moment of inertia and radius of gyration of discs, hollow cylinders, blocks, spheres and rods, with a parallel-axis transfer to an offset rotation axis.
Applies this method
Mass Moment of Inertia: Cone, Torus, Ellipsoid and Shells
Gives the moment of inertia of thin-walled cylinders, cones, tori, ellipsoids, shells and rings, used here for a roller tube treated as a thin wall at its mean diameter.
Applies this method
Moment of Inertia, Section Modulus and Momentum Converter
Converts motor and servo datasheet inertias between kg·m², kg·mm², g·cm², lb·ft², lb·in², oz·in² and slug·ft², keeping mass moment of inertia apart from second moment of area.
Applies this method
Reflected Inertia and Motor Acceleration Torque
Reflects a load's inertia and torque through a reduction to the motor and gives the inertia ratio, motor speed, accelerating torque and peak motor torque for a move.
Used in Inertia through gears and belts
Applies this method
RMS Torque for a Motor Duty Cycle
Gives the RMS and peak torque of a cycle of up to five segments against a motor's continuous and peak ratings, for cutting lists the trapezoid cannot describe.
Used in Move profiles and RMS torque
Applies this method
Rotating Disc Stress and Burst Margin
Gives the largest centrifugal stress in a solid or bored disc, its rim speed, its safety factor against yield and the speed at which yield is reached.
Used in Spinning safely
Applies this method
Rotor Run-Up Time (Inertia, Acceleration Time, Start Energy)
Gives a heavy rotor's run-up time, turns and stored energy on a direct-on-line start, and the energy dissipated in the motor's rotor per start and per hour against limits.
Used in Starting a heavy rotor
Applies this method
Servo or Motor Duty Cycle Check (Move Profile, RMS and Peak)
Builds a trapezoidal move's acceleration, run, deceleration and dwell torques and checks RMS torque, peak torque, inertia ratio and energy returned against a motor's ratings.
Used in Move profiles and RMS torque
Applies this method
Torque to Accelerate a Rotating Load
Gives the angular acceleration, accelerating and total torque, turns, energy change and peak power to start or stop a rotor such as a saw blade in a set time.
Used in Torque to speed up and slow down
Related Tools and Examples
Useful alongside
Torque, Power and Speed
Converts a motor's rated power and speed into its rated torque, for comparison with the torque a start or stop needs.
Related to Torque to speed up and slow down
References and source notes
The manuscript and worked examples are original KEVOS® writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 1 November 2026.
- C. J. Pethick and H. Smith, Cambridge University Press. Bose–Einstein Condensation in Dilute Gases (2002). In copyright. Used only as a map of ideas: the moment of inertia of a rotating body, angular momentum and energy held in rotation, rotation viewed from a turning frame, and the release of energy as a system's speed changes, which this book applies to machine rotors and flywheels. No text, equations, figures or problems are reproduced, and all cases and calculations are KEVOS® originals.
- R. G. Budynas and J. K. Nisbett, McGraw-Hill Education. Shigley's Mechanical Engineering Design, tenth edition (2015). Background for flywheel energy and the coefficient of speed fluctuation, and for stresses in rotating rings and discs, used in Chapters 8 to 10.
- S. P. Timoshenko and J. N. Goodier, McGraw-Hill. Theory of Elasticity, third edition (1970). Background for the stresses in a rotating disc of uniform thickness, solid and with a central hole, used in Chapters 9 and 10.
- W. C. Young, R. G. Budynas and A. M. Sadegh, McGraw-Hill. Roark's Formulas for Stress and Strain, eighth edition (2012). Background for rotating disc formulas and their limits, used in Chapter 10.
- International Electrotechnical Commission. IEC 60034-1 Rotating electrical machines, Part 1: Rating and performance (2022). Named by number and scope only in Chapters 6 and 7, as the standard that defines motor duty types and rating conditions. No values are quoted from it; motor limits in this book are the case's working figures and are labelled as illustrations.
- Safe Work Australia. Model Code of Practice: Managing the risks of plant in the workplace (2022). Named in Chapters 3 and 10 for the duty to manage the risks of plant under the model WHS laws, including guarding and stopping rotating parts.