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Whirling Shafts by the Numbers
Critical speeds, supports, unbalance and balancing, centrifugal and gyroscopic loads, starting and vibration checks for rotating machinery
How rotors behave at speed: critical speeds by Dunkerley and Rayleigh, foundation stiffness, balance grades, one- and two-plane balancing, centrifugal and gyroscopic loads, starts and spectra.
Start readingAbout this book
A spinning shaft is a spring, and every rotor carries a little unbalance that pushes it sideways once a revolution. Near one particular speed, the shaft’s critical or whirling speed, those pushes arrive in step with the shaft’s own natural frequency and its whirl grows many times over. Most of what goes wrong with rotating machines, from a prototype that shakes on its first run to a bearing that fails in a year, traces back to that resonance, to the unbalance it amplifies, or to the loads that speed itself creates.
This book shows how to calculate and control them. It starts with the single-disc rotor and the critical speed, then estimates real critical speeds by Dunkerley’s and Rayleigh’s methods, sizes a shaft for strength, twist and whirl together, and adds the flexibility of bearings, frames and foundations. It then turns to unbalance: balance grades, balancing in one plane and in two, from known masses or with trial weights. It finishes with the loads that speed creates, centrifugal and gyroscopic, the start of a heavy rotor, and how to read a machine’s vibration in service.
It is for mechanical designers, machinery and reliability engineers, service technicians and fitters. It assumes school physics, algebra and some familiarity with vectors. 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 machine. This is an illustration. An agricultural equipment maker in Wagga Wagga, New South Wales, designs and builds the HM-30, a 30 kW hammer mill for grinding stock feed, direct-coupled to a two-pole motor at 2,950 rpm.
- Rotor: seven 520 mm steel discs on the shaft between two pillow-block bearings 1,000 mm apart, with four hammer pins at 230 mm radius carrying 48 swinging hammers of 0.75 kg; 180 kg in all, spread over the middle 400 mm of the span.
- Shaft: 80 mm on the prototype, 90 mm in production, 100 mm on the 60 Hz export variant at 3,550 rpm.
- Around it: a narrow discharge fan wheel, balanced in one plane; a truck-mounted variant used in feedlots; and the mill’s customers, foundations and maintenance crews.
Masses, dimensions, stiffnesses, vibration readings, motor and bearing data, prices and times are invented for the case or calculated from stated assumptions. None of it describes KEVOS® work or any real business, product or manufacturer’s rating.
How to use the book
Read Chapters 1 and 2 first: the critical speed and its estimates are used throughout. Chapters 3 and 4 complete the design of the shaft and its supports. Chapters 5 to 7 cover unbalance and balancing and are best read in order. Chapters 8 to 10 cover centrifugal loads, gyroscopic loads and starting, and can be read in any order. Chapter 11 brings the book together in service.
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. The hand methods used are the classical ones; they show which changes matter and give sound first estimates, but they assume rigid rotors and simple supports, and each chapter says where they stop being reliable.
Source limits and professional responsibility
This book is educational. Its methods estimate critical speeds, balance tolerances and loads for rigid rotors on simple supports. Machines that run near or above a critical speed, flexible rotors, high-speed or high-energy rotors and machines whose failure could injure people need rotordynamic analysis, testing and design review by competent people. Balancing and vibration work on running machinery must be done with the machine guarded, isolated between runs and under a safe system of work, as the model WHS laws and the maker’s instructions require. Decisions about real machines remain with the competent people responsible for them.
Who this book is for
Mechanical designers, machinery engineers, maintenance and reliability engineers, service technicians and fitters who design, install, balance or look after fans, mills, pumps and other rotating machines.
What you will learn
- Explain why a shaft whirls near its critical speed, and estimate the first critical speed of a real rotor by Dunkerley's and Rayleigh's methods with a stated margin.
- Size a shaft for strength, twist and critical speed together, and allow for the stiffness of bearings, pedestals, frames and foundations.
- Set a balance tolerance from a balance grade, and balance narrow rotors in one plane and long rotors in two, from known masses or with trial weights.
- Check the centrifugal loads on discs, pins and attachments, the gyroscopic loads on a rotor whose axis turns, and the run-up time and heating of a heavy rotor's start.
- Measure and read vibration in service, recognise the signatures of unbalance, resonance, misalignment and bearing defects, and keep the records that make diagnosis quick.
Inside this book
Chapters
- 01What whirling isWhy a spinning shaft bows into a rotating curve near one particular speed: the critical speed, the single-disc rotor, amplification of unbalance and why rotors settle above it.
- 02Critical speeds of real shaftsEstimating the first critical speed of a shaft carrying several masses: Dunkerley's lower bound, Rayleigh's energy method, how close each is to the exact answer, and how margins are defined.
- 03Sizing a shaft for strength, twist and whirlSizing a rotating shaft against three criteria at once, combined bending and torsion, twist and critical speed, and why on fast, heavy rotors the critical speed usually sets the diameter.
- 04Supports, frames and stiffnessHow flexible bearings, pedestals, frames and mountings lower a shaft's critical speed: springs in series, a bump test that found a soft mezzanine, overhung parts and machines on isolators.
- 05Unbalance and balance gradesWhat unbalance is, the rotating force it makes, how balance quality grades turn a rotor's mass and speed into a permissible residual unbalance, and what one lost hammer does to a hammer mill.
- 06Balancing in one planeBalancing a narrow rotor in one plane: adding unbalances as vectors, sizing and placing a correction weight, the residual left by weight steps and angle errors, and trial-weight field balancing.
- 07Balancing long rotors in two planesWhy a long rotor needs corrections in two planes: resolving unbalances by moments, checking each plane against its grade, weighing hammer sets, and why a static balance left a couple behind.
- 08Centrifugal loads on discs and hammersThe loads that speed itself puts on a rotor: centrifugal force on hammers and pins, hoop stress in rotating discs, why both rise with the square of speed, and a mill run 20% fast on a drive.
- 09Gyroscopic loadsWhy a spinning rotor pushes back when its axis is turned: angular momentum, the gyroscopic couple and bearing reactions, applied to a truck-mounted mill that lifted a bearing on tight turns.
- 10Starting, stopping and torsionGetting a heavy rotor up to speed: inertia, run-up time and the energy a direct-on-line start puts into a motor, resonances passed through on the way, and torsional checks of the drive.
- 11Vibration checks in serviceReading a rotating machine's vibration: levels, trends and spectra, what 1×, 2× and other peaks point to, bearing defect frequencies, and the wrong bearing geometry that hid a failing race.
Apply This in Practice
Use these KEVOS® tools to try the method from this reading with your own numbers.
Applies this method
Bearing Defect Frequency Check (BPFO, BPFI, Ball Spin, Cage, Match to a Measured Peak)
Calculates a rolling bearing's outer race, inner race, roller spin and cage defect frequencies and matches a measured spectrum peak to them.
Used in Vibration checks in service
Applies this method
Centrifugal Force on a Rotating Mass
Gives the centrifugal force, acceleration in g and tangential speed of a mass rotating at a radius, such as a hammer on its pin.
Applies this method
Gyroscopic Bearing Load on a Turning Rotor
Works out a turning rotor's angular momentum, gyroscopic couple and bearing reactions, adds its weight, and flags a bearing load that reverses.
Used in Gyroscopic loads
Applies this method
Permissible Residual Unbalance (Balance Grade)
Turns a rigid rotor's mass, speed and balance grade into its permissible residual unbalance, per plane, and checks a measured residual against it.
Used in Unbalance and balance grades
Applies this method
Rotating Disc Stress and Burst Margin
Gives the maximum centrifugal stress in a flat solid or bored disc, its rim speed, factor against yield and the speed at which it would yield.
Applies this method
Rotor Balancing Check (Grade, Plane Unbalance and Corrections)
Checks each correction plane's unbalance against the grade, gives the corrections and the rotating bearing forces, and checks the weights fit.
Applies this method
Rotor Run-Up Time (Inertia, Acceleration Time, Start Energy)
Gives a rotor's inertia, run-up time, revolutions and stored energy, and the heat a direct-on-line start puts into the motor per start and per hour.
Used in Starting, stopping and torsion
Applies this method
Shaft Critical (Whirling) Speed
Gives the first critical speed of a uniform shaft, alone and with one disc combined by Dunkerley, for simple, overhung or fixed supports, against running speed.
Used in What whirling is
Applies this method
Shaft Critical Speed with Several Masses (Dunkerley and Rayleigh)
Estimates a simply supported shaft's first critical speed with up to three masses by Dunkerley's method, with a masses-only Rayleigh value for comparison.
Used in Critical speeds of real shafts
Applies this method
Shaft Design: Strength, Twist and Critical Speed
Sizes a shaft for combined bending and torsion with shock factors, for a twist limit, and for a critical speed ratio, and says which criterion governs.
Applies this method
Single-Plane Balance Correction Check (Unbalance, Correction Mass, Residual, Grade)
Adds known unbalances as vectors, sizes and places a single-plane correction, and checks the residual left by weight steps and angle error against the grade.
Used in Balancing in one plane
Applies this method
Two-Plane (Dynamic) Balance Corrections
Resolves up to three unbalance masses along a rotor into correction masses and angles in two chosen planes by taking moments.
Related Tools and Examples
Useful alongside
Drive Shaft Torsional Check (Natural Frequency, Resonance Margins, Stress, Twist)
Finds the torsional natural frequency of a two-inertia drive joined by a shaft and checks it against running-speed orders; coupling stiffness is added by hand.
Related to Starting, stopping and torsion
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 31 October 2026.
- H. Kleinert, World Scientific, Singapore. Gauge Fields in Condensed Matter, Volume I: Superflow and Vortex Lines (Disorder Fields, Phase Transitions) (1989). In copyright. Used only as a map for the idea, in Chapter 2, of reading a system's behaviour from the expansion of its energy about a minimum, which underlies Rayleigh's method. No text, equations, figures or problems are reproduced, and all cases and calculations are KEVOS® originals.
- H. H. Jeffcott, Philosophical Magazine, Series 6, volume 37. The lateral vibration of loaded shafts in the neighbourhood of a whirling speed: the effect of want of balance (1919). Named in Chapter 1 as the origin of the single-disc rotor model and its whirl response above and below the critical speed.
- S. Dunkerley, Philosophical Transactions of the Royal Society of London, Series A, volume 185. On the whirling and vibration of shafts (1894). Named in Chapter 2 as the origin of Dunkerley's method of combining critical speeds.
- J. W. Strutt, Lord Rayleigh, Macmillan, London. The Theory of Sound (1877). Named in Chapter 2 as the origin of Rayleigh's energy method for natural frequencies.
- M. I. Friswell, J. E. T. Penny, S. D. Garvey and A. W. Lees, Cambridge University Press. Dynamics of Rotating Machines (2010). Background for the single-disc rotor, support flexibility, gyroscopic effects, forward and backward whirl and rotordynamic models in Chapters 1, 2, 4 and 9.
- International Organization for Standardization. ISO 21940-11 Mechanical vibration — Rotor balancing — Part 11: Procedures and tolerances for rotors with rigid behaviour (2016). Named by number and scope only in Chapter 5 as the standard that defines balance quality grades and the allocation of permissible unbalance to correction planes. No table or value from it is quoted; the case's choice of grade is its own.
- International Organization for Standardization. ISO 21940-12 Mechanical vibration — Rotor balancing — Part 12: Procedures and tolerances for rotors with flexible behaviour (2016). Named by number and scope only in Chapter 7 for the balancing of flexible rotors. No clause or value from it is quoted.
- International Organization for Standardization. ISO 21940-32 Mechanical vibration — Rotor balancing — Part 32: Shaft and fitment key convention (2012). Named by number and scope only in Chapter 5 for the convention for keys during balancing. No clause from it is quoted.
- International Organization for Standardization. ISO 20816-1 Mechanical vibration — Measurement and evaluation of machine vibration — Part 1: General guidelines (2016). Named by number and scope only in Chapter 11 as general guidance on measuring and evaluating machine vibration. The case's acceptance, alarm and trip levels are the case maker's own settings, not values from the standard.
- Safe Work Australia. Model Code of Practice: Managing the risks of plant in the workplace (2022). Background for the safety notes on guarding rotating parts, isolating machines between runs and safe field balancing, under the model WHS laws.