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Random Vibration by the Numbers

Spectra, RMS levels, Miles' equation, isolation, peaks, fatigue and shaker testing, worked through the electrics on a mobile screening plant

Specify, analyse and test equipment for random vibration: power spectral density, RMS and peaks, single-degree-of-freedom response, isolation, fatigue under random loading and test time compression.

9 chapters · About 90 minutes · Free to read · Intermediate mechanical and product engineering

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KEVOS® LibraryENGINEERINGRandom Vibrationby the NumbersKEVOS® ORIGINAL BOOK9 CHAPTERS

About this book

A machine on a truck, a box on a vibrating screen, a product in a delivery van: none of them is shaken by a single tidy sine wave. They are shaken by a jumble of frequencies whose amplitudes rise and fall at random, the combined effect of rough roads, engines, gears, impacts and structures ringing in sympathy. That jumble has a precise description, the power spectral density, and from it an engineer can predict how hard each part of the equipment will be shaken, how far it will move, how often its peaks will reach any level, and how long it will last before fatigue cracks it.

This book works through that chain of ideas. It starts with random vibration as a random variable and with the spectrum that describes it, shows how a spectrum is measured and how uncertain the measurement is, and then follows the vibration into a resonant part, through isolators and into the boards and brackets inside a box. It finishes with peaks, fatigue under random loading, and the shaker tests that compress years of service into hours.

It is for mechanical, electrical and product engineers, test engineers, technicians and students who meet vibration requirements and want to understand and check them. It assumes the idea of a natural frequency, basic statistics and logarithms. Each chapter works the running case with real numbers, includes one plausible mistake and how it was found, links the KEVOS® tool that performs the chapter’s calculation where one exists, and ends with an exercise and its worked answer.

The running case

Every chapter follows one fictional installation. This is an illustration. A mobile screening plant for quarries carries its control equipment on its own frame: a diesel engine running at 1,500 rpm drives the vibrating screen and a generator, and the electrics sit in a steel enclosure, CB-7, bolted near the engine.

  • The enclosure: 18 kg with its contents, on four mounts; inside, a main circuit board whose first natural frequency is about 180 Hz.
  • The vibration specification: a random profile from 5 to 500 Hz, rising at 3 dB per octave from 0.0025 g²/Hz at 5 Hz to 0.02 g²/Hz at 40 Hz, flat at 0.02 g²/Hz to 200 Hz, then falling to 0.005 g²/Hz at 500 Hz: 2.55 g RMS in all.
  • A sensor bracket: a steel bracket carrying a proximity sensor and junction box, bolted directly to the screen frame, with a natural frequency of about 120 Hz.
  • Service: about 2,000 operating hours a year for ten years.

The plant, its profile, its frequencies and any material data are invented. None describes KEVOS® work or a real product. Accelerations are in g, where 1 g = 9.807 m/s².

How to use the book

Read Chapters 1 to 4 in order: they set out the random-variable view, the spectrum, measurement and the single-degree-of-freedom response on which everything else rests. Chapters 5 and 6 follow the vibration into the enclosure and should be read together. Chapters 7 and 8 deal with peaks and fatigue, and Chapter 9 with testing.

The methods are the classical hand methods: Miles’ equation, transmissibility, the Rayleigh distribution of peaks and Miner’s rule. They are the right tools for first sizing and for checking finite element and test results, and the book states the assumptions behind each.

Source limits and professional responsibility

Real structures have many modes, non-linear joints and damping that varies with amplitude, and real environments are not always stationary or Gaussian. The hand methods here give estimates whose errors can be large, and fatigue predictions in particular can differ by a factor of ten or more depending on assumptions the data cannot settle. Equipment whose failure could cause harm must be analysed and tested by qualified people to the standards that apply to it.

Shaker testing uses large forces and electrical power, and test articles can break and release parts. Test laboratories manage those hazards under work health and safety law. This book teaches how to understand and calculate random vibration. It is educational and is not a design or a test specification for any product.

Who this book is for

Mechanical, electrical and product engineers, test engineers, technicians and students who must design or check equipment that is shaken by vehicles, machines or transport, and plan or read vibration tests.

What you will learn

  • Describe random vibration by its power spectral density, calculate the RMS level of a profile and read its peaks from the normal distribution.
  • Estimate a power spectral density from measured data and state the statistical uncertainty of the estimate.
  • Predict the response of a resonant part or a mounted box with Miles' equation, and its relative displacement.
  • Select isolators that avoid machine forcing frequencies and reduce random vibration, and check sway space.
  • Estimate fatigue damage under random loading with the Rayleigh peak distribution and Miner's rule, and plan a time-compressed shaker test.

Inside this book

Chapters

  1. 01What random vibration isRandom vibration as a random variable: stationary signals, why amplitudes are Gaussian, RMS as the standard deviation, how often peaks reach 2σ, 3σ and beyond, and how random differs from sine.
  2. 02Power spectral density: where the energy sitsThe acceleration spectral density in g²/Hz, why its area is the mean square, reading log-log profiles in dB per octave, calculating RMS by segments, band levels, and displacement from a spectrum.
  3. 03Measuring a spectrum and trusting itFrom accelerometer to PSD: sampling and aliasing, resolution and record length, averaging and the statistical error of a spectrum, resolution bias, and checks for Gaussian, stationary data.
  4. 04One resonance and Miles' equationHow a single-degree-of-freedom part responds to random base vibration: transmissibility, Q and bandwidth, the response spectrum, Miles' equation for RMS response, relative displacement and its limits.
  5. 05Isolating the enclosureChoosing isolators against both a machine tone and broadband random vibration: frequency ratio and transmissibility, the isolator tool, Miles' response on soft mounts, sway space, damping and run-up.
  6. 06Inside the enclosure: boards and nested resonancesFollowing vibration through mounts, panels and circuit boards: transmissibility in series, board deflection, why nested resonances should be an octave apart, and heavy components and their mounting.
  7. 07Peaks, cycles and the Rayleigh distributionHow often a resonant response reaches each level: zero-crossing rate, the Rayleigh distribution of peaks, the largest peak to expect in a given time, and the three-band approximation.
  8. 08Fatigue under random loadingFrom RMS stress to fatigue life: the S–N line, Miner's rule with three bands and with the Rayleigh distribution, and how the endurance limit assumption swings the predicted life.
  9. 09Shaker testing and time compressionPlanning a random vibration test: what a test proves, compressing years into hours with the fatigue exponent, the limits of exaggeration, shaker force and stroke, fixtures, control and safety.

Apply This in Practice

Use these KEVOS® tools to try the method from this reading with your own numbers.

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 30 October 2026.

  • D. S. Lemons, Johns Hopkins University Press. An Introduction to Stochastic Processes in Physics (2002). In copyright. Used only as a map of the physics behind the book: random variables and the normal distribution, the central limit theorem, the Langevin equation and the stochastically driven damped harmonic oscillator. No text, structure, figures, tables or problems are reproduced.
  • J. W. Miles, Journal of the Aeronautical Sciences 21(11), 753–762. On structural fatigue under random loading (1954). Named in Chapter 4 as the origin of Miles' equation for the RMS response of a single-degree-of-freedom system to broadband random input.
  • R. Simmons, NASA Goddard Space Flight Center. The Miles Equation (FEMCI workshop poster) (2001). Cited in Chapter 4 for the equation's single-degree-of-freedom and flat-input assumptions and the errors that follow when a spectrum is shaped near the resonance.
  • S. O. Rice, Bell System Technical Journal 23 and 24. Mathematical analysis of random noise (1944–1945). Named in Chapter 7 as the origin of the expected rate of zero crossings and the peak distribution of a Gaussian random process.
  • J. S. Bendat and A. G. Piersol, John Wiley & Sons. Random Data: Analysis and Measurement Procedures, 4th edition (2010). Named in Chapter 3 as the standard text on estimating power spectral densities and their statistical error. No text or tables are reproduced.
  • D. S. Steinberg, John Wiley & Sons. Vibration Analysis for Electronic Equipment, 3rd edition (2000). Named in Chapters 6 and 8 as the source of the three-band approach to random fatigue and of the practice of separating the natural frequencies of nested assemblies. No text or tables are reproduced.
  • National Aeronautics and Space Administration. NASA-HDBK-7005 Dynamic Environmental Criteria (2001). Named as a public handbook on predicting and testing for dynamic environments, including random vibration. No values are quoted.
  • International Electrotechnical Commission; US Department of Defense. IEC 60068-2-64 Environmental testing, Part 2-64: Tests, Test Fh: Vibration, broadband random and guidance; MIL-STD-810H Method 514.8 Vibration (current editions). Named by number and scope in Chapter 9 as the test standards most often specified for random vibration testing. No clauses, profiles or values are quoted; the book's profile is an illustration.
  • Standards Australia. AS 4100 Steel structures, Section 11 Fatigue (current edition). Named by number and scope in Chapter 8 for the treatment of fatigue in welded steel details with S–N curves that change slope below the constant-amplitude fatigue limit. No values are quoted.

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