1、BSI Standards PublicationBS ISO 18431-3:2014Mechanical vibration and shock Signal processingPart 3: Methods of time-frequency analysisBS ISO 18431-3:2014 BRITISH STANDARDNational forewordThis British Standard is the UK implementation of ISO 18431-3:2014. The UK participation in its preparation was e
2、ntrusted to TechnicalCommittee GME/21/2, Mechanical vibration, shock and condition monitoring - Vibration and shock measuring instruments and testing equipment.A list of organizations represented on this committee can be obtained on request to its secretary.This publication does not purport to inclu
3、de all the necessary provisions of a contract. Users are responsible for its correct application. The British Standards Institution 2014.Published by BSI Standards Limited 2014ISBN 978 0 580 77279 5ICS 17.160Compliance with a British Standard cannot confer immunity from legal obligations.This Britis
4、h Standard was published under the authority of the Standards Policy and Strategy Committee on 31 May 2014.Amendments/corrigenda issued since publicationDate T e x t a f f e c t e dBS ISO 18431-3:2014 ISO 2014Mechanical vibration and shock Signal processing Part 3: Methods of time-frequency analysis
5、Vibrations et chocs mcaniques Traitement du signal Partie 3: Mthodes danalyses en temps et frquence et par chelle de tempsINTERNATIONAL STANDARDISO18431-3First edition2014-03-15Reference numberISO 18431-3:2014(E)BS ISO 18431-3:2014ISO 18431-3:2014(E)ii ISO 2014 All rights reservedCOPYRIGHT PROTECTED
6、 DOCUMENT ISO 2014All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior written permission. Permissio
7、n can be requested from either ISO at the address below or ISOs member body in the country of the requester.ISO copyright officeCase postale 56 CH-1211 Geneva 20Tel. + 41 22 749 01 11Fax + 41 22 749 09 47E-mail copyrightiso.orgWeb www.iso.orgPublished in SwitzerlandBS ISO 18431-3:2014ISO 18431-3:201
8、4(E) ISO 2014 All rights reserved iiiContents PageForeword ivIntroduction v1 Scope . 12 Normative references 13 Terms and definitions . 14 Symbols 25 Time-frequency transforms 25.1 Short-time Fourier transform 25.2 Generalized Wigner-Ville transform 35.3 Wavelet transform 4Annex A (informative) Anal
9、ysis of gear tooth fault using Wigner distribution . 5Bibliography 7BS ISO 18431-3:2014ISO 18431-3:2014(E)ForewordISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is norm
10、ally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in t
11、he work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.The procedures used to develop this document and those intended for its further maintenance are described in the ISO/IEC Directives, Part 1. In particular the
12、 different approval criteria needed for the different types of ISO documents should be noted. This document was drafted in accordance with the editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives). Attention is drawn to the possibility that some of the elements of this docum
13、ent may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of any patent rights identified during the development of the document will be in the Introduction and/or on the ISO list of patent declarations received (see www.iso.org
14、/patents). Any trade name used in this document is information given for the convenience of users and does not constitute an endorsement.For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment, as well as information about ISOs adherence to the WTO pr
15、inciples in the Technical Barriers to Trade (TBT) see the following URL: Foreword - Supplementary informationThe committee responsible for this document is ISO/TC 108, Mechanical vibration, shock and condition monitoring.ISO 18431 consists of the following parts, under the general title Mechanical v
16、ibration and shock Signal processing: Part 1: General introduction Part 2: Time domain windows for Fourier Transform analysis Part 3: Methods of time-frequency analysis Part 4: Shock-response spectrum analysisiv ISO 2014 All rights reservedBS ISO 18431-3:2014ISO 18431-3:2014(E)IntroductionTime-frequ
17、ency analysis is used to quantitatively display a vibration or shock in terms of time and frequency. This is useful for analysing vibrations in a machine at varying speeds, e.g. in an automobile at varying engine rotational frequencies. Time-frequency analysis is also used to quantitatively display
18、impulsive responses from machinery, e.g. the response of an impact. The duration of the impact as well as the frequency response is displayed. The frequency response can be displayed in terms of frequency, RPM, or octaves. The four methods included in this part of ISO 18431 are the short-time Fourie
19、r transform, the Wigner-Ville transform, the Choi-Williams transform, and the wavelet transform. When any of these methods is used with the correctly specified parameters, time and frequency components of shocks and vibrations are displayed quantitatively. Quantitative display enables quantitative s
20、pecification of the machinery. ISO 2014 All rights reserved vBS ISO 18431-3:2014BS ISO 18431-3:2014Mechanical vibration and shock Signal processing Part 3: Methods of time-frequency analysis1 ScopeThis part of ISO 18431 specifies methods for the digital calculation of a time-frequency analysis of a
21、given sampled measurement of a physical or engineering quantity, such as acceleration, force, or displacement, over an interval of time. Several mathematical formulations of time-frequency transformations are given with requirements for recording of parameters and recommendations.The data can be obt
22、ained experimentally from measurements of a mechanical structure or obtained from numerical simulation of a mechanical structure. This category of data is very broad because there is a wide variety of mechanical structures, e.g. microscopic instruments, musical instruments, automobiles, manufacturin
23、g machines, buildings, and civil structures. The data can determine the response of machines or of humans to mechanical vibration and shock.2 Normative referencesThe following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For d
24、ated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.ISO 2041, Mechanical vibration, shock and condition monitoring VocabularyISO 18431-1, Mechanical vibration and shock Signal processing Part 1: Gen
25、eral introduction3 Terms and definitionsFor the purposes of this document, the terms and definitions given in ISO 2041, ISO 18431-1, and the following apply.3.1short-time Fourier transformFourier-related transform used to determine the time dependence of the sinusoidal frequency and phase content of
26、 a time-varying vibration3.2Wigner-Ville distributionquadratic time-frequency description of a vibration based on the autocorrelation of a signal3.3generalized Wigner-Ville distributiontime-frequency description of a vibration based on a filtered autocorrelation of a signal3.4Choi-Williams transform
27、generalized Wigner-Ville distribution of a vibration using a specific kernel3.5wavelet transformtime-frequency description of a vibration based on the scaled frequency transformation of a signalINTERNATIONAL STANDARD ISO 18431-3:2014(E) ISO 2014 All rights reserved 1BS ISO 18431-3:2014ISO 18431-3:20
28、14(E)4 Symbolsfs= 1/t, sampling frequency; the inverse of the sampling periodK number of samples in window w(k)m index of scaled frequencyN data block length; the number of sampled points that are transformedn index of timeSx(m,n) short-time Fourier transform of sampled data x(n)T = Nt, total time o
29、f a block of sampled datat = nt, timeu units of the sampled signal, e.g. displacement or velocityV(n,m) Cohen class filter for smoothing the Wigner distributionX(m) discrete Fourier transform of x(n)Xm()namely Xm()= 0 for N/2 mx(n) sampled physical data in the time domainxn()analytic sampled signalf
30、 frequency resolutionn increment of index of timet sample periodx(m,n) Choi-Williams distribution filter parameter of Choi-Williams distributionx(m,n) Wigner distribution of signal x(n)x(k,m) wavelet transform of x(n)5 Time-frequency transforms5.1 Short-time Fourier transformThe short-time Fourier t
31、ransform of a signal is the Fourier transform of segments shorter than a given data block. It shows how the spectrum changes with time. It is defined by Formula (1) and detailed in Reference 3.Smnfxn kwkixkK,exp()=+()() =1201spimkK(1)for0 m K/2 and 0 n N K2 ISO 2014 All rights reservedBS ISO 18431-3
32、:2014ISO 18431-3:2014(E)The following parameters shall be specified: the short-time integration window, K, window function w(k), 0 k K 1, and the increment of the index of time, n.The quantity |Sx(m,n)|2is plotted for quantitative analysis. The units are u2/Hz2.NOTE 1 The increment of the index of t
33、ime is often 1, K/2, or K.NOTE 2 The value of K depends on the time and frequency dependence of the signal to be analysed. Larger values of K show finer frequency details and smaller values of K show finer time details.5.2 Generalized Wigner-Ville transform5.2.1 Wigner-Ville transformThe Wigner-Vill
34、e time-frequency transform of a signal is defined by Formula (2).xmnfNXkrmkXmkiNnk,*exp() ()()=+s 022pi(2)for0 m (N/2) 1, 0 n N, r = min(N/2) 1 m,nThe absolute value of x(m,n) is plotted for quantitative analysis. The units are u2s.NOTE 1 The Wigner-Ville transform is also expressed in terms of the
35、autocorrelation with Formula (3).xknmnfxnkxnkimkn,Re exp*()=+ =222202spi=+ 222202fxnkxnkimkNnkNnsRe exp*()pi()forformn nNmNnNnN,0 22,(3)NOTE 2 The Wigner distribution is derived from the autocorrelation of the complete data block. This allows greater accuracy in the resolution of time and frequency
36、components, but also generates spurious components in the distribution.NOTE 3 The sample period of time in the Wigner distribution is twice that of the sample period of the original data x(n) because the time increments of the autocorrelation have twice the period. Therefore, the time corresponding
37、to sample n is t = n2t.NOTE 4 An example of the resolution and spurious components are shown in Annex A.5.2.2 Choi-Williams transformThe Choi-Williams transform is defined by Formulae (4) and (5).xkNlkNkmnfimkNVkln xl k,exp ,;()=()+=24011spi() ()xlk*(4)whereVklnklnkVln,;exp ,;pi()=()()=14400222and (
38、5)The value of parameter is specified. This parameter is dimensionless. ISO 2014 All rights reserved 3BS ISO 18431-3:2014ISO 18431-3:2014(E)The absolute values of x(m,n) are plotted for quantitative analysis. The units of |x(m,n)| are u2s.NOTE This transform decreases the specific types of spurious
39、components.5.3 Wavelet transform5.3.1 Definition of continuous wavelet transformThe continuous wavelet transform is defined by Formula (6).Okmfxn nkxmnrm,()= () ( )=12220s (6)forr 2m(N 1)where(n) is the mother wavelet used in the analysis.The specific mother wavelet shall be stated. The absolute val
40、ue squared |x(k,m)|2is plotted. The units are u2/Hz2.NOTE The wavelet transform is based on performing analysis over scaled frequencies. Therefore, the frequency resolution is coarser at lower frequencies and finer at higher frequencies. The time resolution is less at lower frequencies and greater a
41、t higher frequencies.5.3.2 Definition of mother waveletThe mother wavelet, (n), determines the appearance of the transform. The choice of the proper mother wavelet is based on the time and frequency characteristics of the signal to be analysed. The definition of the Gaussian enveloped sinusoid is de
42、fined by Formula (7). nc niffn()=( )expexppi22 02s(7)where the constant c is imaginary so that the real part of is real with a magnitude that normalizes over the range of the summation.4 ISO 2014 All rights reservedBS ISO 18431-3:2014ISO 18431-3:2014(E)Annex A (informative) Analysis of gear tooth fa
43、ult using Wigner distributionFor the example of a pinion gear in Reference 6, we consider an input gear with 24 teeth and a pinion gear with 16 teeth. The rotation frequency of the pinion gear is 37,5 Hz, resulting in a mesh frequency of 600 Hz. A gear signal is numerically simulated with 2 048 samp
44、les with a sampling frequency of 6 600 Hz. The upper time half of the signal is set to zero to prevent aliasing in the time domain. The analytic signal representation of the signal is used in Formula (2) to prevent aliasing the frequency domain. The case of perfect gear teeth is modeled as a periodi
45、c series of unit impulses with period 1/600 s or 11 samples. Therefore, x(n) = 1000000000010. The absolute value of the Wigner-Ville transform of the perfect case is shown in Figure A.1. The horizontal axis of the plot is frequency and the vertical axis is pinion rotation angle. The harmonics of the
46、 mesh frequency are seen at multiples of 600 Hz. The positions of the gear meshes occur at multiples of 22,5. The greyscale is chosen so that large values are darker and the differences between the perfect and defective pinion gears are apparent.Keypinion anglef frequencyFigure A.1 The Wigner transf
47、orm of an impulse sequence simulating the vibration of gears meshing Perfect pinion ISO 2014 All rights reserved 5BS ISO 18431-3:2014ISO 18431-3:2014(E)The fourth tooth of the pinion gear is modeled to be defective with a vibration by replacing the fourth impulse and two adjoining zeros in x(n) = 01
48、0 with three data samples so that xdefective(n) = 05 2,0 0,5 . The absolute value of the Wigner-Ville transform is shown in Figure A.2 with the same axes and greyscale. Note that there is a modification in the image at the angular position of the fourth tooth, namely 90. The detection of the defective tooth can then be measured using image processing techniques.Keypinion anglef frequencyFigure A.2 The absolute value of the Wigner transform of a pinion gear with a defective tooth6 ISO 2014 All rights reservedBS ISO 18431-3:2014ISO 18431-3:2014(E)Bibliography1 Mallat S. A wavelet tour of si
