Fundamentals Of Signal Analysis - Iran University Of .

Transcription

Vibration Measurement SystemsFundamentals ofSignal AnalysisH.AhmadianCovering TopicsFundamentals of Signal Analysis Introduction Time and Frequency Domains: A matter of Perspective The Time DomainThe Frequency DomainUnderstanding Dynamic Signal Analysis See Section 83.1 Spectrum Analysis and CorrelationBasics of Discrete Fourier Transform (DFT)AliasingLeakageWindowingFilteringImproving ResolutionH. AhmadianMeasurement Systems2

Introduction The measured vibrationsignals are in timedomain.The signals are digitizedby an A/D converterAnd recorded as a set ofN discrete values evenlyspaced in the period TH. AhmadianMeasurement Systems3H. AhmadianMeasurement Systems4

Basics of DFT The spectral properties of the recordedsignal can be obtained using DiscreteFourier Transform/Series (DFT/DFS): The DFT assumes the signal x(t) is periodicIn the DFT there are just a discrete number ofitems of data in either form There are just N values xkThe Fourier Series is described by just N valuesH. Ahmadian5Measurement SystemsBasics of DFTx (t )x (t )Znanbnx (t T )a0 an cos(Z n t ) bn sin(Z n t )2 n12Sn,T2 Tx (t ) cos(Z n t )dtT ³02 Tx (t ) sin(Z n t )dtT ³0orx (t )N 12xka02Snk2Snk) bn sin() an cos(NN2 n1an2Nfbn X eiZ n tnX nH. Ahmadian1 Tx (t )e iZnt dtT ³0X n*cos(k 0ksin(N 1 xk 0N 1fk x2Snk)N2Snk)Norxkn fXn2NN 1XnMeasurement Systems2 ink / N X e Sn 01NnN 1 x ek 0 2Sink / Nk6

Basics of DFT 2S ink1 fXnx ke N N k 1 X 0 ½ X1 . ¾. . X N 1 ¿ªn««n«.««.«.«« nH. Ahmadian. . . .nn.o . . . . . . . . . . . . . . . . .Measurement Systems x 0 ½ n º x 1 » » . . » . » ¾. » . » .» .»¼ x N 1 ¿7Basics of DFT The sampling frequency:1 N2S 2SN Zsts TtsT The range of frequency spectrum:fsf max fs Zmax2ZsSN2Nyquist FrequencyTThe resolution of frequency spectrum:'fH. Ahmadian1, 'ZT2STMeasurement Systems8

Basics of DFT There are a number of features of DFanalysis which if not properly treated, cangive rise to erroneous results: Aliasing Mis-interoperating a high frequency component as alow frequency oneLeakage Periodicity of the signalH. Ahmadian9Measurement SystemsAliasing Digitizing a ‘low’ frequencysignal produces exactly thesame set of discrete values asresult from the same processapplied to a higher frequencysignalH. AhmadianMeasurement SystemsZZs2Zs Z10

AliasingCompare :kksin(2Sp ) sin(2S ( N p ) N )N2Spksin(2Sk )N2Spk) sin(Np N2H. AhmadianMeasurement Systems11Measurement Systems12AliasingH. Ahmadian

Aliasing The solution to theproblem is to use an antialiasing filter Subjecting the originalsignal to low pass withsharp filterFilters have a finite cut-offrate; it is necessary toreject the spectral rangenear Nayquist frequencyZ ! (08 1.0)H. AhmadianZs2Measurement Systems13Leakage A direct consequence oftaking a finite length oftime history coupled withassumption of periodicityEnergy is leaked into anumber of spectral linesclose to the truefrequency.H. AhmadianMeasurement Systems14

H. AhmadianMeasurement Systems15Measurement Systems16LeakageH. Ahmadian

Leakage To avoid the leakage there are number ofscenarios: Increasing the record time TWindowing Multiply the time record by a function that is zero atthe ends of the time record and large in the middle,the FFT content is concentrated on the middle of thetime recordH. AhmadianMeasurement Systems17Measurement Systems18WindowingH. Ahmadian

WindowingH. AhmadianMeasurement Systems19H. AhmadianMeasurement Systems20

Windowing Windowing involves the imposition of aprescribed profile on the time signal prior toperforming the FTxc(t ) w(t ) u x(t ) a0 a1 cos(Z0t ) a1 cos(Z0t ) a2 cos(2Z0t ) 0 t T,w(t ) a3 cos(3Z0t ) a4 cos(4Z0t ) 0elsewhereZ0 2S TH. Ahmadian21Measurement Hanning11---Kaser-Bessel11.2980.244 0.003Flat top11.9331.286 0.388 0.032H. AhmadianMeasurement Systems-22

WindowingH. AhmadianMeasurement Systems23Measurement Systems24WindowingH. Ahmadian

Improving Resolution (Zoom) There arises limitations of inadequatefrequency resolution at the lower end of the frequency rangeFor lightly-damped systemsA common solution is to concentrate allspectral lines into a narrow band Within fmin-fmaxInstead of 0-fmaxH. AhmadianMeasurement Systems25Zoom Method 1: Shifting the frequency origin of the spectrumx(t ) A sin(Zt )xc(t ) A sin(Zt ) u cos(Zmint ) H. AhmadianA sin(Z Zmin )t sin(Z Zmin )t @2The modified signal is then analyzed in therange of 0-(fmax-fmin)Measurement Systems26

Zoom Method 2: A controlled aliasingeffect Applying a band passfilterBecause of the aliasingphenomenon, thefrequency componentbetween f1 and f2 willappear aliasedbetween 0-(f2-f1)H. AhmadianMeasurement Systems27Covered TopicsFundamentals of Signal Analysis Introduction Time and Frequency Domains: A matter of Perspective The Time DomainThe Frequency DomainUnderstanding Dynamic Signal Analysis See Section 83.1 Spectrum Analysis and CorrelationBasics of Discrete Fourier Transform (DFT)AliasingLeakageWindowingFilteringImproving ResolutionH. AhmadianMeasurement Systems28

Vibration Measurement SystemsVibration Measurements;ApplicationsH.AhmadianCovering TopicsFundamentals of Rotating Machinery Diagnostics Ch 1-2 IntroductionVibration Signal FrequencyAmplitudePhaseVibration of MachinesRotation and PrecessionPhase Measurement The Keyphasor EventAbsolute PhaseRelative PhaseDifferential PhaseH. AhmadianMeasurement Systems2

Introduction Examples of vibration analysisapplications: predictive maintenance,acceptance testing,quality control,loose part detection,noise control,leak detection,engine analyzers,machine design and engineeringH. AhmadianMeasurement Systems3Introduction A mechanical equipment inmotion generates a vibrationprofile, or signatureVibration signature reflects itsoperating condition regardless ofspeed or the mode of operation(rotation, reciprocation, linearmotion).Vibration profile analysis is auseful tool for predictivemaintenance, diagnostics, andmany other uses.H. AhmadianMeasurement Systems4

Introduction All machinery that hasrotating or moving elementsallows vibration-basedanalysis techniques to beused for predictivemaintenance.ANALYSIS TECHNIQUES: FREQUENCY-DOMAINRESONANCE/CRITICALSPEED ANALYSISREAL-TIME ANALYSISH. AhmadianMeasurement Systems5Vibration Signal A (non-contact) displacement transducer measures therelative position of an objectThe primary characteristics of the signal are frequency andamplitude.Complex signals contain several frequencies of vibrations andamplitudes.Rotating position vectorH. AhmadianMeasurement SystemsObject displacement 6

Vibration Signal: Frequency Sub-synchronous: anyfrequency less than 1X(0.37X,1/4X)Sub-harmonic: integerfraction of 1X(1/2X,1/3X, )H. Ahmadian Super-synchronous: anyfrequency grater than 1X(1.4X, 4X)Super-harmonic: integermultiple of 1X (2X,3X, )7Measurement SystemsVibration Signal: Amplitude Amplitude is the magnitude ofvibration expressed in termsof signal level.Amplitude can be measuredusing several methods: Peak-to-PeakPeak methodRoot-mean-squareRMS1T³T02A sin(Zt ) dtIn a sign wave the RMSamplitude is equal to 0.707 PKand 0.354 PP.H. AhmadianMeasurement Systems8

Vibration of Machines Transducers mounted onthe casing observe theshaft motion: In machines with stiffcasing support themeasurement is a goodapproximation of the shaftabsolute motionIn presence of casingvibration the shaft relativemotion is measured.H. AhmadianMeasurement Systems9Rotation and Precession Rotation is the angular motion of the rotor about itsgeometric centerPrecession is the lateral motion of the geometric center(forward/reverse precession) The concept of forward/reverse precession have powerfulapplication in full spectrum and in the diagnosis of certaintypes of malfunctions.It is possible for a rotor to rotate without precession andvice versaH. AhmadianMeasurement Systems10

Phase Vibration never occurs in isolation; there is a rootcause of vibration in a machine.In identifying this root cause on the basis offrequency and amplitude alone is difficult.One piece of information that can be very usefulis the timing difference, or phase, betweenevents. Why Is Phase Important?The Keyphasor EventPhase MeasurementAbsolute PhaseRelative PhaseDifferential PhaseH. AhmadianMeasurement Systems11What is Phase? The two signals reach thepositive peaks at differenttimes i.e. phase difference.The phase difference ofequivalent events on differentvibration signals is calledrelative phase.Absolute phase comparesthe timing of an event on thevibration waveform to amarker on a shaft.H. AhmadianMeasurement Systems12

Why Is Phase Important?A healthy machine should operate andvibrate with a repeatable pattern dayafter day. Changes in vibration that break thepattern indicate that something may bewrong with the machine. Changes in phase are just as importantas changes in vibration amplitude orfrequency, and one may changeindependently of the others. H. AhmadianMeasurement Systems13Why Is Phase Important? When the vibration is 1X, the pointon the shaft which is on the outsideof the deflected shaft is called thehigh spot.The timing of the rotor high spotpassage under a transducer providesimportant information about rotorbehavior.High Spot Passage can be comparedto the timing at different axialpositions in the same machine.The amplitude and phase informationcan be combined to produce apicture of the deflection shape, ormode shape:The rotor at running speed, The casing or structure.H. AhmadianMeasurement Systems 14

Why Is Phase Important?As vibration propagates away from thesource location, it experiences a timedelay (phase lag). By measuring the relative phasebetween different axial positions in amachine and looking for the earliestsignal, we can sometimes determinethe location closest to the source of theproblem. H. AhmadianMeasurement Systems15The Keyphasor Event Common vibration in a rotor is due to unbalance: Acts as a one-cycle-per-revolution rotating force This 1Xforcing produces a 1X, or synchronous, vibrationresponse in the machine.It is desirable to have a fixed, timing reference signal sothat we can make phase measurements.An eddy current displacement transducer looking at akeyway or key serves this purpose perfectly.H. AhmadianMeasurement Systems16

The Keyphasor Event The Keyphasor event can be used to measure theelapsed time between the Keyphasor event andan event on another signal.This once-per-turn event is the timing referenceused by instrumentation to measure the absolutephase of vibration signals.It is also used to measure rotor speed and otherimportant characteristics of the dynamicresponse of the rotor.H. AhmadianMeasurement Systems17Phase Measurement In order to make meaningful phase measurement The signals being used must consist of a single primaryfrequencyIn the case of the Keyphasor signal, one clearlyidentifiable reference event.For this reason, signals are usually filtered to thefrequency of interest before making themeasurementUnfiltered signals can be used if they aredominated by one frequency.H. AhmadianMeasurement Systems18

Phase Measurement The convention used in mostvibration measurementinstrumentation is to measurephase lag with a positive number,sometimes called positive phaselag.H. AhmadianMeasurement Systems19Absolute Phase Absolute phase is thephase angle measuredfrom the Keyphasorevent to the firstpositive peak of thewaveform.H. AhmadianMeasurement Systems20

Absolute Phase The 1X signal has one Keyphasorevent per cycle of vibrationIn the 2X signal, the absolute phaseis measured to the first positivepeak; the second peak is ignored.Absolute phase can not bemeasured on vibration signals whentheir frequency is not a harmonicmultiple of running speed (thesignal is not 1X, 2X, 3X, etc.).The phase measurement from eachsuccessive Keyphasor dot produces adifferent result.H. AhmadianMeasurement Systems 21Relative Phase Relative phase is the time delaybetween equivalent events on twoseparate signals, Peaks, zero crossings, etc.Doesn't use the Keyphasor eventThe two vibration signals havebeen filtered to the samefrequency and represent thedisplacement vibration at differentaxial positions on a machineVibration transducers should havethe same radial orientation if theyare in different axial planes.H. AhmadianMeasurement Systems22

Relative Phase Relative phase measurements can be madebetween transducers with different orientations,as long as they are in the same plane, todetermine the direction of precession of a rotor.H. AhmadianMeasurement Systems23Differential Phase Differential phase is a special application of relative phasemeasurement.It can be used to locate the source of a machine problem, Several vibration measurements, filtered to the frequency ofinterest, are taken at different axial locations in a machine.Relative phase measurements can be made between thesignals.The signal with the earliest phase will be from the transducerthat is mounted closest to the source of the problem.For this kind of measurement, all the transducers musthave the same radial mounting orientation.This technique can be used on vibration signals of anyfrequency, like those that result from fluid-inducedinstability.Significant phase changes can occur across nodal pointsthat can produce misleading results.H. AhmadianMeasurement Systems24

Covered TopicsFundamentals of Rotating Machinery Diagnostics Ch 1-2 IntroductionVibration Signal FrequencyAmplitudePhaseVibration of MachinesRotation and PrecessionPhase Measurement The Keyphasor EventAbsolute PhaseRelative PhaseDifferential PhaseH. AhmadianMeasurement Systems25

H. Ahmadian, Modal Testing Lab, Mech Eng., IUSTVibration signalFrequency/AmplitudeDisplacement, Velocity and AccelerationVibration of MachinesRotation and PrecessionFr

Vibration Measurement Systems H.Ahmadian Fundamentals of Signal Analysis H. Ahmadian Measurement Systems 2 Covering Topics Fundamentals of Signal Analysis Introduction Time and Frequency Domains: A matter of Perspective The Time Domain The Frequency Domain Understanding Dynamic Signal Analysis See Section 83.1 Spectrum Analysis and Correlation Basics of Discrete