Spectral Analysis. This week in lab. Next classes: 3/26 and 3/28. Your next experiment Homework is to prepare

Transcription

1 Spectral Aalysis This week i lab Your ext experimet Homework is to prepare Next classes: 3/26 ad 3/28 Aero Testig, Fracture Toughess Testig Read the Experimets 5 ad 7 sectios of the course maual

2 Spectral Aalysis What is spectral aalysis? How do we estimate the spectrum of a sigal? Gettig the Spectrum right

3 What is Spectral Aalysis? Cosider a fluctuatig sigal output by a trasducer of some kid, e.g. usteady lift force o a wig (strai gage balace), velocity from a hot-wire aemometer i a turbulet flow, positio of a vibratig structure (proximeter) We ofte eed to aswer the questio What frequecies are i the sigal? (e.g. may wat to kow size of eddies, motio of wig, effects of structure motio o coected objects). This questio is more artificial tha it souds. A more accurate restatemet of the questio would be - If we were to fit the shape of the sigal to a sum of siusoids of differet frequecies, what would be the distributio of amplitudes ad phases as a fuctio of the frequecy?

4 What is Spectral Aalysis? Also called Fourier Aalysis Defiitio: Decompositio of a sigal ito a series of compoet siusoids, e.g. A cos( 2π f t + ε ) ε 1/f Each siusoid will have a differet frequecy f, amplitude A ad phase ε. With such a decompositio we ca plot A t May be a plot of the amplitude itself or the RMS amplitude (0.7071A) May be i degrees or radias Amplitude vs. Frequecy Phase vs. Frequecy ½Amplitude 2 vs. Frequecy Amplitude Spectrum Phase Spectrum Power Spectrum

5 decibels (db) Logarithmic scale Amplitude i db is 20log 10 A RMS amplitude i db is 20 log 10 (0.7071A ) Power spectrum i db is 10 log 10 ( ½A 2 ) Used i may applicatios, always has the form 10log 10 ( cost. quatity 2 )

6 How to estimate the spectrum Most sigals, ad the pheomea they represet, are cotiuous ad go o for a very log time (if ot forever). To measure the sigal, however, we must take discrete samples of it at a fiite rate, ad we ca oly do that for a limited amout of time. This process places 2 limits o the frequecy iformatio we ca uambiguously extract from the measured sigal. 1. Measure the sigal Origial sigal Sampled Sampled & widowed = 1 record v(iδt) v(t) Total samplig time T Highest frequecy = Nyquist = 1/(2Δt) (limited by samplig rate) Lowest frequecy = Frequecy that lasts etire samplig time = 1/T = 1/(NΔt) (limited by samplig time) = Frequecy resolutio t Samplig period Δt No. of samples N = T/Δt t t Widowig - process by which a timelimited sectio of sigal is extracted

7 How to estimate the spectrum Origial sigal (as Σ sies) Where: a = 2. Calculate the spectrum Amplitude Phase Frequecy v( t) 2 N 2 i v( iδt)cos π N i= 1 N ε = A 2 o + 2 N / 2 = 1 A = a + b A 2 = arcta( b f = /(NΔt) b = cos / a A cos( 2π f t + ε ) ) 2πt NΔt + ε N = o. of samples Δt = samplig period 2 N 2 i v( iδt)si π N i= 1 N This set of relatios is kow as the DISCRETE FOURIER TRANSFORM. Numerical calculatios of spectra are usually performed usig a short-cut algorithm (that produces idetical results) kow as the FAST FOURIER TRANSFORM (FFT).

8 How to estimate the spectrum Results g(t) Origial sigal (1 record) t Amplitude spectrum A ε 90 Frequecy resolutio Δ=1, Δf = 1/(NΔt) Phase spectrum Max frequecy =N/2, f = 1/(2Δt)

9 How to estimate the spectrum 3. Usig stadard software I Matlab, if v with idices 1 to N is a array of samples cotaiig the sampled time sigal, the c=fft(v) gives a complex array c where a +ib = 2i*c +1 /N. From this all eeded quatities ca be calculated. I LabView there are may fuctios that deal with computig FFTs, spectra ad related quatities. The simplest to use is Spectral Measuremets

10 Spectral Measuremets VI Sigal i (Voltage samples from A/D) Output form depeds o iteral settigs of VI Amplitudes out Phases out

11 Output is actual cosie amplitudes A Output is A Output is power spectrum ½A 2 (o phase output) Iteral Settigs Amplitude output i decibels (or ot) Phase output i degrees (otherwise radias)

12 Gettig the Spectrum Right 1. Fiite frequecy resolutio. Frequecy resolutio is limited by the total samplig time for the record. Solutio: sample for a loger time (i.e. icrease the record legth N) the Δf = 1/(NΔt) will be smaller 2. Aliasig. Origial sigal may cotai frequecies higher tha the Nyquist. Whe we sample the sigal these appear as though they are lower frequecies ad thus corrupt the spectra at these frequecies. Solutios: icrease samplig rate to twice highest frequecy, or, filter out all frequecies higher tha the Nyquist before samplig ( ati-aliasig filter ).

13 Gettig the Spectrum Right 3. Broadeig. Sharp features i the spectrum may be smoothed (ad thus iaccurately represeted) i the measured spectrum. Explaatio: There is a cotradictio i what we have doe. We have decomposed oe record (lastig oly for time T) ito a sum of ifiitely log siusoids. This ca oly be doe by implicitly treatig the measured record as though it were oe period of a periodic sigal... v(t) Sudde jumps resultig from implied periodicity itroduce ew frequecies ot i origial sigal = Broadeig Origial sigal Periodic sigal implied by Fourier Trasform t Actual Measured record

14 Gettig the Spectrum Right Broadeig. Solutio: Multiply record by a smooth fuctio that is zero, or early so at the start ad ed poits of the record. There are differet types of these widowig fuctios oe ca choose from. These fuctios do ot substatially chage the spectrum, but they ted to miimize the broadeig. Commo widowig fuctios

15 Gettig the Spectrum Right 4. Radomess i the sigal. a) Due to oise corruptig the sigal (e.g cotamiatio of respose sigal from structure due to electrical oise or floor vibratios) b) Due to radomess i the physical quatity beig measured (turbulece i the velocity sigal from a cylider wake) Itroduces ucertaity ito the measured spectrum Solutio: Average the spectrum. How you average depeds o which of the above situatios is preset

16 Averagig To elimiate the effects of uwated oise i the sigal. Take may measuremets of the same sigal ad calculate their spectra Average the a ad b coefficiets (= vector averagig i Spectral Aalysis VI) If you wat the resultig phase spectrum to be meaigful each measuremet of the sigal must be sychroized with the same features of the sigal (i.e. it must be triggered). To average the effects of radom fluctuatios i the measured quatity Take may measuremets of the same sigal ad calculate their spectra Average the power spectral values ( ½A 2 ) No phase spectrum ca be usefully calculated (radom sigal = radom phase)