Frequency Domain Analysis

Transcription

1 1 Frequency Domain Analysis Concerned with analysing the frequency (wavelength) content of a process Application example: Electromagnetic Radiation: Represented by a Frequency Spectrum: plot of intensity vs frequency

2 3 Solar Radiation Spectrum 0.5 Solar radiation spectrum ( Irradiance [W/cm /µm] nm violet Visible 700 nm red Near IR Infrared Wavelength [nm] 4 Why frequency domain analysis? Consider the following processes: Vibration on a transport vehicle

3 5 Why frequency domain analysis? Consider the following processes: Whale sound 6 Why frequency domain analysis? Consider the following processes: Apache helicopter flyover

4 7 Why frequency domain analysis? Consider the following processes: Ocean surface level fluctuations (waves) 8 Why frequency domain analysis? Consider the following processes: Axle load of passenger vehicle on test track

5 9 Why frequency domain analysis? Consider the following processes: Vibration of structure due to explosion load (pyrotechnic shock) 10 Why frequency domain analysis? Consider the following processes:

6 11 Why frequency domain analysis? Consider the following processes: 1 Why frequency domain analysis? Consider the following processes:

7 13 All continuous signals can be shown to be comprised of a summation of individual harmonic (Fourier) components of various frequencies, amplitudes and phases. Common methods to compute the frequency spectrum of measured data: Filter (octave band) analysis Filter analysis consists of a number of band-pass frequency filters (analogue or digital) These filters allow only a (narrow) band of frequencies to pass A number of filters with adjacent frequency bands are used to generate a frequency spectrum 0-10 db k k 4k 8k 16k (fractional) Octave bands 14 Digital Fourier Transform (DFT) or Fast Fourier Transform (FFT) Transforming a signal from the time to the frequency domain can be achieved via the Fourier Transform (also called Fourier Analysis). Information is neither gained or lost when transforming signals from the time domain to the frequency domain via the FFT. The Fourier transform is reversible Inverse Fourier Transform (IFT) Although Fourier theory applies strictly to periodic signals, the periodicity of sampled or measured signals is assumed resulting in a estimate of he frequency spectrum Signal (real) Complex Magnitude Spectrum FFT Phase Spectrum IFT

8 15 Random signals (Dadisp: Freqa) Broad-band signals contain more sinusoid components than narrow-band signals A sinusoid can be considered as a very-narrow-band random signal. The phase of the sinusoids which make up random signals are values uniformly-distributed between 0 and π. The phase of the sinusoids which make up pulse signals are equal. Effect of phase on frequency spectrum An infinite number of sinusoidal components with equal phase produces the Delta function (very sharp pulse). 16 Signal bandwidth examples: Uniformly-distributed random phase.

9 17 Signal bandwidth examples: Zero bandwidth signals 18 Signal bandwidth examples: Constant phase.

10 19 40 Example of constant phase signal: Gaussian wave packet Elevation [mm] Time [sec.] Energy Spectrum [mm.s] Phase [rad] Frequency [Hz] Frequency [Hz] 0 Digital Fourier Transform (DFT) or Fast Fourier Transform (FFT) Random signals (Dadisp: Apache_PSD & Spectral_averag) Each observation is unique sample of process one physical realisation of the process The frequency spectrum of each sample is an estimate of the frequency spectrum of the entire process The estimate of the true frequency spectrum is improved by computing spectral averages. Important issues when computing the Fourier Transform: Bandwidth: Frequency range to be analysed Frequency Resolution [Hz] = 1/sub-record duration [secs] Spectral estimate accuracy (random error): Std. Deviation of error = 1/ # averages. Spectral error is reduced by: Identifying sub-records within the measured record Computing the spectrum of each sub-record Computing the average spectrum Given a fixed record length, a compromise has to be reached with respect to frequency resolution and spectral error.

11 1 Effects of frequency resolution & spectral averaging. Example: Helicopter fly-by (sound) Effects of frequency resolution & spectral averaging. Example: Heavy vehicle vibrations.

12 3 Spectral averaging must be used carefully When signals are strongly non-stationary (ie. Evident variations in vital characteristics such as RMS levels or frequencies) spectral averaging will conceal these non-stationary properties. 4 Effects of frequency content variation. Example: Whale cry.

13 5 Effects of frequency content variation. Example: Whale cry. 6 Effects of frequency content variation. Example: Whale cry.

14 7 Nyquist Frequency and Sampling Rate (Dadisp: Shannonsine) Leakage and the effects of widowing functions (Dadisp: leakage_sin & leakage_rnd) Overlapping Zero-padding 8 Effects of spectral leakage and windowing. Example: sinusoid.

15 9 Effects of spectral leakage and windowing. Example: Heavy vehicle vibrations. 30 Influence of signal clipping on frequency spectrum No clipping Clipped

16 31 Influence of broad-band and narrow-band (power line) noise on frequency spectrum Clean spectrum + Broad-band noise + Narrow-band noise 3 Influence of intermittent noise (switchgear interference) on frequency spectrum Clean spectrum + Intermittent noise (sharp pulses)

17 33 System Analysis (Excitation Response Relationships) 34 Frequency analysis is useful in determining the frequency characteristics of systems: relationship between output and input as a function of frequency. Real systems are often assumed to approximate an ideal system. Ideal systems: Have constant parameters (no variation in system characteristics wrt time) Are linear (ie. additive and homogeneous): Additive: Response (output) to sum of excitations (inputs) = sum of responses due to each individual input: f (x1+ x ) = f(x 1) + f(x ) Homogeneous: Response from excitation x constant = response x constant from excitation: f( kx) = kf(x)

18 35 Cross Spectrum Excitation signal H(f) System FRF j( π ft + ΦA ) j( π ft + ΦB ) A= Ae, B= Be * S AB(f) = A(f) B(f) = Ae Be = A Be Response signal jπ ft jπ ft Y { } Y { } A( f ) a(t) a(t)e dt = = The cross spectrum of A wrt B is defined as: j(π ft + ΦA ) j(π ft + ΦB ) j( ΦB ΦA ) B( f ) b( t ) b( t )e dt = = Where A*(f) is the complex conjugate of the instantaneous spectrum of a(t) and B(f) is the instantaneous spectrum of b(t) The amplitude of the cross spectrum is the product of the two amplitudes The phase of the cross spectrum is the difference between the phase of B relative to A. The cross spectrum S BA has the same amplitude but opposite phase. Auto spectra and cross spectra are generally expressed in one-sided form: A( f) = Re(f) + iim(f) * A (f) = Re(f) iim(f) 36 Cross Spectrum Excitation signal H(f) System FRF Response signal jπ ft jπ ft Y { } Y { } A( f ) a(t) a(t)e dt = = B( f ) b( t ) b( t )e dt = = The auto spectrum is obtained in the same way: AA * S (f) = A(f) A(f) j( π ft + ΦA ) j( π ft + ΦA ) = A e Ae = A The autospectrum is the power spectrum which has additive properties useful for averaging. Auto spectra and cross spectra are generally expressed in one-sided form: G ( f ) = 0 G ( f ) = 0 f < 0 G (f) = S (f) G (f) = S (f) f= 0 G ( f ) = S ( f ) G ( f ) = S ( f ) f > 0 AA AB AA AA AB AB AA AA AB AB As for the auto spectrum the cross spectrum of stationary random signals is best estimated by averaging over a number of records.

19 37 Coherence The coherence is used to determine the level of linear dependence b/w two signals as a function of frequency The coherence is defined as: G AB( f ) γ (f) = G AA( f ) G BB( f ) And can be viewed as a squared correlation coefficient which quantifies the linear relationship between two variables: ρ xy = 1 ρ xy < 1 ρ xy σ xy = σx σ y Covariance Variance Note that for a single set of records (no averaging) the coherence is 1. ρ xy < 1 ρ xy = 0 38 Coherence In practical cases, reasons for obtaining coherences of less that unity are: 1. Contamination of either excitation or response signal with noise. The presence on nonlinearities in the relationship between the excitation and response 3. Spectral leakage due to insufficient resolution or unsuitable windowing function 4. Time delay between the excitation and response signals Signal:noise ratio The coherence can be used to determine the signal:noise ratio of the measurement which is defined as: γ S:N= 1 γ If noise contamination of the response signal is assumed to be the only factor influencing the coherence, then γ is proportional to the coherent power while (1- γ ) represents the non-coherent power which is due to the noise in the response signal.

20 39 Frequency Response Function The system is excited with a signal of suitable bandwidth (Swept sine [not suited to FFT], band limited random signal or impulse) and the system response is measured The excitation and response signals are each transformed to the frequency domain (FFT) to obtain a complex, instantaneous spectrum which are averaged to produce a mean power spectrum (autospectrum). Further frequency-domain functions such as the cross-spectrum are then computed and the (linear) relationship between the excitation and response signals as a function of frequency is established giving the Frequency Response Function (FRF). a( t ) Y A( f) H(f) System FRF b(t) = a(t) h(t) Y B(f) = A(f) H(f) Relationship between excitation and response for an ideal system without noise contamination In the time domain the response is obtained by convolving the system impulse response function h(t) with the excitation. In the frequency domain, the response spectrum is obtained by multiplying the system Frequency Response Function H(f) with the excitation spectrum. 40 Frequency Response Function a( t ) Y A( f) H(f) System FRF b(t) = a(t) h(t) Y B(f) = A(f) H(f) Relationship between excitation and response for an ideal system without noise contamination For an ideal system without noise, the Frequency Response Function can be determined by: B( f ) H( f ) = A( f ) A= Ae B= B e j( π ft + ΦA ) j( π ft + ΦB ) B H( f ) = e e A B H( f ) = e A j( π ft + ΦB ) j( π ft + ΦA ) j( ΦB ΦA ) Which is a complex function in terms of the magnitude ratio and the phase difference.

21 41 Frequency Response Function In practice it has been shown that when the response signal is contaminated by broad-band random noise (transducer noise), a better estimate of the FRF is obtained by normalising the cross spectrum by the auto spectrum of the input: B(f) A*(f) S AB(f) G AB(f) H( f) = = = = H(f) 1 A(f) A*(f) S (f) G (f) B(f) B*(f) S BB(f) G BB(f) H( f) = = = = H (f) A(f) B*(f) S (f) G (f) AA BA AA If the complex conjugate of the output spectrum is used the resulting FRF is improved when noise contamination is present in the excitation signal: It is interesting to note that the ratio H 1 :H always gives the coherence: BA 1 AB BA AB AB G AB( f ) = = = AA BB AA BB AA BB 1 = γ (f) H(f) G (f) G (f) G (f) G *(f) H (f) G (f) G (f) G (f) G (f) G (f)g (f) H(f) H ( f ) Although the magnitude of H 1 and H will be different for noise contaminated measurements, their phase is always the same. 4 Frequency Response Function effects of noise Consider an ideal system where the measured response signal b(t) is contaminated by extraneous uncorrelated noise n(t). The noise signal may include some component generated by the system but not caused by the excitation signal a(t). n(t) Given sufficient averaging, the measured cross spectrum G AB will approximate the true cross spectrum G AV and the measured response auto h(t) v(t) spectrum is the sum of the clean output and a(t) H(f) the uncorrelated noise: Σ b(t) G (f) = G (f) AB AV BB = VV + NN = AA + NN G (f) G (f) G (f) H(f) G (f) G (f) G (f) = G (f) + G (f) = G (f) = H(f)G (f) AB AV AN AV AA G AB(f) G AV (f) H(f) 1 = = = H(f) Optimum FRF (effects of noise minimised) G (f) G (f) AA AA

22 43 Frequency Response Function effects of noise n(t) while a(t) h(t) v(t) H(f) G BB(f) G VV (f) + G NN (f) H(f) G AA(f) + G NN (f) H ( f ) = = = G (f) G (f) H*(f)G (f) BA VA AA Σ b(t) G NN ( f ) H(f) G NN (f) = H( f ) + = H( f ) + H*(f)G (f) H*(f)G (f) AA VV G NN ( f ) H(f) = H(f)1 + G VV ( f ) FRF magnitude overestimated (phase OK) G wher NN ( f ) e G vv is the coherent output power spectrum and is the noise : signal ratio. G ( f ) VV 44 Frequency Response Function effects of noise When the measured excitation signal is contaminated by extraneous noise m(t) (measurement noise), it can be shown that: u(t) h(t) H(f) b(t) while H( f ) H(f) 1 = FRF magnitude underestimated (phase OK) G MM ( f ) 1 + G UU ( f ) m(t) H(f) = H(f) Optimum FRF (effects of noise minimised) Σ a(t) When there is extraneous noise in both the measured excitation and response signals H 1 and H give the lower and upper bound to the true FRF.

23 45 Frequency Response Function Examples: Excitation: Wave height Excitation: Gusts Excitation: Pavement topography Excitation: Engine vibrations Excitation: Aerodynamic loads System: Ship System: Tower System: Road vehicle System: Military submarine System: Aeroplane wing Response: Pitch or roll Response: Sway or deflection Response: Vertical vibration Response: Sound Response: Stresses or deflection 46 Frequency Response Function Excitation: vertical vibrations System: Packaged product Response: Product vibration Dead Weight Signal Analyser Response accel. signal Guided platen Test sample Vibration Controller Input accel. signal (control) Vibration table Servo-hydraulic Actuator

24 47 Frequency Response Function Linear bearings housing Response accelerometer Guided dead weight Fibreboard cushion sample Guide rod Vibration table Input (table) accelerometer 48 Frequency Response Function 10 steady-state Random Nd = 5 Transmissiblity Frequency [Hz]

25 49 Frequency Response Function Accelerometer Dead Weights Charge Amplifier Accelerometer Charge Amplifier Packaged Unit Table Servo Amplifier Hydraulic Servoactuator PC with ADC and DAC Modules. Function Generator with externally-controlled frequency & amplitude. 50 Frequency Response Function

26 51 Response acceleration Sensor Frequency Response Function Excitation acceleration Sensor Frequency Response Function Before test After test Gain Phase Difference [Deg] Before test -30 Frequency [Hz] After test Frequency [Hz]