# Basics of Spectrum Analysis/Measurements and the FFT Analyzer

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Basics of Spectrum Analysis/Measurements and the FFT Analyzer

Mechanical Engineering - 22.403 ME Lab II ME 22.403 Mechanical Lab II Basics of Spectrum Analysis/Measurements and the FFT Analyzer Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 1 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Transformation of Time to Frequency Many times a transformation is performed to provide a better or clearer understanding of a phenomena. The time representation of a sine wave may be difficult to interpret. By using a Fourier series representation, the original time signal can be easily transformed and much better understood. Transformations are also performed to respresent the same data with significantly less information. Notice that the original time signal was defined by many discrete time points (ie, 1024, 2048, 4096 …) whereas the equivalent Fourier representation only requires 4 amplitudes and 4 frequencies. Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 2 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II The Anatomy of the FFT Analyzer The FFT Analyzer can be broken down into several pieces which involve the digitization, filtering, transformation and processing of a signal. ANALOG SIGNAL ANALOG FILTER ADC DISPLAY DIGITAL FILTER FFT Several items are important here: Digitization and Sampling Quantization of Signal Aliasing Effects Leakage Distortion Windows Weighting Functions The Fourier Transform Measurement Formulation DISCRETE DATA Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 3 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II The Anatomy of the FFT Process ANALOG SIGNALS OUTPUT INPUT ANTIALIASING FILTERS Actual time signals Analog anti-alias filter AUTORANGE ANALYZER ADC DIGITIZES SIGNALS OUTPUT INPUT APPLY WINDOWS INPUT OUTPUT COMPUTE FFT LINEAR SPECTRA LINEAR OUTPUT SPECTRUM LINEAR INPUT SPECTRUM Digitized time signals Windowed time signals Compute FFT of signal AVERAGING OF SAMPLES COMPUTATION OF AVERAGED INPUT/OUTPUT/CROSS POWER SPECTRA INPUT POWER SPECTRUM OUTPUT POWER SPECTRUM CROSS POWER SPECTRUM Average auto/cross spectra COMPUTATION OF FRF AND COHERENCE Compute FRF and Coherence FREQUENCY RESPONSE FUNCTION Dr. Peter Avitabile COHERENCE FUNCTION University of Massachusetts Lowell Spectrum Analysis 082702 - 4 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Aliasing (Wrap-Around Error) WRAP-AROUND ACTUAL SIGNAL OBSERVED ACTUAL ALIASED SIGNAL f max Aliasing results when the sampling does not occur fast enough. Sampling must occur faster than twice the highest frequency to be measured in the data - sampling of 10 to 20 times the signal is sufficient for most time representations of varying signals However, in order to accurately represent a signal in the frequency domain, sampling need only occur at greater than twice the frequency of interest Anti-aliasing filters are used to prevent aliasing These are typically Low Pass Analog Filters Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 5 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Anti-Aliasing Filters Anti-aliasing filters are typically specified with a cut-off frequency. The roll-off of the filter will determine how quickly the signal will be attenuated and is specified in dB/octave FILTER ROLLOFF G dB = 20 log10 G = 20 log10 Vout Vin fc The cut-off frequency is usually specified at the 3 dB down point (which is where the filter attenuates 3 dB of signal). Butterworth, Chebyshev, elliptic, Bessel are common filters Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 6 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Digitization of a Signal Sampling rate of the ADC is specified as a maximum that is possible. Basically, the digitizer is taking a series of “snapshots” at a very fast rate as time progresses Digital Representation Analog Signal ADC Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 7 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Sampling Each sample is spaced delta t seconds apart. Sufficient sampling is needed in order to assure that the entire event is captured. The maximum observable frequency is inversely proportional to the delta time step used Digital Sample Rayleigh Criteria fmax = 1 / 2 ∆t ∆t spacing Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 8 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Sampling Theory In order to extract valid frequency information, digitization of the analog signal must occur at a certain rate. Shannon's Sampling Theorem states fs > 2 fmax That is, the sampling rate must be at least twice the desired frequency to be measured. For a time record of T seconds, the lowest frequency component measurable is ∆f = 1 / T With these two properties above, the sampling parameters can be summarized as fmax = 1 / 2 ∆t ∆t = 1 / 2 fmax Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 9 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Sampling Parameters Due to the Rayleigh Criteria and Shannon’s Sampling Theorum, the following sampling parameters must be observed. T=N ∆ t Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 10 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Sampling Parameters Due to the Rayleigh Criteria and Shannon’s Sampling Theorum, the following sampling parameters must be observed. PICK THEN AND ∆t fmax = 1 / (2 ∆t) T = N ∆t fmax ∆t = 1 / (2 fmax ) ∆f = 1/(N ∆t) ∆f T = 1 / ∆f ∆t = T / N T ∆f =1 / T fmax = N ∆f / 2 If we choose ∆f = 5 Hz and N = 1024 Then T = 1 / ∆f = 1 / 5 Hz = 0.2 sec fs = N ∆f = (1024) (5 Hz) = 5120 Hz fmax = fs = (5120 Hz) / 2 = 2560 Hz Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 11 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Sampling Relationship An inverse relationship between time and frequency exists T BW Given delta t = .0019531 and N = 1024 time points, then T = 2 sec and BW= 256 Hz and delta f = 0.5 Hz TIME DOMAIN FREQUENCY DOMAIN T BW Given delta t = .000976563 and N = 1024 time points, then T = 1sec sec and BW = 512 Hz and delta f = 1 Hz TIME DOMAIN T FREQUENCY DOMAIN BW Given delta t = .0019531 and N = 512 time points, then T = 1 sec and BW = 256 Hz and delta f = 1 Hz TIME DOMAIN Dr. Peter Avitabile FREQUENCY DOMAIN University of Massachusetts Lowell Spectrum Analysis 082702 - 12 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Quantization Error Sampling refers to the rate at which the signal is collected. Quantization refers to the amplitude description of the signal. A 4 bit ADC has 24 or 16 possible values A 6 bit ADC has 26 or 64 possible values A 12 bit ADC has 212 or 4096 possible values ADC BIT STEPS Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 13 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Quantization Error Quantization errors refer to the accuracy of the amplitude measured. The 6 bit ADC represents the signal shown much better than a 4 bit ADC Dr. Peter Avitabile A D C A D C M A X M A X R A R A N G E N G E University of Massachusetts Lowell Spectrum Analysis 082702 - 14 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Quantization Error Underloading of the ADC causes amplitude errors in the signal 10 volt range on ADC All of the available dynamic range of the analog to digital converter is not used effectively 0.5 volt signal This causes amplitude and phase distortion of the measured signal in both the time and frequency domains Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 15 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II AC Coupling A large DC bias can cause amplitude errors in the alternating part of the signal. AC coupling uses a high pass filter to remove the DC component from the signal All of the available dynamic range of the analog to digital converter is dominated by the DC signal 10 volt range on ADC The alternating part of the signal suffers from quantization error This causes amplitude and phase distortion of the measured signal Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 16 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Clipping and Overloading Overloading of the ADC causes severe errors also The ADC range is set too low for the signal to be measured and causes clipping of the signal 1 volt range on ADC A D C M A X R A N G E Dr. Peter Avitabile 1.5 volt signal University of Massachusetts Lowell This causes amplitude and phase distortion of the measured signal in both the time and frequency domains Spectrum Analysis 082702 - 17 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II The Fourier Transform Forward Fourier Transform +∞ Sx (f )= ∫ x ( t )e − j2 πft dt −∞ and Inverse Fourier Transform +∞ x ( t )= ∫ Sx (f )e j2 πft df −∞ Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 18 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Discrete Fourier Transform Even though the actual time signal is continuous, the signal is discretized and the transformation at discrete points is +∞ Sx (m∆f )= ∫ x ( t )e − j2πm∆f t dt −∞ This integral is evaluated as Sx (m∆f )≈∆t +∞ ∑ x(n∆t )e− j2πm∆f n∆t n = −∞ However, if only a finite sample is available (which is generally the case), then the transformation becomes N −1 Sx (m∆f )≈∆t ∑ x(n∆t )e − j2πm∆f n∆t n =0 Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 19 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Fourier Transform and FFT Actual Time Signal ACTUAL Captured Time Signal CAPTURED DATA DATA T Reconstructed Time Signal RECONTRUCTED DATA Frequency Spectrum Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 20 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Fourier Transform and FFT Actual Time Signal ACTUAL Captured Time Signal CAPTURED DATA DATA T Reconstructed Time Signal RECONTRUCTED DATA Frequency Spectrum Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 21 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Leakage F R E Q ACTUAL DATA CAPTURED DATA T I M E Periodic Signal T RECONTRUCTED DATA Non-Periodic Signal U E N C Y ACTUAL DATA CAPTURED DATA T RECONTRUCTED DATA T Leakage due to signal distortion Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 22 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Leakage When the measured signal is not periodic in the sample interval, incorrect estimates of the amplitude and frequency occur. This error is referred to as leakage. Basically, the actual energy distribution is smeared across the frequency spectrum and energy leaks from a particular ∆f into adjacent ∆f s. Leakage is probably the most common and most serious digital signal processing error. Unlike aliasing, the effects of leakage can not be eliminated. Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 23 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Windows - Minimize Leakage In order to better satisfy the periodicity requirement of the FFT process, time weighting functions, called windows, are used. Essentially, these weighting functions attempt to heavily weight the beginning and end of the sample record to zero - the middle of the sample is heavily weighted towards unity ACTUAL DATA CAPTURED DATA T I M E Periodic Signal T RECONTRUCTED DATA Non-Periodic Signal ACTUAL DATA CAPTURED DATA T RECONTRUCTED DATA Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 24 F R E Q U E N C Y Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Windows - Rectangular/Hanning/Flattop In order to better satisfy the periodicity requirement of the FFT process, time weighting functions, called windows, are used. Essentially, these weighting functions attempt to heavily weight the beginning and end of the sample record to zero - the middle of the sample is heavilty weighted towards unity Rectangular - Unity gain applied to entire sample interval; this window can have up to 36% amplitude error if the signal is not periodic in the sample interval; good for signals that inherently satisfy the periodicity requirement of the FFT process Hanning - Cosine bell shaped weighting which heavily weights the beginning and end of the sample interval to zero; this window can have up to 16% amplitude error; the main frequency will show some adjacent side band frequencies but then quickly attenuates; good for general purpose signal applications Flat Top - Multi-sine weighting function; this window has excellent amplitude characteristics (0.1% error) but very poor frequency resolution; very good for calibration purposes with discrete sine Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 25 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Windows - Rectangular/Hanning/Flattop Time weighting functions are applied to minimize the effects of leakage AMPLITUDE ROLLOFF Rectangular WIDTH General window frequency characteristics Hanning Flat Top and many others Windows DO NOT eliminate leakage !!! Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 26 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Windows - Rectangular The rectangular window function is shown below. The main lobe is narrow, but the side lobes are very large and roll off quite slowly. The main lobe is quite rounded and can introduce large measurement errors. The rectangular window can have amplitude errors as large as 36%. 0 -10 -20 Amplitude -30 -40 -50 -60 dB -70 - 80 - 90 -100 -16 -14 -12 Dr. Peter Avitabile -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 15.9375 University of Massachusetts Lowell -3 -2.5 -2 -1.5 -1 -0.5 0 Spectrum Analysis 082702 - 27 0.5 1 1.5 2 2.5 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Windows - Hanning The hanning window function is shown below. The first few side lobes are rather large, but a 60 dB/octave roll-off rate is helpful. This window is most useful for searching operations where good frequency resolution is needed, but amplitude accuracy is not important; the hanning window will have amplitude errors of as much as 16%. 0 -10 -20 Amplitude -30 -40 -50 -60 dB -70 - 80 - 90 -100 -16 -14 -12 Dr. Peter Avitabile -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 15.9375 University of Massachusetts Lowell -3 -2.5 -2 -1.5 -1 -0.5 0 Spectrum Analysis 082702 - 28 0.5 1 1.5 2 2.5 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Windows - Flat Top The flat top window function is shown below. The main lobe is very flat and spreads over several frequency bins. While this window suffers from frequency resolution, the amplitude can be measured very accurately to 0.1%. 0 -10 -20 Amplitude -30 -40 -50 -60 dB -70 - 80 - 90 -100 -16 -14 -12 Dr. Peter Avitabile -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 15.9375 University of Massachusetts Lowell -3 -2.5 -2 -1.5 -1 -0.5 0 Spectrum Analysis 082702 - 29 0.5 1 1.5 2 2.5 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Windows Rectangular Hanning Flat Top 0 0 0 -10 -10 -10 -20 -20 -20 -30 -30 -30 -40 -40 -40 -50 -50 -50 -60 -60 -60 dB dB dB -70 -70 -70 - 80 - 80 - 80 - 90 - 90 - 90 -100 -16 -14 -12 -10 -8 -6 Dr. Peter Avitabile -4 -2 0 2 4 6 8 10 12 14 15.9375 -100 -16 -14 -12 -10 -8 University of Massachusetts Lowell -6 -4 -2 0 2 4 6 8 10 12 14 15.9375 -100 -16 -14 -12 -10 Spectrum Analysis 082702 - 30 -8 -6 -4 -2 0 2 4 6 8 10 12 14 15.9375 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Windows - Force/Exponential for Impact Testing Special windows are used for impact testing Force window Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 31 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Windows - Force/Exponential for Impact Testing Special windows are used for impact testing Exponential window Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 32 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Hammer Tips for Impact Testing METAL TIP HARD PLASTIC TIP Real Real -976.5625us TIME PULSE 123.9624ms dB Mag -976.5625us 123.9624ms dB Mag 0Hz 6.4kHz FREQUENCY SPECTRUM 0Hz FREQUENCY SPECTRUM 6.4kHz RUBBER TIP SOFT PLASTIC TIP Real Real -976.5625us TIME PULSE -976.5625us 123.9624ms TIME PULSE 123.9624ms dB Mag dB Mag 0Hz Dr. Peter Avitabile TIME PULSE FREQUENCY SPECTRUM University of Massachusetts Lowell 6.4kHz 0Hz FREQUENCY SPECTRUM Spectrum Analysis 082702 - 33 6.4kHz Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Pretrigger Delay and Double Impacts Pretrigger delay used to reduce the amount of frequency spectrum distortion DOUBLE IMPACT Real t=0 NO PRETRIGGER USED -976.5625us 998.53516ms TIME PULSE dB Mag DOUBLE IMPACT t=0 0Hz FREQUENCYRealSPECTRUM 800Hz PRETRIGGER SPECIFIED -976.5625us Double impacts should be avoided due to the distortion of the frequency spectrum and force dropout that can occur Dr. Peter Avitabile University of Massachusetts Lowell TIME PULSE 998.53516ms dB Mag 0Hz Spectrum Analysis 082702 - 34 FREQUENCY SPECTRUM 800Hz Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Exponential Window If the signal does not naturally decay within the sample interval, then an exponentially decaying window may be necessary. However, many times changing the signal processing parameters such as bandwidth and number of spectral lines may produce a signal which requires less window weighting Dr. Peter Avitabile University of Massachusetts Lowell T = N∆ t T = N∆ t Spectrum Analysis 082702 - 35 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Measurement - Linear Spectra x(t) INPUT Sx(f) Dr. Peter Avitabile h(t) y(t) TIME SYSTEM OUTPUT FFT & IFT H(f) Sy(f) FREQUENCY x(t) - time domain input to the system y(t) - time domain output to the system Sx(f) - linear Fourier spectrum of x(t) Sy(f) - linear Fourier spectrum of y(t) H(f) - system transfer function h(t) - system impulse response University of Massachusetts Lowell Spectrum Analysis 082702 - 36 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Measurement - Linear Spectra +∞ x ( t )= ∫ Sx (f )e j2 πft df −∞ +∞ y( t )= ∫ S y (f )e j2 πft h ( t )= ∫ H (f )e Sx (f )= ∫ x ( t )e − j2 πft dt −∞ +∞ df −∞ +∞ +∞ S y (f )= ∫ y( t )e − j2 πft dt −∞ +∞ j2 πft df −∞ H(f )= ∫ h ( t )e − j2 πft dt −∞ Note: Sx and Sy are complex valued functions Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 37 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Measurement - Power Spectra Rxx(t) INPUT Gxx(f) Ryx(t) Ryy(t) TIME SYSTEM OUTPUT FFT & IFT Gxy(f) Gyy(f) FREQUENCY Rxx(t) - autocorrelation of the input signal x(t) Ryy(t) - autocorrelation of the output signal y(t) Ryx(t) - cross correlation of y(t) and x(t) Gxx(f) - autopower spectrum of x(t) G xx ( f ) = S x ( f ) • S*x ( f ) Gyy(f) - autopower spectrum of y(t) G yy ( f ) = S y ( f ) • S*y ( f ) Gyx(f) - cross power spectrum of y(t) and x(t) G yx ( f ) = S y ( f ) • S*x ( f ) Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 38 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Measurement - Power Spectra R xx (τ)=E[ x ( t ), x ( t + τ)]= lim 1 x ( t )x ( t + τ)dt ∫ T → ∞TT +∞ G xx (f )= ∫ R xx (τ)e − j2 πft dτ=Sx (f )•S*x (f ) −∞ R yy (τ)=E[ y( t ), y( t + τ)]= lim 1 y( t )y( t + τ)dt ∫ T → ∞T T +∞ G yy (f )= ∫ R yy (τ)e − j2 πft dτ=S y (f )•S*y (f ) −∞ R yx (τ)=E[ y( t ), x ( t + τ)]= lim 1 y( t )x ( t + τ)dt ∫ T → ∞TT +∞ G yx (f )= ∫ R yx (τ)e − j2 πft dτ=S y (f )•S*x (f ) −∞ Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 39 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II The Frequency Response Function and Coherence S y =HSx H1 formulation - susceptible to noise on the input - underestimates the actual H of the system S y •S*x =HSx •S*x Other formulations for H exist Sy •S*x G yx H= = * Sx •Sx G xx COHERENCE γ Dr. Peter Avitabile 2 xy (S y •S*x )(Sx •S*y ) G yx / G xx H1 = = = * * (Sx •Sx )(S y •S y ) G yy / G xy H 2 University of Massachusetts Lowell Spectrum Analysis 082702 - 40 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Typical Measurements Measurements - Auto Power Spectrum Measurements - Cross Power Spectrum x(t) y(t) AVERAGED INPUT AVERAGED OUTPUT POWER SPECTRUM POWER SPECTRUM Gyy (f) G xx (f) OUTPUT RESPONSE INPUT FORCE G xx(f) G yy (f) AVERAGED INPUT AVERAGED OUTPUT POWER SPECTRUM POWER SPECTRUM AVERAGED CROSS POWER SPECTRUM G yx (f) Measurement Definitions 12 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Measurements - Frequency Response Function Measurement Definitions 13 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Measurements - FRF & Coherence Coherence 1 Real AVERAGED INPUT AVERAGED CROSS AVERAGED OUTPUT POWER SPECTRUM POWER SPECTRUM POWER SPECTRUM Gyx(f) G yy(f) G xx(f) 0 0Hz AVG: 5 200Hz COHERENCE Freq Resp 40 dB Mag -60 0Hz AVG: 5 200Hz FREQUENCY RESPONSE FUNCTION FREQUENCY RESPONSE FUNCTION H(f) Measurement Definitions Dr. Peter Avitabile 14 Dr. Peter Avitabile Modal Analysis & Controls Laboratory University of Massachusetts Lowell Measurement Definitions 15 Spectrum Analysis 082702 - 41 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Hammers and Tips 40 COHERENCE dB Mag FRF INPUT POWER SPECTRUM -60 0Hz 800Hz COHERENCE 40 FRF dB Mag INPUT POWER SPECTRUM -60 0Hz Dr. Peter Avitabile University of Massachusetts Lowell 200Hz Spectrum Analysis 082702 - 42 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Leakage and Windows for Impact Testing ACTUAL TIME SIGNAL SAMPLED SIGNAL WINDOW WEIGHTING WINDOWED TIME SIGNAL Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 43 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Simple time-frequency response relationship RESPONSE increasing rate of oscillation WOW !!! FORCE time frequency Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 44 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Sine Dwell to Obtain Mode Shape Characteristics MODE 1 MODE 2 Dr. Peter Avitabile University of Massachusetts Lowell MODE3 MODE 4 Spectrum Analysis 082702 - 45 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Mode Shape Characteristics for a Simple Beam MODE # 1 MODE # 2 MODE # 3 DOF # 1 DOF #2 DOF # 3 Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 46 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Mode Shape Characteristics for a Simple Plate MODE 2 2 1 4 3 6 MODE 1 5 2 4 1 3 6 5 Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 47 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II Why and How Do Structures Vibrate? OUCH !!! Motor or disk unbalance can cause unwanted vibrations or worse OOOPS !!! INPUT TIME FORCE OUTPUT TIME RESPONSE f(t) y(t) FFT Dr. Peter Avitabile IFT INPUT SPECTRUM FREQUENCY RESPONSE FUNCTION OUTPUT SPECTRUM f(jω ) h(j ω) y(j ω) University of Massachusetts Lowell Spectrum Analysis 082702 - 48 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II HP 35660 FFT Dual Channel Analyzer Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 49 Copyright © 2001 Mechanical Engineering - 22.403 ME Lab II HP 35660 FFT Dual Channel Analyzer Dr. Peter Avitabile University of Massachusetts Lowell Spectrum Analysis 082702 - 50 Copyright © 2001