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Matthew S. Allen Jerry H. Ginsberg Georgia Institute of Technology George W. Woodruff School of

On the Accuracy of Modal Parameters Identified from Exponentially Windowed, Noise Contaminated Impulse Responses for a System with a Large Range of Decay Constants. Matthew S. Allen Jerry H. Ginsberg Georgia Institute of Technology George W. Woodruff School of Mechanical Engineering

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Matthew S. Allen Jerry H. Ginsberg Georgia Institute of Technology George W. Woodruff School of

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  1. On the Accuracy of Modal Parameters Identified from Exponentially Windowed, Noise Contaminated Impulse Responses for a System with a Large Range of Decay Constants Matthew S. Allen Jerry H. Ginsberg Georgia Institute of Technology George W. Woodruff School of Mechanical Engineering November, 2004

  2. Outline • Background: • Introduction to Experimental Modal Analysis • Measuring Frequency Response Functions • Persistent vs. Impulsive Excitations • Difficulties in testing a system with a range of decay constants in the presence of noise. • Exponential Windowing • Experiment: Noise contaminated data • Effect of exponential window on accuracy • Conclusions

  3. Experimental Modal Analysis • A Linear-Time-Invariant (LTI) system’s response is a sum of modal contributions. • wr zr fr • Natural Frequency • Damping Ratio • Mode Vector (shape) • In EMA we seek to identify these modal parameters from response data. F …

  4. EMA Applications • Applications of EMA • Validate a Finite Element (FE) model • Characterize damping • Diagnose vibration problems • Simulate vibration response • Detect damage • Find dynamic material properties • Control design • …

  5. H() H() FFT EMA Theory – Measuring FRFs • Two common ways of measuring the Frequency Response • Periodic or Random Excitation • Impulse Excitation. • Impulse method is often preferred: • Doesn’t modify the structure • Cost • High force amplitude • Noisy Data Y U H()

  6. Noise + Range of Decay Constants: (rr) Response +

  7. Fast Noise Slow + + Range of Decay Constants: (rr) • Noise dominates the response of the quickly decaying modes at late times. Early Response Late Response

  8. Range of Decay Constants: (rr) Slow Fast Noise + +

  9. Exponential Windowing • Exponential Windows (EW) are often applied to reduce leakage in the FFT. • Effect on modal parameters: • Adds damping – (can be precisely accounted for) • Other windows (Hanning, Hamming, etc…) have an adverse effect. • An EW also causes the noise to decay, reducing the effect of noise at late times. • Could this result in more accurate identification of the quickly decaying modes? 

  10. Range of Decay Constants • Prototype System: • Modes 7-11 have large decay constants. • The FRFs in the vicinity of these modes are noisy. Frame Structure

  11. Noisy Data Window Windowing Experiment • Apply windows with various decay constants to noise contaminated analytical data. • Estimate the modal parameters using the Algorithm of Mode Isolation (JASA, Aug-04, p. 900-915) • Evaluate the effect of the window on the accuracy of the modal parameters. • Repeat for various noise profiles to obtain statistically meaningful results. FFT AMI Modal Parameters

  12. Mean Standard Deviation Sample Results: Damping Ratio • Two distinct phenomena were observed. • Increase in scatter – (Lightly damped modes.) • Decrease in bias – (Heavily damped modes.) • These are captured by the standard deviation and mean of the errors respectively.

  13. Results: Damping Ratio • Largest errors were the bias errors in modes 8-11. • These decreased sharply when an exponential window was applied. % Scatter in Damping Ratio % Bias in Damping Ratio

  14. Results: Natural Frequency

  15. Noise Level vs. Exponential Factor • Bias errors are related to the Signal to Noise Ratio. • Bias is small when the signal is 20 times larger than the noise. • SNR attains a maximum when the window factor equals the modal decay constant.

  16. Conclusions • Exponential windowing improves the SNR of the FRFs in the vicinity of each mode, so long as the window factor is not much larger than the modal decay constant. • Damping Ratio: • Bias Errors in the damping estimates are small so long as the SNR is above 20 (see definition.) • Natural Frequency: • EW has a small effect so long as the exponential factor is smaller than the modal decay constant. • Similar Results for Mode Shapes & Modal Scaling.

  17. ? Questions? ? ? ? ? ?

  18. Results: Damping Ratio • Observations: • Exponential windowing did not decrease the scatter significantly for modes 8-11. • The scatter for modes 1-7 increased sharply for large exponential factors. • Exponential factors as large as the modal decay constant could be safely used. % Scatter in Damping Ratio % Bias in Damping Ratio

  19. EMA Theory • Equation of Motion • Frequency Domain • Frequency Response • Modal Parameters • Two common ways of measuring the Frequency Response • Apply a broadband excitation and measure the response. • Apply an impulsive excitation and record the response until it decays.

  20. Effect of Exponential Window on SNR • Damping added by the exponential window decreases the amplitude of the response in the frequency domain. • The amplitude of the noise also decreases. • The net effect can be increased or decreased noise. Increasing Damping

  21. Range of Decay Constants: (rr) Early Response • Noise dominates the response of the quickly decaying modes at late times. • A shorter time window reduces the noise in these modes, though it also results in leakage for the slowly decaying modes. + + Noise Slow Fast Late Response

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