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

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 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 H(  ) U Y FFT

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

7 Range of Decay Constants: (  r  r )  Noise dominates the response of the quickly decaying modes at late times. + + Slow Fast Noise 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 FFT 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 )  Evaluate the effect of the window on the accuracy of the modal parameters.  Repeat for various noise profiles to obtain statistically meaningful results. AMI Modal Parameters  Noisy Data Window

12 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. Standard Deviation Mean

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

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 % Bias in Damping Ratio % Scatter in Damping Ratio  Observations: Exponential windowing did not decrease the scatter significantly for modes 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.

19 EMA Theory  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.  Equation of Motion  Frequency Domain  Frequency Response  Modal Parameters

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 )  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. ++ Slow Fast Noise Early Response Late Response