On the Accuracy of Modal Parameters Identified from

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

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

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

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

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

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

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

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

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

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

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

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

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

Results: Natural Frequency 14

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

Conclusions w 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. w Damping Ratio: n Bias Errors in the damping estimates are small so long as the SNR is above 20 (see definition. ) w Natural Frequency: n EW has a small effect so long as the exponential factor is smaller than the modal decay constant. w Similar Results for Mode Shapes & Modal Scaling. 16

Questions? 17

Results: Damping Ratio % Bias in Damping Ratio n n n 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 w Observations: 18

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

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

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