Input-Output Peak Picking Modal Identification & Output only Modal Identification and Damage Detection of Structures using

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1 Input-Output Peak Picking Modal Identification & Output only Modal Identification and Damage Detection of Structures using Time Frequency and Wavelet Techniquesc Satish Nagarajaiah Professor of Civil and Mechanical Engineering Rice University, i Houston, TX, USA Funding: National Science Foundation

2 Input-Output Peak Picking Modal Identification

3 Natural Frequency, ω Modal Parameters the frequency at which a system vibrates when set in free vibration Damping Ratio, ζ a dimensionless measure describing how oscillations in a system decay after a disturbance Mode Shape, Φ a pattern of motion in which all parts of the system move sinusoidally with the same frequency and in phase 3

4 Typical Experimental Setup 4

5 Impulse Response (Free Vibrations) After impact excitation, the system vibrates freely 1.5 x 10-4 Time History of Acc 1 4 x 10-3 FFT of Acc X: Y: Amplitude(v) FFT Amplitude X: 11.8 Y: X: Y: Time(s) Frequency(Hz) Free Vibration Response after Impulse Fourier Response Spectrum 5

6 Frequency Response Functions (FRF) (Forced Vibrations) Known Excitation and Response Signals, p i, u j Mathematical model for N-DOF i t Mu + Cu + Ku = pe ω FRF H ( ω) H ( ω) φφ N ir jr ij ui / p = j r= 1 K 2 r r i rr φ 1 ( 1 ) + ( 2ζ ) where r ω, is undamped r th mode shape = ωr r 6

7 FRF Experimental Results FRF Calculation Averaged da Auto & Cross Power Spectrum Navg 1 AA n n G = A A N avg n= 1 N 1 avg * ( ω) ( ω) * ( ω) ( ω) G = A B FRF AB n n N avg n= 1 H = G G u / p uu u p i j i i i j 7

d, n but not too small so that peak value

8 Peak Picking Method SDOF or widely separated natural frequencies Accuracy depends on the reliability of the peak value Damping must be small (so that ω n ω d, n but not too small so that peak value has high measurement uncertainty Peak Picking Example 8

9 Natural Frequency & Damping Ratio FFT or FRF Amplitude Plot Natural frequency ω n = ω k Damping Ratio ζ n = ω2 ω1 2ω k 9

10 Mode Shape Theory Exciting the structure at one of the undamped natural frequency, the complex FRF: φ H f = f i φ = i ( ) jr ir ai/ pj r ir jr 2 Kr ζr 2 Kr ζr Thus, mode shape r can be obtained by examing the imaginary part of the FRFs for any output point i. When it is widely separated natural frequencies system, peaks of the FRF could be directly used. φ φ 10

11 Mode Shape Four Points Cantilever First peak at 35Hz φ φ = Values at Imaginary Plot Tip Normalized Value 11

12 Output only Modal Identification and Damage Detection of Structures using Time Frequency and Wavelet Techniques

13 Motivation Modal identification key step in structural identification Signal identification with Fourier analysis is inherently global in nature, unable to capture time varying nature of the frequency content of the signal New signal processing techniques developed for nonstationary signals such as Empirical Mode Decomposition (EMD), Hilbert Transform (HT) can be used for modal identification Pi Primary objective is to develop output tonly modal identification and structural damage detection

14 Introduction Overview Time frequency methods: STFT, EMD and HT Modal identification of LTI systems using STFT and EMD/HT Modal identification of LTI systems using Wavelets Experimental and Numerical Validation Conclusions

15 Analytic Signal Analytic Signal, s a (t) Hilbert Transform, H[s(t)] given by Analytic signal also written as Where A(t) instantaneous amplitude φ(t) instantaneous phase

16 Instantaneous Frequency Instantaneous Frequency, ω i (t)

17 Short-term Fourier Transform (STFT)

18 Short-term Fourier Transform (STFT) Three Story 1:10 Scale Building Model STFT of the Measured Third Floor Freevibration Displacement Response

19 LTV system Using STFT? Spectrogram of the Third Floor Acceleration Response to White Noise Excitation

20 Empirical Mode Decomposition Empirical Mode Decomposition (EMD) developed by Huang (Huang 1998) adaptively decomposes a signal into intrinsic mode functions which can then be converted into analytical signals using HT. For example, signal x j (t) j th degree of freedom displacement of a MDOF system can be decomposed.

21 Model identification of LTI systems using EMD/HT Equation of motion of MDOF is substituting leads to m uncoupled equations of motion The transfer function summing over all modes is The inverse transform of above equation gives impulse response function

22 Model identification of LTI systems using Impulse response function is EMD/HT Where, damped frequency of k th mode is The IRF can also be rewritten

23 Model identification of LTI systems using Leading to analytical signal EMD/HT that can be written as Damping can be obtained from

24 Empirical Modes : Intrinsic Mode Functions (IMF) Three Story 1:10 Scale Building Model (a) Third Floor Freevibration Displacement Response; (b) IMF Compoents

25 Hilbert Transform: Inst. Frequency and Damping Ratio Frequencies and Damping Ratios Estimated using EMD/HT Mode Free Vibration Tests Identified Identified Frequency Damping (Hz) Ratio (%) White Noise Tests Identified Frequency (Hz) Identified Damping Ratio (%) Mode Shapes Estimated using EMD Mode-1 Mode-2 Mode-3 Storey Storey Storey HT of IMF3 of the Third Floor Acceleration Free Vibration Response Analytical Mode Shapes Mode-1 Mode-2 Mode-3 Storey Storey Storey

26 STFT vs. Wavelets: Resolution in Time and Frequency Comparing STFT and Wavelet Transform Comparing STFT and Wavelet Transform Resolution in Time and Frequency Domain

27 Modal identification of LTI systems using Wavelet function is defined as wavelets The inverse of the wavelet transform is given by Multiple component signals can be written as Response of an under-damped SDOF system can be expressed as

28 Modal identification of LTI systems using wavelets Assuming a slowly varying envelope A(t) The response of the MDOF system is calculated as for a given value of a i related to natural frequency f ni modulus of wavelet transform leads to and is used for identifying damping.

29 Modal parameter identification using wavelets To detect the bands of frequencies in which natural frequencies lie, energy corresponding to each band is calculated for a particular state of response. For j k th band, the mother basis for the packet, ω s (t) is formed with frequency domain description Its corresponding time domain description is The frequency band for the p th sub-band b within the original i jk th band dis in the interval

30 Modal parameter identification using wavelets Using corresponding basis functions and wavelet coefficients of and natural frequencies are obtained more precisely. The k th mode shape is obtained using scale parameter j k and sub-band band parameter p.

31 Wavelet based online monitoring of Linear Time Varying systems Considering a linear time varying system where the damping and stiffness matrices vary with respect to time. The r th component of the time varying k th mode is represented by the ratio of wavelet coefficients of the considered states at time instant t = b. 5WSCSM, Tokyo, Japan

Experimental Validation using Wavelets Three storey

Instantaneous frequencies and damping ratios

Scalogram (view from below the x-y plane showing

32 Experimental Validation using Wavelets Three storey scaled structure subjected to free vibration tests Instantaneous frequencies and damping ratios extracted using wavelets. Scalogram (view from below the x-y plane showing existence of three modes and free vibration decrement) of the Measured Third Floor Free Vibration

33 Experimental validation Wavelet Coefficients of the Measured Free Vibration Displacement Response of All Three Floors

34 Experimental validation Mode Shapes Estimated using Wavelets Mode 1 Mode 2 Mode 3 Storey Storey Storey First Mode Damping and Frequency Estimation using Wavelet Coefficient/Hilbert Transform

35 Numerical validation of LTV system Validation of wavelet technique using numerical simulation of 2DOF LTV system Original system Changed system Stiffness of the system changed at 5.72s and restored to its original value at 12.48s.

36 Numerical Validation Validation of wavelet technique using numerical simulation of 2DOF LTV system (rad/s) ω n Actual Estimated t (s) 1.8 φ 21 /φ t (s)

37 Experimental Validation of LTV system using Wavelets Measured Third Floor Acceleration Response to White Noise Excitation Fourier Spectrum of Third Floor Acceleration

38 Experimental Validation of LTV system using Wavelets Scalogram of Third Floor Acceleration Response: Note the Shift in The First Mode Frequency (Ridge) of 5.5 Hz to 4.9 Hz after 10 seconds

39 Experimental Validation of LTV system using Wavelets Wavelet Coefficients of the Measured Third Floor Acceleration Response

40 Experimental Validation of LTV system using Wavelets First Mode Frequency Estimation using Wavelet Coefficient/Hilbert Transform (Note the Change in Frequency Before and After Damage at 10 sec)

41 Experimental Validation of LTV system using Wavelets First Mode Frequency Estimation Before Damage using Random Decrement/HT

42 Experimental Validation of LTV system using Wavelets First Mode Frequency Estimation After Damage using Random Decrement/HT

43 Experimental Validation of LTV system Using STFT Spectrogram of the Third Floor Acceleration Response to White Noise Excitation

44 Experimental Validation of LTV system Using EMD/IMF IMFs of the Third Floor Acceleration Response to White Noise Excitation

45 Conclusions The effectiveness of developed time-frequency algorithms for modal identification of MDOF systems has been demonstrated for experimental and simulated results. The EMD/HT and wavelet algorithms applied to MDOF are suitable for output only modal identification.

46 Acknowledgement This material was prepared as a part of course taught at Rice University by Professor Satish Nagarajaiah Professor Biswajit Basu, visiting iti professor, Rice University, it Associate Professor, Trinity College, University of Dublin, contributed to the wavelets section. References: Nagarajaiah, S. "Adaptive Passive, Semiactive, Smart Tuned Mass Dampers: Identification and Control using Empirical Mode Decomposition, Hilbert transform, and Short-Term Fourier transform," Structural Control and Health Monitoring, 16(7-8), (2009). Nagarajaiah, S., and Basu, B. " Output only Modal Identification and Structural Damage Detection using Time Frequency & Wavelet Techniques," Earthquake Engineering and Engineering Vibration, 8(4), (2009). B. Basu, S. Nagarajaiah, A. Chakraborty Online Identification of Linear Time-Varying Stiffness of Structural Systems by Wavelet Analysis, International Journal of Structural Health Monitoring, 7(1), (2008).

Introduction to structural dynamics

Introduction to structural dynamics p n m n u n p n-1 p 3... m n-1 m 3... u n-1 u 3 k 1 c 1 u 1 u 2 k 2 m p 1 1 c 2 m2 p 2 k n c n m n u n p n m 2 p 2 u 2 m 1 p 1 u 1 Static vs dynamic analysis Static