Multiple outputs operational modal identification of time-varying systems Mathieu BERTHA University of Liège (Belgium) Friday, February 23, 2018
The global framework of this research is the modal identification of structures It is a pretty large field of structural analysis It can be decomposed in: ◮ Experimental or Operational Modal Analysis ◮ Linear or Nonlinear Systems ◮ Time Invariant or Time-Varying ◮ Single or Multiple Outputs ◮ And any of their combinations... Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 1
The global framework of this research is the modal identification of structures Nonlinear Well assessed Not fully exploited Under development Output only Time variant Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 2
Why time-varying behaviour can occur ? Several possible origins : ◮ Structural changes ◮ Operating conditions ◮ Damage occurrence Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 3
MDOF linear time-varying mechanical systems are considered M ( t ) ¨ y ( t ) + C ( t ) ˙ y ( t ) + K ( t ) y ( t ) = f ( t ) The dynamics of such systems is characterized by : ◮ Non-stationary time series ◮ Instantaneous modal properties ◮ Frequencies : ω r ( t ) ◮ Damping ratio’s : ζ r ( t ) ◮ Modal deformations : v r ( t ) Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 4
Outline of the presentation Several identification methods are proposed in the thesis The presentation is organized as follows: Non-parametric approach ◮ Presentation of the experimental setup Combined parametric and non-parametric approach Fully parametric approaches Applications to more complex cases Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 5
Some classical signal processing techniques can deal with nonstationary signals Time-frequency representations: Short-time Fourier transform Wavelet analysis Wigner-Ville distribution Signal decomposition methods: Hilbert-Huang Transform (HHT) Hilbert Vibration Decomposition (HVD) Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 6
The Hilbert Transform The Hilbert transform H of a signal x ( t ) is the convolution product of 1 this signal with the impulse response h ( t ) = π t x ( t ) ˜ = H ( x ( t )) = ( h ( t ) ∗ x ( t )) � + ∞ = p.v. x ( τ ) h ( t − τ ) dτ −∞ � + ∞ 1 x ( τ ) = π p.v. t − τ dτ −∞ It is a particular transform that remains in the same domain as the original signal It corresponds to a phase shift of π 2 of the signal Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 7
The Hilbert transform is used to build the complex analytic form of a signal The analytic signal z is built as z ( t ) = x ( t ) + i H ( x ( t )) A ( t ) e iφ ( t ) = In the frequency domain, the analytic signal becomes a one-sided signal Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 8
The analytic signal can be seen as a rotating phasor in the complex plane x Time i × H ( x ) H ( x ) x Time Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 9
It is suitable to find the envelope of the signal The instantaneous envelope of the signal is given by the absolute value of the analytic signal A ( t ) = | z ( t ) | x(t) A(t) x Time Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 10
It also gives information about the instantaneous phase The instantaneous phase angle of the signal is given by the argument of the analytic signal φ ( t ) = ∠ z ( t ) The time derivative of the phase angle gives the instantaneous frequency ω ( t ) = d φ ( t ) d t Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 11
The Hilbert transform is powerful but only meaningful for monocomponent signals Let us consider a 2-component signal mixture Time Time Time Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 12
Signal decomposition methods In order to get meaningful instantaneous frequencies, the signals have to be decomposed. Two good candidates exist: The Empirical Mode Decomposition method ◮ Successive extraction of the highest instantaneous frequency component The Hilbert Vibration Decomposition Method ◮ Successive extraction of the highest instantaneous amplitude component Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 13
The Empirical Mode Decomposition sifting process The EMD method iteratively removes the mean of the upper and lower envelopes computed by spline fitting Iteration 1 Iterations Iteration n IMF 1 Time Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 14
The Hilbert Vibration Decomposition sifting process x ( t ) Main signal Dominant component Secondary component Analytic signal z ( t ) = x ( t ) + i H ( x ( t )) Frequency extraction ω ( t ) = dφ ( t ) = d ∠ z ( t ) dt dt Lowpass filtering ω ( t ) → ω k ( t ) Synchronous demodulation x k ( t ) Sifting process x ( t ) := x ( t ) − x k ( t ) Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 15
The HVD method in that scheme has some drawbacks It is applicable to single channel measurement The application on multiple channels has to be done in parallel In a multivariate case, all the modes have to be excited at each time instant on all the channels The method will always follow the instantaneous dominant mode Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 16
Example: a simple 2–DoF time–variant system System properties: = 3 [kg] ◮ m 1 ◮ m 2 = 1 [kg] = 20000 [N/m] ◮ k 1 c 1 = 3 [N.s/m] ◮ = 25000 ց 5000 [N/m] ◮ k 2 Frequency [Hz] 40 20 0 40 20 0 0 5 10 15 Time [s] Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 17
Application of the HVD method on each channel A simple application of the HVD method on each channel leads to mode mixing 10 -3 10 -3 5 5 0 0 -5 -5 Frequency [Hz] Frequency [Hz] 40 40 20 20 0 0 Frequency [Hz] Frequency [Hz] 40 40 20 20 0 0 0 5 10 15 0 5 10 15 10 -3 10 -3 5 5 0 0 -5 -5 10 -3 10 -3 2 2 0 0 -2 -2 0 5 10 15 0 5 10 15 Time [s] Time [s] Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 18
In the case of multiple channel measurements, a source separation step is introduced in the algorithm x ( t ) The sources are used as references to get Source separation x ( t ) → s ( t ) the instantaneous frequencies Analytic signal z ( t ) = s 1 ( t ) + i H ( s 1 ( t )) Phase extraction φ ( t ) = ∠ z ( t ) A trend extraction method computes the Trend extraction φ ( t ) → φ ( k ) ( t ) phase of the dominant mode A Vold-Kalman filter (VKF) is used for VKF φ ( k ) ( t ) → x ( k ) ( t ) , v k ( t ) component extraction Sifting process x ( t ) := x ( t ) − x ( k ) ( t ) Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 19
Introducing a source separation method can help to avoid the mode mixing phenomenon A Blind Source Separation method (BSS) separates a set of signals in a set of uncorrelated or independent sources 0.04 0.04 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 Frequency [Hz] Frequency [Hz] 40 40 20 20 0 0 Frequency [Hz] Frequency [Hz] 40 40 20 20 0 0 0 5 10 15 0 5 10 15 10 -3 10 -3 5 2 0 0 -5 -2 10 -3 10 -3 5 2 0 0 -5 -2 0 5 10 15 0 5 10 15 Time [s] Time [s] Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 20
The experimental setup The experimental setup is an aluminum beam with a moving mass. The whole system is supported by springs and excited by a shaker. ◮ 2 . 1 meter long and 8 × 2 cm for the cross section ◮ 9 kg for the beam and ≈ 3 . 5 kg for the moving mass (ratio of 38 . 6 %) ◮ The excitation and measurements are performed with a Siemens LMS system Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 21
Time invariant modal identification of the beam subsystem 9.8 Hz 30.43 Hz 39.23 Hz 53.32 Hz 99.22 Hz CMIF ε f : 1% s s s s s 40 s s s s s 39 ε ζ : 1% s s s s s 38 s s s s s ε V : 1% 37 s s s s s 36 v s s s v 35 s s s s s 34 s s s s s 33 s s s s s 32 s s s s s 31 s s s s s 30 v s s s v 29 v s s s s 28 v s s s s 27 s s s s s 26 v s s s s 25 v s s s s 24 v s s s s 23 s s s s s 22 v s s s v 21 s s s s s 20 s s s s s 19 v s s s s 18 v s s s s 17 v s s s s 16 v s s s s 15 o s s s s 14 s v s s 13 s s s s 12 v s s s 11 v v s v 10 v v s s 9 o o v v 8 v v 7 v v 6 o s 5 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Frequency [Hz] Mathieu BERTHA (ULg) Multiple outputs operational modal identification of time-varying systems 22
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