A compressive MUSIC spectral approach for identification of closely-spaced structural natural frequencies and post-earthquake damage detection

TL;DR: Results suggest that the adopted approach for identifying resonant frequencies of white-noise excited structures using acceleration measurements acquired at rates significantly below the Nyquist rate is robust and noise-immune while it can reduce data transmission requirements in acceleration wireless sensors for natural frequency identification and damage detection in engineering structures.

About: This article is published in Probabilistic Engineering Mechanics.The article was published on 2020-02-07 and is currently open access. It has received 15 citations till now. The article focuses on the topics: Sampling (signal processing) & Frequency domain.

Summary

1 Introduction and motivation

• Moreover, over the past two decades, wireless sensors/accelerometers have been heavily explored to further support the considered aim within the OMA framework as they enable rapidly deployable and low up-front cost field instrumentation compared to arrays of wired sensors [11- 12].
• Along these lines, herein, a sparse-free structural system identification approach is put forth to estimate natural frequencies of existing linearly vibrating structures exposed to unmeasured broadband/white noise, within the OMA framework, from response acceleration measurements sampled at rates significantly below the nominal Nyquist rate.

2 Mathematical Background of Proposed Method

• 1 Co-prime sampling and auto-correlation estimation of stationary stochastic processes Let x(t) be a real-valued wide-sense stationary band-limited stochastic process assuming a spectral representation by a superposition of R sinusoidal functions with frequencies fr, real amplitudes Br, and uncorrelated random phases θr uniformly distributed in the interval [0, 2π].
• This signal model is motivated by the fact that response time-histories of linear vibrating structures under low-amplitude ambient excitation have well-localized energy in the frequency domain centered at the structural natural frequencies (e.g., [34]) and, in this respect, the model proved to be adequate for CS-based modal analysis in a previous study [14].
• Co-prime sampling assumes that the process x(t) is simultaneously acquired by two sampling units, operating at different (sub-Nyquist) sampling rates, 1/(N1Ts) and 1/(N2Ts), where N1, N2 are coprime numbers (N1 < N2), and 1/Ts= 2fmax is the Nyquist sampling rate with fmax being the highest frequency component in Eq. (1) [24].
• In the following section, the latter matrix is used as input to the MUSIC super-resolution spectral estimator to detect the R frequencies fr, (r= 1,2,…,R), of the considered stochastic process x(t).
• The first term in Eq. (8) represents the signal sub-space with R eigenvalues 2( )i + , i=1,…,R, and R principal eigenvectors spanning the same subspace with the signal vector in Eq. (5).

3 Identification of closely-spaced natural frequencies from noisy acceleration data

• The proposed co-prime sampling with MUSIC spectral estimator approach is numerically assessed to estimate closely-spaced resonant frequencies of white-noise excited structures modelled as multi-degree-of-freedom (MDOF) dynamic systems.
• The derived noisy acceleration response signals, x[q], are then and co-prime sampled as detailed in section 2.1 and the full-rank autocorrelation matrix in Eq. (7) is constructed.
• For the other sub-Nyquist sampling cases in Table 1 the pertinent coprime sampling parameters and correlation estimators are defined in a similar manner as above.
• MUSIC pseudo-spectra of structure 2 in Fig.2(b) obtained for co-prime sampling specifications of Table 1 and for 5 different SNR values, also known as Figure 4.

4 Application for natural frequency-based post-earthquake damage detection

• An additional numerical study is undertaken to demonstrate the applicability and usefulness of the proposed system identification method in detecting relatively light structural damage induced to buildings by earthquakes.
• Herein, much more flexible structural systems than those examined in the previous section (Fig.2) are considered being representative of large-scale engineering structures for which wireless-sensor assisted OMA is practically mostly relevant [11].
• To this aim, the proposed approach is applied to estimate natural frequencies before (healthy state) and after (potentially damaged state) a seismic event within the standard OMA context (i.e., stationary excitation and linear structural response assumptions apply).
• Notably, in this setting, the consideration of wireless sensors in conjunction with the proposed co-prime sampling plus MUSIC approach leading to reduced sensor energy consumption is practically quite beneficial as long-term/permanent structural monitoring deployments are required for the purpose.
• In such deployments reducing battery replacement frequency, and thus maintenance costs, becomes critical and may be a main criterion for installing a monitoring system in the first place (e.g., [13]).

4.1 Adopted structure and seismic action

• The planar 3-storey single-bay reinforced concrete (r/c) frame in Figure 5 is considered as a case-study structure with beams and columns longitudinal and transverse reinforcement as indicated in the figure.
• Lo is the shear span taken herein as half the structural member length, dbl is the diameter of the longitudinal reinforcement, and fyk, fuk/fuk are the steel strength and strain hardening ratio, respectively, given in the previous sub-section.
• Specifically, two equivalent linear FE models are defined, corresponding to the two different damage states, in which the earthquake-induced damage is represented by means of the flexural stiffness reduction factors of Table 3.
• Further, the pre-eartquake/“healthy” state of the considered structure is modelled by a linear FE model with the secant flexural rigidities at yield presented in Table 2 assigned to the full length of structural members.

4.3 Post-earthquake damage detection

• Linear RHA is undertaken for the three FE models defined in the previous sub-section (healthy plus two damaged states), using the same low amplitude white noise base excitation of 80s duration.
• It is noted that a certain level of overlapping between the considered time blocks occurs, given that the structural response acceleration signals are only 8000 Nyquist samples long.
• Compared to Fourier-based spectral estimators, MUSIC yields a pseudo-spectrum with sharp peaks corresponding to the natural frequencies of the white-noise excited 3-storey frame (following standard OMA and linear random vibrations considerations), while filtering out additive broadband noise.
• In all plots, a shift of the natural frequencies towards smaller values is seen indicating structural damage.

5 Concluding Remarks

• A novel natural frequency identification and damage detection approach has been established utilizing response acceleration measurements of white-noise excited structures sampled at rates significantly below the Nyquist rate supporting reduced data transmission in wireless sensors for vibration-based structural monitoring.
• Acceleration time-histories are treated as realizations of a stationary stochastic process without posing any sparse structure requirements.
• It was shown that the adopted co-prime MUSIC-based strategy is a potent tool for natural frequency identification within the operational modal analysis context, capable to efficiently address the structural modal coupling effect even by treating response signals buried in noise.
• The effectiveness and applicability of the proposed approach was numerically evaluated using a white-noise excited linear reinforced concrete 3-storey frame in a healthy and two damaged states caused by two ground motions of increased intensity.
• The numerical results demonstrate that the considered approach is capable to detect very small structural damage directly from the compressed measurements even for high noise levels at SNR=10dB.

Acknowledgments

• This work has been partly funded by EPSRC in UK, under grant No EP/K023047/1: the second author is indebted to this support.
• The first author further acknowledges the support of City, University of London through a PhD studentship.

Figures

Table 3. Flexural rigidity reduction factor (EIeff/EIy) at critical member zones of the structure in Figure 5 for the two different damage states considered due to different seismic intensity excitation

Figure 7. Moment-curvature (M-φ) hysteretic curves at the left plastic hinge of the 1st storey beam for (a) damage state 1 and (b) damage state 2.

Figure 3. MUSIC pseudo-spectra of structure 1 in Fig.2(a) obtained for co-prime sampling specifications of

Figure 1. Simulation-based assessment framework for the proposed natural frequencies identification approach in OMA applications

Figure 8. Spectrum estimation from noisy acceleration response signals with SNR=10dB at the (a) first, (b) second, and (c) third floor of the structure in Fig. 5 (healthy state) subject to 80s duration white noise base excitation

Table 1. Co-prime sampling specifications

Figure 2. Normalized target PSDs in Eq.(13) for the two adopted 3-DOF structural systems.

Table 2. Average secant flexural rigidity at yielding, EIy, at the ends of the frame structural members of Fig. 5

Figure 6. Horizontal ground motion component recorded at “Sanjo Shinbori” station during the Mw=6.8 Chuetsu-oki event (16.7.2007) in Japan: (a) Time-history, (b) Squared amplitude of Fourier spectrum [23] 4.2 Finite element modelling of earthquake-induced

Table 4. Assessment of MUSIC spectra from co-prime sampled noisy measurements for damage detection based on structural natural frequency shifts: damage state 1

Figure 10. MUSIC pseudo-spectra with co-prime sampling of noisy acceleration response signals with SNR=10dB at the (a) first, (b) second, and (c) third floor for the healthy and the damaged state 2 structure

Table 5. Assessment of MUSIC spectra from co-prime sampled noisy measurements for damage detection based on structural natural frequency shifts: damage state 2

Figure 9. MUSIC pseudo-spectra with co-prime sampling of noisy acceleration response signals with SNR=10dB at the (a) first, (b) second, and (c) third floor for the healthy and the damaged state 1 structure