# 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.

Abstract: Motivated by practical needs to reduce data transmission payloads in wireless sensors for vibration-based monitoring of engineering structures, this paper proposes a novel approach for identifying resonant frequencies of white-noise excited structures using acceleration measurements acquired at rates significantly below the Nyquist rate. The approach adopts the deterministic co-prime sub-Nyquist sampling scheme, originally developed to facilitate telecommunication applications, to estimate the autocorrelation function of response acceleration time-histories of low-amplitude white-noise excited structures treated as realizations of a stationary stochastic process. Next, the standard multiple signal classification (MUSIC) spectral estimator is applied to the estimated autocorrelation function enabling the identification of structural natural frequencies with high resolution by simple peak picking in the frequency domain without posing any sparsity conditions to the signals. This is achieved by processing autocorrelation estimates without undertaking any (typically computationally expensive) signal reconstruction step in the time-domain, as required by various recently proposed in the literature sub-Nyquist compressive sensing-based approaches for structural health monitoring, while filtering out any broadband noise added during data acquisition. The accuracy and applicability of the proposed approach is first numerically assessed using computer-generated noise-corrupted acceleration time–history data obtained by a simulation-based framework examining white-noise excited structural systems with two closely-spaced modes of vibration carrying the same amount of energy, and a third isolated weakly excited vibrating mode. Further, damage detection potential of the developed method is numerically illustrated using a white-noise excited reinforced concrete 3-storey frame in a healthy and two damaged states caused by ground motions of increased intensity. The damage assessment relies on shifts in natural frequencies between the pre-earthquake and post-earthquake state. Overall, numerical results demonstrate that the considered approach can accurately identify structural resonances and detect structural damage associated with changes to natural frequencies as minor as 1% by sampling up to 78% below Nyquist rate for signal to noise ratio as low as 10dB. These results suggest that the adopted approach 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.

## Summary (3 min read)

### 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.

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##### Citations

136 citations

^{1}, University of Camerino

^{2}, University of Strathclyde

^{3}, City University London

^{4}, Imperial College London

^{5}, University of Surrey

^{6}, Towers Watson

^{7}, University of Bristol

^{8}, AXA

^{9}

7 citations

### Additional excerpts

...Recent approaches for reducing wireless data transmission tailored for seismic V-SHM include the consideration of smart sensor triggering for on-demand measurements at the onset of seismic events, using programmable on-board event-based switching [338] as well as the consideration of compressive sampling schemes for accumulating and transmitting measurements at a small fraction of the Nyquist rate to detect natural frequency shifts due to earthquake damage [339]....

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5 citations

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...Accurate identification of the natural frequencies of large-scale (civil) engineering structures and structural components is key to several important practical applications such as: the design verification of structural systems sensitive to resonance with external loading frequencies [1,2]; the detection of structural damage [3-5]; the tuning/designing of resonant vibration absorbers [6], meta-structures [7], and dynamic energy harvesters [8] for suppressing structural vibrations; the performance assessment of structures equipped with dynamic vibration absorbers [9]....

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...(5), whose number increases with increasing N1 and/or N2, are systematically eliminated by extending the approach discussed in [26] for the case of spatial processes (direction of arrival problem), to treat the herein addressed problem of temporal frequency estimation....

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