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

## 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

144 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}

63 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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42 citations

11 citations

10 citations

##### References

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...In this regard, note that, with the exception of the approach in [21], all sub-Nyquist system identification techniques rely on the compressive sensing (CS) paradigm [23] involving randomly sampled in time measurements whose (sub-Nyquist) sampling rate for faithful time/frequency domain information recovery or modal properties extraction depends on the acceleration signals sparsity....

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### "A compressive MUSIC spectral approa..." refers background in this paper

...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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