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Real-time cable tension estimation from acceleration measurements using wireless sensors with packet data losses: analytics with compressive sensing and sparse component analysis

Journal of Civil Structural Health Monitoring volume 12, pages 797–815 (2022)Cite this article

Abstract

Stay-cables in the cable-stayed bridge are the most vital components as they carry the bridge deck’s load and transmit the force to the bridge pylons. However, dynamics loads due to vortex-induced vibration, ambient wind excitation, and even vehicular vibration cause fatigue in the stay-cable. Hence continuous real-time performance monitoring of such cables is necessary for maintenance to avoid any kind of damage to the cable. Wireless sensors are contact-based sensor that provide accurate measurement, and it does not involve any wiring cost like conventional wired sensors. Monitoring cable health using such wireless sensors is a good choice provided packet loss (which occurs while transmitting the measured data to the base station) that invariably occurs is addressed by data processing. Such discontinuity in data (due to packet loss) may interrupt the real-time/online cable health monitoring process - depending on the window length of the data loss. In general, online health monitoring using multiple sensors reduces the estimation errors. In this paper, we propose a framework that takes the wireless sensor data as the input, then reconstructs the packet lost samples (if any), and finally, provides a real-time tension estimation as an output. The novel framework, first adopts compressive sensing algorithm to reconstruct the data due to packet loss. Subsequently, we synthesize the reconstructed responses from multiple sensors to estimate the real-time frequency variation using Blind Source Separation (BSS) Technique. As the cable response due to ambient vibration contains a large number of modes, the dominant modal response or the corresponding dominant frequency is estimated from very few measurements using a variant of the BSS technique named Sparse Component Analysis (SCA). Finally, real-time cable tension is estimated from the frequency variation using the taut-string theory. The proposed technique is applied to a real full-scale cable-stayed bridge. The mean tension obtained from the framework is comparable with the cable’s actual design tension. The accurate estimation of real-time stay-cable tension by the proposed algorithm shows great potential in the field of structural health monitoring.

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Acknowledgements

This research was made possible by financial support from Science and Engineering Research Board of India (SERB) and Rice University to Debasish Jana and Satish Nagarajaiah. The financial support by SERB-India is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Rice University, Houston, TX, 77005, USA

    Debasish Jana, Satish Nagarajaiah, Yongchao Yang & Shunlong Li

  2. Department of Mechanical Engineering, Rice University, Houston, TX, 77005, USA

    Satish Nagarajaiah

  3. Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, 49931, USA

    Yongchao Yang

  4. Bridge and Tunnel Engineering, Harbin Institute of Technology, Harbin, 150090, China

    Shunlong Li

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  1. Debasish Jana
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  2. Satish Nagarajaiah
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Correspondence to Satish Nagarajaiah.

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Appendices

Appendix A: Selection of time-window size for real-time tension estimation

In the real-time tension estimation framework, the penultimate step is to estimate the real-time frequency, which is the dominant frequency in a particular sliding window. If the window size is very small, that small window may not preserve the very low-frequency content of the signal. Conversely, a larger window increases the processing time of Fast Fourier Transform (FFT) calculation as the computational complexity of FFT is \({\mathcal {O}}(n\log n)\) for n samples. In addition, a large window will produce the real-time estimation with a larger time lag. In this paper, the length of the sliding window is chosen as 300. The fundamental frequency of the cable is 0.609 Hz (Table 1). So to capture this frequency, one needs at least \(1/0.609=1.642\) seconds, in this case, 165-time samples as the sampling frequency is 100 Hz. A sliding window of 300 ensures approximately two waves of the fundamental frequency.

To study the effect of the window size in real-time tension estimation, we choose window sizes of different lengths as 200, 300, 400, and 500. Here the window length of 300 is selected for the study mentioned before. For these different window sizes, the estimated real-time tension history is shown in Fig. 14 and the statistics of the tension is mentioned in Table 4. The estimated tension shown in Fig. 14 is very noisy for the window size of 200; window size of 300 offers a trade-off between noise and accuracy. We have chosen the window size of 300 for this study since from Table 4 higher window length do not improve the mean estimation.

Fig. 14
figure 14

Real-time tension from the signals in Fig. 13a, b for different window sizes

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Table 4 Real-time tension statistics for different window sizes
Full size table

Appendix B: Comparison of estimated real-time tension for various size of missing data

To study the effectiveness of the proposed framework, in this paper, we have considered maximum packet loss as \(20\%\). The commonly accepted standards [89] specify the quality levels for data packet loss. The levels are set at \(0-1\%\) = good, \(1-5\%\) = acceptable, \(5-12\%\) = poor, and greater than \(12\%\) = bad. In general wireless sensor with a high percentage of data packet loss is discarded. We have already shown that real-time tension can be accurately estimated using the data with packet loss with 20%, the acceptable packet loss scenarios (loss \(<12\%\)) should exhibit better accuracy.

In this section, we numerically study the real-time tension estimation with various sizes of missing data. We considered the loss percentages as \(5\%\), \(10\%\), and \(15\%\) and compared with the \(20\%\) loss and no loss input. The original data is obtained from Fig. 13a, b where t=5 to t=20 seconds is considered. Different missing data percentages are numerically introduced for this study. For these different sizes of missing data, the estimated real-time tension history is shown in Fig. 15 and the statistics of the tension is mentioned in Table 5. It is observed that as the loss percentage increases, the estimation uncertainty is increased, which is obvious as the reconstruction error is less for low packet loss data.

Fig. 15
figure 15

Real-time Tension from the Signals in Fig. 13a, b (from \(t=5\) to 20) for various size of missing data

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Table 5 Real-time tension statistics for various size of missing data
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Jana, D., Nagarajaiah, S., Yang, Y. et al. Real-time cable tension estimation from acceleration measurements using wireless sensors with packet data losses: analytics with compressive sensing and sparse component analysis. J Civil Struct Health Monit 12, 797–815 (2022). https://doi.org/10.1007/s13349-021-00526-4

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Keywords

  • Real-time frequency estimation
  • Time-varying cable tension
  • Cable-stayed bridge
  • Wireless sensors
  • Packet loss
  • Compressive sensing
  • Data reconstruction
  • Sparse component analysis
  • Blind source separation

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