A new signal reconstruction for damage detection on a simply supported beam subjected to a moving mass

Abstract

A new signal reconstruction is proposed for damage detection on a simply supported beam using multiple measurements of displacement induced by a moving sprung mass. The new signal is constructed from the difference between the spatially integrated deflection for the intact (baseline) and damaged beams under quasi-static loading. To that end, it is shown that the static component of displacement from the dynamic moving mass experiment may be extracted very effectively using a robust smoothing technique and that this outperforms some comparable techniques. It is shown that by measuring displacement at a modest number of points on the beam the new reconstructed signal is able to detect the location of the damage more accurately than methods that use only a single-point data. In particular, the technique is able to detect damage present simultaneously at multiple locations and can do so with a highly variable moving mass velocity. In order to construct an a posteriori baseline, the strain data from the same traverse could be used to recover the displacement-time history of the intact beam, which could enhance the method by enabling the baseline to be determined from the same experiment, further eliminating effects of experimental conditions if required. However, a Monte Carlo simulation is run to consider the effect of signal noise, showing that the proposed damage detection strategy locates damage even in the presence of noise of 50% in the measured signals (\({\text {SNR}} =7\, {\text {dB}}\)).

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

References

  1. 1.

    Lakshmi K, Rao ARM (2019) Detection of subtle damage in structures through smart signal reconstruction. Procedia Struct Integr 14:282–289

    Article  Google Scholar 

  2. 2.

    Ghannadi P, Kourehli SS (2019) Data-driven method of damage detection using sparse sensors installation by serepa. J Civ Struct Health Monit 9:1–17

    Article  Google Scholar 

  3. 3.

    Yu Y, Zhao X, Shi Y, Ou J (2013) Design of a real-time overload monitoring system for bridges and roads based on structural response. Measurement 46(1):345–352

    Article  Google Scholar 

  4. 4.

    Li J, Hao H (2016) Health monitoring of joint conditions in steel truss bridges with relative displacement sensors. Measurement 88:360–371

    Article  Google Scholar 

  5. 5.

    Jin Q, Liu Z (2019) In-service bridge shm point arrangement with consideration of structural robustness. J Civ Struct Health Monit 9:1–12

    Article  Google Scholar 

  6. 6.

    He W-Y, Ren W-X, Zhu S (2017) Damage detection of beam structures using quasi-static moving load induced displacement response. Eng Struct 145:70–82

    Article  Google Scholar 

  7. 7.

    He W-Y, Zhu S (2016) Moving load-induced response of damaged beam and its application in damage localization. J Vib Control 22(16):3601–3617

    Article  Google Scholar 

  8. 8.

    He W-Y, He J, Ren W-X (2018) Damage localization of beam structures using mode shape extracted from moving vehicle response. Measurement 121:276–285

    Article  Google Scholar 

  9. 9.

    Zhang W, Li J, Hao H, Ma H (2017) Damage detection in bridge structures under moving loads with phase trajectory change of multi-type vibration measurements. Mech Syst Signal Process 87:410–425

    Article  Google Scholar 

  10. 10.

    Sun Z, Nagayama T, Nishio M, Fujino Y (2018) Investigation on a curvature-based damage detection method using displacement under moving vehicle. Struct Control Health Monit 25(1):e2044

    Article  Google Scholar 

  11. 11.

    Yang Q, Liu J, Sun B, Liang C (2017) Damage localization for beam structure by moving load. Adv Mech Eng. https://doi.org/10.1177/1687814017695956

    Article  Google Scholar 

  12. 12.

    Ono R, Ha TM, Fukada S (2019) Analytical study on damage detection method using displacement influence lines of road bridge slab. J Civ Struct Health Monit 9:1–13

    Article  Google Scholar 

  13. 13.

    Huang NE, Shen Z, Long SR, Wu MC, Shih HH, Zheng Q, Yen N-C, Tung CC, Liu HH (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc Lond A Math Phys Eng Sci 454:903–995. https://doi.org/10.1098/rspa.1998.0193

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Meredith J, González A, Hester D (2012) Empirical mode decomposition of the acceleration response of a prismatic beam subject to a moving load to identify multiple damage locations. Shock Vib 19(5):845–856

    Article  Google Scholar 

  15. 15.

    Roveri N, Carcaterra A (2012) Damage detection in structures under traveling loads by Hilbert–Huang transform. Mech Syst Signal Process 28:128–144

    Article  Google Scholar 

  16. 16.

    OBrien EJ, Malekjafarian A, González A (2017) Application of empirical mode decomposition to drive-by bridge damage detection. Eur J Mech A Solids 61:151–163

    Article  Google Scholar 

  17. 17.

    Khorram A, Bakhtiari-Nejad F, Rezaeian M (2012) Comparison studies between two wavelet based crack detection methods of a beam subjected to a moving load. Int J Eng Sci 51:204–215

    Article  Google Scholar 

  18. 18.

    Mao Q, Mazzotti M, DeVitis J, Braley J, Young C, Sjoblom K, Aktan E, Moon F, Bartoli I (2019) Structural condition assessment of a bridge pier: a case study using experimental modal analysis and finite element model updating. Struct Control Health Monit 26(1):e2273

    Article  Google Scholar 

  19. 19.

    Moughty JJ, Casas JR (2017) A state of the art review of modal-based damage detection in bridges: development, challenges, and solutions. Appl Sci 7(5):510

    Article  Google Scholar 

  20. 20.

    He W-Y, Ren W-X, Zhu S (2017) Baseline-free damage localization method for statically determinate beam structures using dual-type response induced by quasi-static moving load. J Sound Vib 400:58–70

    Article  Google Scholar 

  21. 21.

    Garcia D (2010) Robust smoothing of gridded data in one and higher dimensions with missing values. Comput Stat Data Anal 54(4):1167–1178

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Caddemi S, Morassi A (2007) Crack detection in elastic beams by static measurements. Int J Solids Struct 44(16):5301–5315

    MATH  Article  Google Scholar 

  23. 23.

    Yang Y, Lin C (2005) Vehicle–bridge interaction dynamics and potential applications. J Sound Vib 284(1–2):205–226

    Article  Google Scholar 

  24. 24.

    Hu N, Dai G-L, Yan B, Liu K (2014) Recent development of design and construction of medium and long span high-speed railway bridges in China. Eng Struct 74:233–241

    Article  Google Scholar 

  25. 25.

    Zhu X, Law S (2006) Wavelet-based crack identification of bridge beam from operational deflection time history. Int J Solids Struct 43(7–8):2299–2317

    MATH  Article  Google Scholar 

  26. 26.

    Kurata M, Kim J-H, Lynch JP, Law KH, Salvino LW (2010) A probabilistic model updating algorithm for fatigue damage detection in aluminum hull structures. In: ASME 2010 conference on smart materials, adaptive structures and intelligent systems. American Society of Mechanical Engineers, pp 741–750

  27. 27.

    Zhang B, Qian Y, Wu Y, Yang Y (2018) An effective means for damage detection of bridges using the contact-point response of a moving test vehicle. J Sound Vib 419:158–172

    Article  Google Scholar 

  28. 28.

    Agostinacchio M, Ciampa D, Olita S (2014) The vibrations induced by surface irregularities in road pavements—a Matlab® approach. Eur Transp Res Rev 6(3):267–275

    Article  Google Scholar 

  29. 29.

    Worden K, Farrar CR, Manson G, Park G (2007) The fundamental axioms of structural health monitoring. Proc R Soc A Math Phys Eng Sci 463(2082):1639–1664

    Article  Google Scholar 

  30. 30.

    Tukey JW (1977) Exploratory data analysis, vol 2. Addison-Wesley, Reading

    Google Scholar 

  31. 31.

    Dragomiretskiy K, Zosso D (2014) Variational mode decomposition. IEEE Trans Signal Process 62(3):531–544

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Park J-H, Lim H, Myung N-H (2012) Modified Hilbert-Huang transform and its application to measured micro doppler signatures from realistic jet engine models. Progr Electromagn Res 126:255–268

    Article  Google Scholar 

  33. 33.

    Doyle JF (1997) A wavelet deconvolution method for impact force identification. Exp Mech 37(4):403–408

    Article  Google Scholar 

  34. 34.

    Martin M, Doyle J (1996) Impact force identification from wave propagation responses. Int J Impact Eng 18(1):65–77

    Article  Google Scholar 

  35. 35.

    Xu Y, Chen J (2004) Structural damage detection using empirical mode decomposition: experimental investigation. J Eng Mech 130(11):1279–1288

    Article  Google Scholar 

  36. 36.

    Yang JN, Lei Y, Lin S, Huang N (2004) Hilbert–Huang based approach for structural damage detection. J Eng Mech 130(1):85–95

    Article  Google Scholar 

  37. 37.

    Pines D, Salvino L (2006) Structural health monitoring using empirical mode decomposition and the Hilbert phase. J Sound Vib 294(1–2):97–124

    Article  Google Scholar 

  38. 38.

    Cheraghi N, Taheri F (2007) A damage index for structural health monitoring based on the empirical mode decomposition. J Mech Mater Struct 2(1):43–61

    Article  Google Scholar 

  39. 39.

    Rezaei D, Taheri F (2011) Damage identification in beams using empirical mode decomposition. Struct Health Monit 10(3):261–274

    Article  Google Scholar 

  40. 40.

    Peng ZK, Peter WT, Chu FL (2005) An improved Hilbert-Huang transform and its application in vibration signal analysis. J Sound Vib 286(1):187–205

    Article  Google Scholar 

  41. 41.

    Yang W-X (2008) Interpretation of mechanical signals using an improved Hilbert–Huang transform. Mech Syst Signal Process 22(5):1061–1071

    Article  Google Scholar 

  42. 42.

    Daubechies I, Lu J, Wu H-T (2011) Synchrosqueezed wavelet transforms: an empirical mode decomposition-like tool. Appl Comput Harmon Anal 30(2):243–261

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Gilles J (2013) Empirical wavelet transform. IEEE Trans Signal Process 61(16):3999–4010

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Dominique Zosso (2020) Variational mode decomposition. MATLAB Central File Exchange. Retrieved March 21, 2019 from https://www.mathworks.com/matlabcentral/fileexchange/44765-variational-mode-decomposition

  45. 45.

    Weaver W Jr, Timoshenko SP, Young DH (1990) Vibration problems in engineering. Wiley, New York

    Google Scholar 

Download references

Acknowledgements

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The authors declare that they have no conflict of interest.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Damien Holloway.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix 1: Signal decomposition techniques

Empirical mode decomposition (EMD)

As mentioned in the main body of the paper, the static part of vibration data recorded at some point on the beam is used for damage detection. However, one can only measure the total response of the beam, which has both the static and dynamic parts in it. Therefore, a decomposition technique can be used to decompose the signal into its constructive components. The EMD algorithm has been shown to be very effective in decomposing non-stationary and nonlinear signals and, therefore, is recognised as an effective method for the purpose of this paper. EMD has been also used in other context of SHM by many researchers so far [35,36,37,38,39].

The EMD algorithm was first introduced by Huang et al. in order to decompose a signal into its oscillation modes, termed intrinsic mode functions (IMFs) [13]. IMFs are fundamentally different to the mode functions in traditional linear modal analysis in that they can be non-stationary, i.e. they can be modulated in both amplitude and frequency. However, in common with linear modal analysis, each IMF is narrow band and only involves one mode of oscillation.

Huang et al. [13] made some assumptions for a signal in order that the sifting process can be applied to it:

  1. 1.

    the signal has at least two extrema (at least one maximum and one minimum);

  2. 2.

    the characteristic time scale is defined by the time lapse between the extrema;

  3. 3.

    in the case that the signal has no extrema but contains only inflection points, it can be differentiated several times until the extrema appear. Then, the results can be obtained by integration(s).

Considering the above preliminary discussions the EMD algorithm is applied to a signal X(t) as follows:

  1. 1.

    First find all local maxima and interpolate a cubic spline curve through them; do the same for the minima.

  2. 2.

    Take the mean of the two curves (envelopes) obtained from the first step and call it \(m_1\).

  3. 3.

    Compute \(h_1=X(t)-m_1\) and check if \(m_1\) complies with the definition of the IMF.

  4. 4.

    If not, repeat the steps 1 to 3 for \(h_1\) and compute \(m_{11}\) so that \(h_{11}=h_1-m_{11}\). If still \(h_{11}\) is not an IMF, repeat steps 1 to 3 to obtain \(h_{1k}=h_{1k-1}-m_{1k}\) so that \(c_1=h_{1k}\) is an IMF.

  5. 5.

    Obtain the first residual \(r_1=X(t)-c_1\) and repeat steps 1 to 5 for \(r_1\).

  6. 6.

    continue the sifting process until no IMF can be derived from \(r_n\). In this case, \(X(t)=\sum _{i=1}^{n}c_i+r_n\).

Huang et al. introduced a termination rule for the above algorithm by limiting SD, the standard deviation of two consecutive sifting results, which is calculated as

$$\begin{aligned} {\text {SD}}=\sum _{t=0}^{T} \frac{|h_{1(k-1)}(t)-h_{1k}(t)|^2}{h_{1(k-1)}^2(t)}. \end{aligned}$$
(14)

The flowchart of the basic EMD algorithm applied to an arbitrary signal X(t) is shown in Fig. 16. Refinements and improvements to the EMD algorithm have been introduced by several researchers [40, 41]. In this paper the function emd, available in Matlab (2018a and later versions) is used.

Fig. 16
figure16

Flowchart of the EMD algorithm

Variational mode decomposition (VMD)

Like EMD, VMD seeks to decompose a real-valued signal X(t) into its component modes \(u_k(t)\), but based on a new definition of an IMF. In the previous section, the basic definition for an IMF in EMD is discussed. However, thereafter, the criteria for a mode to be considered as an IMF slightly changed [42, 43]. Accordingly, in VMD an IMF is an amplitude-modulated-frequency-modulated (AM-FM) sinusoid which has the following additional characteristics:

  1. 1.

    the phase is a non-decreasing function;

  2. 2.

    the envelope is non-negative;

  3. 3.

    both the envelope and the instantaneous frequency vary much more slowly than the phase;

i.e. the IMF is written as

$$\begin{aligned} u_k(t)=A_k(t)\cos (\phi _k(t)), \end{aligned}$$
(15)

where the instantaneous frequency \(\omega _k(t)={\text {d}}\phi (t)/{\text {d}}t\ge 0\), and \(A_k(t)\) and \(\omega _k(t)\) vary much more slowly than \(\phi _k(t)\).

Generally, for a given signal X(t), VMD solves the following variational optimisation problem on k IMFs \(\{u_k\}=\{u_1, u_2, \ldots , u_k\}\) with center frequencies \(\{\omega _k\}=\{u_1, u_2, \ldots , w_k\}\) ,

$$\begin{aligned} \underset{\{u_k\}\; \& \;\{\omega _k\}}{\min }~ \sum _k\;\bigg \Vert \partial _t\;\left( \delta (t)+\frac{j}{\pi t}\;*\;u_k(t)\right) \;{\text {e}}^{-j\omega _kt}\bigg \Vert ^2, \end{aligned}$$
(16)

where in the above equation, \(*\) is the convolution operator and \(X(t)=\sum _k u_k\). However, the authors of the VMD paper add two further terms to the goal function of the optimisation problem of Eq. 16. These are a quadratic penalty at finite weight, and a Lagrangian multiplier to strictly enforce the constraint, which further guarantees the achievement of convergence in the presence of noise in the signal. The reader is referred to the original paper for further study [31].

A Matlab code can be found for VMD in Ref. [44]. However, it is essential to know that there are some parameters which must be tuned when decomposing a signal using VMD. The most important ones are the number of the modes k into which the user wishes to decompose the original signal and the Lagrangian multiplier \(\lambda\) that determines how much noise is allowed in the decomposition process.

Appendix 2: Static and dynamic parts of the vibration of a beam subjected to a moving mass

Fig. 17
figure17

Simply supported beam subjected to a moving load

A key component of the damage detection procedure proposed in Sect. 2 is to obtain the static part of the vibration data. We, therefore, present the analytic solution for the response of an undamped simply-supported uniform beam of length L subjected to a moving load \(P=m_v g\), shown in Fig. 17. In this analytical model the static and dynamic components of response are mathematically discrete expressions.

The beam is assumed to have a continuous cross section with flexural rigidity of EI and mass per unit length \(\rho A\). The response of the beam to arbitrary force f(xt) may be obtained by modal superposition as

$$\begin{aligned} y(x,t) = \sum _{n=1}^\infty \phi _n(x)\,y^*_n(t), \end{aligned}$$
(17)

in which \(\phi _n(x)\) is the nth mode shape and \(y^*_n(t)\) denotes the solution to the modal differential equation,

$$\begin{aligned} \ddot{y}^*_n(t)+\omega _n^2 y^*_n(t)=\frac{1}{\rho A b}f^*_n(t), \end{aligned}$$
(18)

where in Eq. 18, \(\omega _n\) is the natural frequency corresponding to mode \(\phi _n(x)\) and

$$\begin{aligned} f^*_n(t)= & {} \int _0^L \phi _n(x) f(x,t) \,{\text {d}}x , \end{aligned}$$
(19)
$$\begin{aligned} b= & {} \int _0^L \phi _n^2(x) \,{\text {d}}x . \end{aligned}$$
(20)

In the case of an intact simply-supported beam the nth frequency and mode shape of the beam are, respectively, \(\omega _n=\frac{n^2\pi ^2}{L^2} \sqrt{\frac{EI}{\rho A}}\) and \(\phi _n (x)=\sin \left( \frac{n\pi }{L}x\right)\). Accordingly, \(f^*_n(t)\) and b in Eqs. 19 and 20 are

$$\begin{aligned} f^*_n(t)= & {} -\int _0^L \sin \left( \frac{n\pi }{L}x \right) P\, \delta (x-Vt)\,{\text {d}}x =-P \sin \left( \frac{n\pi V}{L}t \right) \end{aligned}$$
(21)
$$\begin{aligned} b= & {} \int _0^L \sin ^2\left( \frac{n\pi }{L}x \right) \,{\text {d}}x =\frac{L}{2}, \end{aligned}$$
(22)

where the minus sign for the force in Eq. 21 reflects the fact that P is is pointing downwards in Fig, 17.

The modal differential equation 18 can, therefore, be solved subject to the initial conditions \(y^*_n(0)=\dot{y}^*_n(0)=0\) and the response of the beam can be written in the form of Eq. 17. Following [45], the total response of the beam is then

$$\begin{aligned} y(x,t)= & {} -\frac{2PL^3}{\rho A\pi ^2} \sum _{n=1}^\infty \frac{\sin \left( \frac{n\pi }{L}x\right) }{n^2(n^2\pi ^2a^2 - V^2L^2)}\nonumber \\&\times \left\{ \sin \left( \frac{n\pi V}{L}t\right) - \frac{VL}{n\pi a}\sin (\omega _n t) \right\} \end{aligned}$$
(23)

in which

$$\begin{aligned} a=\sqrt{\frac{EI}{\rho A}}. \end{aligned}$$

The first term in the last bracketed section of Eq. 23 contains the moving load pseudo-frequencies \(\frac{n\pi V}{L}\) and combines to give the static deflection component, while the second term contains the natural frequencies of the beam \(\omega _n\) and is the dynamic part [10].

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mousavi, M., Holloway, D. & Olivier, J.C. A new signal reconstruction for damage detection on a simply supported beam subjected to a moving mass. J Civil Struct Health Monit (2020). https://doi.org/10.1007/s13349-020-00414-3

Download citation

Keywords

  • Vehicle bridge interaction
  • Moving mass
  • Vibration
  • Damage detection
  • SHM
  • EMD
  • Residual IMF