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 quasistatic 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 singlepoint 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 displacementtime 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 (\({\text {SNR}} =7\, {\text {dB}}\)).
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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 nonstationary 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 nonstationary, 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.
the signal has at least two extrema (at least one maximum and one minimum);

2.
the characteristic time scale is defined by the time lapse between the extrema;

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

2.
Take the mean of the two curves (envelopes) obtained from the first step and call it \(m_1\).

3.
Compute \(h_1=X(t)m_1\) and check if \(m_1\) complies with the definition of the IMF.

4.
If not, repeat the steps 1 to 3 for \(h_1\) and compute \(m_{11}\) so that \(h_{11}=h_1m_{11}\). If still \(h_{11}\) is not an IMF, repeat steps 1 to 3 to obtain \(h_{1k}=h_{1k1}m_{1k}\) so that \(c_1=h_{1k}\) is an IMF.

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

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
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.
Variational mode decomposition (VMD)
Like EMD, VMD seeks to decompose a realvalued 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 amplitudemodulatedfrequencymodulated (AMFM) sinusoid which has the following additional characteristics:

1.
the phase is a nondecreasing function;

2.
the envelope is nonnegative;

3.
both the envelope and the instantaneous frequency vary much more slowly than the phase;
i.e. the IMF is written as
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 \(\{\omega _k\}=\{u_1, u_2, \ldots , w_k\}\) ,
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
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 simplysupported 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(x, t) may be obtained by modal superposition as
in which \(\phi _n(x)\) is the nth mode shape and \(y^*_n(t)\) denotes the solution to the modal differential equation,
where in Eq. 18, \(\omega _n\) is the natural frequency corresponding to mode \(\phi _n(x)\) and
In the case of an intact simplysupported 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
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
in which
The first term in the last bracketed section of Eq. 23 contains the moving load pseudofrequencies \(\frac{n\pi V}{L}\) 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].
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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/s13349020004143
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Keywords
 Vehicle bridge interaction
 Moving mass
 Vibration
 Damage detection
 SHM
 EMD
 Residual IMF