## 2 Materials and Methods
### Second-Order Approximation Procedure for Damping Factor Estimation  
There are two issues, which require further investigation before attempting to adapt the contactless diagnosis principle of [19] to the current application. First, the possibility of a failing joint quite far from the sensing element, which was not a problem in [19] since only one slab was involved; second, a significant variability in dynamics (following occurrence of a failing joint) of a structure formed from a group of slabs as opposed to dynamics of a single (continuous) slab. When system testing and recording of experimental data is completed, examination of the data frequency content usually leads to analyzing the most prominent modes, approximately corresponding to the most visible peaks in the plots (as already verified via finite element analysis in [19]). According to [20], chapter 10, such peaks in Frequency Response Function (FRF) plots may be attributed to complex pairs of system poles with very low values of damping factors. In essence, mapping peaks to pairs of lightly damped poles means that the response of the linear system is due to the superposition of responses of elementary under-damped second order systems. In other words, this principle is similar to partial fraction expansion (see [23], p. 61) usually performed in linear systems, with higher order systems decomposed to a series of first and second order systems for calculating the system response signal. Thus, for such cases [20] proposes an empirical method to estimate the value of damping factor \(\zeta\) for such pairs of lightly damped poles, based on the value \(\omega n\) of the mode's frequency, its peak magnitude in dB, and the half-power frequency values. These are two frequency values for which the mode's magnitude reaches 70.7% of its peak, before (\(\omega n_{\rm b}\)) and after (\(\omega n_{\rm a}\)) frequency \(\omega n\). Using this set of points, the unknown damping factor value is computed as equal to 0.5 \(\times\) (\(\omega n_{\rm a}-\omega n_{\rm b}\))/\(\omega n\) (see [20] for further details). This method is nicely suited to data characterized by low noise-to-signal ratio, but hardly suitable for many real-life noisy signals. In such cases, only peaks of modes of lightly damped pole pairs may be visible over the underlying noise in plots. Then, half-power frequencies \(\omega n_{\rm b}\) and \(\omega n_{\rm a}\) cannot be easily defined, since they are masked by the noise. Another issue is that very often the excitation input is not available. This is typically the case of recorded system responses not resulting from a dedicated experiment, but rather from system operation in typical real-life conditions. Then, for such output only systems, only power spectrum plots of the output signal may be available for estimation. Therefore, a modification of this procedure is postulated in order to circumvent the noise masking effects. Each monitored peak of magnitude Mn and frequency \(\omega n\) is still considered as resulting from a lightly damped complex pair of poles. Let the group of \(q\) spectrum points with frequencies be defined as [\(\omega 1\), \(\omega 2\),..., \(\omega n\),... \(\omega q\)] and let [M1, M2,..., Mn,..., Mq] be the group of the associated magnitudes. Spectrum points \(\omega i\) and M\(i\) with \(i=1\)... \(q\) and \(I\
eq n\), are all points preceding and following the peak [\(\omega n\), M\(n\)] of interest, that are still visible over the noise layer in spectrum plots. The effort, now, is to identify the second order system producing a peak that matches the monitored one at all points \(\omega i\) and M\(i\) with \(i=1\)... \(q\). This second-order approximation procedure is formulated as an iterative nonlinear optimization problem, as follows:  
1.