## 3 Modal Identification
### Signal Processing Procedure to Limit the Disturbance of Measurement Noise on Modal Identification  
In order to process the numerous measurement points resulting from DIC, a signal processing procedure is proposed. Indeed, the number of points being much more numerous (several hundreds) than classically with accelerometers (at best several tens of points), some of them are located near the nodes of the different mode shapes and, thus, disturb a global procedure as they contain low information due to a poor signal-to-noise ratio (SNR) due to their location, especially for the higher modes. The process using the CWT method proposed here, for each mode, is as follows:  
* First of all, the length of the signals to be processed has to be fixed, i.e., the beginning and the end of the signals have to be defined. There is no particular difficulty for the beginning of the signal. For the end of the signal, a first wavelet transform is applied to all the measurement points on a narrow frequency bandwidth close to the frequency of the mode considered ([0.99, 1.01]\(\times f_{i}\)) with a factor \(Q=(Q_{min}+Q_{max})/5\). Then, after a first approximation of the mode shape, the points whose modal amplitude is less than 0.1 (the mode shape is normalized to the maximum value, so its maximum is equal to 1) are not used for determining the end of the signals. For each of the remaining points, the curve \(\log\left(A_{u_{ij}}(t)\right)\) is normalized to the maximum and the time from which the curve is under the arbitrarily fixed threshold of 0.2 is saved. The final time for all the points to be analyzed is defined as the minimum of the times saved previously. * Once the length of the signals is determined, a second wavelet analysis on the same bandwidth as before is performed using a factor \(Q\) equal to \(0.95\times Q_{max}\) in order to minimize the time fluctuation on the ridge. The values obtained for the modal parameters (frequency, modal damping ratio, and mode shape) are the averages of these parameters for the time interval not subject to edge effects (inside the two vertical red lines represented in Figure 8). * In order to limit the noise disturbance on the identification of the modal parameters, a singular value decomposition (SVD) is applied to the wavelet scalogram as proposed in [26]. The scalogram is recombined from matrices obtained using SVD after setting the lowest singular values to zero. Since it is time-consuming and not needed for the first wavelet transform used to determine the final duration of the signals, the SVD is only applied for the second wavelet transform used to obtain the modal parameters. The noise present in the data has no effect on the determination of the frequency, a small effect on the mode shape (Figure 9a,b (zoom)), and a quite important effect on the values of the modal damping ratio (Figure 9c). Another point to pay attention to is the analysis of the results in terms of statistics, in particular the dispersion of the results due to the large number of measurement points. Indeed, as the mode shapes obtained are quite noisy, if they are normalized to the maximum without precaution, a higher relative standard deviation (RSD) would result (Figure 10a). Figure 9: Comparison of the results obtained for the clamped–clamped beam with (red dash curves or circles) and without (blue curves or asterisks) the use of SVD for denoising: (**a,b**) mode shape of Mode 6 and (**c**) modal damping ratio of Mode 4. To avoid this, the mode shapes are filtered with a low-pass filter to remove the noise disturbance and the normalization factors are deduced from these filtered mode shapes. The low-pass filter is applied by performing zero-phase digital filtering by processing the input data, i.e., the mode shapes, in both the forward and reverse directions (filtfilt function in MATLAB(r)).