**4. Discussion**

The identification of the material parameters of a real historic structure, based on modal identification results, is discussed in this paper. The efficacy of the modal identification is crucial in obtaining true material parameters. The efficacy appeared to be different for the natural frequencies and the mode shapes. The three different methods considered in this study provide convergent results for the natural frequencies. The values are between 2.24 and 2.26 Hz, and 2.74 and 2.80 Hz, for the first pair of natural frequencies (see Table 1). Both modes are the first two bending mode shapes in two orthogonal directions. The obtained values of the fundamental frequencies are similar to the characteristics of other masonry towers. In the paper [29], the values for 30 masonry towers with heights of 16–46 m are summarized. They are between 0.61 Hz and 5.28 Hz. The damping coefficient corresponding to the first natural frequency varies between 0.0226 and 0.0244. The values are included in the most common range of damping coefficient determined for masonry towers, which is 0.02–0.03, according to [30]. The mode shapes identified by the application of the three OMA techniques to the three kinds of input signals are not always repeatable. First of all, the PP-CA method, which is dedicated to ambient excitations, does not produce a reliable mode shape for the SA signals (see Figure 7). The application of the ambient signals enhanced by the sheet peel hammering significantly improves the result, however untypical changes in the mode shapes curvatures are visible in the upper parts of the modes. The most reasonable results were obtained for the combined signals, examples of which are presented in Figure 4. In this case, the mode shapes show the curvature of one sign along the whole length; moreover, they show the typical shapes of the modes of a cantilever supported by a rotational spring—such a simplified model can approximate the considered tower. It is worth mentioning that the efficacy of the mode shapes' identification does not relate to the signal amplitudes in general. The average root mean square (RMS) of all signals in subsequent groups equals 0.0014 for SA, 0.0039 for SH and 0.0027 for SC signals. The SH signals have the biggest amplitudes, related to the hammering frequency. The identification of modes related to other frequencies is enhanced somehow, but the best solutions are obtained when the combined signals are considered with moderate RMS and with different dynamic influences. Limited results are obtained by the NExT-ERA technique. Only two modal vectors are obtained based on the SA signals, and both of them are in directions perpendicular to the wall planes (Figure 8). The SH signals produced no results, and the SC signals provide only one modal vector in the *x* axis of the tower (Sensors 1, P1–P5 line). Contrary to the PP-CA case, the application of SC signals does not improve the results obtained for the SA signals with the NExT-ERA algorithm. The different efficiencies of various OMA techniques in the practical cases of six British lighthouses are also reported in the paper [3].

The application of the ERA technique to the short-term free-decay parts of SA signals leads to a set of modal forms. Then, the averaged mode shape vector is found as the final ERA solution. The most considerable results obtained using three identification methods are compared in Figure 10, together with the numerical results. The PP-CA results were obtained for the case of the SC signals, while the NExT-ERA results were obtained for the SA signals. The comparison of the results shows the most repeatable forms in the line P1–P5, measured by sensor 1. This suggests the best quality of the signals collected by these sensors. The line is situated along the tower wall on the river side, and sensor 1 measures the accelerations in the direction perpendicular to the river (direction *x*, see Figure 2). This line and these sensors receive the highest excitation from the river's undulations. This may be the factor affecting the signal's quality. Besides, the structure looks integral, and no damages are visible. However, maybe the renovations after World War II play some role in the signal deteriorations. It is worth mentioning that the different lengths of signal considerations in the NExT-ERA and the ERA techniques did not improve the presented results.

The enhanced OMA analysis validated the data for the FEM model and the material parameters' identification via the validation process. The results obtained and presented in Table 2 are included in the range of material parameters reported for other masonry structures. In the papers [4,31,32], the investigations of masonry bell towers are described. The elastic modulus of the masonry tower described in [4] was assessed by a simplified global analysis based on the first natural frequencies obtained through experimental analysis. The results were, depending on direction, 3.579 and 4.746 GPa, with frequencies of 1.294 and 1.489 Hz. However, in the FE model the elastic modulus value of 2.819 GPa was implemented. A similar tower is the object of another study [31], where the elastic modulus is stated as 1.8 GPa with a first frequency of 0.67 Hz. In the paper [32], the elastic modulus of masonry varied from 1.8 GPa to 2.5 GPa, relative to the wall height. The first frequencies were assumed as 0.585 Hz and 0.709 Hz. In every case, stairs were unimportant from a structural point of view, and were thus not included in any numerical model. According to [28], the elastic modulus of granite varies from 2.59 GPa to 88.79 GPa, and in the case of sandstone from 3.4 GPa to 71.7 GPa. In another

paper [33], 48.8 GPa is the identified value of the elastic modulus of stone masonry. The values of the elastic moduli of granite masonry applied in three different historic structures are presented in [34]. They belong in the range of 20.8–39.2 GPa, with a unit weight in the range of 24.1–26.4 kN/m3. All those values are similar to the material parameters identified in the present study.

An interesting and rare result concerns the shape response identification, which is induced by the sheet peel hammering (see Figure 12). This shape could be identified for the *x* direction only (perpendicular to the river), which corresponds to the direction of wave propagation in the ground generated by the hammer action. This particular response could be identified using the ERA and the NExT-ERA methods. This shape is similar to the eigenmode of the numerical model (third bending one) (see Figure 12). The related numerical and experimental frequencies differ considerably. The distant values of the frequencies prove the safety of the peel hammering works in the vicinity of the lighthouse.

**Figure 12.** The response shape induced by the peel hammering and similar numerical mode shape: (**a**) ERA results; (**b**) experimental response shape (NExT-ERA and ERA mean results) in comparison with numerical third mode shape (**c**) the third numerical bending mode shape (Abaqus).
