## 3 Dynamic Identification: Results and Discussion  
On December 2 2019, the authors measured the response of the masonry tower to ambient excitation. The covariance-driven Stochastic Subspace Identification (SSI) method yield the estimate of the experimental modal parameters [15; 16; 17]. The authors used a set of 10 Force-Balance monoaxial accelerometers (SARA Instruments SA10). They have a dynamic range higher than 165 dB in the frequency interval 0.1-20 Hz and a 5 V/g sensitivity. The acquisition system consists of two measurements chains, each one driven by a master recorder unit, synchronized by a GPS sensor. In the current application, two accelerometers provided the reference location in two orthogonal directions [18], while the other eight were placed in the lower levels in several measurement points, see Figure 10. The data were sampled at a rate of 200 Hz. The cut-off frequency of the anti-aliasing filter was set to 20 Hz. The number of samples was set to N = 720,000, which resulted in a measurement time of 1 h. Since two accelerometers were fixed, the remaining eight were each one rotated and placed in two different locations, resulting in four measurement setups, which increased the mode shape resolution. Data from multi-setup measurements (4 setups) were then merged following the approach recalled in [19; 20]. The identification was carried out under environmental conditions; additionally, the authors recorded the response of the three top accelerometers to the ringing bells to derive an elementary analytical model of the coupled bell-tower dynamics.  
Figure 10: Experimental setup with the different configurations and the two measuring directions.  
Two modes arise from the stabilization diagram, see Figure 11a and Table 1.  
Unfortunately, the ambient excitation did not sufficiently excite the upper modes. The root-mean-square of the recorded acceleration signals is very low, approximately equals to \(2\times 10^{-4}\) g. Both modes are translational in two orthogonal directions and resemble the first mode of a plane cantilevered beam. The two directions of the identified modes likely indicate the two inertia principal axis of the tower cross-section.  
However, the principal axis of inertia of the belfry cross-section do not match with the directions of the two translational modes. The constraint provided by the facade and the naive walls likely determine the observed axis rotation. Hence, the authors attempted to assess the degree of restraint offered by the neighbouring structures, by estimating an equivalent cross-section comprehensive of parts of the facade and the naive wall, which yield the observed translational directions. The two directions uncouple the belfry dynamics, which oscillates like two plane clamped-free planar beams along the two principal axes of inertia of the tower.  
The mode shapes further provide information on the constraint exhibited by the neighbouring facade and have walls along with the height. The mode shape can be possibly divided into three sections, see Figure 12: AB almost moves with the ground; BC involves the dynamics of the tower interacting with the facade and the naive wall; CD represents the dynamics of the tower with the facade. Nevertheless, as a whole, the mode shapes resemble the ones of a cantilevered beam, as evidenced in  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode** & **Frequency [Hz]** & **Damping [\%]** \\ \hline
1 st & 2.367 & 3.727 \\
2 nd & 2.749 & 1.957 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results of Operational Modal Analysis (OMA).  
Figure 11: **(a)** Stabilization diagram; **(b,c)** View of the first two experimental mode shapes. The black double arrows indicate the two inertia principal axis of the tower cross-section, while the red ones the directions of the mode shapes.  
the next section. The estimated damping ratios range between 2% and 4%; The values are consistent with the damping of masonry. However, a reliable estimate of damping requires further investigations.