## 6 Discussion: Response to Ringing Bell
These peaks are equidistant and decreasing in the same proportion as if they were higher harmonics of the ringing bell. This aspect is quite peculiar: [6; 7] did not observe the repetition of resonance peaks from the FFT of the experimental responses. Regrettably, the authors had not the chance to further examine this aspect via experimental testing. Figure 16 presents the frequency response functions of the mechanical system in Equation (2), tuned to the two detected experimental frequencies, respectively. In each plot, two FRFs show the effects of the bell: the FRFs of the system without the bell coincide with the same system with the mass \(m_{2}\) for frequency values higher than 0.8 Hz; The mass \(m_{2}\) determines an additional peak by the resonance frequency of the bell. The ratio between the two masses is approximately \(m_{2}/m_{1}\approx 0.2\%\): it is expectable that the bell does not influence the seismic performance of the tower. However, the effect of the more massive bell over the seismic performance requires further investigations, despite the frequency content of earthquakes generally exceeds 0.4 Hz. The authors chose a set of seven ground motion earthquakes in Table 5, currently used in seismic analysis, to numerically assess the effect of the more massive bell over the maximum displacement of the lower substructure [30; 31]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(A\) [m\({}^{2}\)] & 25.16 \\ \(I_{I}\) [m\({}^{4}\)] & 120.22 \\ \(I_{II}\) [m\({}^{4}\)] & 115.07 \\ \(h\) [m] & 12.3 \\ \(ro\) [kg/m\({}^{3}\)] & 1800 \\ \(E\) [MPa] & 991.3 \\ \(M\) [Kg] & \(\approx\)1000 \\ \(I_{p}/g\)\(d\) [N/m] & 6825.2 \\ \(xi_{1}\) [\%] & 3.7 \\ \(xi_{2}\) [\%] & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the model in Equation (2). The bar plot in Figure 17 shows the results of the time-history analysis: the maximum displacements of the lower substructure coincide in both the considered cases, without and with the mass \(m_{2}\) contribution. This aspect proves that the more massive bell of the San Silvestro belfry can be left free to swing and does not need to be constrained: its oscillation does not affect the seismic performance of the masonry tower. It is worth to notice that a cantilevered beam with a free length equals to the maximum height of the belfry (\(\approx\)25 m) would have a natural frequency approximately equals to 0.5 Hz: The lack of connections to the facade and the naive walls would have lowered the natural frequency of the tower to values similar to the oscillation frequency of the bell, with likely unfortunate consequences. Figure 16: Estimate of the Frequency Response Function (FRF) of the mechanical system in Figure 15 without and with the contribution of the \(m_{2}\) mass. The lower substructure is tuned to the first (**a**) and second natural frequency (**b**) respectively. Figure 17: Maximum displacement of the substructure with mass \(m_{1}\) tuned to the first (**a**) and second (**b**) bending mode respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Name** & **Areas Affected** & **Year** & **Mw** & **PGA [m/s\({}^{2}\)]** \\ \hline
**1** & El Centro & United States, Mexico & 1940 & 6.9 & 3.50 \\
**2** & Erzican & Erzincan Province, Turkey & 1939 & 7.8 & 5.03 \\
**3** & Kobe & Japan & 1995 & 6.9 & 6.76 \\
**4** & L’Aquila & Italy & 2009 & 6.3 & 6.63 \\
**5** & Northridge & Southern California, United States & 1994 & 6.7 & 5.51 \\
**6** & Loma Prieta & San Francisco, United States & 1989 & 6.9 & 6.55 \\
**7** & Parkfield & California, United States & 2004 & 6.0 & 4.96 \\ \hline \end{tabular}
\end{table} 
Table 5: List of the earthquakes adopted in the analysis.