## 7 Validation of the Methods
### Bell Oscillations and Clapper Strikes of Three Reference Bells  
As recalled before, the clapper strikes on one cycle of a free oscillating bell have been extensively studied in [18], where results are diagrammatically shown for the three reference bells. Assuming the properties already recalled in Table 1, the cycles, starting at an angle of \(171^{\circ}\), have been reproduced with the proposed algorithm, considering two cases, according as the bell is starting ringing right or wrong. The bell is said to be ringing right if the clapper impacts are \"in phase\" with the bell oscillations, i.e., the clapper impacts the bell on the upper part of the mouth, or it is said ringing wrong if the clapper impacts are \"in opposition of phase\" with the bell oscillations, i.e., the clapper impacts the bell on the lower part of the mouth. Analytically, the bell starts ringing right if the first clapper impact satisfies the condition:  
\[\text{sgn }(\theta\,\overline{\varphi})=1, \tag{50}\]  
it starts ringing wrong, if the first clapper impact satisfies  
\[\text{sgn }(\theta\,\overline{\varphi})=-1, \tag{51}\]  
instead. In the considered cases, the corresponding initial conditions are thus  
\[\theta(0)=171^{\circ};\ \dot{\theta}(0)=0;\ \varphi(0)=26^{\circ};\ \dot{ \varphi}(0)=0\, \tag{52}\]  
for the bell ringing right and  
\[\theta(0)=171^{\circ};\ \dot{\theta}(0)=0;\ \varphi(0)=-26^{\circ};\ \dot{ \varphi}(0)=0\, \tag{53}\]  
for the bell ringing wrong. According to the assumption made in [18], in both cases the kinetic coefficient of restitution has been hypothesized \(K=\sqrt{0.2}\), independently on the impact sequence. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Great St. Mary’s Tenor**} & \multicolumn{2}{c}{**Bell 1**} \\ \cline{2-5}  & \(T_{b}\)\([\mathbf{s}]\) & \(T_{c}\)\([\mathbf{s}]\) & \(T_{b}\)\([\mathbf{s}]\) & \(T_{c}\)\([\mathbf{s}]\) \\ \hline Bell excitation & 2.330 & — & 1.225 & — \\ Clapper excitation & — & 1.762 & — & 0.953 \\ Theoretical & 2.330 & 1.762 & 1.225 & 0.953 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between theoretical and numerical results—periods of free oscillations. Figure 6: Small free oscillations of the bell 1: (**a**) Oscillation of bell in case of bell excitation; (**b**) Oscillation of bell and clapper in case of clapper excitation. Figure 7a,b, adapted from [18], illustrates reference results in terms \(\theta=\theta(t/T)\), \(\varphi=\varphi(t/T)\) and \(\dot{\varphi}=\dot{\varphi}(t/T)\) for bell \(1\) ringing right or wrong, respectively. Figure 7. First cycle of bell oscillations and clapper strikes for bell 1: (**a**) Reference results for bell ringing right (adapted from [18]); (**b**) Reference results for bell ringing wrong (adapted from [18]); (**c**) Bell ringing right: \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves obtained with the proposed algorithm; (**d**) Bell ringing wrong: \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves obtained with the proposed algorithm; (**e**) Bell ringing right: \(\dot{\varphi}=\dot{\varphi}(t)\) curve obtained with the proposed algorithm; (**f**) Bell ringing wrong: \(\dot{\varphi}=\dot{\varphi}(t)\) curve obtained with the proposed algorithm. It must be considered that in [18] specific information about the period of the cycles as well as on the magnitude of the angular velocity of the clapper is not given, therefore they have been considered in relative sense. In Figure 7c,d they are reported the \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) diagrams obtained with the proposed algorithm in the two considered cases, while the corresponding \(\dot{\varphi}=\dot{\varphi}(t)\) clapper angular velocity diagrams are shown in Figure 7e,f. Adopting a time step of \(\Delta t=0.025\) ms, the calculated periods are \(T=2.883\) s for the bell ringing right and \(T=2.706\) s for the bell ringing wrong.