## 7 Validation of the Methods
### Damped Oscillations of St. Mary Old Tenor Bell
In addition, the influence of the clapper on the bell behavior has been neutralized assuming the clapper as massless and \(\overline{c}_{c}=0\); while the influence of the bell on the clapper behavior has been minimized suitably increasing \(\overline{c}_{b}\):  
\[\overline{c}_{b}=\min\biggl{(}\frac{2.5\;\mathrm{m}}{\Delta t};2500\;\frac{ \mathrm{m}}{\mathrm{s}}\biggr{)}. \tag{60}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\Delta t\) **[ms]** & - & - & - \\  & \(\overline{c}_{cr,b}\) **[m s\({}^{-1}\)]** & \(\overline{c}_{cr,b}\) **[m s\({}^{-1}\)]** & - & - \\ \hline
100 & 7.459 & 6.785 & 5.368 & 4.755 \\
50 & 7.313 & 7.032 & 5.390 & 5.199 \\
10 & 7.275 & 7.226 & 5.543 & 5.318 \\
5.0 & 7.274 & 7.253 & 5.532 & 5.366 \\
1.0 & 7.274 & 7.272 & 5.505 & 5.515 \\
0.5 & 7.275 &The first condition in Equation (60) is dictated by the convergence condition of both methods, which result in the adoption of a constant time step \(\overline{\Delta t}\) satisfying the inequality  
\[\overline{\Delta t}\;\overline{c}_{b}\leq 2.6\;\text{m}. \tag{61}\]  
In the present exercise, initial conditions have been set to:  
\[C_{cr}=2\;\omega_{n}\;I=2\;\sqrt{\frac{\delta}{L}}\;I=2\;\sqrt{g\;M\;a_{G}\;I}, \tag{54}\]0  
for small amplitude bell oscillations and to  
\[C_{cr}=2\;\omega_{n}\;I=2\;\sqrt{\frac{\delta}{L}}\;I=2\;\sqrt{g\;M\;a_{G}\;I}, \tag{54}\]1  
for small amplitude clapper oscillations. In Figure 10a the time\(-\)rotation curve of the critically damped bell in semilogarithmic scale is represented, while in Figure 10b the time\(-\)rotation curve of the critically damped clapper is represented. Results clearly demonstrate that, if time step complies with Equation (61), the proposed methods can manage stiff equations too. To further stress that conclusion, also two overdamped cases have been studied, for the initial values  
\[C_{cr}=2\;\omega_{n}\;I=2\;\sqrt{\frac{\delta}{L}}\;I=2\;\sqrt{g\;M\;a_{G}\;I}, \tag{54}\]2  
or  
\[C_{cr}=2\;\omega_{n}\;I=2\;\sqrt{\frac{\delta}{L}}\;I=2\;\sqrt{g\;M\;a_{G}\;I}, \tag{54}\]3  
Damping coefficients have been set to:  
\[C_{cr}=2\;\omega_{n}\;I=2\;\sqrt{\frac{\delta}{L}}\;I=2\;\sqrt{g\;M\;a_{G}\;I}, \tag{54}\]4  
for bell and clapper, respectively; corresponding to damping ratios:  
\[C_{cr}=2\;\omega_{n}\;I=2\;\sqrt{\frac{\delta}{L}}\;I=2\;\sqrt{g\;M\;a_{G}\;I}, \tag{54}\]5  
\[C_{cr}=2\;\omega_{n}\;I=2\;\sqrt{\frac{\delta}{L}}\;I=2\;\sqrt{g\;M\;a_{G}\;I}, \tag{54}\]6 Time\(-\)rotation curves of the St. Mary tenor bell in critically damped conditions: (**a**) \(\log\theta-t\) curve of critically damped bell; (**b**) \(\theta-t\) curve of critically damped clapper. To enrich the study, the solutions obtained with a rather coarse time step, \(\Delta t_{1}=10\) ms, nearly corresponding to the limit condition for convergence, are compared with the reference solution, obtained with a significantly smaller time step, \(\Delta t_{2}=0.025\) ms, in Figure 11a,b, which refers to initial values given by Equations (64) and (65), respectively. In Figure 11, the dashed lines, referring to \(\Delta t_{1}\), practically coincide with the solid lines, referring to \(\Delta t_{2}\), thus confirming the soundness and the reliability of the algorithms in handling stiff equations.