## 8 Some Relevant Case Studies Referring to St. Mary's Tenor Bell
### Viscous and Friction Damped Free Oscillations  
Aiming to compare \"equivalent\" damping phenomena, case I.a has been studied considering viscous or friction damping as well. Preliminarily, the equivalence between viscous and friction damping has been assessed. As known, the period, \(T_{d}\), of viscous damped small free oscillations is constant over time; it depends only on the damping ratio \(\zeta\) and on the period, \(T\), of undamped oscillations, being independent on the exponentially decreasing amplitude of oscillations  
\[T_{d}=\frac{T}{\sqrt{1-\zeta^{2}}}. \tag{71}\]  
On the contrary, the apparent period of friction damped oscillations rigorously depends, besides on the friction coefficient, also on the amplitude of the oscillations; consequently, it varies over time, although slightly and the system generally does not retrieve the rest position. Given that the decay rate of friction damped oscillations is not exponential, their logarithmic decrement is variable, therefore the parameters of the equivalent viscous damping depend on the equivalence criterion. Thanks to its simplicity, widely used procedures to determine the damping coefficient are based on the measurement of the period of damped oscillations or on the evaluation of the logarithmic decrement \(\delta^{*}\). The damping ratio \(\zeta\) can be thus calculated using Equation (71) or by the formula:  
\[\zeta=\left[1+\left(\frac{2\;\pi}{\delta^{*}}\right)^{2}\right]^{-0.5}. \tag{72}\]  
Following that procedure, the equivalence between friction and viscous damping has been established equalizing the logarithmic decrement of the viscous case and the apparent logarithmic decrement \(\delta^{*}_{\alpha\beta}\) of the friction case determined over five oscillations, using the first Equation (68), accordingly deducing damping ratios and damping coefficients. The amplitude of the first oscillation has been assumed, in turn, \(\theta_{0}=5^{\circ}\) or \(\varphi_{0}=5^{\circ}\). For the St. Mary's tenor bell, assuming \(\mu_{b}=0.02\) and \(r_{b}=50\) mm, at the bell pivot, and \(\mu_{c}=0.04\) and \(r_{c}=10\) mm, at the clapper pivot, the equivalent damping ratios and the relative damping coefficient indicated in the last two columns of Table 8 have been derived. As already said, the equivalence of the clapper has been derived assuming \(r=0\). The corresponding \"equivalent\" time histories of bell and clapper rotation are illustrated in Figure 17a,b, respectively: dashed lines refer to friction cases, and continuous lines to viscous cases. Looking at the figures, it is evident that, although first oscillations of friction and viscous curves are nearly coincident, over time friction decay tends to prevail. Comparison of Large Damped Free Oscillations in Case I (\(\theta_{0}=171^{\circ}\))  
Cases I.a and I.b have been studied also in damped conditions, adopting the damping equivalent parameters previously determined (Table 8), assuming the initial values given by Equations (64) and (65). The results are compared in Figure 18 in terms of time histories of rotations, in Figure 19 in terms of \(\dot{\theta}-\theta\) curves, and in Figure 20 in terms \(\dot{\varphi}-\varphi\) curves. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Pivot Radius [m]** & **Friction Coefficient** & \(\delta^{*}_{eq}\) & \(\zeta\) & \(\dot{\epsilon}\) [m s\({}^{-1}\)] \\ \hline Bell & 0.05 & 0.02 & 8.077\(\cdot 10^{-2}\) & 1.285\(\cdot 10^{-2}\) & 9.354\(\cdot 10^{-2}\) \\ Clapper & 0.01 & 0.04 & 3.413\(\cdot 10^{-2}\) & 5.432\(\cdot 10^{-2}\) & 2.993\(\cdot 10^{-2}\) \\ \hline \end{tabular}
\end{table} 
Table 8: Friction parameters and \(5^{\circ}\) equivalent viscous damping parameters of St. Mary’s tenor bell. Figure 17: Comparison of friction damped (dashed lines) and equivalent viscous damped (solid lines) for small amplitude (\(5^{\circ}\)) free oscillations of St. Mary’s tenor bell: (**a**) Bell oscillations; (**b**) Clapper oscillations.