## 8 Some Relevant Case Studies Referring to St. Mary's Tenor Bell
### Viscous and Friction Damped Free Oscillations
This aspect is not surprising since the energy dissipation due to friction depends,  
Figure 21: Energy dissipation\(-\)time curves for friction (dashed lines) and “equivalent” viscosity (solid lines) for large oscillations of St. Mary’s tenor bell (\(\theta_{0}=171^{\circ}\)): (**a**) “5\({}^{\circ}\) equivalent” viscosity parameters; (**b**) “171\({}^{\circ}\) equivalent” viscosity parameters. apart from the oscillation amplitude, on the increase of the friction forces due to the increase of radial forces, while the energy dissipation due to viscosity depends, besides the oscillation amplitude, on the angular velocity. Owing the fact that the mean modulus of the radial acceleration at the pivot \(A\), which is \(\overline{|a_{rA}|}\approx g\) for \(\theta_{o}=5^{\circ}\) oscillations, becomes \(\overline{|a_{rA}|}\approx 2.2\ g\) for \(\theta_{o}=171^{\circ}\) oscillations, and that the mean angular velocity, which is \(\overline{\dot{\theta}}\approx 0.15\ \mathrm{s}^{-1}\) for \(\theta_{o}=5^{\circ}\) oscillations, becomes \(\overline{\dot{\theta}}\approx 2.5\ \mathrm{s}^{-1}\) for \(\theta_{o}=171^{\circ}\) oscillations, the energy dissipations in \(\theta_{o}=171^{\circ}\) oscillations can be roughly estimated, starting from the dissipation evaluated considering \(\theta_{o}=5^{\circ}\) oscillations. As the mean value of the dissipated energy for \(\theta_{o}=5^{\circ}\) oscillations is around \(\overline{E}_{dis}(5^{\circ})\approx 4\ \mathrm{J}/\mathrm{cycle}\) (Figure 20a), the dissipation caused by friction in \(\theta_{o}=171^{\circ}\) oscillations, which is approximately \(\overline{E}_{dis,f}(171^{\circ})\approx 42.234\approx 300\ \mathrm{J}/\mathrm{ cycle}\), results sensibly smaller than the energy dissipated by \"\(5^{\circ}\) equivalent\" viscosity, which is around \(\overline{E}_{dis,\rho}(171^{\circ})\approx 41634\approx 2200\ \mathrm{J}/\mathrm{ cycle}\). It must be remarked that these values, although roughly estimated, satisfactorily fit the calculated values for the first cycles (Figure 21a). To conclude the analysis, the \"\(171^{\circ}\) equivalent\" viscous damping parameters have also been determined, reported in Table 9. These values, which are one order of magnitude lower than the ones determined for small oscillations (Table 8), lead to the time\(-\)dissipated energy curve illustrated in Figure 21b (solid line), substantially coinciding with the dashed curve regarding the friction case. In addition, the entity of the dissipative forces can play an important role in discriminating the aptitude of the bell to ring right or wrong, as demonstrated by the St. Mary's tenor bell behavior when large free oscillations are considered. In fact:  
* on the one hand, as long as dissipative forces are low, over the time the bell tends to maintain the initial ringing state (right or wrong) (see Figure 18a,c),
* on the other, if dissipative forces increase, the bell tends to switch, independently on the initial state, into a ringing wrong state, in which each cycle is characterized by a single instantaneous impact (see Figure 18b,d). Time Histories of Excitations at the Pivots in Case of Large Oscillations \((\theta_{0}=171^{\circ})\)  
Adopting the approach already described in Section 8.1.2. for case II--undamped free oscillations with \(\theta_{0}=40^{\circ}\), the time histories of the excitations at the bell and clapper pivots in case of full circle oscillations \((\theta_{0}=171^{\circ})\) have also been determined, considering undamped and friction damped oscillations. Recalling that the \"equivalent\" viscous damping case, whose parameters accord with Table 9, leads to outcomes practically coinciding with the ones obtained for friction damping, this case has not been explicitly considered. The time histories of the accelerations are compared in Figure 22, where solid lines refer to the undamped situation and dashed lines to the damped situation.