## 8 Some Relevant Case Studies Referring to St. Mary's Tenor Bell
### Large Undamped Free Oscillations
#### 8.1.1 Case I Undamped Free Oscillations \((\theta_{0}=171^{\circ})\)  
Relevant curves regarding undamped oscillations of the bell in cases La and I.b, both starting from \(\theta_{0}=171^{\circ}\), are compared in Figure 12. Figure 12a,b illustrates \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves, Figure 12c,d show \(\varphi-\dot{\varphi}\) curves, and Figure 12e,f \(\theta-\dot{\theta}\) curves.  
It must be stressed that, despite that damping is nil, energy is not conserved, since in both cases, as anticipated, the clapper impact is not perfectly elastic \(\left(K=\sqrt{0.2}\right)\).  
The analysis of the results shows that, in the given conditions, the bell tends to maintain the initial ringing condition. More precisely, if the bell starts ringing wrong, it continues to ring wrong. Nevertheless, as clearly results from Figure 12b, while in a first phase the bell ringing wrong exhibits two well distinct impacts per semicycle, over time the impacts become more regular, and only one instantaneous and clear contact occurs per semicycle. Moreover, the number of significant impacts (Figure 12a,b) and the angular relative velocity \(\dot{\varphi}\) at the time of impacts (Figure 12c,d) are higher in the ringing wrong case I.b than in the ringing right case I.a; case I.b is thus characterized by more pronounced dissipation and transfer of energy, as the comparison of Figure 12e,f clearly demonstrates.  
In addition, said \(\theta_{i}^{*}\) the amplitude and \(T_{i}^{*}\) the \"apparent period\" of the \(i-\)th cycle, defined as the time interval between \(\dot{\theta}_{i}^{*}\) and \(\theta_{i+1}^{*}\), apparent logarithmic decrements of amplitude, \(\delta_{\theta}^{*}\), and period, \(\delta_{T}^{*}\), can be defined. In the following, these apparent logarithmic decrements are evaluated considering the first five apparent periods, according to:  
\[\delta_{\theta}^{*}=\frac{1}{5}\ln\left(\frac{\theta_{1}^{*}}{\theta_{5}^{*}} \right)\!;\;\;\delta_{T}^{*}=\frac{1}{5}\ln\left(\frac{T_{1}}{T_{5}^{*}}\right)\!. \tag{68}\]  
The outcomes of the analysis are summarized in Table 6, which confirms that case I.a is characterized by smaller apparent logarithmic decrements.  
Although marginal in the present paper, another important question regards the aptitude of the bell to maintain the initial ringing condition, especially when initial condition is wrong. As already remarked, the adoption of a coefficient of restitution \(K=\sqrt{0.2}\) implies that in case I.b the bell tends to ring wrong, but this is not always the case.  
Truly, hypothesizing higher values of the coefficient of restitution the result can be the opposite. For example, assuming \(K=0.8\), the curves \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) resulting in cases La and I.b become those illustrated in Figure 13a,b, respectively. Curves in Figure 13a confirm that, like in Figure 12a, the bell lasts in ringing right state over time; on the contrary, curves in Figure 13b show that, after some cycles, three in that case, the ringing state switches from wrong to right.  
The tendency to stay in \"wrong\" condition thus depends not only on the relevant ratios \(r/L_{\rm c}\) and \(L_{b}/L_{c}\) but also on the COR. Moreover, as shown in the following, that tendency depends not only on the quotas of energy dissipated or transferred outside the system during the impact but also on the extent and on the nature of other dissipative phenomena.