## 8 Some Relevant Case Studies Referring to St. Mary's Tenor Bell
### Large Undamped Free Oscillations
#### 8.1.2 Case II Undamped Free Oscillations (\(\theta_{0}=40^{\circ}\))  
Results regarding case II, undamped oscillations of the bell starting from \(\theta_{0}=40^{\circ}\), are reported in Figure 14a,b in terms of \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves, in Figure 14c,d in terms of \(\varphi-\dot{\varphi}\) curves, and in Figure 14e,f in terms of \(\theta-\dot{\theta}\) curves, in analogy with the criterion adopted for case I.  
It must be underlined that in case II.b there are no impacts, while in case II.a, on average, impacts occur about every two oscillations. That results even more clearly from Figure 15a,b, which illustrates the time\(-\)angular acceleration curves \(\dot{\theta}=\ddot{\theta}(t)\) and \(\ddot{\varphi}=\ddot{\varphi}(t)\) obtained in case II.b adopting \(\Delta t=0.1\) ms. In the figures, impacts obviously correspond to sudden variations, whose extents, recalling Equation (23) or (24), are inverse function of the adopted time step.  
These remarks are further corroborated by Table 7, evidencing that in case II.a part of energy is lost, while in case II.b, lacking any kind of dissipation, the energy is conserved.  
The latter observation confirms that the proposed methods are symplectic integrators and that the energy drift is practically nil.  
Recalling Equations (33) and (34), the radial accelerations at the clapper pivot \(B^{\prime}\) and at the bell pivot \(A\), including gravity acceleration, can be finally obtained as  
\[\stackrel{{\rightarrow}}{{a}}_{rB^{\prime}}=\left(\begin{array}[] {c}b\left(\dot{\theta}+\dot{\varphi}\right)^{2}\sin(\theta+\varphi)\\ b\left(\dot{\theta}+\dot{\varphi}\right)^{2}\cos(\theta+\varphi)+g\end{array} \right), \tag{69}\]  
\[\stackrel{{\rightarrow}}{{a}}_{rA}=\frac{1}{M+m}\left(\begin{array} []{c}\left(M\,a+m\,r\right)\,\dot{\theta}^{2}\sin\theta+m\,b\left(\dot{\theta} +\dot{\varphi}\right)^{2}\sin(\theta+\varphi-\alpha)\\ \left(M\,a+m\,r\right)\,\dot{\theta}^{2}\cos\theta+m\,b\left(\dot{\theta}+ \dot{\varphi}\right)^{2}\cos(\theta+\varphi-\alpha)+\left(M+m\right)\,g\end{array} \right). \tag{70}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(\theta_{1}^{*}\) & \(\theta_{5}^{*}\) & \(\delta_{\theta}^{*}\) & \(T_{1}\) [s] & \(T_{5}\) [s] & \(\delta_{T}\) \\ \hline Case I.a & 0.7066 & 0.7047 & 5.18\(\cdot 10^{-4}\) & 2.402 & 2.400 & 1.83\(\cdot 10^{-4}\) \\ Case I.b & 0.6981 & 0.6981 & — & 2.400 & 2.400 & — \\ \hline \end{tabular}
\end{table}
Table 7: Max amplitudes \(\theta_{1}^{*}\), \(\theta_{5}^{*}\), apparent periods \(T_{1}^{*}\) and \(T_{5}^{*}\) and apparent logarithmic decrements in Cases II.a and II.b.  
Figure 13: Undamped free oscillations of St. Mary’s tenor in cases Ia and Ib (\(K=0.8\)): (a) \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves for case Ia; (b) \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves for case I.b.  
Referring again to case II.b, the resulting time histories of the dimensionless horizontal and vertical component of the radial accelerations in pivots \(A\) and \(B^{\prime}\) are illustrated in Figure 16. More precisely, Figure 16a,b refers to horizontal components \(a_{rA,x}/g\) and \(a_{rB^{\prime},x}/g\), while Figure 16c,d refers to vertical components \(a_{rA,y}/g\) and \(a_{rB^{\prime},y}/g\).  
As expected, that amplitude does not induce significant excitation on the supports.