## 8 Some Relevant Case Studies Referring to St. Mary's Tenor Bell
### Synchronized Resonant Forced Oscillations of the Bell in Transient Phase
Aiming to reproduce a manual excitation, it has been adopted a piecewise-constant bidirectional forcing function, which is the more realistic one:  
\[M_{e1}\!\left(\theta,\dot{\theta}\right)=\left\{ \begin{array}{ll}250\ \mathrm{sgn}\ \dot{\theta}\ \ Nm&\mathrm{if}-\frac{\pi}{4}\leq\theta\leq\frac{\pi}{4}\\ \\ 0&\mathrm{if}\ \ \theta\!\left\langle-\frac{\pi}{4}\ \ \vee\ \ \theta\right\rangle\!\frac{\pi}{4}\end{array} \right.\, \tag{74}\]  
or a truncated sinusoidal bidirectional forcing function:  
\[M_{e2}\!\left(\theta,\dot{\theta}\right)=\left\{ \begin{array}{ll}250\ \mathrm{sgn}\ \dot{\theta}\cos\ \theta\ Nm&\mathrm{if}-\frac{\pi}{4}\leq\theta\leq\frac{\pi}{4}\\ \\ 0&\mathrm{if}\ \ \theta\!\left\langle-\frac{\pi}{4}\ \vee\ \ \theta\right\rangle\!\frac{\pi}{4}\end{array} \right.. \tag{75}\]  
It must be remarked that the peak modulus of \(M_{e1}\) and \(M_{e2}\), 250 Nm, matching the torque easily exerted by a person, is significantly smaller than the peak of the torque needed to lift statically the St. Mary's tenor bell in the upward position, \(M_{max}\approx 10\) kNm. Since \"equivalent\" viscous parameters strongly depend on the amplitude of the oscillations, the damped case refers to friction damping. Friction coefficients and pivot radii are those already recalled in Tables 8 and 9, leading to very low level of damping, corresponding to a damping ratio \(\zeta\approx 0.1\%\). Relevant curves obtained in the undamped and in the friction damped case are compared in Figures 23-25. In the figures, panel (**a**) refers to \(M_{e1}\) (Equation (74)), and panel (**b**) to \(M_{e2}\) (Equation (75)); curves referring to undamped case are generally in blue and curves referring to damped case are generally in red, except \(\varphi-t\) curves in Figure 23, whose colors are opposites. Looking at the time histories of the rotations of bell and clapper (Figure 23), it results that:  
* in the undamped case, the time needed to reach the maximum amplitude, \(t_{d}\), i.e., the time during which the bell is driven, is \(t_{d}\approx 83\) s, when the forcing function is \(M_{e1}\), increasing to \(t_{d}\approx 93\) s, when the forcing function is \(M_{e2}\),
* in the friction damped case, the time needed to reach the maximum amplitude is \(t_{d}\approx 110\) s, when the forcing function is \(M_{e1}\), increasing to \(t_{d}\approx 121\) s, when the forcing function is \(M_{e2}\),
* also considering the transient phase, and without stopping it in the upward position before the free oscillation phase, the bell tends to ring wrong in all considered cases. Figure 23: Comparison of time histories in the transient phase (\(\theta=\theta(t)\) curves in red in the undamped case, in blue in the damped case; \(\varphi=\varphi(t)\) curves in blue in the undamped case, in red in the damped case): (**a**) \(M_{e1}\) forcing function (Equation (74)); (**b**) \(M_{e2}\) forcing function (Equation (75)). Further information can be derived from the time histories of the horizontal and of the vertical component of the acceleration at the bell pivot \(A\), illustrated in Figures 24 and 25, respectively. Considering forced oscillations, the comparison of the diagrams obviously confirms that the number of oscillations needed to reach a given state in the transient phase is bigger in damped conditions than in undamped conditions. Anyhow, it is interesting to remark that the amplitude and the period of these forced damped oscillations are generally smaller than those pertaining to forced undamped oscillations, even if, clearly, as already anticipated, the apparent period of a given large oscillation is higher in damped condition. The latter observation is diagrammatically illustrated in Figure 26, where, referring to the present case study, the apparent periods \(T^{*}\) of the forced resonant oscillations are plotted as a function of the maximum amplitude of the oscillations themselves \(\theta^{*}_{\text{max}}\). In the diagram, dashed line represents the damped case, solid line the undamped case.