## 7 Validation of the Methods
### Damped Oscillations of St. Mary Old Tenor Bell  
As previously remarked, one of most important features to be assessed in studying numerical methods for solution of ODEs is their ability to manage \"stiff\" equations. Although there is no precise definition of equation stiffness, it is commonly agreed that differential equations manifest stiffness if their integration requires unnecessarily short time steps. Since equation stiffness increases raising the damping coefficient, to assess the capability of the proposed methods to manage this kind of equations, they have been applied to St. Mary tenor bell to evaluate the value of the critical damping coefficient; to determine the structural response in critical or overdamped conditions as well as to estimate the relevant conditions for convergence. In the analysis, time steps have been varied in the range \([0.025-10\) ms]. Figure 9: First cycle of bell oscillations and clapper strikes for bells 2 and 3 ringing wrong: (**a**) \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves obtained with the proposed algorithm for bell 2; (**b**) \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves obtained with the proposed algorithm for bell 3; (**c**) \(\dot{\varphi}=\dot{\varphi}(t)\) curve obtained with the proposed algorithm for bell 2; (**d**) \(\dot{\varphi}=\dot{\varphi}(t)\) curve obtained with the proposed algorithm for bell 3. As known, said \(\omega_{n}\) its natural angular frequency, the critical value, \(C_{cr}\), of the angular damping coefficient of a pendulum characterized by mass \(M\), equivalent length \(L\), and moment of inertia \(I\), is  
\[C_{cr}=2\;\omega_{n}\;I=2\;\sqrt{\frac{\delta}{L}}\;I=2\;\sqrt{g\;M\;a_{G}\;I}, \tag{54}\]  
Being that \(a_{G}\) is the distance between the center of gravity of the pendulum and the pivot. From practical point of view, the damping coefficient can be more conveniently represented by a relative damping factor, \(\overline{c}\),  
\[\overline{c}=\frac{C}{M\;a_{G}}, \tag{55}\]  
whose critical value,  
\[\overline{c}_{cr}=2\;\sqrt{g\;L}, \tag{56}\]  
depends only on \(L\), or by the dimensionless damping ratio \(\zeta\)  
\[\zeta=\frac{\overline{c}}{\overline{c}_{cr}}, \tag{57}\]  
which is the ratio between the actual damping coefficient and the critical one. As known, underdamped oscillators, characterized by \(\zeta<1\), tend to the equilibrium position by damped oscillations, overdamped systems, for which \(\zeta>1\), smoothly return to the equilibrium position never overpassing it, critically damped systems, \(\zeta=1\), require the minimum time to regain, without oscillations, a pre-established position. Applying Equations (54) and (55) to the bell and the clapper of [PERSON]'s tenor bell (Table 1), it results  
\[C_{cr,b}=7069.44\;\mathrm{kg}\;\mathrm{m}^{2}\mathrm{s}^{-1},\;\;\;\;\;\; \overline{c}_{cr,b}=7.277\;\mathrm{m}\;\mathrm{s}^{-1}\;, \tag{58}\]  
for the bell and  
\[C_{cr,c}=78.44\;\mathrm{kg}\;\mathrm{m}^{2}\mathrm{s}^{-1},\;\;\;\;\;\; \overline{c}_{cr,c}=5.503\;\mathrm{m}\;\mathrm{s}^{-1}, \tag{59}\]  
for the clapper. The critical damping obtained for the St. Mary's bell and clapper small oscillations with the proposed methods are reported in Table 5.  
In the elaborations, damping has been considered in turn on the bell and on the clapper.