## 8 Some Relevant Case Studies Referring to St. Mary's Tenor Bell
### Viscous and Friction Damped Free Oscillations
Figure 18: Comparison of friction damped and “5” equivalent” viscous damped large amplitude free oscillations of St. Mary’s tenor bell: (**a**) \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves—case Ia (friction damping); (**b**) \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves—case Ia (viscous damping); (**c**) \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves—case I.b (friction damping); (**d**) \(\theta=\theta(t)\) and \(\varphi=\varphi(t)\) curves—case I.b (viscous damping). Figure 19: Comparison of friction damped curves (in blue) and \"5\" equivalent\" viscous damped curves (in red) for large amplitude free oscillations of St. Mary’s tenor bell: (**a**) \(\dot{\theta}-\theta\) curves—case 1a; (**b**) \(\dot{\theta}-\theta\) curves—case 1.b; (**c**) \(\dot{\varphi}-\varphi\) curves—case 1a; (**d**) \(\dot{\varphi}-\varphi\) curves —case 1.b. Figure 20: Comparison of energy dissipation caused by friction (dashed lines) and by “5” equivalent” viscosity (solid lines): (**a**) \(\theta_{0}=5^{\circ}\) damped oscillations; (**b**) case 1a–\(\theta_{0}=171^{\circ}\) damped oscillations. Figure 18a,b shows the time histories of bell and clapper rotations derived in case La--bell ringing right--considering friction damping and viscous damping, respectively; Figure 18c,d refer to the corresponding situations in case Lb--bell ringing wrong. Figure 19a,b compares \(\dot{\theta}-\theta\) curves in case I.a e Ib, respectively, while Figure 19c,d compares the corresponding curves \(\dot{\varphi}-\varphi\). In these diagrams, curves pertaining to friction damping are in blue and curves pertaining to \"5\({}^{\circ}\) equivalent\" viscous damping are in red. The diagrams demonstrate that equivalent viscous damping coefficients determined considering small oscillations cannot be adopted for large oscillations. In fact, the comparison of diagrams in Figures 18 and 19 is a crystal-clear evidence that amplitude and period decrease more quickly for \"5\({}^{\circ}\) equivalent\" viscous damped oscillations than for friction damped oscillations. The reason for the latter remark is that, as soon as the initial amplitude of the oscillations increases, viscous dissipation increases much more rapidly than friction dissipation. The observation can be further corroborated comparing the energy dissipation caused by friction or by viscosity in a group of cycles or in a given time interval. Since energy dissipation is nothing else than the work done by the corresponding dissipative force, its calculation is not particularly difficult. The energy dissipated by friction damping and the energy dissipated by \"5\({}^{\circ}\) equivalent\" viscous damping are compared in the diagrams of Figures 20 and 21a. In Figure 20 the dissipated energies \(E_{dis}\) are plotted as a function of the bell rotation, \(\theta\), cycle by cycle: Figure 20a refers to the small oscillation case considered to assess the equivalence, Figure 20b to the case I.a. To facilitate the reading, curves in Figure 20b are parameterized considering different time intervals: the dashed magenta curve is the energy dissipated by friction in the time interval \(0-25\) s, while the solid blue line is the energy dissipated by \"5\({}^{\circ}\) equivalent\" viscosity in the time interval \(0-6\) s. Finally, the \(E_{dis}-t\) curves obtained for case La are compared in Figure 21a, in which solid line refers to \"5\({}^{\circ}\) equivalent\" viscosity and dashed line to friction. Figure 20a clearly shows that, as expected, for 5\({}^{\circ}\) oscillations the energy dissipations due to friction or equivalent viscosity practically coincide, but Figures 20b and 21a underline that using \"5\({}^{\circ}\) equivalent\" viscosity leads to huge overestimations of the dissipated energy. That confirms that the equivalence parameters between friction and viscous dissipation is sensibly dependent on the initial amplitude.