## 3 Modelling the Clapper Impact  
Over the years, several approaches have been proposed to model the clapper to bell impact, also depending on the aim of the specific study. In classical approaches, the impact is modelled from macroscopic point of view, based on the coefficient of restitution [41]. The coefficient of restitution is a simple and synthetic macroscopic parameter, which summarizes not only the intricate interactions occurring during the impact, but also the influence of the clapper and bell geometry and of the ringing style on the impact itself. The coefficient of restitution can be alternatively defined:  
* in kinematic terms, as ratio between the relative orthogonal velocities after and before the impact, according to the Newton definition,
* in kinetic terms, as ratio between the normal impulses during the restitution phase and the compression phase, according to the [PERSON] definition, or
* in energetic terms, as recently suggested in [42], as ratio between the works done during the restitution phase and the compression phase. In case of the bell, the COV \"measures\" also the efficiency of the energy transfer between clapper and bell, in exciting bell vibrations. Manifestly, different definitions of COR are equivalent only if the impact is frictionless; on the contrary, as remarked in [42], if friction between the impacting bodies is relevant, hypothesizing the equivalence leads sometimes to illogical results. Recently, more sophisticated models have been proposed, particularly focusing on the influence of the impact on the integrity [43] or the acoustic performance of the bell, but a careful examination of these sophisticate models is outside the scope of the present work. In the present case, the kinematic approach in terms of relative velocities appears particularly comprehensible. The impact phase is studied applying the law of conservation of angular momentum of the whole system with respect to the bell pivot \(A\) in the configuration assumed by the system when the impact occurs (Figure 4). According recurrent assumptions and considering that the impact is quite instantaneous, the impulse of ordinary forces, if present, the energy losses at the pivot, and the displacements of body particles, have been disregarded during the impact phase. Figure 3: Swing intervals of the clapper. Said \(K\) the kinematic coefficient of restitution of the clapper, it results  
\[\dot{\varphi}_{f}=-K\,\dot{\varphi}_{i}\qquad\quad-1\leq K\leq 1, \tag{14}\]  
where \(\dot{\varphi}_{i}\) and \(\dot{\varphi}_{f}\) are the relative angular velocities of the clapper at the time of impact and at the end of the impact, respectively. As known, \(K=-1\) corresponds to an impact against massless bell, \(K=0\) to the perfectly plastic case, and \(K=1\) to an ideal perfectly elastic impact against a fixed surface, in the latter case, the kinetic energy of the system is obviously conserved. The kinematic COR can be easily measured in practice by means of high-speed video cameras, using super slow-motion video recording capabilities of such kind of devices. As already said, the clapper strikes the bell each time it holds one of the conditions  
\[\varphi=\varphi^{*}\;\;\vee\;\;\varphi=-\varphi^{**}\;, \tag{15}\]  
holds. For example, referring to the case \(\varphi=\varphi^{*}\), the angular momentum \(L\) of the system around \(A\) is:  
\[L=\left(I_{b}+I_{c}-m\,b^{2}+m\,b^{\prime 2}\right)\,\dot{\theta}\;+\left[I_{c} +m\,r\,b\cos(\varphi^{*}-\alpha)\right]\dot{\varphi}. \tag{16}\]  
In Equation (16) \(b^{\prime}\) is the distance \(\overline{AG_{c}}\) between the bell pivot and the clapper center of gravity in the impact position (Figure 4),  
\[b^{\prime 2}=b^{2}+r^{2}+2\;r\,b\cos(\varphi^{*}-\alpha)\;, \tag{17}\]  
and \(b+r\,\cos(\varphi^{*}-\alpha)\) is the measure of its projection on the clapper axis.