## 3 Modelling the Clapper Impact
Said \(I_{CA}\) the moment of inertia of the clapper around \(A\) in that position,  
\[I_{CA}=I_{c}+m\,\left[\,r^{2}+2\;r\,b\cos(\varphi^{*}-\alpha)\right], \tag{18}\]  
\(I_{lt}\) the moment of the inertia around \(A\) of the whole system,  
\[I_{lt}=I_{b}+I_{CA}, \tag{19}\]  
Figure 4: Geometry of the bell in the clapper impact phases. and \(I_{c*}\) the term pertaining to \(\dot{\varphi}\) in (16)  
\[I_{c*}=I_{c}+m\;r\;b\cos(\varphi^{*}-\alpha)\;, \tag{20}\]  
the law of conservation of the angular momentum for an isolated system gives  
\[I_{bt}\;\dot{\theta}_{i}+I_{c*}\;\dot{\varphi}_{i}=I_{bt}\;\dot{\theta}_{f}+I_{c* }\;\dot{\varphi}_{f}, \tag{21}\]  
being \(\dot{\theta}_{i}\) and \(\dot{\theta}_{f}\) the angular velocities of the bell before and after the impact, respectively. Recalling Equation (14), \(\dot{\theta}_{f}\) is easily obtained from Equation (21) as  
\[\dot{\varphi}_{f}=-K\,\dot{\varphi}_{i}\qquad\quad-1\leq K\leq 1, \tag{14}\]0  
Finally, the variations of the angular velocities during the impact result:  
\[\left\{ \begin{array}{l}\Delta\dot{\varphi}=\dot{\varphi}_{f}-\dot{\varphi}_ {i}=-(1+K)\;\dot{\varphi}_{i}\\ \\ \Delta\dot{\theta}=\dot{\theta}_{f}-\dot{\theta}_{i}=(1+K)\;\frac{I_{c*}}{I_{ bt}}\;\dot{\varphi}_{i}\end{array} \right.. \tag{23}\]  
Equations (14) and (22) allow to consider the impact effects in a very effective way, since they constitute, together with the pertinent condition on \(\varphi\) (Equation (15)) and the corresponding value of \(\theta_{f}\) at the end of the impact, the initial conditions for the subsequent \"piece\" of the piecewise-defined solution. From another point of view, these variations can be simulated setting the angular accelerations, \(\ddot{\theta}_{j}\) and \(\ddot{\varphi}_{j}\), at the \(j\) - th step in the form  
\[\left\{ \begin{array}{l}\ddot{\varphi}_{j}=\frac{\Delta\;\dot{\varphi}_{j}}{ \Delta t}=-\frac{(1+K)\;\dot{\varphi}_{j}}{\Delta t}\\ \\ \dot{\theta}_{j}=\frac{\Delta\dot{\theta}_{j}}{\Delta t}=\frac{(1+K)\;I_{c*} \;\dot{\varphi}_{j}}{I_{bt}\;\Delta t}\end{array} \right.\;, \tag{24}\]  
in which \(\dot{\varphi}_{j}\) is the relative angular velocity of the impacting clapper and \(\Delta t\) the time step adopted in numerical integration. Similar conclusions can be derived from the impulse-momentum theorem. As said before, refined evaluation of the coefficient of restitution would require much more sophisticated modelling of the impact phase, nevertheless the simplified approach proposed above can be easily improved and corrected; on one hand, to avoid physical paradoxes, on the other, to properly consider additional phenomena associated with the impact. First, it must be remarked that the bounds on \(K\) given in Equation (14) can be finer tuned, since, on the light of what previously discussed, in a swinging bell \(K=1\) infringes the law of conservation of energy. For physical coherence, the effective relative angular velocity between the clapper and the bell after the impact, \(\Delta\omega\), involving not only \(\dot{\varphi}_{f}\), but also \(\Delta\dot{\theta}\), should satisfy the conditions  
\[-\dot{\varphi}_{i}\leq\Delta\omega=\dot{\theta}_{f}-\dot{\theta}_{i}+\dot{ \varphi}_{f}\leq\;\dot{\varphi}_{i}, \tag{25}\]  
that, recalling Equations (14) and (22), finally provides  
\[-1\leq K\leq\;\frac{I_{bt}-I_{c*}}{I_{bt}+I_{c*}}, \tag{26}\]  
in which the upper bound corresponds to a perfectly elastic impact. It must be stressed that, while in the first clapper stroke(s) a considerable part of energy is spent to induce vibrations in the bell, in the subsequent ones the transfer is less relevant, being only responsible for maintaining the vibrations themselves. The first impacts are thus characterized by lower coefficients of restitution. Furthermore, in some cases bell ringers cover a side of the clapper with felt pads to muffle the sound as a function of the side of impact. In any case, these aspects can be simply managed, varying the COR as a function of the strike sequence or of the impact side.