## 2 The Governing Equations  
In the classical Lagrange-Euler approach, the equations of the bell and clapper dynamics can be easily derived starting from the Lagrangian function \(\mathcal{L}\) of the two degrees of freedom system,  
\[\mathcal{L}\Big{(}\theta,\varphi,\dot{\theta},\dot{\varphi},t\Big{)}=T-U, \tag{1}\]  
where,  
* \(\theta=\theta(t)\) is the angle between the axis of the bell and the vertical axis,
* \(\varphi=\varphi(t)\) the relative angle between the clapper axis and the bell axis (Figure 1),
* \(\dot{\theta}=\dot{\theta}(t)\) and \(\dot{\varphi}=\dot{\varphi}(t)\) their first derivatives with respect to the time \(t\), i.e., their angular velocities, respectively,
* \(T\) the kinetic energy of the system,
* and \(U\) its potential energy. Referring again to Figure 1, let \(A\) be the pivot of the bell, \(B\) the pivot of the clapper, and \(r\) their distance, \(r=y_{B}-y_{A}\), being \(y_{A}\) and \(y_{B}\) the ordinates of \(A\) and \(B\), respectively. An error affecting the position of the pivot of the clapper is hypothesized. Be \(\delta\) such an error, which is the distance \(\overline{BB^{\prime}}\) between the actual position \(B^{\prime}\) of the clapper pivot and its theoretical position. Obviously, the positioning error can be expressed in other terms by means of the angle \(\alpha\),  
\[\alpha=\text{atan}\Big{(}\frac{\delta}{r}\Big{)}. \tag{2}\]  
Although the positioning error, typically ranging in the interval \(\delta=10-20\) mm, is significant only for small bells; it is considered here for the sake of generality. * \(G_{b}\) the center of gravity of the bell,
* \(G_{c}\) the center of gravity of the clapper,
* \(M\) the mass of the bell,
* \(m\) the mass of the clapper,
* \(a\) the distance between the center of gravity of the bell and the bell pivot \(A\),  
Figure 1: Bell and clapper scheme. \(b\) the distance between the center of gravity of the clapper and the clapper pivot \(B^{\prime}\),  
\(I_{b}\) the moment of inertia0 of the bell about \(A\),  
\(I_{c}\) the moment of inertia of the clapper about \(B^{\prime}\),  
\(\stackrel{{\rightarrow}}{{v}}_{gc}\) the velocity of \(G_{c}\),  
it results:  
\[T=\frac{1}{2}I_{b}\;\dot{\theta}^{2}+\frac{1}{2}\Big{(}I_{c}-m\;b^{2}\Big{)} \left(\dot{\varphi}+\dot{\theta}\right)^{2}+\frac{1}{2}\;m\;\Big{|}\stackrel{{ \rightarrow}}{{v}}_{gc}\Big{|}^{2}\;, \tag{3}\]  
where  
\[\stackrel{{\rightarrow}}{{v}}_{gc}=\left(\begin{array}{c}- \Big{(}r\dot{\theta}  
After some manipulations, omitting variables in the moment expressions, Equation (8) becomes,  
\[\left\{ \begin{array}{l}\left[I_{b}+I_{c}+m\;r^{2}+2\;m\;r\;b\cos(\varphi-\alpha) \right]\ddot{\partial}+\left[I_{c}+m\;r\;b\;\cos(\varphi-\alpha)\right]\ddot{ \varphi}-m\;r\;b\left(2\;\dot{\theta}\;\dot{\varphi}+\;\dot{\varphi}^{2}\right) \\ \sin(\varphi-\alpha)+M\;g\;a\sin\theta+m\;g[\;r\sin(\theta+\alpha)+b\;\sin( \theta+\varphi)]=M_{e}+M_{dis}-m_{dis}\\ \\ \left[I_{c}+\;m\;r\;b\cos(\varphi-\alpha)\right]\ddot{\partial}+I_{c}\;\ddot{ \varphi}\;+m\;r\;b\sin(\varphi-\alpha)\dot{\theta}^{2}+m\;g\;b\sin(\theta+ \varphi)=m_{e}+m_{dis}\end{array} \right. \tag{9}\]  
which is a system of two nonlinear second order ordinary differential equations (ODEs), whose solution depends on the given initial conditions.