## 6 Discussion: Response to Ringing Bell  
The experimental modes match with the modes of a clamped-free Euler Bernoulli beam. The authors estimated the effect of the bells on the tower dynamics by adopting a two-degrees-of-freedom mechanical system, depicted in Figure 15. The lower substructure represents the tower, while the upper substructure the more massive bell. Concerning Figure 15, [PERSON]'s equations of motion for the two-degrees-of-freedom model can be directly written and expressed in the following matrix form  
\[\left[ \begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array} \right]\left\{ \begin{array}{cc}\dot{x}_{1}(t)\\ \dot{x}_{2}(t)\end{array} \right\}+\left[ \begin{array}{cc}c_{1}+c_{2}&-c_{2} \\ -c_{2}&c_{2}\end{array} \right]\left\{ \begin{array}{cc}\dot{x}_{1}(t)\\ \dot{x}_{2}(t)\end{array} \right\}+\left[ \begin{array}{cc}k_{1}+k_{2}&-k_{2} \\ -k_{2}&k_{2}\end{array} \right]\left\{ \begin{array}{cc}x_{1}(t)\\ x_{2}(t)\end{array} \right\}=-\left\{ \begin{array}{c}m_{1}\\ m_{2}\end{array} \right\}\dot{x}_{g}(t), \tag{2}\]  
where \(k_{1}\) and \(m_{1}\) are the stiffness and mass values of the lower substructure, \(k_{2}\) and \(m_{2}\) are the stiffness and mass values of the upper substructure, \(c_{1}\) and \(c_{2}\) are the damping coefficient, \(x_{g}(t)\) is the base excitation. \(c1\) and \(c_{2}\) are expressed in terms of modal damping \(c_{1}=2\,xi_{1}\,omega_{1}\) and \(c_{2}=2\,xi_{2}\,omega_{2}\), where \(xi_{1}\) and \(xi_{2}\) are the modal damping. Two mechanical systems approximately reproduce the tower dynamics, tuned to the first and second natural frequencies respectively. The estimate of the mass matrix derives, as follows, from the first natural frequency of the tower and the bell, by assuming the uncoupling of the bell-tower dynamics. Specifically, Equation (3) yields the mass and the elastic constant of the spring from the natural frequency of a cantilevered Euler-Bernoulli beam,  
\[omega_{1}^{2}=\frac{k_{1}}{m_{1}}=\frac{(beta\,h)^{4}}{h^{4}}\frac{EI}{roA} \to k_{1}=\frac{(beta\,h)^{4}\,EI}{h^{3}},\,\,m_{1}=ro\,A\,h, \tag{3}\]  
Figure 15: **(a)** FFT of a top accelerometer response to the bell swinging **(b)** Main 2-DOF archetype system; **(c)** Superposition of the experimental and analytical response to ringing bells. where _omega1_ is the natural pulsation of the substructure. The first solution of the eigenvalue problem of a clamped-free beam is _beta_\(h=1.875\). Equation (4) returns the mass and the elastic constant of the spring from the natural frequency of a physical pendulum, representative of the more massive bell. If the amplitude is limited to small swings, the pulsation is [29]  
\[omega_{2}^{2}=\frac{k_{2}}{m_{2}}=\frac{M\,g\,d}{I_{p}}\to k_{2}=\frac{I_{p}}{g \,d},\,\,\,m_{2}=M, \tag{4}\]  
where _omega2_ is the natural pulsation of the substructure, \(k_{2}\) and \(m_{2}\) the stiffness and mass values of an equivalent SDOF oscillator, \(I_{p}\) is the moment of inertia of the pendulum about the pivot point, \(M\) is the mass of the pendulum, \(d\) is the distance between the pivot point and the centre of mass and \(g\) is the gravity acceleration. The modal viscous dampings of the substructure and the bell are set equal to 3.7% (see Table 1) and 1% respectively. Table 4 lists the parameters adopted in the mechanical model. The experimental response of the masonry tower to the swinging of the more massive bell agrees with the response of the mechanical system in Equation (2) to the oscillation of the upper substructure, Figure 15b. As noticed by [PERSON] et al. [7], this evidence further confirms that a simple pendulum reproduces the effect of swinging bells over the towers adequately. The FFT of the acceleration time-history of a top accelerometer, reported in Figure 15a, clearly present the peaks corresponding to the oscillation frequency of the bell at \(f=0.416\) Hz and the natural frequencies of the tower at \(f=2.367\) Hz and \(f=2.749\) Hz. Nevertheless, there are three additional peaks at 0.828 Hz, 1.282 Hz and 1.652 Hz.