## 1 Introduction
As described in [5], generally, they can be distinguished:  
* the \"flying clapper\" ringing technique, mainly used in continental Europe, except the Iberic peninsula and north Italy,
* the full circle techniques, like the English full circle ringing or change ringing technique [6; 7; 8; 9], and the Italian \"Bolognese\", \"Veronese\", and \"Ambrosiano\" styles,
* the Spanish system, typically used in Iberic peninsula. In the \"flying clapper\" technique, the maximum oscillation of the bell, which is not counterweight, is around 90\({}^{\circ}\), although in some cases it can attain 135\({}^{\circ}\), and the clapper strikes the bell on the upper part of the mouth. Occasionally, in France, Belgium, and the Netherlands also the opposite \"falling clapper\" scheme is adopted, where the clapper strikes the lower part of the mouth. The full circle techniques are based on alternate full circle oscillations, which facilitate the sound control. The bells are firstly driven in the upward position, where they are briefly stopped with the help of suitable devices or even manually, and then put in oscillations, which are maintained accurately adjusting the excitation. The English full circle and the Bolognese style use unbalanced bells, on the contrary \"Veronese\" and \"Ambrosiano\" styles adopt balanced bells. In the Spanish system, balanced bells continuously rotate full circle in one direction [10; 11; 12; 13]. From the above considerations, it clearly emerges that study of swinging bells can involve:  
* the assessment of the acoustic characteristics of the bell itself, intended as a musical instrument,
* the quality and the precision of the sound, depending on the ease of controlling the clapper impacts during the oscillations, and
* the interactions between the swinging bell and the supporting structures: belfry and bell tower. The acoustics of the bells is a very specialized topic, outside the scope of the present study, therefore it will not be further discussed. The interested reader can find some basic information about theoretical, experimental, or numerical acoustic studies in [14; 15; 16; 17]. Truly, the ease of controlling the impacts significantly influences the acoustic performance of the bell; in fact, as remarked in [18], \"the \"ringability\" of bells depends not only on the linear acoustics of the modes and natural frequencies of the bells but also on the rotational dynamics of the bell and clapper, interacting through impact each time the clapper strikes the bell surface.\"  
The inertia forces due to the swinging of the bell play a crucial role in dynamic identification [19] or assessment of existing bell towers [8; 9; 10; 11; 12; 13; 20; 21; 22; 23; 24; 25], especially when the frequency of the bell oscillations is in the neighborhood of the relevant natural frequencies of the tower. Some analytical solutions for bell-tower interaction are given in [13], considering free oscillations and different mounting systems. However, some studies focus on the seismic response of the bell itself [26]. The time histories of bell and clapper rotations and support reactions are commonly studied in the framework of the Lagrangian mechanics, following the approach proposed by [PERSON] [27; 28], which is described in detail in every modern book of theoretical mechanics [29; 30; 31]. The analytical model of the bell, based on rigid body dynamics, originally suggested by [PERSON] [32] in the 19 th century, was recovered in the late 20 th century by [PERSON] and [PERSON] [33], who also suggested a simple experimental method to derive the inertia forces transmitted by full circle ringing bells. Since then, several studies have been carried out, shortly discussed in the following. A recurrent simplification disregards the presence of the clapper; the bell motion is thus governed by the ordinary differential equation (ODE) of the physical pendulum, depending on two independent initial conditions. But, as better illustrated in Section 2, more refined approaches lead to a system of two second-order nonlinear ODEs.