## 1 Introduction
The solutions of that system are the time histories of bell and clapper rotations, provided that four independent initial conditions are given. It must be underlined that, except in the case of small amplitude oscillations, when the physical pendulum equation can be linearized, the ODEs are always nonlinear, even if springs and dampers are not present. That peculiarity significantly distinguishes the swinging bell model from the simple harmonic oscillator, whose behavior is nonlinear only if nonlinear terms are explicitly introduced, like, for example, nonlinear restoring forces or springs in the Duffing oscillator [34]. Sometimes Duffing oscillator has been also used to approximate the nonlinear pendulum equation, based on the McLaurin expansion of the sinus function, but these aspects are outside the scope of the study. To solve the swinging clapper bell problem, many different approaches have been proposed, alternatively concentrating the attention on the bell-clapper interactions or on the bell-tower excitation. Frequently, aiming to obtain convenient expressions of the equations, drastic simplifications are introduced, so arriving at suitable analytical or numerical solutions. Noteworthy simplifications, often combined, may include:  
* neglecting the clapper, so that the bell reduces to the already cited physical pendulum,
* neglecting the damping, thus considering only undamped oscillations,
* considering only viscous damping,
* disregarding external forces, thus considering only free oscillations,
* disregarding the transient phase, dealing with forced oscillations,
* suitably modifying the forcing functions to obtain \"workable\" equations from the analytical [22], or the numerical point of view [35; 36; 37; 38],
* adopting simplified models to describe the impact of the clapper on the bell. However, if analytical solutions are available only for special cases, also usual numerical approaches can result unsatisfactory. In fact, numerical solutions reported in the relevant literature are commonly obtained adopting built-in algorithms for the solutions of ODEs, available in advanced software packages, like Matlab(r). These algorithms are frequently based on Runge-Kutta methods. The availability of such sound and validated algorithms, characterized by clearly defined limitations, is certainly beneficial, but resorting to them involves rather evident disadvantages, because these general methods are scarcely flexible and not easily adaptable to handle specific problems. Moreover, hypothesizing viscous damping only and assuming special forcing functions, the field of application of these \"traditional\" approaches sensibly reduces. With regards to the previous remarks, it must be stressed, on one hand, that other kinds of damping, like friction damping, can play a key role and on the other, that different forcing functions can be envisaged in stationary or transient phases, also depending on the type of excitation\(-\)manual or automatic, and on the type of the motorization: linear, or rotating, or even contactless, if any. Since the pivots are generally equipped with bushes or roller bearings, in which dissipative forces depend on the radial reaction forces, friction damping appears as the most appropriate model. Nevertheless, it is common practice to adopt an \"equivalent\" viscous damping model, so drastically simplifying the equations. Unfortunately, as confirmed by the outcomes of the study, the equivalence holds on a limited range of oscillation amplitudes, so that the parameters of the equivalent model should be continuously adjusted and adapted, depending on the actual situation. Regarding the forcing functions, it is noteworthy that a typical transient is the starting phase, in which the bell is set in motion and suitably excited until a pre-established configuration is achieved.