## 1 Introduction
Especially in the case that the bells are manually driven or contactless excitations are adopted, the forcing functions are often optimized to be \"resonant\", i.e., continuously and suitably tuned by an external device or by a trained bell ringer to match the instantaneous natural frequency of the system, which is a function of the amplitude of the oscillation. For these reasons, an ideal and fully featured method to solve the ODEs of motion should allow:to take into account both viscous and friction damping,  
* to consider \"arbitrary\" forcing function,
* to determine the solution in both stationary and transient phases,
* to introduce or to deduce \"resonant\" excitations, automatically tuned on the instantaneous natural frequency of the system. In addition, the method should work also with \"stiff equations\". The aim of the study is the setup of an original all-purposes general procedure to solve the system of ODEs describing the motion of the bell and of the clapper. The proposed algorithms discussed in Section 6 are based on explicit-implicit predictor-corrector integration methods with constant time step, directly operating on the ODEs system. They do not require any preliminary manipulation of the equations themselves, with evident advantages. To highlight their flexibility, two slightly different versions of the algorithms are presented:  
* a moderately simplified version, in which the prediction-correction phase works only on the rotations of bell and clapper,
* the complete, more refined, version, in which the convergence is improved thanks to an additional prediction-correction phase regarding the angular velocities and the angular accelerations too. The paper is organized as follows:  
* Section 2 illustrates the deduction of the ODEs governing the problem. * Section 3 refers to the model of the clapper to bell impact and how it is considered in the equations. * Section 4 illustrates the friction and viscous damping models, while
* Section 5, considering the dissipative moments corresponding to damping phenomena, describes the general form of the nonlinear system of ODEs. * Section 6 illustrates in detail the implementation of the proposed algorithms. * Section 7 deals with the validation of the proposed method, checking its capability to reproduce some relevant theoretical or experimental results. To enrich the comparison, the convergence of the proposed algorithms is assessed, and its dependence on the time step and other influencing parameters is critically discussed, also considering \"stiff\" equations. Of course, that section is not essential for the understanding of the remaining parts of the paper. * Section 8 considers a significant case study, consisting in the old tenor bell of the Great St. Mary's church in Cambridge, whose characteristics are precisely reported in [17]. Several relevant situations are analyzed, considering damped or undamped conditions; free oscillations; forced oscillations; stationary or transient phases; different initial conditions; demonstrating the main features and the potentialities of the proposed algorithms. * Section 9 summarizes the main conclusions, envisaging further developments. The novel approach and the algorithms proposed in the paper allow one to overcome the limitations affecting the \"traditional\" approaches proposed until now; in fact, besides satisfying all the previously recalled requisites (a\(-\)c), they are also able to derive or to define resonant excitations and to solve stiff equations, as well. Independently on the oscillation amplitude and on the duration of the considered time interval, the algorithms can successfully manage undamped oscillations; friction and viscosity damped oscillations; free oscillations in transient and stationary phases; and can be applied also to solve stiff equations. Furthermore, the capability of the proposed methods to deal with arbitrary forcing functions is particularly innovative.