## 2 Materials and Methods
### Numerical Simulations  
The numerical calculations of electromagnetic induction and magnetic force values were performed using the finite element method. Namely, the FEMM software (Ph.D. [PERSON], Waltham, MA, USA) was used, which consider magnetic, electrostatic, heat flow and current flow problems. The first simulation task was the change of magnetic induction value in the angular and linear position change of the additional magnet inside the coils and mounted on the pendulum. Then, we received results in the form of a 3D surface presenting the change of electromagnetic phenomena in the system. The estimation was performed for a coil, an additional permanent magnet and the permanent tip magnet mounted on the rotating pendulum keeping the geometrical and magnetic features of applied magnets (NdFeB 37) and coils (28 AWG); also, the repulsive orientation of magnets was taken into account. The aim of the simulation is to check how the change of additional magnet position influence on magnetic induction arose in the coil [40; 41; 42]. In the Figure 4 the layout of the magnets and coil and exemplary results in FEMM software are presented. In total, created surface will consist of 900 results, changing the position of magnet inside coil by 0.5 mm and position of magnet on pendulum by 2.5\({}^{\circ}\). The resultant value of electromagnetic induction is taken from the coil in the \(x\)-axis. The magnetic induction results surface is presented in Figures 5 and 6. The position of the additional magnet was limited by the geometry of the plastic cylindrical channels on the sides of the system. Value discrepancies of magnetic induction are the result of magnetic interactions between magnets. It must be emphasized that only two pairs of parallel coils with additional magnets were installed. In other coils without additional magnets, the value of electromagnetic induction would be smaller. Figure 4: (**a**) Magnets and coil layout in Finite Element Method Magnetics (FEMM) pre-processor, (**b**) Exemplary results in FEMM solver. Figure 5: Magnetic induction results in \(x\)-axis in additional magnet position and angular magnet position domain. The second simulation task using the FEMM software was to find the value of magnetic force between acting magnets and use such value for modelling of mechanical system. The manual of FEMM software predicts the application of co-energy of entire energetic system for two different close positions [43]. The magnetic force equation has the following form:  
\[F_{mag}\ =\ \frac{-\delta W_{c}}{\delta l}, \tag{1}\]  
where \(\delta W_{c}\) is variation of co-energy value of entire energetic system for two different close positions, \(F_{mag}\) is the magnetic force acting on the pendulum, \(\delta l\) is the variation of the additional magnet position. To compare the magnetic forces' values in the system for different distances between additional magnet and the pendulum, Figure 7 shows the curves for x = 1 mm and x = 13 mm. The differences between the curves in Figure 7 are caused by the magnetic interactions. The shape with three local maxima results from the rectangular shape of cross-section of the magnets in 2D simulation. The amplitude and distribution of local maxima changes with the distance between the magnets. In Figure 7b, the local maxima disappear due to the decrease in amplitude of the magnetic force for the 13 mm distance. Two-dimensional simulation provided very important information concerning change of magnetic force depending on the distance between magnets and the obtained values can be used in the equation of motion. The calculated resultant value of magnetic force acting on the pendulum was applied to the equation of motion of the system. The aforementioned magnetic force comprised in some way the damping of the pendulum's movement. That is why its value is so important in modelling.