# Energy Harvester Based on a Rotational Pendulum Supported with FEM
## 4 Numerical Simulations  
The analysis of the influence of magnetic forces on the dynamics of the rotational pendulum started with the FEM computations conducted in the COMSOL (v5.3) Software. In the system, there were two unstable equilibrium points where the axes of moving and additional magnets coincided. The amplitude and the distribution of the magnetic force with the angular position of the rotating pendulum depended on the geometry of the magnets and the distance of their alignment, which were considered in this research. It was observed in the prior work [7] during experiments that the system's frame needed excitation in the specific range of frequencies, i.e., \(f\) = 10-15 Hz, in order to obtain a completely rotational solution of the pendulum to attain the highest voltage/power performance. Outside the mentioned range of frequencies, the pendulum became stuck at the equilibrium  
Figure 6: (**a**) The torque from magnet–magnet coupling \(F_{mm}\) [Nm] (see Equation (1)) calculated with the FEM model. (**b**) The torque from magnet–coil \(F_{mc}\) coupling [Nm] (see Equation (1)) during the clockwise rotation of the pendulum with a constant rotational velocity (1 revolution per second). The initial angle in FEM calculations was 0.25\(\pi\). point and did not move. Additionally, it was observed that changing the distance between additional magnets mounted on the sides influenced the potential barrier at the equilibrium positions. In this work, the proposed model considers the influence of varying magnetic forces with the alignment distance on the magnetic pendulum. We also focused on solutions with complete rotation of the pendulum. In the equation of motion, the magnet-magnet and magnet-coil interactions are important terms influencing significantly the dynamics of the pendulum. To estimate these interactions, the 3D model of the energy harvester was meshed for the FEA (Figure 4) and a series of torques was obtained, which are depicted in Figure 6. The FEA results showed the influence of the magnet-magnet interaction was significantly stronger than that of the coil-magnet coupling on the dynamics of the pendulum. Regarding the interactions between the magnet-magnet in the system (Figure 6a), mutually opposite areas with the strongest magnetic interactions were found. Such a repulsive configuration of magnets helps the pendulum to escape from this area if the potential energy is enough to go through the potential barrier created by the magnetic coupling. This is shown by the red dotted lines in Figure 7. For comparison, gravitational potential is also shown by black dotted lines in Figure 7. It had an almost 10 times weaker influence on the pendulum. The magnet-coil couplings (Figure 6b) had smaller amplitude than the magnet-magnet couplings; however, their influence on the pendulum was also considered in this study. We considered the following distinct configurations of the harvester for revealing additional details about the dynamics:  
1. System without coils and without magnets, only pendulum with magnet (only gravitational acceleration): \(\alpha=0\), \(\beta=0\). 2. System with coils and pendulum with magnet (only magnet-coil coupling): \(\alpha=0\), \(\beta=1\),
3. System with all magnets and coils. In Figure 8, the dynamic response of the system is illustrated with the change of angular position \(\Phi\) and angular velocity \(\omega\) for the abovementioned different configurations. In all considered cases, the transient state can be observed in the beginning of the system's operation. The reason for this is that the pendulum must first synchronize with the external excitation, and then the steady state operation is observed. For energy harvesting studies, the steady state response is considered from which the highest output voltage/power is observed. In the first case, the system without additional magnets and without coils tended to maintain constant rotations after the synchronization of the pendulum and the vertical excitation.