# Energy Harvester Based on a Rotational Pendulum Supported with FEM
## 4 Numerical Simulations
The variations in the angular velocity between positive and negative values depict the pendulum's oscillations around the equilibrium point. Further, the pendulum  
Figure 7: The potential energy of the magnetic force interaction was obtained for one revolution of the pendulum and is shown by the red-dotted line. The potential of the gravity force is shown by a black-dotted line. was able to pass this equilibrium point after around 50 s. Adding induction coils (Figure 8b) established the coil-magnet coupling. The pendulum at first started oscillating, but later when its energy was dissipated by coils, the angular velocity was reduced to zero. Mounting additional magnets strongly influenced the dynamics of the pendulum, which can be seen in Figure 8c. The magnetic force was adjusted by adjusting the axial distance between one of the pair of magnets. Initially, the magnetic potential barrier could not be passed by the pendulum to perform a full rotation. However, after increasing the magnetic force, the repulsive force assisted in easier passing of the magnetic barrier and the pendulum oscillated between the two positions. Figure 8: Selected solutions of the dynamic system in terms of the instantaneous rotation angle (left panel—black plots) and angular velocity (right panel—blue plots) of pendulum are shown here. Solutions for (**a**) a system without magnets and coils, (**b**) a system with coils but without additional magnets, and (**c**) a system with additional magnets and coils are presented. In cases b-c, the voltage in the coils was induced due to the electromagnetic transduction during the parallel passing of the magnet-pendulum around the vicinity of the coils. In Figure 9, the power generated by the pendulum's movement is illustrated. The level of generated power increased as the magnetic force interaction became stronger. The mean value of power was estimated by the moving mean algorithm (see Figure 9--black line), whose value was in the range 0.01-0.02 [W] for case (b). For case (c), the mean harvester power was around 0.5 [W] because of larger and magnetic force-assisted oscillations. The phase portraits of the considered three cases are shown in Figure 10, which illustrates the trajectories drawn by the excited pendulum. Excitation of pendulum without any source of damping caused free rotation under the influence of gravitational potential only (see Figure 10a). After adding coils into the system, its damping caused small amplitude oscillations around the central equilibrium point, as shown in Figure 10b. This was caused by the transduction of energy between the pendulum magnet and coils. Installation of additional magnets led to inter-well oscillations around the offset equilibrium point, where magnetic interactions had maximum impact (see Figure 10c). The phase portrait took a square-based shape and depicted the smaller potential well formed at the specific angular position. It was observed that, as the strength of magnets increased, it became easier to pass through the magnetic barrier at the offset pendulum. Figure 10c shows oscillation of the pendulum between the central and offset equilibrium positions. Figure 10: _Cont._  
Figure 9: Power output plots corresponding to the solutions of cases **(a,b)** (shown in Figure 8b,c, respectively) are shown here. Note, there is no power output in the case shown in Figure 8a, because no electromagnetic transduction occurred in the absence of coils.