# Energy Harvester Based on a Rotational Pendulum Supported with FEM
## 3 Design and Modeling of an Energy Harvesting System with a Rotational Pendulum
### Electromagnetic FEM Model  
In comparison to the previous FEA work in [7], the current system was analyzed in 3D space keeping the existing coil layout and the geometry of the energy harvester (Figure 4). The motivation behind achieving precise calculation using 3D FEA was to obtain the information about the magnetic force between magnets during rotation of the pendulum and its influence on the dynamics of the pendulum with magnet. Once this was obtained, the magnetic torque vector was utilized in the mathematical model. Such magnetic interactions strongly affected the dynamics and changed the energetic potential of the energy harvester [49]. In the finite element (FE) model, the influence of housing can be neglected as the entire housing is made of non-magnetic materials and there are no undesirable external magnetic interactions.  
\begin{table}
\begin{tabular}{c c} \hline \hline  & Mechanical Part \\ \hline Mass of the system (g) & \text  
The current problem is a typical multi-body dynamics (MBD) problem dealing with the combination of the dynamics and electromagnetism. Additionally, an electrical circuit was added to analyze the power output of the system. Moreover, couplings between the magnet-magnet and magnet-coil pairs were investigated in the form of acting forces and torques. Thus, the obtained torque vector using FEA was crucial for the analysis of the pendulum's dynamics. As the FEA model was linear, the system could be analyzed for one rotation of the pendulum. Hence, the results were obtained for one rotation per second of the pendulum.  
In order to obtain magnetic force interaction between the pendulum magnet and the additional fixed magnet using FEA, a stationary model was considered, as shown by a meshed view in Figure 4a. In this view, the relative size and positions of the magnets as well as the surrounding air space considered for the magnetic interaction is shown. The linear position of the pendulum magnet relative to the additional magnet-coil pair was changed along the x-axis (see Figure 4b), from which the magnet force in terms of x-, y-, and z-axis components was computed. A detailed description of this procedure is mentioned in the attached Supplementary Materials to this article. Further, using these  
Figure 4: The physical concept of the electromagnetic energy harvesting system with rotational pendulum modeled in COMSOL Multiphysics. (**a**) The entire domain considered for the analysis of magnetic force interaction between pendulum magnet and fixed additional magnet in meshed state. (**b**) Enlarged meshed view of the two interacting magnets mentioned above. (**c**) Meshed geometry considered for the MBD analysis of the system, and (**d**) the distribution of the magnetic field during interactions between the magnets shown in the top planer view of (**b**) (directly taken from the COMSOL results report).  
calculations, resultant magnet forces and torques for magnet-magnet and magnet-coil pairs were estimated. These force and torque quantities along with the electrical variables were then considered for the MBD analyses of the system shown Figure 4c. A top view of Figure 4b showing magnetic flux interaction between the magnets is shown in Figure 4d. Readers can find the geometric dimensions and other physical variables used for these FEM simulations in the Supplementary Materials attached with this article.