# Energy Harvester Based on a Rotational Pendulum Supported with FEM
## 3 Design and Modeling of an Energy Harvesting System with a Rotational Pendulum
### Reduced Mathematical Model  
In Figure 5, the scheme of the mathematical model for the energy harvester with terms applied in the equation of motion is presented. Vertical excitation coming from the shaker initiated the movement of the rotational pendulum around the pivot in the form of oscillations or full rotations, depending on the applied strength of excitation and the strength of magnetic interactions in the system. The mentioned terms had significant influence on the system's response, which is discussed in the next section. In Table 2, the values of the terms used in the mathematical model are listed.  
In comparison with the previously derived mathematical model of the rotational pendulum [7], additional terms were added, which consider magnet-magnet and magnet-coil couplings. Relative displacement between the pendulum magnet and permanent magnet pair in its proximity created strong magnetic interactions in the form of two opposite potential wells, affecting the acting magnetic forces in the system, as can be observed in Figure 6a. During the relative displacement between magnets, the peak-to-peak amplitude of the pendulum's torque changed drastically, creating a direct influence on its dynamics. The coupling between the pendulum's magnet and the coils had a  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Term** & **Value** \\ \hline \(I\) [kg-m\({}^{2}\)] & 0.00377 \\ \(m\) [kg] & 0.0186 \\ \(g\) [m/s\({}^{2}\)] & 9.81 \\ \(h\) [m] & 0.03325 \\ \(\alpha\), \(\beta\) & 0 and 1 \\ \(\lambda\) & 0.048 \\ \(h\) [m] & 0.034 \\ \(\omega\) [rad/s] & 15 \\ \(A\) [m] & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters used in the mathematical model (see Equation (1)).  
Figure 5: Variables in the mathematical model are shown in the simplified configuration of the energy harvester with magnetic pendulum.  
periodic response (Figure 6b) in each of the 12 coils, showing a relatively smaller impact on its dynamics. Thus, the governing equation of motion (Equation (1)) for the pendulum considering these additional forces can be expressed in the following form:  
\[\ddot{\varphi}+\frac{1}{I}(c\dot{\varphi}-\alpha F_{mc}(\varphi)\dot{\varphi}+ mgh\sin(\varphi)-\beta F_{mm}(\varphi)-m\omega^{2}Ah\sin(\varphi)\cos(\omega t))=0, \tag{1}\]  
where \(I\)--moment of inertia of the pendulum, \(m\)--mass of pendulum and additional inertial masses, \(g\)--gravitational acceleration, \(h\)--distance from center of gravity to pivot axis, \(F_{mc}\)--torque between magnet-coil couplings obtained with FEA, \(\omega\)--angular velocity of the pendulum, \(A\)--amplitude of the vertical excitation, \(F_{mm}\)--torque between magnet-magnet couplings in the system obtained with FEA, \(c\)--coefficient of mechanical damping (in the calculations we assumed \(c=0\) for simplicity), \(\alpha\), \(\beta\) = 0 or 1--are the coefficients of magnetic force that are dependent on the distances between magnets and coils. We have neglected the macroscopic eddy currents in the magnets.  
The output power of the system in time domain \(t\) is calculated using the following formula (Equation (2)):  
\[P=\frac{1}{R_{l}}\ \frac{1}{t}\int_{0}^{t}U^{2}(t)dt=(1-\lambda)\alpha F_{mc}( \varphi)\dot{\varphi}^{2}, \tag{2}\]  
where \(U\) is the voltage output estimated on the resistor and \(\lambda=R_{c}/(R_{l}+R_{c})\) is the coil resistivity part of the electrical circuit (\(R_{c}\) and \(R_{l}\) denote coil and load resistances, respectively).