# Two-Degrees-of-Freedom (2 DOF) Magnetic Spring-Based Electromagnetic Energy Harvester
## 6 Dynamic Analysis of the 2 DOF Energy Harvester
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Units** \\ \hline M\({}_{2}\) & 370 & gm \\ \hline M\({}_{3}\) & 327.28 & gm \\ \hline \(\beta_{1}\) & 0.98 & Ns/m \\ \hline \(\beta_{2}\) & 0.235 & Ns/m \\ \hline \(\beta_{3}\) & 0.51 & Ns/m \\ \hline \(k_{1}\) & 524.37 & N/m \\ \hline \(k_{2}\) & 112.875 & N/m \\ \hline \(k_{3}\) & 298.62 & N/m \\ \hline \(\alpha_{1}\) & 4918.6 & N/m\({}^{2}\) \\ \hline \(\alpha_{2}\) & 715.55 & N/m\({}^{2}\) \\ \hline \(\alpha_{3}\) & 3487.5 & N/m\({}^{2}\) \\ \hline \(\lambda_{1}\) & 291,867 & N/m\({}^{3}\) \\ \hline \(\lambda_{2}\) & 92,334 & N/m\({}^{3}\) \\ \hline \(\lambda_{3}\) & 107,199 & N/m\({}^{3}\) \\ \hline B1 and B2 & 0.35 & T \\ \hline R1 and R2 & 5.48 & ohm \\ \hline
11 and l2 & 23.5 & m \\ \hline L1 and L2 & 0.005546 & H \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Required parameters. The natural frequencies of the mechanical parts were almost similar for both generator systems with and without external load. However, the natural frequencies of the electrical parts were much higher in the generator system with external load than in the system without external load. Compared to the generator system with or without external load, the system with external load showed better dynamic results. The generator system's displacements, velocities, and induced voltages were analyzed with an external force. If the external force was applied to any of the floating magnets or  
Figure 8: Frequency response of the generator system in the equilibrium position. Figure 9: Frequency response of the generator system with external load in the equilibrium position. both floating magnets, then both floating magnets started moving relative to each other. Moreover, the velocity of the second floating magnet always remained higher than that of the first floating magnet. After adding the winding coils, the system was analyzed. Because of the movement of the floating magnets, the magnetic flux densities of the floating magnet cut the winding coils, which created an induced voltage inside the winding coils. The applied external harmonic force (Fb) amplitude was 25 N, and the frequency (f) was 0.1 Hz. The values of linear stiffness, nonlinear stiffness, and damping constants are presented in Table 1. The state-space model equations were solved using the Ode23t solver in MATLAB (version R2020b) to find the system's displacements, velocities, and induced voltages. Both floating magnets' excitations were assumed to have initial displacements, and their corresponding velocities were zero. As expected, the displacements and the velocities were sinusoidal and 90\({}^{\circ}\) out of phase. At first, the generator system was analyzed by applying harmonic force to both floating magnets. Figure 10 presents the displacement and velocity of the first floating magnet, and Figure 11 displays the displacement and velocity of the second floating magnet. Figure 11: Displacement and velocity of the 2 nd floating magnet. Figure 10: Displacement and velocity of the 1 st floating magnet. Because of the applied harmonic force, the maximum displacement of the first floating magnet toward the second floating magnet was around 40 mm, and toward the bottom magnet, it was about 30 mm. The frequency of the applied harmonic force we considered was 0.1 Hz; therefore, Figure 10 shows two complete cycles. The maximum velocity of the first floating magnet was around 0.04 m/s. On the other hand, the maximum displacements of the second floating magnet toward the fourth (top fixed magnet) were around 50 mm and 40 mm towards the first floating magnet, as shown in Figure 11. The maximum velocity of the second floating magnet was around 0.05 m/s.