# Two-Degrees-of-Freedom (2 DOF) Magnetic Spring-Based Electromagnetic Energy Harvester
## 6 Dynamic Analysis of the 2 DOF Energy Harvester  
The dynamics of the 2 DOF energy harvester were analyzed by solving Equations (12)-(15) using linear and nonlinear stiffness, calculated from the magnetic restoring forces. The eigenvalues and frequency responses were first determined to understand the system. The diameter of the copper coil was 31 mm, and the number of turns of the winding coils was 100. The inside diameter and height of the winding coils were 75 mm and 10 mm, respectively. The total length of the winding coil was 23.5 m, and the inner resistance was 5.48 ohm. The average magnetic flux density we considered was 0.35 T (calculated from ANSYS MAXWELL version 2020 R2). Because of electromechanical coupling, the generator system consists of electrical and mechanical parts. Two winding coils were added outside the two floating magnets; therefore, the system has two resonance frequencies in the electrical part. Because of the two floating magnets, the system has two mechanical resonance frequencies. The electrical resonance frequencies did not change when changing the position of the floating magnets, but they changed when changing the total length of the winding coil (number of turns) and magnetic flux density. Moreover, the entire  
Figure 7: Free-body diagram of the two-degrees-of-freedom system. length of the winding coil does not affect the mechanical resonance frequency. All required parameters are presented in Table 1.  
The natural frequencies of the electrical parts were 953.46 rad/s and 948.77 rad/s when both floating magnets were in an equilibrium position. The natural frequencies for the mechanical parts were 44.17 rad/s and 33.75 rad/s. The natural frequencies of the mechanical part were almost similar for both generator systems with and without electrical-mechanical coupling. The imaginary parts of the eigenvalues (electrical parts) were zero, but the real parts were \(-\)953.47 and \(-\)948.78 in the equilibrium position. The eigenvalues of the mechanical part were \(-\)0.1954 \(\pm\) 0.3962i (first floating magnet) and \(-\)0.2022 \(\pm\) 0.2703i (second floating magnet) in the equilibrium position. The generator system's frequency response in the equilibrium position is shown in Figure 8. Due to the electromechanical coupling, the frequency response graph does not show the peak amplitude. However, the peak resonance is shown to be at 40.1 rad/s. The generator system's eigenvalues were analyzed by changing the floating magnets' position. The frequency resonance was analyzed by connecting the external load or resistance parallel to the winding coils. Different external loads were connected to explore the frequency resonance of the generator system. The external load was varied from 500 ohms to 50,000 ohms to determine the resonance frequency. Figure 9 displays two resonances, and the values are 33 rad/s and 44.4 rad/s in the equilibrium position, which are almost similar to the natural frequencies of 44.17 rad/s and 33.75 rad/s without an external load, which were determined using eigenvalues. It can be seen in Figure 8 that the resonance frequency graph does not show the peak amplitude, but in Figure 9, the generator system with external load shows two peak resonant frequencies.