# Two-Degrees-of-Freedom (2 DOF) Magnetic Spring-Based Electromagnetic Energy Harvester
## 6 Dynamic Analysis of the 2 DOF Energy Harvester
For the same applied harmonic forces in both floating magnets, the second floating magnet achieved more displacement and velocity than the first floating magnet, as shown in Figure 12.  
It was considered during the analysis that the first winding coil was placed outside of the first floating magnet, and the second winding coil was outside of the second floating magnet. The positions of both winding coils were not considered. As is known, due to the movement of the floating magnets, an induced voltage will be generated in the winding coil. The generated induced voltages were determined, as shown in Figure 13. The displacement and velocity of the second floating magnet were higher than those of the first floating magnet. Therefore, the second winding coil showed a higher induced voltage than the first. The maximum induced voltage in coil 2 was around 0.45 volts and in coil 1 was 0.3 volts. The generator system was analyzed by applying a harmonic force only on the second floating magnet. That applied harmonic force determined the displacements and velocities of both floating magnets. Moreover, the induced voltages for both coils were calculated. As the harmonic force was applied to the second floating magnet, the first floating magnet also started moving when the second floating magnet started moving. For this applied harmonic force (25 N amplitude), the second floating magnet achieved a higher displacement and velocity than the first. Figures 14 and 15 present the displacement and velocity of the first and second floating magnets, respectively. Figure 12: Comparison of displacement and velocity of (**a**) 1 st floating magnet and (**b**) 2 nd floating magnet. Figure 8:0 Displacement and velocity of the 1 st floating magnet. Figure 8:1 Induced voltages in (**a**) 1 st winding coil and (**b**) 2 nd winding coil. Figure 8:2 Displacement and velocity of the 2 nd floating magnet. The first floating magnet moved 13 mm and 9 mm toward the second floating magnet and the first fixed magnet (bottom magnet), respectively. The maximum velocity was around 0.009 m/s during this movement. On the other hand, the second floating magnet moved 47 mm and 35 mm toward the top fixed magnet (fourth magnet) and the first floating magnet, respectively. The maximum velocity for the second floating magnet was around 0.045 m/s during this movement. Figure 16 compares the displacement and velocity of the first floating magnet with those of the second floating magnet. Figure 17 shows the induced voltages of both winding coils. The relative displacements of both floating magnets were determined. The displacement and velocity of the second floating magnet were higher than those of the first floating magnet, as seen in Figure 8:3 Therefore, the induced voltage of coil 2 was higher than that of coil 1. The maximum induced voltage in coil 1 was around 0.075 vol  
Figure 8:4 Comparison of displacement and velocity of (**a**) 1 st floating magnet and (**b**) 2 nd floating magnet. Figure 8:5 Induced voltages in (**a**) 1 st winding coil and (**b**) 2 nd winding coil. highest displacement and velocity compared to the first floating magnet for this applied harmonic, as shown in Figure 8:6 Thus, the induced voltage of coil 1 was higher than that of coil 2. The maximum induced voltage in coil 2 was around 0.068 volts and in coil 1 was 0.22 volts, as can be seen in Figure 8:7  
Figures 22 and 23 show the displacements and velocities of the first and second floating magnets when the force was applied on the first floating magnet and second floating magnet separately and both floating magnets simultaneously. Overall, the first floating magnet showed the minimum displacement and velocity during all different applications compared  
Figure 8:8 Calculated induced voltages in (**a**) 1 st winding coil and (**b**) 2 nd winding coil. Figure 8:9 Comparison of displacement and velocity of (**a**) 1 st floating magnet and (**b**) 2 nd floating magnet. to the second floating magnet. Therefore, the first winding coil produced less induced voltage than the second.