# Two-Degrees-of-Freedom (2 DOF) Magnetic Spring-Based Electromagnetic Energy Harvester
## 5 Mathematical Model of the 2 DOF Energy Harvester  
When an external force is applied to a floating magnet or any floating magnet moves up and down, it creates an elastic restoring force in the magnetic spring. Figure 6 presents the free-body diagram of the SDOF magnetic spring-based energy harvester system. The relative displacement of the first floating magnet is \(y_{2}\) and the relative velocity and acceleration of the first floating magnet are \(\dot{y}_{2}\) and \(\ddot{y}_{2}\), respectively. Moreover, the relative displacement, velocity, and acceleration of the second floating magnet are \(y_{3}\), \(\dot{y}_{3}\), and \(\ddot{y}_{3}\), respectively. The magnetic flux density of the second magnet (first floating magnet) is \(B_{1(y)}\) and the total length of the first winding coil is \(l_{1}\). Similarly, the magnetic flux density of the third magnet (second floating magnet) is \(B_{2(y)}\) and the total length of the second winding coil is \(l_{2}\). The electromagnetic coupling coefficients are \(a_{1}\) (\(a_{1}=B_{1(y)}l_{1}\)) and \(a_{2}\) (\(a_{2}=B_{2(y)}l_{2}\)). Moreover, \(F_{e1}\) and \(F_{e2}\) are the electromagnetic forces written as \(F_{e1}=a_{1}I_{1}\) and \(F_{e2}=a_{2}I_{2}\), respectively. Figure 5: (**a**) Magnetic restoring force of the 2 nd floating magnet. (**b**) Relative errors of the fitting curves versus the analytical force. The masses of the second (first floating magnet) and third (second floating magnet) magnets are M\({}_{2}\) and M\({}_{3}\), respectively. The damping force of the first floating magnet is \(F_{\beta 1}=\beta_{1}\dot{y}_{2}\) and that of the second floating magnet is \(F_{\beta 3}=\beta_{3}\dot{y}_{3}\). In Figure 6, the linear stiffnesses are represented by \(k_{21}\), \(k_{23}\), \(k_{32}\), and \(k_{34}\). The damping constants are represented by \(\beta_{21}\), \(\beta_{23}\), \(\beta_{32}\), and \(\beta_{34}\). The nonlinear coefficients are presented by \(\alpha_{21}\), \(\alpha_{23}\), \(\alpha_{32}\), and \(\alpha_{34}\). In addition, the other nonlinear stiffnesses are \(\lambda_{21}\), \(\lambda_{23}\), \(\lambda_{32}\), and \(\lambda_{34}\). In Figure 7, the linear stiffnesses are \(k_{21}\) = \(k_{1}\), \(k_{23}\) = \(k_{2}\), and \(k_{34}\) = \(k_{3}\). The damping constants \(\beta_{21}=\beta_{1}\), \(\beta_{23}=\beta_{32}\) = \(\beta_{2}\), and \(\beta_{34}=\beta_{3}\). The nonlinear coefficient \(\alpha_{21}\) is equal to \(\alpha_{1}\). Moreover, \(\alpha_{23}\) and \(\alpha_{32}\) are equal to \(\alpha_{2}\), and \(\alpha_{34}\) is equal to \(\alpha_{3}\). In addition, the other nonlinear stiffnesses are \(\lambda_{21}\) = \(\lambda_{1}\); \(\lambda_{23}\) = \(\lambda_{32}\) = \(\lambda_{2}\); and \(\lambda_{34}\) = \(\lambda_{3}\). The dynamic equation of the motion of the system can be written as  
\[ \begin{split} M_{2}\ddot{y_{2}}+\beta_{1}\dot{y_{2}}-\beta_{2}& \big{(}y_{3}-\dot{y_{2}}\big{)}+k_{1}y_{2}-k_{2}(y_{3}-y_{2})+ \alpha_{1}y_{2}{}^{2}-\alpha_{2}(y_{3}-y_{2})^{2}+\lambda_{1}y_{2}{}^{3}- \lambda_{2}(y_{3}-y_{2})^{3}+B_{1}l_{1}\frac{V_{1}}{R_{1}}\\ &=F_{1}Sin(\omega t)\end{split}  \tag{10}\]  
\[ \begin{split} M_{3}\ddot{y_{3}}+\beta_{3}\dot{y_{3}}+\beta_{2}& \big{(}y_{3}-\dot{y_{2}}\big{)}+k_{3}y_{3}+k_{2}(y_{3}-y_{2})+ \alpha_{3}y_{3}{}^{2}+\alpha_{2}(y_{3}-y_{2})^{2}+\lambda_{3}y_{3}{}^{3}+ \lambda_{2}(y_{3}-y_{2})^{3}+B_{2}l_{2}\frac{V_{2}}{R_{2}}\\ &=F_{2}Sin(\omega t)\end{split}  \tag{11}\]  
\[V_{1}+\frac{L_{1}}{R_{1}}\dot{V}_{1}=B_{1}l_{1}\dot{y}_{2} \tag{12}\]  
\[V_{2}+\frac{L_{2}}{R_{2}}\dot{V}_{2}=B_{2}l_{2}\dot{y}_{3} \tag{13}\]  
Figure 6: Two-degrees-of-freedom magnetic spring-based energy harvester system. Equation (10) can be expressed as  
\[