# A Vibrational Energy Harvesting Sensor Based on Linear and Rotational Electromechanical Effects
## 3 Magnetically Coupled Double-Spring Design
\begin{array}{c}0\\ \frac{1}{m_{1}}\Big{[}(k_{11}l_{11}-k_{12}l_{12})\sin\theta_{1}+\frac{3\mu_{ 0}\ V^{2}\big{(}2M_{\text{J}}^{2}-M_{\text{J}}^{2}\big{)}}{4\pi\ |x_{1}-x_{2}|^{2}}(x_{1}-x_{2})-\frac{l\ N\ B_{12}}{2\pi}i\Big{]}\\ 0\\ \frac{1}{m_{2}}\Big{[}(k_{21}l_{21}-k_{22}l_{22})\sin\theta_{2}-\frac{3\mu_{ 0}\ V^{2}\big{(}2M_{\text{J}}^{2}-M_{\text{J}}^{2}\big{)}}{4\pi\ |x_{1}-x_{2}|^{2}}(x_{1}-x_{2})+\frac{l\ N\ B_{12}}{2\pi}i\Big{]}\\ 0\\ \frac{1}{l_{1}}\big{[}(k_{11}l_{11}-k_{12}l_{12})x_{1}\cos\theta_{1}-(k_{11}l _{11}^{2}+k_{12}l_{12}^{2})\cos\theta_{1}\sin\theta_{1}\big{]}\\ \frac{1}{l_{2}}\big{[}(k_{21}l_{21}-k_{22}l_{22})x_{2}\cos\theta_{2}-(k_{21}l _{21}^{2}+k_{22}l_{22}^{2})\cos\theta_{2}\sin\theta_{2}\big{]}\\ \frac{2\ N\ B_{12}}{L_{\text{c}}}(\dot{x}_{2}-\dot{x}_{1})\end{array} \right] \tag{21}\]  
and  
\[\vec{u}(t)=\left[ \begin{array}{c}0\\ 1\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \right]\vec{y}(t). \tag{22}\]  
Note, as derived in Appendix B,  
\[B_{12} = \frac{\mu_{0}\ V}{4\pi}\ \sqrt{\frac{4M_{\text{J}}^{2}+M_{\text{J}}^{ 2}}{\left(x_{1}-x_{2}\right)^{6}}}, \tag{23}\]Following [PERSON]'s law,  
\[F=-kx, \tag{24}\]  
one may identify the linear spring constant and the nonlinear spring function within Equations (20) and (21),  
\[F_{1}=-\ \underbrace{(k_{11}+k_{12})}_{\text{Linear spring constant}}\ x_{1}+ \underbrace{\frac{3\mu_{0}\ V^{2}\Big{(}2M_{x}^{2}-M_{y}^{2}\Big{)}}{4\pi\left| x_{1}-x_{2}\right|^{5}}}_{\text{Nonlinear spring function}}(x_{1}-x_{2}), \tag{25}\]  
with a similar expression for \(x_{2}\). The highly nonlinear nature of the spring function results in a system which is sensitive to driving functions which are orders of magnitude less than the acceleration due to gravity, \(g\). The driving acceleration used in the current model is of the order \(10^{-2}\) g [22]. To understand the nonlinear nature of the system, consider an order of magnitude analysis of Equation (25):  
\[x_{n} \sim 10^{-5}\text{m}, \tag{26}\] \[k_{n,1}+k_{n,2} \sim 10^{2}\text{N}\,\text{m}^{-1},\] (27) \[\mu_{0} \sim 10^{-6}\text{H}\,\text{m}^{-1},\] (28) \[V \sim 10^{-8}\text{m}^{3},\] (29) \[\Big{(}2M_{x}^{2}-M_{y}^{2}\Big{)} \sim 10^{9}\text{A}\,\text{m}^{-1},\] (30) \[\left|x_{1}-x_{2}\right| \sim 10^{-5}\text{m}. \tag{31}\]  
So, the force order of magnitude is  
\[F \sim \left(\underbrace{10^{-3}}_{\text{Linear}}\pm\underbrace{10^{7}} _{\text{Nonlinear}}\right)\text{N}. \tag{32}\]  
The conclusion is the nonlinearity is ten times stronger than the linear components. Static ConditionThe system of equations described by Equation (19) through (22) represent a boundary value problem which is defined for all \(t>0\). However, the initial conditions are based on an equilibrium condition between the static forces from the magnetic fields and the forces due to the static deflection of the supporting springs. Hence, when there is zero input, there must be zero output. Thus, the state-space model of Equation (19) must be identically zero for all time when \(\mathbf{u}(t)\equiv 0\):  
\[F=-kx, \tag{24}\]0  
This is the static condition. Consider holding two magnets, one in each hand, orienting them with identical poles facing each other and bringing one's hands together. The force of the same pole arrangement will push the hands apart in distance and angle. The same occurs in the static case: The springs will displace vertically and rotate to avoid alignment of the magnets. It is not possible to solve analytically for the static physical values of \(\mathbf{x}(0)\), particularly the initial displacements and angles, because of the nonlinear nature of the system. Rather, the static values are determined by minimizing,  
\[F=-kx, \tag{24}\]1over \(\vec{x}(0)\), where the linear and rotational velocities were constrained to be zero. The static values are listed in Table 1 and \[F=-kx, \tag{24}\]2  
It is suspected there are multiple local minima in the static condition objective function. Certain combinations of the known and guessed parameters of Table 1 resulted in static angles in excess of tens of degrees.