# A Hybrid Vibration Isolator Based on Elastomeric and Electromagnetic Restoring Force
## 4 Analytical Solution of the Electrical Model  
The values and the variations in the mutual inductances are strictly dependent on the direction of the flux generated by the coils; in fact, the different manner of connecting and feeding the coils determines a variation in the sign and in the trend of mutual inductances and, according to (16), also a variation in self-inductance values. Considering the direction of magnetic flux density that maximizes the damping effect of the isolator, the characterization of the model requires the calculation of the electrical parameters, which are dependent on the relative position of the upper and bottom coils. In particular, the maximization of force values is obtainable using a connection of the bottom coils able to create a discordant magnetic flux density between the two coils, obtaining the possibility of describing the trend of self and mutual inductances using the following relations:  
\[\left\{ \begin{array}{l}L_{0}=a_{0,d}x^{2}+b_{0,d}x+c_{0,d}\\ L_{3}=a_{3,d}x^{2}+b_{3,d}x+c_{3,d}\\ M_{03}=b_{m,d}x+c_{m,d}\end{array} \right. \tag{19}\]  
where the coefficients \(a_{0,\text{d}}\), \(a_{3,\text{d}}\), \(b_{0,\text{d}}\), \(b_{3,\text{d}}\), \(b_{\text{m,d}}\), \(c_{\text{m,d}}\), \(c_{0,\text{d}}\), \(c_{3,\text{d}}\), can be obtained through numerical fitting. The expressions show that the minimum values of the inductances happen when the upper coil is positioned at the center between the two bottom coils, while the maximum values (in terms of absolute values) are located when the axis of the upper coil overlaps with one of the bottom coils. The knowledge of the self and mutual inductance trends allows us to solve the time-varying model described in (14) and allows us to know the magnetic force (17). Using a numerical approach, it is possible to obtain with good accuracy the value of the magnetic force. However, when a real-time control is needed, the numerical methods are not suitable for an easy implementation, and it is preferable to obtain a closed-form solution of quantities that can be implemented directly in the control. As previously mentioned, the relations (18) give the expression of the currents for the model with time-varying parameters, and using some simple hypotheses, their values can be easily determined by solving a linear algebraic system. Considering a relative harmonic displacement between the isolator plates \(x=Asin(\omega t)\), the relationship (19) becomes a periodic function, and the system (14) can be written as  
\[\left\{ \begin{array}{l}v_{0}=[R_{0}+H_{1}(\omega,t)]i_{0}+H_{2}(\omega,t)i_{ 3}+H_{3}(\omega,t)\frac{di_{0}}{dt}+H_{4}(\omega,t)\frac{di_{3}}{dt}\\ v_{3}=[R_{3}+H_{5}(\omega,t)]i_{3}+H_{2}(\omega,t)i_{0}+H_{6}(\omega,t)\frac{di _{3}}{dt}+H_{4}(\omega,t)\frac{di_{0}}{dt}\end{array} \right. \tag{20}\]  
System (20) can be solved with a numerical approach, based, for example, on the use of Moore-Penrose inverse or using the Fourier series approach. The latter approach is adopted in this paper because the method allows a closed-form solution to be obtained, which is useful for control applications. Approximating the currents with the Fourier series, as reported in Equation (18), and supposing two DC supply voltages \(V_{0}\) and \(V_{3}\), it is possible to collect the constant terms with respect to the time-dependent ones, obtaining the system of Equation (21), where the expressions of terms H\({}_{1}\)(\(\omega\),\(t\)), \(\ldots\), H\({}_{6}\)(\(\omega\),\(t\)) and \(\Lambda_{1}\)(\(\omega\),\(t\)), \(\ldots\), \(\Lambda_{4}\) (\(\omega\),\(t\)) are reported in Appendix A:  
\[ \begin{array}{l}0=V_{0}-R_{0}I_{0}\\ 0=V_{3}-R_{3}I_{0}\\ 0=R_{0}\Lambda_{1}(\omega,t)+H_{1}(\omega,t)[I_{0}+\Lambda_{1}(\omega,t)]+H_{2 }(\omega,t)[I_{3}+\Lambda_{3}(\omega,t)]+H_{3}(\omega,t)\Lambda_{2}(\omega,t) +H_{4}(\omega,t)\Lambda_{4}(\omega,t)\\ 0=R_{3}\Lambda_{3}(\omega,t)+H_{5}(\omega,t)[I_{3}+\Lambda_{3}(\omega,t)]+H_{2 }(\omega,t)[I_{0}+\Lambda_{1}(\omega,t)]+H_{6}(\omega,t)\Lambda_{4}(\omega,t) +H_{4}(\omega,t)\Lambda_{2}(\omega,t)\end{array}