# A Hybrid Vibration Isolator Based on Elastomeric and Electromagnetic Restoring Force
## 4 Analytical Solution of the Electrical Model
 \tag{21}\]  
For the considered isolator, a good agreement is achieved when considering only the first two harmonics of the currents. In this case, the system (21) can be written as system (22). \[\left\{ \begin{array}{l}0=V_{0}-R_{0}I_{0};\\ 0=V_{3}-R_{3}I_{0};\\ \\ 0=R_{0}\left[I_{0,a,1}\cos\omega t+I_{0,b,1}\sin\omega t+I_{0,a,2}\cos 2\omega t+I_{0, b,2}\sin 2\omega t\right]+\ldots\\ \ldots+H_{1}(\omega,t)[I_{0}+I_{0,a,1}\cos\omega t+I_{0,b,1}\sin\omega t+I_{0, a,2}\cos 2\omega t+I_{0,b,2}\sin 2\omega t]+\ldots\\ \ldots+H_{2}(\omega,t)[I_{3}+I_{3,a,1}\cos\omega t+I_{3,b,1}\sin\omega t+I_{3, a,2}\cos\omega t+I_{3,b,2}\sin\omega t]+\ldots\\ \ldots+H_{3}(\omega,t)[-\omega I_{0,a,1}\sin\omega t+\omega I_{0,b,1}\cos \omega t-2\omega I_{0,a,2}\sin 2\omega t+2\omega I_{0,b,2}\cos 2\omega t]+\ldots\\ \ldots+H_{4}(\omega,t)[-\omega I_{3,a,1}\sin\omega t+\omega I_{3,b,1}\cos \omega t-2\omega I_{3,a,2}\sin 2\omega t+2\omega I_{3,b,2}\cos 2\omega t];\\ \\ 0=R_{3}\left[I_{3,a,1}\cos\omega t+I_{3,b,1}\sin\omega t+I_{3,a,2}\cos 2 \omega t+I_{3,b,2}\sin 2\omega t\right]+\ldots\\ \ldots+H_{5}(\omega,t)[I_{3}+I_{3,a,1}\cos\omega t+I_{3,b,1}\sin\omega t+I_{3, a,2}\cos 2\omega t+I_{3,b,2}\sin 2\omega t]+\ldots\\ \ldots+H_{2}(\omega,t)[I_{0}+I_{0,a,1}\cos\omega t+I_{0,b,1}\sin\omega t+I_{0,a,2}\cos 2\omega t+I_{0,b,2}\sin 2\omega t]+\ldots\\ \ldots+H_{6}(\omega,t)[-\omega I_{3,a,1}\sin\omega t+\omega I_{3,b,1}\cos \omega t-2\omega I_{3,a,2}\sin 2\omega t+2\omega I_{3,b,2}\cos 2\omega t]+\ldots\\ \ldots+H_{4}(\omega,t)[-\omega I_{0,a,1}\sin\omega t+\omega I_{0,b,1}\cos \omega t-2\omega I_{0,a,2}\sin 2\omega t+2\omega I_{0,b,2}\cos 2\omega t]\\ \end{array} \right. \tag{22}\]  
The expansion of (22) leads us to obtain all the terms in sine and cosine until the fourth harmonic; by collecting the terms with the same harmonic in the system, it is possible to separate all the terms that obtain an overdetermined system. To use an accurate closed-form solution instead of a least square method, it is possible to neglect the equations related to the third and fourth harmonics because their energetic contribution, in the currents, is very low. The final determined algebraic system, reported in (23), is composed of 10 equations with 10 unknown variables, which correspond to the mean values of currents and the amplitudes of Fourier series coefficients of the first and second harmonic. The expression of terms \(\gamma_{\rm i}\) and \(\delta_{\rm i}\) are reported in Appendix B.  
\[\left[\begin{array}{l}V_  
which are sufficient for the implementation of the model represented in (17)-(23). Starting from Equation (17), the derivatives of self-inductances are evaluated as  
\[\left.\frac{dL_{0}}{dx}\right|_{i_{3}=0}=\frac{2F_{M,0}}{i_{0}^{2}} \tag{24}\]  
\[\left.\frac{dL_{3}}{dx}\right|_{i_{0}=0}=\frac{2F_{M,3}}{i_{3}^{2}} \tag{25}\]  
using two different experimental tests carried out by cancelling the current in the upper (24) and lower coils (25), respectively and then applying a harmonic motion and calculating the magnetic forces \(F_{M,0}\) and \(F_{M,3}\) (evaluated as the difference between the total measured force and the friction force). The derivative of mutual inductance is calculated by means of the third component of the expression (17), which can be obtained as the difference between the total magnetic force obtained by the measurements of experimental tests where all the coils are fed and by the magnetic force due to the self-inductances of the coils, as reported in the following expression:  
\[\frac{dM_{03}}{dx}=\frac{F_{M}-F_{M,0}-F_{M,3}}{i_{0}i_{3}} \tag{26}\]