# A Hybrid Vibration Isolator Based on Elastomeric and Electromagnetic Restoring Force
## 6 Results and Discussions
### Identification of the Electrical Parameters  
As previously mentioned, the model can be characterized using many approaches. In this paper, the indirect method reported in the previous chapter was used. The derivative of the self-inductances in the upper and lower coils are evaluated at the rated current with the aim of detecting a possible effect of magnetic saturation. Figures 9 and 10 show the results obtained for \(dL_{0}/dx\) and \(dL_{3}/dx\). In both cases, the obtained values are fitted with a first-order polynomial according to the relations (19). Using these results, it is possible to determine the value of the derivative of the mutual inductance between the coils, as reported in (27). The expected trend of the derivatives should not depend on the relative displacement between the plates, but due to the constructive and assembly tolerance, there is an unavoidable variable misalignment of the axes of the coils that causes the trend depicted in Figure 11 (blue line), fitted with a horizontal line (red line). Figure 8: (a) Elastomeric pads; (b) force–displacement cycle. The discrepancy between the theoretical and the experimental trend has been attributed to the imperfect parallelism between the axes of the coils, whose arrangement is also variable due to the slight deformability of the coil support. This effect has been theoretically confirmed by means of a finite element model in which the axes of the coils have been defined as not parallel to each other. This result leads to the need for an accurate positioning of the coils on a sufficiently rigid support. Therefore, although in Figure 9 to Figure 11 the experimental results assume a different trend from the theoretical ones, the proposed approach can be considered reliable; in fact, as shown below, the simulations performed considering the numerical trend fit produce results close to the experimental ones. Figure 10: Values of \(dL_{3}/dx\) (blue line) and the numerical fit (red line) with a current of 3 A in the upper coil. Figure 9: Values of \(dL_{0}/dx\) (blue line) and the numerical fit (red line) with a current of 3A in the lower coils. The following coefficient values were identified (Table 2) for the test rig:  
The variable trend of \(dM_{03}/dx\) was attributed to the mechanical misalignments after a series of sensitivity tests were performed using FEM software. By shifting the axis of the bottom coils of 1 and 5 mm, with respect to the center of the isolator, the curves reported in Figure 12 were obtained. It can be noted that the depicted trends have a parabolic trend very similar to the experimental ones. The misalignment is a common problem in real applications, but, as demonstrated in the following paragraph, the error introduced in the model has negligible effects. The electromechanical behavior of the insulator was tested by adjusting the supply voltages of the coils to obtain the desired current value. A comparison between the magnetic forces obtained with the proposed model and those obtained experimentally is reported below. The tests were performed with a current (\(\mathbf{i}_{0}\)) in the lower coil of 3A and a variable current (\(\mathbf{i}_{3}\)) in the upper coil and by imposing a relative harmonic motion with an amplitude of 10.5 mm at a frequency of 0.25 Hz. The comparison is reported as a function of both time (Figure 13) and displacement (Figure 14). Both diagrams show that the proposed model provides a force trend in good agreement with that obtained experimentally, with an approximation that is similar to that found in papers by other researchers [18, 19]. Figure 11: Values of \(dM_{03}/dx\) (blue line) and the numerical fit (red line) with a current of 3A in the coils. \begin{table}
\begin{tabular}{c c} \hline \(a_{0,d}\) & \(-382\) H/m\({}^{2}\) \\ \(b_{0,d}\) & \(-0.449\) H/m \\ \(a_{3,d}\) & \(-333\) H/m\({}^{2}\) \\ \(b_{3,d}\) & \(-0.757\) H/m \\ \(b_{m,d}\) & \(-9.35\) H/m \\ \hline \end{tabular}
\end{table} 
Table 2: Values of the coefficients obtained with the fitting of the trend of the derivatives.