# A Hybrid Vibration Isolator Based on Elastomeric and Electromagnetic Restoring Force
## 6 Results and Discussions
### Transmissibility  
The theoretical model, including the parameters identified through laboratory tests, allows us to define the transmissibility of the system, expressed as the ratio between the amplitude \(X_{m}\) of the displacement of the body and the amplitude \(X_{b}\) of the displacement of the base (ground):  
\[T=\frac{X_{m}}{X_{b}} \tag{28}\]  
Indicating the relative displacement of the mass with \(x=x_{\text{m}}-x_{b}\) with respect to the base (Figure 19), the equation of motion of the isolated body is  
\[m\ddot{x}_{m}+\sigma\dot{x}+kx+fWsign\big{(}\dot{x}\big{)}+F_{M}=0 \tag{29}\]  
where \(m\) is the mass of the body.  
Figure 17: Force isovalues (N) map for the upper coil current of 5 A and the bottom coils’ currents in the range \(-3\) A to 3 A.  
Figure 18: Force value for the bottom coil current of 3 A and the upper coil’s current in the range from \(-5\) A to 5 A.  
Equation (29) contains the reaction of the elastomeric pad, defined in (2); the rolling resistance of the BTUs, defined in (1); and the magnetic force defined in (17) and (18), which can be expressed as  
\[F_{M}=\frac{1}{2}i_{0}^{2}(2a_{0,d}x+b_{0,d})+\frac{1}{2}i_{3}^{2}(2a_{3,d}x+b_{ 3,d})+i_{0}i_{3}b_{m,d} \tag{30}\]  
Table 2 reports the coefficients in Equation (30).  
By forcing the base with a harmonic motion of constant amplitude and variable frequency, it is possible to obtain the ratio defined in (29) as a function of the excitation frequency. This evaluation can be carried out by powering the three coils in different ways. Figure 20 reports several curves, each obtained by considering that the lower coils (C1 and C2) are powered in the same way and keeping the power supply of the coils constant. The blue curve was obtained without feeding the coils. By increasing the current intensity, the peak of the curves assumes a higher value and is reached for higher values of the forcing frequency as the magnetic force determines a stiffening effect of the insulation system.  
If the intensity of the current is controlled with the aim of keeping the body at rest in its initial position by properly defining the maximum values of the power supply, the curves no longer show areas of amplification of the transmissibility. In this case, the magnetic force acts in contrast to the elastic force, making the isolation system less stiff. Figure 21 shows the transmissibility curves obtained with a PID control for several values of the maximum intensity current; by increasing it, the transmissibility curve assumes a descending trend. In this condition, the trend of the amplitude displacement \(\mathrm{x_{b}}\) is imposed on the base (with constant amplitude and increasing frequency) and the amplitude displacement \(\mathrm{x_{m}}\) of the isolated body, which does not show the time intervals during which the amplitude of the motion is amplified with respect to that of the base (Figure 22).  
Figure 19: Dynamic scheme.  
Figure 20: Transmissibility with a constant current feeding the coils.  
Compared to other controlled isolators, the proposed one allows higher relative displacements with transmissibility lower than one. By way of example, it should be noted that the isolator presented in [18] was tested for relative displacements of less than 1 mm, while the proposed one was tested for relative displacements of 20 mm.