# A Hybrid Vibration Isolator Based on Elastomeric and Electromagnetic Restoring Force
## 3 The Isolator Restoring Force
### The Magnetic Restoring Force  
Due to their relative movement (Figure 4), the three coils convert the input electrical energy into mechanical energy, an increment of the energy stored in the electric and magnetic field, and into heat. Below, considering the energy converted into heat negligible and indicating the work done by the magnetic force \(F_{M}\) with with \(F_{M}\,dx\), the energy balance takes the following form:  
\[dW_{e}=F_{M}dx+dW_{m} \tag{3}\]  
where \(dW_{e}\) is the input electric energy and \(dW_{m}\) is the magnetic field energy. As far as the electrical balance equations are concerned, each coil voltage is balanced by the resistive drop added to the induced back electromotive force within the coils themselves:  
\[\left\{ \begin{aligned} & v_{1}+e_{1}=R_{1}i_{1}\\ & v_{2}+e_{2}=R_{2}i_{2}\\ & v_{3}+e_{3}=R_{3}i_{3}\end{aligned} \right. \tag{4}\]  
where \(R_{i}\) and \(i_{i}\) are the resistance and the current of the \(i\)-th coil, respectively. Figure 4: Reference system. The terms \(e_{1},e_{2}\), and \(e_{3}\) refer to the self-inductances of the coils and the mutual inductances relating to the interaction with the other coils. The corresponding analytical expressions are  
\[\left\{ \begin{aligned} & e_{1}=-\frac{d}{dt}[L_{1}(x)i_{1}+M_{12}(x)i_{2 }+M_{13}(x)i_{3}]\\ & e_{2}=-\frac{d}{dt}[L_{2}(x)i_{2}+M_{21}(x)i_{1}+M_{23}(x)i_{3} ]\\ & e_{3}=-\frac{d}{dt}[L_{3}(x)i_{3}+M_{31}(x)i_{1}+M_{32}(x)i_{2 }]\end{aligned} \right. \tag{5}\]  
where \(L_{i}\) is the self-inductance and \(M_{ij}\) is the mutual inductance, which depend on the displacement \(x=x(t)\). Carrying out the derivatives and substituting the expressions in (4), it follows that  
\[\left\{ \begin{aligned} & v_{1}=R_{1}i_{1}+L_{1}\frac{d i_{1}}{dt}+i_{1}\frac{dL_{1}}{dt}+i_{2}\frac{dM_{12}}{dt}+M_{12}\frac{d i_{2}}{dt}+i_{3}\frac{dM_{12}}{dt}+M_{13}\frac{d i_{3}}{dt}\\ & v_{2}=R_{2}i_{2}+L_{2}\frac{d i_{2}}{dt}+i_{2}\frac{dL_{2}}{dt}+i_{1}\frac{dM_{13}}{dt}+M_{21}\frac{d i_{1}}{dt}+i_{3}\frac{dM_{23}}{dt}+M_{23}\frac{d i_{3}}{dt}\\ & v_{3}=R_{3}i_{3}+L_{3}\frac{d i_{3}}{dt}+i_{3}\frac{dL_{3}}{dt}+i_{1} \frac{dM_{31}}{dt}+M_{31}\frac{d i_{1}}{dt}+i_{2}\frac{dM_{23}}{dt}+M_{32}\frac {d i_{2}}{dt}\end{aligned} \right. \tag{6}\]  
Making explicit the dependence on the relative position \(x\), Equation (6) can be written as  
\[\left\{ \begin{aligned} & v_{1}=R_{1}i_{1}+L_{1}\frac{d i_{1}}{dt}+i_{1}\frac{dL_{1}}{dt}+i_{2}\frac{dM_{12}}{dx}\frac{dx}{dt}+M_{12} \frac{d i_{2}}{dt}+i_{3}\frac{dM_{12}}{dx}\frac{dx}{dt}+M_{13}\frac{d i_{3}}{dt}\\ & v_{2}=R_{2}i_{2}+L_{2}\frac{d i_{2}}{dt}+i_{2}\frac{dL_{2}}{dt}+i _{1}\frac{dM_{23}}{dx}\frac{dx}{dt}+M_{22}\frac{d i_{1}}{dt}+i_{3}\frac{dM_{23}}{dx}\frac{dx}{dt}+M_{23}\frac{d i_{3}}{dt}\\ & v_{3}=R_{3}i_{3}+L_{3}\frac{d i_{3}}{dt}+i_{3}\frac{dL_{3}}{dt}+i_{1}\frac{dM_{13}}{dx}\frac{dx}{dt}+M_{31} \frac{d i_{1}}{dt}+i_{2}\frac{dM_{23}}{dx}\frac{dx}{dt}+M_{32}\frac{d i_{2}}{dt}\end{aligned} \right. \tag{7}\]  
According to the theory of the mutual coupling between two or more coils, it is assumed that \(M_{ij}=M_{ji}\). Furthermore, in the analyzed case, all the coil resistances are equal; therefore \(R_{1}=R_{2}=R_{3}=R\) and the relation (7) can be rewritten as  
\[\left\{ \begin{aligned} & v_{1}=Ri_{1}+L_{1}\frac{d i_{1}}{dt}+i_{1}\frac{dL_{1}}{dt}\frac{dx}{dt}+i_{2}\frac{dM_{12}}{dx} \frac{dx}{dt}+M_{12}\frac{d i_{2}}{dt}+i_{3}\frac{dM_{13}}{dx}\frac{dx}{dt}+M_{13 }\frac{d i_{3}}{dt}\\ & v_{2}=Ri_{2}+L_{2}\frac{d i_{2}}{dt}+i_{2}\frac{dL_{2}}{dt}\frac{dx}{dt}+i_{1}\frac{dM_{12}}{dx} \frac{dx}{dt}+M_{12}\frac{d i_{1}}{dt}+i_{3}\frac{dM_{23}}{dx}\frac{dx}{dt}+M_{23}\frac{d i_{3}}{dt}\\ & v_{3}=Ri_{3}+L_{3}\frac{d i_{3}}{dt}+i_{3}\frac{dL_{3}}{dx} \frac{dx}{dt}+i_{1}\frac{dM_{13}}{dx}\frac{dx}{dt}+M_{13}\frac{d i_{1}}{dt}+i_{2}\frac{dM_{23}}{dt}\frac{dx}{dt}+M_{23}\frac{d i_{2}}{dt}\end{aligned} \right. \tag{8}\]  
Consequently, Equation (3) may be written considering that  
1.