# A Hybrid Vibration Isolator Based on Elastomeric and Electromagnetic Restoring Force
## 3 The Isolator Restoring Force
### The Magnetic Restoring Force
The input energy is the difference between the total energy absorbed by the system and the Joule losses in the coils: \[dW_{e}=i_{1}d(L_{1}i_{1}+M_{12}i_{2}+M_{1,3}i_{3})+i_{2}d(M_{12}i_{1}+L_{2}i_{ 2}+M_{2,3}i_{3})+i_{3}d(M_{1,3}i_{1}+M_{2,3}i_{2}+L_{3}i_{3})\] (9) 2. The magnetic energy is the sum of all the magnetic energy contributions due to the self and mutual inductances: \[W_{m}=\frac{1}{2}L_{1}i_{1}^{2}+\frac{1}{2}L_{2}i_{2}^{2}+\frac{1}{2}L_{3}i_{ 3}^{2}+M_{12}i_{1}i_{2}+M_{1,3}i_{1}i_{3}+M_{2,3}i_{2}i_{3}\] (10) The electromagnetic force \(F_{M}\) can be deduced from Equation (3) after differentiating Equation (10): \[F_{M}=\frac{1}{2}i_{1}^{2}\frac{dL_{1}}{dx}+\frac{1}{2}i_{2}^{2}\frac{dL_{2}} {dx}+\frac{1}{2}i_{3}^{2}\frac{dL_{3}}{dx}+i_{1}i_{2}\frac{dM_{12}}{dx}i_{3} \frac{dM_{13}}{dx}+i_{2}i_{3}\frac{dM_{23}}{dx}\] (11) If the two lower coils are powered in series, the electrical equilibrium equations take the following form: \[\left\{ \begin{aligned} & v_{0}+e_{0}=R_{0}i_{0}\\ & v_{3}+e_{3}=R_{3}i_{3}\end{aligned} \right.\] (12)where the subscript 0 indicates the quantities relating to the coils C\({}_{1}\) and C\({}_{2}\) powered in series. Under the same hypotheses highlighted at the beginning of this section and with \(M_{03}\) as the mutual inductance between the upper and lower coils, the back electromotive forces can now be expressed as  
\[ \begin{cases}e_{0}=-\frac{d}{dt}[L_{0}(x)i_{0}+M_{03}(x)i_{3}]\\ e_{3}=-\frac{d}{dt}[L_{3}(x)i_{3}+M_{03}(x)i_{0}]\end{cases}  \tag{13}\]  
The voltage equations become  
\[ \begin{cases}v_{0}=R_{0}i_{0}+L_{0}\frac{d{i_{0}}}{dt}+i_{0}\frac{d{L_{0}}}{dx} \frac{dx}{dt}+i_{3}\frac{d{M_{0}}}{dx}\frac{dx}{dt}+M_{03}\frac{d{i_{3}}}{dt}\\ v_{3}=R_{3}i_{3}+L_{3}\frac{d{i_{3}}}{dt}+i_{3}\frac{d{x}}{dx}\frac{dx}{dt}+i_ {0}\frac{d{M_{0}}}{dx}\frac{dx}{dt}+M_{03}\frac{d{i_{0}}}{dt}\end{cases}  \tag{14}\]  
where  
\[R_{0}=R_{1}+R_{2}=2R \tag{15}\]  
\[L_{0}(x)=L_{1}+L_{2}-2M_{12} \tag{16}\]  
The way to connect the coils and the resultant flux path determine the sign of the mutual inductance \(M_{03}\) and the value of series self-inductance \(L_{0}\). Finally, the magnetic force expression is  
\[F_{M}=\frac{1}{2}\hat{q}\frac{dL_{0}}{dx}+\frac{1}{2}\hat{q}\frac{dL_{3}}{dx} +i_{0}i_{3}\frac{d{M_{0}}_{3}}{dx} \tag{17}\]  
The system of time-variant differential Equation (14) is usually difficult to solve analytically, but an accurate approximating solution can be found considering a periodic relative displacement. In addition to the analytical simplification, this assumption makes it possible to compare the results of the analytical model with the experimental ones in a simple way. In this case, the approximate solution can be expressed as the sum of a steady-state solution in a Fourier series expansion [24] and a unidirectional component as follows  
\[ \begin{cases}i_{0}=A_{0}\,e^{-\frac{t}{t_{0}}}+\left[I_{0}+\sum\limits_{n=1} ^{k}I_{0,\alpha,n}\text{cos}\,n\omega t+\sum\limits_{m=1}^{k}I_{0,b,m}\text{ sin}\,m\omega t\right]\\ i_{3}=A_{3}\,e^{-\frac{t}{t_{3}}}+\left[I_{3}+\sum\limits_{n=1}^{k}I_{3,a,n} \text{cos}\,n\omega t+\sum\limits_{m=1}^{k}I_{3,b,m}\text{sin}\,m\omega t \right]\end{cases}  \tag{18}\]  
where \(A_{0}\) and \(A_{3}\) are the amplitude of the unidirectional components, \(\tau_{0}\) and \(\tau_{3}\) are the time constants related to the coils, \(\omega\) is the fundamental angular frequency of the displacement, \(n\) is the harmonic index, and \(k\) is the order of harmonics that must be considered for the approximation of the currents. The values \(A_{0}\) and \(A_{3}\) are calculated by setting the initial conditions of the currents, while the time constants are derived from those of a resistive-inductive load. It is important to note that the dependency between the electrical parameter values (e.g., the self-inductance) and the displacement also introduces a relation between the time constant value and the position of the coils. The use of an average value of self-inductance allows the error of the approximate solution considered to be reduced.