## 2 Numerical Formulation
Coupling between electrical and mechanical equations is achieved by the terms \(\upsilon_{k}(t)\times\mathbf{B}_{k}(t)\) that, once integrated on the elementary volumes, are assembled to form the matrix of the motional terms \(\mathbf{K}\), and by the terms \(\mathbf{j}_{k}(t)\times\mathbf{B}_{k}(t)\) which are integrated to provide the forces and the torques on the moving parts of the device. The mechanical equation for the armature of the launcher is:  
\[\mathbf{m}\ddot{\mathbf{x}}_{G}=\mathbf{F}(t,\mathbf{C}(t)) \tag{2}\]  
while the equation for the compulsator are:  
\[\mathbf{I}_{G}\dot{\omega}+\dot{\omega}\dot{\mathbf{I}}_{G}\omega=M_{G}(i,\mathbf{C}(t)) \tag{3}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\mathbf{I}_{k}\) & Current density in the \(k\)-th conductive element \\ \(\mathbf{E}_{k}\) & Electric field in the \(k\)-th conductive element \\ \(V_{k}\) & Electric potential in the \(k\)-th conductive element \\ \(\mathbf{A}_{k}\) & Vector potential in the \(k\)-th conductive element \\ \(\mathbf{B}_{k}\) & Flux density in the \(k\)-th conductive element \\ \(\mathbf{\upsilon}_{k}\) & Velocity of the \(k\)-th conductive element \\ \(i_{k}\) & Current in the \(k\)-th conductive element \\ \(u_{k}\) & Voltage drop across the \(k\)-th conductive element \\ \(L_{k,j}\) & Mutual induction coeff. between conductive elem. \\ \(K_{k,j}\) & Motional voltage coeff. between conductive elem. \\ \(R_{k}\) & Resistance of the \(k\)-th conductive element \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. Figure 1: Conceptual flow chart of the numerical formulation. (Reproduced with permission from [30], IEEE, 2017). In the above equations \(F\) represents the resultant force on the launcher armature and \(m\) is its mass, \(M_{G}\) is the resultant torque on the rotor of the compulsator and \(I_{G}\) is its inertia tensor. The coupled differential equations for electrical and mechanical equilibrium are time varying, and the resulting system is nonlinear; integration is carried out as described in [27; 28]. It is worth to mention that the currents in the branches of the equivalent network are not fictitious currents. The current in a branch of the equivalent network is in correspondence with one component of the current density of an elementary volumes as shown by the insets in the upper right and in the lower left corners in Figure 1. By the above described equivalent network the distribution of all the relevant electromechanical quantities can be evaluated everywhere in the whole system. In particular, the expression of the force by the term \(j_{k}(t)\times B_{k}(t)\) allows to evaluate its distribution in each elementary volumes and to determine the contribution to the total dynamical action in materials that are usually heavily stressed when used in EML technology. Simultaneous analysis of interacting devices simply requires building a bigger equivalent network constituted by the properly connected equivalent networks of the components. This implements a \"strong interaction\" between devices for which a \"strong coupled\" analysis is performed. If the devices can be assumed magnetically uncoupled (i.e., leakage magnetic fluxes from a device that links the others are negligible) independent networks are built, linked together at the common terminals and simultaneously solved. It is worth to remark that the topology of the resultant network is the same than that of the more general case of magnetically coupled devices. The only difference lies in the filling of the inductance matrix \(L\) and of the motional terms matrix \(K\), which are more densely populated in the latter case. Coupling of the equivalent network so far described with external circuits is straightforward and this will be performed in the next section to investigate the \"weak interaction\" between the rail launcher and the compulsator. The presence of sliding contacts has been taken into account by introducing an auxiliary network [20].