## 4 Strong Interaction Analysis
### Results  
By using the proposed numerical formulation we are able to obtain the simultaneous evolution of the both the electrical and mechanical dynamics of the compulsator and of the rail launcher. The results of the strong interaction analysis are compared with those of a weak interaction where the launcher is substituted by a lumped equivalent network whose topology is shown in Figure 7 and parameters are characterized by the simplified expressions discussed in Section 3. These parameters have been evaluated by running EN4 EM on a model of the rail launcher characterized by a very coarse discretization which subdivides the rails along the direction of the motion only; this implies that the current is uniformly distributed in the cross section of the rails. As far the discretization of the armature we assumed that the current was concentrated in its most backward quarter (i.e., in the width \(\Delta z=3.5\leavevmode\
obreak\ mm\)). We choose this value analyzing the current distribution (affected by the VES) in the armature of the standing alone rail launcher as described in Section 3 in the range of the speeds obtained by the strong interaction analysis. Figure 11 shows the current delivered to the railgun. As known, the pulse shape can be adjusted by properly varying the angular position and the extension of the shield. It is important that the zero crossing of the current waveform occurs at the end of the launch, i.e., when the armature exits the launcher. This allows obtaining an increased efficiency in the electromechanical conversion and at the same time to avoid arcing between the armature and the muzzle. In fact, if, at the end of the launch, the current in the system is zero also the magnetic energy stored at the same instant is zero, this means that all the energy delivered from the generators t is converted in kinetic energy of the armature (plus, of course, the energy losses in the resistance). Residual energy stored in the magnetic fields of the system (i.e., in the launcher and in the compulsator) is lost in the arch at the muzzle of the launcher. The weak interaction analysis predicts a greater current delivered by the compulsator, which in turn produces greater thrust force and speed; the acceleration time is reduced and the zero crossing of the current changes accordingly. As observed, the accurate prediction of the zero crossing allows improving the performance of the system. The thrust force waveforms on the armature by the two analyses are shown in Figure 12. The ripple superimposed to the thrust force profile predicted by the strong iteration analysis is an artifact due to the commutation at discrete times of the branches of auxiliary network used for the sliding contacts modelling. It is worth noting that the currents do not present this ripple. This is due to the total inductance of the system, i.e., the equivalent inductance of the compulsator and the one of the launchers. A further insight to the cause of the ripple shows that it is due to the discrete variation of the \"active\" length of the rails, i.e., the portion of the rails behind the armature. Considering the configuration as schematized in Figure 2b, we see that as soon as the contact between element _(a)_ in the rail and element _(a')_ on the armature is interrupted, the current in _(a)_ instantaneously loses its transverse component. Similarly, at some later instant, a contact is set between element _(e)_ in the rail and element _(c')_ in the armature and the current in element _(d)_ of the rail will assume a longitudinal component. The magnitude of the flux density in the armature accordingly changes; also, the terms \(\mathbf{j}_{k}(t)\times\mathbf{B}_{k}(t)\), related to the elementary volumes of the armature, suddenly change and produce the discontinuity in the trust force. The ripple is absent in the force profile predicted by the weak interaction analysis since the parameters of the lumped equivalent vary with continuity.