*4.2. 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 EN4EM 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 Δ*z* = 3.5 *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 *jk*(*t*) × *Bk*(*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.

**Figure 11.** Time waveforms of the current delivered by the compulsator to the launcher in the strong and weak interaction.

**Figure 12.** The thrust force on the armature in the strong and weak interaction. The ripple is an artifact due to the commutations that happen in the auxiliary network used to manage the sliding contacts in the strong interaction analysis.

Let us now consider the dynamic quantities on the compulsator. Figure 13 shows the torque acting on the rotor. As expected the torque is mostly negative, producing a decrease of the speed of the rotor. The figure shows that in the last portion of the launch time, the torque assumes positive values so increasing the velocity and the kinetic energy of the rotor, with respect to their minima.

**Figure 13.** The torque on the rotor of the compulsator. The portion of the curve with positive value corresponds to recovering of magnetic energy as kinetic energy.

The instant when the torque changes its sign is roughly the same as the one when the delivered current reaches its maximum. A decreasing current implies a reduction of the magnetic energy stored in the system. Part of this magnetic energy is converted in mechanical energy by increasing the speeds of the rotor of the compulsator and of the armature of the launcher. The remainder increases the temperature of the conductors. The weak interaction analysis produces a smoother waveform than that of the strong interaction one.

The comparison of the speeds of the armature obtained by the two models is reported in Figure 14. The weak interaction analysis overestimates the speed of about 10%. The ripple in the thrust force is cancelled by the integration and does not affect the speed waveform produced by the strong interaction analysis.

If a lumped equivalent circuit was used for the compulsator, further errors would appear. These errors will be more relevant if components made up of massive conductors are present in the compulsator (e.g., a conductive shield). In this case, the actual distribution of the currents cannot be predicted a priori. Anyhow, the errors are expected to be lower when compensating concentrated windings are used.

The errors in the exit speed, lead to a wrong estimate of the launch time and therefore on the length of the current pulse. If this happens, the exit of the armature from the launcher could occur in correspondence of a non-zero value of the current, with consequent reduction of the system performance (low efficiency and arcing between armature and rail at the muzzle).

Despite the complexity of the problem EN4EM was able to complete the strong interaction analysis in about 150 min on a desktop computer based on an intel i7 6 core and equipped with 20 GB RAM. The maximum allocated memory was about 6 GB. The weak interaction analysis took about 45 min and required about 3 GB.

**Figure 14.** Comparison of the armature speed during the launch as predicted by weak and the strong interaction analysis.

#### **5. Conclusions**

The use of lumped equivalent circuits in modeling the coupled electro-mechanical behavior of a rail launcher and its feeding compulsator may produce results whose accuracy is not always satisfactory. The causes are due to the presence of eddy currents in the compensation shield of the compulsator and in the uneven current distribution in the rails and in the armature of the launcher. Coupled 3D electro-mechanical analysis is needed if accurate results are required. The paper has compared the results by the strong and the weak interaction analysis by the research code EN4EM. The availability of such a numerical tool could represent a valuable resource in the design of the launcher and of its feeding compulsator since it allows to determine the more important parameters of the launch.

In particular, it will be possible to prepare a look-up table to arrange the operative parameters of the compulsator (e.g., the excitation current, the initial speed of the rotor, its angular position at the instant of firing) to achieve a designed muzzle speed on a given payload.

**Author Contributions:** Conceptualization, A.M. and L.S.; methodology A.M., V.C.; software, R.R., A.M.; validation, A.M., and L.S.; formal analysis, V.C., R.R.; investigation, R.R., L.S.; writing—original draft preparation, L.S., R.R.; writing—review and editing, V.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to thank the NVIDIA's Academic Research Team for the donation of two NVIDIA Tesla K20c GPUs that have been extensively exploited for the simulations.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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