A Railgun Secondary Propulsion System for High-Speed Hyperloop Transportation

Secondary propulsion methods for high-speed hyperloop transportation are sparsely researched. Secondary propulsion methods are essential to quickly, efficiently, and safely get a hyperloop pod up to its target speed from a stationary state. In this article, we propose and analyze the feasibility of a form of electromagnetic (EM) secondary hyperloop propulsion, called a railgun, commonly used in modern-day artillery technology for high-speed ammunition launching. We assess the feasibility of two different materials and three different geometries for a railgun armature to propel a hyperloop pod. Inverse design of multiphysics simulation of multibody dynamics, magnetic fields, and electric currents is used for material selection of the armature that minimizes rail current energy requirements and the armature geometry that maximizes structural integrity.

. Hyperloop railgun system assembly. The railgun system consists of two electrified rails and a magnetized armature that induces high-speed linear motion of the hyperloop pod along the rails. Fig. 2. Discontinuous three-circuit railgun system. Each rail and the armature are electrified by an independent circuit. Armature motion is induced along the rails by the EM force generated within the armature by the B-field produced by the applied currents. ground node, which generates a resultant magnetic field (Bfield) and electromagnetic (EM) force to induce linear motion in an armature [5], [6]. In a conventional railgun system, the armature acts both as the load to accelerate ammunition to high speeds and as a conductive bridge between the two rails to complete the circuit [7]. In our proposed railgun system, we electrify each rail and the armature using three independent circuits that generate the B-field and EM force to induce motion, as shown in Fig. 2 [8], [9]. By leveraging these discontinuities in the circuit, an electrically insulating air gap may exist between the armature and the rails, reducing the friction experienced by the armature during motion.
In this study, we analyze the following: 1) the applied currents necessary to induce high-speed armature motion at a target acceleration; 2) the heating within the system resulting from the applied currents; and 3) various armature geometries 0093 that minimize the mechanical stress within the system during propulsion.

A. Electric Current Equations
Electric currents flowing through the rails and armature are the basis for inducing the linear motion in a railgun. Motion is derived from the EM force acting on the armature. Both the B-fields and EM force are a function of the applied current where E is the electric field vector, and V is the voltage obtained from the applied current by applying Ohm's law V = I R to the rail and armature material. The density of current within the armature and rail is computed where σ is the electrical conductivity of a material, and J e is an external current density.

B. Magnetic Field Equations
The induced electric field and current density drive the generation of the magnetic field where H is the magnetic field, and J is the current density. The magnetic flux density (B-field) is computed where A is the magnetic vector potential. The rails and the armature are simulated as homogenized linear single-turn coils with a cross-sectional area of 1.16 × 10 −2 m 2 . The coil applies an external current density in the direction of the rails where I coil is the total current in the coil, A is the coil crosssectional area, and e coil is a unit vector pointing in the direction of the rails.

C. Multibody Dynamics Equations
The EM force can now be computed as the boundary integral over the material surfaces where n is the vector normal to the surface, and T is the stress tensor. Moreover, the EM and mechanical forces must balance where σ M is the mechanical stress tensor, T EM is the EM stress tensor, and ρ g is the gravitational force. Finally, we simply compute the acceleration and velocity vectors where m is the total mass. Velocity and acceleration are the target design variables used in the railgun system analysis.

D. Heat Transfer
The applied current causes a temperature rise due to joule heating. It gives rise to a heat source, Q, as follows: which interacts with the solid body as follows: where ρ is the material density, c p is the specific heat capacity, ∇T is the thermal gradient, and q = −κ∇T is the conductive heat flux with thermal conductivity κ. On boundaries, cooling primarily occurs by conductive heat flux acting normal to the boundary where n is the vector normal to the surface, and this heat is dissipated as follows: for an environmental temperature T ext and heat transfer convection coefficient h.

A. Target Design Variables
The target design variables for the railgun system are an acceleration G-force of 2G (19.62 m s −2 ) and a velocity of 100 mph (44.70 m s −1 ). These target variables are selected to achieve high speeds quickly while also maintaining human safety within the hyperloop pod [10], [11]. Inverse design is employed to compute the system specifications required to meet these target design variables. Fig. 3 illustrates the flow of calculations between physics modules. To obtain the required rail current for the railgun system, we instantiate the multiphysics model with a railgun armature geometry and material.
In this article, two armature materials are analyzed: iron and aluminum alloy 6063, along copper rails with ≈2.5-ft spacing between the rails. Copper is selected as the rail material in this study, as it is a common and standard EM rail material [12], [13], [14]. Iron [9], [15], [16] and aluminum [17], [18] are selected as armature materials in this study, as they have been tested in prior literature as viable linear actuator or armature cores; iron and aluminum also have vastly different material properties, which makes for interesting comparative analysis, and have never been analyzed for hyperloop propulsion applications. Three different armature geometries are analyzed: convex, concave, and topology optimized. Convex and concave railgun armatures [12], [19] have been extensively analyzed in the literature; however, topology optimized armatures [20] have not been analyzed in detail.

B. Multibody Dynamics
In Fig. 4, a material parametric sweep is computed for two different armature materials at two different EM force magnitudes (2200 and 5270 N). These EM-force magnitudes account for the minimum force required to overcome the mass of the pod and the armature, as well as air resistance and any static friction. The pod has a fixed mass of 153 kg and the maximum dimensions of 8 × 2 × 2 ft 3 . The armature mass and dimensions vary based on the material and geometry analyzed. Across all geometry and material permutations, the armature mass varies between 82 and 218 kg with the average approximate dimensions of 3.75 × 2.5 × 0.5 ft 3 .
Through the multibody dynamics simulation, both iron and aluminum 6063 alloy armatures are demonstrated to achieve the target velocity of 100 mph. However, applying various magnitude EM-forces results in achieving the target speed at different rates. For human safety, we set an upper bound to the allowable G-force experienced to 2G of acceleration [10], [11]. By taking the derivative of the velocity curves, the permutations iron 2200 N, iron 5270 N, aluminum 2200 N, and aluminum 5270 N have the following approximate G-forces 1G, 2G, 2G, and 5G, respectively. Therefore, given the G-force constraints, iron 5270 N and aluminum 2200 N achieve the target velocity in ≈2.3 s.
The linear motion of the armature along the rails is simulated as a rigid prismatic joint with motion in 1 degree of  freedom (DOF), as shown in Fig. 5. 1 The linear motion of the armature via prismatic joint is computed as follows: where u c,arm is the displacement vector for the center of armature, R arm is the rotation matrix of the armature, I is the total current vector, X c is the joint center, X c,arm is the Fig. 7. Induced current density for the three-circuit railgun armature. The current density induced by a 43-kA current applied to a convex iron armature is illustrated by the color bar. The flow direction of the current is illustrated by the cyan arrow heads. A small air gap exists as an electrical insulator between the armature and the rails. armature centroid position, and u arm is the displacement at the armature centroid.

C. Magnetic and Electric Fields
To achieve the target acceleration and velocity design variables computed in the previous section, EM field simulations are run. Following the inverse design flow from Fig. 3, the EM fields are computed to achieve the target EM-force vectors for each material: F = 5270 N for an iron armature and Fig. 9. Armature topology optimization. The computed armature geometry (right) is optimized to maximize strength when applied with a load. The direction of the applied force is illustrated by the red arrow, and fixed constraints are illustrated by the red arrow heads. F = 2200 N for an aluminum 6063 alloy armature. For a given armature geometry, the iron armature is ≈3× more massive than the aluminum armature; however, iron has more favorable magnetic properties than aluminum under an external B-field induced by the rails [21], [22]. Thus, these density and magnetism differences between the materials result in the different applied EM-forces required to propel either railgun armature to the target design speed [22], [23]. The B-field generated to achieve an EM force magnitude of 5270 N on an iron armature is shown in Fig. 6(a). The B-field is used to determine the rail current necessary to achieve the target EM force. Fig. 6(b) illustrates the rail current required to achieve a given EM force on the railgun armature for both iron and aluminum 6063 alloy materials. An applied current of 57 kA is required for the aluminum 6063 alloy armature, and an applied current of 43 kA is required for the iron armature to achieve the target acceleration and velocity design variables. We aim to minimize the power needed to electrify the hyperloop; thus, the iron armature achieves the target design variables given the lower input current.

D. Computed Electric Currents
Leveraging the flow of calculations between multibody dynamics and EM field models, the required rail currents are computed to achieve a target velocity of 100 mph (44.70 m s −1 ) at 2G acceleration for a railgun method of secondary hyperloop pod propulsion. In a standard convex railgun armature, input rail currents of 43 kA are required to achieve the target acceleration and velocity. Fig. 7 illustrates the current density and flow direction induced within the armature by the 43-kA input. It is demonstrated that using an iron armature, rather than an aluminum armature, reduces the rail current required to achieve these target variables by 14 kA-resulting in significant cumulative energy savings over the operating lifetime of the railgun system.

IV. HEAT TRANSFER ANALYSIS
In Fig. 8, the influence of joule heating due to applied currents is computed for a range of heat transfer convection Fig. 10. Stress analysis of three armature geometries. Surface mapping of stress on iron armatures from an external load of 5270 N. The direction of the applied force is illustrated by the red arrow, and fixed constraints are illustrated by the red arrow heads. It should be noted that (a) and (b) are plotted over the same range of stresses, while the range of (c) is plotted with a maximum value two orders of magnitude less due to the highly optimized structure experiencing significantly less stress than its counterparts. coefficients for both armature materials under consideration, given a small air gap between circuits. Under the rail current of 43 kA, no phase change is expected in the copper rails or iron armature with moderate convection [24]. However, even under high-temperature conditions that do not induce melting, the mechanical performance of the structural armature and rails may degrade. Thus, to achieve sufficient rail current, increased air convection is required to reduce the effect of structural degradation. Fig. 8 demonstrates the amount of convection required at a target rail current of 43 kA for the armature-rail boundary to be sufficient below the material melting points. Although rail currents below 43 kA would induce less Joule heating, they would not supply enough EM propulsion to overcome the system mass, air resistance, and friction. Due to lower melting points, an aluminum 6063 alloy armature would require greater minimal convection to prevent phase change due to joule heating and, hence, is a less practical material to select for the armature compared to iron [6], [21].

V. STRUCTURAL MECHANICS ANALYSIS
The railgun armature experiences high reactive forces from pushing the pod at high velocities, as illustrated in the previous sections. To accelerate convex iron armature and pod to 100 mph at 2G acceleration, an EM force of 5270 N is required to act on the armature. Hence, an equal magnitude and opposite direction force are acted on the armature by the propelled pod. These forces can be detrimental to the structural integrity of the armature.
The stress of two common railgun armature geometries, convex and concave [12], [19], is analyzed in addition to a novel armature geometry obtained via topology optimization. Fig. 9 demonstrates the topology optimization of the railgun armature to maximize the strength of the structure when applied with a load. The armature optimized in this article to maximize mechanical strength matches the optimized armature proposed by Guo et al. [20]. We redesign the computed armature geometry to more closely resemble that of Guo et al., such that it is feasible to manufacture. Fig. 10 shows the stress fields for the convex, concave, and optimized armatures when applied with the maximum design load of 5270 N. For the conventional armatures, the maximum stress experienced by the convex armature is 1.43 × 10 6 N m 2 , and the maximum stress experienced by the concave armature is 0.98 × 10 6 N m 2 . The maximum stress experienced by the topology optimized armature is 1.70 × 10 4 N m 2 -two orders of magnitude lower than the conventional armature geometries. The impressive load-bearing properties of the optimized armature arise due to the presence of material along the load-to-boundary path, which increases the strength of the armature and minimizes deformation.

VI. FUTURE WORK
In this article, we analyze the physical feasibility of iron and aluminum as railgun armature materials for hyperloop propulsion. In future work, we would like to expand this analysis to cover several additional materials from the literature, such as sintered NdFeB, Fe 78 B 13 Si 9 , and GO FeSi [12], [25].

VII. CONCLUSION
We conclude that with proper armature material and geometry selection, a railgun system is capable of supporting both high speed and safe propulsion of a hyperloop pod. Both iron and aluminum material armatures were analyzed to determine which material requires a lower rail current to propel the pod to its target speed of 100 mph at 2G acceleration. The iron armature minimizes the rail current required to generate a sufficient external magnetic field for target propulsion, requiring 43 kA of rail current. Three armature geometries, convex, concave, and topologically optimized, were analyzed to determine which geometry can be accelerated to the target speed while minimizing structural stresses. The topologically optimized armature with a load-to-boundary path achieves the lowest maximum stress under the propulsion forces, experiencing 1.70 × 10 4 -N m 2 maximum stress. Therefore, from the analyzed armature materials and geometries in this study, an iron armature with a load-to-boundary path geometry is found to minimize the rail current energy requirement for target hyperloop pod propulsion while maximizing structural integrity.