Lyall Cooper Dr. Dann ASR – D Block 10 May 2012 the Mighty Railgun I. Abstract the Idea of This Project Is to Build a Railgun
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Lyall Cooper Dr. Dann ASR – D Block 10 May 2012 The Mighty Railgun (All Bark no Bite) I. Abstract The idea of this project is to build a railgun capable of firing a small metal projectile at substantial velocity. The original plan was to build iterative prototypes capable of firing small projectiles, and using them to figure out the ideal design for the final version, but the smaller versions were not very successful due to their relative lack of power, so it was difficult to use them to learn what to do. The final, largest scale railgun is powered by a bank of capacitors with an equivalent capacitance of 24.6mF charged to around 350V. This produces 3013.5J of energy discharged over approximately 58.75 µs (it varies for each firing), drawing a peak of about 1425A when firing a projectile (it once again varies) and about 1600A when not firing a projectile (short circuiting the rails). The railgun is capable of consistently firing a small piece of metal (usually aluminum or copper); however the projectile does not usually travel very far, although this is hard to measure due to the nature of the gun and the speed at which the firing takes place. II. Introduction Electricity is often seemingly mysterious, but we have come to accept and understand how through the interaction of electric and magnetic fields we can create a simple motor, as we did in the first semester. A railgun is just a linear electric motor, at very high speeds. What makes it different, however, is that it uses neither magnets nor coils of wire, and relies entirely on the induced magnetic field in the rails due to the extremely large current to produce a Lorentz Force to propel the projectile (which will be discussed in greater depth in the theory section). [2] The idea for a railgun first came from trying to figure out a way to fire a small object into space, but after some estimations and early tests, that seemed like it would be impossible given current resources (not to mention safety issues), but is still theoretically feasible, and in many ways more economical than a traditional space launch. The large currents and large forces involved create some interesting problems that need to be overcome, and leaves a lot of room to determine the best materials and design. This means that when building the various scales and iterations of the railguns, there are many things that need to be taken into account and many choices that need to be explored and made. The railgun combines precise craftsmanship with a complex physics background to make a very challenging yet interesting (if not at least dangerous) project. Firstly, the basic design of the railgun has to be sturdy enough to withstand the large forces involved, and designed in such a way that it can be safely operated. The materials have to be chosen Figure 1: A Diagram of a simple railgun [3] to perform well in a number of categories, including conductivity, heat resistance, and overall strength and reusability. Everything must be constructed precisely enough that the projectile will always make good contact with the rails, but without hindering its movement. Capacitors have to be chosen and wired in such a way that will maximize the electrical potential energy transferred into kinetic energy while still maintaining a degree of safety and practicality. Other things, like an injection system may be required to stop the projectile from welding itself to the rails if the power is connected and the projectile is stationary between the rails. [3] This project is also very relevant to the real world, despite the fact that the final product may not be. The history of the railgun actually goes back to the mid to late 1800s, but the first proof of concept railgun wasn’t built until 1944, which could fire a 10g projectile to around 1 km/s. [4] The railgun has been continuously developed in many different places since then into what it is today, for both scientific and military use. The US Navy is in the progress of developing and testing railguns that may eventually replace traditional missiles found on naval vessels for a fraction of the cost, as the projectile is just Figure 2: An image from the Navy's railgun test [2] a hunk of metal, as opposed to explosives and a guidance computer, among other expensive sensors and materials. [5] It is not only a weapon of the future, but also could be used for scientific and industrial testing at significantly less cost than something like a rocket sled once heat dissipation and other issues are worked out. III. CAD Drawings (Measurements in millimeters) Figure 4: A perspective view of the railgun Figure 3: Front view of the railgun with measurements Figure 5: A side view of the railgun with measurements (1000 mm, 300 mm) IV. Theory Force: As mentioned previously, a railgun works by inducing massive magnetic fields in the rails, then the current through the projectile causes a Lorentz Force to act on the projectile, propelling it down the track in the direction of the current in the right rail. The basic principle is that an electrically charged particle moving through a magnetic field experiences a force perpendicular to the direction of travel. It is this simple concept that allows a railgun to function, but the math behind it is far from simple. The magnitude of a Lorentz Force is given by the following equation: → → → Where q is the charge of a particle, v is the velocity, and B is the magnetic field. Note how the vector cross product operator shows that the force acts perpendicularly to the direction of travel of the particle. [6] This equation can be rewritten to give the force on a wire in a magnetic field, given known current and dimensions: Where I is the current and d is the length of the wire, or projectile in our case. The magnetic field can then be determined using the Biot–Savart Law [7], which states that the magnetic field induced by current through a wire is the following: Where is the permeability constant, I is the current, and s is the distance from the wire. From here, we can determine the magnetic field between two wires (the two rails) using the following equation: ( ) Where d is the distance between the two wires. Then to average the magnetic field, we have to integrate (r is the radius of the wire): ∫ ∫ ( ) → ∫ ∫ → → → So finally we plug that into our force equation to get the force on the projectile: To find the total work done on the projectile, we then have to integrate again, taking into account that the current decreases exponentially as the capacitors discharge: ∫ ∫ ( ) ( ) Where R is the resistance of the entire circuit, and C is the total capacitance of the capacitors used. Note that these equations will not necessarily lead to a good approximation of how fast the projectile will go due to the friction and intense heat created by the huge current. From these equations, one can gather that the wider the projectile, the greater the force on it, but there are diminishing returns because the force is proportional to the log of the width, so there is a point where the tradeoff between width and added mass becomes a factor. Also the force is proportional to the square of the current, so doing small things to decrease the resistance of the entire system or increase the voltage of the capacitors can actually have a large effect on the theoretical power of the railgun. Heat: We know that most of the resistance of wiring and the rails dissipate electrical energy into heat. Heat is also generated by the friction of the projectile on the rails. We can make an estimate of how much heat is generated by using the following equations: ( ) ∫ ( ) ∫ ( ) ( ) → Since power dissipated is proportional to the resistance, it is very important to try and minimize the resistance in order to minimize the amount of energy lost to heat instead of to exerting a force on the projectile. Now if we assume we are using capacitors charged to 300 Volts with an equivalent capacitance of .01 Farads, we can make some estimates [8]: We now know that the amount heat generated must be less than this amount, because at least some of the energy (ideally most of it) is turned into kinetic energy as opposed to heat energy. The problem is that the heat is not dissipated across all of the material as it is created so quickly, heat is mainly created at the points of contact. The ideal material then is a good conductor and has a high melting point. I chose copper for my rails because as you can see from the chart below, it has a good combination of low resistivity and high melting point. Some possible materials [8] [9]: Material Resistivity (µΩm) Melting point (°C) Copper 1.72 1084.62 Brass (avg.) 3.9 ~920 Gold 1.62 1064.18 Steel (avg.) 17 1434 Stainless Steel 72 1434 Capacitors: The capacitors are one of the most important parts of the build, as they provide the power, and consequently the force. Simply put, a capacitor works by storing charge on parallel (or semi-parallel) plates separated by an insulator known as the dielectric. The dielectric could be air, but often things like mineral oil are used in order to increase the amount of charge that can be stored before charge jumps from one plate to the other (i.e.