Effect of Projectile Design on Coil Gun Performance

Jeff Holzgrafe, Nathan Lintz, Nick Eyre, & Jay Patterson

Franklin W. Olin College of Engineering

December 14, 2012

Abstract

In this study we provide an analysis of the effect of projectile material on the exit velocity of a coilgun: an electromagnetic actuator which uses a pulse of current to accelerate magnetically activeprojectiles to high velocity. We derive an approximate closed form solution for the exit velocityof the projectile. As a more accurate model, we also derive a non-linear differential equationfor projectile position from first principles. We validate these analytical results with both finiteelement models and experimental results. The analytical calculations tend to predict larger thenmeasured velocities, which is expected given the assumptions made. From our analytical resultswe show that the exit velocity is proportional to the magnetic susceptibility of the projectilematerial. We found that iron projectiles with large cross sectional area produced the highest exitvelocities. Steel and hollow projectiles produced lower velocities and length was found to notmatter substantially. Overall, the investigation was successful and useful results were obtained.

1 Introduction

A coil gun is a device which uses electromagnetsto accelerate magnetically active projectiles tohigh velocity. In a coil gun, a current is passedthrough a solenoid, creating a magnetic fieldwhich draws a projectile to the center of thecoil. If the current is stopped quickly, it contin-ues out of the barrel at rapid speed. Many coilguns involve multiple stages of coils which aretriggered in series for maximum acceleration.

Coil guns have many useful applications. Forexample, orbital satellite launch with coil gunshas been proposed but has of yet been deemedimpractical due to large stresses on the pay-load. Coil guns have also been proposed asreplacements for chemical firing weapons fornaval purposes, although the similar rail gunis generally preferred. Coil guns are relativelyquiet, wear very well and can be used with avariety of ammunition types.

The goal of this project is to investigate theeffect of varying projectile material and geom-etry on coil gun performance. To accomplishthis goal, we built a working coil gun prototypewhich served as a test platform for a variety ofprojectiles.

In this paper, we begin by deriving theoret-ical models for a coil gun. The design of ourcoil gun is then presented, followed by numer-ical simulations for model validation. Finally,

experimental results are presented for a numberof projectiles and the results are analyzed.

2 Coil Gun Theory

The coil gun works by quickly discharging a largeamount of energy from a capacitor through acoil, creating a strong magnetic field throughthe coil. This strong field induces magnetizationin the slug, causing microscopic dipoles to alignwith the field in a lower energy state. Becausethe coil is a solenoid of finite length, the pro-duced magnetic field decays in strength axiallyaway from the edge of the coil. The aligneddipoles feel a force proportional to the gradientin the magnetic field:

F = µ0∇(M ·H) (1)

Where M is the projectile magnetization andH is the applied field.

If the current were sustained indefinitely, af-ter being accelerated from one side of the coil,the projectile would encounter an opposite gra-dient on the other side which would apply aforce back towards the center of the coil. Thus,a coil gun with constant current would simplycause the projectile to oscillate within the coil.If the slug reaches the other side of the coil whilesignificant current is still running in the coil, theslug will experience a force toward the center

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of the coil, slowing its motion. This effect iscommonly called suckback, and is undesirablefor coil gun operation. However, if all of thecharge is drained from the capacitors quickly,before the projectile reaches the other side ofthe coil, there will be no repelling force and theprojectile will continue on to exit the barrel.

Order of Magnitude Estimation

We can get a rough estimate for the exit velocityof the slug by comparing the energy state of theslug outside and inside the coil. Before the coilfires, the magnetic field on the slug is negligible.We will make the assumption that the currentin the coil is constant until the slug reaches thecenter, at which point it drops to zero. Whenthe slug is inside the coil, we can apply thecommon Ampere’s law estimation of the appliedfield H and assume the slug is within a constantmagnetic field. This will have a lower potentialenergy state because the induced magnetizationM align with the applied field.

Initial Final

Figure 1: The two states of the slug consideredin the order of magnitude estimation. The slugstarts in neglible field and when it reaches thefinal state the field completely cuts off.

We define the potential energy state of theslug outside the coil to be zero: Eoutside = 0J.The magnitude of the applied field inside thecoil can be approximated by assuming an in-finitely long coil and applying Ampere’s law.The well-known result is

H = nIx (2)

Where n is the turn count per length, I isthe current through the coil, and x is an axial

unit vector. Assuming the slug is completelywithin the uniform field and a linear model ofmagnetization with magnetic susceptibility χm,the induced magnetization in the slug will be:

M = χmH = χmnIx (3)

The energy per unit volume of a magnetizedmaterial in an applied field is ρe = −µ0M ·H.In our case, the applied field and magnetizationare parallel, because the former creates the lat-ter, and thus the dot product can be renderedas multiplication. The potential energy of theslug inside the coil is thus:

Einside = −µ0MH = −µ0χmn2I2 (4)

This is a lower potential energy state thanthe slug outside the coil. Assuming that all po-tential energy is transferred into kinetic energy,the exit velocity is:

vexit =

√2

mV µ0χmn2I2 (5)

Where V is the volume of the slug and mis the mass of the slug. Using reasonable val-ues for our coil gun with a commercial gradeiron slug[3], µ0 = 4π × 10−7, n = 500, I = 50,χm = 100, m = .03 and V = 3 × 10−6 (all inSI units) this approximation predicts an exitvelocity of about 4m/s.

This order of magnitude calculation makesa number of simplifying assumptions which im-pair the accuracy of the model. The assumptionthat the slug lies completely within a uniformmagnetic field in the final state overestimatesthe amount of potential energy loss, and hencecauses a larger exit velocity prediction. Theassumption that the current dies out instantlywhen the slug reaches the center of the coil ig-nores the effect of suckback, and causing a largerexit velocity prediction. Furthermore, all the po-tential energy will not be transferred into kineticenergy: much of it will go toward frictional work.This too, will overestimate the exit velocity.

A More Complete Model

To improve the model, we will take into accountthe effects of non-uniform fields and suck back.

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To do this we will derive the axial applied fieldof a solenoid and use it to calculate the force onthe slug.

The Applied Field of a Solenoid

In the following analysis, we will use the systemdefinition presented in Figure 2. Our goal is tofind the magnetic field at a test point p which isa distance x from the left face of the coil. Thederivation will first be made in terms of θ, theangle formed by the x-axis and a line from p to apoint on the coil, as it proves to be algebraicallyeasier. Later, we will convert this into a functionof x, to create a differential equation of x.

P

θ

θ2θ1

x

L

R

r

Figure 2: The system parameters used in thederivation of the force on the slug. Note that inthe figure the value x, the position of point p, isa negative value.

The applied field due to one loop of wireis[1]:

H(θ) =I sin θR

2r2=I sin θ3

2R

Where R is the radius of the loop and r isthe distance from p to the edge of the loop.

If we consider the coil to be made of manyinfinitesimal loops, we can use the equivalentsurface current density K = nI to calculate thecurrent through those loops as a function of θ:

dI = Kdx = −KR csc θ2dθ (7)

The contribution to the applied field fromone of these loops is thus, using equation ??:

dH =−1

2K sin θdθ (8)

We can now integrate over the angles sub-tended by the line from p to the coil to find thetotal applied field due to all loops:

H(θ1, θ2) =

∫ θ2

θ1

−K2

sin θ dθ =nI

2(cos θ2−cos θ1)

(9)This can then be converted into a function

of x:

H(x) =nI

2(

L− x√(L− x)2 +R2

+x√

x2 +R2) x

(10)

The Force on the Slug

Equation 10 can now be used to calculate themagnitude of the force on the slug. We willconsider the slug to be made up of many in-finitesimally thin disks, sliced axially. We willassume that the applied field on these disks isconstant and equal to the field through the axis.We can then calculate the force on each diskand integrate over the disks to find the totalforce. The force on a small volume object in anapplied field is [4]:

dF = µ0∇(M ·H) dV = µ0∇(M ·H)Adx (11)

Where A is the cross sectional area of theslug. For our case, the magnetization and ap-plied field are assumed to be parallel to thex-axis. If we also assume a linear model, theforce on a slice is:

dF = µ0d

dx(χmH

2)Adx = µ0χm2HdH

dxAdx

(12)This infinitesimal equation parallels the em-

pirical formula for the force on a small object inan applied field [5]:

F = V χmµ0HdH

dx(13)

This similarity gives credence to our derivedresult. We can now integrate over the slicesto get the total force on our slug. If x is theposition of the right side of the slug, and l is thelength of the slug the total force is:

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F =

∫ x

x−lµ0

d

dx(χmH

2)Adx (14)

F (x) = µ0Aχx(H(x)2 −H(x− l)2) (15)

This equation for the force on the slug givesa non-linear second order ordinary differentialequation. In the Simulation section, we submitthis differential equation to numerical solution.

While this model is more accurate than theestimation, there are still a number of simpli-fying assumptions. The assumption of uniformfield for a given x-position ignores the radialdependence of the field. The field close to thewires would be larger than the center field, mean-ing the predicted velocity would be too small.This model still assumes frictionless travel ofthe slug, although it would not be difficult toinclude a frictional force in the simulation. Thelinear model of magnetization is also not en-tirely accurate, especially given the large fieldswe produce in the experimental setups: in factwe may be reaching saturation magnetization.This would mean the expect exit velocity is toohigh. Furthermore, the calculation of the fieldassumes a thin solenoid - that is, the wires areinfinitely thin and do not stack on one another.This makes the expected field larger, and theexit velocity larger.

RLC Analysis

In order to reduce suckback, the RLC systemdefined by the capacitor bank and coil should betuned to provide the minimum discharge time.The faster the discharge time, the lower the cur-rent will be when the slug exits the coil, givingit a higher exit velocity. We developed a math-ematical model of the RLC system in order totune the parameters before creating it.

Resistor

Inductor

Capacitor

Figure 3: An RLC Circuit.

The RLC circuit shown in Figure 3 has aKirchoff’s Voltage Law of:

VR + VC + VL = 0 = IR+ VC + LdI

dt(16)

This can be rewritten in terms of VC only:

RCdVCdt

+ VC + LCd2VCdt

= 0 (17)

The characteristic polynomial of this homo-geneous second order linear differential equationis:

LCλ2 +RCλ+ 1 = 0 (18)

Which gives eigenvalues of:

λ1,2 =−RC ∓

√R2C2 − 4LC

2LC(19)

The solution to the equation is thus:

VC(t) = c1eλ1t + c2e

λ2∗t (20)

Applying the initial conditions I(0) = 0 andVC(0) = V0, we can solve for the constants:

c2 = V0/(1−λ1λ2

) (21)

c1 = V0 − c2 (22)

The current through the inductor can thenbe expressed as:

I(t) = C ∗ dVCdt

(23)

We used this equation to find a reasonableset of parameters for our RLC system in orderto create a relatively fast discharge time. Thissame analysis was also used in our dynamicmodel to find the current though the coil as afunction of time.

3 Prototype Design

A prototype coil gun and projectiles were de-signed to allow us to test the effect of projectilematerial on performance. The design, shown inFigure 4, was manufactured by team membersin the Olin College machine shop. The gun’stransparent polycarbonate barrel has an inner

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diameter of 5⁄8” and is one foot long. The bar-rel is mounted on a clamping support structuremanufactured from black ABS plastic.

Figure 4: Coil gun prototype design.

Projectiles

Eight different styles of projectile were manu-factured for testing. All projectiles are 5⁄8” indiameter and slide easily in the plastic barrel.Iron projectiles, which we predict will shoot thebest, were made in three different lengths (1”,11⁄2” and 3”) as well as in hollow and solid con-figurations. 1” long solid projectiles were alsomanufactured out of 6061-T6 Aluminum, A36Steel, 1018 Steel and 4130 Steel.

In addition to machined projectiles, a pur-chased magnet and a rolled sheet of mu-metalwill be launched. Mu-metal is a very magneti-cally active material and is commonly used forelectromagnetic shielding. The purchased mag-net has a surface field of 7157 Gauss1. The sizesand masses of all projectiles are given in Table1.

Material Style Diameter Length MassIron Solid 5⁄8” 1” 30 gIron Hollow 5⁄8” 1” 18 gIron Solid 5⁄8” 13⁄4” 51 gIron Solid 5⁄8” 3” 93 g

Steel A36 Solid 5⁄8” 1” 36 gSteel 4130 Solid 5⁄8” 1” 36 gSteel 1018 Solid 5⁄8” 1” 36 g

Magnet Solid 1⁄2” 1” 24 gMu-Metal Rolled 5⁄8” 4” 18 gAluminum Solid 5⁄8” 1” 12 g

Table 1: Projectile Parameters

It is important to note that the three differ-ent steels were chosen for their iron content as

well as their availability. The iron content ofeach of the steel alloys is given in Table 2. TheA36 steel, which is a hot-rolled alloy, has thehighest iron content of the three followed by the1018 and the 4130 on bottom, both cold rolledalloys.

Alloy Iron ContentA36 99.0%1018 98.8% - 99.3%4130 97.0% - 98.2%

Table 2: Iron Composition of Steel Alloys [6]

Coil

The coil gun’s coil is made from 20 turns of 14AWG wire. The coil was designed to have a lowresistance and inductance so that the capacitorwould discharge quickly. The parameters for thecoil are provided in Table 3. As shown, the resis-tance of the coil is extremely low at .116 Ω andthe inductance is also quite low at 4.62×10−3

H.

Length 30 mmWire Gauge 14 AWGTurns 20Inner Diameter 19 mmOuter Diameter 33 mmAverage Inductance 4.62×10−3 HInductance Standard Deviation 3.74×10−5 HAverage Resistance 1.16×10−1 ΩResistance Standard Deviation 9.05×10−3 ΩSamples Taken 7

Table 3: Coil Parameters

Circuit Design & Characterization

The electrical components of our coil gun areshown in Figure 5. We used a power supply tocharge the capacitors before firing and a largeblock of metal with an insulated handle to dis-charge the capacitors for storage. Note thatwhile there is no resistor in this circuit, the in-ductor and capacitors provide some resistance.This equivalent series resistance is omitted forsimplicity here but is utilized in calculationselsewhere.

1K&J Magnetics Product #D8X0DIA

5

CoilCapacitor Bank

SCR6V Reg

R1

1 kΩ

R2

1 kΩTrigger

Trigger Circuit Coil Gun Circuit

Figure 5: Coil gun electrical system schematic

The system is triggered by a silicon-controlled rectifier (SCR). The SCR we are usingis triggered by the application of 1.7-5V refer-enced to the SCR’s anode. So the system wouldbe self-contained, we wanted the triggering cir-cuit to be driven off of the firing capacitors. Weused a 6V voltage regulator which accepts inputvoltages between 7 and 40V and an additional1000 Ω voltage divider to produce the desired3V for the trigger. We attached the SCR to theoutput of the voltage divider and use a linearSPDT switch to control the voltage regulatorstate and fire. By using a switch to connect anddisconnect the voltage divider, we eliminatedcapacitor voltage bleed through the voltage reg-ulator which some other trigger circuits have.

The system’s charge is stored in a bank ofthirty two 18mF capacitors in parallel. The ca-pacitors were characterized with a capacitancemeter to determine their actual capacitance andequivalent series resistance (Table 4). Duringcharacterization, the capacitors were found tohave capacitance much lower than rated. How-ever, each of the capacitors has a ±20% toler-ance so this lower capacitance is not unreason-able.

Number of Capacitors 32Rated Unit Capacitance 1.80×10−2 FRated Capacitor Tolerance ± 20 %Average Measured Unit Capacitance 1.53×10−2 FCapacitance Standard Deviation 8.33×10−4 FUnit Capacitance Measurements Taken 10Average Measured equivalent Resistance 8.35×10−2 ΩResistance Standard Deviation 7.11×10−4 ΩSamples Taken 7

Table 4: Capacitor Parameters

As the capacitors discharge into the coil, thecircuit acts as an RLC circuit. To characterizethe circuit, we measured its discharge curve with

an oscilloscope (Figure 6). After 50ms, 87% ofthe charge had been dissipated; after 100ms,over 98% of the charge was gone. This exper-imental data matched up with our theoreticalcalculations quite well.

Note that the values for resistance, induc-tance and capacitance obtained via dischargecharacterization do not match up perfectly withthe results from direct measurement. The char-acterization suggests that the values of resis-tance, inductance and capacitance are .13 Ω,1× 10−5 H and .2 F, respectively. We suspectthis is because the properties of the elementschange somewhat when run at such high cur-rents. The equivalent series resistance, for ex-ample, may be dependent on current. We arenot truly sure of the cause of this discrepancybut the characterization results presented herereflect how the circuit will behave in practice.Thus these are the values we use in simulation.

−0.05 0 0.05 0.1 0.15 0.2

0

5

10

15

Time (s)

Vol

tage

(V)

MeasuredTheoretical

Figure 6: Capacitor Voltage Discharge Curve

Experimental Test Setup

To obtain results on coil shooting performance,a Vernier system photogates was used to mea-sure the time required for the projectile to pass.The known length of the projectile could thenbe used to calculate the exit velocity.

4 Simulation

In order to assist in field visualization, severalcomputer simulations of the system were made.We used two finite element modeling packages to

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calculate the field induced by the coil with andwithout the slug. We used a MATLAB modelto numerically integrate equation 15, the forceon the projectile.

Finite Element Models

First, to visualize the fields surrounding the coil,a simple coil model was made in COMSOL. Thismodel uses the peak discharge current whichwas measured at 60 A. The resultant fields areshown in Figure 7. As shown, the fields vary,but not significantly, off the axis of the coil. OurMATLAB simulation assumes the on-axis fieldis constant through the projectile and becauseof this simulation result, we are confident thatthis assumption is not throwing off the resultssignificantly.

Figure 7: Simulated magnetic field around coilfrom COMSOL

Next, a secondary model was created in Fi-nite Element Method Magnetics (FEMM), afree simulation package specifically designed formagnetic systems. FEMM is ideal for the coilgun system because it is scriptable, allowing usto calculate the work done by the coil on theprojectile as it moves through the length of thebarrel.

To validate our FEMM results, we replicatedthe magnetic field plot from COMSOL usingFEMM (Figure 8). The numerical value of thefield is slightly different with the values fromFEMM being lower than the COMSOL valuesby a factor of 4⁄3. However, this is quite close,within less than an order of magnitude and thequalitative shape of the field is very similar be-

tween the two models so we are confident in theresults obtained from FEMM.

Figure 8: Simulated magnetic field around coilfrom FEMM

Next, the iron projectile was added to theFEMM model. The addition of the projectilechanges the fields significantly, with extremelyintense fields concentrated within the projectile(Figure 9).

Figure 9: Simulated magnetic field around coilwith Iron Projectile Present from FEMM

Using the FEMM model, the force on theprojectile as it moves through the field gradientwas calculated at thirty five points through thefield gradient of the coil. These forces were nu-merically integrated to derive the work done bythe field on the projectile. When combined withthe actual masses of the projectiles, the barrelexit velocity of each was calculated. Note thatthis simulation assumes that the current is at

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its peak for the entire transit of the projectileand disregards friction in the barrel.

The results of this FEMM simulation aregiven in Table 5. Note that two of the chosensteels were not in FEMM’s material databaseand were substituted with similar steels. “HotRolled Low Carbon Steel” was used for A36 and“Cold Drawn Carbon Steel” was used for 4130.Furthermore, the Mu-Metal was modeled as asolid 1” Long Slug, which differs from the actualgeometry of our projectile.

Material Length Style Exit VelocityIron 1” Solid 0.988 m⁄s

Steel A36 1” Solid 0.245 m⁄sSteel 4130 1” Solid 0.205 m⁄sSteel 1018 1” Solid 0.989 m⁄sMu-Metal 1” Solid 0.180 m⁄sAluminum 1” Solid 0.001 m⁄s

Table 5: Velocity Results in m⁄s

It is interesting to see that according to theFEMM simulation, the 1018 steel ended witha higher exit velocity than the pure iron, anunexpected result. Furthermore, the two steelsfor which similar steels were substituted bothperformed very poorly with vastly different re-sults than the 1018 steel. As expected, becausealuminum has low magnetization, the forces onthe aluminum projectile were very low and wesuspect that in experimental testing, it will notovercome friction and will remain stationary.Finally, the exit velocity of the mu-metal wassurprisingly low given the high magnetizationof the material.

Dynamic Model

We submitted equation 15 to numerical inte-gration in Matlab, using equation 23 to findthe current through the coil at each time-step.The results of a simulation with parameters thatmatch our experimental setup are shown in Fig-ure ??.

5 Experimental Results

Each of the ten projectiles was launched threetimes out of the coil gun barrel and its velocitymeasured by the sensors. For each test, the lead-ing edge of the projectile was aligned with the

tail edge of the coil. The velocity data is given inTable 6. Note that the aluminum projectile didnot move at all in the barrel, as expected dueto its extremely small magnetic susceptibility.

Material Length Style Average St. Dev.Iron 1” Solid 1.72 0.04Iron 1” Hollow 1.00 0.26Iron 1.75” Solid 1.92 0.05Iron 3” Solid 1.68 0.05

Steel A36 1” Solid 1.49 0.12Steel 4130 1” Solid 1.23 0.35Steel 1018 1” Solid 1.24 0.12

Magnet 1” Solid 1.62 0.16Mu-Metal 4” Rolled 1.37 0.02Aluminum 1” Solid 0 0

Table 6: Velocity Results in m⁄s

A plot of the deviation of the data is shownin Figure 10. For certain materials, the launchvelocities were rather inconsistent leading towider than ideal spreads. We attribute this in-consistency to poor surface finishes on some ofthe machined slugs, causing friction to vary sig-nificantly depending on the orientation of theslug in the barrel.

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

Laun

ch V

eloc

ity (m

/s)

Figure 10: Deviation of launch velocity resultsfor different projectiles

6 Analysis

Comparison of Results

An analysis of ideal theoretical calculations, nu-merical simulations and experimental resultsleads to some interesting conclusions.

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First of all, we see that the order of mag-nitude theoretical calculation which was per-formed matches up surprisingly well with theexperimental results. The results from this cal-culation were off from the final results by afactor of about 3, quite close considering all ofthe assumptions that were made.

Furthermore, the FEMM numerically inte-grated velocity data lines up with the data quitewell and is off by less than a factor of two forthe iron slug. For some of the steels, however,the results differ significantly from the FEMMsimulation, a fact which we believe can be at-tributed to differences between the simulatedmaterials and the actual used materials.

Our Matlab dynamics model predicts exitvelocities on the order of 4m/s, about twice asfast as our experimental results showed. The dy-namic model makes several assumptions whichwould increase the exit velocity. It assumes thereis no friction or air resistance in this system, thata linear model of magnetization holds true andthat the field is radially uniform and equal inmagnitude to the axial field. The finite elementmodels showed that the radial uniformity ap-proximation was not a terrible, but the actualfield generally decreases in the radial direction.The Matlab model predicts peak magnetic fieldsinternal to the slug in excess of 3T using thelinear model. The saturation magnetization ofmost soft irons is around 1 or 2T. Thus theactual magnetization may be much lower thanthe linear model predicts. The combination ofthese three phenomena could explain the dis-crepancy between the dynamic model and theexperimental results.

Material Performance

Although the collected launch data for severalof the materials had high standard deviation,some definite conclusions can be drawn from thedata.

First of all, it is quite clear that solid pro-jectiles perform better than their hollow coun-terparts. Although the hollow projectiles haveless air resistance, the 1” long iron slugs clearlydemonstrate that the increased magnetizationfrom the extra material makes a significant dif-ference. This demonstrates that cross-sectionalarea is quite important in projectile design. The

1” long hollow slug reached an exit velocity of1.00 ± 0.26 m⁄s, much lower the 1” long solidslug which reached 1.72 ± 0.04 m⁄s. In fact, thehollow slug may have been the slowest of allof the projectiles tested. This much lower exitvelocity for the hollow projectile helps to corrob-orate our theoretical results: both the order ofmagnitude and detailed calculations show thatthe force is roughly proportional to the crosssectional area of the slug. Thus a hollow slugwould be expected to have a lower exit velocity,which we see in the experimental data.

Subsequently, it is interesting that length ofiron projectile did not significantly affect thelaunch performance of the coil gun. This is be-cause the larger projectiles obtain higher levelsof magnetization but also have higher massesand normal forces in the barrel, leading to in-creased friction. This suggests that a coil gunof a fixed design may be able to launch a largerange of projectiles with good results. In fact, inthe order of magnitude approximation the massof the slug has no effect on the exit velocity,using equation 5:

vexit =

√2

mV µ0χmn2I2 =

√2ρmµ0χmn2I2

(24)That is, the exit velocity depends only on

the density of the object.Next, the data shows that the iron projectiles

performed better than their steel counterparts.This was expected because iron is a strongerferromagnetic material than most steels. How-ever, there is not a correlation between percentiron content in a steel and the performance ofthe steel, leading us to believe that the solutesof the alloy have a greater affect its magneticsusceptibility than the iron content. It is alsopossible that the process by which the steelwas formed affects its magnetic performance.The data suggests that the hot-rolled A36 steelperformed better than its cold-rolled 1018 coun-terpart. However, the data from the 4130 cold-rolled steel is inconclusive.

The both analytical results suggest that theexit velocity is proportional to the magneticsusceptibility of the material. This is the mostlikely explanation for the

Finally, the launch performance of the rolled

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mu-metal and the permanent neodymium mag-net raises interesting conclusions. In tests, themu-metal performed about as well as the testedsteels. However, in its rolled configuration andwith half the mass of the steel projectiles, itsperformance is nonetheless impressive. We pos-tulate that if we had a solid mu-metal projectile,it would have performed better than both theiron and steel projectiles. However, mu-metal isa rather rare material and is not commonly pro-duced in forms other than sheets. Furthermore,the performance of the permanent magnet wasdisappointing, with its launch velocities strug-gling to keep up with the solid iron slugs. Wesuspect this disparity is because of the high mag-netization of the iron projectiles which, whencombined with an intense magnetic field, causedthe projectiles to reach magnetic saturation, aspredicted by our dynamics model. The magnethas a lower saturation magnetization of 1.3 Twhen compared to the Iron’s 2.3T[3]. Further-more, the magnet had a smaller diameter thanthe slugs and this decreased cross-sectional areamay have affected the performance.

7 Conclusion

In conclusion, we successfully built a coil gunand tested its performance with a wide variety ofprojectiles. In the process, we used several meth-ods to predict our gun’s performance includingthe development of a custom, remarkably ac-curate mathematical model. Furthermore, con-clusive results were reached for a number ofdifferent classes of projectiles. It was found thatsolid iron projectiles perform very well in a coilgun and that projectile length is not very im-portant to launch performance. Furthermore,permanent magnets perform well in a coil gunbut not substantially different from their ironcounterparts. Additionally, steel projectiles canperform well, but not as well as iron slugs. How-ever, steel is a harder and stronger materialand may be more useful than iron in real-worldapplications.

8 Further Work

Further work in the topic could be done in a num-ber of areas. In order to improve the dynamicmodel, an expression for the off-axis field of afinite solenoid could be derived, and further in-vestigation could be done into non-linear magne-tization models. The experimental setup couldbe improved by investigating barrel materialsand lubricants to reduce friction. Furthermore,improved finish on the projectiles would helpto reduce the exit velocity variance and reducefriction. Further study could be done to ana-lyze a wider class of projectiles. Attributes likeprojectile shape, a wider range of alloys, typesof permanent magnets and projectile lengths.Second, exploration could be done into how dif-ferent barrel materials affect performance andpossibly into lubrication as a method of decreas-ing barrel friction. Nevertheless, we feel thateven without including some of these additionalfactors, our results are conclusive and useful.

References

[1] Griffiths, David J. Introduction to Electrody-namics (3rd Edition). Upper Saddle River,New Jersey: Prentice-Hall, 1999.

[2] Hansen, Barry. Barry’s Coilgun Designs.http://coilgun.info, 2012.

[3] Miner, Douglas F. and John B. Seastone.Handbook of Engineering Materials (1st Edi-tion). New York: John Wiley & Sons, 1955.

[4] Livingston, James D. Electronic Propertiesof Engineering Materials. New York: JohnWiley & Sons, 1999.

[5] Hummel, Rolf E. Electronic Properties ofMaterials (3rd Edition). New York: Springer-Verlag, 2001.

[6] MatWeb, Online Materials Information Re-source. http://matweb.com, 2012.

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