1. Theoretical and Experimental Analysis of a Newly
Designed Copper Electromagnetic Rail Gun
Connor Walton
May 1, 2016
Abstract
For my senior project, I set out to produce a 400V+ electromagnetic rail gun with
high muzzle velocity capabilities. I significantly modified a previous student’s work on
an electromagnetic rail gun, by improving the design of the projectile, rails, injection
system, and electrical system. I applied Ampere’s Law to predict the magnitude of
the magnetic field in between the rails and utilized this concept and the Lorentz Force
Law in order to calculate the kinematics of the projectile. To test the theory behind
the newly designed rail gun as well as the firing capability, I launched a T-shaped
aluminum projectile and measured its exit velocity.
1
4. 1 Introduction
A rail gun is an electromagnetic device that propels a particle at high speeds by converting
electrical energy to kinetic energy. A DC voltage is produced by an electrical system and
then a driving current flows through two rails and projectile. Abiding by basic laws of
electromagnetism, the electric current creates a Lorentz force that accelerates the projectile
away from the device at high speeds.
This technology has a definite advantage over other types of particle-accelerating
devices/fire-arms. The acceleration of a bullet fired from gas powered guns is limited to how
quickly the gas expands. But for rail guns, the conversion of electric energy into kinetic
energy is effectively instantaneous, so these devices are not limited a maximum acceleration
[digitalcommons]. Rail guns have an advantage because the driving force acting on rail
gun projectile is directly proportional to the amount of current passing through the projec-
tile [carrollcollege]. Despite these strengths in the rail gun design, there are significant
drawbacks. A rail gun is very susceptible to damage when the projectile is fired. A common
flaw that occurs during the operation of a rail gun is that the projectile tends to electrically
weld to the rails when it is moving down the length of the rails. This causes drastic loss in
the projectile’s kinetic energy and it damages the projectile and rails. One way to mitigate
or eliminate this flaw is to make sure that the projectile is in constant contact with the
rails. While there are flaws in this technology and room for improvement in design, there is
theoretically no limit to how much electrical energy that can be put into the rail gun device
and thus no limit to the projectile’s force, acceleration, and maximum velocity.
1.1 Device Composition
Rail guns are typically made of four main components: an injector, an electric power supply,
supported rails, and a projectile [digitalcommons].
A. Power Supply The power supply is the main part that makes a rail gun
work. This system’s function is to output a high DC voltage in order to fire the gun. The
subsystems consist of a initial voltage source, a charging system, and a device to deliver the
voltage. The starting voltage source delivers power to the charging system through wires
until a desired amount of voltage is reached. Then the charging system delivers current to
the rails and projectile and the projectile is accelerated down the length of the rails and
away from the device.
B. Rails The rails are conductive bars and serve as the barrel of the rail gun.
These bars are electrically connected to the power supply and they create the driving force
that propels the projectile. The bars are fixed in place parallel to one another on a supporting
material. In for the projectile to successfully be accelerated, the rails must be completely
4
5. parallel; this insures constant contact between the projectile and the rails which allows
electricity to flow through the projectile easily and without arcing.
C. Projectile The projectile is the object that is fired from the rail gun via the
magnetic force. This piece is electrically conductive in order for it to experience the desired
magnetic force. It is ideal for the projectile to have as little mass as possible. The less
massive the projectile, the greater acceleration it will experience. The projectile must also
be designed to have constant contact with the rails. If the projectile is not consistently
electrically connected to the rails during its period of acceleration, it will arc weld to the
rails and lose significant amounts of kinetic energy.
D. Injector This piece gives the projectile an initial velocity before it is acceler-
ated down the metal rails. For high voltage rail guns, this is a vital component that enables
the projectile to be effectively accelerated away from the rails. If the projectile already has
initial motion prior to moving down the length of the rails, it is less likely for the projectile
to experience arc welding.
The concept of a rail gun has been around for quite awhile; it existed for decades
in the imagination of video game creators and writers of science fiction as well as military
visionaries. It first appeared as a weapon in the video games “Quake 3” and “Metal Gear
Solid 2”[dartmouthchien]. The first real rail gun was built in 1944 by Joachim H¨ansler and
it could launch a 10 gram projectile at 1 km/s [woosteremaccel]. The fastest projectile
speed achieved so far is 5,600 mph (2.5 km/s) by the U.S. Navy. Right now, the Navy is
highly involved in rail gun development and they are planning test fire their rail gun at
sea in the summer of 2016 [navy]. So far, rail guns have primarily been used as weapons.
However, as this technology improves, it may have other important applications, including
future space launches.
2 Theory
2.1 The Underlying Launching Force: Lorentz Force
When an electric current flows through the rails and projectile, this produces net magnetic
field orthogonal to the direction of current. The magnetic field and current then induce an
orthogonal force, the Lorentz force.
5
6. Figure 1: Railgun diagram [doityourself] (with edited coordinate axes)
As shown in Figure 1, the driving electric current flowing through the left-most
rail in the x-direction induces a magnetic field that rotates counterclockwise around the
rail. Similarly, the current flowing in the other rail in the negative x-direction will induce a
magnetic field that rotates clockwise around the wire. The magnetic fields created by the
rails add together as a net magnetic field that points in the z-direction between the rails. As
for the projectile, current is flowing in the y-direction through the projectile. Since current
flows through the projectile in the positive y-direction and the induced magnetic field is
in the z-direction, the Lorentz force on the projectile is created and points in the positive
x-direction.
The Lorentz force on a segment of wire dl is represented by the vector product of
the current I and the magnetic field B below [e&mgriffiths]:
Fmag = (I × B)dl. (1)
If we apply this law to the projectile, we can assume that the current I and magnetic field
B do not vary with position. The equation then simplifies to:
Fmag = Il × B
where l is the length of one of the projectile. I is perpendicular to B, so the equation reads:
Fmag = IlB. (2)
2.2 Subprinciples
Rail gun circuitry often consists of one or more capacitors to store electrical energy. The
circuit can be assumed as an RC circuit by treating the projectile as a resistor of resistance R
6
7. assuming that the resistance of the rest of the electrical circuit is negligible compared to the
projectile [princeton]. So then the current flowing through the circuit is then represented
by the following time-dependent relationship:
I(t) = I0e−t/RC
=
V0
R
e−t/RC
(3)
where V0 is the initial voltage of the capacitor. This relationship will be used for calculations
throughout the paper.
Since the direction of current flow in each rail is opposite from the other, this causes
a repulsive magnetic force to be exerted on the rails. This is derived from Ampere’s Law
for two parallel wires 1 and 2 with different currents I1 and I2 flowing in opposing directions
[giancoli]:
F2 =
µ0
2π
I1I2
d
l2 (4)
where µ0 is the permeability of free space and l2 is the length of wire 2. We can assume that
the current on each rail has the same magnitude, so the repulsive force on each rail is:
F =
µ0
2π
I2
d
lr (5)
where lr is the length of each rail. It is important to be aware of this force as it can damage
the rail set-up if the rails are not fastened securely. The greater the electrical current, the
stronger the reinforcement will be required in order to compensate for the repulsive force.
2.3 Specific Calculations
2.3.1 Average Magnetic Field, Lorentz Force, and Exit Velocity
In order to calculate the Lorentz force acting on the projectile, the average magnetic field
between the two rails must be determined first. To find the average magnetic field, I used
Amp`ere’s Law for the magnetic field due to a long wire of radius r carrying current I:
B =
µ0I
2πr
.
Treating the rails of the rail gun as long wires gives that the magnetic fields pro-
duced by the rail are B1(s) = µ0I
2π(s)
and B2(s) = µ0I
2π(b−s)
where s is the distance from the
center of a rail as seen in the figure below.
7
8. Figure 2: Diagram of rails producing magnetic field
So the total magnetic field as a function of the distance s is:
B(s) = B1(s) + B2(s) =
µ0I
2π(1
s
+ 1
b−s
)
.
Then the average magnetic field is,
Bavg =
1
2(b − 2a)
b−a
a
B(s)ds =
µ0I
2π(b − 2a)
ln
b − a
a
=
µ0 ln b−a
a
2π(b − 2a)
V0
R
e−t/RC
.
By assuming b a, the average magnetic field is,
Bavg(t) =
µ0 ln b−a
a
2π(b − 2a)
V0
R
e−t/RC
. (6)
8
9. By plugging in Equation 1 for I and Equation 6 for B in equation 3, we obtain the
Lorentz force:
Fmag(t) =
µ0V 2
0 ln(b/a)
2πR2
e−2t/RC
. (7)
3 Equipment
• Variac
• Transformer
• Computer fan - used to cool off the transformer
• 2 6V batteries
• Rectifier
• 1 2200 µF 500V Capacitor, 1 6300µF 400V Capacitor
• 600V-rated wires
• Electrical box 15.88in x 15.88in x 5.5in
• Large Resistor
• 2 Copper Alloy C11000 Bars 3/8 in x 3/4 in 1.5’
• Graphite Dry Film Lubricant
3.1 Pnuematic and Rail Support Components
• Push-to-Connect Tube Fitting for Air Straight Adapter for 1/8” Tube OD x 1/8 NPT
Male
• Ultra-Chemical-Resistant Versilon PVC Tubing
• Impact-Resistant UHMW Polyethylene Rod 1/2” Diameter, 2’ Length
• Impact-Resistant UHMW Polyethylene Rectangular Bar 3/8” Thick, 3/4” Width, 2’
Length
• Impact-Resistant UHMW Polyethylene Rectangular Bar 2” Thick, 2” Width, 1’ Length,
White
• Impact-Resistant UHMW Polyethylene Rectangular Bar 3/8” Thick, 2” Width, 2’
Length
• Brass Compression Tube Fitting Straight Adapter for 1/4” Tube OD x 1/8 Male Pipe
• Air compressor
9
10. 4 Design and Building Process
4.1 Rails and Support System
My rail system is composed of two 1.5’ bars made of Copper Alloy C11000. I decided on this
type of metal because of its high conductivity and temperature range. To support the bars
so I mounted them on to a 3/8” x 2” x 2’ piece of UHMW that was fixed on top of three 2”
wide, 2” blocks of UHMW shown in Figure 3.
Figure 3: Copper Rails and UHMW Polyethylene support. This photo was taken at an
earlier stage of the overall design.
Next I drilled three holes into each bar - one on each end of the bar and one in
the middle of the bar. Alum screws were then screwed into the holes and I placed bolts on
the ends of the screws. The holes in the bars were made larger than the screws as to allow
the rail separation distance to be adjustable to the width of projectile being used. After
this, I adjusted the rails to have a uniform separation distance that had the same distance
as the projectile’s width (see Figure 4). and fastened them to the UHMW fixture. Once I
got the right separation distance at every point in the barrel, I fixed the rails into place, by
10
11. screwing the bolts up snuggly against the bottom of the UHMW piece as shown in Figure
4. Adjusting the rails to be fixed parallel to one another was essential as it allowed the
projectile to be in maximum contact with the rails as it is accelerated down the barrel. As
a safety feature, I acquired a plastic piece to place on top of the rails. I drilled six holes into
the plastic that were at least as large as the head of the alum screws so that the piece would
easily fit on top of the rails. The width of the projectile wing required the plastic piece to
be elevated at least 1 mm above the rails, so I had a friend cut out 4 aluminum pieces that
had a thickness that was slightly greater than the projectile’s wing width and then I taped
the shims on to the rails. I also placed random heavy metal objects on the plastic piece in
the locations of the shims to keep the plastic piece in contact with the shims. This created
a small distance between the top of the projectile’s wing and the bottom of the plastic piece
which allowed the projectile to slide down the length of the rails, but still be able to maintain
good electrical contact.
4.2 Electrical
For the electrical aspect of the rail gun, I used a previous student’s electrical system and
added a second capacitor and other components, and I did significant rewiring of the circuit.
There are two key circuits that make up the rail gun’s electrical system, a charging circuit
and a discharging circuit.
A. Charging The charging circuit consists of a source of AC current, a voltage
multiplier, and a device to convert the voltage from AC to DC. First, a cord is plugged into
a wall outlet and is connected to a Variac, a device that allows the user to select a specific
amount of AC voltage (see Figure 4). Next, the voltage travels through a transformer and
increases by a factor of 10. The transformer put out a lot of heat and started charring
the box at one point, so I screwed in a computer fan on to top of the bottom left ledge
and connected to two 6V batteries and a switch attached to the outside of the wooden box
(see Figure 5). Adding this source of heat and air flow required there to be another hole
in the box for air to flow. Luckily, the previous student that constructed the box left a
hole on the right side of the box, so this allowed my computer fan to effectively cool off the
transformer. The voltage is then converted into DC voltage by flowing through a rectifier.
Then the DC voltage goes through the two blue wires and charges a set of two capacitors
that are connected in series, one that is 400V 6300 µF and the other that is 500V 2200
µF. Connecting two capacitors The capacitors are hooked up to two test points that are
attached to the hinged plastic covering of the box. To measure the total voltage across both
capacitors, I used a multimeter and stuck two probes into the test points.
B. Discharging The discharging circuit is a more intricate system. This system
consists of the capacitors, the rails and projectile, a large resistor, and a switch to drain
the voltage into the resistor. As shown in both Figure 4 and 5, one white wire is connected
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12. to a terminal (positive terminal) on the left capacitor and one black wire is connected to a
terminal (negative terminal) on the right capacitor. The black and white wires come together
in a larger black tube and on the other end of the tube, the black and wires stick out and
are connected to the ends of the rails. I stripped the ends of the wires and crimped ring
terminals to the loose wire to establish a tight connection. The right side of Figure 8 shows
the connection between the rail and the white wire. I fit the ring terminal on to the thread of
the alum screw and then I screwed on a bolt to insure a good electrical connection between
the rail and the wire as well as to tighten that end of the rail to the UHMW fixture.
I needed to be able to manually drain the voltage from the capacitors. To do this,
I connected two wires to the green resistor to the wire junctions at the capacitor terminals
and connected two more wires from the wire junctions to a switch (see top of Figure 4 and
5). I would connect the capacitors to the large green resistor as seen on the right side of
Figure 4 and 5. Wires coming from the two wires junctions on the capacitors are also hooked
up to a big green resistor as seen on the right side of Figure 4 and 5.
Figure 4: Top down view of electrical system with lid open
12
13. Figure 5: Top down view of electrical system with lid closed
13
14. Figure 6: View of electrical system and switch to computer fan
4.3 Projectile
There are many design aspects that I needed to consider when deciding on a projectile. The
goal was to design a projectile that was as lightweight as possible, highly conductive, a high
melting point, able to maintain constant contact with the rails, and have structural stability
to withstand high electrical current. My first design was a thin strip of slightly springy
copper metal and it was not very successful. The elasticity of the projectile insured constant
electrical connection between the rails, but it couldn’t handle the voltage and parts of the
projectile fragmented during a shot. The best projectile design that I built and tested my
rail gun with was a T-shaped slug made of aluminum(see Figure 7). The base width of the
T-shaped slug was able to slide smoothly between the rails and the projectile wing was in
constant contact with the rails due to gravity. To reduce the mass of the projectile, I cut
out a small section from the bottom of the slug (see Figure 8).
14
15. Figure 7: Front view of projectile
Figure 8: Bottom of projectile
15
16. 4.4 Injection System
For the injector, I acquired a aluminum metal trough that was 2 feet long and 0.375 inches
wide. I placed the trough onto the UHMW fixture about 5 mm away from the rails. With
this design, the core of the projectile was able to hang down into the trough and the wings
would rest on the sides of the trough. In order to fix the trough into place, I drilled two
holes in the end of the trough and drilled two holes into the UHMW fixture. Then I screwed
the trough on to the fixture. To propel the projectile, I used an air compressor. I attached a
push-to-connect tube fitting for 1/8 inch OD (outer diameter) tubing on the air compressor’s
hose and then attached a 1/8 inch OD air tube on the fitting. I fixed the other end of the
tube inside of the trough, but the blast of air unfortunately had aerodynamic effects on the
projectile and the projectile frequently ascended up into the air before reaching the rails.
The way I corrected this error was by fitting a pipe that had the same length as the trough
and that was 0.375 inch wide x 0.375 inch thick rectangular pipe onto the trough. I made
about a 2 inch cut on the bottom of the trough so that the pipe would snug fit into the
UHMW. To fix the pipe into place, I acquired an aluminum block and cut out a 0.1 inch
deep, 0.375 in wide section and placed it on top of the trough. Then I drilled and tapped
two holes into the UHMW that were on both sides of the pipe as shown in Figure 9. Finally
I drilled one hole on each side of the metal trough fastener and screwed alum screws into
the holes of the metal and UHMW, restricting the pipe from moving. I had to make another
adjustment because the projectile would often ascend after exiting the injector right before
reaching the rails and it would collide with the insulating plastic piece on the rails. To keep
the projectile from ascending after it exited the injector, I placed two blocks of aluminum
on the end of the rails next to the injector. This raised the plastic covering at that end of
the rails to the height of the square pipe, so it was able to channel the projectile into the
spacing between the rails. To make sure that the plastic piece did not move, I placed a
random heavy piece of metal on the plastic in the locations of the metal blocks.
Figure 9: Close-up of injector fastened to the UHMW support.
On the other end of the pipe, I cut out a small section on the top of the pipe (see
Figures 10 and 11). This allowed me to easily load the projectile in the trough at a consistent
distance from the rails. Finally, to propel the projectile into the rail system, I inserted the
16
17. air tube inside of the pipe right up against the projectile and released the air from the air
compressor.
Figure 10: Front view of injector. The projectile is resting on top of the trough inside of the
tube.
Figure 11: Top-down view of the projectile loaded into the injector.
17
18. 5 Device Testing
5.1 Average Magnetic Field and Projectile Kinematics
From constructing my rail gun, the quantities that I was most interested in measuring were
the average magnetic field between the rails, the Lorentz force acting on the projectile, and
the exit velocity of the projectile. I was not able to obtain a sufficient method to measure
the average magnetic field nor Lorentz force, but I was able to obtain predicted quantities
and plots for the average magnetic field and Lorentz force using Wolfram Mathematica (see
Data). I was able to measure the exit velocity and make quantitative interpretations of the
data. I wanted to quantify the projectile’s exit velocity dependence on the air flow from
the air compressor, so I recorded the exit velocity at various air pressures with no voltage
across the rails. Then I wanted to determine how effective my rails were in accelerating the
projectile, so I shot the projectile at different voltages at a constant air pressure of 100 PSI.
A. Projectile Dimensions and Rail Separation I took measurements for the
exit velocity of the T-shaped projectile using a 1.869 g projectile with a wing span of 14.8
mm, wing thickness of 1.33 mm, and base width of 8.98 mm and base length of 15 mm.
To allow the projectile to slide smoothly through between the rails but still have contact
on the sides, I adjusted the rails to have a separation distance of 9.00 mm. I also applied
Graphite Dry Film Lubricant to the tops and sides of the rails as well as the projectile to
reduce friction, increase conductivity, and improve electrical contact, which would mitigate
arc-welding.
B. Set-up and Operation of EMRG For my experimental set-up, I placed my
rail system and injector on a counter and put a wooden box with a soft absorbing material
about a foot away from the device to capture the launched projectile (see Figure 12). I stored
my electrical box and air compressor on a cart away from the rails and injector. When I
wanted to record the exit velocity of the projectile with zero voltage, I inserted the projectile
into the injector and then plugged in the air compressor and powered it until it reached
a desired air pressure. Then I held the air tube up to the projectile and opened the air
compressor’s valve to release air and launch the projectile. (As I was testing the injection
system, I observed that the projectile was hitting a part of the left rail and lost a noticeable
amount of momentum at that location. The shape of the right rail was slightly different than
the left rail and it allowed for the projectile to slide smoothly from the injector to the barrel
even if it were to touch the right rail first. So, I attempted to eliminate the interference
between the rails and projectile by placing sheets of metal under the left side of the UHMW
fixture to incline the left side of the injector and rails at a slight angle). To measure the
muzzle velocity with an applied voltage, I plugged the Variac’s cord into a wall outlet. Then
I inserted the multimeter’s probes into the test points of on the outside of the electrical box.
I adjusted the voltage that the Variac was producing until it reached the value I needed.
After that I charged the air compressor to 100 PSI and then I launched the projectile.
18
19. C. Velocity Recording Equipment To measure the exit velocity of the pro-
jectile, I used a Photogate and recorded the data using LoggerPro software and a Vernier
LabPro. I fixed the Photogate on to a ring stand to a height in which the projectile would
intercept the laser beam that travels from one arm of the Photogate to the other (see Figure
12). I used the “Gate Timing” mode in the LoggerPro software to record the exit velocity.
The Photogate recorded the amount of time that laser beam was intercepted and divided it
by the length of the projectile.
Figure 12: Testing set-up
19
20. Figure 13: Rail system on test day. Plastic covering is weighed down in two locations and
the rail system is propped up at an angle.
20
21. 6 Data
6.1 Exit Velocity Measurements
Table 1: Exit Velocity for Different Air Pressures
Air Pressure (PSI) Trial 1 (m/s) Trial 2 (m/s) Trial 3 (m/s) Average (m/s)
20 2.308 2.151 2.027 2.162
30 3.142 2.901 3.329 3.124
40 3.309 3.358 4.126 3.597666667
50 4.17 5.008 5.185 4.787666667
60 5.14 6.233 4.725 5.366
70 6.813 6.233 6.233 6.426333333
80 5.979 7.146 5.979 6.368
90 7.71 8.138 7.71 7.852666667
100 7.324 6.975 7.324 7.207666667
Figure 14: Zero voltage exit velocities fit to a linear curve
Table 2: Exit velocities at 100 PSI for different voltages
Voltage (V) Exit Velocity (m/s)
50 6.716
100 7.813
150 6.658
200 6.369
250 6.369
21
22. Figure 15: Plot of measured exit velocities at different voltages
6.2 Electrical Quantities
• Resistance of projectile: R = 10−6
Ω
• Calculated equivalent capacitance of the two capacitors in series: Ceq = 1630µF
• Calculated rail “radius”:a = 0.00760m
• Calculated separation distance between centers of rails: b = 0.2921m
22
23. 6.3 Predicted Magnetic Field
Figure 16: Mathematica file and code with Bavg(t) plot.
6.4 Predicted Lorentz Force and Exit Velocity Plots and Values
To predict the exit velocity, I could not derived this quantity from the Lorentz force because
the formula yielded very impractical results. I applied a different theoretical approach and
used the predicted exit velocity derived from this theory [princeton]:
vf =
6Uc
Rmc
ln(b/a)
where Uc is the potential energy of both of my capacitors, R is the resistance (assumed to
be constant) of the projectile, and c is the speed of light.
23
25. Figure 18: Energy of capacitors at an equivalent capacitance, predicted vf at 100V and
vf(R) plot with the projectile used for testing. I got my injection velocity v0 = 7.21 m/s by
using the zero voltage exit velocity data point (100 PSI, 7.21m/s). I used this exit velocity
as the injection velocity for the exit velocity at 100 V by assuming the friction between
the rails and projectile for the zero voltage shots to be negligible and that the projectile is
experiencing negligible force from the air during it’s travel through the rails compared to its
journey through the injector.
25
26. 7 Results/Analysis
7.1 Zero Voltage Exit Velocity
Until 80 PSI, the data points of the projectile’s average exit velocity steadily fit a linear
curve. The average exit velocities at 80 PSI, 90 PSI, and 100 PSI suggest the possibility that
the slope of the curve levels off with pressures higher than 80 PSI, but there are not enough
data points to support that hypothesis. The method of manually holding the air tube up to
the projectile was inconsistent and prone to human error, so that likely affected the data as
well. However, I noticed in several trials that the projectile was still bumping into the left
rail even though I tried to account for that error by tilting the rail component at an angle.
This interference caused the projectile to exit the barrel at a lower velocity than trials with
no interference as seen in Table 1 and Figure 14. Figure 14 also shows that the data points
fit the curve less and less as the air pressure increased. This indicates that the effect of the
interference intensified as the air pressure increased.
7.2 Exit Velocity with Voltage
7.2.1 Experimental Aspects
Table 2 and Figure 15 both indicate that the voltage had little if not any effect on the
projectile. The data points I obtained was affected by several factors.
B. Injection Velocity In order for the projectile to successfully transition down
the barrel, it needs to have sufficient speed to overcome electric arc welding. The zero voltage
trial shots indicated that the projectile was traveling at a maximum of about 8 m/s at 100
PSI. Because of the high potential difference across the rails, it is likely that the projectile’s
injection speed was not high enough.
Electrical Contact While the bottom of the projectile’s wings did maintain
constant contact with the top surface of the rails with the help of gravity, the sides of the
projectile’s base was not in consistent contact with the side of the rails. After every shot, I
noticed circular patches on the projectile indicating arc welding.
A. Operation Error At some time after I fired the rail gun at 100V and recorded
the data for that shot, one of the shims elevating the plastic frame on top of the rails slid
toward the space between the rails, blocking the projectile path. This problem was unfortu-
nately unnoticed before I fired the rail gun again at 150 V and the shim halted the projectile
and caused it to weld to the rails. I had to make repairs during my experiment by sanding
26
27. the the projectile and recoating it with the graphite lubricant. Due to time constraints on
finishing the project, I repaired the rails as best as I could during the experiment by just
sanding the damaged parts with out reapplying the dry graphite spray. Unfortunately, this
introduced more friction between the rails and projectile at the damaged spots and very
likely intensified arc welding effects. As shown in the graph, I was able to achieve data for
150V, 200 V, and 250 V shots, but it is highly probable for the data to have been affected
by the damage that occured when I fired the rail gun at 150V. Any attempts to shoot the
gun above 250 V were unsuccessful as the projectile welded itself to the rails in mid launch.
7.3 Predicted Quantities
The theoretical values that I obtained for the average magnetic field, Lorentz force acting on
the projectile, and final velocity of the projectile act was found by assuming the resistance of
the circuit to be negligible compared to the projectile and for the resistance of the projectile
to be constant as it moves down the barrel of the rail gun. Assuming R as constant lead to
predictions of extremely high initial currents flowing through the projectile (I = V0/R) and
it predicted that the magnetic field drops from 300 T to 0 T in a after a few nanoseconds.
Similarly, these assumptions lead to predict a maximum Lorentz force of 108
N and it claimed
that this force would rapidly approach zero in a matter of nanoseconds.
On the other hand, calculating the exit velocity vf with an initial voltage V = 100V
and injection velocity of v0 = 7.21 m/s resulted in 30 m/s, which isn’t totally unreasonable.
However, it appears that assuming vf to depend on R yielded a more likely model for pre-
dicting vf. Plotting the exit vf as a function of R claims that vf gets very close to v0 if
the resistance increases to a 0.0002 Ω, which is much more supported by the data in Table
2. This plot and the extremely high predicted quantities for the average magnetic field and
Lorentz force strongly suggest that that assuming the projectile to have constant resistance
as it is traveling down the length of the rails is not a valid assumption.
8 Conclusion
The process of almost entirely redesigning an EMRG and figuring out the theory behind
this device has been a fun and very challenging experience. From testing the rail gun, I
see that there are several areas to improve on, both in the theory and design of the device.
In the experiment, the Lorentz force appeared to have little to no effect on the projectile’s
maximum speed for most shots. It is possible that the data for 150 V and above may have
been different if one of the shims hadn’t blocked the spacing between the rails and caused
damage to the projectile and rails. However, it is most likely that the Lorentz force appeared
to have very little influence on the projectile due to the arc welding between the sides of the
projectile and sides of the rails. A possible design improvement would be to cut out slits on
both sides of the rails and create projectile that fits in both slits. This would insure much
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28. better contact and less arc welding effects.
Another aspect to focus on regarding design improvement is the projectile’s mass;
Making lighter projectiles would likely improve the overall performance of the gun. The
projectile would achieve greater acceleration in the injection system as well as in the rail
configuration. The projectile would also have a higher initial speed before sliding before
reaching the rails and this would lessen arc welding. One possible approach to achieve a
lighter projectile would be to decrease the separation distance between the rails. This would
require a less wide and therefore, less massive projectile.
The theory used for calculations of the average magnetic field between the rails
as well as the Lorentz force and projectile exit velocity presented very impractical models.
This is likely due to the assumptions used, such as assuming that the projectile’s resistance
is constant. As demonstrated in the experiment, the resistance of the projectile is certainly
not constant. The resulting equations also neglected inductive effects that occur when the
projectile slides down the rails and completes the electrical circuit. One possible way to
improve the theory would be to account for induction and to treat the projectile’s resistance
as a function of temperature.
9 Acknowledgments
I would like to thank Steve Ward for his support in the completion of my design by providing
guidance for picking materials to use, and helping me to figure out appropriate dimensions.
I would also like to thank Jonathan Langton for machining projectiles for me and pushing
me to independently figure out most of the problems that I encountered with the project. I
would also like to give a big thanks to my best friend Matthew Herman for taking the time
to help me redesign a large amount of the electrical system and his assistance with taking
data.
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