eodm-rep-si-002-02

EARTH ORBITING DEBRIS MITIGATION

DESIGN PROJECT

Subsystem

Earth Gun Ballistic Model and Space Gun Reservoir Design

EODM-REP-SI-002-02

April 8, 2015

Sultan Islam

Approved By: __________________

Checked By: Theo Ridley

Carleton University EODM-REP-SI-002-02 April 2014


Sultan Islam (100822163) Earth Orbiting Debris Mitigation ii

Revision Version Date Comments

EODM-REP-SI-002-01 5th April 2015 Original

EODM-REP-SI-002-02 8th April 2015 Peer Edited by Group

Member

Abstract

This year final report discusses the work done over the whole year (from Fall 2014 to Winter

2015). A method for determining muzzle velocity was created and using this method with a

reservoir pressure of 20 psi the initial muzzle velocity was determined to be 31.62 m/s. Utilizing

the measured projectile mass of 0.3 grams and a diameter of 1.25 cm the quadratic drag factor

was determined to be 0.0192 Ns2/m2. Preliminary design was done on the space net gun. Sizing

was done in an 87.5 by 64 by 10 cm rectangular volume and using the Darcy-Wiesbach equation

along with the loss factors and other pipe lengths and volume, a required reservoir pressure of

39.68 psi was determined. The report also discusses the problems that have arisen and the

possible future work that can be done from this point on.



Sultan Islam (100822163) Earth Orbiting Debris Mitigation iii

Table of Contents

1.0 Introduction .......................................................................................................................... 1

2.0 Earth Net Gun Ballistic ........................................................................................................ 1

2.1 Problem Statement ........................................................................................................... 1

2.2 Work Performed ............................................................................................................... 1

2.2.1 Determining Initial Muzzle Velocity ........................................................................ 2

2.2.2 Determining the air drag factor ................................................................................. 3

2.2.3 Mathematical model of the earth net gun ballistic profile ........................................ 4

3.0 Space Net Gun Reservoir and Piping Design ...................................................................... 6

3.1 Problem Statement ........................................................................................................... 6

3.2 Work Performed ............................................................................................................... 6

3.2.1 Pressure Regulator .................................................................................................... 6

3.2.2 Reservoir and Piping Design .................................................................................... 6

3.2.3 Valves and Connectors ............................................................................................. 7

3.2.4 Pipe Flow Calculation and Schematics ..................................................................... 9

4.0 Summary ............................................................................................................................ 11

5.0 References .......................................................................................................................... 12

6.0 Appendices ......................................................................................................................... 13

6.1 Appendix A: Frame Analysis Used to Determine Muzzle/Initial Velocity ................... 13

6.2 Appendix B: Frame Analysis to determine drag coefficient .......................................... 15

6.3 Appendix C: MATLAB Code For Ballistic Profile ....................................................... 16

6.4 Appendix D: Fluid Flow Calculation Results ................................................................ 17

6.5 Appendix E: Preliminary Space Net Gun Hand Sketch ................................................. 18

6.6 Appendix F: Tech Memo 1 to 4 ..................................................................................... 19

List of Figures

Figure 1 Ballistic Profile of Earth Net Gun 5

Figure 2 Latch Valve Schematic [6] 7

Figure 3 Launch Velocities of Space Gun 9

Figure 4 Launch Velocities of Space Gun (Condensed) 10

Figure 5 Space Net Gun Schematic 11

Figure 6 Frame 1 14



Sultan Islam (100822163) Earth Orbiting Debris Mitigation iv

Figure 7 Frame 10 14

Figure 8 Frame 1 of Drag Experiment 15

Figure 9 Frame 2 of Drag Experiment 15

Figure 10 Rough Sketch of Space Net Gun Design 18

List of Tables

Table 1 Initial Velocity Analysis Result ......................................................................................... 3

Table 2 Drag Factor Determination Results ................................................................................... 4

Table 3 Fitting and Valve Losses [9] .............................................................................................. 8

Table 4 Tractor Velocity Vs. Pressure Differential ...................................................................... 17

Table 5 Tractor Velocity Vs. Pressure Differential (Condensed) ................................................. 17

Abbreviations

FPS Frames per Second

EODM Earth Orbit Debris Mitigation

gHE Gaseous Helium

List of Symbols

Vmuzzle Muzzle Velocity (Initial)

C2 Quadratic Drag Coefficient

g Gravitational Acceleration of 9.81 m/s2

m Mass of Projectile

D Diameter of Projectile

to Starting time

T time of flight (time elapsed)

θ Angle between velocity vectors

γ Quadratic Drag Factor

u Fluid Velocity



Sultan Islam (100822163) Earth Orbiting Debris Mitigation v

K Minor Flow Loss

(L/D)eq Equivalent Length to Diameter Ratio

𝑉�̇� Acceleration of projectile for the nth direction

Vter Terminal Velocity



Sultan Islam (100822163) Earth Orbiting Debris Mitigation 1

1.0 Introduction

EODM (Earth Orbiting Debris Mitigation) design project is aimed to design a spacecraft in order

to deorbit space debris. CAPTURE (Carleton Aerospace Project to Undertake Re-entry) is a

CubeSat based on a modular bus and payload design, and is designed to deorbit RADARSAT-1

and the DELTA-II second stage. The method in capturing and deorbiting is done using a

pneumatic net gun, this report focuses on the net gun itself.

This report outlines the contributions and work done for the design and analysis of CAPTURE

over the Fall 2014 and Winter 2015 term. Majority of the work is done on the net gun subsystem.

This report encompasses the work done on both the earth and space model of the net gun. This

introduction section outlines the topics covered within the report and in which order they are

discussed.

Section 2 covers the work done on the earth based gun, specifically ground testing. A

mathematical model was designed to describe the ballistic profile of the net gun projectile,

specifically one with drag and one without drag. Section 3 covers the work done on the

preliminary design of the space gun, specifically the reservoir and piping specs from the pressure

regulator up until the launch tubes. Section 4 summarizes the work done in the previous two

sections and discusses the problems that were encountered and any future work required for each

of the topics discussed previously. The details of each contribution and a discussion are included

in their respective sections.

2.0 Earth Net Gun Ballistic

2.1 Problem Statement

One of the few things that were not considered last year (probably due to time restraints) is a

mathematical model of the ballistics of the earth net gun that was designed previously. A model

for the ballistics is needed using experimental data such that this model can be used to verify and

correct errors in the theoretical ballistic model of the gun based on fluid flow between the

reservoir and launch tube. The next few subsections will look into addressing this problem.

2.2 Work Performed

Before a ballistic model can be discussed, work is needed on a method to experimentally

determine the initial muzzle velocity of the projectile, section 2.2.1 discusses this further.

Secondly a method is needed to determine the air drag factor of the projectile, section 2.2.2

discusses this further. Having both the initial muzzle velocity and the drag factor determined, a

MATLAB script was created to compare the ballistic profile on the projectile for both the drag

and no drag case. This is discussed in section 2.2.3 which rounds off the work performed for the

earth net gun.




2.2.1 Determining Initial Muzzle Velocity

In order to determine the initial muzzle velocity, a 1000 frames per second high speed camera

was used to capture the motion of the projectile at a small rate such that the speed can be

determined. In order to determine the speed, a 50 by 50 mm grid was used, aligned with the

inclination plane of the launch tubes. The camera was focused on the grid and then the

projectiles were launched over the grid such that the displacement of the projectile can be

measured.

In order to determine velocity, the time of flight of the projectile and displacement is needed.

The time of flight of the projectile is given by the equation bellow:

𝑇 =𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐹𝑟𝑎𝑚𝑒𝑠

𝐹𝑟𝑎𝑚𝑒 𝑅𝑎𝑡𝑒 EQ1

The displacement of the projectile then is given by the following equation:

𝐷 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒 × 50 𝑚𝑚 EQ2

Utilizing both Equations 1 and 2, the velocity (in the X and Y direction) can be determined by:

𝑉 =𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒 ×50 𝑚𝑚

𝑇 EQ3

It is important to note that due to the fact that the grid is aligned with the inclination plane and

not in the direction the launch tube is firing, the angle in between the velocity vector is not the

same as the angle of the launch tube, thus the angle between the velocity vectors can be

determined as:

𝜃 = 𝑇𝑎𝑛−1(𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑌

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑋) EQ4

Thus knowing the angle of the velocity vectors and the velocity of each component, the initial

muzzle velocity can be determined using the equation bellow:

𝑉𝑚𝑢𝑧𝑧𝑙𝑒 =𝑉𝑥

𝐶𝑜𝑠(𝜃) 𝑂𝑟

𝑉𝑦

𝑆𝑖𝑛(𝜃) EQ5

Since the high speed camera outputs each frame in a bitmap (.bmp) files, each of the photo were

analyzed. Out of the 30 frames which exhibited motion, 10 were used to calculate the muzzle

velocity. The frames used can be found in Appendix A while a more detailed explanation

including all the analyzed frames can be found in EODM-TM-SI-002 [2] which can also be

found in the appendix at the end of this report. Utilizing this method, an initial muzzle velocity

of 31.62 m/s was determined for the projectile launched at a rated pressure of 20 psig. Table 1

bellow list the results found for this section.




Table 1 Initial Velocity Analysis Result

Parameters Value

Measured Velocity in the X-direction (m/s) 30

Measured Velocity in the Y-direction (m/s) 10

Approximated Angle of the Velocity Vector

(deg)

18.43

Approximated Muzzle Velocity (m/s) 31.62-31.63

2.2.2 Determining the air drag factor

The next step is to determine the air drag factor. In order to find the air drag factor lets first recall

the drag force equation.

EQ6

As seen in Equation 6, the quadratic drag force is given as the drag coefficient “c” and the

velocity. The drag coefficient “c” can be found using the quadratic drag factor (based on

geometry of the projectile) and the diameter of the projectile itself. The gamma term in the above

equation is the quadratic drag factor that needs to be found in order to complete the ballistic

profile for the gun. This was done using a high speed camera as well.

In this experiment a 240 frames per second camera was used. The camera was pointed directly at

a grid which is measured to be a 50 by 50 mm grid. The projectile was held above the grid and

then was dropped from rest at a pre-determined height. Setting the zero datum point at the rest

position, the terminal velocity achieved by the projectile can be measured by determining the

time of flight and the displacement of the projectile.

Due to the fact that the footage recorded was in a video format (.mp4) the equations used will be

different from the equations used in the previous section. The time of flight can be determined

using this equation:

𝑇 =𝑉𝑖𝑑𝑒𝑜 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛


The displacement of the projectile can be found using equation 2 from the previous section. Thus

utilizing the solutions of equation 2 and 7 a terminal velocity can be found for the projectile. The

terminal velocity of an object dropped from rest can also be expressed using the drag coefficient

and the force gravity, this is given as:

EQ8 c

mgv ter




Since the drag coefficient is also defined in equation 6, using equations 6 in conjunction with

equation 8, the quadratic drag factor can be determined. The first and last frame of the video

used can be found in Appendix B (The total video duration was about 2 sec). Using a time of

flight of 0.008 sec, a projectile displacement of 0.25 m with a projectile mass of 0.3 grams and a

projectile diameter of 1.25 cm, the quadratic drag factor was determined as 0.0192 Ns2/m2. Table

2 (found bellow) tabulates the results of the image processing in this section.

Table 2 Drag Factor Determination Results

Parameters Value

Time of Flight (sec) 0.008

Measured Terminal Velocity (m/s) 31.25

Mass of Projectile (kg) 0.0003

Diameter of Projectile (m) 0.0125

Drag Coefficient 3 x 10-6

Drag Factor (Ns2/m2) 0.0192

2.2.3 Mathematical model of the earth net gun ballistic profile

Now that there is a method to determine the initial muzzle velocity of the projectile and with the

quadratic drag factor, the ballistic profile can be determined mathematically. There are two

models, one which includes the drag and one without drag.

The first model which includes drag can be determined using Newton law of motion and force

balance of the projectile itself. Looking at both X and Y direction, the force balance equations

become:

EQ9

EQ10

Both equations 9 and 10 include the effects of drag on the projectile. This equation was derived

from the use of equation 6, since we are only dealing with a quadratic drag factor and not a linear

one. Equations 9 and 10 need to be solved together as they are coupled for projectile motion.

Preliminary work on this during the beginning of the fall term did not account for this, and thus

the analysis found in the first technical memo (EODM-TM-SI-001) computed the motion in the x

and y direction individually. This was corrected and now the code found in Appendix C solves

equations 9 and 10 simultaneous using ODE 45.

For the model without drag force, force balance was taken again without the drag force. Thus

equations 9 and 10 are changed and expressed without the drag coefficient bellow:

𝑚𝑉�̇� = 0 EQ11

yyxy vvvcmgvm 22

xyxx vvvcvm 22




𝑚𝑉�̇� = −𝑚𝑔 EQ12

Since the equations output a velocity and time array, integrating the velocity array (which

include data for Vx and Vy) to give the displacements in their respective directions gives the

ballistic profile of the projectile once launched from the gun at a given speed. Thus using

equation 9 through 11 with the results from section 2.2.1, 2.2.2 and a desired time of flight, a

ballistic profile can be plotted as displacement in the x vs. displacement in the y direction. Note

that this model is set such that the height at which the gun is set at is considered the zero datum

point. Appendix C includes the main body of the code plus the functions used to calculate the

ballistic profile.

Thus for an initial muzzle velocity of 31.62 m/s shot out of a launch tube angled at 26.56 degree

with a mass of 0.3 grams and a quadratic drag coefficient of 3 x 10-6, the resulting ballistic

profile is shown in the plot bellow for a time of flight of 3 seconds.

Figure 1 Ballistic Profile of Earth Net Gun

The above profile was done for a random time of flight to illustrate the code works, with an

experimentally or pre-determined time of flight, the code will output the corresponding profile

for that case. This thus rounds up the work needed to be done for the earth net gun.




3.0 Space Net Gun Reservoir and Piping Design

3.1 Problem Statement

Since there was no work done for the space net gun design, majority of the winter semester was

spend on preliminary designs for the space gun. This report focuses on the design of the

reservoir, piping, valves and regulators which connect to a helium gas tank (used in ADCS and

Propulsion) to supply necessary fluid forces to ensure tractor launch and thus net launch in space.

Note that for the space model, there are two net guns, one for each target. The only requirement

given was that the whole design for the reservoir, piping between the helium tank and the launch

tube must fit inside an 87.5 by 64 by 10 cm rectangular volume. The launch tube connectors are

located at (87.5, 0, 10) cm and at (87.5, 64, 10) cm Cartesian co-ordinates. Using these

constrains along with keeping 90 degree turns at minimal a reservoir, piping, regulator and valve

design were chosen. The process of the design is discussed in detail in the section 3.2.

3.2 Work Performed

Most of the work performed was preliminary. Sections 3.2.1 will discuss the pressure regulator

choice and some design requirements, section 3.2.2 will discuss the reservoir and piping design

while section 3.2.3 will discuss connectors and the losses associated with them. Section 3.2.4

will put all the other section together and discuss the flow analysis done to compute the required

reservoir pressure.

3.2.1 Pressure Regulator

Since majority of the suppliers whom supply pressure regulators used in spacecraft and design to

the design points given by the customer, the specs of the regulator must be determined first.

Since the gaseous Helium (gHe) in the propulsion tank is pre-pressurized at 9 bar psia, the

regulators input must be able to handle that pressure. Both airbus space and defense and Moog

Inc supply pressure regulators which work with gHe. Moog Single stage regulator can handle an

inlet max pressure of 6.9-34.5 bar [3] while Airbus RDS2000 Helium Pressure Regulator can

handle an inlet pressure of 25-310 bar [4]. The limiting factor here for the inlet pressure far

exceeds the actual inlet pressure so these two options are viable. The outlet diameter is

determined by the diameter of piping that is being used. It is discussed in section 3.2.2. The

outlet pressure range typically for each of these regulators are 140-180 psig (with a +/- of 5 psig)

and 17-17.5 bar (+/- approx. +4/-2.5%) [3] [4]. Since the outlet pressure can change based on

manufacturing constraints given by the customer, the outlet pressure isn’t a requirement for

choosing a pressure regulator at this stage.

3.2.2 Reservoir and Piping Design

Cited from EODM-REP-TR-002-00, the diameter of the launch tube connectors were designed

to be 3.5 cm with a pipe thickness of 4 mm [5]. The piping in between the regulator, reservoir

and launch tubes are designed to the same specs. Thus the piping has an internal diameter of 3.5

cm with a thickness of 8 mm. The total length of piping (including the Wye bend and the wide

angle 90 degree bend, discussed in section 3.2.3) is about 34.85 cm. The reservoir design chosen




was to use a section of the pipe that was thicker. Since pressure determines the volume but the

same is true vice versa, an estimated pipe sizing was chosen for the reservoir design. The pipe

section used for the reservoir is about 57 cm, the internal diameter was expanded from 3.5 cm to

4.5 cm (the diameter is then contracted back to 3.5 cm after the reservoir though, more detail in

the next section) with a thickness increase from 4 mm to about 8 mm. Thus the total length of

piping used is about 91.85 cm per gun.

3.2.3 Valves and Connectors

Two requirements were driving the connectors and valves are that, one the gas must not outgas

and two the flow must not be broken in order for safe net launch. Thus the preliminary design

suggested was to use 4 latching valves. 3 before the reservoir and one at the reservoir outlet into

the launch tubes. The idea is that they will open one by one until the reservoir is full, then the

valves behind the reservoir are closed and when a firing command is sent, the valve at the outlet

of the reservoir opens causing all the stored gas to flow into the launch tube and thus accelerating

the tractors to launch the net.

The valves chosen (as discussed above) are latching valves. These valves work using a poppet

which lifts open the valve to let flow through, an electronic signal (typically a solenoid) lifts the

poppet in the open position and keeps it there by latching it in the open position. Hence the name

of the valve. Below is a schematic of the valve.

Figure 2 Latch Valve Schematic [6]

Since this valve works basically like a lift check valve (i.e. the valve remains closed or open

until a signal is sent to either open or close it depending on its starting position) this gives us the

estimated loss factor as since many manufactures quote losses in pressure drop only and those

change based on flow rate among other things. The losses will be discussed later on in this

section. Similar to the pressure regulator, there are two options in the valve manufacturers (both




of which have space heritage). Moog manufactures a solenoid actuated latching isolation valve

which weigh less than 0.3 lb with an operating pressure range of 0-186 bar and an operating

voltage of 20 to 42 vdc [7]. Airbus Space and Defense Helium Latching Valves don’t have an

operating pressure listed but state that they can handle most pressure ranges and operate with a

voltage range of 22-36 vdc [8]. From the given information, the range in which they can operate

is greater than what we need thus this should not be a problem.

It was mentioned in the previous section that a Wye bend and a wide 90 degree bend were used

in the design. Since there will be two space net guns, instead of having two different pipe

systems a wye bend is used to split the flows between the two different guns. Flow to the each

gun is isolated by the first latching valve along the piping, thus one gun will still work if the

other one fails. Since the launch tubes connections are 90 degree to the piping (as the piping

travels in the x direction of that rectangular volume while the connectors are along the y

direction of that volume) a 90 degree bend is needed to connect the reservoir to the launch tube.

A wide 90 degree angle turn (i.e. a Radius of Curvature/Diameter of Pipe is equivalent to 1) is

used to connect each reservoir to their respective launch tubes. Also since the reservoir is bigger

and thicker than the rest of the pipe, an expansion in the pipe internal diameter and thickness was

used to connect the piping to the reservoir inlet and a contraction in the pips internal diameter

and thickness was used to connect the piping to the reservoir outlet.

Since each of the valves and connectors will cause losses in the flow, the loss factors are needed

to be determined. The losses are given as either an equivalent (L/D) ratio or by a k factor. K

factor can be related to the (L/D)eq by multiplying the Darcy Friction Factor. Most of the friction

factor tabulated bellow are found from experimental data [9]. Please note that since the latch

values function just like a lift check valve, the loss factor of the lift check valve was used.

Table 3 Fitting and Valve Losses [9]

Fitting’s and Valves Diameter (cm) Loss Factor

90º Bend (r/d=1) 3.6 (L/D)e= 20

Latch Valves 3.6 (L/D)e= 600

Expansion (B=0.8) 3.6 to 4.6 K= 0.32

Contraction (B=0.8) 4.6 to 3.6 K= 0.44




3.2.4 Pipe Flow Calculation and Schematics

Now that all the piping length and reservoir volume has been determined along with all the

losses in the fittings and valves. A flow calculation can be done to determine how much pressure

is required to launch the tractors. Flow calculation is done using the Darcy-Weisbach equation:

𝑢2 = 2∆𝑃

𝜌(𝑓𝐿

𝑑+ ∑ 𝐾)

Where 𝐾 = 𝑓𝑑 (𝐿

𝐷) EQ13

Equation 13 gives the fluid velocity given a pressure drop and knowing the losses and friction

factors. The calculation was done under the assumption that its steady state, no changes in

temperature or density. As it is assumed as incompressible (for ease of calculation). The L/D

term that is multiplied with the friction factor is found from section 3.2.2 where the length of

pipe is from the contraction of the reservoirs to the launch tube. Which has a length of 5.65 cm.

The loss factors used were that of the contraction and that of the 4th latching valve in between the

reservoirs and the launch tube. Since the pressure differential in space is just the pressure in the

reservoirs as the pressure downstream is 0 as it’s in a vacuum, a pressure range of 0 to 150 psi

was used as the input. The volume of the reservoir as also used. The losses in the launch tubes

and the friction factor along with relating fluid velocity to tractor velocity was done by the

second individual in the net gun subsystem, details on those values can be found in EODM-REP-

TR-001-00. Taking those values in, resulted in a tractor velocity versus reservoir plot.

Figure 3 Launch Velocities of Space Gun

Since the required tractor velocity is about 1.031 m/s for a launch angle of 14 degree, the plot is

then condensed to a smaller range.

0

0.5

1

1.5

2

2.5

0 50 100 150

BA

RR

EL V

ELO

CIT

Y (M

/S)

RESERVOIR PRESSURE (PSI)

Launch velocities




Figure 4 Launch Velocities of Space Gun (Condensed)

Looking closely between data points for a barrel velocity of 1.03 and 1.032 m/s from figure 4, a

required reservoir pressure 39.68 psi (or about 2.74 bar) is obtained. Thus for the current net gun

design a reservoir pressure of 2.74 bar is needed, which is easily supplied by the 9 bar source

tank.

Now that the layout and its estimated specs have been discussed, a schematic for the current

design can be created. Figure 5 illustrates what the current net gun design is. This concludes the

work done on the space net gun for this year.

1.024

1.026

1.028

1.03

1.032

1.034

1.036

1.038

1.04

39.2 39.3 39.4 39.5 39.6 39.7 39.8 39.9 40 40.1 40.2

BA

RR

EL V

ELO

CIT

Y (M

/S)

RESERVOIR PRESSURE (PSI)

Launch velocities




Figure 5 Space Net Gun Schematic

4.0 Summary

Analysis of the earth net gun and test firing lead to the creation a ballistic model for the

projectile. A ballistic profile of the projectile can now be found given initial muzzle velocity and

time of flight for cases which include drag effects and without drag effects. One thing to note




that this model assumes that the position of the net gun is where the zero datum point is located.

In order to actually create such an algorithm, experiments were done in order to determine an

initial muzzle velocity and to find the drag coefficient of air for said projectile. Using a reservoir

pressure of 20 psi, an initial muzzle velocity was determined to be 31.63 m/s. It is important to

note that the theoretical fluid flow algorithm approach (found in EODM-REP-TR-002-00) the

achieved initial muzzle velocity for 20 psi was about 32 m/s, which is pretty close. With a

projectile mass of 0.3 g and a projectile diameter of 1.25 cm, a quadratic drag coefficient of

0.0192 Ns2/m2.

One of the difficulties during the semester was trying to get a high speed camera to finish the

experiment and determine drag and then do several additional firings in order to further optimize

the fluid flow model from EODM-REP-TR-002-00. In the end a high speed camera rated at 240

fps was sufficient enough to determine the drag factor but it was not sufficient to do more test

firings to determine initial muzzle velocities. In addition during the design phase for the space

net gun, there was difficulty in finding a starting point. One of the issues was that sizing of the

components such as reservoir was dependant on the pressure but pressure in the reservoir was

also dependant on the sizing, thus leading to a spiralling problem. It was decided that actual

design would take an iterative process and thus the sizing was estimated based on the size

restrictions imposed by the volume allocated for the piping and reservoir.

The work done on the earth space gun provides a foundation for more testing in order to

accurately tune the fluid flow simulation to output the correct velocity given the pressure and

mass of projectile. This also allows testing for future tractor designs as well and testing its

performance on earth and extrapolating that to its performance in space. The preliminary space

net gun design provides a starting point for the design cycle and allows for an iterative process to

begin with trade studies and such to get a solid sizing for the net gun components and the

required pressures needed to launch the net in space. One of the important thing is to be able to

test the flow in a vacuum in order to remove assumptions such as steady state and incompressible

flow. As well considerations of the flow becoming choked downstream due to so much

resistance given by the valves and connection must be taken to fully be confident in the design.

All the work done on this year lays the ground work for more specialized work to be done in the

net gun payload design.

5.0 References

[1] Sultan Islam, “Technical Memo #1 EODM-TM-SI-001-01,” Carleton U., Ottawa, ON, Rep.

001, Oct. 2014

[2] Sultan Islam, “Technical Memo #2 EODM-TM-SI-001-02,” Carleton U., Ottawa, ON, Rep.

001, Oct. 2014




[3] Moog.com, 'Spacecraft Components - Single Stage Regulator (Fluid) | Moog', 2015. [Online].

Available: http://www.moog.com/products/propulsion-

controls/spacecraft/components/regulators/single-stage-regulator-fluid-/. [Accessed: 08- Apr-

2015].

[4] Cs.astrium.eads.net, 'Pressure Regulators for Space Propulsion Systems', 2015. [Online].

Available: http://cs.astrium.eads.net/sp/spacecraft-propulsion/valves/pressure-regulators.html.

[Accessed: 08- Apr- 2015].

[5] Theo Ridley, “Final Report EODM-REP-TR-002-00,” Carleton U., Ottawa, ON, Rep. 002,

Apr. 2015

[6] Wiki.umn.edu, 2015. [Online]. Available:

https://wiki.umn.edu/pub/Asteroid/WebHome/Schematics_and_Valves.ppt. [Accessed: 08- Apr-

2015].

[7] Moog.com, 'Spacecraft Components - Solenoid Actuated Latching Isolation Valve | Moog',

2015. [Online]. Available: http://www.moog.com/products/propulsion-

controls/spacecraft/components/latching-isolation-valves/solenoid-actuated-latching-isolation-

valve/. [Accessed: 08- Apr- 2015].

[8] Cs.astrium.eads.net, 'Latch Valves for Space Propulsion Systems', 2015. [Online]. Available:

http://cs.astrium.eads.net/sp/spacecraft-propulsion/valves/latch-valves.html. [Accessed: 08- Apr-

2015].

[9] Roymech.co.uk, 'Fluid Engineering Flow in pipes', 2015. [Online]. Available:

http://www.roymech.co.uk/Related/Fluids/Fluids_Pipe.html. [Accessed: 08- Apr- 2015].

6.0 Appendices

6.1 Appendix A: Frame Analysis Used to Determine Muzzle/Initial Velocity




Figure 6 Frame 1

Figure 7 Frame 10




6.2 Appendix B: Frame Analysis to determine drag coefficient

Figure 8 Frame 1 of Drag Experiment

Figure 9 Frame 2 of Drag Experiment




6.3 Appendix C: MATLAB Code For Ballistic Profile % Solving Coupled equation using quadratic drag function vdot = quad_drag(t,v)

m = 0.0003; % Mass of projectile, in kg g = 9.8; % Acceleration of gravity, in m/s gamma = 0.0192; %Quandratic Drag Constant dia = 0.0125; %Diameter of Projectile c = gamma*dia^2; %Drag Force Vdot_x = -(c/m)*sqrt(v(1)^2+v(2)^2)*v(1); Vdot_y = -g-(c/m)*sqrt(v(1)^2+v(2)^2)*v(2); vdot = [Vdot_x; Vdot_y]; end

% Solving Coupled equation without using quadratic drag function vdot1 = noquad_drag(t1,v1)

m = 0.0003; % Mass of projectile, in kg g = 9.8; % Acceleration of gravity, in m/s Vdot_x = 0; Vdot_y = (-g); vdot1 = [Vdot_x; Vdot_y]; end

% Main Body of Code, calling in functions and plotting profile clc tspan = linspace (0, 3, 100); %Time of Flight Array [T,V] = ode45('quad_drag',[tspan],[26.83,4.47]); %Array of velocity and their

respective time with drag included [T1, V1] = ode45('noquad_drag',[tspan],[26.83,4.47]); %Array of velocity and

their respective time without drag included

%Integrating velocity into distance pos = cumtrapz(T,V); pos2 = cumtrapz(T1,V1);

%Plotting Both Cases figure plot(pos2(:,1),pos2(:,2),'color','blue'); % Plot without Drag Case hold on plot(pos(:,1),pos(:,2),'color','red'); % Plot Quadratic Drag Case hold off




6.4 Appendix D: Fluid Flow Calculation Results

Table 4 Tractor Velocity Vs. Pressure Differential

Pressure Differential

(Psi)

Tractor Velocity

(m/s)

0 0

5 0.366

10 0.5176

15 0.6339

20 0.732

25 0.8184

30 0.8965

35 0.9683

40 1.0351

50 1.1572

60 1.2676

75 1.4171

100 1.636

125 1.8286

150 2.0026

Table 5 Tractor Velocity Vs. Pressure Differential (Condensed)

PRESSURE

DIFFERENTIAL (PSI)

TRACTOR

VELOCITY (M/S) 39.2 1.0247

39.3 1.026

39.4 1.0273

39.5 1.0286

39.6 1.0299

39.7 1.0312

39.8 1.0325

39.9 1.0338

40 1.0351

40.1 1.0364

40.2 1.0377




6.5 Appendix E: Preliminary Space Net Gun Hand Sketch

Figure 10 Rough Sketch of Space Net Gun Design




6.6 Appendix F: Tech Memo 1 to 4

Date: Monday, October 13th, 2014 Document Number: 001

To: Burce Burlton

From: Sultan Islam (100822163)

Subject: Computation for the Ballistics of the Projectile from the Net Gun & Preliminary Matlab

Model

1.0 Purpose

This report is for our project lead Bruce Burton and any other lead engineers regarding current

work done on the CAPTURE Capstone Project running at Carleton University. The purpose of

this memo is to show and discuss preliminary ballistics analysis of a projectile being launched

from a pneumatic net gun. As well to discuss a preliminary MATLAB model that analyzes the

ballistics of the net gun in Earth environment with variable inputs.

2.0 Discussion

In order to determine a ballistic profile for net gun, there was a test fire of the net gun done

before hand and several approximate measurements were taken. First of all for the testing, all

tubes accept for 1 were partially sealed. At this point a method of determining muzzle velocity

wasn’t available and thus muzzle velocity was estimated using assumed pressure values. The

mass of the projectile used was 1.2 ounce or 0.034 kg and the pressure supplied to the main tank

was 20 psi or 137.895 kPa with the diameter of the launch tube being measured at ½ in or

0.0127m. During the testing, the range of the projectile was measured and estimated to be 18 m

(approx.). Next in order to figure out the ballistics, the initial/muzzle velocity was calculated.

2.1 Pressure Calculation

In order to figure out how much pressure is really being fed into the launch tub, previous years

design points were referenced. The following calculations were done under the assumptions that

the flow (when valves are open) are assumed to be steady state and thus the pressure ratio across

two modules are equal, to express this in mathematical terms:

.'2

'1

2

1Const

P

P

P

P EQ1




Thus referencing previous years design points (for the space version since ground version was

not available at this time):

Table 6 Previous Years Design Point [1]

And utilizing the given information with equation 1:

.00797.0146.137895

'Pr

247990

1977

es

Therefore,

Paes 34.1099)146.137895(*)00797.0('Pr

Similarly, for the net and tractor launch tubes: Pnet_tube= 332.53Pa & Ptractor_tube=716.2Pa.

2.2 Initial/Muzzle Velocity Determination

In order to calculate the muzzle velocity for the projectile in the launch tubes, a couple of

assumptions are made. Firstly it is assumed that friction (in the tube) is accounted for by adding

3% to the total applied force (taken from last year final reports) [1]. As well the force applied on

the projectile by the pressure is assumed to be an impulsive force which is given by Ft=P*A

(Force of P*A applied at 1sec). Thus using the impulse equation:




m

APVmuzzle

VoVmuzzlemAP

VmtF

**03.1

)(***03.1

**

EQ2

Note that since friction is accounted for by adding 3% of the applied force, hence the total force

was scaled by 1.03 (3% additional). Thus using Eq 2 along with m=0.034, we get:

smVmuzzle

Vmuzzle

m

APVmuzzle

/99.10

034.0

)2)^0127.0(*(*)2.716(*03.1

**03.1

Once the muzzle velocity was found, using the angle of the launch tube cited from previous year

final reports [1] [2], the X and Y directional velocities were computed:

Vx=VmuzzleCos(26.56)=9.5471m/s

Vy=VmuzzleSin(26.56)=4.7725m/s

It is noted that since there was no muzzle velocity given and the method of determining a muzzle

velocity from pressure was not completed by the other party, the muzzle velocity was estimated

using vague assumptions. The above calculations for muzzle velocity and pressure can be

ignored and it can be assumed that for 20 psi case, muzzle velocity is estimated at 10.99 m/s.

2.3 Corrected Terminal Velocity Due to Air Resistance

Since this experiment was done on earth where air resistance is still influential, the next step is to

calculate the initial projectile velocities corrected with air resistance. Referring to Analytical

Mechanics by Fowles 4th Editon, fluid resistance is given by:

𝐹𝑜 + 𝐹(𝑣) = 𝑚𝑑𝑣

𝑑𝑡

𝐹𝑜 + 𝐹(𝑣) = 𝑚𝑣𝑑𝑣

𝑑𝑥

Equation 3 Terminal Velocity Equation [3]

Thus, by first realizing that, the ballistic analysis is done in two parts, one horizontal and the

other vertical, terminal velocity can easily be found in both direction. The resistance force is

given by:

𝐹(𝑣) = −𝑐1 − 𝑐2𝑣|𝑣| = −𝑣(𝑐1 + 𝑐2|𝑣|)

Equation 4 Resistance Force [3]




Next, we assume the projectile to be spherical and thus solving ratio of constants, the dominating

constant can be determined (either C1 or C2).

0.22𝑣|𝑣|𝐷2

1.55 × 10−4𝑣𝐷= 1.4 × 103|𝑣|𝐷

Equation 5 Ratio of Constant [3]

Using the above equations with D as in the diameter of the projectile (i.e assumed to be the same

as the diameter of the launch tube), a ratio is calculated to be 17.78|v| and thus equating this to be

1, v can be solved to be v=0.056m/s. Based on the theory in the book, the quadratic term C2

dominates above the calculated value with the linear term C1 dominating bellow that. Since our

muzzle velocity for both direction is higher than 0.0056m/s, the quadratic term dominates. Next

the constants can be found using this approximation:

𝑐1 = 1.55 × 10−4𝐷

𝑐2 = 0.22𝐷2

Equation 6 Approximated Constants [3]

Thus, using the diameter of the tube, C2 can be calculated as C2=0.0000355. Next we

differentiate equations 3 and using equation 4 for each direction and obtain a terminal velocity

expression:

For the x-direction, calculation is computed after force has been applied and muzzle velocity has

been reached, thus with Fo=0 & F(v)=-C2v|v|, we obtain:

𝑣 =𝑣0

1+𝑘𝑡

Equation 7 Terminal Velocity for Horizontal Direction [3]

A few caveats, first of all, since the equation is given in both differential terms of distance and

time, we can differentiate the above equations for distance and get the similar equation with x

replacing t. Also note k= C2vo/m, thus equation 7 is simplified as:

xm

VoC

VoV

**2

1

EQ8

Where x is the distance traveled horizontally (x=18m) and m is the mass (m=0.034kg) and C2 as

found above with the initial velocity as the velocity in the x-direction, a corrected muzzle

velocity is found to be V= 8.0953m/s.

Similarly for the vertical component, the only force being applied is gravitational force, thus,




𝑉 = √𝑚𝑔

𝐶2tan[

𝑡0−𝑡

√𝑚

𝐶2𝑔

+ tan−1 𝑉0

√𝑔

𝐶2

] (Rising)

𝑉 = −√𝑚𝑔

𝐶2tanh[

𝑡−𝑡0′

√𝑚

𝐶2𝑔

− tanh−1 𝑉0

√𝑚𝑔

𝐶2

] (Falling)

Equation 9 Terminal Velocity for Vertical Direction [3]

As seen above, the vertical portion is differentiated w.r.t time and given for both rising and

falling component. Since the ballistic analysis is treated like projectile problem, both rising and

falling portions maybe used, for simplicities sake, only the rising part is used. Similarly like the

previous case, inserting known values into Equation 9 (specifically equation 2.30 from the book),

a terminal velocity of v= 4.5817m/s is found.

2.4 Ballistics’ approximated as Projectile Motion

Now that, the corrected velocity is found, kinematic equations of motion for both the x and y

direction can be applied to figure out characteristics of the motion.

For the x direction, the given range of 18m is known, as well acceleration in the x is known to be

0, thus applying this equation:

2^**2

1* tatVid

EQ10

And rearranging for t, we can obtain the time of flight,

sec226.20853.8

18

t

Vi

dt

Since the time of flight is now computed, max height of projectile is the only parameter missing

which will complete the full description for the ballistics of the net gun. Thus for the vertical

case, we know initial velocity, acceleration and to an extent, the time it takes to reach max

height, i.e. since the motion is parabolic, max height is obtained at 1.113sec. Looking through

the available equations, max height can be calculated using:




daViVf **22^2^ EQ11

It is known that V final is 0m/s, thus rearranging for distance gives us:

md

a

Vid

0699.1)81.9(*2

2)^5817.4(max

2

2^max

And thus, the maximum height of the projection is calculated as 1.0699m.

2.5 MATLAB Preliminary Model for Net Gun

Since the ballistic characteristics were computed for a specific case, utilizing the above equations

and using mass as a variable from 0.0034-1.2kg with an increment of 0.001kg, a graphic model

can be computed. Please note that as an edit, this work was done under the assumption that

pressure was an impulsive force and that this is wrong. The main point of this part was to be able

to create a MATLAB code to perform trade studies of each of the ballistic parameters. The code,

which is found in the appendix can be modified to do trade studies for any parameter and thus is

still relevant to our case.

3.0 Summary

To summarize, with the above assumptions, a ballistic profile for the net gun is determined to

test experimental results. As such these are the results bellow:

Table 7 Calculated Results

Parameters Value

Muzzle Velocity (m/s) 10.99

Corrected Velocity in X (m/s) 8.0853

Corrected Velocity in Y (m/s) 4.5817

Time of Flight (sec) 2.226

Maximum Distance (m) 1.0699

And the MATLAB Code plotted:




Figure 11 Matlab Plot 1





These results do match what was observed in the lab, though more work is needed to refine the

model to generate more accurate values. It is also important to note that while programming in

MATLAB it was difficult to smooth out the plots and thus the resolution of the data isn’t quite

large, further work needs to be done to increase resolution of the solutions. In addition referring

to MATLAB plot 5, it can be seen that due to the nature of the equation and utilizing corrected

velocity for drag, the maximum height is almost a parabola. So for future simulations, a

restriction should be posed to only take a muzzle velocity that are above zero.

For future work, these assumptions made needs to be corrected/removed, an actual test of how

much pressure is distributed between two chambers is needed. A more detailed look into the

dynamics of the projectile is needed, for example to see if there is any difference in applying the

terminal velocity equations during or after the pressure forces been applied. Since most of the

code in MATLAB can be modified, the initial parameters could be changed and thus if any

changes are made, the code can be updated easily in order to be relevant. A theoretical model for

the ballistics in space is also required, though from a top down design perspective, for space

there would be no air resistance but have to verify if any perturbations from orbit will cause large

effects to projectile. Ballistics is only a small part of the net gun design, more work is needed in




other area’s but work is needed to be done on the net closure, net materials and other subsystems

within payload before ballistic analysis can continue.

4.0 References

[1] Sarah Mackenzie-Picot, “Final Report: Payload, Propulsion, and Research EODM-REP-

SMP-002-00,” Carleton U., Ottawa, ON, Rep. 002, April. 2014

[2] Nathan Cole, “Net Launcher Design & Testing and Preliminary Concept Research EODM-

REP-NCO-002-00,” Carleton U., Ottawa, ON, Rep. 002, April. 2014

[3] Fowles, “2.5 Velocity Dependant Forces, Fluid Resistance and Terminal Velocity,”

Analytical Mechanics 4th ed. Sounders College Publishing, 1986, 2.5, pp. 48-52

5.0 Appendix: MATLAB Code

%Assumed/Known/Tested values: d1=18; %lateral distance in m m=0.034; %mass of projectile in kg g=9.81; %Gravitational Const on earth ay=(-1)*g; %Acceleration of Projectile r=0.00635; %radius of launch tube in m D=(2)*r; %diameter of launch tube in m Ptank=137895; %Pressure in the Main tank used (20psi) in Pa Theta=26.56; %Angle of Tube

%Pressure Calculations:

%Assumed at Steady State and that the pressure ratio over difference

componants are constant

%Design point values: Pdt=247990; %Pressure in main tank in Pa Pdr=1977; %Pressure in Reservoir (Secondary Tank) in Pa Pdn=598; %Pressure in net tube in Pa Pdp=1288; %Pressure in projectile tube (tractor) in Pa

%Estimated Pressure Ratio: Pr1=(Pdr)/(Pdt); %Ratio for Reservoir to Main Tank Pr2=(Pdp)/(Pdr); %Ratio for Projectile tube to Reservoir Pr3=(Pdn)/(Pdr); %Ratio for Net tube to Reservoir

%Experimental Pressure values: Pres=(Pr1)*(Ptank); %Experimental Reservoir Value Pnet=(Pr3)*(Pres); %Experimental Net Tube Value Pproj=(Pr2)*(Pres); %Experimental Projectile/Tractor Tube Value

%Intial/Muzzle Velocity calculations: %Since doing a case where, all but one tube is sealed, P=Proj,




%otherwise if all tubes loaded, individual tubes will have P=(1/4)Proj

%Assuming the force due to pressure is an impulse, A=(pi)*(D^2); %Area of Launch tube Vmuz=((Pproj)*(A))/m; %Muzzle Velocity of Projectile Vx=(Vmuz)*cosd(Theta); %X-Directional Velocity Vy=(Vmuz)*sind(Theta); %Y-Directional Velocity

%Projectile Motion corrected for air resistance (Earth only):

%Air resistance Constants: C2=(0.22)*(D^2);

%Lateral/Horizontal Motion: Vx1=Vx/(1+((C2*Vx)/m)*d1); %Corrected Velocity t1=(d1)/(Vx1); %Time of Flight

%Transverse/Vertical Motion: to=0; %Intial time of launch t2=(1/2)*t1; %Time of flight to max height vt=sqrt((m*g)/C2); %intermediate step tau=sqrt(m/(C2*g)); %Intermediate step Vy1=(vt)*tand(((to-t2)/tau)+atand(Vy/vt)); %Corrected Velocity for rise Vy2=((-1)*vt)*tanh(((to-t2)/tau)-atanh(0)); %Corrected Velocity for drop d=((Vy1)^2)/(2*g); %Max height

%Ballistics Simulation for Variable Mass: m2=0.034:0.1:1.2; %Mass Ranges Vmuz1=((Pproj)*(A))./m2; %Muzzle Velocity of Projectile Vxx=(Vmuz1)*cosd(Theta); %X-Directional Velocity Vyy=(Vmuz1)*sind(Theta); %Y-Directional Velocity66 Vx2=Vxx./(1+((C2*Vxx)./m2)*d1); %Corrected Velocity t11=(d1)./(Vx2); %Time of Flight t22=(1/2)*t11; %Time of flight to max height vt1=sqrt((m2*g)/C2); %intermediate step tau1=sqrt(m2./(C2*g)); %Intermediate step Vy11=(vt1).*tand(((to-t22)./tau1)+atand(Vyy./vt1)); %Corrected Velocity for

rise Vy22=((-1).*vt1).*tanh(((to-t2)./tau1)-atanh(0)); %Corrected Velocity for

drop d1=((Vy11).^2)./(2*g); %Max height

%Graphing: figure plot(m2,Vmuz1,'O-') title('Muzzle Velocity vs. Variable Mass') xlabel('Mass of Projectile (in KG)') ylabel('Muzzle Velocity (in m/s)') legend('Muzzle Velocity')

figure plot(m2,Vxx,'rO-',m2,Vyy,'gO-') title('Velocity Componants vs. Variable Mass') xlabel('Mass of Projectile (in KG)') ylabel('Muzzle Velocity (in m/s)')




legend('Muzzle Velocity in the X','Muzzle Velocity in the Y')

figure plot(m2,Vx2,'rO-',m2,Vy11,'gO-') title('Corrected Velocity Componants vs. Variable Mass') xlabel('Mass of Projectile (in KG)') ylabel('Corrected Muzzle Velocity with Air resistance (in m/s)') legend('Muzzle Velocity in the X','Muzzle Velocity in the Y')

figure plot(Vx2,t11,'O-') title('TOF vs. Muzzle Velocity') xlabel('Muzzle Velocity in the X (in m/s)') ylabel('Time Of Flight (in s)') legend('Time of Flight')

figure plot(Vy11,d1,'O-') title('Maximum Height vs. Muzzle Velocity') xlabel('Muzzle Velocity (in m/s)') ylabel('Max Height (in m)')

Date: Monday, November 24th, 2014 Document Number: 002

To: Bruce Burlton


Subject: Image Analysis to Determine Projectile Velocity of Ground based Net Gun

1.0 Purpose

This the second technical memo for the lead engineers for the Capstone Capture project outlining

further progress in the payload specifically net gun sub group. This report is aimed at explaining

the method used to determine muzzle velocity using a high speed camera and image processing

techniques. This work will be used to determine a correct model for net gun ballistics in a Earth

environment.

2.0 Discussion

In order to determine a value for the muzzle velocity experimentally, we need a way to take

measurements of the projectile flight path and behaviour. One of the methods that was chosen is

to use a high speed camera to capture the projectile in flight and determine its ballistic profile

using image/video processing technique. For this purpose a high speed camera from the

engineering department has been procured thanks to the lead engineer. Before the experimental

setup and procedure is outline, there are a couple of assumptions made as well as a few

adjustments that is needed to be noted. The pressure used to drive the projectile was about 20 psi,




the projectile mass was measured as 1/10 of a gram. As well from previous testing the distance

traveled by the projectile horizontally is about 18 m.

2.1 High Speed Camera Set-Up and Image Processing Method

The high speed camera used can record at various speeds, for our testing the footage was

recorded at a 1000 frames per sec. It should be noted I was not present during this test firing so

lighting conditions and such are still unknown, further test done with better lighting is to be

expected. The experimental set-up used to measure the projectile displacement from frame to

frame is a white cardboard with a grid drawn on the center of it using a ruler and a black marker.

The grid size measured in as 50x50 mm per square. The board was placed along the azimuth

plane of the two launch tubes on one side. The video footage then is captured via software and

outputted as a raw video file and frame by frame bit map images. 3 trials were done and on the

last trial only 20 frames with actual movement was taken.

During the experimentation one of the projectiles was already 2/3 of the way out of the barrel

before the other projectile actually fired. So only one projectile was taken into account. The

leading edge of the projectile was used for this analysis. It is seen that the projectile motion is

easily seen and measured between frames 9 to 19 (pictures can be found in the appendix). The

duration of the projectile flights (in sec) is given using this equation:



Thus for our case, the frame rate used as 100 Frames Per Section and the number of frames

measured was 9 frames which would equate to 0.01 sec for the projectiles time of flight. Then

once the duration is calculated the velocity measured (in mm/s) is given by (for this case):

𝑉 =𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒 ×50 𝑚𝑚

𝑇 EQ2

Since the grid was placed along the azimuth plane, the problem is treated as a 2D and thus the

component of the velocity vector (both X and Y) can be determined using EQ2. For the X

direction (the vertical direction in the bitmap) the leading edge traveled 6 squares in 10 frames,

therefore the velocity for the x direction is determined to be 30000 mm/s or 30 m/s. Similarly for

the Y-direction the leading edge travelled 2 squares in 10 frames thus resulting in a velocity of

10000 mm/s or 10 m/s in the y direction. It is noted that for the duration of 0.01 sec, the effect of

drag is ignored. Unfortunately the grid is not aligned with the barrel and thus the grid and the

barrel are 26.56 degree apart, thus in order to calculate actual muzzle velocity, the experiment

must be done with both the grid and the barrel aligned 26.56 degree to the horizontal. Though an

estimated angle can be measured and thus a muzzle velocity can be determined. The angle at

which the projectile flew over the grid can be approximated as,




𝜃 = 𝑇𝑎𝑛−1(𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑌

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑋) EQ3

Which would lead to an angle of 18.43 degree. Thus using basic trigonometry, muzzle velocity

can be estimated as,

𝑉𝑚𝑢𝑧𝑧𝑙𝑒 =𝑉𝑥

𝐶𝑜𝑠(𝜃) 𝑂𝑟

𝑉𝑦

𝑆𝑖𝑛(𝜃) EQ4

Which using the Vx gives 31.62 m/s and using Vy gives 31.63 m/s. Which is only 0.01 m/s

difference. The experiment needs to be repeated using a more accurate grid set along one of the

launch tubes, the footage need to be taken either at a higher resolution, or in a better lighting

environment and the projectile needs to be a better quality so a more accurate result can be

determined.

2.2 Ballistic Profile Using Projectile Motion

The next step is to use the experimentally measured velocity and plot out a time of flight,

maximum height and distance traveled and check that it matches the experimental values. Due to

the fact that the current velocity is still an estimation, the next step should be to set up a more

accurate experiment and measure velocity that way before creating a ballistic profile and

determining the drag force behind the projectile theoretically. Thus for this technical memo the

ballistic profile and projectile motion is not calculated.

3.0 Summary

To Summarize, using a high speed camera capturing at 1000 frames per sec and using a hand

drawn grid with a 50 by 50 mm grid spacing the velocities in the X and Y were determined.

Using trigonometry the angle between the X and Y velocity vectors was estimated and then the

actual muzzle velocity was calculated using the estimated angle.

Table 8 Results

Parameters Value

Measured Velocity in the X-direction (m/s) 30

Measured Velocity in the Y-direction (m/s) 10

Approximated Angle of the Velocity Vector

(deg)

18.43

Approximated Muzzle Velocity (m/s) 31.62-31.63

For future work a 12 by 18 inch gridded board was purchased to use for measurement, the

experimental apparatus needs to be changed so the board is along the 26.56 degree line just like




the barrel to get a more accurate measurement. A better projectile needs to be used and the

experiment needs to be done using better lighting. Also a further experiment needs to be done to

observe and measure the real world ballistic profile on the projectile along with distance travel,

height launched, maximum height, time of flight and any other parameter needed to verify the

theoretical model and obtain a drag value which agrees with the experimental data.

4.0 References







[4] Sultan Islam, “Technical Memo #1 EODM-AS-SI-001-02,” Carleton U., Ottawa, ON, Rep.

001, Oct. 2014




5.0 Appendix

Figure 16 Frame #9




Figure 17 Frame #10




Figure 18 Frame #11




Figure 19 Frame #12




Figure 20 Frame #13




Figure 21 Frame #14




Figure 22 Frame #15




Figure 23 Frame #16




Figure 24 Frame #17




Figure 25 Frame #18




Figure 26 Frame #19

Date: Saturday, January 31th, 2015 Document Number: 003

To: Bruce Burlton


Subject: Experimentation for Drag Coefficient and Preliminary Design for Space Based Gun

1.0 Purpose

This is the third technical memo for the lead engineers for the Capstone Capture project outlining

further progress in the payload (specifically net gun) sub group. This report is aimed at

explaining what steps are needed to complete the experiment to find the drag coefficient (for air

drag) and give a preliminary overview on a conceptual net gun design for space.




2.0 Discussion

In order to round up the work for the ground based net gun, we must determine the air drag

factor “C2” thus we can use the ballistic equation found in EODM-TM-SI-003-00 to create a

mathematical model which can provide another way to determine the muzzle velocity via

experimental data (given time of flight and distance traveled both horizontal and vertically).

Section 2.1 will discuss this further into detail. The next component of this tech memo is to

outline a preliminary net gun design for space, which will also be discussed into detail in Section

2.2.

2.1 Experimental Set-Up for Drag Determination

In order to determine the drag coefficient “C2” we need to have measured the projectile diameter,

initial muzzle velocity, time of flight and its distance travelled (height or range). If you recall, the

last tech memo (EODM-TM-SI-002) covered how the muzzle velocity was determined using a

grid and a high speed camera for 20 psi of tank pressure. Thus we have 1 of 4 required values.

The diameter of the projectile was measured in the first tech memo so we know have 2 of the 4

required values. We now only need to measure time of flight and the distance traveled either in

the horizontal and vertical direction.

The experimental set-up is shown below in figure 1. Setting the net gun to a predetermined

height and then shooting the projectile from 4 of the barrels (for simplicity each projectile will be

observed independently). Using a high speed camera rated at 240+ fps (1000 fps would be ideal

as the granularity of the time step allows for higher accuracy) we observe when the projectiles hit

the ground. Then using EQ1 we can stipulate the time of flight for each projectiles.






Figure 27 Experimental Set-up Idea

Then we measure the distance of the point of landing from the net gun using a measuring tape or

a ruler. Using trigonometry and knowing the angle of the launch tube we can determine the range

w.r.t the azimuth plane using equation 2.

𝑅 =𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝐶𝑜𝑠(𝜃 𝑜𝑓 𝑙𝑎𝑢𝑛𝑐ℎ 𝑡𝑢𝑏𝑒) EQ2

Now we have all of the required values for calculating the drag factor. Since the ballistics is a 2-

D problem, we can use either the X or Y direction. For simplicity we are using the X direction.

We can use Equation 3 to calculate the corrected velocity in the X.

2^**2

1* tatVid

EQ3

Now that we have the corrected velocity, we can use the drag formula (equation 4) to figure out

the drag coefficient “C2” knowing that the initial velocity is the muzzle velocity measured in the

last tech memo.

xm

VoC

VoV

**2

1

EQ5

2.2 Preliminary Design of the space based Net gun

Since the high speed camera needed to complete the experiment hasn’t been procured yet due to

unforeseen circumstances, work on the net gun has changed gears towards the space based

version of the gun. For preliminary design, we have set some requirements on the gun itself. One

of the requirements is that the pipes of the gun must not have a 90 degree bend as that causes




stress in the pipe and severely decreases flow of the pipe which would in turn affect the muzzle

velocity of the tractors. The other requirement is that each pipe (supply tank, launch tube, etc.)

should have a two stage valves, this would ensure that the each pipe is filled with the gas before

moving onto the next stage such that if any failures occur or a valve is lagging behind the

mission would not be jeopardized. In order to get started, one of the lead engineers suggested to

look at valve technologies used in space to get some ideas. Taking this suggestion into

consideration, I have found a Superfluid Helium Valve made by Airbus Defense and Space. A

figure of its schematic can be found bellow.

Figure 28 Superfluid Helium Valve [6]

This valve can take both liquid and gaseous helium, and has been used in Herschel and Plank

space programs before. These valves come in a range of sizes between 3 to 150 mm flow

diameters, can take pressures up to 250 bar in a vacuum and has an operating temperature

between 1.5 to 350 K. This valve also has filters built into both the inlet and outlet which would

filter out any debris found left from machining, also has a failsafe for when power is cut off

(valve will remain either open or closed depending on last configuration). As well the valve is

leak proof thus we can be assured that the helium may not leak through during flight (may as the

operative word). More research needs to be done on other possible options.

3.0 Summary

To Summarize, an experimental set-up for calculating the drag factor has been designed, all we

need is a high speed camera to do the experiment. This would allow us to finalize a mathematical

model for muzzle velocity determination through experiments which would be used as a proof of

concept for our space based gun as well as a way to check the theoretical model being worked on

by They Ridley. Since the work on the design of a space based gun was just started, not a lot of

work has been done on it. The next step will be to look into other options for valves (such as




valves which can work with nitrogen gas) and weigh the pros and cons of each. Also more work

on the sizing of the gun is required, which is the next step after valve choices.

4.0 References








001, Oct. 2014


001, Sept. 2014

[6] Airbus Defense and Space (2015). Superfluid Helium Valve [Online]. Available at:

http://cs.astrium.eads.net/sp/launcher-propulsion/propellant-valves/Herschel/index.html




Date: Saturday, February 28th, 2015 Document Number: 004

To: Bruce Burlton


Subject: Experimental Drag Model and Continuation of the Space Based Gun Design

1.0 Purpose

This is the fourth technical memo for the lead engineers for the Capstone Capture project

outlining further progress in the payload (specifically net gun) sub group. This report is aimed at

further explaining what steps are needed to complete the experiment to find the drag coefficient

(for air drag) and provide an update for the space based net gun design.

2.0 Discussion

In order to finalize the work done for the ground based net gun, the quadratic drag coefficient

“C2” needs to be obtained experimentally such that calculating muzzle velocity from high speed

camera method (found in EODM-TM-SI-002) can be done accurately for variations of pressure

settings. This data will be forwarded to my partner Theo Ridly such that he can verify and

correct his flow model for the current ground based gun. In order to verify that the calculated

drag coefficient is accurate, a firing test will be done where time of flight, height and distance

travelled will be measured and then it will be compared to a MATLAB script file that outputs

those values theoretically based on the drag factor that was found. This will be discussed further

in section 2.1. Next an update on the space based gun will be discussed, a top level design layout

will be shown and the method of sizing will be briefly discussed.

2.1 Experimental Set-Up for Drag Determination

In order to determine the quadratic drag coefficient “C2” we must first recall the drag force

equation incorporating the drag coefficient and the terminal velocity (shown in equation 1).

EQ1

Where gamma is the quadratic drag coefficient of air (which in our case is C2) and D is the

diameter of the projectile. Equating the drag force to the force of gravity, one can solve for the

terminal velocity, equation 2 illustrates this.

EQ2

Thus using equation 2 knowing the force of gravity, the terminal velocity and the diameter of the

projectile, we can solve for the quadratic drag coefficient. This is done using the high speed

camera method [5], the projectile is dropped from rest and using the 50 by 50 cm grid and a high

c

mgv ter




speed camera, the terminal velocity can be found. Since the mass, g and D is known, solving for

gamma gives us the quadratic drag factor of the projectile in the air. In order to verify that the

drag factor found is accurate, a test fire is needed to measure the time of flight and range the

projectile traveled (Figure 1 illustrates an experimental layout). Once that is measured, using

equations 3 and 4 and solving them simultaneously using ODE 45 on matlab to output the range

of the projectile given the time of flight and initial velocity inputs.

EQ3

EQ4

It is important to note that the initial velocity used will be the velocity found in the second tech

memo. Once MATLAB determines these values (script found in Appendix A), hopefully the

difference in value is minimal. After the quadratic drag coefficient is verified, repeating the same

experiment found the second tech memo for multiple pressure values the actual muzzle velocity

is determined by eliminating the effects of air drag. These values will be given to my partner

Theo to use in correcting his flow code.

Figure 29 Experimental Layout

2.2 Design of the space based Net gun

Since the propulsion subsystem is pressurizing helium on board the s/c, the idea is to get the

source for the net gun from the propulsion subsystem. It is quoted from the propulsion subsystem

that the tank they are using is holding approximately 9 bars of pressure, though he is able to

create up to 30 bar of pressure. The plan is to connect a pipe to the pressurizing system or the

tank and use a helium pressure regulator (Model number RDS20000) [6] to regulate the pressure

to what the net gun needs to launch the net. Then using Helium valve to transport the fluid into

the net gun own supply tank. From there another valve is used to transport the gas into the launch

tubes. Apparently Theo is working on how he wants to angle the tractor tubes so I will probably

yyxy vvvcmgvm 22

xyxx vvvcvm 22




will have to look into how long they need to be. Sizing is done by utilizing Bernoulli equation of

flow between two points. Equation 5 illustrates the Bernoulli Energy Equation.

EQ5

Applying Equation 5 for each step in the net gun allows us to account for losses in turns,

expansions/contractions and frictional losses. Other forms of energy losses (i.e. heat transfer) is

going to be ignored for the sizing. Given the required velocity, desired pressure drop, minor loss

factor “K”, friction loss factor, the length of the pipe and the diameter of the pipe can be

calculated.

3.0 Summary

To Summarize, an experimental set-up for calculating the drag factor has been designed, all we

need is a high speed camera to do the experiment. This would allow us to finalize a mathematical

model for muzzle velocity determination through experiments which would be used as a proof of

concept for our space based gun as well as a way to check the theoretical model being worked on

by They Ridley. All is left to do is figure out a way to get rid of drag factor by solving the

differential equations backwards to find the initial conditions. In terms of the space gun sizing, I

need to obtain certain values such as the velocities needed, desired pressure drop and find the

loss factors from literature I can create a matlab code which will output the length of the pipe

given the diameter and vice versa. The sizing of the net gun and the experiment should cover this

semester’s work on the payload subsystem (specifically net gun).

4.0 References








001, Oct. 2014


001, Sept. 2014

[6] Airbus Defense and Space (2015). Helium Pressure Regulator [Online]. Available at:

http://cs.astrium.eads.net/sp/spacecraft-propulsion/valves/pressure-regulators.html




5.0 Appendix A

function vdot = quad_drag(t,v) … m = value; % Mass of projectile, in kg g = 9.8; % Acceleration of gravity, in m/s diam = value; % Diameter of projectile, in m gamma = value; % Coefficient of drag in air at STP, in Ns^2/m^2 c = gamma*diam^2; Vdot_x = -(c/m)*sqrt(v(1)^2+v(2)^2)*v(1); Vdot_y = -g-(c/m)*sqrt(v(1)^2+v(2)^2)*v(2); … vdot = [vdot_x; vdot_y];

[T,V] = ode45('quad_drag',[0 time of flight],[V0x; V0y]);

function y = int_yp(t,yp) %Function for finding trajectory

n = length(t);

y = yp;

y(1,:) = [0 0];

for i=1:n-1

dt = t(i+1)-t(i);

dy = yp(i,:)*dt;

y(i+1,:) = y(i,:)+dy;

end

y = y(1:n-1,:);

pos = int_yp(T,V);

hold on

plot(pos(:,1),pos(:,2),'color','red'); % Overplot quadratic drag case

hold off

%Note: Certain values aren’t entered in yet, will be filled in after experiment.

eodm-rep-si-002-02

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