AAE 450 Spring 2008
Elizabeth HarknessTrajectory Design Group Contact
Preliminary Trajectory Model:
Vertical Launch
AAE 450 Spring 2008
Comparison of Launch System Performance
Trajectory Optimization
Comparison of altitudes using vertical launch trajectory.-Plot generated by:
Elizabeth Harkness
-From code generated by:
Junichi Kanehara and
Amanda Briden
-with assistance from:
Elizabeth Harkness
AAE 450 Spring 2008
Comparison of Launch System Performance
Trajectory Optimization
4786 km
4686 km
4372 km
hmax
9.5 %1821 shi = 16.8 km
vi = 56.58 m/sAircraft
7.2 %1791 shi = 30.5 km
vi = 0 m/sBalloon
NA1707 shi = 0 km
vi = 0 m/sGround
% Improvement from GroundthmaxI.C.s1Launch
System
1. Initial conditions for Balloon and Aircraft Launches found in cited articles.
AAE 450 Spring 2008
Model Assumptions� Values input into code are from Vanguard
data Table 20. Payload of 100 lbs, orbit at 180 statute miles.
� All stages are modelled vertically.
– Poor assumption
• Does not take into account the lengthy portion of trajectory through the atmosphere.
� Preliminary test of trajectory code and comparison of three of the launch methods
AAE 450 Spring 2008
ReferencesHeister, S., “AAE 539 atmosphere4”
[https://engineering.purdue.edu/AAE/Academics/Courses/aae539/2008/std.%20atmosphere%20matlab%20code%20for%20HW1%20Prob2.]
Howell, K., “AAE 340 Dynamics and Vibrations," Numerical Integration and Spherical Coordinates, Purdue University, West Lafayette, IN, 2006.
Klauans, B., and Raughards, J., “The Vanguard Satellite Launching Vehicle An Engineering Summary," The Martin Company, Engineering Report No. 11022, Baltimore, MD, April 1960. (pp.170)
Kloster, K., and Longuski, J., Personal Conversation, January 15-16, 2008.
Longuski, J., “AAE 508 Optimization in Aerospace Engineering Lectures," Lecture #10, Purdue University, West Lafayette, IN, 2008, pp. 62.
Wade, M., “Cape Canaveral," Encyclopedia Astronautica.[http://www.astronautix.com/sites/capveral.htm. Accessed 1/16/08.]
Trajectory Optimization
AAE 450 Spring 2008
Elizabeth HarknessTrajectory Design Group Contact
Parametric Study of Steering Law and Drag Effects
AAE 450 Spring 2008
Effects of changing the initial launch angle.
Trajectory Design
63.4°
Trial #1 (mPL = 5 kg)Assumptions:-No wind model-Linear tangent steering law
Plot generated by E. Harknessfrom code written by A. Briden, E.
Harkness and K. Donahue
AAE 450 Spring 2008Plots generated by E. Harknessfrom code written by A. Bridenand E. Harkness
∆VDrag versus end-of-first-stage launch angle ∆VTotal versus end-of-first-stage launch angle
• Trends•∆VDrag decreases with steering angle
• ∆VTotal increases with steering angle
AAE 450 Spring 2008
Trajectory Design
b1Psi_0
(deg)
Psi_1_end
(deg)
Psi_2_end
(deg)
Psi_3_end
(deg)r_final (km)
Tangential velocity
(r_final*theta_dot_final)
(km/s)
Radial
Velocity
(r_dot_final)
(km/s)
deltaV_2circ
(km/s)
delta V Drag
(km/s)
delta V Total
(km/s)
Percent
Drag
(%)
2 63.4349 60 0 -20 306 4.4 8.8 9.2 1.8 21.6 8.33
2 63.4349 50 0 -20 302 4.8 8 8.4 2.3 20.8 11.06
2 63.4349 40 0 -20 304.6 5 7.1 7.4 3 19.9 15.08
2 63.4349 30 0 -20 302.4 4.9 5.7 6.2 4.1 18.7 21.93
2 63.4349 20 0 -20 300.8 3.7 2.9 5.1 6.6 17.4 37.93
2 63.4349 10 0 -20 did not go reach 300 km
b1Psi_0
(deg)
Psi_1_end
(deg)
Psi_2_end
(deg)
Psi_3_end
(deg)r_final (km)
Tangential velocity
(r_final*theta_dot_final)
(km/s)
Radial
Velocity
(r_dot_final)
(km/s)
deltaV_2circ
(km/s)
delta V Drag
(km/s)
delta V Total
(km/s)
Percent
Drag
(%)
2 63.4349 60 -20 -50 304.6 3.2 9.2 10.2 1.8 22.5 8.00
2 63.4349 50 -20 -50 304.3 3.5 8.5 9.4 2.3 21.7 10.60
2 63.4349 40 -20 -50 306.5 3.6 7.6 8.6 3 20.8 14.42
2 63.4349 30 -20 -50 303.5 3.5 6.3 7.5 4.1 19.6 20.92
2 63.4349 20 -20 -50 300.6 2.5 2.9 6 6.9 17.9 38.552 63.4349 10 -20 -50 did not go reach 300 km
Parametric Study of Launching Angles and Steering Angleby Elizabeth Harkness and Amanda Briden30 January 2008
AAE 450 Spring 2008
ReferencesHowell, K., “AAE 340 Dynamics and Vibrations," Numerical Integration
and Spherical Coordinates, Purdue University, West Lafayette, IN, 2006.
Kloster, K., and Longuski, J., Personal Conversation, January 15-16, 2008.
Longuski, J., “AAE 508 Optimization in Aerospace Engineering Lectures," Lecture #10, Purdue University, West Lafayette, IN, 2008, pp. 62.
Trajectory Design
AAE 450 Spring 2008
Elizabeth HarknessTrajectory Design Group Contact, MAT Trajectory Contact
14 February 2008
Trajectory Input/ Output Interfacing
with the MAT Codes
AAE 450 Spring 2008
Interfacing with MAT� MAT Outputs to Trajectory Inputs
– ID, Trajectory Output Array
• MAT Output Array includes: type of launch, number of stages, thrusts, burn times, mass flow rates, exit areas, exit pressures for all stages.
� Trajectory Inputs (from Trajectory)
– Steering Angles and other steering law parameters (for manual run version)
� Initial Conditions defined by type of launch
AAE 450 Spring 2008
Procedure
� Call MAT Code main_loop.m>>[ID,Trajectory_Input]= main_loop
� Produce Trajectory Input File>>AAE450_Trajectory_run(ID,Trajectory_Input)
� Run Trajectory Main Code>>AAE450_Trajectory_Main
Output
� ∆v (all components and total), orbital characteristics, maximum acceleration (G’s), plots of trajectory.
AAE 450 Spring 2008
Input Variable Definitions (MAT)
diam_1 Diameter of first stage. [m]
diam_2 Diameter of second stage. [m]
diam_3 Diameter of third stage. [m]
diam_4 Diameter of fourth (payload) stage. [m]
Pe_1 Exit pressure of 1st stage engine. [Pa]
Pe_2 Exit pressure of 2nd stage engine. [Pa]
Pe_3 Exit pressure of 3rd stage engine. [Pa]
Ae_1 Exit area of 1st stage engine. [m^2]
Ae_2 Exit area of 2nd stage engine. [m^2]
Ae_3 Exit area of 3rd stage engine. [m^2]
Variable Name Description
ID Run Identifier. This is the code for each case being run.
E.g. ‘XX-XX-XX’
type If the case is running an air launch or ground launch.
E.g. Ground: 1, Balloon: 2, Aircraft: 3.
N Number of stages. E.g 1, 2, 3
payload_mass Mass of Payload [kg] E.g. 0.2, 1, 5
m1 Mass of the 1st Stage [kg]
m2 Mass of the 2nd Stage [kg]
m3 Mass of the 3rd Stage [kg]
T1 Average Thrust of first stage. [N]
T2 Average Thrust of second stage. [N]
T3 Average Thrust of third stage. [N]
t_vertical Time of vertical flight [s]
t_burn_1 First stage burn time. [s]
t_burn_2 Second stage burn time. [s]
t_burn_3 Third stage burn time. [s]
m_dot_1 Mass flow rate of 1st Stage. [kg/s]
m_dot_2 Mass flow rate of 2nd Stage. [kg/s]
m_dot_3 Mass flow rate of 3rd Stage. [kg/s]
AAE 450 Spring 2008
Input Variable Definitions (User)� Ground and Balloon Launch
� Aircraft Launch
Variable Name Description
psi1_0 Angle at the end of 1st stage vertical [deg]
psi1_1 Starting angle for end of first stage variation [deg]
psi1_2 Final angle for end of first stage variation [deg]
psi_increment Number of increments between psi1_1 and psi1_2
psi2 Angle at the end of 2nd stage [deg]
psi3 Angle at the end of 3rd stage [deg]
Variable Name Description
angle_tclimb Steering angle at the end of climb [deg]
tb2sec1_percent_tb2 Percent of first steering law for stage two [s]
psi2 Angle at the end of 2nd stage [deg]
psi3 Angle at the end of 3rd stage [deg]
AAE 450 Spring 2008
Initial Conditions
Variable Value
h_i 0 m
psi1_0 87°
r_dot_i_1 0 m/s
theta_dot_i_1 0 rad/s
launch_lat 28°
launch_long -80°
phi_dot_i_1 0 rad/s
R_Earth 6376000 m
Variable Value
h_i 30000m
psi1_0 87°
r_dot_i_1 0 m/s
theta_dot_i_1 0 rad/s
launch_lat 28°
launch_long -80°
phi_dot_i_1 0 rad/s
R_Earth 6376000 m
Variable Value Variable Value
h_i 15000m launch_lat 28°
angle_tclimb 87° launch_long -80°
r_dot_i_1 0 m/s phi_dot_i_1 59.72/(h_i+R_Earth) rad/s
theta_dot_i_1 0 rad/s R_Earth 6376000 m
� Ground:
� Aircraft:
� Balloon:
AAE 450 Spring 2008 17AAE 450 Spring 2008
Trajectory Design
17
Elizabeth HarknessTrajectory Design Group Contact
MAT Trajectory Contact28 February 2008
Trajectory Optimisation
with Model Analysis
AAE 450 Spring 2008 18
Cost Comparison
$0
$500,000
$1,000,000
$1,500,000
$2,000,000
$2,500,000
$3,000,000
$3,500,000
0.2 kg 1 kg 5 kg
Payload Mass
To
tal
Co
st
Aircraft
Balloon
Ground
GLOW Comparison
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0.2 kg 1 kg 5 kg
Payload Mass
GL
OW
[kg
]
Aircraft
Balloon
Ground
∆ V Comparison
0
2000
4000
6000
8000
10000
12000
0.2 kg 1 kg 5 kg
Payload Mass
∆ V
[m
/s]
Aircraft
Balloon
Ground
Effect of ∆V on GLOW and Cost
AAE 450 Spring 2008
Trajectory Design
18
Plots Created by Elizabeth Harkness, from numbers generated with the MAT code
Eye Candy
AAE 450 Spring 2008 19AAE 450 Spring 2008
Trajectory Design
19
0.2 kg Model Name ∆ V GLOW Cost ∆ V GLOW Cost ∆ V GLOW Cost
Aircraft SA-SA-DT-DT 12000 7855 $3,951,564 8200 1466 $2,107,448 -32% -81% -47%Balloon SB-SA-DA-DA 12000 9354 $4,100,226 8500 1720 2,157,403$ -29% -82% -47%
Ground SG-SA-DT-DT 12000 7855 $3,951,564 9051 2069 $2,477,500 -25% -74% -37%
1 kg Model Name ∆ V GLOW Cost ∆ V GLOW Cost ∆ V GLOW Cost
Aircraft MA-SA-DA-DA 12000 9292 $4,104,172 8500 1723 $2,247,300 -29% -81% -45%
Balloon MB-SA-DA-DT 12000 11497 $4,125,954 8760 2883 $2,524,942 -27% -75% -39%Ground MG-SA-DA-DT 12000 9292 $4,143,640 9500 4342 2,699,178$ -21% -53% -35%
5 kg Model Name ∆ V GLOW Cost ∆ V GLOW Cost ∆ V GLOW Cost
Aircraft LA-SA-DA-DT 12000 11572 $4,144,163 8360 1650 2,487,533$ -30% -86% -40%
Balloon LB-SA-DA-DA 12000 12678 $4,224,938 8900 2661 2,752,318$ -26% -79% -35%Ground LG-SA-DT-DT 12000 11572 $4,188,728 10000 4133 $3,231,298 -17% -64% -23%
Starting Point Finish Point Percent Decrease
• With a moderate decrease in ∆V (15 – 30%):
• Gross lift-off weight decreases significantly (50 – 85%)
• Large cost decreases (20 – 50%)
First Round Final Analysis: Final Analysis:
0.2 kg ∆ V GLOW Cost
7000 583 1,274,787$
1 kg ∆ V GLOW Cost
8100 1383 1,855,641$
5 kg ∆ V GLOW Cost
9800 5151 2,603,219$
Numbers generated by MAT and Traj codes, compiled by E. Harkness.
AAE 450 Spring 2008 20
Psi1
(deg.)
**
Psi2
(deg.)
Psi3
(deg.)
∆ V 1st
Stage
(m/s)
∆ V 2nd
Stage
(m/s)
∆ V 3rd
Stage
(m/s)
∆ V Total
Thrust
(m/s)0.2 kg Model Name ∆ V GLOW Cost ∆ V GLOW Cost ∆ V GLOW Cost
Aircraft SA-SA-DT-DT 12000 7855 $3,951,564 8200 1466 $2,107,448 -32% -81% -47% 10% 5 -5Balloon SB-SA-DA-DA 12000 9354 $4,100,226 8500 1720 2,157,403$ -29% -82% -47% 25 -20 -20 3979 2346 4510 10834Ground SG-SA-DT-DT 12000 7855 $3,951,564 9051 2069 $2,477,500 -25% -74% -37% 0 -10 -10 4461 1779 4519 9589
1 kg Model Name ∆ V GLOW Cost ∆ V GLOW Cost ∆ V GLOW Cost
Aircraft MA-SA-DA-DA 12000 9292 $4,104,172 8500 1723 $2,247,300 -29% -81% -45% 20% -5 -20 4475 2266 4333 11074
Balloon MB-SA-DA-DT 12000 11497 $4,125,954 8760 2883 $2,524,942 -27% -75% -39% -2 -20 -20 4608 1734 4383 10725Ground MG-SA-DA-DT 12000 9292 $4,143,640 9500 4342 2,699,178$ -21% -53% -35% 20 -30 -30 4144 3037 4318 11500
5 kg Model Name ∆ V GLOW Cost ∆ V GLOW Cost ∆ V GLOW Cost
Aircraft LA-SA-DA-DT 12000 11572 $4,144,163 8360 1650 2,487,533$ -30% -86% -40% 10% 2 -6 3814 3169 3644 10627Balloon LB-SA-DA-DA 12000 12678 $4,224,938 8900 2661 2,752,318$ -26% -79% -35% 0.2 -16 -16 3640 3346 3870 10856Ground LG-SA-DT-DT 12000 11572 $4,188,728 10000 4133 $3,231,298 -17% -64% -23% 34 -26 -26 3809 3605 4240 11654
5 kg Balloon ∆ V GLOW Cost
Hybrid LB-HA-DA-DT 8450 1794 2,132,874$ -23% -13% -84% -4 -15 -15 4522 2264 3723 10509Hybrid MB-HA-DA-DA 8900 2325 2,212,691$ -20% -24% -69% -14 -24 -24 4781 2396 4007 11184Solid LB-DA-DA-DA 9900 3115 2,558,951$ -7% -73% -8% -15 -23 -23
(Continued)
∆ V
LEO
(m/s)
∆ V
Grav
(m/s)
∆ V Drag
(m/s)
∆ V
Earth
Assist
(m/s)
∆ V
Total
(m/s)
∆ V to Circ
(m/s)
Made
to
orbit
(Y/N)
Apoge
e (km)
Perig
ee
(km)
e
MAT
inert
Mass
Frac
1st
MAT
inert
Mass
Frac
2nd
MAT
inert
Mass
Frac
3rd
MAT ∆V
Dist 1st
(m/s)
MAT ∆V
Dist 2nd
(m/s)
MAT ∆V
Dist 3rd
(m/s)
MAT
∆V
Total
(m/s)
0.2 kg Model Name
Aircraft SA-SA-DT-DT 0.18 0.24 0.27
Balloon SB-SA-DA-DA 7727 1239 8 411 9287 562 Y 3262 394 0.175 0.18 0.24 0.24 0.45 0.15 0.40 8500Ground SG-SA-DT-DT 7727 1664 607 409 9589 353 Y 3222 301 0.180 0.15 0.24 0.24 0.40 0.20 0.40 9051
1 kg Model Name
Aircraft MA-SA-DA-DA 7727 1373 75 410 8765 809 Y 2449 398 0.132 0.18 0.24 0.24 0.4 0.2 0.4 8500
Balloon MB-SA-DA-DT 7727 1493 7 411 8816 448 Y 718 604 0.008 0.15 0.30 0.24 0.45 0.15 0.40 8760Ground MG-SA-DA-DT 7727 1886 1335 409 9841 569 Y 1104 491 0.043 0.12 0.27 0.24 0.5 0.15 0.35 9500
5 kg Model Name
Aircraft LA-SA-DA-DT 7727 1591 77 410 8985 161 Y 2062 404 0.109 0.18 0.21 0.24 0.35 0.30 0.35 8360Balloon LB-SA-DA-DA 7727 1323 6 411 8645 961 Y 5406 301 0.276 0.18 0.18 0.24 0.35 0.30 0.35 8900Ground LG-SA-DT-DT 7727 2187 539 409 10043 249 Y 689 452 0.017 0.15 0.18 0.21 0.35 0.30 0.35 10000
Starting Point
(Percent decrease
from LB-SA-DA-DA)
Finish Point Percent Decrease
AAE 450 Spring 2008 21
Psi1
(deg.)
**
Psi2
(deg.)
Psi3
(deg.)
∆ V 1st
Stage
(m/s)
∆ V 2nd
Stage
(m/s)
∆ V 3rd
Stage
(m/s)
∆ V
Total
Thrust
(m/s)
∆ V
LEO
(m/s)
∆ V
Grav
(m/s)
∆ V
Drag
(m/s)
∆ V
Earth
Assist
(m/s)
∆ V
Total
(m/s)
∆ V to
Circ
(m/s)
0.2 kg Model Name ∆ V GLOW Cost
SB-HA-DA-DA 7000 583 1,274,787$ 9 0 0 2928 1466 4516 8910 7727 1737 9 411 9062 740
1 kg Model Name ∆ V GLOW Cost
MB-HA-DA-DA 8100 1383 1,855,641$ -13 -15 -15 4376 1739 4320 10434 7727 1205 8 411 8529 760
5 kg Model Name ∆ V GLOW Cost
LB-HA-DA-DA 9800 5151 2,603,219$ -20 -20 -20 6411 1983 3720 12114 7727 1283 6 411 8605 663
(Continued)
Made
to
orbit
(Y/N)
Apoge
e (km)Perigee (km) e
MAT
inert
Mass
Frac
1st
MAT
inert
Mass
Frac
2nd
MAT
inert
Mass
Frac
3rd
MAT ∆V
Dist 1st
(m/s)
MAT ∆V
Dist 2nd
(m/s)
MAT ∆V
Dist 3rd
(m/s)
MAT
∆V
Total
(m/s)
0.2 kg Model Name
SB-HA-DA-DA Y 782 305 0.035 0.27 0.30 0.24 0.35 0.15 0.50 7000
1 kg Model Name
MB-HA-DA-DA Y 7554 303 0.352 0.21 0.3 0.24 0.45 0.15 0.40 8100
5 kg Model Name
LB-HA-DA-DA Y 1854 439 0.094 0.12 0.27 0.24 0.55 0.15 0.30 9800
Final Winners