trajectory design group contact preliminary trajectory ...€¦ · aae 450 spring 2008 input...

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


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.]


AAE 450 Spring 2008


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


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]


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


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%


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.


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


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


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


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

trajectory design group contact preliminary trajectory ...€¦ · aae 450 spring 2008 input...

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