04/12/08
AAE 450 Spring 2008

 Results:
 Compare to Shuttle (2029633 kg GLOM): 107 m/s
Titan IV/Centaur (886420 kg GLOM): 156 m/s
(from SMAD)
AAE 450 Spring 2008
Trajectory Optimization
Chua – 01/31

Feb 7, 2008
Trajectory group, Delta V
Delta V determination, Saturn V comparison
AAE 450 Spring 2008

 All stages masses modified to Saturn V
 Engine thrust and exit area modified
 Burn Time and mass flow rate modified
 Stage diameters modified
 Assume ΔV_Leo is same
*All Saturn V specs provided by http://www.nasm.si.edu/collections/imagery/apollo/saturnV.htm
AAE 450 Spring 2008
Trajectory

Launch Type ΔV_Grav ΔV_Drag ΔV_Total
Default Inputs
Saturn V Inputs
1310 m/s 293 m/s ~15000 m/s
2362 m/s 36 m/s ~11000 m/s
 Continue on Trajectory Model
AAE 450 Spring 2008
Trajectory

02/21/08
Trajectory Analyst
Modeling Drag
Assistance provided by Jayme Zott, Kyle Donohue
AAE 450 Spring 2008 6<#>

 Assumptions:
– Atmosphere molecular weight is constant
– Angle of Attack is zero
 Speed of Sound: a = [ γRT] 1/2
 Use Mach Number to get C
D
 Apply Equation for Drag
D = C
D
* q * S
AAE 450 Spring 2008
Trajectory Optimization
7<#>

 With function, notice drag behavior
 Over most
Mach numbers, drag without function is higher
Drag Force v. Mach Number
25000
20000
W/O Function
Cd Function
15000
10000
5000
0
0 1
AAE 450 Spring 2008
Trajectory Optimization
2 3
Mach Number
4 5
Figure by Brad Ferris
8<#>

 Without Function
– 762 / 232710 km
(periapsis / apoapsis)
– Eccentricity: 0.942
– Delta V Drag: 461 m/s
– Delta V Total: 10760 m/s
– Steering Angles:
6,-28,-28 deg.
 With Function
– 807 / 232477 km
(periapsis / apoapsis)
– Eccentricity: 0.942
– Delta V Drag: 384 m/s
– Delta V Total: 10672 m/s
– Steering Angles:
6,-28,-28 deg.
AAE 450 Spring 2008
Trajectory Optimization
9<#>

25000
20000
15000
10000
5000
0
0
Drag v. Time
W/O Function
Cd Function
50 100
Time (s)
150 200 250
AAE 450 Spring 2008
Trajectory Optimization
Figure by Brad Ferris
10<#>

02/21/2008
Trajectory
Validation of Thrust in the Trajectory Codes
AAE 450 Spring 2008 11

 T m dt
Procedure
• Set Drag = 0 >> Assume No Atmosphere
• Calculate

V thrust
  T m dt for each stage, using the data from Ariane 4, Saturn V and Pegasus
3. Compare with the historical data.
Results
The calculated values matched with the historical data!!
(3-5 or more significant figures)
Special Thanks to Kevin & Mr. Tsohas for helping us
AAE 450 Spring 2008
Trajectory
12

•Full
Atmosphere
•Exit
Pressure and
Exit Area were calculated.
Thrust in Vacuum Condition
Thrust in Sea Level Condition
AAE 450 Spring 2008
Trajectory
13

Backup
Slides
# of engines: 4
Ariane 4
1st Stage 2nd Stage 3rd Stage
Thrust
Isp
Sea Level
Vacuum
Sea Level
Vacuum
Pressure chamber
Nozzle Expansion Ratio (epsilon)
676.9
758.5
248.5
278.4
5.85
10.48
785
293.5
5.85
30.8
62.7
445.1
3.50
62.5
[kN]
[kN]
[s]
[s]
[MPa]
Special Thanks to Mr. Tsohas for providing the data
AAE 450 Spring 2008
Trajectory
14

Backup
Slides
Saturn V, First Stage
Thrust
Isp
Pressure
Nozzle Expansion Ratio
Sea Level
Vacuum
Sea Level
Vacuum chamber
(epsilon)
6,747.50
7,740.50
265
304
7.0
16
AAE 450 Spring 2008
Trajectory
[kN]
[kN]
[s]
[s]
[MPa]
15

Backup
Slides
 
1 
A
*

A e
 
1
2
1

 
1 

 p p e
0



1




1

1




1


 p p
0 e





1




Solve for
SL
 *
T A p
0
 p e and get p e p
0





2

2

1

 
1





1

1



1


 p p
0 e








1




1
2
  p e
 p a
 
A
*
Solve for A
* and get A e Special Thanks to D.Lattibeaudiere
for co-working on prop_test.m
AAE 450 Spring 2008
Trajectory
16

Backup
Slides
 Ariane 4: 10,120[m/s]
 Saturn V: 13,470 [m/s]
 Pegasus: 8,360 [m/s]
AAE 450 Spring 2008 17

2/28/08
APM, Trajectory Group
Ballistic Coefficient Analysis
‘measure of its ability to overcome air resistance in flight’ 1
AAE 450 Spring 2008 18

Ballistic Coefficient Definition
BC
 m
C
D
S where m - total mass
C
D
- drag coefficient f(M)
- calculated by Aerothermal solve_cd.m
S – reference area; stage diameter
Expect:
- Larger BC for more massive launch vehicles
10s vertical flight our region
AAE 450 Spring 2008
Trajectory
BC: Large Launch Vehicles transonic regime
M = 1.19 @ t = 70s out of atmosphere
19

BC: Sample MATs Launch Vehicles end of 1 st stage
S changes
Conclusions:
- Trends are as expected
- Our vehicle cannot easily overcome air resistance
Thank you to Jun Kanehara, Elizabeth Harkness, Alan Schwing, and Kevin Kloster for all of their help this week!
AAE 450 Spring 2008
Trajectory
20

Backup Slides
Future Work
 Optimize final design cases
 Write report
References:
1. "Ballistic coefficient." Wikipedia January 18, 2008 February 27, 2008
<http://en.wikipedia.org/wiki/Ballistic_coefficient>.
2. Longuski, Prof. J. "AAE 450 Spacecraft Design Lecture #6." Purdue University, West
Lafayette, IN.
3. Longuski, Prof. J. Ballistic Coefficient interview. February 20, 2008.
4. Kloster, Kevin. Ballistic Coefficient interview. February 20-27, 2008.
AAE 450 Spring 2008
Trajectory
21

Steering Angles Used In Analysis
Vehicle
Pegasus
Saturn V
Ariane 4
SB-HA-DA-DA
(8700 km/s)
Traj Ver 5.3
SG-SA-DT-DT
(9051 km/s)
Traj Ver 5.3
LG-SA-DT-DT
(10,000 km/s)
Traj Ver 5.3
Alt @ end of stage 1 (km)
94.43
109.75
57
194.09
71.44
56.44
mdot1 (kg/s)
206.136
13,360.24
1,112.19
6.847
14.208
18.391
tburn1 (s)
73
161
205
196.5
182.4
171.2
0
34 psi1e (deg)
87
87
87
-14
-10
-26 psi2e (deg)
-25
40
40
-20
-10
-26
Psi3e (deg)
-30
0
0
-20
AAE 450 Spring 2008
Trajectory
22