Brad Ferris - engineering.purdue.edu

Brad Ferris01/24/0801/24/08

Trajectory AnalystModeling ThrustModeling Thrust

Assistance Provided by Daniel Chua

AAE 450 Spring 2008

ModelingA tiAssumptions– Constant mass flow rate, exit velocity– No flow separation

New code structure– Call a function to get thrust at a given time– Apply Thrust Equationpp y q

T = m_dot * ve + ( pe – pa) * Ae

AAE 450 Spring 2008Group Name (i.e.Trajectory Optimization)

ResultsVelocity at 320 km circular orbit: 7 715 km/sVelocity at 320 km, circular orbit: 7.715 km/sVelocity at 320 km w/o pressure thrust: 5.296 km/sVelocity at 320 km w/ pressure thrust: 6 911 km/sVelocity at 320 km w/ pressure thrust: 6.911 km/sVelocity difference due to pressure thrust: 1.615 km/s

C l i d F t W kConclusions and Future WorkCode can now model thrust more accuratelyCode can now model thrust more accuratelyIncorporate data from Propulsion group

AAE 450 Spring 2008Group Name (i.e.Trajectory Optimization)

Velocity Plots–Without Pressure ThrustRadial Velocity Tangential Velocityy g y

Blue – Simulation VelocityGreen – Circular Orbit Velocity

AAE 450 Spring 2008Group Name (i.e.Trajectory Optimization)

Trajectory Plots–Without Pressure ThrustAltitude Trajectoryj y

Red – Simulation AltitudeGreen – Circular Orbit Altitude

AAE 450 Spring 2008Group Name (i.e.Trajectory Optimization)

Velocity Plots–With Pressure ThrustRadial Velocity Tangential Velocityy g y

Blue – Simulation VelocityGreen – Circular Orbit Velocity

AAE 450 Spring 2008Group Name (i.e.Trajectory Optimization)

Trajectory Plots–With Pressure ThrustAltitude Trajectoryj y

Red – Simulation AltitudeGreen – Circular Orbit Altitude

AAE 450 Spring 2008Group Name (i.e.Trajectory Optimization)

Brad Ferris02/07/08

Trajectory AnalystEffect of Varying Vehicle Parameters

AAE 450 Spring 2008

Nominal OrbitN i l C ditiNominal Conditions– Jupiter C Vehicle Model

3 Stage Ground– 3 Stage, Ground Launch

– 1 kg Payload Massg y

Orbit Parameters– Eccentricity: 0.650Eccentricity: 0.650

(0 for a circular orbit)– Periapsis: 95 km

Image by: Brad Ferris

AAE 450 Spring 2008Trajectory Optimization

Image by: Brad FerrisTraj. Code by: Traj. Group

Eff t f V i P tEffects of Varying ParametersChange Effect Effect of Varying Parameters on Orbit Shape

0 8

0.9

1

Change EffectIncrease 1st

Stage Prop M

Less Eccentric

0.4

0.5

0.6

0.7

0.8

Ecce

ntri

city 1st Stage Prop Mass

1st Stage Burn Time1st Stage Thrust

MassIncrease 1st

Stage Burn Time

More Eccentric

0

0.1

0.2

0.3

65 75 85 95 105 115 125 135

ETime

Increase 1st

Stage ThrustMore Eccentric

65 75 85 95 105 115 125 135

Percent Nominal Value

Increase Payload Mass

Less Eccentric Chart and Data by: Brad Ferris

Traj. Code by: Traj. Group

AAE 450 Spring 2008Trajectory Optimization

V i P l d MVarying Payload MassEccentricity v. Payload Mass

0.6

0.7

0.3

0.4

0.5

Ecc

entri

city

0

0.1

0.2

E

00 2 4 6 8 10 12

Payload Mass (kg)

Chart and Data by: Brad Ferris

AAE 450 Spring 2008Trajectory Optimization

Chart and Data by: Brad FerrisTraj. Code by: Traj. Group

Brad Ferris02/21/0802/21/08

Trajectory AnalystModeling DragModeling Drag

Assistance provided by Jayme Zott, Kyle Donohue

AAE 450 Spring 2008

ModelingA tiAssumptions:– Atmosphere molecular weight is constant– Angle of Attack is zero

Speed of Sound: a = [γRT]1/2

Use Mach Number to get CD

Apply Equation for DragApply Equation for DragD = CD * q * S

AAE 450 Spring 2008Trajectory Optimization

V lid tiValidationDrag Force v. Mach NumberWith

20000

25000

W/O F ti

function, notice drag

15000ra

g (N

)W/O FunctionCd Function

behaviorOver most M h

5000

10000DMach numbers, drag

00 1 2 3 4 5

Mach Number

drag without function is

AAE 450 Spring 2008Trajectory Optimization

higher Figure by Brad Ferris

Orbit parametersWithout Function With F tiWithout Function– 762 / 232710 km

(periapsis / apoapsis)

With Function– 807 / 232477 km

(periapsis / apoapsis)(p p p p )– Eccentricity: 0.942– Delta V Drag: 461 m/s

(periapsis / apoapsis)– Eccentricity: 0.942– Delta V Drag: 384 m/s

– Delta V Total: 10760 m/sSteering Angles:

g– Delta V Total: 10672

m/sSt i A l– Steering Angles:

6,-28,-28 deg.– Steering Angles:

6,-28,-28 deg.

AAE 450 Spring 2008Trajectory Optimization

D d TiDrag and TimeDrag v. Time

20000

25000

15000

rag

(N)

W/O FunctionCd Function

5000

10000Dr

00 50 100 150 200 250

Time (s)

AAE 450 Spring 2008Trajectory Optimization

Figure by Brad Ferris

Brad Ferris03/06/08

Trajectory AnalystEffect of Launch Angle

AAE 450 Spring 2008

DescriptionS tSetup– Case: V125, 200 g (SB-HA-DA-DA)– Constant Steering Law : 26, -2, -2 (deg.)

Method– Vary Launch Angle– Examine Effect on Orbit Obtained

AAE 450 Spring 2008Trajectory Optimization

C iComparisonsP i i & A i L h A l

As launch angle decreases

Periapsis & Apoapsis v. Launch Angle

2500

3000

apoapsisperiapsisdecreases,

orbit becomes more eccentric 1500

2000

eigh

t (km

)

p p300 (km)

As launch angle decreases

500

1000 He

decreases, total ΔV required d

07980818283848586878889

Launch Angles (deg.)

Figure by: Brad Ferris

AAE 450 Spring 2008Trajectory Optimization

decreases Figure by: Brad Ferris

ΔV T t l C iΔV Total ComparisonsΔV Total v. Launch Angle

9180

9190

9150

9160

9170

otal

(km

/s)

9120

9130

9140

ΔV

To

91107980818283848586878889

Launch Angle (deg.)

AAE 450 Spring 2008Trajectory Optimization

Figure by: Brad Ferris

E t i it C iEccentricity ComparisonsEccentricity v. Launch Angle

0.14

0.16

0.08

0.1

0.12

entri

city

0.02

0.04

0.06 Ecc

07980818283848586878889

Launch Angle (deg.)

AAE 450 Spring 2008Trajectory Optimization

Figure by: Brad Ferris

Brad Ferris03/27/08

Trajectory AnalystForces, Equations of Motion

AAE 450 Spring 2008

Free Body DiagramForcesForces

Thrust (T)Drag (D)Weight (W)

AnglesAnglesFlight path angle (γ)Thrust offset (α΄)

Figure by Brad Ferris

AAE 450 Spring 2008Trajectory Optimization

Modeling ForcesDrag (D)g ( )• D = CD * q * S• Speed of sound: a = [γRT]1/2Speed of sound: a [γRT]• CD - function on Mach number (Aerothermal

group)g p)Thrust (T)• T = m dot * ve + ( pe – pa) * AeT m_dot ve ( pe pa) Ae

• Topt= m_dot * ve, pe, Ae (Propulsion group)Weight (W=m(t)*g)

AAE 450 Spring 2008Trajectory Optimization

Weight (W m(t) g)

ee

Vector BasesBasesBases

ei frame: Earth fixedEarth fixedai frame: rotates along ez by θ(longitude)b frame:bi frame: rotates along ay by Φ

AAE 450 Spring 2008Trajectory Optimization

y y(latitude) Figure by Amanda Briden

Equations of Motion

Given in body-fixed unit vectors (bi)

AAE 450 Spring 2008Trajectory Optimization