astrodynamics - ltas- · pdf filegaëtan kerschen space structures & systems lab (s3l)...

Gaëtan KerschenSpace Structures & Systems Lab (S3L)

7. Launch Vehicle Dynamics

Astrodynamics(AERO0024)

2

Course Outline

THEMATIC UNIT 1: ORBITAL DYNAMICSChapter 2: The Two-Body Problem

Chapter 3: The Orbit in Space and Time

Chapter 4: Non-Keplerian Motion

THEMATIC UNIT 2: ORBIT CONTROLChapter 5: Orbital Maneuvers

Chapter 6: Interplanetary Transfer

THEMATIC UNIT 3: ORBITAL LAUNCHChapter 7: Launch Vehicle Dynamics

3

Motivation: Severe Constrains due to Launch

Launch vehicle:

Payload mass and volume.

Attainable orbit.

Mechanical vibrations and acoustics.

Cost per kilogram.

Launch site: attainable inclination.

4

Example: Payload Mass and/or Destination

Soyuz ST v2-1b (Kourou Launch)

5

STK Astrogator

6


7.1 Ascent flight mechanics

7.2 Staging

7


7.1 Ascent flight mechanics

7.1.1 Kinematics and dynamics

7.1.2 Rocket performance

7.1.3 Ascent trajectory

8

Kinematics

2

ˆ ˆ

ˆ ˆ ˆ ˆ

, radius of curvature

t t

t t n n t n

s v

va a vρ

ρ

= =

= + = +

v u u

a u u u ur

a P

path

s

v


9

Kinematics

, center of curvature

vds d s

C

ρ φ ρφ φρ

= ⇒ = ⇒ =

dφ

ˆ tu

ˆ nuCρ


10

Flight over a Flat Earth

, flight path angle

vd dγ φ γ φρ

γ

= − ⇒ = − = −

C

dφ ρP

g

dγ

γ2

cosnva g γρ

= =

cosgvγγ = −


11

Account for the Earth Curvature


12

Tangential and Normal Accelerations

2

2

Flat Earth:

With curvature: cos

t

n

nE

dvadt

v da vdtd va vdt R h

γρ

γ γ

=

= = −

= − ++


13

What Are the Forces Acting on a Vehicle ?

Rocket-powered ascent vehicles bridge the gap between

1. Flight in the atmosphere (governed by gravitational and aerodynamic forces.

2. Space flight, shaped principally by gravitational forces.

14

Newton’s Second Law

2

sin

cos cos

t

nE

dv T Da gdt m m

d va v gdt R h

γ

γ γ γ

= = − −

= − + =+

2

sin

cosE

dv T D gdt m md vv gdt R h

γ

γ γ

= − −

⎛ ⎞= − −⎜ ⎟+⎝ ⎠


15

Downrange Distance and Altitude

cos

sin

E

E

Rdx vdt R hdh vdt

γ

γ

=+

=


16

Assumptions Made ?

2

sin

cos

cos

sin

E

E

E

dv T D gdt m md vv gdt R h

Rdx vdt R hdh vdt

γ

γ γ

γ

γ

= − −

⎛ ⎞= − −⎜ ⎟+⎝ ⎠

=+

=

Nonrotating Earth

Incidence & pitch

Lift neglected

Incidence & pitch


17

Is α the aerodynamic angle of attack ?

P. Fortescue et al., Spacecraft Systems Engineering, Wiley.


18

Pitch Angle

Direction of thrust vector.

It is a control variable that allows vehicle steering. The vehicle should adhere to a predetermined flight path.

Space shuttle example: each solid rocket booster has two independent hydraulic power units that gimbal the rocket's nozzle to provide the primary means of steering the shuttle during launch.


19

Lifting Force

Launch vehicles are designed to be strong in lengthwise compression.

To save weight, they are made relatively weak in bending, shear and torsion, which are the kind of loads induced by lifting surfaces.

Lifting loads are held closely to zero during powered ascent through the atmosphere by maintaining small angles of attack.


20

Lifting Force: Ariane 5 – Flight 501

http://www-rocq.inria.fr/qui/Philippe.Deschamp/divers/ariane_501.html

… le lanceur a commencé à se désintégrer à environ H0 + 39 secondes sous l'effet de charges aérodynamiques élevéesdues à un angle d'attaque de plus de 20° …


Video

http://www-rocq.inria.fr/qui/Philippe.Deschamp/divers/ariane_501.html

21

Nonrotating Earth

Predictions of position and velocity relative to the surface are in error.

The atmosphere rotates with the Earth.

Planetary rotation aids the launch by providing an initial velocity in the direction of rotation.


22

Equations with Lift and Pitch Angle

2

cos( ) sin

sin( ) cos

cos

sin

E

E

E

dv T D gdt m md T L vv gdt m m R h

Rdx vdt R hdh vdt

α δ γ

γ α δ γ

γ

γ

+= − −

⎛ ⎞+= + − −⎜ ⎟+⎝ ⎠

=+

=

These equations are not solvable in closed form. In addition, g, ρ, CL, CD, etc. are not constant.

⇒ Numerical integration is required.7.1.1 Kinematics and dynamics

23

Newton’s Third Law

Newton’s balance of momentum principle dictates that when mass is ejected from a system in one direction, the mass left behind must acquire a velocity in the opposite direction.

Example: a diver keaping off a small boat at rest in the water


24

Newton’s Third Law

A rocket motor uses chemical energy of solid or liquid propellants to steadily and rapidly produce a large quantity of high pressure gas which is then expanded and accelerated through a nozzle.

This mass of combustion products flowing out of the nozzle at supersonic speed possesses a lot of momentum and causes the vehicle itself to acquire a momentum in the opposite direction.


25


( ) ( )( ) e e a em m v v mv mv p p A t−Δ + Δ −Δ − = − Δ⎡ ⎤⎣ ⎦

ImpulseLinear momentum


26


( ) ( )( ) e e a em m v v mv mv p p A t−Δ + Δ −Δ − = − Δ⎡ ⎤⎣ ⎦

( )( )e e e a edvm m v v p p Adt

− + = −

0tΔ →

( ) ( )( )e e e e a em m t v v m t v mv p p A t− Δ + Δ − Δ − = − Δ⎡ ⎤⎣ ⎦

em m tΔ = Δ

( ) ( )e a e e edvm p p A m v v Tdt

= − + + =


27

Thrust Equation

( ) 0( )e a e e e e eq e spT p p A m v v m V m I g= − + + = =

Veq is the equivalent exhaust velocity.

The portion of Veq due to the pressure term will nearly always be small relative to ve.

Thrust losses: degradation in specific impulse or thrust if the nozzle flow is not ideally expanded.


28

Rocket Performance

0 0sp e spdmT I g m I gdt

= = −

( )cossin

Tdv D gdt m m

α δγ

+= − −

( )0

0 0 0 0

cosdt dt dt sin dtf f f f

spt t t t

dmI gdv Ddt gdt m m

α δγ

− += − −∫ ∫ ∫ ∫


29

Rocket Performance

( )0 0

0 0 0

0 0

(1 cos )dt dt dt

dt sin dt

f f f

f f

sp spt t t

t t

dm dmI g I gdv dt dtdt m m

D gm

α δ

γ

− − − += − −

−

∫ ∫ ∫

∫ ∫

Drag loss Gravity loss

Steering lossIdeal velocity

increment


30

Ideal Velocity Increment: Tsiolkovsky

0

00dt lnf

spt isp

f

dmI g mdtv I gm m

− ⎛ ⎞Δ = = ⎜ ⎟⎜ ⎟

⎝ ⎠∫

This equation gives the maximum theoretically obtainable velocity increment from a single stage.

Clearly, high Isp is desired.

The design goal is to have a vehicle consisting, as much as possible, of payload and propellant only.


31

Specific Impulse


32

Steering Loss

( )0

0

(1 cos )dtf

spt

S

dmI gdtV

m

α δ− − +Δ = ∫

This term is nonzero when thrust is not aligned with the absolute velocity. The thrust component normal to the direction of travel fails to add to the vehicle velocity.

Caused by the need to steer the launch vehicle.

If one accounts for Earth’s rotation, a zero angle of attack means nonzero steering loss !


33

Steering Loss

Vertical takeoff vehicles have much less steering losses as compared to horizontal takeoff vehicles since a horizontal vehicle must pitch upward 90 degrees to change its direction of flight to vertical.

LEO: 34 m/s LEO: 365 m/s


http://fr.wikipedia.org/wiki/Fichier:Delta_II_7925_(2925)_rocket_with_Deep_Impact.jpg

http://en.wikipedia.org/wiki/File:STS-79_rollout.jpg

34

Drag Loss

Caused by friction between the launch vehicle and the atmosphere.

A long slender cylinder with a pointed nose is a favored shape to reduce drag losses since over three-quarter of drag losses are caused by supersonic drag.

Only significant during the first 2 minutes of flight.

0dtft

DDVm

Δ = ∫


35

Gravity Loss

Arises because part of the rocket engine's energy is wasted holding the vehicle against the pull of Earth's gravity.

Is not influenced by the shape of the launch vehicle.

Dependence on the time of flight and on the trajectory.

0sin dtft

Gv g γΔ = −∫


36

Launch Losses



37

Launch Losses

Ariane A-44L: 1576 m/s 135 m/s

Atlas I: 1395 m/s 110 m/s

Shuttle: 1222 m/s 107 m/s

Saturn V: 1534 m/s 40 m/s

Titan IV/Centaur: 1442 m/s 156 m/s

Drag lossGravity loss


38

Launch Losses


39

Objectives of the Ascent Trajectory

1. Desired orbital plane: inclination and RAAN.

2. Injection into orbit: fly horizontally at burnout.

3. Efficiency: minimize losses !

4. Structural design: limit the angle of attack to decrease the loads applied to the launch vehicle.


40

Orbit Inclination

The plane of the orbit must contain the center of the earth as well as the point at which the satellite is inserted into orbit.

If the launch direction is not directly eastward, the orbit will have an inclination greater than the launch latitude.

Launch azimuth is the flight direction at insertion measured clockwise from north on the local meridian.

A launch azimuth equal to 90º is due east and takes full advantage of the Earth’s rotational velocity.


41

Launch Azimuth

Vallado, Fundamental of Astrodynamics and Applications, Kluwer, 2001.


42

Launch Azimuth

0 50 100 150 200 250 300 3500

50

100

150

200

Launch azimuth, degrees

Incl

inat

ion,

deg

rees

Lat 0 degLat 20 degLat 40 degLat 60 deg


43

STK Example


44

Launch Azimuth Restrictions

Vallado, Fundamental of Astrodynamics and Applications, Kluwer, 2001.


45

KSC: No Polar and SSO Satellites !

Larson and Wertz, Space Mission Analysis and Design, Microcosm, 3rd Edition.7.1.3 Ascent trajectory

46

RAAN

The UT for launch should be selected according to the desired orbit’s initial nodal location.


47

Minimize Losses

( )0 0

0 0 0

0 0

(1 cos )dt dt dt

dt sin dt

f f f

f f

sp spt t t

t t

dm dmI g I gdv dt dtdt m m

D gm

α δ

γ

− − − += − −

−

∫ ∫ ∫

∫ ∫

Drag loss Gravity loss

Steering lossIdeal velocity

increment


48

Drag Loss

Vertical trajectory at take-off to go above the dense lower layers of the atmosphere.

Slow ascent to minimize the effect of the squared velocity in regions of higher density.

0dtft

DDVm

Δ = ∫


49

Gravity Loss

Attain horizontal flight as soon as possible.

A high-thrust-to-weight ratio is desirable to minimize the time of flight.

0sin dtft

Gv g γΔ = −∫


50

Steering Loss

Any turning of the vehicle at all is undesirable … but launch vehicles take off vertically but must be flying parallel to the earth’s surface at injection into orbit.

Turn early at low speeds (α↑ if v↑):

( )0

0

(1 cos )dtf

spt

S

dmI gdtV

m

α δ− − +Δ = ∫

21sin cos /

E

d L v Tv gdt m R h mγα γ− ⎛ ⎞⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ ⎝ ⎠⎝ ⎠⎝ ⎠


51

Incompatible Requirements !

Drag: vertical trajectory at take-off and slow ascent.

Gravity: horizontal flight and small time of flight.

Steering: don’t turn !


52

Gravity Can Steer the Vehicle

gcosγ produces a component normal to the flight path. A gradual turn toward the horizontal will be executed for any case other than a vertical ascent. No thrust is wasted.

Because gravity does the steering, the angle of attack can be maintained at zero: no transverse aerodynamic stress on the launch vehicle.

2

2

sin( )Gravity turn : cos

cos

E

E

d T L vv gdt m m R h

vgR h

γ α δ γ

γ

⎛ ⎞+= + − −⎜ ⎟+⎝ ⎠

⎛ ⎞= − −⎜ ⎟+⎝ ⎠


53

Gravity Turn

Gravity turns are very useful when departing from an airless planet (e.g., Lunar Module ascent flight).

For a mission around the Earth, gravity turns may comprise portions of an ascent profile, but are rarely used for a complete mission (higher fuel consumption).


54

Complex Trajectory Optimization

Objective: accurate orbital injection

Criterion: maximize payload mass or minimize fuel consumption

Constraints:Angle of attack during atmospheric phase

Maximum dynamic pressure

Maximum acceleration

Thermal fluxes

Visibility (radar)

Safety


55

Possible Ascent Trajectory

Vertical liftoff for a few hundred feet (γ=90º).

After clearing the launch pad, the rocket must roll to the desired launch azimuth.

Then, a pitch program is initiated to turn the vehicle (γ<90º)⇒ Zero-incidence trajectory (gravity turn).

Once outside the atmosphere (~50km,~2 minutes)⇒ Prescribed ascent profile (angular rate vs. time).

Ideally, the vehicle is flying horizontally at orbital speed (γ=0º).


56

Typical Ariane Flight Profile (GTO)



57

Typical Ariane Flight Profile (GTO)



58

Ariane 1 Trajectory using LVSim

100 200 300 400 500 600 700 8000

2

4

6

8

Time (s)

Spee

d x

(km

)


59


100 200 300 400 500 600 700 800-40

-20

0

20

40

60

80

Time (s)

Flig

ht p

ath

angl

e (o

)


60


100 200 300 400 500 600 700 800-10

-5

0

5

10

15

20

Time (s)

Angl

e of

atta

ck (o

)


61

Ariane 5 Typical Sequence of Events

62

Ariane 5 Typical GTO: Ground Track

64

Space Shuttle Typical Sequence of Events

65

Roll Program and Sound Barrier

Video

66


7.2 Staging

67

Dimensionless Quantities

0

1PL P E

PL P E

m m m mπ π π= + +

= + +

PL

P E

mm m

λ =+

E

P E

mm m

ε =+

Payload ratio Structural ratio

7.2 Staging

68

Tsiokolvsky’s Equation

0 01ln lni

sp spf

mv I g I gm

λε λ

⎛ ⎞ +⎛ ⎞Δ = =⎜ ⎟ ⎜ ⎟⎜ ⎟ +⎝ ⎠⎝ ⎠

The advantage of a light structure is clear. The designer should therefore keep the structural ratio as small as possible.

Space Shuttle external tank: ε=0.0361 (mE=27000 kg and mP=721000 kg)

7.2 Staging

69

No Payload / No Drag and Gravity Losses

01ln

7.8 3451 09.8ln

0.1 0

sp

sp

vIg

I s

λε λ

Δ=

+⎛ ⎞⎜ ⎟+⎝ ⎠

= =+⎛ ⎞

⎜ ⎟+⎝ ⎠

Launch vehicle staging is a necessity !

7.2 Staging

70

Launch Vehicle Staging

Series staging Parallel staging

7.2 Staging

71

Series Staging

Series staging

The total delta-v is merely computed by summing up the contributions of the different stages.

7.2 Staging

72

Series Staging (All Stages Similar)

7.2 Staging

73

Optimal Staging (Different Stages)

For given specific impulses and structural ratios of each stage, the objective is to seek the minimum mass multistage vehicle that will carry a given payload to a specified burnout velocity.

7.2 Staging

74

Another Advantage of Staging

Nonideal expansion of nozzle flow.

The multistage rocket offers a solution to this problem.

The first stage can be designed for best performance in the lower atmosphere.

The upper stages can be designed to perform best in vacuum.

7.2 Staging

75

7.1 ASCENT FLIGHT MECHANICS




7.2 STAGING


76

Launch Vehicle: The Future (Now the Past !)

http://upload.wikimedia.org/wikipedia/commons/6/63/Size_Comparison.png

Gaëtan KerschenSpace Structures & Systems Lab (S3L)


Astrodynamics(AERO0024)

astrodynamics - ltas- · pdf filegaëtan kerschen space structures & systems lab (s3l)...

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