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MAE 4262: ROCKETS AND MISSION ANALYSIS

Single and Multi-Stage Rockets

September 6, 2012

Mechanical and Aerospace Engineering DepartmentFlorida Institute of Technology

D. R. Kirk

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SINGLE-STAGE SOUNDING ROCKET SUMMARY

Want to reduce burn time as much as possible while accelerating against a gravity field

Short burn time reduces energy consumed in lifting propellants

Very short burn time implies very high accelerations Structural limitations

High mass flows, lots of weight for nozzles, turbo-machinery, cooling, etc.

Drag goes as V2

Is there an optimum acceleration for a given rocket configuration?

In limit of no drag and no gravity, burn time has no influence on velocity increment

1ln12

ln

2

1

1

ln

111ln

22

max

2

RRRtu

gRuh

gtR

Rtutuh

gtt

t

RutV

bee

bbebeb

b

eVelocity during

Powered flight

Height at burnout

Maximum altitude

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PRELIMINARY DEFINITIONS

Total mass of rocket, Mo, may be written as sum of 3 primary components:

Payload mass, ML

Propellant mass, MP Structural mass, MS

Includes everything but payload and propellant

Engines, tanks, controls, etc.

If rocket consumes all its propellant during firing, burnout mass consists ofstructure and payload:

NOTE: Other texts and references will breakdown rocket components in variousways and into many more parts (i.e., Sutton, Kerrebrock, Turner, Humble)

SPLo MMMM

SLb MMM

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DEFINITIONS

Symbol Ratio Description

RSL

o

b

o

MM

M

M

MR

Mass Ratio: initial mass / mass at the

end of the thrust period. Want this

ratio large.

SP

L

Lo

L

MM

M

MM

M

Payload Ratio: ratio of payload to

everything but payload. Want this

large, but larger the payload, the

lower maximum attainable velocity.

Lo

Lb

SP

S

MM

MM

MM

M

Structural Coefficient: ratio of the

structural weight to everything but

the payload. Want this small.

Lo

P

SP

P

MM

M

MM

M

Propellant Ratio: Ratio of propellant

to everything but the payload.

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PICTURES OF DEFINITIONS

Payload

Propellant

Structure

Rocket Initial

Propellant is Full

SPLo MMMM

Mo= + +

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PICTURES OF DEFINITIONS

Payload

Propellant

Structure

Rocket Final

Propellant is Empty

Mass at Burnout

SLb MMM

Mb= +

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MASS RATIO DEFINITION

Payload

Propellant

Structure

SL

o

b

o

MM

M

M

MR

R=

+ +

+

RocketInitial R

ocketFinal

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PAYLOAD RATIO DEFINITION

Payload

Propellant

Structure

=

+

SP

L

Lo

L

MM

M

MM

M

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STRUCTURAL COEFFICIENT DEFINITION

Payload

Propellant

Structure

=

+

Lo

Lb

SP

S

MM

MM

MM

M

10


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PROPELLANT FRACTION DEFINITION

Payload

Propellant

Structure

=

+

Lo

P

SP

P

MM

M

MM

M

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SUMMARY: SINGLE-STAGE ROCKETS

Payload

Propellant

Structure

Lo

P

SP

P

MM

M

MM

M

=

+

Lo

Lb

SP

S

MM

MM

MM

M

=

+

SP

L

Lo

L

MM

M

MM

M

=

+

SL

o

b

o

MM

M

M

MR

R =

+ +

+

1

R 1 12


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MULTISTAGE ROCKETS

Main idea is to discard empty tanks and extra structure as rocket travels, so that

this mass is not subjected to gravity losses

Large engines used for initial high thrust phase, may produce excessiveaccelerations when propellant is nearly consumed

Multistage rocket is a series of individual vehicles or stages, each with its own

structure, tanks and engines

Each stage accelerates payload before being detached

Two points:

1. Stages are ordered in number of firing

2. Analysis of multistage rockets is similar to that for single stage Payload for an particular stage is the mass of all subsequent stages

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MULTISTAGE ROCKET EXAMPLE

1

ML

3

2

Total Mass 1: Mo1=MP1+MS1+Mo2


Total Mass 3: Mo3=MP3+MS3+ML

Total Mass i: Moi=MPi+MSi+Mo(i+1)

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MULTISTAGE ROCKET EXAMPLE

1

ML

3

2


Payload for Stage 1: ML1=Mo2


Payload for Stage 2: ML2=Mo3

Total Mass 3: Mo3=MP3+MS3+MLPayload for Stage 3: ML3=ML

Total Mass i: Moi=MPi+MSi+Mo(i+1)Payload for Stage i: MLi=Mo(i+1)

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PAYLOAD RATIO: MULTISTAGE ROCKETS

1

ML

21

2

)1(

)1(

oo

o

iooi

io

Lo

L

iMM

M

MM

M

MM

M

The payload ratio for stage 1 is:

1

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ML

32

3

)1(

)1(

oo

o

iooi

io

Lo

Li

MM

M

MM

M

MM

M


2

2

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ML

Lo

L

iooi

io

Lo

L

i

MM

M

MM

M

MM

M

3)1(

)1(


3

3

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STRUCTURAL COEFFICIENT: MULTISTAGE ROCKETS

1

ML

The structural coefficient for stage 1 is:

1

21

1

)1( oo

S

iooi

Si

SP

S

i MM

M

MM

M

MM

M

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ML


2

32

2

)1( oo

S

iooi

Si

SP

S

i MM

M

MM

M

MM

M

2

20


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ML


3

Lo

S

iooi

Si

SP

S

i MM

M

MM

M

MM

M

3

3

)1(3

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SUMMARY: MULTISTAGE ROCKETS

)1(

)1(

iooi

io

i

MM

M

)1(

iooi

si

i

MM

M

)1(

)1(

iooi

iobi

i

MM

MM

bi

oi

iM

MR

ii

i

iR

1

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SOME EXAMPLES: SATURN V

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PROTON (SOVIET) First Launch: July 1965

Flight Rate: 13 per year

Capability: 44,100 lb to LEO; 12,100 lb to GTO; 4,850 lb to GEO

Originally intended as a ballistic missile but converted to a space launchvehicle during development

Two, three, and four-stage versions were developed

Used to launch satellites into GEO, interplanetary spacecraft, and mannedspace stations such as Salyut and Mir

Three or four-stage liquid-fueled vehicle

Stage 1 has six strap-on boosters with RD-253 engines burning N2O4fed from the core stage 1 tank with UDMH fuel carried in the strap-ontanks, generating a total of 1,986,000 lb of thrust

Stage 2 has four RD-0210 sustainer engines burning N2O4/UDMH fedfrom stage 2 tank, generating a total of 540,000 lb of thrust

Stage 3 has one RD-473 engine with four verniers burningN2O4/UDMH, generating a total thrust of 142,000 lb

Stage 4 has one RD-58 burning LO2/kerosene, generating a total

thrust of 19,100 lb

Length: 197 ft

Launch Weight: 1,550,000 lb

Diameter 22.6 ft

Liftoff Thrust: 1,986,000 lb

Payload Fairing: 24.6 ft x 12 ft

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