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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
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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 2: Mo2=MP2+MS2+Mo3
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
Total Mass 1: Mo1=MP1+MS1+Mo2
Payload for Stage 1: ML1=Mo2
Total Mass 2: Mo2=MP2+MS2+Mo3
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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PAYLOAD RATIO: MULTISTAGE ROCKETS
ML
32
3
)1(
)1(
oo
o
iooi
io
Lo
Li
MM
M
MM
M
MM
M
The payload ratio for stage 2 is:
2
2
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PAYLOAD RATIO: MULTISTAGE ROCKETS
ML
Lo
L
iooi
io
Lo
L
i
MM
M
MM
M
MM
M
3)1(
)1(
The payload ratio for stage 3 is:
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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STRUCTURAL COEFFICIENT: MULTISTAGE ROCKETS
ML
The structural coefficient for stage 2 is:
2
32
2
)1( oo
S
iooi
Si
SP
S
i MM
M
MM
M
MM
M
2
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STRUCTURAL COEFFICIENT: MULTISTAGE ROCKETS
ML
The structural coefficient for stage 3 is:
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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