MAE 4262: ROCKETS AND MISSION ANALYSIS

Single and Multi-Stage Rockets

September 9, 2014

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

1

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

RR

Rtu

g

Ruh

gtR

Rtutuh

gtt

t

RutV

bee

bbebeb

be Velocity during

Powered flight

Height at burnout

Maximum altitude

2

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 of structure and payload:

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

SPLo MMMM

SLb MMM

3

DEFINITIONS

Symbol Ratio Description

R SL

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.

4

PRELIMINARY DEFINITIONS

• Using previous definitions, we can write mass ratio as:

• Also note that propellant ratio and structural coefficient are related by:

1

R

1

5

PICTURES OF DEFINITIONS

Payload

Propellant

Structure

Rocket InitialPropellant is Full

SPLo MMMM

Mo = + +

6

PICTURES OF DEFINITIONS

Payload

Propellant

Structure

Rocket FinalPropellant is EmptyMass at Burnout

SLb MMM

Mb = +

7

MASS RATIO DEFINITION

Payload

Propellant

Structure

SL

o

b

o

MM

M

M

MR

R=+ +

+

Roc

ket I

niti

alR

ocket Final

8

PAYLOAD RATIO DEFINITION

Payload

Propellant

Structure

=

+

SP

L

Lo

L

MM

M

MM

M

9

STRUCTURAL COEFFICIENT DEFINITION

Payload

Propellant

Structure

=

+

Lo

Lb

SP

S

MM

MM

MM

M

10

PROPELLANT FRACTION DEFINITION

Payload

Propellant

Structure

=

+

Lo

P

SP

P

MM

M

MM

M

11

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

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 excessive accelerations 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

13

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)

14

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+ML

Payload for Stage 3: ML3=ML

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

Payload for Stage i: MLi=Mo(i+1) 15

PAYLOAD RATIO: MULTISTAGE ROCKETS

1

ML

21

2

)1(

)1(

oo

o

iooi

io

Lo

Li MM

M

MM

M

MM

M

The payload ratio for stage 1 is:

1

16

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

17

PAYLOAD RATIO: MULTISTAGE ROCKETS

ML

Lo

L

iooi

io

Lo

Li MM

M

MM

M

MM

M

3)1(

)1(

The payload ratio for stage 3 is:

3

3

18

STRUCTURAL COEFFICIENT: MULTISTAGE ROCKETS

1

ML

The structural coefficient for stage 1 is:

1

21

1

)1( oo

S

iooi

Si

SP

Si MM

M

MM

M

MM

M

19

STRUCTURAL COEFFICIENT: MULTISTAGE ROCKETS

ML

The structural coefficient for stage 2 is:

2

32

2

)1( oo

S

iooi

Si

SP

Si MM

M

MM

M

MM

M

2

20

STRUCTURAL COEFFICIENT: MULTISTAGE ROCKETS

ML

The structural coefficient for stage 3 is:

3

Lo

S

iooi

Si

SP

Si MM

M

MM

M

MM

M

3

3

)1(

3

21

SUMMARY: MULTISTAGE ROCKETS

)1(

)1(

iooi

ioi MM

M

)1(

iooi

sii MM

M

)1(

)1(

iooi

iobii MM

MM

bi

oii M

MR

ii

iiR

1

22

SOME EXAMPLES: SATURN V

23

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 launch vehicle during development

• Two, three, and four-stage versions were developed • Used to launch satellites into GEO, interplanetary spacecraft, and manned

space 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 N2O4

fed from the core stage 1 tank with UDMH fuel carried in the strap-on tanks, generating a total of 1,986,000 lb of thrust

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

– Stage 3 has one RD-473 engine with four verniers burning N2O4/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 24