Parallel Staging • Multistaging Rocket...

Rocket PerformancePrinciples of Space Systems Design

Rocket Performance

• The rocket equation• Mass ratio and performance• Structural and payload mass fractions• Multistaging• Optimal Delta-V distribution between stages• Trade-off ratios• Parallel Staging• Modular Staging

© 2001 David L. Akin - All rights reservedhttp://spacecraft.ssl.umd.edu


Derivation of the Rocket Equation

• Momentum at time t:

• Momentum at time t+∆t:

• Some algebraic manipulation gives:

• Take to limits and integrate:

M mv=

M m m v v m v Ve= − + + −( )( ) ( )∆ ∆ ∆

m v mVe∆ ∆= −

dmm

dvVeV

V

m

m

initial

final

initial

final

= −

∫∫


The Rocket Equation

• Alternate forms

• Basic definitions/concepts– Mass ratio

– Nondimensional velocity change“Velocity ratio”

∆V Vm

mV re

final

initiale= −

= −ln ln( )r

m

mefinal

initial

V

Ve≡ =−

∆

∆VVe

rm

mor

mm

final

initial

initial

final

≡ ℜ =


Rocket Equation (First Look)

1

0.001

0.01

0.1

1

0 1 2 3 4 5 6 7 8

Velocity Ratio, (∆V/Ve)

Mas

s Ra

tio,

(Mfi

nal/

Min

itia

l)

Typical Rangefor Launch to

Low Earth Orbit


Sources and Categories of Vehicle Mass

• Payload• Propellants• Inert Mass

– Structure– Propulsion– Avionics– Mechanisms– Thermal– Etc.


Basic Vehicle Parameters

• Basic mass summary

• Inert mass fraction

• Payload fraction

• Parametric mass ratio

m initial mass

m payload mass

m propellant mass

m inert mass

L

p

i

0 =

=

=

=δ = = + +mm

mm m m

i i

L p i0

r = +λ δ

λ = =+ +

mm

mm m m

L L

L p i0

m m m mL p i0 = + +


Rocket Equation (including Inert Mass)

0.001

0.01

0.1

1

0 1 2 3 4 5 6 7 8

0.000.050.100.150.20


Payl

oad

Frac

tion

, (M

payl

oad/

Min

itia

l) Typical Rangefor Launch toLow Earth Orbit

Inert MassFraction δ


The Rocket Equation for Multiple Stages

• Assume two stages

• Assume Ve,1=Ve,2=Ve

∆V Vm

mV re

final

initiale1 1

1

11 1= −

= −,

,

,,ln ln( )

∆V Vm

mV re

final

initiale2 2

2

22 2= −

= −,

,

,,ln ln( )

∆ ∆V V V r V r V r re e e1 2 1 2 1 2+ = − − = −ln( ) ln( ) ln( )


Continued Look at Multistaging

• Converting to masses

• Keep in mind that mfinal,1~minitial,2

• r0 has no physical significance!

∆ ∆V V V r r Vm

m

m

me efinal

initial

final

initial1 2 1 2

1

1

2

2

+ = − = −

ln( ) ln ,

,

,

,

∆ ∆V V V r r Vm

mV re e

final

initiale1 2 1 2

2

10+ = − = −

= −ln( ) ln ln( ),

,


Multistage Vehicle Parameters

• Inert mass fraction

• Payload fraction

• Payload mass/inert mass ratio

δ δ λ00 1

1

1

= =

∑ ∏∑=

−

=

m

mi j

j

j

j

n stages,

ll

λδ

0

0

λ λ00 1

= ==

∏mm

Li

i

n stages


Effect of Staging

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.5 1 1.5 2 2.5

1 stage2 stage3 stage4 stage

Payl

oad

Frac

tion

, (M

payl

oad/

Min

itia

l)


Inert Mass Fraction δ=0.2


Effect of ∆∆∆∆V Distribution

-0.200

0.000

0.200

0.400

0.600

0.800

1.000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

lambda0delta0lamd0/del0

1st Stage Delta-V (m/sec)

1st Stage: LOX/LH2 2nd Stage: LOX/LH2


Lagrange Multipliers

• Given an objective function

subject to constraint function

• Create a new objective function

• Solve simultaneous equations

y f x= ( )

z g x= ( )

y f x g x z= + −[ ]( ) ( )λ

∂∂

yx

= 0∂∂λ

y= 0


Optimum ∆∆∆∆V Distribution Between Stages

• Maximize payload fraction (2 stage case)

subject to constraint function

• Create a new objective function

➥Very messy for partial derivatives!

λ λ λ δ δ0 1 2 1 1 2 2= = − −( )( )r r

∆ ∆ ∆V V Vtotal = +1 2

λ δ δ0 1 2 1 21

1

2

2= − − + + −[ ]− −

( )( )e e K V V VV

V

V

VTotal

e e

∆ ∆

∆ ∆ ∆


Optimum ∆∆∆∆V Distribution (continued)

• Use substitute objective function

• Create a new constrained objective function

• Take partials and set equal to zero

ln( ) ln( ) ln( )

ln( ) ln( )

λ δ δ0 1 1 2 2

1 1 2 2

= − + −

+ + +[ ]r r

K V V r V rTotal e e∆

max( ) max ln( )λ λ0 0⇔ [ ]

∂ λ∂

ln( )01

0r

=∂ λ

∂ln( )0

2

0r

=∂ λ

∂ln( )0 0

K=


Optimum ∆∆∆∆V Special Cases

• “Generic” partial of objective function

• Case 1: δ1=δ2 Ve1=Ve2

• Case 2: δ1≠δ2 Ve1=Ve2

• More complex cases have to be done numerically

∂ λ∂ δ

ln( )0 1 0r r

KVri i iei

i

=−

+ =

r r V V1 2 1 2= ⇒ =∆ ∆

r r11

2

2δ δ=


Trade-off Ratios

∂∂

∂

mm

Vm m

Vm m

L

i k V

e jj f jj

k

ef

N

Total,

,, ,

,, ,

∆ =

=

=

=

− −

−

∑

∑001

01

1 1

1 1l

l ll

∂∂

∂

mm

Vm

Vm m

L

p k V

e jjj

k

ef

N

Total,

,,

,, ,

∆ =

=

=

=

−

−

∑

∑001

01

1

1 1l

l ll


Trade-off Ratios (continued)

∂∂

∂

mV

m

m

Vm m

L

e k V

k

f kj

k

ef

N

Total,

,

,

,, ,

ln

∆ =

=

=

=

−

∑

∑0

0

1

01

1 1l

l ll


Parallel Staging

• Multiple dissimilarengines burningsimultaneously

• Frequently a result ofupgrades to operationalsystems

• General case requires“brute force” numericalperformance analysis


Parallel-Staging Rocket Equation

• Momentum at time t:

• Momentum at time t+∆t:(subscript “b”=boosters; “c”=core vehicle)

M mv=

M m m m v v

m v V m v Vb c

b e b c e c

= − − +

+ − + −

( )( )

( ) ( ), ,

∆ ∆ ∆

∆ ∆


Modular Staging

• Use identical modules toform multiple stages

• Have to cluster modules onlower stages to make up fornonideal ∆V distributions

• Advantageous fromproduction and developmentcost standpoints

Parallel Staging • Multistaging Rocket...

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