Rocket Performance
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The rocket equation
Mass ratio and performance
Structural and payload mass fractions
Regression analysis
Multistaging
Optimal ΔV distribution between stages
Trade-off ratios
Parallel staging
Modular staging
© 2009 David L. Akin - All rights reserved
http://spacecraft.ssl.umd.edu
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Derivation of the Rocket Equation
Ve
• Momentum at time t:
Δm
M = mv
m-Δm
v
m
v+Δv
v-Ve
• Momentum at time t+Δt:
M = (m − ∆m)(V + ∆v) + ∆m(v − Ve )
• Some algebraic manipulation gives:
m∆v = −∆mVe
• Take to limits and integrate:
!
mf inal
dm
=−
m
minitial
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2
!
Vf inal
Vinitial
dv
Ve
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
The Rocket Equation
• Alternate forms
mf inal
− ∆V
r≡
= e Ve
minitial
!
"
mf inal
∆v = −Ve ln
= −Ve ln r
minitial
• Basic definitions/concepts
– Mass ratio
mf inal
minitial
r≡
or " ≡
minitial
mf inal
– Nondimensional velocity change
∆V
“Velocity ratio”
Ve
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Rocket Equation (First Look)
Mass Ratio, (Mfinal/Minitial)
1
Typical Range
for Launch to
Low Earth Orbit
0.1
0.01
0.001
0
2
4
6
8
Velocity Ratio, (ΔV/Ve)
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4
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Sources and Categories of Vehicle Mass
Payload
Propellants
Inert Mass
Structure
Propulsion
Structure
Avionics
Propulsion
Power
Avionics
Mechanisms
Power
Thermal
Mechanisms
Etc.Thermal
Etc.
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Basic Vehicle Parameters
• Basic mass summary
mo = mpl + mpr + min
• Inert mass fraction
mo =initial mass
mpl =payload mass
mpr =propellant mass
min =inert mass
min
min
δ≡
=
mo
mpl + mpr + min
• Payload fraction
mpl
mpl
λ≡
=
mo
mpl + mpr + min
• Parametric mass ratio
r =λ+δ
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Rocket Equation (including Inert Mass)
Payload Fraction, (Mpayload/Minitial)
1
Typical Range
for Launch to
Low Earth Orbit
Inert Mass
Fraction δ
0
0.1
0.05
0.1
0.15
0.01
0.2
0.001
0
2
4
6
8
Velocity Ratio, (ΔV/Ve)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Limiting Performance Based on Inert
Asymptotic Velocity Ratio, (ΔV/Ve)
5
4.5
Typical Feasible
Design Range for
Inert Mass Fraction
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Inert Mass Fraction, (Minert/Minitial)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Regression Analysis of Existing Vehicles
Veh/Stage
Delta 6925 Stage 2
Delta 7925 Stage 2
Titan II Stage 2
Titan III Stage 2
Titan IV Stage 2
Proton Stage 3
Titan II Stage 1
Titan III Stage 1
Titan IV Stage 1
Proton Stage 2
Proton Stage 1
average
standard deviation
prop mass gross mass
(lbs)
(lbs)
13,367
15,394
13,367
15,394
59,000
65,000
77,200
83,600
77,200
87,000
110,000
123,000
260,000
269,000
294,000
310,000
340,000
359,000
330,000
365,000
904,000 1,004,000
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T y p e P r o p e l l a n t s Isp vac isp sl s i g m a
eps
delta
(sec) (sec)
Storable
N2O4-A50
319.4
0.152 0.132 0.070
Storable
N2O4-A50
319.4
0.152 0.132 0.065
Storable
N2O4-A50
316.0
0.102 0.092 0.087
Storable
N2O4-A50
316.0
0.083 0.077 0.055
Storable
N2O4-A50
316.0
0.127 0.113 0.078
Storable
N2O4-A50
315.0
0.118 0.106 0.078
Storable
N2O4-A50
296.0
0.035 0.033 0.027
Storable
N2O4-A50
302.0
0.054 0.052 0.038
Storable
N2O4-A50
302.0
0.056 0.053 0.039
Storable
N2O4-A50
316.0
0.106 0.096 0.066
Storable
N2O4-A50
316.0 285.0 0.111 0.100 0.065
312.2 285.0 0.100 0.089 0.061
8.1
0.039 0.033 0.019
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Inert Mass Fractions for Existing LVs
0.18
Inert Mass Fraction
0.16
0.14
0.12
LOX/LH2
0.10
LOX/RP-1
0.08
Solid
0.06
Storable
0.04
0.02
0.00
0
1
10
100
1,000
10,000
Gross Mass (MT)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Regression Analysis
• Given a set of N data points (xi,yi)
• Linear curve fit: y=Ax+B
– find A and B to minimize sum squared error
N
!
error =
(Axi + B − yi )2
i=1
– Analytical solutions exist, or use Solver in Excel
• Power law fit: y=BxA
error =
N
!
2
[A log(xi ) + B − log(yi )]
i=1
• Polynomial, exponential, many other fits possible
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Solution of Least-Squares Linear
N
!
∂(error)
=2
(Axi + B − yi )xi = 0
∂A
i=1
A
N
!
x2i + B
i=1
N
!
xi −
i=1
N
!
N
!
∂(error)
=2
(Axi + B − yi ) = 0
∂B
i=1
xi yi = 0
i=1
A
N
!
xi + N B −
N
!
i=1
A=
N
!
!
!
xi yi − xi yi
! 2
! 2
N xi − ( xi )
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B=
!
yi = 0
i=1
!
!
yi x2i −
! 2
N xi −
!
xi xi yi
! 2
( xi )
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Inert Mass Fraction
Regression Analysis - Storables
0.10
0.08
0.06
0.04
2
0.02
R = 0.0541
0.00
1
10
100
1,000
Gross Mass (MT)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Regression Values for Design Parameters
Vacuum Ve
(m/sec)
Inert Mass
Fraction
Max ΔV
(m/sec)
LOX/LH2
4273
0.075
11,070
LOX/RP-1
3136
0.063
8664
Storables
3058
0.061
8575
Solids
2773
0.087
6783
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
The Rocket Equation for Multiple Stages
• Assume two stages
∆V1 = −Ve1 ln
!
∆V2 = −Ve2 ln
!
mf inal1
minitial1
"
= −Ve1 ln(r1 )
mf inal2
minitial2
"
= −Ve2 ln(r2 )
• Assume Ve1=Ve2=Ve
∆V1 + ∆V2 = −Ve ln(r1 ) − Ve ln(r2 ) = −Ve ln(r1 r2 )
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Continued Look at Multistaging
• There’s a historical tendency to define r0=r1r2
∆V1 + ∆V2 = −Ve ln (r1 r2 ) = −Ve ln (r0 )
• But it’s important to remember that it’s really
∆V1 + ∆V2 = −Ve ln (r1 r2 ) = −Ve ln
!
mf inal1 mf inal2
minitial1 minitial2
• And that r0 has no physical significance, since
mf inal1 != minitial2
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"
mf inal2
⇒ r0 =
!
minitial1
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Multistage Inert Mass Fraction
• Total inert mass fraction
min,1 + min,2 + min,3
min,1
min,2
min,3
δ0 =
=
+
+
m0
m0
m0
m0
min,1
min,2 m0,2
min,3 m0,3 m0,2
δ0 =
+
+
m0
m0,2 m0
m0,3 m0,2 m0
• Convert to dimensionless parameters
δ 0 = δ1 + δ 2 λ1 + δ 3 λ 2 λ 1
• General form of the equation
n
δ0 =
" j−1 $
stages
#
!
δj
λ!
j=1
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!=1
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Multistage Payload Fraction
• Total payload fraction (3 stage example)
mpl
mpl m0,3 m0,2
λ0 =
=
m0
m0,3 m0,2 m0
• Convert to dimensionless parameters
λ0 = λ3 λ2 λ1
• Generic form of the equation
n
λ0 =
stages
!
λj
j=1
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Effect of Staging
Inert Mass Fraction δ=0.15
0.25
Payload Fraction
0.20
1 stage
0.15
2 stage
3 stage
0.10
4 stage
0.05
0.00
0
1
2
3
4
5
Velocity Ratio (ΔV/Ve)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Effect of ΔV Distribution
1st Stage: Solids
Normalized Mass (kg/kg of payload)
70
2nd Stage: LOX/LH2
60
50
40
Total mass
Stage 2 mass
30
Stage 1 mass
20
10
0
2000
3000
4000
5000
6000
7000
8000
9000
10000
Stage 2 ΔV (m/sec)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
ΔV Distribution and Design Parameters
1st Stage: Solids
0.45
2nd Stage: LOX/LH2
0.4
0.35
0.3
delta_0
0.25
lambda_0
0.2
lambda/delta
0.15
0.1
0.05
0
2000
3000
4000
5000
6000
7000
8000
9000
10000
Stage 2 ΔV (m/sec)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Lagrange Multipliers
• Given an objective function
y = f (x)
subject to constraint function
z = g(x)
• Create a new objective function
z = f (x) + λ[g(x) − z]
• Solve simultaneous equations
∂y
=0
∂x
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∂y
=0
∂λ
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Optimum ΔV Distribution Between Stages
• Maximize payload fraction (2 stage case)
λ0 = λ1 λ2 = (r1 − δ1 )(r2 − δ2 )
subject to constraint function
∆Vtotal = ∆V1 + ∆V2
• Create a new objective function
λo =
!
e
−∆V1
Ve,1
" ! −∆V
"
2
Ve,2
e
− δ1
− δ2 + K [∆V1 + ∆V2 − ∆Vtotal ]
➥Very messy for partial derivatives!
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Optimum ΔV Distribution (continued)
• Use substitute objective function
max (λo ) ⇐⇒ max [ln (λo )]
• Create a new constrained objective function
ln (λo ) = ln (r1 − δ1 ) + ln (r2 − δ2 ) + K [∆V1 + ∆V2 − ∆Vtotal ]
• Take partials and set equal to zero
∂ [ln (λo )]
=0
∂r1
∂ [ln (λo )]
=0
∂r2
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∂ [ln (λo )]
=0
∂K
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Optimum ΔV Special Cases
• “Generic” partial of objective function
∂ [ln (λo )]
1
Ve,i
=
+K
=0
∂ri
ri − δ i
ri
• Case 1: δ1=δ2 Ve,1=Ve,2
∆Vtotal
r1 = r2 =⇒ ∆V1 = ∆V2 =
2
• Same principle holds for n stages
r1 = r2 = · · · = rn =⇒
∆Vtotal
∆V1 = ∆V2 = · · · = ∆Vn =
n
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Sensitivity to Inert Mass
ΔV for multistaged rocket
∆Vtot =
where
n stages
!
∆Vk =
n
!
Ve,k ln
k=1
k=1
mo,k = mpl + mpr,k + min.k +
n
!
"
mo,k
mf,k
#
(mpr,j + min,j )
j=k+1
mf,k = mpl + min.k +
n
!
(mpr,j + min,j )
j=k+1
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Finding Payload Sensitivity to Inert Mass
• Given the equation linking mass to ΔV, take
∂(∆Vtot )
∂(∆Vtot )
dmpl +
dmin,j = 0
∂mpl
∂min,j
and solve to find
!
∂mpl !!
∂min,k !
"k
∂(∆Vtot )=0
#
$
− j=1 Ve,j m1o,j − m1f,j
$
#
= "
N
1
1
−
V
"=1 e," mo,!
mf,!
• This equation shows the “trade-off ratio” Δpayload resulting from a change in inert mass for
stage k (for a vehicle with N total stages)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Trade-off Ratio Example: Gemini-Titan II
Stage 1
Stage 2
Initial Mass
(kg)
150,500
32,630
Final Mass (kg)
39,370
6099
Ve (m/sec)
2900
3097
dmpl/dmin,k
-0.1164
-1
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Payload Sensitivity to Propellant Mass
• In a similar manner, solve to find
!
∂mpl !!
∂mpr,k !
∂(∆Vtot )=0
"k
#
$
1
V
j=1 e,j mo,j
$
#
1
1
−
V
"=1 e," mo,!
mf,!
−
="
N
• This equation gives the change in payload mass as
a function of additional propellant mass (all other
parameters held constant)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Trade-off Ratio Example: Gemini-Titan II
Stage 1
Stage 2
Initial Mass
(kg)
150,500
32,630
Final Mass (kg)
39,370
6099
Ve (m/sec)
2900
3097
dmpl/dmin,k
-0.1164
-1
dmpl/dmpr,k
0.04124
0.2443
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Payload Sensitivity to Exhaust Velocity
• Use the same technique to find the change in
payload resulting from additional exhaust velocity
for stage k
!
∂mpl !!
∂Ve,k !
∂(∆Vtot )=0
="
N
"k
j=1
"=1 Ve,"
ln
#
#
mo,j
mf,j
1
mo,!
−
$
1
mf,!
$
• This trade-off ratio (unlike the ones for inert and
propellant masses) has units - kg/(m/sec)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Trade-off Ratio Example: Gemini-Titan II
Stage 1
Stage 2
Initial Mass
(kg)
150,500
32,630
Final Mass (kg)
39,370
6099
Ve (m/sec)
2900
3097
dmpl/dmin,k
-0.1164
-1
dmpl/dmpr,k
0.04124
0.2443
dmpl/dVe,k
2.870
6.459
(kg/m/sec)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Parallel Staging
• Multiple dissimilar engines
burning simultaneously
• Frequently a result of
upgrades to operational
systems
• General case requires
“brute force” numerical
performance analysis
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Parallel-Staging Rocket Equation
• Momentum at time t:
M = mv
• Momentum at time t+Δt:
(subscript “b”=boosters; “c”=core vehicle)
M = (m − ∆mb − ∆mc )(v + ∆v)
+∆mb (v − Ve,b ) + ∆mc (v − Ve,c )
• Assume thrust (and mass flow rates) constant
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Parallel-Staging Rocket Equation
• Rocket equation during booster burn
∆V = −V̄e ln
!
mf inal
minitial
"
= −V¯e ln
!
min,b +min,c +χmpr,c +m0,2
min,b +mpr,b +min,c +mpr,c +m0,2
where χ= fraction of core propellant remaining
after booster burnout, and where
V̄e =
Ve,b ṁb +Ve,c ṁc
ṁb +ṁc
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=
"
Ve,b mpr,b +Ve,c (1−χ)mpr,c
mpr,b +(1−χ)mpr,c
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Analyzing Parallel-Staging Performance
Parallel stages break down into pseudo-serial stages:
• Stage “0” (boosters and core)
∆V0 = −V̄e ln
!
min,b +min,c +χmpr,c +m0,2
min,b +mpr,b +min,c +mpr,c +m0,2
• Stage “1” (core alone)
∆V1 = −Ve,c ln
!
min,c +m0,2
min,c +χmpr,c +m0,2
• Subsequent stages are as before
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"
"
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Parallel Staging Example: Space Shuttle
• 2 x solid rocket boosters (data below for single SRB)
–
–
–
–
Gross mass 589,670 kg
Empty mass 86,183 kg
Isp 269 sec
Burn time 124 sec
• External tank (space shuttle main engines)
–
–
–
–
Gross mass 750,975 kg
Empty mass 29,930 kg
Isp 455 sec
Burn time 480 sec
• “Payload” (orbiter + P/L) 125,000 kg
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Shuttle Parallel Staging Example
Ve,b = gIsp,e
m
m
= (9.8)(269) = 2636
Ve,c = 4459
sec
sec
480 − 124
= 0.7417
χ=
480
m
2636(1, 007, 000) + 4459(721, 000)(1 − .7417)
= 2921
V̄e =
1, 007, 000 + 721, 000(1 − .7417)
sec
m
862, 000
= 3702
∆V0 = −2921 ln
3, 062, 000
sec
m
154, 900
= 6659
∆V1 = −4459 ln
689, 700
sec
m
∆Vtot = 10, 360
sec
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Modular Staging
• Use identical modules to form
multiple stages
• Have to cluster modules on
lower stages to make up for
nonideal ΔV distributions
• Advantageous from production
and development cost
standpoints
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Module Analysis
• All modules have the same inert mass and
propellant mass
• Because δ varies with payload mass, not all
modules have the same δ!
• Introduce two new parameters
min
min
ε≡
=
min + mpr
mmod
min
σ≡
mpr
• Conversions
δ
ε=
1−λ
δ
σ=
1−δ−λ
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Rocket Equation for Modular Boosters
• Assuming n modules in stage 1,
nε +
n(min ) + mo2
r1 =
=
n(min + mpr ) + mo2
n+
mo2
mmod
mo2
mmod
• If all 3 stages use same modules, nj for stage j,
where
n1 ε + n2 + n3 + ρpl
r1 =
n1 + n2 + n3 + ρpl
mpl
ρpl ≡
; mmod = min + mpr
mmod
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Example: Conestoga 1620 (EER)
•
•
•
•
•
•
Small launch vehicle (1 flight, 1 failure)
Payload 900 kg
Module gross mass 11,400 kg
Module empty mass 1,400 kg
Exhaust velocity 2754 m/sec
Staging pattern
–
–
–
–
1st stage - 4 modules
2nd stage - 2 modules
3rd stage - 1 module
4th stage - Star 48V (gross mass 2200 kg,
empty mass 140 kg, Ve 2842 m/sec)
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Conestoga 1620 Performance
• 4th stage ∆V
mf 4
900 + 140
m
∆V4 = −Ve4 ln
= −2842 ln
= 3104
mo4
900 + 2200
sec
• Treat like three-stage modular vehicle; Mpl=3100 kg
min
1400
!=
=
= 0.1228
mmod
11400
mpl
3100
ρpl =
=
= 0.2719
mmod
11400
n1 = 4; n2 = 2; n3 = 1
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Constellation 1620 Performance (cont.)
n1 ! + n2 + n3 + ρpl
4 × 0.1228 + 2 + 1 + 0.2719
r1 =
=
= 0.5175
n1 + n2 + n3 + ρpl
4 + 2 + 1 + 0.2719
n2 ! + n3 + ρpl
2 × 0.1228 + 1 + 0.2719
r2 =
=
= 0.4638
n2 + n3 + ρpl
2 + 1 + 0.2719
n3 ! + ρpl
1 × 0.1228 + 0.2719
r3 =
=
= 0.3103
n3 + ρpl
1 + 0.2719
m
m
V1 = 1814
; V2 = 2116
sec
sec
m
m
V3 = 3223
; V4 = 3104
sec
sec
m
Vtotal = 10, 257
sec
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Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
Discussion about Modular Vehicles
• Modularity has several advantages
– Saves money (smaller modules cost less to develop)
– Saves money (larger production run = lower cost/
module)
– Allows resizing launch vehicles to match payloads
• Trick is to optimize number of stages, number of
modules/stage to minimize total number of
modules
• Generally close to optimum by doubling number
of modules at each lower stage
• Have to worry about packing factors, complexity
UNIVERSITY OF
MARYLAND
45
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design
OTRAG - 1977-1983
UNIVERSITY OF
MARYLAND
46
Rocket Performance
ENAE 483/788D - Principles of Space Systems Design