A Generalized Staging Optimization Program For Space Launch Vehicles Ezgi Civek-Co�kun, MSc
Prof. Dr. Kemal Ozg6ren
System Design Department Roketsan Missile Industries Inc. Ankara, Turkey
[email protected]
Mechanical Engineering Department Middle East Technical University Ankara, Turkey
[email protected]
Abstract-This
paper
addresses
the
staging
Another advantage of staging is that launch vehicle configuration can be optimized for the requirements of a particular mission by adjusting the amount of propellant and engine thrust, and using different types of engines, propellants and structural materials for various stages. Stages can also be designed for best performance considering their operating conditions.
optimization
problem for multistage rockets which carry payloads from the Earth's surface into the Earth orbits. In the early design phases, requirements are not so strict, there are many unknowns and problem arises as to what is the optimum staging to achieve the given
mission.
providing
Therefore,
a quick
insight
designers on the
need
vehicle
simplified
tools
performance
with
minimum basic vehicle data. For this purpose, a Matlab® based computer
program
has
been
written
to
determine
The optimum stage mass distribution between stages for a multistage launch vehicle and the propellant and structural masses for each individual stage can be determined by staging optimization for a given set of technology options. By optimal staging, launch vehicle can achieve its specified mission with minimum gross lift-off mass which can be considered as a key driver of both performance and cost.
staging
parameters (number of stages, mass distribution between stages, and the propellant and structural masses for each individual stage) which minimize the gross lift-off mass of the launch vehicle for a specific mission. In
this
study,
staging
optimization
problem
has
been
formulated based on Delta-V equations and solved by method of
Till today, many efforts have been made to optimize launch vehicle staging for minimum gross lift-off mass [1-9]. In the early papers, problem has been solved with many simplifying assumptions. Malina and Summerfield [1] were the first to optimize staging; however, the solution was limited to stages having equal propellant exhaust velocities and structural ratios. Vertregt [2] extended the solution to the case when all stages have different exhaust velocities. Goldsmith [3] offered a solution for two stage rockets when the structural masses are proportional to propellant masses. Weisbord [4], Subotowicz [5], Hall and Zambelli [6] have all presented general solutions for minimum gross lift-off mass with non-homogenous stages and the solution holds for arbitrary number of stages.
Lagrange Multipliers. The problem has been stated in a general form to handle launch vehicles having arbitrary number of stages and with various configurations involving serial, parallel and clustered
stages;
and
with
different
structural
ratios
and
propellant exhaust velocities in each stage. Staging optimization program developed in this study has been verified for different missions using available data of existing launch vehicles. Thus, a quick and effective tool to find optimal vehicle configurations in the conceptual design phase of a generic multistage launch vehicle has been achieved. Keywords-multistage rockets; space launch vehicles, staging optimization; minimizing gross lift-off mass; conceptual design
I.
All of these authors used the method of Lagrange Multipliers, which was proven many times, and they ignore the effects of gravity, drag and steering in order to obtain the derivatives analytically. Later on, Srivastava [7], Tawakley [8] and Adkins [9] have examined the isolated effects of gravity, steering and drag, respectively.
INTRODUCTION
Exploration and utilization of space for the benefit of mankind require space launch vehicles, which carry payloads from the Earth's surface into the Earth orbit and beyond. Space launch vehicles are multistage rockets composed of two or more stages, each of which contains its own propellant and structure. The idea behind staging is to improve performance by reducing the vehicle's mass on the way to orbit. Once the propellant of a stage is consumed, the empty stage which is no longer useful and only adds weight to the vehicle is discarded and the next stage is ignited. This stage then accelerates the rest of the vehicle much faster. Thus, less propellant is required to achieve the desired orbit.
978-1-4673-6396-9/13/$3l.00 ©2013 IEEE
This paper presents a staging optimization method well suited to be used in the conceptual design phase of a generic multistage launch vehicle having different structural ratios and propellant exhaust velocities in each stage. This method allows considering the gravitational, aerodynamic and propulsive loss factors with proper margins under the simplest possible conditions without determining the flight trajectory.
857
II. A.
MATHEMATICAL MODEL
Tsiolkovsky's Rocket Equation
The famous fundamental rocket equation derived from Newton's second law of motion governs the relationship between the propellant consumption and the velocities that can be attained by the rocket vehicle and is given by . L'1 Vvehicle = C In A
mO,2
(1)
where:
L'1 Vvehicle : maximum change of speed : exhaust velocity : mass ratio
C
A
mO,1
Mass ratio (A) is just the ratio of the initial mass to the final mass A
_
!!!.::... mj
_
mo
m,+mp +mpl
mo-mp
m,+mpl
Fig, I, Mass definitions for serial staging,
(2)
In line with the single stage rocket, mass of the kth stage and the initial and fmal masses before and after the operation of the kth stage are
where: mo : initial mass ml : final mass ms : structural mass mp : propellant mass mp/ : payload mass
(8) (9) m I,k =ms,k + mO,k+1
Exhaust velocity (C) is defined by
1) Serial staging: Several stages stacked on top of each other and one stage burns alone until its propellant is exhausted. It is then jettisoned and the next stage is ignited (Fig. 1).
(3) where:
lsp : specific impulse
go : gravity at seal level, go
B.
=
(10)
2 9.81 mls
Mass ratio of the kth stage (Ak) is
Multistage Rocket Parameters
(11)
Ideal velocity increment for an N-stage rocket is the sum of the velocity increments of the individual stages. N
L'1v"ehlcle = L Ck k=1
·In A k
Structural ratio of the kth stage (ck) is
(4)
E:k =
In the analysis of N-stage rocket, payload of any particular stage (k) can be considered as the mass of subsequent stages (k+1, ... , N) as illustrated in Fig. l. m pi,k =mO,k+l
m pl,N
=
(5)
ms,k
(12)
(13)
A, C and A are not independent. If we express ms and mp/ in terms of mp from (12) and (13), respectively and insert those in (11), then we get the mass ratio for the kth stage (Ak) as
(6)
Total payload ratio (A,) is defmed as m mp A = --.!!!... = __1 r
__
Payload ratio of the kth stage (Ak) is
And the payload of the last (Nth) stage is the actual payload of the launch vehicle. =m pI
m_...:.. ,,k
__
1+ A
k Ak = --E:k + Ak
(7)
(14)
2) Parallel staging: Several stages are mounted in parallel and burn in simultaneously. Some launch vehicles use both serial and parallel staging (Fig. 2).
where: mo,! : gross lift-off mass (GLOM) of the launch vehicle
858
/\
mO,N -�
C
mp,IO
Orbital Velocity Equation
Velocity of the satellite at any point in an orbit can be calculated using the following vis viva equation derived from orbital energy conservation equation,
m l= mc - mp,IO
mp,11
mO,1
Vorbit.
mO= mb+ mp,IO
mb
mo,o
Fig, 2, Mass definitions for parallel staging,
The analysis of a parallel staged rocket is quite similar to the one presented above, the main difference is the stage numbering and the need for calculation of average exhaust velocity,
0
A0
m o, o ms,b + mO,l
=
ms,b mb + mp,iO mO.l
=
mb + mp,iO
(15)
go
!sp,b ,mp,b + !sp,c ,mp,1O
mp,b + mp,1O
(17)
Approximated values obtained from real data samples can be used for rough estimations of gravity and drag losses, In this study, IIVg and L1V d variations versus T/W presented by [11] have been used (Fig, 3), which is valid for vertical take-off vehicles, For horizontal take-off, thrust losses will be higher, but gravity loss is much lower, For horizontal take-off highest value of T/W can be used to approximate this behavior,
(18)
(19)
1
==
m ,-,-s,c,-_
__
m
c
-m
1 p, O
3
Gravity losses (llVg) and drag losses (llV d) are the most signifIcant loss terms, and they are primarily dependent upon the initial thrust-to-weight ratio (T/W), TIW needs to be greater than unity for the vehicle to leave the launch pad, and typical lift-off T/W values are in the range 1,3 to 2 [10],
Similarly, the equivalent ratios for the fIrst stage are
&'
=
where: L1Vg : velocity loss due to gravity L1V d : velocity loss due to aerodynamic drag L1Vp : propulsive losses due to steering and pressure change L1Vgain: velocity gain due to Earth's rotation or initial altitude or initial velocity
(16)
Average exhaust velocity of the zeroth stage is =
a
Velocity losses/gains can also be included separately in the calculation of the total velocity increment needed to get into orbit (L1Vmission) '
where: : subscript for core fust stage c : subscript for boosters b mp,IO : propellant mass burned in parallel with the boosters in zeroth operation
Co
r
Velocity change of an N-stage launch vehicle (L1Vvelticle) can be calculated from rocket equation (4) knowing the fact that this is the ideal velocity change, and gravitational and aerodynamic forces, flight maneuvers and all other velocity losses/gains are neglected, Tewari [12] proposed to add a total of 1,5 km/s margin for velocity losses/gains for a launch to the low earth orbit and 2 km/s for a launch to the geosynchronous orbit
According to Fig, 2, mass, structural and payload ratios of the zeroth stage equivalent to a serial rocket are given by =
(22)
�GM(�-�)
D, Delta- V Calculations
When the parallel boosters and the core fIrst stage are burning simultaneously, they are taken together and called the zeroth stage, while the propellant remaining in the core's fIrst stage after discarding the parallel boosters is called the fust stage of the rocket
A0
=
where: VOrbll : orbital velocity at radial distance r : Earth's gravitational parameter, GM 398600 Ian /S2 : radial distance from the Earth's center to the satellite r a : semimajor axis
GM
[;
(21)
m c -m p.1O
Launch vehicles also experience propulsive losses due to the maneuvering and static pressure difference at the nozzle exit during their flight These losses are smaller compared to IIVg and L1V d especially for vertical take-off vehicles and it is diffIcult to estimate the magnitude without having the flight trajectory,
(20)
859
1000 900
"'"'-
800
�'" '"'" .2
700 600
OJ
500
>
400
0
(43)
(46)
Structural mass of each stage is found from (12) (47) And propellant mass of each stage is
(39)
861
(48)
So far, staging has been optimized based on equations of serial staging. But, many launch vehicles have two to six number of boosters strapped on to the first stage to provide more thrust at lift-off. For launch vehicles with parallel staging, an equivalent launch vehicle with serial staging can be defmed. After staging has been optimized based on serial staging, optimal staging data can be converted to a parallel configuration using the defmitions given in Section II-B-2.
300,000 ,------
250,000 +----------F---
�
-;;: 200,000 1') E � 150,000 +-------/-'------::;£=---
,; � 100,000 �
50,000
A Matlab® script has been written to determine the optimal staging of an N-stage rocket for a set of given technology options. Thus, a quick and effective tool to find optimal vehicle configurations in the conceptual design phase of a generic multistage launch vehicle has been achieved.
=
km/s
10
km/'
......V-llkm/s
1000
2000
3000
4000
pavload mass (kg)
5000
6000
=
REFERENCES [I] [2]
F.J. Malina and M. Summerfield. 'The problem of escape from the earth by rocket." Journal of the Aeronautical Sciences 14 (1947): 471. M. Vertregt. "A method of calculating the mass ratio of step rockets." Journal of the British interplanetary Society 15 (1956): 95.
300,000
mp'= 500 kg
� 250,000
[3]
+----\----------
___ v
150,000
3 number of
stages
6
-+-v
= =
9 km/s
10km/,
...... V - 11
km/s
7
1. Weisbord. "A generalized optimization procedure for n-staged missiles." Jet Propulsion 28 (1958): 164-167.
[5]
M. Subotowicz. 'The optimization of the n-step rocket with different construction parameters and propellant specific impulses in each stage." Jet Propulsion 28 (1958): 460-463.
[6]
H.H. Hall and E.D. Zambelli. "On the optimization of multistage rockets." Jet Propulsion 28 (1958): 463-465.
[7]
[9]
Fig. 6 and Fig. 7 illustrates the variation of GLOM with Ll Vmission and payload mass (mp/), respectively. Figures show that GLOM increases exponentially with increasing Ll Vmission and increases linearly with the increasing mpl. 60,000 ..,------
50,000 +----------,1''----;;: 40,000 1') E � 30,000 +------�=------:----
,; �
20,000 +----11=--------:;£---- ::; e
C.N. Adkins. "Optimization of multistage rockets including drag." Journal of Spacecraft (1970): 751-755.
[10] H.D. Curtis. Orbital mechanics for engineering students. Burlington, MA: Elsevier Butterworth-Heinemann, 2005.
1st.
[11] J.P. Loftus and C. Teixeira. "Chapter 18: Launch systems." Space mission analysis and design. Ed. W.J. Larson and J.R. Wertz. 3rd. EI Segundo, CA & New York, NY: Microcosm Press & Springer, 1999. 719-744.
[13] D.n. Burghes. "Optimum staging of multistage rockets." international Journal of Mathematical Education in Science and Technology 5.1 (1974): 3-10.
___ 100 kg -+-SOOkS
...... lOOOkg
1-----..=---11
V.B. Tawakley. "On the calculation of optimum mass distribution of a multi-stage rocket vehicle." Defence Science Journal 17 (1967): 53-66.
[12] A. Tewari. Atmospheric and space flight dynamics: Modeling and simulation with Matlab and Simulink. Boston: Birkhaeuser, 2007.
�
10
T.N. Srivastava. "Optimum staging with varying thrust attitude angle." Defence Science Journal 16 (1966): 153-164.
[8]
mission d.ltaV (km/s)
Goldsmith. "On the optimization of two stage rockets." Jet
[4]
Fig. 5. Variation of GLOM with number of stages.
8
M.
Propulsion 27 (1957): 415-416.
� +----+--[100.000 +--....-\---------
10,000
9
It is necessary to point out that the method proposed in this study for estimating gravitational, aerodynamic and propulsive losses during flight is a poor approximation, but useful for preliminary evaluation during the conceptual design phase. It is always necessary to carry out trajectory simulations by solving equations of motion for detailed performance analysis in order to check the validity of the assumptions.
350,000 ,-----
aD
=
Fig. 7. Variation of GLOM with payload mass.
Fig. 5 illustrates the variation of GLOM with the number of stages. The decrease in GLOM between N 2 and N 3 is quite appreciable, but for higher values of N the decrease becomes very small.
200,000
=
+-__."'-:.A....-""='-----. a
Staging optimization program developed in this study has been verified for different missions using available data of existing launch vehicles. In order to illustrate the method of application, variation of GLOM with N, Ll Vmission and mp/ have been shown graphically in Fig. 5 to Fig. 7.
� ;::
-+-V
DISCUSSION AND CONCLUSION
V.
�
:; -�It-+-------7----:;£'--
___ V
12
Fig. 6. Variation of GLOM with mission Delta-V.
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