A Tool for Preliminary Design of Rockets

Diogo Marques Gaspar ∗

July 2014

Abstract

The only way mankind can explore space is with the use of space launch vehicles, commonly knownas rockets, which can carry payloads from Earth into Space. The main goal in the rocket design isto reduce the gross lift-off weight (GLOW) and increase the payload ratio, giving them the capacityof having the maximum amount of payload, within its range. The trajectories of spacecrafts andlaunch vehicles are permanently being optimized, due to the demand of increased payload and reducedpropellant requirements. The main objective of this study is to find the best design of a rocket for aspecific orbit and payload mass, considering the best trajectory that can be used. A tool was developedallowing the study of different configurations, varying the number of stages and boosters in order toachieve the best rocket design. Beside the different configurations the tool allows the variation of designparameters, calculating for each configuration and parameters the different masses and dimensionsthat characterize a rocket, and then iteratively with the trajectory will achieve the minimum GLOWand obtain the maximum payload ratio. In order to validate the code comparisons with Vega andProton K launchers were made. At last an optimization of the Ariane 5 launcher is made by varyingits configuration and some design parameters.

Keywords: Multistage Rockets, Gravity Turn, Trajectory optimization, Rocket optimization,Maximize Payload Ratio

1. Introduction

The main objective of rocket design is to reduceGLOW and increase the payload ratio for a specificmission. In this work, a tool is developed for thepreliminary design of rockets, where the user canmake decisions about the configuration and definethe interval of variation of the design parameters.Then, it will obtain a new launcher optimized withrespect to the goals desired.

The preliminary design and optimization of rock-ets has been the focus of several research studies.The minimum GLOW and maximum payload ra-tio can be obtained by staging and trajectory op-timization [6, 7]. The collaborative optimizationis an alternative design architecture, whose charac-teristics are well suited for launch vehicle design [5].The Multidisciplinary Optimization (MDO) strate-gies are based in the exploration on the interactionof sub-systems representative models and on the ex-ploitation of their coupling, in order to find the op-timal conception parameters. Several works usingMDO have been developed in the preliminary de-sign and optimization of rockets [3, 9]. Recently,one work in multi-attribute evaluation provided aninteresting approach in the conceptual design ofrockets with a cost model [12].

The calculation of the most adequate trajectory

will increase the launchers efficiency. By minimizingdrag and gravity losses, a trade-off analysis needsto be performed [1, 7]. For the atmospheric flight,several works consider the use of gravity turn (GT)[1, 8, 10]. For the exo-atmospheric flight, there areseveral methods for trajectory optimization used inworks of the rocket design [2, 11].

There are two methods of numerical approachesin trajectory optimization, direct and indirectmethods. In a first overview the direct methodsconsist in discretizing the state and the control andthus reduce the problem to a nonlinear optimiza-tion with constraints. The indirect methods consiston solving numerically boundary value problem de-rived from the application of the Pontryagin Max-imum Principle and lead to the shooting methods[4]. The main advantages of using indirect methodsare their high solution accuracy and the guaranteeof satisfying the optimal conditions of the solution[11].

2. Rocket DesignIn order to launch the satellites in orbit it’s nec-essary to escape from the Earth atmosphere andgravity. To accomplish that, a large amount of∆V is necessary and with the current technology

∗. Under supervision of Paulo J. S. Gil

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only rockets are able to achieve that. Currently, aSSTO is still impossible and only multistage con-figurations can achieve space, that allow optimizethe structure and propulsion of each stage for dif-ferent conditions. Often, more than 90% of therocket is propellant, meaning that rockets need astrong structure for accommodate all the propel-lants and its powerful engines to burn them. Theirstructure is long and thin allowing the reduction ofDrag during the trajectory. However this will bringstructural problems when normal loads are appliedin the structure.

2.1. Delta-V calculations

The ∆V budget of a mission is represented as

∆VDesign = ∆Vorbit + ∆Vgravity + ∆Vdrag, (1)

where ∆Vorbit is the injection velocity required forthe desired orbit, ∆Vgravity and ∆Vdrag are respec-tively the total gravity and drag losses. This equa-tion allows us to make an estimate on the required∆V consumption and thus the required fuel neededto reach the trajectory and LEO. Other losses dueto the manoeuvring and static pressure differenceat the nozzle exit during flights are considerablysmaller compared to gravity and drag losses. This∆V budget must be larger or in exceptional casesequal to ∆V of the mission, otherwise the pay-load/spacecraft wouldn’t be successful reaching thedesired orbit.

The speed of a satellite in a circular orbit is de-fined by

Vorbit =

õ

R. (2)

where R is the radius of the orbit and µ the gravi-tational parameter of the planet.

The gravity loss is by definition

∆Vgravity =

∫g sin γdt, (3)

where g is the gravitational acceleration γ and theflight path angle. The drag loss is by definition

∆Vdrag =

∫D

Mdt, (4)

where D is the drag force and M the mass of rocketat each time.

A preliminary estimation of gravity and draglosses is necessary for the development of the tool.These heuristics are defined by:

∆Vgravity = 0.08 < Vorbit < 0.12 (5)

∆Vdrag = 0.008 < Vorbit < 0.012 (6)

2.2. Rocket equationsThe mass of the rocket is the sum of the payloadmass with the structural and propellant masses ofeach stage. It’s possible to define the following ra-tios using mentioned masses.

The payload of a specific stage is the mass ofeverything above that stage.

λN =M0N+1

M0N, (7)

where M0N is the total mass and M0N+1 is the massof everything above that stage.

Total Payload Ratio is equal to the product of allpayload ratios of all rocket sections and is equal to

λtot =

N∏i=1

λi. (8)

The Structural Ratio measures how much of thelaunch vehicle is structure, depends exclusively ofthe stage.

εN =MsN

MsN +MpN, (9)

where MsN and MsN are respectively the structuraland propellant mass of the N stage.

The Propellant Mass Ratio measures how muchof the vehicle is propellant.

ϕN =MpN

M0N= (1 − εN )(1 − λN ), (10)

where MpN and M0N are respectively the propel-lant mass and the total mass of the N stage.

The Mass ratio is a measure of the efficiency of arocket for any given efficiency a higher mass ratiotypically permits the vehicle to achieve higher ∆V .

ΛN =M0N

M0N −MpN=

1

1 − ϕN(11)

where the MpN and M0N are respectively the pro-pellant mass and the total mass of the N stage andϕ is the propellant mass ratio.

3. TrajectoryThe rocket flight can be divided in two main phases:the atmospheric flight and the exo-atmosphericflight. The frontier between them is not well de-fined, but can be linked to an altitude around 120kilometres, where atmosphere forces can be ne-glected.

3.1. Ascent PhaseA typical launcher starts its mission locked to theLaunchpad. The engines are not yet ignited or thethrust provided is still lower than the vehicle weight.The fist phase is a vertical lift off, required to gainvelocity before the gravity turn phase and avoid anycontact with the launchpad tower. The duration of

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the ascent phase varies from vehicle to vehicle andthe current tendencies is to decrease, because of thedevelopments made in space technology.

In the figure 1 is represented the launch sequencebetween lift-off and the final orbit, and the divisionof altitude in two parts, one where the atmospherecan’t be neglected and other where the atmosphereis neglected.

Figure 1: Launch Sequence.

3.2. Gravity TurnThe gravity turn phase, also named ”zero lift tra-jectory” is defined as a non-guided trajectory. Pri-marily, the launch vehicle crosses the atmosphere,only influenced by the gravity force. The angle ofattack must be zero, because even a small angle ofattack can lead to structural failure of the vehicle.Most launch vehicles have requirements of strength,regarding axial direction, that are fulfilled by com-promising the structural strength in the transversedirection. As a result, they cannot fly at any signif-icant angle of attack, without risks of catastrophicfailure of the whole structure. The trajectory dataincludes altitude, flight path angle variation, thrust,drag and total system mass. The equations of mo-tion, for an object in flight with gravity turn tra-jectory, are shown below:

X = V cosγ, (12)

H = V sinγ, (13)

mV = T −D − (mg − mV 2

Re+ h)sin(γ), (14)

γ = −1

v(g − V 2

Re+ h)cos(γ), (15)

where X, H, V, γ, T, D, m, g and Re are respectivelydownrange distance, altitude, velocity, flight pathangle, thrust, drag, mass, gravity and earth radius.

3.3. Free-flight phaseAs the vehicle emerges from the atmosphere a bet-ter trajectory than the gravity turn can be used.

It has a (present) position and velocity, and it isdesired to achieve a different position and velocityat thrust termination. The only two forces thatcan cause the vehicle to accomplish the desired po-sition and velocity changes are thrust and gravity.Continuous problems of trajectory optimization aretraditionally solved by direct or indirect methods.We want to minimize the final time i.e. the dura-tion of the free-flight phase with a TPBVP, less timemeans less propellant because we assumed constantthrust, then this will result in minimize the GLOW.

For minimum time problems, the cost functioncan be written as

J = tf (16)

the initial conditions of TPBVP are the final con-ditions of gravity turn.

t0 = t (17)

x0 = x (18)

y0 = y (19)

Vx0 = Vx (20)

Vy0 = Vy (21)

the final conditions, for the particular case of circu-lar orbit are

yf = h (22)

Vxf = V c (23)

Vyf = 0 (24)

Applying transversality condition and the Min-imum Principle, which states that the Hamilto-nian must be minimized. We have a λ1 = c1 andλ1f = c0, so c1 = 0.

We have a well defined TPBVP, with four stateequations, and four costate equations.

x = Vx (25)

y = Vy (26)

Vx =F

m0 − mt(

−λ3√λ23 + λ24

) (27)

Vy =F

m0 − mt(

−λ4√λ23 + λ24

) − g (28)

λ1 = 0 (29)

λ2 = 0 (30)

λ3 = −λ1 (31)

λ4 = −λ2 (32)

Thus there are eight differential equations.

x = f(t, x, λ) (33)

λ = g(t, x, λ) (34)

Now there are five initial conditions and five finalconditions.

The steps that shooting method used are:

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1. Guess λ10 , λ20 , λ30 , λ40 and tf

2. Integrate the x and λ forward to t = tf

3. Compute the final conditions:

Ψ1 = yf − hcirc +REarth (35)

Ψ2 = Vxf − Vcirc (36)

Ψ3 = Vyf (37)

Ψ4 = λ1f (38)

Ψ5 = Hf + 1 (39)

where λ and tf are changed iteratively

4. Until the convergence on Ψi(λ0,tf ) = 0 for i =1, .., 5

In the figure 2 is possible to observe, the initialconditions, and a few iterations of Shooting Methoduntil the convergence has achieved.

Figure 2: Shooting Method for TPBVP

3.4. Drag ModelThe drag coefficient is a function of the atmosphericconditions and the geometry of the rocket. An as-sumption was made, the drag force is independentof the length of the vehicle (assuming skin frictiondrag to be negligible).

D =1

2ρCDSrefV

2, (40)

where ρ is the density of the atmosphere, the Cd isthe drag coefficient, Sref is the cross-sectional areaof the body and V is the velocity of the vehicle. Asimple model for the Cd was adopted and its varia-tion only depends on the Mach number.

3.5. Atmospheric ModelIn this work, the U.S. Standard Atmosphere Con-vention of 1976 was used as a reference for altitudeslower than 86 kilometres, and the 1962 U.S. Stan-dard Atmosphere was followed for altitudes above86 kilometres, it models atmosphere up to 2000 kilo-metres.

The Knudsen number will define the bound-ary between the atmospheric and exo-Atmosphericflight, i.e. it will define the end of gravity turn andthe beginning of the free-flight phase of the flight.The Knudsen number is dimensionless defined as

Kn =λknL, (41)

where the λkn is the mean free path of free-streammolecules and L is a length characteristic, the valueassumed was the cone radius that it’s the same oflast stage radius.

3.6. Gravitational ModelThe external force caused by gravity is the mainforce that acts on the launch vehicle. Gravity is thenatural phenomenon by which physical bodies areattracted to each other with a force proportional totheir masses.

The local gravity is calculated with the followingequation

g =g0

1 + hRe

, (42)

where h is the current altitude, Re the radius ofEarth and g0 the surface gravity.

4. Rocket Knowledge DatabaseFor a better understanding of the differences be-tween launchers, and to compare converging pa-rameters between them, a database of well-knownlaunchers was developed. It gathers informa-tion about their different masses, dimensions andpropulsion, to understand if the concept of new de-signed rockets is ”real”, and if their values can becompared with similar launchers.

For each stage of each launcher a lot of informa-tion is available in its user’s manual but only areselected the principal characteristics that define arocket. In the table 1 is a list of the informationgathered.

Some information wasn’t available for all launch-ers, for example the nozzle area ratio is one char-acteristic that is very difficult to obtain, such asaerodynamic performance. The most important in-formation gathered is about the structural factor ofeach stage, that allows us to know how much massof each stage is structure and to give an appropri-ate range of values of Structural factor of each stagefor mass model developed in this work. The aver-age value obtained was used in mass model as firstvalue for the iterations of Structural factor.

5. Tool for Rocket DesignThe computer simulation program was designed toperform a rocket optimization. This means that itoptimizes all the masses and the dimensions of thevehicle as well as the last part of the trajectory. Todevelop this work a Matlab code was developed, in

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Characteristics Units

Class Small/Medium/HeavyYear -

Engine -Length m

Diameter mTake-off mass kg

Propellant mass kgStructural mass kgStructural factor kg

Propellants -Burning duration s

Thrust (vac) kNIsp (vac) s

Table 1: Characteristics gathered for each Stage ofeach Launcher.

order to design the rocket and simulate trajectoriesinto orbit. In this section is described how the toolachieves the best configuration for one combinationof parameters and configuration. This algorithm isrepeated for all the combination of parameters andconfigurations.

First, the user inputs his objectives: the orbitaltitude and the payload mass. After this he has tomake basic decisions about the configuration: howmany stages there will be, and if boosters will beused in the first stage and how many there will be.It is assumed that the launch system starts fromground, with velocity equal to zero and launch fromvertical.

The first ∆V evaluation assumes an estimate forgravity and drag losses. A good preliminary esti-mate of ∆V losses increase the velocity of program,i.e. the tool will need less iterations to find the bestsolution.

Before the user enters the mass model, he had al-ready selected the configuration, some parameters,shown in table 2, and its variation for each stage,

Characteristics Units

Number of Stages -Number of Boosters -Diameter Booster m

Diameter mNozzle Area Ratio m

Propellant Selection -Thrust kN

Thrust Booster kN∆V division -

Table 2: Configuration and Parameters introducedin the tool.

The propulsion parameter selected was the thrust

instead of burn time. Due to the creation ofdatabase, the user can have an insightful idea of thevalues of thrust for each stage and different payloadmasses. The burn time is obtained after definingthe thrust. The dimension parameter selected wasthe diameter instead of length, because defining thediameter will give an idea to the user of how muchdrag will be generated.

Afterwards, in Mass Model the program will com-pare the masses obtained with heuristic equationsi.e. Mass Estimation Relationships from the Akinmodel, and will iterate the structural ratio, untilit converges. During the mass model the programcalculate in parallel the dimensions of the vehicle.The user already defined the diameter of each stage.Then the volume is calculated and will iterate insideMass Model. After this all the masses and dimen-sions of the launcher are specified, the launcher isready to trajectory simulations.

Then the trajectory has to be run, according tothe three distinct phases of flight, firstly Verticalflight, secondly Gravity turn and then the free-flightphase, until the objective is fulfilled. After know-ing the best trajectory new values of ∆V drag andgravity and how much Propellant mass of the laststage remains or need to be added, the program willtransform those propellant in a ∆V , and then williterate with those new values, regarding the Massmodel. The program will iterate until it finds asolution, i.e. when the ∆V losses used in the de-velopment of the launcher, are equal to the lossesobtained in the trajectory and the payload reachesits objective with the right amount of propellant.The Launch Vehicle is optimized considering theinitial objectives.

In the figure 3 it’s represented the algorithm ofthe tool only for one combination of parameters andconfiguration. For each combination of these thetool will run the algorithm described earlier andpresented next. In the end, it will select the combi-nation of parameters and configuration with mini-mum GLOW and maximum payload ratio.

Figure 3: Algorithm for the Design of the rocket

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5.1. Mass ModelThe mass model of this tool was developed tocalculate all the masses, dimensions and parame-ters of the rocket needed to perform the trajectoryfor each combination of parameters previously se-lected. When the mass model starts, the tool hasalready defined the parameters and the configura-tion. Those parameters will be used later for theHeuristics MERs. For given stage velocities, themass of the structure and propellants can be esti-mated using the ideal velocity equations, known asthe Tsiolkovsky’s Equation. The Lift-off weight isthe sum of payload with all the calculated masses.An initial estimate of the structural factor is usedbased on heuristics selected from database. Thestructural and propellant mass of each stage are cal-culated with the following two equations:

msn =εn(kn − 1)

1 − εnknmpl, (43)

mpn=

(kn − 1)(1 − εn)

1 − knεnmpl, (44)

kn = e∆V

gIspn , (45)

where εn and kn are respectively the structural fac-tor and the mass ratio for each stage.

Starting from the last stage, and knowing the Ini-tial structural factor, the mass ratio kn and payloadmass, it is possible to estimate the structural andPropellant Mass of that stage. Then a heuristiccomparison is performed using the values generated(propellant and structural mass, volume of thatstage) and the parameters previous defined (Thrustand Nozzle Area Ratio). The Heuristic Mass gener-ated will provide a new Structural factor, then theprogram iterates using this new value. The programwill stop when the iteration of the Structural factorconverges with a maximum tolerance of 0.1%. Thiswill define the last stage. Now, the program hasthe conditions to estimate the masses of next stage,where the Mass of the Payload will be the sum ofmasses of everything above. The same process isapplied until all masses are defined.

When the user chooses to use boosters during thefirst stage, he will select two ∆V , one for the ”ze-roth” stage considering the boosters and first stageburning simultaneously and one only for the firststage, after the boosters were discarded. A specificheuristic comparison for boosters is performed tovalidate the configuration.

In order to get appropriate dimensions for theLauncher, all the main dimensions are calculated.For each iteration of Structural factor in Massmodel, a new propellant mass is generated. Know-ing the density of Propellant, we’re in conditions toknow the volume and calculate the involving struc-ture for each stage.

The figure 4 shows the iterative process of howthe mass model calculates the propellant and struc-tural masses for each stage (rocket section mass),where the ∆V is the ∆V assigned for the specificstage.

Figure 4: Mass model loop

5.2. Trajectory

After the lift-off the launcher follows a vertical tra-jectory until a designed altitude where Gravity Turnstarts. Although the altitude was fixed for run-ning the simulations presented in this thesis, it canbe changed and be a parameter. After the end ofthe gravity turn i.e. the end of considerable atmo-sphere, the free-flight phase starts and the positionand velocity of the vehicle is known, along with itsdestination (velocity and position). Considering thestages and respective thrust, the program will calcu-late how much burn time is necessary to achieve thedesired orbit. Three different results can appear:arriving at a circular orbit at burnout, burn timenecessary is less than was previous calculated by themass model; burn time is not enough to achieve thedesired orbit. These cases still depend on the effec-tive gravity and drag losses determined during thetrajectory. If their values aren’t correct the massmodel will iterate until the both values converge.The propellant mass missing or in excess in the laststage is converted to ∆V by Tsiolkovsky’s equation.This will generate a new ∆V total. Then, the pro-gram will iterate until the value of ∆V convergeswith a tolerance of 0.01%.

When the program finishes, a complete launcherhas been developed, and its data can be saved to trydifferent orbits regarding that the maximum pay-load is the previously used for the initial design.

The figure 5 represents the trajectory part of theprogram described earlier.

6. Results

Before using the tool for its purpose, its validationwas necessary. It was performed in two separatedphases, starting with the validation of the trajec-tory and ending with the validation of the mass

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Figure 5: Trajectory

model. The main objective of the simulation withVega launcher was to place 1500 kilograms of pay-load into a circular orbit, at an altitude of 700 kilo-metres, and the main objective of the Proton K wasto place 19360 kilograms of payload in a circular or-bit at an altitude of 200 kilometres. Posteriorly, aset of simulations was made for the preliminary de-sign and optimization of a new launcher, using asinitial parameters the values of Ariane 5. The op-timization was based on the variation of some ofthe original parameters and the result was a newlauncher.

6.1. Trajectory and mass model validation

To validate the trajectory, firstly, a simulation wasdone with each launcher, using the parameters pre-sented in user’s manual of the Vega and Proton K,to confirm if it managed to use the proposed tra-jectory.

The final times obtained at the end of the trajec-tories were lower than the sum of all burning andcoast times, and in the last stage 34% and 38%respectively of the propellant mass wasn’t burned,meaning that the trajectory used was consideredvalidated for this two launchers. After the success-ful validation of the trajectory for each launcher.It’s time to perform the validation of the massmodel and of the entire tool.

The parameters introduced in the tool for the twosimulations are presented in the table 2, all the pa-rameters were equal to the ones presented in therespective user’s manual.

The deviation obtained in the different massesare shown in the table 3. The most significant dif-ferences were obtained in structural mass, speciallyin the second stage where the difference was almost30%. The structural and propellant mass of laststage are the ones that have more differences, spe-cially the structural mass because last stage usuallyhave ”more structure”, i.e. more avionics, wiringand a complicated thrust attitude controller forthe re-ignitions, so the MERs used in Mass modeldoesn’t have the same accuracy for upper stages as

- mp ms

1 Stage -5.83 15.62 Stage -4.1 29.83 Stage -15.3 8.14 Stage 40.7 57.6

Table 3: % Deviation of the structural and propel-lant masses - Vega.

they have for lower stages.

The program overestimates all the structuralmasses but on the other hand the propellant massescalculated were inferior with the exception of thelast stage once again, that can be explained by thefact that the last stage of the original launcher havemultiple burns and long coast times and that analy-sis isn’t provided in the trajectory and for the differ-ence in the others stages are explained for the nonexistence of the propellant margin. The biggest dif-ference in the propellant masses excluding last stagewas in third stage. The GLOW was a difference of4.8%, that is inferior to 10% and for a tool of pre-liminary design the differences are relatively small.The mass model was considered validated for theVega case.

The deviation obtained in the different massesfor the Proton K are shown in the table 4. All

- mp ms

1 Stage -5.7 19.52 Stage -8.5 5.23 Stage -26.7 26.2

Table 4: % Deviation of the structural and propel-lant masses - Proton K.

the propellant masses calculated were smaller thanthe original Launcher, the biggest difference beingin the last stage and the smallest the first stage.The differences are explained for the non existenceof the propellant margin. The structural mass cal-culated were always larger than the real Launcher:the biggest difference was in the last stage and thesmallest was in the second stage. The GLOW hada difference of 6.2%. For a tool of preliminary de-sign the differences are relatively small and the massmodel was considered validated for this case.

The following figures show different variables ofthe trajectory of Vega for the last simulation. Inthe figure 6 is possible to observe the evolution ofaltitude with time and when the GT ends and thenTPBVP starts. In the figure 7 is possible to observethe Velocity function of time and observe how thevelocity varies in GT and TPBVP and the changethat happens after the end of Gravity turn. The

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Figure 6: Altitude vs Time - Vega

Figure 7: Velocity vs Time - Vega

following figures show different variables of the tra-jectory of the last simulation of Proton K. In figure8 are represented the Altitude of the trajectory vstime and the altitude vs downrange. It’s possibleto observe that the payload reaches the desired or-bit and time. The figure 9 represents velocity of

Figure 8: Altitude vs Time - Proton K

Launcher vs time, where it’s also possible to ob-serve the end of GT and the start of TPBVP andhow the launcher rapidly ascents to the desired or-bit.

Figure 9: Velocity vs Time - Proton K

6.2. Preliminary design and optimizationIn the last case, a set of simulations was done withthe objectives of Ariane 5 for LEO, an altitude of200 km and a payload mass of 19.3 tons, but inthis set of simulations some parameters were variedwith the goal of obtaining an optimal configuration.This means the minimum GLOW that can satisfythe desired objectives.

Three groups of different simulations were run.In the first group, the diameter and Thrust of eachstage were fixed but on the other hand the numberof boosters and its characteristics, Thrust, diame-ter and burn time were varied in a certain range. Inparallel with the first group simulation, the secondgroup simulation was run with all the parametersfixed except the diameter of core stages. The thirdgroup simulation used the best result of the firstgroup simulation for the boosters (its thrust anddimensions) and the best diameter obtained in thesecond group simulation with the variation of ∆Vdivision and respective thrust for each stage. Inthe end, an optimal solution was found and its con-figuration was the one with minimum GLOW thatachieves the objectives early proposed. A total of170 simulations were run.

The objective was to optimize an existinglauncher and not to validate the code, the objectivewasn’t to see if the results match but how differentwill be the new optimized launcher compared withthe original Ariane 5, how lighter will be, and howthe volume will vary. The values achieved for thebest configuration are presented in the table 5.

Parameters Final Value

Diameter [m] 3.9Thrust [kN] 1700 and 58.5

∆V division [%] [25% 49% 26%]Number of boosters 2Thrust booster [kN] 8000

Diameter booster [m] 2.625Burn time Booster [%] 25%

Table 5: Optimal Configuration Achieved.

The deviation obtained in the different masses forthe last simulation are shown in the table 6.

- mp ms

Booster -4.5 -3.51 Stage -27.2 -22 Stage -85 -5

Table 6: % Deviation of the structural and propel-lant masses.

In this optimization all the propellant masses cal-

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culated were inferior than the ones from the originalLauncher. Comparing this differences, with thoseobtained for the validations of the model it’s pos-sible to state that optimization of the parametersdecrease the propellant necessary to achieve the or-bit. The structural masses calculated were slightlyinferior than the ones from the real Launcher, thesedifference can be explained by the reduction of themass of the propellant and its direct influence in de-creasing the mass of the propellant tanks. The goalof this optimization was the reduction of GLOW,the final simulation had a difference of 11%, whichallows to reduce 84 tons the weight of the Launcherat lift-off and compared with the other simulations,where the objective was to validate the tool, it pos-sible to say that the original Ariane 5 was optimizedand consequently its GLOW reduced.

The dimensions obtained were inferior comparedwith Ariane 5 original because the propellantmasses of each stage were inferior due to optimiza-tion. The differences are higher in upper stage be-cause that was the stage that had more propellantreduction. The overall length was 3.1 meters higherthan original Ariane 5, increasing 5.5% the totallength of the Launcher.

The following figures show the evolution of dif-ferent variables of the final trajectory simulation.In the figure 10 is represented the evolution of thealtitude of the trajectory vs time where its possibleto observe the end of GT and the start of TPBVP.The figure 11 represents the velocity of launcher vs

Figure 10: Altitude vs Time - Ariane 5

time, where it’s possible to observe the end of GTand the start of free-flight phase when the velocityinitially raises slowly and then accelerates until thedesired orbit.

7. ConclusionsThe main goal of this research was to develop atool to assist the preliminary design of rockets. Thetool was modelled using a MATLAB code, whichapplies a simplified optimization, and covers twomajor fields: the mass model and the trajectory.

The database created during this study allowedthe understanding of some important variables,such as length, diameter, thrust, burning time, spe-cific masses and others, providing their range of val-

Figure 11: Velocity vs Time - Ariane 5

ues and ensuring the viability of the final result.Some of these values were used in the mass model,as initial structural factors, for the iteration of themass model.

The mass model created, showed that the use ofTsiolkovsky’s equation with the MERs heuristics,provides realistic results for the structural masses ofthe launcher. This model also calculates the dimen-sions of the launcher, with the goal of adding morerealism to the final design. When the mass modelis concluded, all the masses and dimensions neededfor the trajectory of the launcher have been de-fined. The trajectory was divided into three distinctphases. The vertical ascent, the gravity turn and atlast the free flight phase. A simplified drag modelwas used, where the Cd only varies with Mach num-ber. The same happens for gravity model where thegravity only varies with altitude.

The Model validation is important to determinethe accuracy of the results obtained. In this study,the model validation was done regarding two exam-ples, Vega and Proton K, targeting diverse objec-tives (Orbit and Payload Mass) and characterisedby different configurations. Both validations weredone with the values presented in the user’s man-ual. First, an isolated simulation of the trajectorywas done, i.e. using the mass and dimension valuesfrom the user’s manual. Both trajectory simula-tions were successful.

After the validation of the trajectory, it was pos-sible to validate the mass model. All the parame-ters selected were the same of the respective User’smanual. The results showed that almost all thepropellant masses were inferior, when comparedto real launchers, explained by the fact that themass model only adds propellant margin for the laststage, instead of adding it for all stages. The struc-tural masses were almost all superior, when com-pared with real launchers, meaning the mass modelis over estimating the structural weight for the se-lected launchers. The GLOW of each launcher wasinferior, and that is explained by the reasons pre-sented for the deviation regarding the propellantmasses.

After the validation of the developed tool, this

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was used for the preliminary design of a rocket. Theoriginal parameters used were based on launcherAriane 5, and some were modified, to optimize therocket i.e. increase the payload ratio by reduc-ing the GLOW. This allowed the understanding ofwhich parameters influence more the difficult taskof designing a rocket. The objectives of the mis-sion were the same as those defined for Ariane 5and in the end of the optimization all masses anddimensions were compared with the user’s manual.All the masses were smaller; the propellant massesdifferences can be explained by the non-existenceof propellant margin and also by the reduction ofthe diameter, which allowed the free-flight phase tostart earlier; the structural masses were also inferiorbecause of the reduction of all propellants masses,the lighter tanks and the reduction of the diam-eter, which decreased the weight of the structureand allowed the free-flight phase of the trajectoryto start earlier, optimizing the fuel consumption. Inconclusion, the reduction of GLOW was significant,meaning that the optimization was successful.

The ∆V division and thrust are the parametersthat most influence the GLOW. The definition ofdiameter effects drag and trajectory, because of thedefinition of the Knudsen number and must be care-fully selected, due to the fact that in this study astructural analysis wasn’t developed.

7.1. Future work

It would be interesting to include the chamber andexit pressure and the Nozzle areas in the Massmodel, currently its effect is neglected on the nozzlemass. Also in mass model development of a heuris-tic by a student, creating a database for differentcomponents of launchers (Tanks, Engines Nozzles,Interstage adapters, Avionics, Wiring and others).

Implement constrains in the Trajectory model,such as Max. Dynamic Pressure, Heat Flux, Bend-ing Load and Axial Acceleration. Improve the Dragmodel, a more realistic Drag model can be achievedwith analytical equations for each Nose Cone Con-figuration.

Study air-launched rockets (Pegasus), transfer or-bits to GEO and interplanetary trajectories thatwill include long coast phases and improve the tra-jectory. Develop a Cost Model, in order to obtaina realistic insight of Launch Vehicle Design.

References

[1] F. Alonso Zotes and M. Santos Penas. Multi-criteria genetic optimisation of the manoeuvresof a two-stage launcher. Information Sciences,180(6):896–910, Mar. 2010.

[2] D. M. Azimov. Active Rocket Trajectory Arcs:A Review. Automation and Remote Control,66(11):1715–1732, Nov. 2005.

[3] M. Balesdent, N. Berend, P. Depince, andA. Chriette. A survey of multidisciplinary de-sign optimization methods in launch vehicledesign. Structural and Multidisciplinary Op-timization, 45(5):619–642, Sept. 2011.

[4] J. T. Betts. Survey of Numerical Methods forTrajectory Optimization. Journal of Guidance,Control, and Dynamics, 21(2):193–207, Mar.1998.

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[7] R. Jamilnia and A. Naghash. Simultane-ous optimization of staging and trajectory oflaunch vehicles using two different approaches.Aerospace Science and Technology, 23(1):85–92, 2012.

[8] P. Lu and B. Pan. Highly Constrained OptimalLaunch Ascent Guidance. Journal of Guid-ance, Control, and Dynamics, 33(2):404–414,Mar. 2010.

[9] L. F. Rowell, R. D. Braun, and J. R.Oldst. Multidisciplinary Conceptual DesignOptimization of Space Transportation Sys-tems. Journal of Aircraft, 36(1), 1999.

[10] E. D. Sotto and P. Teofilato. Semi-AnalyticalFormulas for Launcher Performance Evalua-tion. Journal of Guidance, Control, and Dy-namics, 25(3), 2002.

[11] E. Trelat. Optimal Control and Applicationsto Aerospace: Some Results and Challenges.Journal of Optimization Theory and Applica-tions, 154(3):713–758, Apr. 2012.

[12] R. Ullah, D.-q. Zhou, P. Zhou, M. Hussain, andM. A. Sohail. An approach for space launchvehicle conceptual design and multi-attributeevaluation. Aerospace Science and Technology,25(1):65–74, 2013.

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