A Model and Sounding Rocket Simulation Tool with Mathematica® Aerospace Engineering

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A model and sounding rocket simulation tool with Mathematica ®

Tiago Miguel Oliveira Pinto

Thesis to obtain the Master of Science Degree in Aerospace Engineering

Supervisor: Prof. Doutor Paulo Jorge Soares Gil

Examination Committee

Chairperson: Prof. Doutor Filipe Szolnoky Ramos Pinto Cunha

Supervisor: Prof. Doutor Paulo Jorge Soares Gil

Member of the Committee: Prof. Doutor Jo˜aoManuel Gon¸calves de Sousa Oliveira

November 2015 ii Acknowledgments

I would like to thank my supervisor, Professor Paulo Gil, for providing me the opportunity to develop this work, for all the guidance and, especially, for sharing his knowledge throughout this journey.

To my fellow friends with whom I spent these years and stood beside me during the toughest and best moments.

To my devoted father, mother and sister who helped me, in several ways, to reach my goals and gave all of their heart.

iii iv Resumo

Durante o lan¸camento de foguetõesde modelismo, a trajectóriaéconsideravelmente influenciada pelo vento jáque estes sãofoguetõesde controlo passivo. Este trabalho tem como objectivo o desenvolvi- mento de um programa em Mathematica® que simula a trajectóriadestes foguetõese de foguetes-sonda sob a influênciade um perfil de vento. E´ realizado um estudo da camada limite atmosféricae consideram- se trêsperfis que levam em conta as condi¸cõesatmosféricase o tipo de superf´ıcieno local do lan¸camento. Mostrando a flexibilidade do programa desenvolvido, érealizada uma simula¸cãoMonte Carlo para se inferir a dispersãodo local de aterragem devida àsincertezas nos parâmetrosdo vento. Com as poten- cialidades do Mathematica®, a ferramenta desenvolvida permite que seja definido um qualquer número de estágios,com boosters externos ou internos no primeiro estágio,e que os modelos necessáriospara o cálculoda trajectóriasejam facilmente modificados ou acrescentados novos modelos na base de dados. Foram efectuadas duas simula¸cõescom foguetõesde potênciadistinta, ambos com dois estágiose sob as mesmas condi¸cões.Tambémse determinou uma trajectóriaem que a direçcãodo perfil de vento varia com a altitude e compararam-se trajectóriassimuladas em estabilidades atmosféricasdiferentes sob a mesma intensidade de vento no lan¸camento. Os resultados obtidos mostram grande dependênciadas trajectóriasno vento, especialmente quando a estabilidade da atmosfera varia. Considerando as incertezas definidas para a velocidade e direçcãodo vento, bem como para o tipo de terreno, estas mostram uma forte influênciana dispersãodos locais de aterragem.

Palavras-chave: Simula¸cãode Trajectória,Modelismo de Foguetões,Camada Limite At- mosférica,Perfil de Vento, Monte Carlo.

v vi Abstract

During the launch of model rockets, the trajectory is considerably influenced by the wind as these are passive-guided rockets. This work aims to develop a program in Mathematica® that simulates the trajectory from model to sounding rockets under the influence of a wind profile. A study of the Atmospheric Boundary Layer (ABL) is carried out and are considered three wind profiles that take into account the atmospheric conditions and the terrain type in the launch site. Presenting the developed tool’s flexibility, a Monte Carlo simulation is performed in order to deduce the dispersion of the landing sites due to the wind parameters’ uncertainties. Owing to the Mathematica®’s potentialities, the developed tool allows to define any number of stages, with external or internal boosters in the first stage, and to easily modify the required models to compute the trajectory or to implement new ones in the database. Two simulations using two-stage rockets provided with a distinct amount of power were predicted under the same conditions. Also, a trajectory in which the wind direction changes with altitude was determined and comparisons were performed between trajectories computed with different atmospheric stabilities under the same wind speed at the launch. The results show a great dependence of the trajectories on the wind profile, specially when the atmospheric stability changes. Considering the defined uncertainties from the wind speed and direction, as well from the surface type, they present a strong influence on the dispersion of the landing sites.

Keywords: Trajectory Simulation, Model Rocketry, Atmospheric Boundary Layer, Wind Pro- ﬁle, Monte Carlo.

vii viii Contents

Acknowledgments ...... iii Resumo ...... v Abstract ...... vii List of Tables ...... xiii List of Figures ...... xv List of Acronyms ...... xvii List of Common Symbols ...... xix

1 Introduction 1 1.1 Motivation ...... 1 1.2 Historical background ...... 1 1.3 Model and high power rocketry ...... 3 1.3.1 Associations and current events ...... 3 1.4 Rocket trajectory simulators ...... 4 1.4.1 OpenRocket ...... 4 1.4.2 RockSim ...... 5 1.5 Work goals and strategies ...... 5

2 Model rocketry concepts 7 2.1 Flight proﬁle ...... 7 2.2 Motors ...... 8 2.2.1 Coding system ...... 9 2.3 Stability ...... 10 2.4 Multi-staging and clustering ...... 11

3 Wind and trajectory 13 3.1 Wind and atmospheric boundary layer ...... 13 3.1.1 ABL proﬁles ...... 14 3.1.2 Atmospheric stability ...... 17 3.2 Kinematics and reference frames ...... 18 3.3 Rocket dynamics ...... 20 3.3.1 Dynamic stability ...... 21

ix 4 Rocket design 25 4.1 Mass ...... 26 4.2 Center of mass ...... 27 4.3 Moments of inertia ...... 30

4.4 Center of pressure and CNα ...... 32 4.5 Drag coeﬃcient ...... 34 4.5.1 Skin friction drag ...... 36 4.5.2 Compressibility eﬀects ...... 37 4.5.3 Recovery device ...... 38

5 Rocket trajectory simulator 39 5.1 Simulator description ...... 39 5.2 Rocket assembly ...... 41 5.3 Atmospheric model ...... 43 5.4 Wind model ...... 45 5.4.1 Wind gusts ...... 47 5.5 Motors data ...... 47 5.6 Drag coeﬃcient model ...... 48 5.7 Validation of the simulator ...... 49

6 Simulation tests and results 53 6.1 Rockets description ...... 53 6.1.1 Model rocket specification ...... 53 6.1.2 Sounding rocket specification ...... 54 6.2 Rockets characteristics ...... 55 6.2.1 Model rocket ...... 55 6.2.2 Sounding rocket ...... 57 6.3 Trajectory simulations conditions ...... 58 6.4 Results from the trajectory simulations ...... 59 6.4.1 Model rocket launch ...... 59 6.4.2 Sounding rocket launch ...... 62 6.4.3 The influence of atmospheric stability ...... 65

7 Stochastic simulations 67 7.1 Simulation rocket and conditions ...... 68 7.2 Landing site uncertainties ...... 69 7.3 Rocket optimization ...... 72

8 Conclusions 73 8.1 Future work ...... 74

x Bibliography 75

A Tangent ogive proﬁle 79

B Connector’s Center of Mass (CM) and inertia 81

xi xii List of Tables

2.1 Rocketry motor total impulse classiﬁcation system [23]...... 9

3.1 Surface roughness values for diﬀerent land surfaces [1]...... 15 3.2 Atmospheric stability classes according to intervals of Obukhov length, L [33]...... 17

3.3 Absolute and relative errors of H0/(ρcp) considering three heights...... 18

4.1 Solid textile parachutes’ projected to reference diameter (dproj/dref ) and respective drag coeﬃcient [52]...... 38

6.1 Model rocket stages’ data (without connector and nose) [55] [56]...... 53 6.2 Model rocket ﬁns’ data...... 54 6.3 Model rocket’s main characteristics...... 54 6.4 Nike and Orion stages’ data (without connector and nose) [58] [59] [60]...... 54 6.5 Sounding rocket ﬁns’ data...... 55 6.6 Sounding rocket’s main characteristics...... 55 6.7 Launch site conditions...... 58

7.1 High power rocket data...... 68 7.2 High power rocket ﬁns’ data...... 68 7.3 Mean (µ) and standard deviation (σ) of the landing site in the two directions (x – Southing; y – Easting) changing each input within a distribution of 20 samples...... 69

xiii xiv List of Figures

1.1 Launch of a V-2 rocket from Blizna in Poland in 1944 [3]...... 2

2.1 Black powder motor design [21]...... 8 2.2 Grain sections showing diﬀerent geometries and respective thrust proﬁles [22]...... 9 2.3 Rocket in stable conditions [24]...... 10 2.4 Air speed components...... 11 2.5 Direct staging method [26]...... 12 2.6 Usual clustering arrangements [29]...... 12

3.1 Inertial (SXYZ) and topocentric-horizon (oxyz) reference frames [25]...... 18 3.2 Elevation-azimuth topocentric coordinate frame [37]...... 19 3.3 Additional angle of attack during rocket’s rotation [40]...... 22

4.1 Multi-stage rocket design...... 25 4.2 Generalized fin geometry and respective division to find its centroid...... 29 4.3 Generalized fin geometry...... 34 4.4 Pressure drag coefficient of wedges, cones and similar shapes as a function of the half-vertex angle (ε) [47]...... 35

5.1 Simulator’s algorithm...... 40 5.2 Rocket assembly list structure...... 41 5.3 Motor list structure...... 42 5.4 Body list structure...... 42 5.5 Fins list structure...... 42 5.6 Connector or nose list structure...... 43 5.7 Delays list structure...... 43 5.8 Committee on Space Research (COSPAR) International Reference Atmosphere 1986 (CIRA- 86) upgraded version structure...... 44 5.9 Temperature profiles for the CIRA-86 model...... 45 5.10 Neutral wind speed profile for the ABL...... 46 5.11 Stable wind speed profile for the ABL...... 46 5.12 Unstable wind speed profile for the ABL...... 46

xv 5.13 Wind gusts list structure...... 47 5.14 Angle of attack until the apogee with a wind gust perturbation...... 47 5.15 Thrust proﬁle for the D12 motor...... 48 5.16 Conical nose drag coeﬃcient with Mach...... 49 5.17 Altitude’s relative error between analytical and numerical solution until the burnout. . . . 50 5.18 Descent velocity from a vertical launch in which the rocket loses mass at 25 s...... 50 5.19 Relative error between analytical and numerical solutions until landing from a projectile launch with 70° of elevation, constant mass, no drag and no wind...... 51 5.20 Flight path angle from the projectile trajectory with the launch angle (elevation) of 70°.. 51

6.1 Model rocket properties’ functions...... 56 6.2 Sounding rocket properties’ functions...... 57 6.3 Model rocket trajectory with a vertical launch and a neutral ABL; Parachute ejection at the apogee...... 59 6.4 Vertical model rocket launch flight data...... 60 6.5 Model rocket trajectory in a 75° launch eastward with the downwind changing from northeastward at the ground to northward at the apogee...... 61 6.6 Sounding rocket trajectory with a vertical launch and wind blowing northeastward; Parachute ejection at about 28 km of altitude...... 62 6.7 Sounding rocket’s dynamic pressure until the parachute ejection...... 63 6.8 Vertical sounding rocket launch flight data...... 64 6.9 Wind profiles in a stable atmosphere for a measured wind speed of 4.3 m s−1 at 6 m of altitude...... 65 6.10 Trajectory profiles of the model rocket launch under three different wind profiles from a stable atmosphere...... 65 6.11 Wind profiles in an unstable atmosphere for a measured wind speed of 4.3 m s−1 at 6 m of altitude...... 66 6.12 Trajectory profiles of the model rocket launch under three different wind profiles from an unstable atmosphere...... 66

7.1 Normal distribution N (0, 1) and respective probability density histogram from 1000 samples. 67 7.2 Landing positions and respective confidence ellipses from the Monte Carlo simulation. . . 71 7.3 Probability density function of the landing coordinates resulting from the Monte Carlo simulation...... 71 7.4 Altitude over time from nine launches with different rocket’s length and diameter configurations...... 72

A.1 Ogive proﬁle...... 79

B.1 Connector’s half section (thick lines)...... 81

xvi List of Acronyms

ABL Atmospheric Boundary Layer

CIRA-86 Committee on Space Research (COSPAR) International Reference Atmosphere 1986

CM Center of Mass

CP Center of Pressure

ISA International Standard Atmosphere

LDRS Large and Dangerous Rocket Ships

NAR National Association Rocketry

NARAM NAR Annual Meeting

NOAA National Oceanic and Atmospheric Administration

SEDS Students for the Exploration and Development of Space

SL Surface Layer

TARC Team America Rocketry Challenge

TRA Tripoli Rocketry Association

xvii xviii List of Common Symbols

Greek letters CDf Skin friction drag coeﬃcient

α Angle of attack CDlam Skin friction drag coeﬃcient in laminar

Γ Adiabatic lapse rate regime C Skin friction drag coefficient in turbu- γ Ratio of specific heats at constant pres- Dturb sure and volume lent regime C Specific heat transfer coefficient δ Latitude H C Normal force coefficient derivative ε Nose half-vertex angle Nα c¯ Fin average chord θ Potential temperature

cp Specific heat at constant pressure θ0 Potential temperature at height z0 c Root chord κ Von Kármánconstant r c Tip chord µ Dynamic viscosity t D Drag µ⊕ Earth’s gravitational parameter d Diameter ξ Damping ratio El Elevation ρ Density

el Launch direction unit vector ψ Stability correction function f Coriolis parameter Ω Earth’s angular velocity g Gravitational acceleration ωn Natural frequency g0 Gravitational acceleration at sea level

Latin letters H0 Sensible heat ﬂux at the surface

Az Azimuth h Atmospheric Boundary Layer (ABL) a Speed of sound depth a Inertial acceleration I Transversal moment of inertia I¯ Principal transversal moment of inertia a0 Linear acceleration of the local reference frame (non inertial) I Total impulse b Fin span Isp Speciﬁc impulse

CD Drag coeﬃcient kf Fin’s interference factor

xix ∗ L Obukhov length u0 Local surface friction velocity

LMBL Length scale in the middle of the ABL V Air ﬂow velocity l Length v Velocity

M Mach number ve Exhaust velocity

Mda Aerodynamic damping moment vt True air speed vector

Mdj Jet damping moment W Weight

Ms Stabilizing moment w Wind velocity m Mass x Position relative to the nose tip m˙ Mass burning rate xcm Center of Mass (CM) position relative N Normal force intensity to the nose tip p Pressure xcp Center of Pressure (CP) position rela-

R Speciﬁc gas constant of the air tive to the nose tip

R¯ Universal gas constant z0 Surface roughness length

R Position relative to the Earth’s center Subscripts for d, l, m, m˙ , r, x or xcm R0 Origin of the local reference frame rel- b Body tube ative to the Earth’s center bt Boat tail ReL Reynolds number c Connector Recr Critical Reynolds number ca Connector aft opening Retr Transition Reynolds number cf Connector fore opening r Radius r Position d Delay charge

S Reference area e Ejection charge s Sweep length ext External

T Temperature f Final motor

T Thrust g Gap

Tp Thrust proﬁle int Internal tb Burnout time m Motor td Delay time from burnout to ejection n Nose tf Fin thickness p Propellant u Wind proﬁle velocity s Structural

xx Chapter 1

Introduction

The goal of the present work is to simulate trajectories from model to sounding rockets taking into account the wind and the Atmospheric Boundary Layer (ABL). The simulation tool is also capable to forecast the most probable landing region and perform rocket design optimization using Monte Carlo methods.

1.1 Motivation

Model rocket launches depend on its features and also on atmospheric conditions that may not be well estimated and change the desired trajectory. Hence, the hobbyists’ goal has better chances to be fulfilled if there is available a simulation tool provided with all the variables influencing the trajectory. The scope of our work focuses on passively controlled rockets, therefore the wind plays an important role during the flight due to its influence on the airflow that stabilizes the rocket. It is known that mean wind velocity profiles change according the land surface (or the surrounding environment) [1] and time of the day [2]. Thus, if the wind profile is foreseen wherever and whenever the launch is performed, the simulations’ errors can be minimized and simulations in different conditions can provide to users important information to help selecting the site and day of the launch to achieve successful flights. Sounding rockets consist of one or more stages (propelled by solid or liquid fuel) carrying a scientific payload in order to study the atmosphere and carry out microgravity research between 40 km to 2000 km height [3]. From 40 km to 200 km, these rockets are specially interesting because neither balloons nor low- earth satellites can reach these altitudes. Hence, allowing the developed tool to simulate trajectories of fin-stabilized sounding rockets up to these heights, brings additional valuable applications to our project.

1.2 Historical background

Rockets were used for the first time in 1232 by the Chinese (who also used gunpowder for fireworks in the first century A.D.) during the war with the Mongols [4]. In this conflict, rocket-like arrows, propelled by gunpowder, were launched by the Chinese through a guiding stick. After the war, this knowledge was also developed by the Mongols who presumably spread the use of rockets to Europe. In the following

1 centuries, rockets keep improving: they started to be launched through tubes (envisioning the modern bazooka) and a surface-running rocket-powered torpedo was used to set ships on fire. Nevertheless, in the 16th century, they fell on disuse as weapons of war but remained as fireworks, which used the multi-stage concept for the first time, employed by Johann Schmidlap, in order to reach higher altitudes. In the end of the 18th century, rockets had a revival as weapons of war due to the Indian rocket barrages against the British that inspired Colonel William Congreve to design rockets for the army [4]. However, they had not still improved very much and the cannons continued to represent a better alternative considering their accuracy and efficiency. In 1898, Konstantin Tsiolkovsky proposed the use of liquid-propelled rockets regarding the space exploration. Then, in 1926, Robert H. Goddard, who developed modern rocketry in the early 20th century, launched the first liquid-fuel rocket [5]. This small rocket, weighing 2.7 kg empty and 4.7 kg fueled, flew for 2.5 seconds reaching 12.5 m high and 56 m away from the launch site. Although it was a small flight, this historical event was the precursor of the new technological advances that enabled the comeback of warfare rockets in World War II. With the development of vehicles that allowed the transportation of launchers, barrage rockets were an ideal battlefield complement due to its simplicity. Therefore, the Nazis developed the Nebelwerfer (“Smoke Thrower”), consisting of six short, wide tubes arranged in a circular cluster, that could launch six 150 mm rockets in less than ten seconds but had a limited range and a poor accuracy [6]. The Soviet Union developed the Katyusha (roughly, “Little Katie”) consisting of eight parallel guiding rails fixed on a steel frame that allowed to launch the rockets in the desired direction. Each rail carried two rockets with 132 mm in diameter that could reach a 8 km range. Although these rockets were also inaccurate, they were an effective weapon when fired in large quantities. The most sophisticated rocket used during the war was the V-2 Vergeltungswaffe (“Vengeance Weapon”), also developed by the Nazis under the directorship of Wernher von Braun [4]. This was the

Figure 1.1: Launch of a V-2 rocket from Blizna in Poland in 1944 [3]. world’s first operational ballistic missile [6] and the first to fly into space. It measured 14.2 m in length, 1.65 m in diameter, and had a range of 330 km [3]. Even during the World War II, this rocket, combined with new flight conditions and technological advances, demanded further atmospheric and space research. Hence, after the war, the V-2 was used as a sounding rocket which boosted the development of modern rockets and new scientific researches in the following decades.

2 1.3 Model and high power rocketry

After the World War II, and during the Space Race, people became interested in amateur rocketry, specially inventors and intellectuals, which led to several imprudent and unsupervised launches [7, 8]. George H. Stine reported these problems and, together with Orville Carlisle, started the ﬁrst model rocket company based on a solid-fuel motor to be safely used in rockets that could be recovered and reused [8]. Model rockets are typically made of safe materials such as cardboard, plastic and balsa wood [7]. They are propelled by a replaceable, small and pre-packaged solid fuel motor that has an ejection charge to deploy the recovery system [9]. This consists of a parachute (or a streamer) attached to the nose cone. High power rockets diﬀer from model rockets in the propulsion power and weight. According to the American National Association Rocketry (NAR), a rocket is considered a high power rocket if it [10]:

Uses a motor with more than 160 N s of total impulse or multiple motors that all together exceed 320 N s; Uses a motor with more than 80 N average thrust; Exceeds 125 g of propellant; Uses a hybrid motor or a motor designed to emit sparks; Weighs more than 1500 g (including motor(s)) or uses any airframe parts of ductile metal.

These rockets are usually made from the same materials used in model rockets but require a construction that ensures resistance to higher stress conditions. They usually carry electronic devices to record flight data or to activate the recovery system. High power rockets fly under the safety code of the local governing organizations and only a qualified user can purchase a motor provided with such power [10].

1.3.1 Associations and current events

The NAR is the oldest and largest sport rocketry organization in the world [11]. Together with the American Tripoli Rocketry Association (TRA), which are the two major associations in the United States, they provide regulations to other organizations around the globe. The NAR and the TRA provide safety codes and are recognized to certify their members. NAR clubs organize several NAR meetings where people launch their rockets and compete with each other. These competitions are divided in four events [12]: altitude, duration, craftsmanship, and miscellaneous events. The altitude and duration events consist in maximize the apogee and the ﬂight time, respectively, for given motor power limits. Within the craftsmanship event, scale models of sounding rockets, missiles and space vehicles are launched with no restriction to the motor power. The miscellaneous events contain other contests that do not ﬁt under the previous events, such as landing in a chosen spot and presenting a written research or engineering project. Annually, more events take place in a week-long national meeting (NAR Annual Meeting (NARAM)) bringing together all the NAR members to decide the national champion. Similarly, the TRA is made up

3 of local groups (“Prefectures”) that schedule events and launches. It also organizes bigger competitions like Large and Dangerous Rocket Ships (LDRS) which has a category to launch bizarre and original rockets. To younger enthusiasts, there is in the United States the Team America Rocketry Challenge (TARC) for students from 7th to 12th grade in order to incite them following an aerospace career [13]. College and university students are allowed to compete in the NASA Student Launch in which it is required to ﬂy a scientiﬁc payload up to a given altitude [14]. Teams from many American institutions participate annually in this competition. A similar event is held by Students for the Exploration and Development of Space (SEDS) opened to any university in the United States.

1.4 Rocket trajectory simulators

Before there was any rocket trajectory simulator, rocketeers had to rely on their experience to figure out the probable trajectory under specific conditions. If the hobbyists were beginners or lacked such skills, they could not find a way to predict their launches and reach their goal. Nowadays, these problems are overcome due to the development of software capable to simulate rocket trajectories within several different conditions. Also, these simulators allow to test what if trajectories in order to determine the outcome from changing a given rocket feature or atmospheric condition and adapt their devices to match the desired solution. Two of the commonly used rocket flight simulators, as they include a rocket design tool, are Open- Rocket and RockSim. The latter is a proprietary software which makes it impossible to validate its methods. Also, it may represent a significant investment for students and rocket hobbyists. However, open-source software (like OpenRocket) can be used to avoid these drawbacks.

1.4.1 OpenRocket

OpenRocket was developed by Sampo Niskanen [15]. In this program it is required to model the rocket before the simulation. The mass is determined by the volume and material density but it can be manually defined due to existent residual materials impossible to model in the program. The Center of Pressure (CP) is found by the Barrowman Method (see Section 4.4) which is based on geometric dimensions of the several parts of the rocket and the aerodynamic forces and moments. OpenRocket simulates from subsonic to supersonic regimes but at transonic speeds interpolation functions are used as it is difficult to determine the flow properties. The simulator also make the following assumptions [15]:

The drag comes from 2 sources: skin friction drag plus pressure distribution drag (body, parasitic and base pressure drag). The shock wave drag is included in the body pressure drag; All the boundary layer on the rocket’s surface is taken as turbulent since the error calculating the apogee altitude is less than 5%; The atmosphere model is the International Standard Atmosphere (ISA) and the user is only allowed to specify the local temperature that is replaced in the ﬁrst layer of the model;

4 The humidity effects are ignored because the difference in air density and speed of sound between dry and saturated air is less than 1%; The wind is determined by summing a constant speed along the altitude with a random, zero-mean turbulence velocity; A flat Earth model is used and the Coriolis effect is ignored; During the recovery simulation all the drag comes from the deployed recovery devices. The default parachute drag is 0.8 and it can be changed by the user.

This simulator can be used for multi-stage rockets and allows to deﬁne instants to ignite the motors or eject the recovery system. Other required inputs are the launch site (longitude, latitude and altitude) and the angles of the launch rod relative to the wind and the vertical direction.

1.4.2 RockSim

RockSim is the standard program used in model rocketry [15]. Like OpenRocket, it is provided with features such as: modeling interface, multi-staging, automatic mass estimation (or deﬁned by the user), atmospheric conditions input at ground level and ejection delay. Some assumptions and features are worth mentioning [16]:

When Mach is larger than 0.8, the drag coefficient is adjusted to match the expected CD increase at transonic regime; it is not specified how it is done; The low wind speed is defined by selecting a value from a list with standardized wind speed ranges, which also sets the high wind speed from National Oceanic and Atmospheric Adminis- tration (NOAA) standards; custom values can also be specified; The wind is considered to move horizontally and may start only after a defined altitude; Wind turbulence is optional and it is modeled by a sine wave oscillating between low and high speeds; The atmosphere is simulated by the 1976 US Standard Atmosphere model; Thermals, consisting of convective circulations due to the warm surface in response to solar heating [17], may be implemented specifying their position, size and strength; Currently, this program ignores the lift produced by cross wind.

1.5 Work goals and strategies

This work is mainly focused on the development of the trajectory simulator in order to provide a tool that extends certain boundaries imposed by other simulators, although it may cost a larger processing time due to the interpreted language of Mathematica®. Therefore, the developed tool aims to: allow the user to simulate trajectories from micro to sounding rockets (and even any type of missiles); give flexibility to change easily a certain input in a given model and figure out its influence in the trajectory; give freedom to edit or implement new functions and values into the simulator’s database; and process the

5 resulting data in order to obtain further conclusions. Besides these features, the developed simulator has all the other important characteristics of the mentioned simulators like: multi-staging, boosters, launch direction and atmospheric, wind and drag coefficient models. Although the developed trajectory simulator does not include a modeling interface, it allows to define a versatile rocket model that is functional to every type of rockets. This is accomplished by using a default scheme of lists structured according the number of stages and boosters. Each major list corresponds to each stage (not counting the first list that is stored for the boosters data); the internal sublists define the respective components in each stage, and so on. Therefore, this structure of lists, allied to the simulator code, allows the rocket model to have any number of stages. However, this scheme of lists must always be respected so that the program runs properly. The simulator is developed under a modular programming philosophy. The function that computes the trajectory receives all the necessary inputs to the calculation and each of them is independent from one another. Hence, all the required parameters can be easily modified without having to worry about the other models. This allows to run several simulations under very different conditions and analyze their impact on the trajectory. For example, compare the outcomes from changing the drag coefficient from a complicated function to a constant value. Two databases are already implemented as default, one for the rocket’s motors and another for the atmospheric models. The databases are defined in a way that simplifies the access to the data and its properties. For instance, a list of data can be totally changed to the respective parameters from another model only by specifying the name of the new model. Also, other models can be joined in the databases to expand the options to use during the simulations. The developed simulator also allows the user to compute trajectories in loop changing a determined group of parameters. These analysis may consist of calculations to determine a probable landing region given a certain parameters’ uncertainties or a rocket design optimization evaluating the combination of possible values for the components that achieve the desired goal in the trajectory. The mentioned methods are explored in the present work.

6 Chapter 2

Model rocketry concepts

The developed tool is capable to simulate trajectories from model to sounding rockets, as already mentioned. However, throughout this work, we focus our attention mainly on model rockets. This is due to the larger influence the wind portrays in the trajectory from smaller rockets, specially within the ABL. Hence, simulations of these trajectories are the best approach to present the results of a rocket launch combined with the wind profile estimation. Model and high power rockets are an assembly of several pieces fitting together to produce the desired shape of the rocket. Basic model rockets consist of a nose and fins attached to a cylindrical body tube. The most common used noses are conical, parabolic or ogive shaped, and at their bottom exists the nose’s shoulder to fit, internally, the nose to the body. To connect the stages of the rocket, or to change the body tube diameter, as an option, conical transition sections can be used. If these parts increase the rocket diameter, they are called shoulders, otherwise are reducers or boat-tails. The latter are used at the back of the rockets to decrease the base drag. The more motors a rocket has, the more pieces it will need to hold them in a fixed position and align the thrust force with the longitudinal axis of the rocket. To ensure this, we must use engine mounts which consist of a mount tube, centering rings and a motor block.

2.1 Flight proﬁle

A model rocket’s ﬂight is divided in ﬁve phases [18]:

1. Ignition and lift-oﬀ; 2. Engine burnout; 3. Coasting phase; 4. Apogee and ejection; 5. Recovery.

Each one of these phases must develop in harmony so the rocket achieves its goals and returns safely to the ground.

7 Before the flight, the rocket is placed on a launch pad in order to be steady at the desired lift-off angle. During the first instants of the launch, and because the motor is not steerable, the rod on the pad guides the rocket while it gains the required speed to become aerodynamically stable. The ignition is usually triggered electrically increasing the safety comparing to a fuse in case of any unexpected problem. At the burnout the rocket has already reached its top speed and the motor no longer produces thrust. Throughout the coasting phase the rocket gains altitude and loses speed until the recovery system is deployed, safely, as the drag forces are significantly reduced. During this phase the delay charge burns inside the motor producing a smoke trail that allows the rocket to be tracked at high altitudes. The deployment of the recovery system, made by an ejection charge inside the motor that splits the nose cone from the body, is desired to occur at the apogee where the rocket reaches its minimum speed. This instant must be carefully predicted in order to choose a motor provided with an adequate delay charge. After the ejection, the rocket’s descending trajectory is very influenced by the wind. This may cause the rocket to fly away from the launch site and, in worst cases, be lost. Usually, a parachute is used as a recovery system but a streamer can be used in order to keep the rocket in sight, since the fall becomes faster. It is also possible to cut a hole in the parachute to produce the same effect.

2.2 Motors

The two types of propellant used in model rockets are black powder and composite propellant [19]. Black powder is a solid propellant using potassium nitrate as the oxidizer and carbon and sulfur as the fuel. Composite propellant is made of ammonium perchlorate as the oxidizer, a powdered metal as the fuel (usually aluminum or magnesium) and an elastomer as the binder. This propellant is more powerful than black powder so it is normally used in motors from larger model rockets. Black powder motors are the most common and consist of a paper case housing a clay nozzle, the solid propellant, a time delay composition, an ejection charge and a clay cap (see Figure 2.1). Composite motors have the same design although the nozzle and the case are made of high-temperature plastic [20].

Figure 2.1: Black powder motor design [21].

The propellant’s burning rate, which is proportional to the chamber pressure, is not constant and changes according to the burning area [19]. Therefore, using different grain configurations, it is possible to build several thrust profiles in order to get the desired propulsive properties. A black powder motor

8 usually uses an end burning configuration which burns from the nozzle toward the delay composition keeping a constant burning area [19]. On the other hand, usually, a composite motor is a core burning which has a central hole running lengthwise the grain. Thus, the burning occurs transversely to the propellant from the interior to the exterior, increasing the burning area and the thrust. A core burning grain is represented by the first geometry illustrated in Figure 2.2 where other configurations are given as examples.

Figure 2.2: Grain sections showing diﬀerent geometries and respective thrust proﬁles [22].

The delay charge does not produce thrust allowing the rocket to coast up to the apogee leaving a smoke trail behind at a slower burning rate than the propellant. After the delay time is over, the ejection charge ﬁres which over-pressurizes the case and bursts the clay cap. This increases the pressure in the rocket’s body and deploys the recovery system.

2.2.1 Coding system

Every motor is labeled with a code which provides information about its propulsive characteristics. This code consists of a letter and two numbers (e.g., “B6-4”). The letter refers to the total impulse produced by the motor and it was deﬁned that 2.5 N s corresponds to an “A” motor [23]. In Table 2.1 are listed the range of the total impulse for several motor classes. Usually, the motors are built to operate at the top limit of these classes [20]. As showed in Table 2.1, the power of the motor doubles if we pick the next letter, and so on. A motor from class G has a maximum total impulse of 160 N s, thus it is the most

Code Total impulse (N s) 1/4A 0 - 0.625 1/2A 0.626 - 1.25 A 1.26 - 2.50 B 2.51 - 5.00 C 5.01 - 10.00 D 10.01 - 20.00 E 20.01 - 40.00 F 40.01 - 80.00 G 80.01 - 160.00 H 160.01 - 320.00 I 320.01 - 640.00

Table 2.1: Rocketry motor total impulse classiﬁcation system [23].

9 powerful to be used in model rocketry, as mentioned in Section 1.3. Consequently, motors from class H to class O (which are not all tabulated) are considered high-power motors. The ﬁrst number after the letter gives us the average thrust (in Newton) produced by the motor. Then, dividing the total impulse of the motor by the average thrust, we can ﬁnd the burning time. Regarding the features of the rocket or the goals of the launch, it is possible to choose the most suitable motor for a given total impulse. For example, a high average thrust motor is convenient if the rocket needs stability at launch, although it will decrease the apogee since the drag increases. The second number tells us the coasting time delay (in seconds) from burnout until the ignition of the ejection charge. Motors marked with “0” as the time delay number are boosters to be used in lower stages of multi-stage rockets [20]. If the delay number is replaced with a “P”, the motor is also a booster but it is plugged, which means without ejection charge to burst the top of the casing.

2.3 Stability

A model rocket is stable if its CP is behind the Center of Mass (CM) (seeing from the nose to the tail of the rocket) [24]. The CP is the point where acts the resultant aerodynamic force produced by the air pressure and moves toward the nose with increasing angle of attack (α). This force, when the angle of attack is diﬀerent from zero, may be decomposed in axial and normal (N) components where the latter creates a moment (M) about the CM of the rocket (see Figure 2.3). In stable conditions, this moment

(a) Normal force (b) Stabilizing moment

Figure 2.3: Rocket in stable conditions [24]. produces a damped oscillating movement about the air ﬂow direction until the rocket angle of attack returns to zero and the normal force vanishes [24] (see Section 3.3.1). The arm of the moment is the length between the CM and the CP, called static margin. Decreasing the static margin until the CP coincides with the CM, makes the rocket unstable because the resulting moment becomes smaller and smaller. If the CP overtakes the CM and a perturbation deviates the rocket from the air ﬂow direction, the aerodynamic moment will amplify the disturbance causing the rocket to spin and crash.

10 With a larger static margin, however, the rocket may become overstable and reaches a lower apogee because it turns sooner to the wind [24]. This eﬀect, called weathercocking, is due to the contributions of the rocket and wind’s velocities (v and w, respectively) that deﬁne the direction and intensity of the true air speed as (see Figure 2.4)

vt = v − w . (2.1)

Considering windy conditions, the angle of attack is maximum when the rocket leaves the guiding rods and is found by (see Figure 2.4)

v · v α = arccos t . (2.2) kvkkvtk

Note that v may describe other angles in respect to the ground, so that Figure 2.4 no longer depicts a right triangle. In a broader sense, if there is a windless atmosphere, the rocket may also present an angle of attack if its velocity is not aligned with the longitudinal axis of the rocket.

v vt α

Figure 2.4: Air speed components.

Weathercocking is also enhanced by strong winds and low rocket’s velocity as it leaves the launch platform. In order to reduce this eﬀect, it is advised to have a static margin lower than one maximum rocket diameter, ﬂy in calm winds and use longer guiding rods or higher average thrust motors so that the rocket gains more velocity to counterbalance the wind speed when leaving the launch pad [24]. Except the former, these advices also contribute to guarantee the stability of the rocket at the moment it leaves the guiding rods.

2.4 Multi-staging and clustering

Multi-staging enables the rocket to reach higher altitudes by discarding parts of structural mass made useless by the spending of rocket propellant. If only one stage is burning at a given time and the previous one is discarded right after its propellant is consumed, it is called serial staging [25]. Otherwise, if some stages are burning at the same time, it is called parallel staging. In model rocketry there are two ways of staging a rocket: direct and indirect staging [26]. With direct staging the upper motor is ignited by the lower stage motor called the booster. These type of motors do not carry delay composition and ejection charge in order to ignite the upper stage nearly instantaneously after the burnout, as it is represented in Figure 2.5. As the motor burns, the remaining propellant wall becomes thinner and bursts due to the pressure, which throws hot gases and burning particles directly to

11 Figure 2.5: Direct staging method [26]. the next nozzle. Immediately, the joint between the motors is pressurized and the lower stage is released [27]. Since the composite propellant is soft and rubbery, it cannot hold the chamber pressure in order to burst and expel the particles. For this reason there are only black powder boosters, leaving composite motors to be used in indirect staging. Direct staging is the simplest and cheapest method because it does not require electronic devices to ignite the upper stage, unlike indirect staging [26]. With serial staging the lower stages must be equipped with fins (with increased area on each added stage) to compensate the shift of the CM back to the tail [27]. These fins must be designed so that the stages become aerodynamically unstable in order to tumble and decrease speed during recovery. A larger fin area, however, over-stabilizes the rocket giving it tendency to weathercock. For this reason, a multi-stage rocket should only fly with calm winds and never use more than three stages [28]. Parallel staging can be achieved if the rocket is provided with external boosters with a lower burnout time than the central motor. In this case, when the core motor and the boosters are burning at the same time, it is called the zeroth stage [25]. Several motors can be used together to provide the rocket with more thrust. This method is called clustering in which, at most, four engines must be used since it makes the ignition less reliable [29]. The cluster must be arranged in a way that the resultant thrust is applied through the longitudinal axis of the rocket and all the motors at the same distance from this axis must develop equal thrust (see Figure 2.6).

Figure 2.6: Usual clustering arrangements [29].

In a direct multi-stage rocket, clustering should only be used in the ﬁrst stage due to the diﬃculty in igniting the upper motors at the same time [29]. In this situation, one motor must be placed at the center (to start the next stage) and the others around it.

12 Chapter 3

Wind and trajectory

3.1 Wind and atmospheric boundary layer

The lowest layer of the troposphere is directly influenced by the ground surface characteristics. Therefore, a wind speed profile is worth to be implemented since its effect can be relevant, mainly, in the trajectory of small and medium model rockets. This layer is known as the ABL and its thickness may change from about a hundred meters to a few kilometers varying in time and with geographic region [30]. The bottom 10% of the ABL is named the Surface Layer (SL) with, in average, 100 m height [31] which makes the ABL about 1 km thick. Above the ABL stays the free atmosphere, a more stable region where the frictional influences of the surface can be ignored and the wind is nearly geostrophic [30]. The geostrophic wind, moving parallel to the isobars, results from the balance between the Coriolis and the pressure gradient forces [17]. From the momentum equations in steady state with no turbulence terms, the geostrophic wind’s components in the eastward and northward directions are given, respectively, by [17]:

1 ∂p u = − , (3.1) g ρf ∂y 1 ∂p v = , (3.2) g ρf ∂x where ρ is the air density, p is the air pressure and f is the Coriolis parameter (f = 2Ω sin δ) which depends on the Earth’s angular velocity (Ω) and latitude (δ). With (3.1) and (3.2), the direction of the geostrophic wind can be found if the horizontal pressure gradient is known for a given region. If there is a temperature gradient, the geostrophic wind intensity changes with height and its direction may vary if the isotherms are not parallel to the isobars [17]. This vertical variation is named thermal wind and can be found using the equation of state [32]

p ρ = , (3.3) RT where T is the air temperature and R = 287.0 J kg−1 K−1 is the speciﬁc gas constant of the air (which

13 is the quotient between the universal gas constant, R¯ = 8.314 J mol−1 K−1, and the molecular weight of −1 air, Mair = 28.97 g mol ), and the hydrostatic equation

dp = −ρgdz , (3.4) in (3.1) and (3.2) resulting, approximately, in [17]:

∂u g ∂T g ' − , (3.5) ∂z fT ∂y ∂v g ∂T g ' . (3.6) ∂z fT ∂x

In this situation, when there are horizontal temperature gradients, the atmosphere is called baroclinic. Considering only geostrophic balance, which is the case of a barotropic atmosphere, the wind is constant with height and its intensity can be assumed as equal to the one at the top of the ABL. As the behavior of the atmosphere depends on the weather conditions, the atmospheric stability is classified in three different classes (neutral, stable and unstable) that help to determine the properties of the atmosphere (including the wind) in different conditions. A neutral atmosphere implies an adiabatic lapse rate and no convection which is the case of a partially or highly cloudy atmosphere that may reduce the insolation at the surface [30]. Stable conditions occur mostly at night but can also appear when the ground surface is colder than the surrounding air. An unstable atmosphere is formed in clear weather during the day when there is high radiation from the sun causing ascending heat transfer. Knowing these atmospheric conditions we can deduce the stability class. They also influence the wind profile and must be taken into account due to convective effects.

3.1.1 ABL proﬁles

From measurements of the wind speed, the literature [33] presents the following wind proﬁles modeled considering the entire ABL and its stability:

" # u∗ z z z z u(z) = 0 ln + − , (3.7) κ z0 LMBL h 2LMBL for neutral conditions,

" # u∗ z 4.7z z z z z u(z) = 0 ln + 1 − + − , (3.8) κ z0 L 2h LMBL h 2LMBL for stable conditions, and

" # u∗ z z z z z u(z) = 0 ln − ψ + − , (3.9) κ z0 L LMBL h 2LMBL for unstable conditions, where h is the ABL depth, z0 is the surface roughness length, κ is the von K´arm´an ∗ constant, LMBL is the length scale in the middle of the ABL for its respective condition, u0 is the local

14 surface friction velocity near the ground, L is the Obukhov length and ψ is a stability correction for the SL given by

! z 3 1 + x + x2 √ 1 + 2x π ψ = ln − 3 arctan √ + √ , (3.10) L 2 3 3 3

1/3 where x = 1 − 12z/L . Above the ABL (z > h), (3.7), (3.8) and (3.9) are no longer valid since it is the region of the free atmosphere. Within the ABL, unlike the free atmosphere, there are surface friction eﬀects which cause the wind direction to deviate from the geostrophic wind. Thus, the wind changes with height until it becomes aligned with the direction of the geostrophic wind at the top of the ABL [17]. The last two terms in (3.7), (3.8) and (3.9) represent the contributions of the middle and upper layers of the ABL [33] and, considering the neutral case when these terms are ignored, (3.7) reduces to

u∗ z u(z) = 0 ln , (3.11) κ z0 which is the simpler and most known log wind proﬁle used in the SL. The parameter h, if it is unknown, may be found from [33]

u∗ h = c 0 , (3.12) |f| in a neutral and unstable ABL, and from [34]

s u∗L h = 0.5 0 , (3.13) |f| in stable conditions, where c is an empirical constant (ranging in the literature from 0.1 to 0.4 [35]). Note that, when f → 0, the thickness of the boundary layer increases signiﬁcantly and h no longer has physical meaning. Therefore, the model is not valid for low latitudes [35].

The surface type inﬂuences z0 which is quantiﬁed and can be obtained from Table 3.1. The value of the empirical constant κ is near 0.35 or 0.4 [30].

Surface type z0 (cm) Sand 0.01...0.1 Cut grass (∼ 0.01 m) 0.1...1 Small grass, steppe 1...4 Uncultivated land 2...3 High grass 4...10 Coniferous forest 90...100 Uptown, suburbs 20...40 Downtown 35...45 Big towns 60...80

Table 3.1: Surface roughness values for diﬀerent land surfaces [1].

15 The length scale, LMBL, which is deduced by the Rossby similarity theory (that relates the geostrophic ∗ wind and u0) applied to (3.7), (3.8) and (3.9) at the top of the ABL (z = h), is found from [33]:

−1  2 1/2  ∗ ! h u0 2 h LMBL =  ln − B + A  − ln  , (3.14) 2  |f| z0 z0  for neutral conditions,

−1  2 1/2  ∗ ! h u0 2 h 4.7h LMBL =  ln − B + A  − ln −  , (3.15) 2  |f| z0 z0 2L  for stable conditions, and

−1  2 1/2  ∗ ! h u0 2 h h LMBL =  ln − B + A  − ln + ψ  , (3.16) 2  |f| z0 z0 L  for unstable conditions, where A ≈ 4.9 and B ≈ 1.9 in a neutral atmosphere [33]. Many empirical functions have been developed in the literature to ﬁnd A and B for the remaining conditions [34]. Show- ing good agreement between theoretically and empirically derived functions, these parameters may be approximated by [34]:

h h A = ln − 2.2 + 2.9 , (3.17) L L h B = 3.5 , (3.18) L for stable conditions, and

h |f| h A = ln + ln ∗ + 1.5 , (3.19) |L| u0 ∗ u0 fh 0.2h/L B = κ + 1.8 ∗ e , (3.20) fh u0 for unstable conditions.

1 ∗ ∗ τ /2 The velocity u0 is defined as u0 = ( /ρ) , where τ is the surface stress and can be found solving the ∗ equations in order to u0 using the measured wind speed at the launch site with the respective height of the measurement. The Obukhov length (L) represents the thickness near the surface in which the shear stress dominates over the buoyancy effects in generating turbulence. It is defined by [17]

u∗3 L = − 0 , (3.21) κ g H0 T0 ρcp where T0 is the temperature at the reference pressure (p0 = 1000 mbar), cp is the speciﬁc heat at constant

16 pressure and H0 is the sensible heat ﬂux at the surface which is given by

H0 = −ρcpCH U(θ − θ0) , (3.22)

where CH is the heat transfer coeﬃcient (ranging from 0 to 0.01 for land surfaces [17]), U and θ are, respectively, the wind speed and potential temperature at the height of their measurements and θ0 is the potential temperature at z0. Potential temperature is deﬁned as

p R/cp θ = T 0 , (3.23) p and the subtraction in (3.22) may be approximated by [17]

θ − θ0 ' T − T |z=z0 + Γ(z − z0) , (3.24) where Γ = 9.8 K km−1 is the adiabatic lapse rate in a dry atmosphere.

3.1.2 Atmospheric stability

The atmospheric stability can be determined from Table 3.2 if the Obukhov length is known. To determine

Obukhov length interval (m) Atmospheric stability class 10 ≤ L ≤ 50 Very stable 50 ≤ L ≤ 200 Stable 200 ≤ L ≤ 500 Near stable |L| ≥ 500 Neutral −500 ≤ L ≤ −200 Near unstable −200 ≤ L ≤ −100 Unstable −100 ≤ L ≤ −50 Very unstable

Table 3.2: Atmospheric stability classes according to intervals of Obukhov length, L [33]. this parameter we can use (3.21) with measurements taken from the atmosphere. In a range of a few meters the temperature is practically constant, therefore, the height of the measurements should not be close to the surface in order to reduce errors. Since to get L we do not need to find H0 but H0/(ρcp) (surface kinematic heat flux), which depends of five measured parameters (xi=1,2...5), its maximum absolute error is determined by [36]

5 X ∂H0/(ρcp) ∆ H0/(ρcp) = ∆xi ∂x i=1 i

= CH U(∆T + ∆T |z=z0 + Γ(∆z + ∆z0) + CH T − T |z=z0 + Γ(z − z0) ∆U, (3.25)

where ∆xi is the absolute deviation of each measurement.

Considering three measured heights with CH = 0.01, ∆z = ∆z0 = 1 mm, ∆T = ∆T |z=z0 = 0.1 K, ∆U = 0.1 m s−1, using a temperature proﬁle from the atmospheric model (developed in Section 5.3) to

17 ∗ −1 get the temperatures and (3.11) to estimate U (with z0 = 7 cm, κ = 0.4 and u0 = 0.35 m s ), the absolute and relative errors from (3.25) are listed in Table 3.3. We can see that, even for 700 m above

z (m) (above z0) Temperature (K) Absolute error (K m/s) Relative error (%) 300 289.34 1.6651 × 10−2 39.1 500 287.85 1.8105 × 10−2 22.1 700 286.43 1.9286 × 10−2 15.1

Table 3.3: Absolute and relative errors of H0/(ρcp) considering three heights. the ground, the relative error is still reasonably high. We could get lower errors taking measurements at higher altitudes but the tabulated ones are already impractical to reach. Therefore, and since the absolute deviations of temperature may be higher, it is not reliable to find L from the previous method. Nevertheless, if the atmospheric stability is deduced qualitatively from the weather conditions and time of day (as described in the last paragraph of Section 3.1), we get to know which wind profile to use and a value for L can be estimated from Table 3.2. Summarizing Section 3.1, there are three ABL profiles that we aim to determine for each atmospheric stability. The inputs relevant to mention are: the roughness length (z0), the Obukhov length (L) and the ∗ friction velocity (u0). The former is estimated by the terrain type from Table 3.1. The parameter L shows large uncertainties when finding it quantitatively from atmospheric measurements. Thus, using Table 3.2, it can be estimated qualitatively from the weather conditions that were described for each stability class. ∗ The velocity u0, after knowing the remaining parameters, is determined solving the respective wind profile equation with the wind speed measured at a given altitude.

3.2 Kinematics and reference frames

Considering the origin of the reference frame at the launch site, powerful rockets that reach high altitudes may suffer some errors calculating the trajectory due to the Earth’s rotation. The alternative is to define the center of the Earth as the origin of the inertial reference frame and set the local frame at the launch site (point o, see Figure 3.1). The local frame is defined as a topocentric-horizon coordinate frame with

Figure 3.1: Inertial (SXYZ) and topocentric-horizon (oxyz) reference frames [25].

18 the x axis pointing southward and the y eastward. Thus, as this is a right-handed coordinate frame, z axis points upward collinear with the Earth’s radius direction (or zenith). The elevation (El) is the angle measured in the vertical plane, at the origin of the local frame, from the horizon up to the rocket (see Figure 3.2). Likewise, the azimuth (Az) is deﬁned as the angle measured in

Figure 3.2: Elevation-azimuth topocentric coordinate frame [37]. the horizontal plane from north (negative x axis direction) to the rocket. The coordinate transformations from spherical coordinates (El, Az, r) to Cartesian coordinates (x, y, z) are:

x = −r cos El cos Az , (3.26)

y = r cos El sin Az , (3.27)

z = r sin El . (3.28)

The total acceleration in the inertial reference frame is given by [25]

∂2r dΩ ∂r a = a + + × r + 2Ω × + Ω × (Ω × r) , (3.29) o ∂t2 dt ∂t where Ω is the Earth’s angular velocity (Ω = 7.2921 × 10−5 rad/s), r is the position of the rocket relative to the local frame and ao is the linear acceleration of this frame given as Ω × (Ω × Ro), where Ro is the position of point o relative to the Earth’s center (see Figure 3.1). The rocket’s velocity seen from ∂2r an observer at the launch site is v = ∂t2 , where the partial derivative stands for the time derivative of dΩ r with reference to the local frame. The term dt × r is the angular acceleration of the moving frame, ∂r 2Ω × ∂t is the Coriolis acceleration and Ω × (Ω × r) is the centripetal acceleration. In order to get the trajectory seen by an observer at the launch site, we must write these vectors in the local frame (oxyz) coordinates.

19 3.3 Rocket dynamics

Besides the forces due to the Earth’s rotation, also act on the rocket the drag (D) and the weight (W). Then, from the momentum conservation, and due to the ejection of propellant, the rocket equation of motion becomes [38]

dm Ve ma = Ve − A(Pe − Pa) + D + W , (3.30) dt Ve where m is the rocket’s mass, Ve is the exhaust velocity relative to the rocket, A is the motor nozzle area, Pe is the exhaust pressure and Pa is the ambient pressure. The ﬁrst two terms on the right-hand side of (3.30) correspond to the force responsible for the acceleration due to the loss os mass [39]: the thrust (T). Thus, using this deﬁnition and replacing (3.29) in (3.30), we get

∂2r dΩ ∂r m = −ma − m × r − 2mΩ × − mΩ × (Ω × r) + T + D + W . (3.31) ∂t2 o dt ∂t

Solving (3.31) the position of the rocket seen from the launch site can be obtained. The thrust is always aligned with the rocket’s longitudinal axis because the rocket is considered as not steerable. Since the rocket may ﬂy with angle of attack until it stabilizes and the thrust becomes collinear with the true air speed vector, the eﬀective propulsive force may be lower in this direction. Therefore, the adopted model to the thrust force is

vt T = Tp cos α (3.32) kvtk where Tp is the propulsion from the motors’ thrust profile. The aerodynamic force is assumed to be merely the drag since the rocket only flies with angle of attack during a short interval of time after the launch (or after any perturbation) and it describes a low amplitude oscillation. The lift that is produced during these moments contributes essentially to stabilize the rocket, its average is considered to be zero, and its effect can be implemented as described in Section 3.3.1. Hence, the aerodynamic force is

1 D = ρkv k SC v , (3.33) 2 t D t where S is the reference area (usually chosen as the maximum sectional area of the rocket or, after the apogee, the reference area of the recovery device) and CD is the drag coeﬃcient (determined in Section 4.5). The weight points to the Earth’s center and is found by

µ⊕m W = − 2 z , (3.34) (R0 + z)

5 3 −2 where µ⊕ = 3.986 × 10 km s is the Earth’s gravitational parameter and z is the unit vector of the

20 vertical coordinate. During the launch, when the rocket is constrained by the guiding rods, the drag and weight normal to the launch direction are counterbalanced by the rods reaction and only the tangential forces to the rods contribute to the launch. Thus, (3.31) becomes

∂2r dΩ ∂r 1 m = −ma − m × r − 2mΩ × − mΩ × (Ω × r) − ρkvk SC v + (T + W sin El)e , (3.35) ∂t2 o dt ∂t 2 D p l where el is the unit vector of the rods’ direction determined by the launch elevation and azimuth as

el = (cos El cos Az, cos El sin Az, sin El) . (3.36)

3.3.1 Dynamic stability

After leaving the platform, and when a disturbance occurs, it is considered that the rocket acquires an angle of attack that will decrease gradually and force it to oscillate about the true air speed direction [24], as mentioned in Section 2.3. To generate this damped oscillating movement, we consider that the rocket is subjected to two moments [40]: a stabilizing (or restoring) moment and a damping moment. The stabilizing moment comes from the normal aerodynamic force acting on the CP which causes the rocket to rotate about the CM since these points must distance from each other by the static margin (see

Figure 2.3). The normal force depends on the normal coeﬃcient, CN , which, assuming small angles of attack, is given by

CN = CNα α , (3.37)

where CNα is the rocket’s normal force coeﬃcient derivative. Therefore, the stabilizing moment is [41]

1 M = N(x − x ) = ρV 2SC α(x − x ) , (3.38) s cp cm 2 Nα cp cm where xcm and xcp are, respectively, the rocket’s CM and CP distances from the reference point (often considered as the nose tip, which we also consider throughout this work) and V is the ﬂow velocity. One source of the damping moment results from the aerodynamic resistance of the air while the rocket is rotating [40]. During the rotation, the angle of attack of each part of the rocket changes due to the tangential velocity of this motion. Considering a component i with its CP at distance xcpi − xcm from the rocket’s CM, the variation of the angle of attack for this component, assuming again small angles, is (see Figure 3.3)

α˙ (x − x ) ∆α = cpi cm , (3.39) V whereα ˙ is the angular velocity. Like the stabilizing moment, only the normal force of this additional resistance contributes to dampen the rotation. Thus, using (3.39), and since the aerodynamic damping

21 Figure 3.3: Additional angle of attack during rocket’s rotation [40].

moment (Mda ) is determined by summing all the elemental moments along the rocket [40], we get [42]

n n X X M = M = N (x − x ) da dai i cpi cm i=1 i=1 n 1 2 X = ρV S CN αi(xcp − xcm) 2 αi i i=1 n 1 X 2 = ρV Sα˙ CN (xcp − xcm) , (3.40) 2 αi i i=1 where n is the total number of components of the rocket that contribute to this moment. The other contribution to the damping moment comes from the Coriolis acceleration due to the change of the gas ﬂow through the nozzle [40], also called jet damping. Knowing the Coriolis acceleration as

ac = 2veα˙ , (3.41)

where ve is the exhaust velocity, and taking an element of propulsive mass (dm) with length dx at distance x from the CM, the elemental jet damping comes as [40]

dMdj = 2veαx˙ dm = 2veαxρ˙ pSnozzle dx = 2αx ˙ m˙ dx , (3.42)

where ρp is the density of the propellant, Snozzle is the nozzle section area andm ˙ is the mass burning rate. Integrating (3.42) from the CM up to the nozzle, we get the jet damping as

Z xnozzle 2 Mdj = 2αx ˙ m˙ dx =α ˙ m˙ (xnozzle − xcm) , (3.43) xcm where xnozzle is the nozzle distance from the reference point. Once the moments acting on the rocket are known, we can ﬁnd the governing equation for the motion of the angle of attack assuming the rotation is two dimensional (in the plane formed by the rocket’s velocity and wind vectors). The angular momentum about the rocket’s CM is [43]

Hcm = Iα˙ , (3.44) where I is the transversal moment of inertia relative to the CM , and the balance of moments about the same point is given by the angular momentum derivative as

X d M = H . (3.45) dt cm

22 Therefore, from (3.44) and (3.45), and knowing that the stabilizing and damping moments counteract the rocket’s rotation, we get

˙ −Ms − Mda − Mdj = Iα˙ + Iα¨ , (3.46) which, applying (3.38), (3.40) and (3.43), leads to

 n  1 X 2 2 ˙ 1 2 Iα¨ +  ρV S CNα (xcpi − xcm) +m ˙ (xnozzle − xcm) + I α˙ + ρV SCNα (xcp − xcm)α = 0 . (3.47) 2 i 2 i=1

The coeﬃcients ofα ˙ and α in (3.47) are named the damping and stabilizing moment coeﬃcients [28, 42]

– CMd and CMs , respectively. As expected, (3.47) represents a damped harmonic oscillator, generically formulated as

2 α¨(t) + 2ξωnα˙ (t) + wnα(t) = 0 , (3.48)

where ξ is the damping ratio and ωn is the natural frequency. If ξ < 1 the rocket is underdamped and it will oscillate until the angle of attack decreases to zero; if ξ = 1 the rocket is critically damped and it returns smoothly to zero angle of attack; if ξ > 1 the rocket is overdamped and, although the disturbance decreases, it always ﬂies with some angle of attack. The natural frequency tells how fast are these oscillations. For the case in study, comparing (3.47) and (3.48), the damping ratio and natural frequency are given by:

CMd ξ = p , (3.49) 2 ICMs r C ω = Ms . (3.50) n I

As we can deduce from (3.50), the oscillations of the rocket become faster if we move the CM toward the nose (increasing the CMs ). On the other hand, increasing the inertia of the rocket makes the oscillations slower, as expected.

23 24 Chapter 4

Rocket design

The proposed structure for a general multi-stage rocket developed in the present thesis is shown in Figure 4.1. Although the figure represents a three-stage rocket, this structure can be extended to any number of stages and was developed with the goal of modeling several types of rockets, specially the most common. For each stage the fundamental components are: a connector (or nose, in the case of the last stage), a body tube and fins. Internally, this structure only considers the motors since data from additional components can be joined with the body tube properties. The last stage includes the nose and in the first stage the rocket can also be supplied with an internal cluster of motors or external boosters in order to achieve parallel staging. The reference from which the position and properties of the components are specified was defined as being the top of the respective component (this will be explained in Section 5.2).

3rd Stage 2nd Stage 1st Stage

Figure 4.1: Multi-stage rocket design.

The rocket’s CM, moment of inertia, CP and drag coefficient may depend and change according to the mass, shape and position of the several components. There are cases where the rocket’s properties are impossible to determine directly due to lack of information. Therefore, this chapter presents methods to find or estimate them for the main elements of the rocket. Both these values and the properties taken directly from measurements of the rocket (that can still be used), must be defined in the respective list inside the general structure developed for the model of the rocket (see Section 5.2). It must be emphasized that, although the following methods are focused in model rocketry, the developed simulator can still be used to process other types of rockets or models if the input data is coherently employed.

25 4.1 Mass

During the ignition and engine burnout phases, the propellant is expelled through the nozzle to generate thrust and, also when the delay charge burns, the mass of the motor decreases over time. Thus, the rocket’s mass, in each instant, is found from

m(t) = ms(t) + mm(t) , (4.1)

where ms(t) is the structural mass (without the motor(s)), changing from stage to stage, and mm(t) is the mass of the motor(s). Considering one motor, structured as represented in Figure 2.1, mm(t) is obtained by

mm(t) = mf + mp(t) + md(t) + me(t)

Z tb Z tb+td

= mf + mp0 − m˙ p(t) dt + md0 − m˙ d dt + me(t) , (4.2) 0 tb

where mf is the motor’s final mass, mp0 and md0 are, respectively, the propellant and delay charge initial masses, tb and td are the burnout and delay times,m ˙ p andm ˙ d are the propellant and delay charge mass burning rates and me is the ejection charge mass. Note that in (4.2) we are considering the entire motor since we have to estimate the mass of the delay charge. However, the structural (casing) and variable masses must be defined in different lists under the rocket structure (see Section 5.2). The propellant’s mass burning rate is given by [39]

Tp(t) m˙ p(t) = , (4.3) g0Isp where g0 is the standard sea level acceleration of the Earth, Tp(t) is the thrust proﬁle of the motor and

Isp is the speciﬁc impulse, which represents the linear momentum produced per unit weight of propellant consumed (measured in seconds) [25]. Therefore, the speciﬁc impulse can be found from

I Isp = , (4.4) mp0 g0 where the total impulse, I, is

Z tb I = Tp(t) dt . (4.5) 0

Again from the motor design in Figure 2.1, the delay charge mass burning rate may be estimated as

md0 mm0 − mp0 − me − mf m˙ d = = , (4.6) td td

where mm0 is the motor’s initial mass.

The ejection charge, consisting of black powder, is consumed instantaneously at t = tb + td and, for

26 model rocket motors, it ranges from 0.3 g to 0.5 g [19]. In alternative, knowing the gases from the burst must over pressurize the rocket and assuming that all the mass turns into gas, we can estimate me from the ideal gas law

me pv = RT¯ bp , (4.7) Mbp

−1 where v is the internal rocket’s volume, Mbp = 34.75 g mol is the molecular weight of black powder and

Tbp = 2600 K its combustion temperature [44]. Since p = F/Sint (where F is the required force to open the rocket and Sint is the interior sectional area) and v = lintSint (being lint the interior length), (4.7) yields

Mbp me = F lint . (4.8) RT¯ c

4.2 Center of mass

We consider the rocket as being an axisymmetric body, so the CM lies on the longitudinal axis and its location from the reference point (nose tip) is given, for n pieces, by [43]

n P xcmi mi i=1 xcm = n , (4.9) P mi i=1

where xcmi is the CM of piece i of the rocket (also from the reference) and mi its respective mass. Considering that the rocket is made from an homogeneous material, the CM of each component matches its centroid [43]. Since these parts are, usually, solids of revolution or other common shapes, their centroids are easily known. Thus, we can get xcmi by adding the CM (or centroid) given from the top of the component with the distance from the nose tip to the same component. Again, we alert that the following methods are only applied to homogeneous bodies.

Nose

The cone’s CM, from the nose’s apex, is [43]

3 x = l , (4.10) cmcone 4 n where ln is the nose length. For other shapes, the nose’s CM is found from

Z ln 1 2 xcmnose = ln − πxr (x) dx , (4.11) vnose 0

27 R ln 2 where vnose = 0 πr (x) dx is the nose volume and r(x) its respective proﬁle given, for the remaining shapes, as (see Appendix A for deduction of the ogive’s proﬁle)

! d x2 rparabolic(x) = 1 − 2 , (4.12) 2 ln s ! l2 d2 l2 d r (x) = n + − x2 − n − , (4.13) ogive d 4 d 4 where d is the nose base diameter which is equal to the body tube diameter in case there is only one stage.

Body tube

The body tube, since it is cylindrically shaped, has its centroid located at half of its length. Hence, its CM is given as

l x = x + b , (4.14) cmb b 2 where xb is the distance from the nose’s apex to the beginning of the tube and lb its length.

Connector

Connectors consist of hollow cones sections. The CM of these parts, relative to the nose tip, is given by (see Appendix B)

! lc dcf + 2dca xcmc = xc + , (4.15) 3 dcf + dca

where xc is the distance of this part from the nose tip, lc is the connector’s length and dcf and dca are, respectively, the fore and aft diameters of the connector.

Fins

The ﬁns’ CM (assumed as ﬂat plates) is

xcmfin = xfin + ¯xfin , (4.16)

where xfin is the distance from the nose tip up to the root of the leading edge and ¯xfin is the fin’s centroid. Since the fins may exhibit several different shapes, the Barrowman Method [45], described and used in Section 4.4 to find the CP, simplifies their geometry to trapezoids with the root and tip edges parallel and the same or slightly less area than the original fin [45] (see Figure 4.2). With this geometry it is possible to get many shapes (including rectangles and triangles) by selecting different values for the fin’s dimensions. Therefore, dividing the generalized shape in two triangles (see Figure 4.2), ¯xfin can be found

28 b

x s cr

42 ct

Figure 4.2: Generalized ﬁn geometry and respective division to ﬁnd its centroid. from the centroids of these two parts as

A41 ¯x41 + A42 ¯x42 ¯xfin = , (4.17) A41 + A42

where A41 and A42 are the areas of the triangles and ¯x41 and ¯x42 their centroids. Considering x1, x2 and x3 as the coordinates of the vertexes of a triangle along the x-axis, its centroid’s x coordinate is given by

x + x + x ¯x = 1 2 3 . (4.18) 4 3

Then, from the generalized ﬁn geometry (see Figure 4.2), and placing a local reference frame at the point where the root chord and the leading edge meet, the centroids of the two triangles are:

s + c ¯x = r , (4.19) 41 3 c + 2s + c ¯x = r t . (4.20) 42 3

Motor

During the burnout and coasting phases, the motor is losing mass. Thus, recalling the motor constituents from Section 2.2, its CM (including the casing mass) is changing over time as

1 h i xcmm (t) = xm + xcmf mf + xcmp (t)mp(t) + xcmd (t)md(t) + xcme (t)me(t) , (4.21) mm(t)

where xm is the distance from the nose tip to the top of the motor, xcmf is the motor’s ﬁnal CM (in this case equal to half of its length) and xcmp , xcmd and xcme are the propellant, delay and ejection charge’s CM from the top of the motor, respectively. These are found from:

xcmp (t) = lg + le(t) + ld(t) + lp(t)/2 , for t ≤ tb , (4.22)

xcmd (t) = lg + le(t) + ld(t)/2 , for t ≤ tb + td , (4.23)

xcme (t) = lg + le(t)/2 , for t ≤ tb + td , (4.24)

29 where lg is a gap considered as the thickness of the clay cap plus the empty space at the top of the motor and lp, ld and le are the propellant, delay and ejection charge lengths, respectively. Since every motor’s casing is cylindrically shaped, with internal radius rintm , these lengths are found from

m = ρv = ρπr2 l , (4.25) intm which gives:

m (t) l (t) = p , (4.26) p ρ πr2 p intm

m (t) l (t) = d , (4.27) d ρ πr2 d intm

m (t) l (t) = e , (4.28) e ρ πr2 e intm

−3 −3 −3 where ρp is the propellant’s density (equal to 1750 kg m for black powder and 1520 kg m to 1750 kg m for composite propellant [44]) and ρd is the delay charge density. Since the ejection charge consists of black powder, its density is equal to 1750 kg m−3 for any of the two types of motors. The delay charge density may be estimated from (4.27) with ld and md evaluated at t = 0, being the ﬁrst predicted by

ld|t=0 ≈ lm − lnozzle − lp|t=0 − le|t=0 − lg , (4.29)

where lm is the motor’s length and lnozzle is the length of the nozzle (from the rear of the motor to the propellant).

If the motor is a core burning, its propellant’s length is constant and found from (4.26) with mp evaluated at t = 0 and r2 = r2 − r2 , where r and r are the initial external and internal intm extp intp extp intp propellant radius, respectively.

4.3 Moments of inertia

From Section 3.3.1, we only need to ﬁnd the transversal moment of inertia relative to the CM (I), and not the longitudinal moment of inertia, since the roll motion is not considered in the attitude of the rocket but just the pitch motion from the developed model of the angle of attack. The moment of inertia may be found from

n X I = Ii , (4.30) i=1 where Ii is the moment of inertia of a given component relative to the rocket’s CM found from the parallel axis theorem [43]

¯ 2 Ii = Ii + mi(xcmi − xcm) , (4.31)

30 where I¯i is the central moment of inertia of the respective component, which is known for simple solids of revolution or can be found by integration.

Nose

The moment of inertia of the nose cone relative to the nose tip is [43]

3 1 I = m r2 + l2 , (4.32) cone 5 n 4 n n where mn is the nose mass and rn its base radius. In order to have its moment of inertia relative to the rocket CM, we must apply (4.31) two times: ﬁrstly to determine its central moment of inertia and, secondly, taking this value, to ﬁnd the moment of inertia relative to the rocket CM. To other nose shapes, the moment of inertia relative to the base diameter axis is determined by [43]

Z ln 1 2 2 2 In = r (x) + x ρnπr (x) dx , (4.33) 0 4 where ρn is the nose density and r(x) is given by (4.12) or (4.13). Their moments of inertia relative to the rocket CM is then found by the same method described to the conical nose.

Body tube

The body tube is a hollow cylinder. Thus, its central moment of inertia of the body tube is [43]

1 h i I¯ = m 3(r2 + r2 ) + l2 , (4.34) b 12 b ext int b where rext and rint are the external and internal radius of the body, respectively, and mb is the body tube’s mass.

Connector

The connector’s transversal moment of inertia relative to the smaller diameter is (see Appendix B)

! 1 2 dcf + 3dca Ic = mclc , (4.35) 6 dcf + dca

where mc is the connector’s mass.

Fins

Since the ﬁns are considered as ﬂat plates made of an homogeneous material, their moments of inertia can be found from [43]

0 Ifin = ρf tf Ifin , (4.36)

0 where ρf is the fin’s density, tf is the fin’s thickness and Ifin is the area moment of inertia. Like in Section 4.2, dividing the fin in two triangular pieces and applying the parallel axis theorem to each one

31 0 of them, Ifin relative to the top of the ﬁn is determined by

I0 = I¯0 + A ¯x2 + I¯0 + A ¯x2 (4.37) fin 41 41 41 41 42 42 where the central area moments of inertia of the two triangular pieces, having in mind Figure 4.2, are given by [46]:

c3b − c2sb + s2c b I¯0 = r r r , (4.38) 41 36

c3b − c2(c + s − c )b + (c + s − c )2c b I¯0 = t t t r t r t . (4.39) 42 36

Motor

In order to ﬁnd the motor’s central moment of inertia, we must add the moments of inertia of each component relative to the motor’s CM applying (4.31), where the principal moments of inertia of the casing, propellant, delay and ejection charges are given from (4.34).

4.4 Center of pressure and CNα

Since its publication, the Barrowman Method provides a series of reliable equations that have been widely used to estimate the aerodynamic characteristics of model and high power rockets. This method makes several assumptions in order to ﬁnd those equations [45]:

1. The angle of attack is near zero; 2. Viscous forces are negligible; 3. The flow is steady, irrotational and subsonic; 4. The rocket is thin compared to its length; 5. The nose of the rocket comes smoothly to a point; 6. The rocket is an axially symmetric rigid body; 7. The fins are thin flat plates.

Although there are some restrictions, model and high power rockets meet these requirements specially during and shortly after lift-off which, not considering wind perturbations, is when the CP and CNα play an important role, according to Section 3.3.1. Also, since model rockets have, usually, a general geometry, the Barrowman Method divides the rocket in simple geometric parts (like in Section 4.2), finds its properties and recombine them to find the final results. The third assumption is invalid for sounding rockets due to the high velocities that they reach. In this case, and as the CP is used to determine the angle of attack, we consider that the rocket oscillations are already damped before reaching the transonic regime. At such high speeds the rocket has to fly without angle of attack in order to minimize the drag forces applied on its structure.

According to the Barrowman Method, for subsonic velocities, CNα depends only on the shape of the

32 rocket and is found by summing all the derivatives of the parts of the rocket as [45]

n X CNα = CNαi . (4.40) i=1

The CP is found from [45]

Pn i=1 CNαi xcpi xcp = , (4.41) Cnα

where xcpi is the CP location of each component of the rocket from the reference point. From these equations we get a constant CP, which is a satisfactory result since the rocket meets the ﬁrst and third assumptions of this method at least until it stops swinging.

Nose

For the most common noses geometries used in model rocketry, the Barrowman Method ﬁnds that [45]:

C = 2 , (4.42) Nαnose 2l x = n , (4.43) cpcone 3 l x = n . (4.44) cpparaboloid 2

xcpogive = 0.466ln , (4.45)

Body tube

Since the cylindrical body has constant cross sectional area, according to the Barrowman Method and its assumptions, it does not produce lift [45]. Thus, C = 0. nαbody

Connector

For connectors, from the Barrowman Method, we have [45]

" 2 2# dca dcf CN = 2 − , (4.46) αc d d where d is the rocket’s diameter at the base of the nose. If we have a reducer or boat-tail, then C Nαc comes out negative. The CP location is given by

  1 − dcf lc  dca  xcpc = xc + 1 + 2  . (4.47) 3 d  1 − cf  dca

Fins

The Barrowman Method only allows the rocket to have 3, 4 or 6 ﬁns [45]. Recalling the ﬁn geometry in

33 b

s cr

lc¯

Figure 4.3: Generalized ﬁn geometry.

Figure 4.3, this method founds that

b 2 4nkf ( d ) CN = , (4.48) αfin q 1 + 1 + ( 2lc¯ )2 cr +ct where n is the number of ﬁns and kf is the ﬁn interference factor, due to the presence of the body, given by:

rext kf = 1 + , for n = 3, 4 , (4.49) rext + b

5rext kf = 1 + , for n = 6 , (4.50) rext + b where rext is the external rocket radius between the ﬁns. Fin’s CP is found by [45]

s(cr + 2ct) 1 crct xcpfin = xfin + + cr + ct − . (4.51) 3(cr + ct) 6 cr + ct

4.5 Drag coeﬃcient

The drag force opposes the true speed direction and appears due to several effects of the flow around the rocket. As seen in (3.33), the drag coefficient contributes to the aerodynamic force and it may change throughout the flight. It can be estimated by adding all the drag coefficients of the external components of the rocket and, for each one of them, the drag may come from different sources: pressure drag, skin friction drag, base drag and wave drag. Pressure drag arises due to the distribution of normal forces on the components and the skin friction drag, on the other hand, is a tangential force generated by the viscosity of the flow that creates a boundary layer on the surface of the rocket [47]. Therefore, assuming a flight without angle of attack, the nose, connectors and fins generate these two kinds of drag and the body tubes of each stage only produce skin friction drag. For conical noses, the pressure drag coefficient variation is represented in Figure 4.4. Since the drag coefficient for cones describe, approximately, a linear variation with the half vertex angle (ε)

34 Figure 4.4: Pressure drag coeﬃcient of wedges, cones and similar shapes as a function of the half-vertex angle (ε) [47]. from 0 to 90°, the pressure drag coeﬃcient may be found from

CDp (ε) = 0.233 + 0.011ε . (4.52)

The total drag coeﬃcient for the nose and connector sections is

S S C = C b + C w ,, (4.53) Dnose/connect Dp S Df S

where CDf is the skin friction drag coeﬃcient (determined in Section 4.5.1), Sb is the nose base area and

Sw is the wetted area. Note that, since the connector sections describe a cone without the apex, Sb, in that case, must represent the difference between the aft and fore areas in order to discard the pressure drag already included in the nose coefficient. The fin’s drag coefficient, is given, for the two surfaces, as [48]

t S C = 2C 1 + 2 f w , (4.54) Dfins Df c¯ S wherec ¯ and tf are the fin’s average chord and thickness, respectively, and the wetted area, Sw, must take into account the two sides of the fin. Considering a flight with angle of attack, there is an extra source of drag created by the fins, the induced drag. It can be estimated by [48]

C2 S C = L f , (4.55) Dind πΛe S where Λ is the ﬁn aspect ratio, e ≤ 1 is the Oswald eﬃciency factor (e = 1 for elliptical wings), Sf is the

fin planform area and CL is the fin’s lift coefficient that, assuming the fins as flat plates at low angles of

35 attack, may be approximated by (3.37). Since the flow around the fuselage interacts with the flow passing through the fins, it is created an interference drag. This extra drag can be interpreted as coming from the air flowing on fins that are extended into the fuselage [48]. Then, the interference drag coefficient is measurable by

c r C = 2C r ext n , (4.56) Dint Df S where n is the number of ﬁns and cr is the ﬁn’s root chord. Due to the lower pressure at the rear of the rocket, the boundary layer separates from the surface and increases the drag. This fact originates the base drag which can be found by [47]

0.029 d 3 C = bt , (4.57) Dbase p CDbody d

where CDbody is the drag coefficient of the fore body (not counting with the fins) and db is the boat-tail aft diameter. If dbt = d, the rocket does not have a boat-tail and the base drag coefficient is maximum, as expected. Model rockets are provided with launch lugs that guide the rocket through the rods. They are an extra source of drag estimated as [48]

1.2S + 0.0045S C = ll wll , (4.58) Dll S

where Sll and Swll are, respectively, the sectional and wetted (inner and outer surfaces) areas of the launch lugs.

4.5.1 Skin friction drag

Skin friction drag changes according the regime of the flow. In turbulent flow, the skin friction drag is higher than in laminar flow (although turbulence, in some cases, is desirable in order to delay separation and decrease the pressure drag). As the air hits the rocket’s surface, the flow is laminar and rapidly turns to turbulent with increasing Reynolds number. Between this two regimes appears a laminar-turbulent transition when critical Reynolds is reached (Recr) and the flow becomes fully turbulent above transition

Reynolds (Retr). Reynolds number, representing the relative order of magnitude between inertial and viscous forces of the ﬂuid, is deﬁned as [49]

ρV L Re = , (4.59) L µ where L is the surface’s length which must be selected according the respective part of the rocket and µ is the dynamic viscosity of the air found from the empirical Sutherland’s law as [49]

T 3/2 µ = 1.458 × 10−6 . (4.60) T + 110.4

36 Since transition is hard to predict, a way to find the skin friction drag coefficient in a flat plate

(considering zero pressure gradient) is to assume Recr ≈ Retr, yielding [49]

Re h i C = C − tr C − C Df Dturb l Dturb x Dlam x ReL tr tr

−0.2 Retr h −0.2 −0.5i = 0.074ReL − 0.074Retr − 1.33Retr , (4.61) ReL

5 6 where Retr ≈ 3 × 10 to 10 . Equation (4.61) determines the turbulent drag coeﬃcient for the entire plate and replaces the turbulent contribution by the corresponding laminar value up to the point (xtr) where the transition occurs [49].

4.5.2 Compressibility eﬀects

Mach number is the dimensionless parameter [49]

V M = , (4.62) a where a is the speed of sound given as

a = pγRT , (4.63) where γ ≈ 1.4 for air is the ratio of specific heats at constant pressure and volume. From M > 0.3, the flow no longer can be assumed as incompressible since there are changes in density and temperature and shock waves begin to increase the drag. The Prandtl-Glauert rule is commonly used to relate incompressible and compressible coefficients (CD and CDc , respectively) for slender and planar bodies [50]. This rule is given as

CD CD = , (4.64) c p|1 − M2| that must be applied between 0.3 < M < 0.7 and 1.2 < M < 5. In transonic and low supersonic regimes (0.7 < M < 1.2), an interpolated function should be used to estimate the compressible drag coeﬃcient and avoid the singularity at M = 1. For a conical nose, the wave drag due to shock waves can be estimated by the empirical equation [51]

0.096 ε 1.69 C = 0.083 + for M > 1 , (4.65) Dw M2 10 where ε is again the half vertex angle in degrees. Equation (4.65) should match the subsonic drag coeﬃcient at M = 0.7 which is when the transonic eﬀects begin to appear.

37 There are also corrections for the skin friction coeﬃcient. These relations are [47]:

C = C , (4.66) Dlamc Dlam

C = C (1 − 0.12M2) , (4.67) Dturbc Dturb for subsonic (M < 1) and

CDlam CD = , (4.68) lamc (1 + 0.045M2)0.25

CDturb CD = , (4.69) turbc (1 + 0.15M2)0.58 for supersonic (M > 1) regimes.

4.5.3 Recovery device

After the apogee, or when it is desired, the recovery system is deployed and the main source of drag comes from the parachute or streamer. Since the respective drag coeﬃcient depends on the shape of the recovery device, Table 4.1 lists several ranges of values for common parachute designs. The tabulated

Shape dproj/dref CD Flat circular 0.67 – 0.7 0.75 – 0.80 Conical 0.7 0.75 – 0.90 Biconical 0.7 0.75 – 0.92 Triconical 0.7 0.80 – 0.96 Hemispherical 0.66 0.62 – 0.77 Annular 0.94 0.85 – 0.95 Cross 0.66 – 0.72 0.60 – 0.85

Table 4.1: Solid textile parachutes’ projected to reference diameter (dproj/dref ) and respective drag coefficient [52]. coefficients are associated to the reference area which is given as the total area of the canopy including the vent or other openings. To find the projected area when the parachute is inflated, are also listed the values for the ratio between the projected and reference diameters. If the recovery device is a streamer, its drag coefficient may be found from the Open Rocket docu- mentation that formulates the empirical function [15]

ρs + 25 ls + 1 CDstreamer = 0.034 , (4.70) 105 ls

3 where ρs is the streamer’s material density (in kg/m ) and ls is the streamer’s length (in m).

38 Chapter 5

Rocket trajectory simulator

The developed rocket trajectory simulator aims to:

determine the rocket trajectory and respective ﬂight data given known/estimated inputs; give the user ﬂexibility to change the default trajectory and rocket’s models; enable to simulate from micro rockets up to sounding rockets’ models; forecast the landing region from a Monte Carlo simulation; optimize the rocket characteristics.

This tool is developed in Mathematica® [53] since it is a powerful language in functional programming and to manipulate lists. Due to these characteristics, this language gives ﬂexibility to implement the features we desire: structure of lists for the rocket, listable databases and modular programming. Al- though Mathematica® becomes slower as it is an interpreted language, the mentioned advantages surpass this handicap. Besides the following sections aim to explain the methods applied to develop the simulator, their goal is also to expose some of the advantages Mathematica® oﬀers while programming and manipulating data.

5.1 Simulator description

The simulator’s main function is RocketTrajectory (see Figure 5.1) which integrates the trajectory equations and receives all the required inputs. This function calls the RocketProperties function that computes every parameter of the rocket over time. These properties are determined by combining the data of the several stages regarding the developed structure that gathers the data of the rocket and considering the instants of the ignitions and ejections (see Section 5.2).

The most relevant inputs are the rocket structure, the atmospheric density, the wind and the CD models that, since they are external parameters, the user is allowed to deﬁne or change them according to his needs. The outputs from the simulation are three functions giving the position (one for each Cartesian coordinate) plus a function giving the angle of attack. These are computed in three steps: one for the constrained trajectory due to the guiding rod, another for the climbing phase and the last one for

39 Inputs

RocketT rajectory RocketP roperties

Outputs

Figure 5.1: Simulator’s algorithm. the recovery trajectory. The angle of attack starts to be computed only in the second phase, since it is when the rocket suffers the first perturbation from the wind. After this process, the three parts of the trajectory are merged together and, having these time dependent functions, it is possible to determine its derivatives and get other flight data such as total velocity, total acceleration, Mach and flight path angle. If there are windy conditions, the rocket needs to speed up until it becomes stable. Therefore, a guiding rod is required to restrain the trajectory during the launch in the desired direction. The rod’s length is needed as input in order to determine the instant after which the rocket’s trajectory becomes unrestricted and the model of the angle of attack starts to be computed. The user must be careful in selecting the rod’s length since he should know the required length of the rod that allows a given thrust to overcome the wind conditions and avoid an abrupt increase of the angle of attack over time. Nevertheless, it is also recommended that the rocket’s static margin is previously determined to assure it stays always positive (stable rocket). During the simulation the rocket is treated as a particle since there are only three degrees of freedom (x, y and z) and the model developed for the attitude is only used to estimate the loss of thrust along the true speed direction. Therefore, since the rocket’s nose is always a steady point in the rocket’s reference frame (unlike the CM), it is used to give the rocket’s position over time and, consequently, the initial nose distance to the ground is required as input. The rockets used in the simulator must always have three or more fins, boosters or cluster motors because we did not consider the case when the rocket has two components of each type. In this situation, the rocket acquires two different transversal inertias which is incompatible with the model developed for the angle of attack that assumes the rocket has only one transversal moment of inertia. As default, the recovery system is deployed at the apogee. However, there is an optional argument that orders the simulator to start to compute the recovery trajectory at the ejection instant determined by the data within the rocket structure.

40 5.2 Rocket assembly

The list concerning the model of the rocket has a speciﬁc structure in order to easily access its several data, to be quickly processed by the RocketProperties function and to allow any number of stages. Every parameter inside this structure may be deﬁned directly by the user (from measurements or other available data) and some of them can be estimated from the functions already included in the simulator that apply the methods presented in Chapter 4. Figure 5.2 shows the scheme in which the assembling of the rocket’s data is structured. The stages

Rocket Assembly List Boosters List Number Position Thrust Data List Structure Data List Delay Booster Stages List Stage 1 List Stage 2 List

Stage i List Motor List Body List Fins List Connector/Nose List Delays List Miscellany List Motor Cluster List Number Motors Motor List Recovery List Reference Area Drag Coefficient

Figure 5.2: Rocket assembly list structure. list contains as many lists as the number of stages. Each one of these lists, having in mind the rocket’s structure from Chapter 4, holds other lists for the rocket’s components plus a list to indicate the time intervals between events (explained later). The motor list is described in Figure 5.3. Its first entry is the distance between the top of the motor and the top of the body tube. Likewise, in the thrust data list, the first element is the distance between the top of the combustion chamber and the top of the motor. This list contains time dependent functions since the propellant is burning over time. The moments of inertia should also be given relative to the top of the combustion chamber in order to have a steady reference. Otherwise, considering the propellant’s moment of inertia, its reference would change over time. Although the simulator does not require the longitudinal moment of inertia, we keep an entry for this property because an improved version may need it in the future. The body list is defined as depicted in Figure 5.4.

41 Motor List Position Thrust Data List Position Burnout Isp Thrust (t) Mass (t) CG (t) Inertia List Transv. Inertia (t) Long. Inertia (t) Structure Data List Mass CG Inertia List Transv. Inertia Long. Inertia Diameter

Figure 5.3: Motor list structure.

Body List

Mass CG Inertia List CP CNα Length Diameter

Transv. Inertia Long. Inertia

Figure 5.4: Body list structure.

In the fins list, showed in Figure 5.5, there is an additional CM in order to find the distance of the fin’s CM from the rocket’s CM, which is required for the parallel axis theorem. Also, the CP and the

CNα in this list are assumed to be the values resulting from the ensemble of ﬁns in each stage, as it was determined in Section 4.4.

Fins List Number Position Fins Data List Mass CG List Axial CG Radial CG Inertia List Transv. Inertia Long. Inertia CP

CNα Root chord Tip chord Span Thickness

Figure 5.5: Fins list structure.

42 The connector (or nose) list is structured as depicted in Figure 5.6. In the delays list, deﬁned in Figure 5.7, the delay of the body represents the time when the respective stage’s body tube is discarded after the burnout; the delay of the connector gives the instant when this component is discarded after the previous delay (if there is no connector, this delay must be zero); the third delay gives the time interval until the next stage is ignited. When the delays list belongs to the nose, the sum of these three delays represent the coasting time until the recovery system is deployed if the ejection charge option is chosen in the RocketTrajectory function.

Connector/Nose List

Mass CG Inertia List CP CNα Length

Transv. Inertia Long. Inertia

Figure 5.6: Connector or nose list structure.

Since the boosters are external components, the structure data list, within the boosters list (see Figure 5.2), contains additional elements: the lengths of the cylindrical tube and nose structures, the CP and the CNα . The last two must give the respective property for the complete structure of the booster. All the boosters must also be identical and, when adding them, the user should pay attention if the new rocket’s CM does not surpass the CP. The delay of the boosters represents the time between their burnout and the instant they are discarded, which, at most, should be the instant when the ﬁrst stage’s body tube is also discarded. If the rocket does not have boosters, their number must be zero and the RocketProperties function ignores the other entries within the boosters list.

Delays List

Delay Body Delay Connector/Nose Delay To Next Ignition

Figure 5.7: Delays list structure.

Inside the miscellany list (see Figure 5.2), the cluster list allows to deﬁne a cluster of identical motors around the ﬁrst stage’s motor and equally spaced between them. The recovery list contains the properties of the recovery device (reference area and CD) to be used during the descending phase of the simulation.

5.3 Atmospheric model

The atmosphere is an important element influencing the trajectory since it is the fluid in which rockets fly. As the atmospheric properties change throughout the flight, they must be known at every point of the rocket trajectory. To simulate trajectories of model rockets, as these reach low apogees, it can be important to take into account some atmospheric variations due to the location of the launch and due to the weather conditions. As this work includes simulation of sounding rockets trajectories, it is also required to know the atmospheric properties until a high altitude. One of the existing atmospheric models is the Committee on Space Research (COSPAR) International

43 Reference Atmosphere 1986 (CIRA-86). This is an empirical model ranging from 0 km to 2000 km height [54]. Based on this model, after its publication, occurred several developments. One set of the developed tables consists of the monthly zonal mean values of temperature from 0 km to 120 km height with almost global coverage (80°N to 80°S) [54]. These tables also show pressure data but it is only changing with altitude. Due to the altitude range, latitude coverage and monthly variation, this is the implemented model to approximate the atmospheric properties up to the lower thermosphere (high enough to small sounding rockets) in every populated places on Earth all year long. Corrections are applied in this model as CIRA-86 gives wrong temperatures (999.9 K) at the two lowest altitude layers (0.1 km and 2.2 km) for each month from 70°S to 80°S. Therefore, interpolating all data above 2.2 km in the corresponding latitudes, these values are replaced by extrapolated temperatures at the respective altitudes. Also, since pressure and temperature at the surface are not provided, the corrected model is improved by including these properties found by extrapolation at 0 km height. After we get this upgraded version of CIRA-86, its data is rearranged in lists structured as depicted in Figure 5.8. With this structure, pressure and temperature are easily interpolated in altitude, month and latitude and

Cira-86 Upgraded List Data 1 List Data 2 List

Data i List Variables List Altitude Month Latitude Pressure Temperature

Figure 5.8: CIRA-86 upgraded version structure. the obtained three variable functions are included in the atmospheric database. Using these functions, the density is found from (3.3) assuming the atmosphere is an ideal gas (as gases often do approach ideal gas behavior [32]). This model is stored in a database in order to be easily explored and get access to its properties and details. Also, this database allows to deﬁne other atmospheric models and new functions representing extra atmospheric parameters. The atmospheric database is a listable function (i.e., applies the function to each element in a list), named AtmosphereData, thus it has not a speciﬁc structure. We can access its data by providing the name of the model and the required properties. For example,

AtmosphereData[ ‘ ‘Cira86 ’ ’ , { ‘‘Temperature’’, ‘‘Density ’ ’ } ], gives a list with the temperature and density functions for the CIRA-86 model. If another model is stored, we just have to change the name of the model to get these properties. Also, if we want to know the units of each property (or another information previously saved), we must type

AtmosphereData[ ‘ ‘Cira86 ’ ’ , { ‘‘Temperature’’, ‘‘Density’’ } , ‘‘Units’’].

44 At the launch site, the local conditions must be taken into account. Therefore, the simulator contains functions that receive the atmospheric models and adjust them in order to match the measured pressure and temperature at the corresponding altitude besides tending exponentially (at a chosen rate) to the original model at higher altitudes. Taking as example the correction for the temperature, this function is given by

z − z T (z) = T (z) + T − T (z ) exp l for z ≥ z , (5.1) c m zl m l k l

where Tc is the corrected temperature, Tm is the temperature from the atmospheric model, Tzl is the measured temperature at the launch altitude (zl) and k is a constant to deﬁne the exponential rate. The CIRA-86 temperature (at 40°N in June) and the corresponding correction for a local temperature of 300 K at 100 m above sea level with k = 4 is depicted in Figure 5.9. After getting the adjusted pressure and temperature functions, we use (3.3) to get the density also dependent on the local weather conditions.

Temperature(K) 300 CIRA-86 CIRA-86 with local conditions 280

260

240

220 Altitude(km) 5 10 15 20 25

Figure 5.9: Temperature proﬁles for the CIRA-86 model.

5.4 Wind model

The developed wind model is a combination between the functions for the ABL and the wind data provided in the CIRA-86 model at higher altitudes. Therefore, to model the ABL according to Section 3.1.1, there is required the local wind speed, its respective measurement altitude, the terrain’s speciﬁcation

(given by the surface roughness length – z0), the launch latitude and the constants κ and c. Since the RocketTrajectory function requires the wind model to be deﬁned by its three wind speed components, the wind direction (downwind angle relative to north in the clockwise direction) is also necessary. This model does not consider vertical wind. The user must be careful when increasing the roughness length because it will extend the zero wind region up to a higher height. If this region is greater or equal than the height of the guiding rods, the simulator gives a zero angle of attack over time since there is no wind at the instant supposed to determine the initial condition of the angle of attack. There are optional arguments in the wind model function in order to determine the stability of the ABL. The default option is a neutral atmosphere. If the atmospheric stability is known, the user must indicate the respective conditions (stable or unstable) and give an estimated guess for the Obukhov

45 length (L) based on Table 3.2. Otherwise, giving qualitative information about the ABL, the wind model function deduces the atmospheric stability (as described in the Section 3.1). This is accomplished through other optional arguments: time of day (day/night), weather (clear/cloudy) and temperature gradient (positive/negative). In this case, the Obukhov length is also required and its possible values must comply with the stability found by the wind model. In Figures 5.10, 5.11 and 5.12 are represented the wind proﬁles for a neutral, stable and unstable

Wind speed(m/s) 6 5 4 3 2 1 Altitude(m) 200 400 600 800 1000 1200

Figure 5.10: Neutral wind speed proﬁle for the ABL.

Wind speed(m/s)

Altitude(m) 50 100 150 200 250 300

Figure 5.11: Stable wind speed proﬁle for the ABL.

Wind speed(m/s) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 Altitude(m) 200 400 600 800 1000

Figure 5.12: Unstable wind speed proﬁle for the ABL.

ABL, respectively. These profiles were modeled considering a local wind speed of 2 m s−1 at 2 m height, in a high grass terrain (z0 = 7 cm) at the latitude of 37°N with κ = 0.4 and c = 0.4. Also, L = 125 m for the stable profile and L = −150 m for the unstable. See Section 6.4.3 in order to observe the influence of L in the wind profiles.

46 Above the ABL, the wind speed remains constant with height until a certain altitude (defined by the user) at which the wind starts to approximate the CIRA-86 values at a chosen rate. Since the CIRA-86 model gives westerly and easterly (negative values) winds, the direction of the wind from the ABL also starts to match the west-east direction. This fit between the two models is accomplished by a vector sum in which the intensity of the ABL vector begins to decrease above the user-defined altitude and the CIRA-86 vector’s intensity increases from zero until its original value.

5.4.1 Wind gusts

During the launch the wind may increase suddenly with height, leading the rocket to suffer some wind gusts. The effect of these perturbations is implemented in the RocketTrajectory function by changing the solution from the equation of the angle of attack to a new one with its initial angle condition determined by the rocket and wind gust’s velocities at the perturbation’s instant. Since the angle of attack in the simulator only affects the thrust force, the effect of the wind gusts can only change the trajectory if the new solution gives a non zero angle during any burning. The wind gusts’ input is defined by a list containing as many lists as the number of gusts (see Figure 5.13). Each one of these lists contains the perturbation’s instant (after the ignition) and the gust

Wind Gusts List

Gust 1 List Gust 2 List Gust i List {Instant 1, Gust 1 vector} {Instant 2, Gust 2 vector} {Instant i, Gust i vector}

Figure 5.13: Wind gusts list structure. speed vector. The eﬀect of a wind gust in the angle of attack at 2 s after the ignition is depicted in Figure 5.14. This perturbation has an intensity of 25 m s−1 along the wind direction.

α(º)

t(s) 1 2 3 4 5 6 7

-5

-10

Figure 5.14: Angle of attack until the apogee with a wind gust perturbation.

5.5 Motors data

Some model rocket motors data are stored, by default, in another database. As example, and as described for the atmospheric database, the motor’s properties may be accessed by

47 MotorData[ ‘ ‘C6−0 ’ ’ , { ‘‘Thrust’’,‘‘Isp’’,‘‘Mass’’,‘‘CG’’,‘‘Inertia ’ ’ } ].

Again, other motors can be stored in order to automatically change all the motor’s properties within the structure of the rocket only by specifying the name of the motor. These new motors may be defined by any mathematical model the user may require, which gives freedom to change a given property and observe its influence in the simulation. For all the model rocket motors already in the database, their properties are determined considering an end burning combustion, a delay charge (for the ones which possess a delay time) and an ejection charge. The thrust profiles are obtained by an interpolated function fitting the points given in the data sheets available on-line at www.thrustcurve.org. As an example, Figure 5.15 represents the obtained

Thrust(N) 30

t(s) 0.5 1.0 1.5 2.0

Figure 5.15: Thrust proﬁle for the D12 motor. thrust curve of the motor D12. Although the main motors’ properties are found from the thrust proﬁle and other info provided in the data sheets by the methods described in Chapter 4, they depend on other data which can only be determined by measurements taken from the motors’ dimensions. Therefore, this data must be inserted in the database in order to automatically compute the properties of the motors.

5.6 Drag coeﬃcient model

The simulator provides a drag coefficient model to be used in the RocketTrajectory function. If desired, another model can be developed which may depend on time, the coordinates variables and their derivatives. It is also possible to define a constant drag coefficient. The developed model requires the list containing the structure of the rocket to determine the drag coefficients for each one of the external components and get the total CD, depending on Mach, Rex and time, in respect to the rocket’s reference area. The coefficients for each component are computed by calling previously defined functions that apply the equations from Section 4.5. Using this drag coefficient model, we are considering that all the components contribute to the skin friction drag but the pressure drag comes only from the nose, connectors and fins (due to the normal surface facing the flow). Also, in this model the rocket’s nose is the only wave drag source. To compute the wave drag coefficient from (4.65), and since it is not defined in the transonic regime bellow M = 1,

48 a fitted function is determined to match the incompressible CD at M = 0.7 and the wave plus the incompressible CD at M = 1. In order to match smoothly the other functions, this fit is performed by a third degree polynom. The final drag coefficient function for the conical nose, including the wave drag, is represented in Figure 5.16.

CD 0.38 0.37 0.36 0.35 0.34 0.33 0.32

Mach 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure 5.16: Conical nose drag coeﬃcient with Mach.

5.7 Validation of the simulator

The developed tool requires validation in order to assure the simulator meets the speciﬁcations and correctly computes the expected output. This may be performed by comparing our results with a known analytic solution from a speciﬁc launch. The altitude over time (until the burnout) from a vertical rocket launch, considering the simplest case without atmosphere (no drag and no wind) and constant thrust, can be found by [39]

ve m0 − mt˙ 1 2 H(t) = (m0 − mt˙ ) ln +mt ˙ − g0t , (5.2) m˙ m0 2 where m0 is the initial mass of the rocket,m ˙ is the propellant burning rate and ve = g0Isp is the exhaust velocity. Equation (5.2) also ignores the effects of the Earth’s rotation and assumes a constant acceleration of gravity. Therefore, using a sounding rocket model, supplied with a motor providing 13 kN of thrust during 32 s (burnout), and specifying the same assumptions in the simulator, we can compare the analytical and numerical solutions from the relative error shown in Figure 5.17. It can be observed that the error presents values in the order of magnitude of 10−6%. Although this error gives reasonable results under the mentioned conditions, more tests should be performed in order to increase the confidence in the solutions from the developed simulator. To test the drag effect, we can make a terminal velocity test for a given rocket in free fall after the apogee. From an equilibrium between the gravitational force and the drag force as

1 mg = ρV 2SC , (5.3) 2 d

49 Relative Error (%)

3.× 10 -6

2.× 10 -6

1.× 10 -6

t(s) 5 10 15 20 25 30

Figure 5.17: Altitude’s relative error between analytical and numerical solution until the burnout. the terminal velocity can be found by

r 2mg V = . (5.4) ρSCd

As an example, from (5.4), we conclude that if the rocket loses mass, it must reach a new velocity. Therefore, we can also verify if the simulator is computing this quasi-static process. Testing a vertical launch under the mentioned forces, for a density of 1.225 kg m−3 (equal as the density at the launch site altitude used in the atmospheric model) and for a model rocket with a drag coeﬃcient of 0.6, a reference area of 4.9 cm2 (2.5 cm in diameter) and a mass of 59.7 g, we get, from (5.4), 57.0 m s−1 for the terminal velocity. Decreasing 30 g in the rocket’s mass, in order to verify the simulator’s behavior, the new terminal velocity comes 40.2 m s−1. In Figure 5.18 it is represented the rocket’s free falling velocity from the simulator, 10 s after the lift oﬀ, under the mentioned conditions. As we can see, the rocket tends asymptotically to the terminal velocity of about −57 m s−1 and, after losing 30 g, it tends to a new terminal velocity of about −40 m s−1. These velocities are consistent with the ones deduced in (5.4) and, as expected, the velocity tends asymptotically to each one of the equilibrium states.

Velocity(m/s)

-10

-20

-30

-40

-50 t(s) 15 20 25 30 35 40

Figure 5.18: Descent velocity from a vertical launch in which the rocket loses mass at 25 s.

In order to test non vertical launches, we may compare them with the solution from a projectile

50 launch. The altitude over time for this type of motion with no drag, no wind and no thrust, is given by

1 z(t) = h + v t − g t2 , (5.5) 0 h0 2 0

where h0 is the initial vertical distance from the ground and vh0 is the initial vertical velocity. Assuming −1 the projectile is launched from the ground (h0 = 0 m) with an initial total velocity of 30 m s and 70° of elevation, we get the relative error between the analytical and numerical solutions represented in Figure 5.19. In this test the input concerning the rod’s length is small enough so we can ignore the errors

Relative Error (%)

0.00020

0.00015

0.00010

0.00005

t(s) 1 2 3 4 5

Figure 5.19: Relative error between analytical and numerical solutions until landing from a projectile launch with 70° of elevation, constant mass, no drag and no wind. due to the constrained trajectory in the rod. From the values of the relative error, in Figure 5.19, we conclude the simulator behaves as expected under these conditions. Another verification can be made from Figure 5.20 which represents the flight path angle during the projectile launch. As expected, the projectile lifts off and lands with a flight path of 70° from the horizontal reference since its trajectory is parabolic.

Flight Path(º)

60 40 20 t(s) 1 2 3 4 5 -20 -40 -60

Figure 5.20: Flight path angle from the projectile trajectory with the launch angle (elevation) of 70°.

The results from the tests presented in this section seem to indicate that the simulator behaves as expected taking into account the respective assumptions for each example. It is foreseen an increase in those errors when these tests are performed under the full conditions with which the tool was developed. In that case, although the errors increase due to the variation in the acceleration of gravity with altitude

51 and due the eﬀects of the Earth’s rotation, the numerical solutions should give better approximations to real-case trajectories. Other tests that can provide more information about the performance of our tool are real rocket launches. These were not considered due to the extensive work they demand. Nevertheless, instrumenting a rocket or taking measurements from the ground to estimate the trajectory through triangulation gives precious informations that can be compared with the predictions from our simulation tool. Furthermore, it helps us to deduce how well we modeled other parameters such as the wind and drag coeﬃcient.

52 Chapter 6

Simulation tests and results

Two types of rockets are used to exemplify the results from the developed simulator: a model rocket and a sounding rocket. The model rocket launch is the most suitable to observe the eﬀect of the wind and the ABL on the trajectory since these rockets reach lower velocities and apogees. On the other hand, due to the wide range of rockets handled by the simulator and to extend the limits of the previous launch, we also present the results from a sounding rocket launch.

6.1 Rockets description

6.1.1 Model rocket speciﬁcation

This model rocket has two stages wherein the first is based upon a default rocket example provided in OpenRocket. The rest of the components are defined in order to make the rocket properly modeled. The second stage is propelled by a C6-0 motor and the first by a D12-0 motor, which is discarded at the first burnout and the connector is discarded a half second later. Another half second later starts the second ignition and after the next burnout the rocket coasts until the apogee where the recovery system is deployed. For each stage, excluding the connector and the nose, Table 6.1 presents their respective data. The connector has a length of 2.5 cm and the nose is an ogive with 10 cm long. In Table 6.2 are defined the fin’s dimensions for the two stages.

1st stage (with D12-0 motor) 2nd stage (with C6-0 motor) Average thrust (N) 10.21 4.74 Burnout (sec) 1.65 1.86 Gross mass (g) 55.62 41.66 Dry mass (g) 18.02 30.86 Length (cm) 8 30 Diameter (cm) 4 2.5 Number of ﬁns 3 3

Table 6.1: Model rocket stages’ data (without connector and nose) [55] [56].

53 1st stage 2nd stage Number of ﬁns 3 3 Root chord (cm) 8 5 Tip chord (cm) 8 5 Span (cm) 3 3.4 Sweep angle (°) 45 39.8

Table 6.2: Model rocket ﬁns’ data.

The parachute is deployed at the apogee and is taken as hemispherical shaped. Thus, from Table 4.1, its drag coeﬃcient is chosen to be 0.695 as it is the center of the respective range. To estimate the parachute’s reference area (the total surface area of the canopy), we choose a terminal velocity of 5 m s−1 since it is below the typical safety landing speed [57]. Therefore, from (5.4), again with a density of 1.225 kg m−3, we get 406.1 cm2 (or 22.7 cm of diameter), approximately. Table 6.3 presents the main properties of the model rocket to be used in the ﬁrst launch.

Model rocket property Value Gross mass (g) 113.87 Mass after second burnout (g) 44.06 Initial length (cm) 50.5 Length at second ignition (cm) 40 First stage’s structure discharge (after launch) (s) 1.65 Connector’s discharge (after launch) (s) 2.15 Second stage ignition (after launch) (s) 2.65

Parachute’s CD 0.695 Parachute’s canopy diameter (cm) 22.7

Table 6.3: Model rocket’s main characteristics.

6.1.2 Sounding rocket speciﬁcation

The second launch is based upon a two-stage solid propellant rocket used in the German MiniTEXUS sounding rocket program. This rocket has a Nike ﬁrst stage and an Orion second stage which ignites 9 s after the liftoﬀ [58]. Table 6.4 gives the data about each stage (excluding the connector and nose). The

1st stage (Nike motor) 2nd stage (Orion motor) Average thrust (kN) 217 13 Burnout (sec) 3.2 32 Gross mass (kg) 599 400 Dry mass (kg) 256 111 Length (m) 3.6 1.8 Diameter (m) 0.42 0.35

Table 6.4: Nike and Orion stages’ data (without connector and nose) [58] [59] [60]. connector is modeled in order to ﬁt each stage’s structure, with the height of 30 cm, and is assumed to be discarded at half of the delay between the two ignitions. The ﬁns in each stage are trapezoidal shaped

54 and the second stage’s ﬁns are half the size of the ones in the ﬁrst stage. Their dimensions are given in Table 6.5. The nose is an ogive with a length of 1.8 m and a mass of 160 kg, which is under the payload limits for the Nike-Orion rocket [58].

1st stage 2nd stage Number of ﬁns [58] 3 4 Root chord (cm) 100 50 Tip chord (cm) 40 20 Span (cm) 80 40 Sweep angle (°) 20.56 20.56

Table 6.5: Sounding rocket ﬁns’ data.

This rocket may use a ﬂat circular parachute with 7.3 m of diameter [61]. Hence, again from Table 4.1, we select a drag coeﬃcient of 0.775, which is the central value of the respective interval. The parachute is deployed with a maximum dynamic pressure of 12.0 kN m−2, approximately [61]. After assembling all the components, the essential data of the sounding rocket is given in Table 6.6.

Sounding rocket property Value Gross mass (kg) 1171.5 Mass after second burnout (kg) 271 Initial length (m) 7.5 Length at second ignition (m) 3.6 First stage’s structure discharge (after launch) (s) 4.65 Connector’s discharge (after launch) (s) 6.1 Second stage ignition (after launch) (s) 9

Parachute’s CD 0.775 Parachute’s canopy diameter (m) 7.3

Table 6.6: Sounding rocket’s main characteristics.

6.2 Rockets characteristics

6.2.1 Model rocket

Taking the model rocket described in Section 6.1.1, we get, from the RocketProperties function, the thrust, mass, CM, CP, transversal inertia and reference area over time presented in Figure 6.1. Due to the first discontinuity shown in Figures 6.1b, 6.1c, 6.1d and 6.1e, we can see that the first stage’s structure (not considering the connector) is discarded at the first burnout. Also in these figures, during the coasting phase (lasting 1 s), we see the connector’s discharge represented by the second discontinuity. Except from these sharp changes, the rocket maintains its properties during coasting, leading the mentioned figures to match with Figure 6.1a when there is no thrust between the burnings. The reference area is defined as the largest cross section of the rocket (without considering boosters, for other cases) at each instant. This definition agrees with Figure 6.1f, where it shows the rocket’s reference area changing nearly 2 s after the launch, which is the instant when the connector is discarded

55 and the rocket loses its largest sectional area. Comparing Figures 6.1c and 6.1d, we conclude this rocket is always stable since the CM, instead of the CP, is closer to the nose all the time. Hence, the rocket is well assembled and able to be used in a trajectory simulation.

Thrust(N) Mass(g) 30 110 25 100 20 90

15 80 70 10 60 5 50 t(s) t(s) 1 2 3 4 5 1 2 3 4 5 (a) Rocket’s thrust. (b) Rocket’s mass.

CM(cm) CP(cm) 40 40

35 35

30 30

25 25 t(s) t(s) 0 1 2 3 4 5 0 1 2 3 4 5 (c) Rocket’s CM relative to the nose tip (reference). (d) Rocket’s CP relative to the nose tip (reference).

Ref. Area(cm 2) Mom. Inertia(g.m 3)

4.0 10

3.5 8

3.0 6

t(s) t(s) 1 2 3 4 5 1 2 3 4 5 (e) Rocket’s transv. mom. of inertia relative to the CM. (f) Rocket’s reference area.

Figure 6.1: Model rocket properties’ functions.

56 6.2.2 Sounding rocket

For the sounding rocket described in Section 6.1.2, its properties are represented in Figure 6.2. During the burnings, in Figure 6.2b, the rocket decreases its mass linearly since its thrust profile is constant for each motor (as we only know the motors’ average thrust, see Figure 6.2a) and, consequently, their burning rates as well. In Figures 6.2b and 6.2e, the discontinuities representing the first stage and the connector’s discard match with the instants defined in Table 6.6, as well the reference area, in Figure 6.2b, that changes only when the connector is discarded. From Figures 6.2c and 6.2d, we deduce the CM is always ahead from the CP and, therefore, this rocket is also stable.

Thrust(kN) Mass(kg) 1200 200 1000 150 800 100 600 50 400

t(s) t(s) 10 20 30 40 10 20 30 40 (a) Rocket’s thrust. (b) Rocket’s mass.

CM(m) CP(m) 5.5 5.5 5.0 5.0 4.5 4.5 4.0 4.0 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 t(s) t(s) 0 10 20 30 40 0 10 20 30 40 (c) Rocket’s CM relative to the nose tip (reference). (d) Rocket’s CP relative to the nose tip (reference).

2 Mom. Inertia(kg.m 3) Ref. Area(dm ) 3500 14 3000 13 2500 2000 12 1500 11 1000 500 10 t(s) t(s) 10 20 30 40 10 20 30 40 (e) Rocket’s transv. mom. of inertia relative to the CM. (f) Rocket’s reference area.

Figure 6.2: Sounding rocket properties’ functions.

57 6.3 Trajectory simulations conditions

To run the simulations we need to set the flight conditions for both launches. Besides the rocket list, the other trajectory inputs that differ for each launch are the initial nose position and the guiding rod/rail’s length. Note that the rod’s length is the distance the rocket is traveling constrained in the launch direction, not its full length from the bottom of the launch platform in a real case scenario. Taking into account the dimensions of the rockets, the defined initial nose position for the model rocket launch is 70 cm from the ground and its guiding rod is considered to be 50 cm long. For the sounding rocket launch, the initial nose position is defined as 9 m from the ground and its guiding rail is considered as 7 m long. The launch site, launch direction and atmospheric conditions (density, temperature and wind model) are the same for both simulations. Since we opt that these hypothetical launches take place around Lisbon, the defined geodetic coordinates are 38.72°N, −9.14°W and 53 m above the reference ellipsoid [53]. Both rockets will be launched vertically in order to be easily observed the deviation from the launch direction. Nevertheless, we also perform another model rocket launch in which the rocket is launched with 75° of elevation and 90° of azimuth (eastward). The density, temperature (needed for the drag coefficient model) and wind are defined by the developed models described in Sections 5.3 and 5.4. The local pressure and temperature inputs correspond to the extrapolated values given from the CIRA-86 profiles at the launch site and considering the launch takes place in mid-April (1018.54 mbar and 14.9 ◦C, respectively). To model the wind profile we assume a neutral atmosphere with a constant wind speed above the ABL, in a high grass terrain, and blowing northeastward (profile identical to the one represented in Figure 5.10). The required local wind speed is defined taking into account the semi-empirical Beaufort scale, which ranks the wind speed basing on its effects over the sea and the land surfaces [62]. This scale assigns to the wind the class gentle breeze when leaves and small twigs move or light flags extend, limiting its speed between 3.4 m s−1 to 5.2 m s−1 at 6 m from the ground. For a slower and stronger wind speed, the adjacent classes are, respectively, slight breeze and moderate breeze. Foreseeing the gentle breeze is the class within the most of the launches may take place, we define a local wind speed of 4.3 m s−1 (value at the center of the class), blowing northeastward, at 6 m above the ground. Defining a high grass surface, we get, from Table 3.1, z0 = 7 cm as it is the center value within the respective range. The previous local conditions are resumed in Table 6.7. Although the wind direction is considered constant for the vertical launches, a wind direction variation with altitude is

Local condition Value Temperature (◦C) 14.9 Pressure (mbar) 1018.54 Wind speed at 6 m from the ground (m s−1) 4.3 Downwind (from north, clockwise direction) (°) 45 (northeastward)

Surface type – z0 (cm) 7 (high grass)

Table 6.7: Launch site conditions.

58 deﬁned for the non vertical model rocket launch. In this case the downwind is assumed as northeastward at the ground, changing until the apogee where it takes a northward direction. For every simulation the drag coeﬃcient is determined by the developed model explained in Section 5.6, which uses the same density, temperature and wind models from the respective trajectory to determine the Mach and the Rex.

6.4 Results from the trajectory simulations

6.4.1 Model rocket launch

The obtained vertical model rocket launch trajectory is represented in Figure 6.3. In this trajectory the

Up(m)

500

400

300

200

100

-400 -200 0 200 Ground Track(m) (b) Trajectory proﬁle. (a) 3D trajectory.

Figure 6.3: Model rocket trajectory with a vertical launch and a neutral ABL; Parachute ejection at the apogee. weathercock effect is observable. Since the wind blows northeastward, the rocket climbs in the opposite direction (southwestward) to follow the true air speed. Although this effect increases smoothly with altitude due to the ABL, we can see a slightly discontinuity in the upward flight path around 200 m of altitude. This happens because the weathercock is enhanced during the coasting time, between the first burnout and the second ignition, as the rocket decreases its velocity. During the recovery phase, the rocket slowly descends toward the downwind direction. Hence, it surpasses the launch site and lands 380 m north and 382 m east away from that place (coordinates taken from the simulator). The influence of the ABL is also observable during the recovery phase in which the trajectory describes a smooth curvature since the wind speed is decreasing as the rocket falls. If the wind was constant with altitude, the recovery trajectory would describe a linear path. In Figure 6.4 it is represented the flight data from this launch. It is deduced, from Figure 6.4a, that the rocket reaches the apogee at about 560 m of altitude and lands 120 s after the lift off. Observing when the vertical velocity turns negative, in Figure 6.4b, it is concluded the apogee is reached at 9.5 s,

59 Altitude(m) Velocity(m/s) 120 500 100

400 80

300 60

200 40

100 20

t(s) t(s) 20 40 60 80 100 120 2 4 6 8 10 12 14 (a) Proﬁle of altitude vs time. (b) Proﬁle of vertical velocity vs time.

Mach Accel.(/g) 0.4 20 0.3 10

0.2 t(s) 2 4 6 8 10 12

-10 0.1

-20 t(s) 2 4 6 8 10 12 14 (c) Longitudinal acceleration vs time. (d) Mach vs time.

α(º) Flight Path(º) 15 100

10 50

5 t(s) 20 40 60 80 100 120 t(s) 0.5 1.0 1.5 2.0 2.5 3.0 -50 -5

(e) Angle of attack vs time. (f) Flight path angle vs time until the landing.

CD 0.9

0.8

0.7

0.6

0.5

t(s) 0 2 4 6 8 (g) Drag coeﬃcient vs time until the apogee.

Figure 6.4: Vertical model rocket launch ﬂight data.

60 approximately. As already mentioned, we also see that the rocket decreases velocity due to the coasting between stages. This is also observable from the Mach number, in Figure 6.4d, which remains subsonic throughout the entire flight. During the burnings, in Figure 6.4c, the acceleration describes a profile similar to the thrust (see Figure 6.1a). However, it is decreasing over time as the rocket gains velocity, which increases the drag opposing the thrust force. In contrast, when the rocket is not burning, the acceleration is negative and increases over time (decreases in absolute value) because the velocity is decreasing. The peak acceleration at 9.5 s results from the parachute ejection. The angle of attack, in Figure 6.4e, shows the rocket rapidly damps its oscillations and follows the true air speed during the majority of the ascending flight. During the first instants the angle of attack is not computed because its model receives the impulse to start the oscillations only when the rocket leaves the guiding rod. From Figure 6.4f, we can see the flight path angle as 90° at the ignition since the launch is vertical. This angle decreases until the apogee due to the weathercock effect that causes a curvature in the trajectory. Reaching the apogee, the flight path angle becomes negative because the rocket starts to descend. There is a peak value of about −90° since the rocket reverses its direction and stabilizes in a 30° slope, approximately, toward the ground due to the drag force on the parachute. As the rocket descends, the flight path decreases, which means the velocity is becoming steeper. This fact agrees with the curvature described by the trajectory due to the ABL, observed in Figure 6.3. The drag coefficient, in Figure 6.4g, is dependent on the Reynolds and the Mach numbers. Therefore, when the velocity is increasing, the drag coefficient decreases its value due to the skin friction drag. In- versely, as the velocity decreases, the drag coefficient starts to increase. When the first stage is discarded, the CD decreases instantaneously because the rocket loses a great source of drag. On the other hand,

(a) Eastward perspective.

(b) Northward perspective.

Figure 6.5: Model rocket trajectory in a 75° launch eastward with the downwind changing from northeastward at the ground to northward at the apogee.

61 when the connector is discarded, although it is lost another source of drag, the CD increases signiﬁcantly.

This happens because this is the instant when the reference area becomes smaller and the CD has to change in order to be given in respect to another section. Launching the model rocket eastward, with 75° of inclination from the ground and changing the downwind from northeastward at the surface to northward at the apogee, we get the trajectory in Figure 6.5. As it is depicted, the upward phase of the trajectory is deviated southward since the weathercock eﬀect toward southeast (as in the vertical launch) is counteracted by the initial velocity pointing eastward. Right after the apogee, during the recovery, the rocket falls northward but, gradually, as it approaches the ground, it glides northeastward due to the 45° variation in the wind direction.

6.4.2 Sounding rocket launch

The trajectory from the vertical sounding rocket launch (again with constant wind direction) is represented in Figure 6.6. Like the other trajectories, we can see this one is also deviated upwind (southwest-

Up(km)

10 20 30 40 50 60 Ground Track(km) (a) 3D trajectory. (b) Trajectory proﬁle.

Figure 6.6: Sounding rocket trajectory with a vertical launch and wind blowing northeastward; Parachute ejection at about 28 km of altitude. ward). However, after the apogee, the rocket continues to veer from the launch site because the parachute is ejected only when the rocket returns to 28 km of altitude, approximately. The dynamic pressure of 12 kN m−2 (mentioned in Section 6.1.2) is reached at this altitude, enabling the rocket to decrease its descent rate until it reaches the safety terminal velocity. Also, with this dynamic pressure, the drag force acting on the parachute is high enough to carry the rocket back northeastward, as observed by the trajectory curvature at the end of the recovery phase. If the parachute was ejected at the apogee, the rocket would describe a similar trajectory due to the low density at these altitudes that decrease the drag force from the parachute (no dynamic pressure, see Figure 6.7).

62 Dyn. Pressure(kN/m 2)

200

150

100

t(s) 50 100 150 200 250

Figure 6.7: Sounding rocket’s dynamic pressure until the parachute ejection.

In Figure 6.8 it is represented the data obtained from this launch. As seen from Figure 6.8a, the rocket reaches an apogee of 80 km at about 150 s after the lift off and the flight lasts 23 min, approximately. Like the model rocket launch, the velocity, in Figure 6.8b, decreases during the delays between stages and until the apogee, as expected. After this instant, the velocity increases (in absolute value) as the rocket continues its trajectory back to the ground without ejecting the parachute. The ejection happens at 250 s, approximately, and the velocity decreases from about 1 km s−1 to nearly the terminal velocity. The Mach number, in Figure 6.8d, behaves accordingly the velocity, as expected, and reaches the supersonic regime during the climbing and the descending trajectories. Although the recovery deployment happens at supersonic speeds, the dynamic pressure suffered by the parachute (see Figure 6.7) lies in the limit range mentioned in the literature [61] as the atmosphere is still rarefied. From Figure 6.8c, we see the acceleration increases during the burnings and decreases in absolute value during the coasting time due to the drag slowing down the rocket. After the burnouts, the absolute acceleration is around 1g since the dynamic pressure is nearly zero. Owed to the parachute ejection, the rocket suffers a 120g deceleration that is not represented in Figure 6.8c due to the graphic scale. The amplitudes of the angle of attack, in Figure 6.8e, are smaller comparing to the ones from the model rocket launch because the effect of the wind is softened by the higher velocity of the rocket leaving the guiding rail. The flight path angle, in Figure 6.8f, starts at 90° since the rocket is launched vertically and, again, due to the weathercock effect, this angle decreases as the rocket follows the air speed direction. After the apogee, the flight path increases (in absolute value) because the rocket continues its downward trajectory without the recovery device. When the parachute is deployed, the flight path decreases instantaneously to −90° as the rocket reverts its direction. Then, the flight path increases until nearly to the end of the recovery phase as the descent rate is slowly decreasing due to the higher density. This fact intensifies the effect of the wind dragging the rocket, as observed in Figure 6.6. The drag coefficient, in Figure 6.8g, behaves as explained for the model rocket with the exception during the first burning, where there is a CD increase. This is explained due to the transonic and low supersonic effects (shock waves) implemented in the drag coefficient model. In Figure 6.8h, the CD begins to increase several times its value until the apogee. This increase is due to the small Reynolds number

(from which the CD is dependent on), as the rocket is losing velocity and the atmosphere is becoming rareﬁed.

63 Altitude(km) Velocity(km/s) 80 1.0

60 0.5

40 t(s) 50 100 150 200 250 300 350

20 -0.5

t(s) 200 400 600 800 1000 1200 1400 -1.0 (a) Proﬁle of altitude vs time. (b) Proﬁle of vertical velocity vs time.

Mach Accel.(/g) 25 3.5

20 3.0 2.5 15 2.0 10 1.5 5 1.0 0.5 t(s) 50 100 150 200 250 t(s) 50 100 150 200 250 300 350 (c) Longitudinal acceleration vs time. (d) Mach vs time.

α(º) Flight Path(º) 6 100

4 50

2 t(s) 200 400 600 800 1000 1200 1400 t(s) 1 2 3 4 5 -50

-2

(e) Angle of attack vs time. (f) Flight path angle vs time until the landing.

CD CD

2.5 0.65 2.0 0.60

0.55 1.5

0.50 1.0 0.45 0.5 t(s) t(s) 5 10 15 20 50 100 150 200 250 (g) Drag coeﬃcient vs time until 20 s after the launch. (h) Drag coeﬃcient vs time from 20 s until the recovery.

Figure 6.8: Vertical sounding rocket launch ﬂight data.

64 6.4.3 The inﬂuence of atmospheric stability

In this section we compare the flight profiles from the model rocket launch with the same local wind speed used in the previous sections but changing the atmospheric stability. For sounding rockets the following results are irrelevant since they leave rapidly the ABL and it covers a small part ot the trajectory. Figure 6.9 presents three wind profiles, under three different stable atmospheres, for a measured wind speed of 4.3 m s−1 at 6 m of altitude. We can see that as the atmosphere becomes more stable, the wind

Velocity(m/s)

Very Stable(L= 30 m) 10 Stable(L= 125 m) Near Stable(L= 350 m) 5

Altitude(m) 100 200 300 400 500 600

Figure 6.9: Wind profiles in a stable atmosphere for a measured wind speed of 4.3 m s−1 at 6 m of altitude. speed gets stronger at higher altitudes and the ABL thickness decreases. Comparing to Figure 6.10, where it shows the respective trajectory for each wind profile, we observe that the ascent phase of the trajectory is negligibly affected as the atmosphere changes from near to very stable conditions. On the other hand, the recovery phase from each trajectory is strongly influenced by these changes. We can see that the rocket diverges more from the landing site under stable conditions as it corresponds to the strongest wind speed profile. Inversely, the rocket lands closer to the launch site in a near stable atmosphere as the wind

Altitude(m) 500 400 300 200 100 Ground Track(m) -1000 -500 0

Very Stable(L= 30 m) Stable(L= 125 m) Near Stable(L= 350 m)

Figure 6.10: Trajectory profiles of the model rocket launch under three different wind profiles from a stable atmosphere. is less intense. Comparing to Figure 6.3b, corresponding to the trajectory in a neutral atmosphere, we also see that, in this case, the rocket is even less deviated from the launch site and the apogee becomes a little higher. This is expected since in neutral conditions the wind speed profile intensifies more smoothly with altitude than in a near stable atmosphere. Figure 6.11 presents three wind profiles, under three different unstable atmospheres, for the same previously measured conditions. As it shows, the wind speed gets less intense after a certain altitude

65 Velocity(m/s)

4 Very Unstable(L=-75 m) Unstable(L=-150 m) 2 Near Unstable(L=-350 m) Altitude(m) 200 400 600 800 1000 1200 1400 -2

Figure 6.11: Wind profiles in an unstable atmosphere for a measured wind speed of 4.3 m s−1 at 6 m of altitude. and may reverse its direction if the atmosphere becomes unstable enough. This is due to the convective effects since the warmer air at the surface rises and the ABL becomes turbulent. Linking these profiles to the respective trajectories in Figure 6.12, we observe the rocket lands further from the launch site as the atmosphere turns less unstable, as expected due to the increase of the wind speed. On the contrary, the upward trajectory only changes a bit near the apogee but it reaches higher altitudes than in a stable or neutral atmosphere. The recovery trajectory under a very unstable atmosphere (see Figure 6.12) stands out owned to its different shape. The curvature presented by this descendant trajectory is explained from the absence of wind at the apogee (see Figure 6.11) that allows the rocket to fall near vertically. As it approaches the ground the wind speed increases and the rocket is progressively carried downwind.

Altitude(m)

600

500

Very Unstable(L=-75 m) 400 Unstable(L=-150 m) 300 Near Unstable(L=-350 m)

200

100

Ground Track(m) -200 -100 0 100 200

Figure 6.12: Trajectory profiles of the model rocket launch under three different wind profiles from an unstable atmosphere.

To summarize this section, we conclude that the ascending phase of the trajectory is more inﬂu- enced when the atmospheric stability changes from stable to unstable (or neutral) than when it presents variations under the same stability class. This is also true for the recovery phase but, in this case, the downward trajectory already presents large diﬀerences when the stability has small deviations but remains under the same class.

66 Chapter 7

Stochastic simulations

Many of the parameters required to determine the trajectory cannot be known (or well deﬁned, at least) and the measured data may vary randomly, presenting some uncertainties. Therefore, in order to analyze the outcome of several uncertain scenarios, it lead us to perform simulations that rely on repeated random sampling and statistical analysis to compute the results, known as Monte Carlo simulations [63]. To get the outcome due to the uncertainties, we must identify a suitable probability distribution for each input parameter from which a set of data is randomly generated. Then, all the possible combinations between the created data correspond to the inputs given in each trajectory simulation. Therefore, the number of trajectories to be determined, for a given Monte Carlo, is given by

n Y Ntraj = Nptsi , (7.1) i=1

where n is the number of parameters under study during the simulation and Nptsi is the number of points generated to the parameter i. From Figure 7.1, we deduce that a sample of 1000 points can approximately represent a normal distribution for a given parameter. However, from (7.1), we ﬁnd that, even for a few

Prob. Density

0.4

0.3

0.2

0.1

0.0 -2 -1 0 1 2 3

Figure 7.1: Normal distribution N (0, 1) and respective probability density histogram from 1000 samples. parameters, the resultant Monte Carlo simulation time turns this hypothetical computation impractical in an interpreted language such as Mathematica®. Since we do not own such great computational resources as to minimize the simulation time, a better approach is to perform what if simulations in which it is

67 analyzed the outcome from the uncertainties of only two or three parameters with smaller samples. In the latter case, the number of points generated to each parameter should be less than the former case in order to get a similar simulation time.

7.1 Simulation rocket and conditions

The simulations presented in the following sections use the single stage rocket described in Tables 7.1 and 7.2, which was based on a high power rocket available for sale [64]. With this choice we can simulate

Rocket property Value Average thrust (N) 1550 Burnout (sec) 3.68 Gross mass (kg) 9.59 Dry mass (kg) 5.47 Body tube length (cm) 80 Nose length (cm) 38.1 Diameter (cm) 7.5 Number of ﬁns 3

Parachute’s CD 0.695 Parachute’s canopy diameter (m) 2.54

Table 7.1: High power rocket data. a kind of rocket that produces an amount of thrust that lies between the power used in the two types of rockets from Chapter 6 and, at the same time, achieves low supersonic speeds. The motor M1550-0 is selected in order to get a ratio between this rocket and the model rocket’s apogees with the same order of magnitude from the ratio between the sounding rocket and the high power rocket’s apogees.

From Table 4.1, considering this rocket is provided with a hemispherical parachute, we assume its CD is known and equal to 0.695. Like in Section 6.1.1, the parachute’s reference diameter is found in order to reach the safety terminal velocity of 5 m s−1. Therefore, from (5.4) with a density of 1.216 kg m−3 found from the conditions at the launch site (mentioned later), we get a diameter of 2.54 m.

Fin’s property Value Nº ﬁns 3 Root chord (cm) 12 Tip chord (cm) 6 Span (cm) 12 Sweep angle (°) 63.4

Table 7.2: High power rocket ﬁns’ data.

The hypothetical launch site for the simulations is the same as the one described in Section 6.3. The guiding rod is deﬁned as placed vertically, measuring 2 m, and the initial nose distance to the ground is 3 m.

68 In order to study the influence of the exponential rates at which the local weather conditions approximate the data from the developed atmospheric model along the altitude, these conditions must differ from the defined values in the previous simulations. Since the old conditions correspond to the data presented by the atmospheric model at the respective altitude and latitude, the constants that define the rates do not affect the pressure and temperature. Therefore, for the following simulations, the new local atmospheric conditions are defined as the maximum averages in April 2015 in Lisbon: 1027 mbar and 21 ◦C [65]. For the wind speed and direction, their conditions are defined to be the same as the ones in Table 6.7, respectively.

7.2 Landing site uncertainties

As already mentioned, we cannot afford the necessary resources to make a full Monte Carlo simulation. Therefore, changing one parameter at a time, we can study its impact in the output in order to choose the most critical to be used in our final simulation. We must have in mind that even if changing one parameter does not present a significant influence in the output, its effect may be enhanced by changing simultaneously another parameter from which the former is dependent on. Hence, the following tests are merely a general approximation to predict the critical inputs that present a great impact in the output when changing inside a restricted interval. The uncertainties in the trajectory inputs may arise from sudden fluctuations in the atmospheric conditions or due to the errors when guessing the value of a given property. In Table 7.3 are presented all the parameters that contribute to these uncertainties in the trajectory (indirectly through the atmospheric and drag coefficient models) and their respective influence in the landing site coordinates. The uniform

Landing site Parameter Distribution µx(m) µy(m) σx(m) σy(m) Local pressure (mbar) N (1027, 12) −7089 7096 0.065 0.065 Pressure rate (m−1) U(0.1, 3) −7090 7096 0.427 0.425 Local temperature (◦C) N (21, 22) −7089 7096 0.537 0.537 Temperature rate (m−1) U(0.1, 4) −7089 7096 0.835 0.840 Local windspeed (m s−1) N (4.3, 12) −7500 7507 2078 2077

Surface roughness length – z0 (cm) U(4, 10) −7095 7102 286 286 Downwind (from north, clockwise) (◦ ) N (45, 72) −7153 6903 956 989

Table 7.3: Mean (µ) and standard deviation (σ) of the landing site in the two directions (x – Southing; y – Easting) changing each input within a distribution of 20 samples. distribution is used to pick up values for the parameters that are unknown within an specific interval. On the other hand, the normal distribution is used to generate values for the parameters that are measured but may present some disturbances. In this case, the mean represents the expected value under the current conditions and the standard deviation specifies how much the measured value may deviate from the mean. The minimum and maximum thresholds in the uniform distributions from the pressure and temperature rates are defined, respectively, in a way that the atmospheric profiles match the CIRA-86

69 model at about 500 m of altitude and at the apogee. The standard deviations for the local pressure and temperature are defined as the greatest variation occurred in a 30 min interval during April 15, 2015 in Lisbon [66]. In the case of the wind speed’s normal distribution, its standard deviation is defined as the necessary amplitude that changes the local wind to another class according the Beaufort scale [62] (see fourth paragraph in Section 6.3). Finally, the standard deviation for the wind direction is defined as the variation occurring at the measured wind speed found from a dataset taken during daytime in a flat grass surface [67]. Comparing the mean landing coordinates with the standard deviations, in Table 7.3, we conclude the uncertainties from the pressure and temperature measured at the launch site, and their respective rates to match the atmospheric model along the altitude, present a negligible impact under the defined conditions. On the other hand, the impact from the remaining parameters’ uncertainties is much more considerable, which leads us to neglect the first four parameters from Table 7.3. With only three parameters left, we still have to compute a billion iterations if we use samples of 1000 values for each parameter, as illustrated in Figure 7.1. Therefore, for the remaining normal distributions (wind speed and direction), we take samples of 200 values since, after testing, we concluded that a larger sample would not improve the landing site dispersion but only saturate the graphic. In respect to the surface roughness length, since we do not know any characteristic that helps to define it other than the terrain type (defined as high grass), it can take any value within the range of 4 cm to 10 cm (see Table 3.1). Hence, in order to reduce the simulation time, we decided this parameter should only take the worst case scenarios, i.e.,

4 cm and 10 cm. Nevertheless, it is still required to replace the CD model by a constant value since it slows down each iteration and it is not expected to impact the results much. Therefore, the CD used in the simulation is the mean value of 0.70 calculated from

Z tapogee CD(t) CD = , (7.2) 0 tapogee where CD(t) is the drag coefficient function found from the developed model after simulating a trajectory under the same conditions used in this chapter. Figure 7.2 shows the outcome of the Monte Carlo simulation resulting from the input conditions mentioned above. Since the rocket follows the wind speed direction, it is expected that the uncertainties from the local wind velocity and the surface roughness length (both only influencing the wind intensity) change the landing site along this direction, describing a straight line. Analogously, the changes in the wind direction, due to its uncertainties, make the landing points to describe an arc of a circumference that gets larger as the distance to the launch site increases. The dispersion of the landing points, in Figure 7.2, clearly shows the influence of the normal distributions used to generate the local wind direction and speed applied in each iteration. Since a mean of 45◦ was defined for the wind direction’s normal distribution, the orange dots are heavily concentrated along the northeast direction. In the same way, the rocket lands less often apart from the northeast direction as there are fewer inputs that deviate more from the mean. Also due to this principle, the wind speed uncertainties contribute to the lower density of points near the closest and furthest distances from the

70 Figure 7.2: Landing positions and respective confidence ellipses from the Monte Carlo simulation. launch site. This pattern can also be seen in Figure 7.3 where the probability density function of the landing sites is represented. Furthermore, this figure enables us to distinguish better the most probable landing area since Figure 7.2 gets saturated in this region due to the great amount of trajectories simulated. The confidence ellipses in Figure 7.2 are rather eccentric. Their major axes (aligned northeastward) measure 10.4 km, for a confidence of 95%, and 9.1 km, for 90% of confidence. As the furthest landing site, collinear with the major axes, is 14.6 km away from the launch location, the major axis from the lowest confidence level represents almost two thirds of the furthest distance. Thus, we conclude the measured wind speed, allied with the surface roughness length range, brings a strong uncertainty to the landing site estimation under the specified conditions. Regarding the minor axes compared to their distances from the launch site, we see that the wind direction also induces a considerable uncertainty.

0.06

0.04

0.02

Figure 7.3: Probability density function of the landing coordinates resulting from the Monte Carlo simulation.

71 7.3 Rocket optimization

Taking the previous simulation from a different approach, we get another tool that allows us to perform optimization. Therefore, after defining an optimization criteria (maximization or minimization of an output) given an universe of input variables, we can deduce the best alternative that matches our goal. Changing the rocket parameters in each Monte Carlo iteration (as opposed to Section 7.2, where the rocket properties are kept constant), we can optimize the rocket characteristics. For example, we are aiming to determine the best body tube’s length and diameter combination so that the rocket reaches the highest apogee keeping constant the other properties. To this example a windless atmosphere is defined (in order to have simpler trajectories) and three options for each tube’s dimension, which are selected randomly from an uniform distribution. The minimum and maximum length limits (constraints) are 1 m and 2.5 m, respectively, and for the diameter are 7.5 cm and 13 cm, respectively. In Figure 7.4 the altitude over time is depicted (until the highest apogee’s instant) for these nine configurations. As shown, the configuration with the smallest dimensions (1.215 m and 8.4 cm) reaches

Altitude(km) l=1.686m d=10.1cm l=1.686m d=8.4cm 5 l=1.686m d=12.2cm 4 l=1.215m d=10.1cm

3 l=1.215m d=8.4cm l=1.215m d=12.2cm 2 l=2.266m d=10.1cm 1 l=2.266m d=8.4cm t(s) l=2.266m d=12.2cm 5 10 15 20 25

Figure 7.4: Altitude over time from nine launches with different rocket’s length and diameter configurations. the highest apogee, as opposed to the rocket with the largest dimensions that reach the lowest apogee. These facts are expected due to the major influence of the rocket’s dimensions on the drag coefficient and the reference area. The longer and larger the rocket is, the greater is the drag that decreases the velocity and forces the rocket to reach sooner a lower apogee. From this result, we may say this method works properly but a more extensive optimization should be performed in the future since the geometric parameters of the rocket also influence the mass of the components. Even though this is a simple and expected example, it gives a glimpse of the many possibilities at our disposal after developing a Monte Carlo simulation to our tool. From this starting point, and having enough computational resources, more complex optimizations may be processed. Besides, other efficient optimization methods (not considered in this work) may be developed and applied to the presented tool so that the expected result can be fast and directly found, saving resources finding the best alternative from the outputs.

72 Chapter 8

Conclusions

The present work focused on the development of a trajectory simulator tool for fin-stabilized model and sounding rockets addressing the effects of an ABL profile. This tool was developed under Mathematica® and supports from model to sounding rockets with any number of stages. This work presented the tool’s flexibility to customize the default models and to compute the outcome from the inputs’ uncertainties (or perform an optimization) from a Monte Carlo simulation. Methods to estimate the rockets properties (including constant values and time dependent functions) were presented in this work. Nonetheless, any rocket input can be defined according the user requirements. The developed atmospheric model takes into account the local conditions at the launch site which tend to the CIRA-86 profile at a specified rate. Likewise, above the ABL, the wind model approximates the values given in the CIRA-86 tables. Within the ABL, as the wind plays an important role in the model rockets trajectory, three profiles were proposed and implemented considering the atmospheric stability, which can be deduced from the Obukhov length. However, we concluded this parameter is hard to determine with low errors using common measuring instruments. Therefore, a qualitative alternative was presented to figure out the atmospheric stability from the weather conditions. Besides the local atmospheric data, the ABL profiles also take into account the type of the surface. In order to improve the simulations considering the rocket’s attitude, as our tool possesses three degrees of freedom, we developed a model to compute the angle of attack over time that can be influenced by wind gusts appearing along the wind direction. A drag coefficient model was also implemented depending on the Mach, the Reynolds and the configuration of the rocket. Two types of simulations were performed to demonstrate the potentiality of the tool: a model rocket and a sounding rocket launch. Both of them showed that the developed trajectory simulator provides reliable data considering its restrictions are followed. These results also revealed a large impact of the wind in the trajectories (due to the weathercock effect), which leads us to reassure the importance to forecast the atmospheric stability and to model a suitable wind profile. Comparing the same launch under different atmospheric stabilities with the same local wind, it was concluded the recovery phase is more sensitive to variations of the Obukhov length (keeping the same stability) than the ascendant trajectory, which is barely affected. However, different stabilities already induce a considerable impact in

73 the climbing trajectory near the apogee. From the results we also conclude the wind profiles must always concern the local conditions at the launch since, from the Monte Carlo simulation, and for the defined uncertainties, they showed a strong influence in estimating the landing site.

8.1 Future work

Although the developed trajectory simulator presented results expected theoretically, it lacks validation from real case scenarios. Therefore, as an extension to this work, the presented model rocketry highlights must be deepened and used to perform rocket launches taking into account the methods in this thesis. Prior to these launches, this tool may be used to estimate the ejection delay after the apogee that enables the rocket to land near the launch site, thus preventing recovery problems. Since the presented trajectory simulator is intended to be a ﬂexible tool, we shall propose future developments in order to expand and improve the simulator’s utilities:

As this work was focused on fin-stabilized rockets, an active control method may be implemented in the future. In that case, the wind remains as an important factor to determine the input controls but it does not influence the trajectory as the rocket follows the predefined path. When we changed the wind direction over the altitude, the trajectory presented a strong dependence on this parameter. It is therefore advised to develop a wind direction profile. The way we implemented the wind gusts requires the user to estimate the instant they occur and only affect the angle of attack. Hence, an approach that joins the gusts within the wind profile can bring better results and reveal itself as a more practical method. Since this tool is primarily aimed to model rocketry, the developed drag coefficient model does not take into account the free molecular flow in a rarefied atmosphere. If further studies reveal it plays

a significant role when finding the CD of sounding rockets, its effects should be added into the existing model.

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78 Appendix A

Tangent ogive proﬁle

Figure A.1 represents a circumference arc from which is found the ogive proﬁle y(x), where L is the ogive length and d is the respective diameter at the base.

d/2 r L

R k

Figure A.1: Ogive proﬁle.

Considering the circumference equation

R2 = x2 + y2 , (A.1) we get for x = 0 (where y = k + d/2)

R0 = k + d/2 , (A.2) and for x = L (where y = k)

p 2 2 RL = L + k . (A.3)

79 Since R0 = RL, we can join (A.2) and (A.3) yielding

L2 d k = − . (A.4) d 4

Substituting k in (A.2), and since R = R0, we get

L2 d R = + . (A.5) d 4

The ogive proﬁle is given by r = y − k, thus

p r = R2 − x2 − k , (A.6) where k and R can be replaced by (A.4) and (A.5), respectively, resulting

s ! L2 d2 L2 d r(x) = + − x2 − − . (A.7) d 4 d 4

80 Appendix B

Connector’s CM and inertia

In Figure B.1 the thick lines represent a connector’s half section where L is the surface length, h is the height and r1 and r2 are, respectively, the fore and aft radius. In order to ﬁnd the CM and inertia by integration, it is used the referential xoy.

o x y L

h θ

Figure B.1: Connector’s half section (thick lines).

Since the connector is a conical surface with thickness t, its elemental volume is

dV = t dA , (B.1) where the elemental area, dA, is

dA = 2π(r1 + x sin θ) dx , (B.2) and sin θ is given by

r − r sin θ = 2 1 . (B.3) L

Therefore, integrating (B.2) from 0 to L, and using (B.3), the total area is

A = πL(r1 + r2) . (B.4)

81 The centroid is found from

1 Z L y¯ = y dV , (B.5) V 0 where y is

h y = x cos θ = x . (B.6) L

Thus, replacing (B.1), (B.2), (B.3), (B.4) and (B.6) in (B.5), yields

1 Z L h x y = x t2π r1 + (r2 − r1) dx = tπL(r1 + r2) 0 L L 2h Z L x = 2 x r1 + (r2 − r1) dx = L (r1 + r2) 0 L h r + 2r = 1 2 . (B.7) 3 r1 + r2

The transversal inertia relative to the radius r1, is found from

Z L I = y2 dm . (B.8) 0 where the elemental mass is given by

dm = ρ dV , (B.9) and ρ is the material’s density. Hence, using (B.1), (B.2), (B.3), (B.6) and (B.9) in (B.8), it becomes

Z L h 2 x I = x ρt2π r1 + (r2 − r1) dx = 0 L L 1 = h2Lπtρ(r + 3r ) . (B.10) 6 1 2

Since, from B.4, the connector’s mass is

m = ρtπL(r1 + r2) , (B.11) then, multiplying (B.10) by r1+r2 , and applying (B.11), the inertia can be found by r1+r2

1 r + 3r I = mh2 1 2 . (B.12) 6 r1 + r2