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 foguet~oesde modelismo, a traject´oria´econsideravelmente influenciada pelo vento j´aque estes s~aofoguet~oesde controlo passivo. Este trabalho tem como objectivo o desenvolvi- mento de um programa em Mathematica® que simula a traject´oriadestes foguet~oese de foguetes-sonda sob a influ^enciade um perfil de vento. E´ realizado um estudo da camada limite atmosf´ericae consideram- se tr^esperfis que levam em conta as condi¸c~oesatmosf´ericase o tipo de superf´ıcieno local do lan¸camento. Mostrando a flexibilidade do programa desenvolvido, ´erealizada uma simula¸c~aoMonte Carlo para se inferir a dispers~aodo local de aterragem devida `asincertezas nos par^ametrosdo vento. Com as poten- cialidades do Mathematica®, a ferramenta desenvolvida permite que seja definido um qualquer n´umero de est´agios,com boosters externos ou internos no primeiro est´agio,e que os modelos necess´ariospara o c´alculoda traject´oriasejam facilmente modificados ou acrescentados novos modelos na base de dados. Foram efectuadas duas simula¸c~oescom foguet~oesde pot^enciadistinta, ambos com dois est´agiose sob as mesmas condi¸c~oes.Tamb´emse determinou uma traject´oriaem que a direc¸c~aodo perfil de vento varia com a altitude e compararam-se traject´oriassimuladas em estabilidades atmosf´ericasdiferentes sob a mesma intensidade de vento no lan¸camento. Os resultados obtidos mostram grande depend^enciadas traject´oriasno vento, especialmente quando a estabilidade da atmosfera varia. Considerando as incertezas definidas para a velocidade e direc¸c~aodo vento, bem como para o tipo de terreno, estas mostram uma forte influ^enciana dispers~aodos locais de aterragem. Palavras-chave: Simula¸c~aode Traject´oria,Modelismo de Foguet~oes,Camada Limite At- mosf´erica,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 tra- jectory 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 tra- jectories 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]. ..