**E178: Assignment 10** |[home](../index.md.html)|[syllabus](../syllabus.md.html)|[assignments](../assignments.md.html)|[labs](../labs.md.html)|[final project](../finalproject.md.html)|[flight data](../FlightData.md.html)|[getting certified](../RocketryCertification.md.html)| # Assignment Using the full Rocket Equation with atmospheric variation, design two rockets to enter low-earth orbit, one with Blue Thunder Aerotech propellant, and one with methalox propellant. You may use any of the three following methods for the assignment. 1. Use the existing MATLAB model for the 1-D Rocket Equation with properties that vary with altitude and modify it for the necessary 2-D equations of motion. 2. Write your own software to implement the full 2-D Rocket Equation with atmospheric variation. 3. Use the Kerbal Space Program and add the mods to replace Kerbin with Earth. Your low-earth orbit must be at an altitude of 2000±200 km, and if applicable, an eccentricity of 0.25. Designing a single-stage-to-orbit rocket is considered to be right on the edge of possibility, so I strongly suggest you design a two- or three-stage to orbit rocket. ## MATLAB Full 1-D Rocket Equation The [Rocket Equation Derivation](../PDF/RocketEquationDerivation.pdf) explains the derivation of the 1-D rocket equation. The model as derived assumes purely vertical ascent, so we don't have to account for the angle of the gravity vector. The modifications to achieve the 2-D calculations are to incorporate an angle, $\theta$, that varies with respect to the gravity vector during the course of the flight to aim the rocket around the earth. The rocket should launch with the gravity vector directly opposed to the flight direction, and should end up with the gravity vector normal to the flight direction. The final tangential velocity should be given by the [orbital velocity equation](https://en.wikipedia.org/wiki/Orbital_speed#Mean_orbital_speed) \begin{equation} v_o = \sqrt{\frac{GM}{r}} \end{equation} where $r$ is the radius above the center of mass, $G$ is the Gravitational Constant, and $M$ is the mass of the Earth. The full 1-D rocket equation with properties that vary with altitude (including the acceleration of gravity) is implemented in the MATLAB package [RocketEquationPackage.zip](../MATLABModels/RocketEquationPackage.zip). The files in the package are: File Name | Purpose/Function ----------------|----------------- atmo_compo.m | High Altitude Atmospheric Composition Calculation atmo_p.m | Atmospheric Pressure Calculation atmo_temp.m | Atmospheric Temperature Calculation atmo.m | 1976 Standard Atmosphere Calculator[0-1000 km] AtmosTable.mat | Table of calculated atmospheric property values c_atm.m | Returns the local speed of sound in m/s at the given altitude f_n.m | Gas Integral Program g_atm.m | Returns the local gravitational acceleration in m/s^2 at z GenAtmosTable.m | Generate US 1976 Standard Atmosphere Tables as matfiles (Generates AtmosTable.m) int_tau.m | Tau Integral Computation for Hydrogen Composition k_atm.m | Returns the thermal conductivity in W/m?K at the given altitude mu_atm.m | Returns the dynamic viscosity in N?s/m^2 at the given altitude n_atm.m | Returns the number density of a species in m^-3 at the given altitude n_sum_atm.m | Returns the total number density in m^-3 at the given altitude nu_atm.m | Returns the kinematic viscosity in m^2/s at the given altitude P_atm.m | Returns the absolute pressure in Pa at a given altitude rho_atm.m | Returns the density of air in kg/m^3 at a given altitude rockeq_var.m | Integration of the rocket equation where $T$, $P$, $\rho$, and $g$ change with altitude. T_atm.m | Returns the absolute temperature in Kelvin at a given altitude tester.m | Test Program for atmosphere with plots ThrustCurveSH.m | Returns the thrust curve of a motor. To use the package, extract it, set the extracted folder as the default in MATLAB and run rockeq_var.m. You will need to alter ThrustCurveSH.m to enter the thrust curve for the motor or engine you are using. You may want to save each thrust curve file with a different name and use that name in rockeq_var.m The SH is short for Sandhawk, one of my favorite sounding rockets. You will also need to edit the rocket parameters in rockeq_var.m. The parameters you are likely to need to vary are: Parameter | MATLAB Variable | Meaning ----------|-----------------|-------- $m_0$ | xv0(1) | Initial rocket mass in kg $v_0$ | xv0(2) | Initial rocket velocity in m/s $z_0$ | xv0(3) | Initial altitude MSL in m $I_{sp}$ | ue | Specific Impulse in m/s aka $v_e$ $D$ | D | Rocket diameter in m for drag calculations $C_D$ | CD | Coefficient of drag for drag calculations You will need to modify the rockeq_var.m file to account for the shift to an orbital trajectory. You will also need to modify the plotting to best account for your trajectory. ## Write Your Own Full 2-D Rocket Equation Package The [Rocket Equation Derivation](../PDF/RocketEquationDerivation.pdf) explains the derivation of the 1-D rocket equation. The modifications to achieve the 2-D calculations are to incorporate an angle, $\theta$, that varies with respect to the gravity vector during the course of the flight to aim the rocket around the earth. The rocket should launch with the gravity vector directly opposed to the flight direction, and should end up with the gravity vector normal to the flight direction. Feel free to use whatever package or language you desire. The variation of atmospheric properties with altitude is given [here](https://ntrs.nasa.gov/api/citations/19770009539/downloads/19770009539.pdf?attachment=true). The variation of gravity with altitude can be calculated from [Newton's law of universal gravitation](https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation) and the mass and radius of the earth. The final tangential velocity should be given by the [orbital velocity equation](https://en.wikipedia.org/wiki/Orbital_speed#Mean_orbital_speed) \begin{equation} v_o = \sqrt{\frac{GM}{r}} \end{equation} where $r$ is the radius above the center of mass, $G$ is the Gravitational Constant, and $M$ is the mass of the Earth. Beyond that, you're on your own. ## Kerbal Space Program The [Kerbal Space Program](https://www.kerbalspaceprogram.com/) (KSP) is a commercial software package (meaning _you_ have to pay for it) with realistic flight physics for rockets and a limited range of aircraft. The base package comes designed for the planet Kerbin. The differences between Kerbin and Earth are illustrated [here](https://imgur.com/gallery/ckadxCa). To do actual rocket design with KSP you need to modify it to accurately account for Earth's (and the Solar System's) properties. Two mods that will help are [Real Solar System](https://forum.kerbalspaceprogram.com/index.php?/topic/177216-173-real-solar-system-v164-26-nov-2019/) and [Realism Overhaul](https://forum.kerbalspaceprogram.com/index.php?/topic/155700-173-181-realism-overhaul-v1281-17-april-2020/). KSP may or may not be the easiest way to design your rockets, but it's certainly the most fun. # Aerotech Blue Thunder The $I_{sp}$ of Blue Thunder can be read from several of the larger Blue Thunder motors on [thrustcurve.org](https://www.thrustcurve.org/motors/search.html?manufacturer=AeroTech&designation=&commonName=&type=&impulseClass=&diameter=&propellantInfo=Blue+Thunder&flameColor=&smokeColor=&caseInfo=&sparky=all&class1=all&hazmatExempt=all&availability=available) or it can be calculated from the definition using the data for several of the larger Blue Thunder motors. Just remember that there are two different definitions for $I_{sp}$ and you need to use the one required by the model you are using. # Methalox Methalox is methane oxidized with oxygen. The upper limit for its $I_{sp}$ is found in [this reference](https://thephysicsofspacex.files.wordpress.com/2016/07/isp-upper-limits.pdf). The actual value will vary somewhat from that value. The expected $I_{sp}$'s for SpaceX's Raptor engines are listed in [this article](https://en.wikipedia.org/wiki/SpaceX_Raptor#Description). You may want to check to see if SpaceX has updated the $I_{sp}$ they obtained in their tests and flights. Just remember that there are two different definitions for $I_{sp}$ and you need to use the one required by the model you are using. # Mass Ratio One of the variables that you implicitly have to include into your design is the [Mass Ratio](https://en.wikipedia.org/wiki/Mass_ratio), the ratio of the wet mass (the vehicle plus propellant) to the dry mass (the vehicle by itself). Assume for your rockets that the mass ratio is 12.5. This mass ratio applies to each stage independently. The first stage does need to lift both the first and second stages. # Deliverables Your deliverables for this assignment are: 1. The design for you two rockets, including 1. The dimensions, at least diameter and length 2. The assumed $I_{sp}$ of the fuel 3. The dry mass and the fuel mass 4. The assumed $C_D$ 2. Plots of the acceleration, velocity, mass, and altitude of your final designs from liftoff to orbit. 3. A summary of the design process or strategy you used. Standard report components, such as introduction, conclusions, and recommendations would be nice, but aren't required.