Assessing the Accuracy of a Rocket's Trajectory Through Space The challenge: To explore the impact of changing atmospheric conditions on the trajectory of a rocket. Maplesoft, a division of Waterloo Maple Inc., 2009 Editor's Notes Introduction Problem Statement 1. Model Description 2. Perform a Monte-Carlo Simulation Results ** This application was developed using Maple and MapleSim www.maplesoft.com/appbriefs Page 1 of 17
Editor's Notes Since the goal of a rocket is to arrive at a particular destination point at a particular moment in time, understanding the trajectory the rocket will follow is an essential aspect of rocket design. Whether you are launching a satellite into space or lighting up the night sky with fireworks, an accurate trajectory is crucial in assuring the projectile is on target. Unfortunately, making sure a rocket adheres to its calculated path can be difficult, since atmospheric conditions such as wind and rain can dramatically change the rocket s path. The challenge: To explore the impact of changing atmospheric conditions on the trajectory of a rocket. The engineer uses MapleSim to: Create a realistic model of the rocket in MapleSim. The model is divided into separate subsystems, each of which is responsible for a particular aspect of the rocket. Incorporate a random variable into the model to simulate the effect that environment conditions have on drag. Perform a Monte Carlo simulation to understand the effects of changes in atmospheric conditions. The results are used to estimate the bounds of the trajectory. Creating a detailed model that takes into account specifications of the rocket as well as environmental conditions allows the engineer to investigate the design parameters and determine the effect atmospheric conditions have on the trajectory. With the results from the Monte Carlo simulation, the rocket manufacturer determines the probable bounds of the rocket s trajectory and then uses this information to create comprehensive operating guidelines for customers. Introduction Whether its raison d'être is to light up the skyline with brilliant colored fireworks or to assist a space shuttle propel into the unknown depths of space, a rocket's trajectory is one of the most important design considerations for rocket designers and manufacturers alike. Unfortunately, it is often very difficult to pinpoint in advance the exact trajectory that will be taken by a rocket in advance. Factors such as atmospheric conditions, including wind and rain, often change the rocket's path dramatically. In this application, a rocket that is typically used in ballistic operations is modeled in MapleSim in terms of its physical characteristics such as initial thrust, mass of fuel, payload, ballistic coefficient, air drag, specific impulse of fuel, and air density. The effect of atmospheric conditions on the trajectory of the rocket is determined by simulating the model with a random disturbance. A Monte-Carlo simulation is then conducted to gauge the probable bounds of the trajectory arising due to changing atmospheric conditions. Problem Statement To create a realistic model of a rocket and measure the effect of atmospheric conditions on the rocket's trajectory through space. www.maplesoft.com/appbriefs Page 2 of 17
Details Pertaining to Each Subsystem Mass(t) subsystem The mass subsystem (shown in figure 2) models the mass of the rocket, in pounds, as a function of time. The equation governing the rocket's mass and burn time as a function of fuel and payload is shown in the equations (1) and (2) mass t = mfuel t mfuelk tburn mpay CmPay t%tburn otherwise (1) tburn = specificimpulse mfuel amax mpay (2) where mfuel = amax = 20 mpay massfraction 1 KmassFraction (3) (4) and specificimpulse = 300 (5) www.maplesoft.com/appbriefs Page 4 of 17
XYAltitude Subsystem The XYAltitude subsystem together with the X Altitude subsystem and the Y Altitude subsystem (see figures 4,4A and 4B) calculate the trajectory and velocity of the rocket based on the differential equations defined in equations (7) through (9). y 0 = 10,x 0 = 0, D y 0 = 0, D x 0 = 0 X Altitude: d 2 x t = thrust t Kdrag t cos θ t 2 dt x 0 = 0, D x 0 = 0 Y Altitude: d 2 y t =thrust t sin θ t KgKdrag t sin θ t 2 dt y 0 = 10, D y 0 = 0 (7) (8) (9) (10) (11) www.maplesoft.com/appbriefs Page 7 of 17
2. Perform a Monte-Carlo Simulation The effects of changing atmospheric conditions on the rocket's trajectory can be determined by changing the random variable describing the atmospheric condition in the drag model. The act of simulating the system with a random variable is commonly referred to as the Monte-Carlo method. By changing the simulations parameters below and clicking the Perform Monte-Carlo Analysis button, it is evident that a random variable value less than 1 causes the rocket to surpass its expected peak X and Y values, while a random value greater than 1 causes the rocket to land short of its final target. Monte-Carlo Simulation Simulation Parameters Mean 0 Variance 3 Number of Iterations 8 Number of Points per Iteration 100 Simulation End Time (s) 400 www.maplesoft.com/appbriefs Page 15 of 17
300000 Monte Carlo Simulation Results 200000 Y altitude 100000 0 0 2.#10 5 4.#10 5 6.#10 5 8.#10 5 1.#10 6 1.2#10 6 1.4#10 6 1.6#10 6 X altitude Main.randomVar = 1 Main.randomVar = 0.076682309 Main.randomVar =K0.79687817 Main.randomVar = 4.3505136 Main.randomVar =K2.9735668 Main.randomVar = 0.51714611 Main.randomVar =K0.41200606 Main.randomVar = 2.3530091 Main.randomVar =K0.72222939 www.maplesoft.com/appbriefs Page 16 of 17
Results Regardless of its intended use, the ability to accurately model the trajectory of a rocket as it moves through space is one of the most important design considerations when building a rocket. In the end, the goal of a rocket is to arrive at a particular destination, at a particular moment in time. All too often, atmospheric changes impede the trajectory of a rocket and causes it to fall too short or too far from the desired destination. This application used Monte-Carlo simulation techniques to identify changes in a rocket's trajectory as a result of atmospheric conditions. With this information in hand, a designer or manufacturer of rocket equipment can offer their clients guidelines and specifications into the probable bounds of a rocket's trajectory. Legal Notice: Maplesoft, a division of Waterloo Maple Inc. 2008. Maplesoft, Maple, and MapleSim are trademarks of Waterloo Maple Inc. All other trademarks are the property of their respective owners. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. www.maplesoft.com/appbriefs Page 17 of 17