MAE 8 MATLAB Project Report
Man-Yeung Tsay A11313074 MAE 8 MATLAB Project Report Overview This project models projectile motion of a ball on a 3-dimensional terrain. The objective was to identify the location of where the ball would hit the terrain. Given an initial position (Xo, Yo, and Zo in meters) and initial velocity (Umag in m/s) with launch directions (theta and phi in degrees), the project computes the time (in seconds), 3 components of position (X, Y, Z in m), and 3 components of velocity (U, V, W in m/s) along the trajectory. Theta is the angle from the positive x-axis in the x-y plane. Phi is the angle from the positive z-axis to the vector Umag. In all tasks, 9.81 m/s^2 is used as the value for gravity. Except for calculating the max height and final position, the time step (dt) of 0.001s was used. Task1 In the first task, air resistance was neglected, and projectile motion was modeled using Euler's method. These equations were used to solve for projectile motion: ππ =π ππ‘ ππ =0 ππ‘ ππ =π ππ‘ ππ =0 ππ‘ ππ =π ππ‘ Using Euler's method, the equations are: π(π + 1) = π(π) + π(π) β ππ‘ π(π + 1) = π(π) + π(π) β ππ‘ π(π + 1) = π(π) + π(π) β ππ‘ ππ = βπ ππ‘ π(π + 1) = π(π) π(π + 1) = π(π) π(π + 1) = π(π) β π β ππ‘ A function was written in file "projectile3d.m" to calculate the equations to solve for the positions and velocities along the trajectories of the ball. The inputs were the 3 components of initial position (Xo, Yo, Zo), the initial velocity (Umag), and the launch directions (theta and phi). A while loop was used to loop through the equations. The condition used for the while loop was W(i)>=0. To find the max height position at where "W" would change signs, an "if" statement was used. The condition for the "if" statement was "if W(i+1)<0 && W(i)>0," and the outputs would be calculated with a new time step. To find where the ball lands on the ground, an "if" statement of "if Z(i)<0" was used to calculate the outputs with a different time step. To find the new time step for max height position and final position, the difference between the previous Z value and current value was divided by the previous W value. To extract the max height values from the outputs, the built-in command "max" was used to find the maximum Z value. A variable was set to find the index for the maximum Z value. The remaining corresponding outputs were called using this index. Man-Yeung Tsay A11313074 Task 2 In the second task, air resistance was included. The process for the second task mirrors the first task, except for including air resistance in the equations. A separate file "projectiletask2.m" was created. This new file mirrors "projectile3d.m", except for "C" (coefficient of friction) as an additional input and the inclusion of air resistance in the equations. Four trajectories (with same initial conditions of Xo=Yo=Zo=0, Umag=40m/s, theta=phi=45 degrees) were launched with different coefficients of friction: C=0, 0.2, 0.4, and 0.6. The equations used are: ππ ππ ππ΄ =π = βπΆ πβπ 2 + π 2 + π 2 ππ‘ ππ‘ 2π ππ ππ ππ΄ =π = βπΆ π βπ 2 + π 2 + π 2 ππ‘ ππ‘ 2π ππ ππ ππ΄ =π = βπ β πΆ π βπ 2 + π 2 + π 2 ππ‘ ππ‘ 2π C=coefficient of friction; A=Ο r2 where r=0.04 m; m=0.15kg, Ο=1.2kg/m3 (air) Using Euler's method, the equations are: π(π + 1) = π(π) + π(π) β ππ‘ π(π + 1) = π(π) + π(π) β ππ‘ π(π + 1) = π(π) + π(π) β ππ‘ ππ΄ β 2 π π + π2 + π2 ) β ππ‘ 2π ππ΄ β 2 π(π + 1) = π(π) β (πΆ π π + π2 + π2 ) β ππ‘ 2π ππ΄ β 2 π(π + 1) = π(π) + (βπ β πΆ π π + π2 + π2 ) β ππ‘ 2π π(π + 1) = π(π) β (πΆ Task 3 The goal of the third to task was to accomplish the main objective of this project. A new file "trackprojectile.m" was created to incorporate the terrain and locate the final positions of the ball on the terrain. This new file mirrors the file from the second task "projectiletask2.m" with some exceptions. The built-in MATLAB command "interp2" was used for interpolation. The condition for the while loop was "Z(i) >= interp2(terrain.x, terrain.y, terrain.z, X(i), Y(i)." Then, a variable "surface" was created by setting it equal to the given value from "terrain.mat" after the while loop. To find the final position of the ball, an "if" statement of "if Z(i)<surface" was created, and a new time step was used to calculate the outputs for the final position. Similarly to the previous tasks, this new time step was calculated by finding the difference between the previous Z value and current value and dividing it by the previous W value. Man-Yeung Tsay A11313074 Figures and Tables Task 1 Initial Conditions for Task 1 Results for Task 1 Man-Yeung Tsay A11313074 Task 2 - Initial conditions of Xo=Yo=Zo=0, Umag=40m/s, theta=phi=45 degrees with C=0, 0.2, 0.4, 0.6 Results for Task 2 Man-Yeung Tsay A11313074 Task 3 Initial Conditions for Task 3 Results for Task 3 Man-Yeung Tsay A11313074 Flow Chart of Programs Start with Initial Inputs (Xo,Yo,Zo,Umag,theta,phi, C (if there is air resistance)) projectile3d.m (for no air resistance) projectiletask2.m (with air resistance) calculates equations of motion using Euler's method until ball has landed when ball has landed, a new time step is calculated then used to find the final outputs for max height, a new time step is calculated when W changes sign, and then this time step is used to find the position, velocities, and time during max height End with Outputs (T, X, Y, Z, U, V, W) Man-Yeung Tsay A11313074 Start with Initial Inputs (Xo,Yo,Zo,Umag,theta,phi, C) trackprojectile.m calculate until ball has landed (using interp2 for interpolation) find where ball has landed find new time step to calculate final outputs End with Outputs (T, X, Y, Z, U, V, W) Man-Yeung Tsay A11313074 Appendix project.m - runs all the commands for tasks projectile3d.m - solves for positions and velocities along trajectories of ball with no air resistance Inputs Xo, Yo, Zo (in meters) are components of initial position Umag (in meters/sec) is initial velocity theta and phi (in degrees) are components of launch direction Outputs X, Y, Z (in meters) are components of position U, V, W (in meters/sec) are components of velocity T=time projectiletask2.m -a modified version of projectile3d.m that solves for positions and velocities along trajectories of ball with air resistance Inputs Xo, Yo, Zo (in meters) are components of initial position Umag (in meters/sec) is initial velocity theta and phi (in degrees) are components of launch direction C is coefficient of friction Outputs X, Y, Z (in meters) are components of position U, V, W (in meters/sec) are components of velocity T=time trackprojectile.m - tracks motion of ball as it lands on terrain and calculates positions and velocities of trajectory Inputs Xo, Yo, Zo (in meters) are components of initial position Umag (in meters/sec) is initial velocity theta and phi (in degrees) are components of launch direction C is coefficient of friction Outputs X, Y, Z (in meters) are components of position U, V, W (in meters/sec) are components of velocity T=time
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