Use Euler\'s equations of motion for principal axes in state variable form. Writ

Question

Use Euler's equations of motion for principal axes in state variable form. Write a MATLAB program to integrate Euler's equations of motion and solve for omega_x (t), omega_y (t), omega_z (t) and for H_x(t), H_y(t), and H_z(t) in body-fixed coordinates for an axisymmetric spacecraft with the following PMOIs: I_xx = 2887 kg-m^2 I_yy = 2887 kg-m^2 I_zz = 5106 kg-m^2 Assume there no moments (so M_x = M_y = M_z = 0) and that the ICs are omega_x (0) = 0.6 radians/second, omega_y (0) = 0, and omega_z (0) = 1.1 radians/second. a. Make plots of omega_x (t), omega_y (t), and omega_z (t) versus time. (You may show all three curves on one plot.) b. Make a plot of omega_y (t) on the vertical axis versus omega_x (t) on the horizontal axis. c. Make plots of H_x (t), H_y (t), and H_z (t) versus time. (You may show all three curves on one plot.) d. Make a plot of H_y (t) on the vertical axis and H_x (t) on the horizontal axis. e. Describe in words the dynamics of omega_x (t), omega_y (t), omega_z(t), H_x(t), H_y(t), and H_z(t) relative to the body axes. Explain why your results make (or don't make) sense.

Explanation / Answer

Main Code:

%----------------BEGIN CODE-----------------------------

clear all

close all

clc

% PMOIs (units: kg*m2)

Ix = 2887;

Iy = 2887;

Iz = 5106;

% Initial Conditions

IC = [0.6 0 1.1];

t = [0 20];

%% Use ode 45 to numerically integrate

options = odeset ('RelTol', 2.22045E-14, 'AbsTol', 2.22045E-14);

[t, state] = ode45(@myfunc, t, IC, options, Ix, Iy, Iz);

% Angular velocities rad/s

omegax = state(:,1);

omegay= state(:,2);

omegaz = state(:,3);

% Angular momentum

Hx = Ix . *omegax;

Hy = Iy . *omegay;

Hz = Iz . *omegaz;

%% Figures

%a

figure(1)

subplot(211)

plot(t, omegax, 'r', t, omegay, '--b', t, omegaz, '-.m', 'LineWidth', 2);

axis( [ 0 20 -0.7 1.2 ] )

title( 'Question 1a')

legend( 'omegax', 'omegay', 'omegaz')

xlabel( 'Time (s)')

ylabel( 'omegai (rad/s)' )

grid on

%b

subplot(212)

plot( omegax, omegay, 'r', 'LineWidth', 2);

title(' Question 1b')

xlabel( 'omegax (rad/s)' )

ylabel(' omegay (rad/s)' )

grid on

axis equal

%c

figure(2)

subplot(211)

plot(t, Hx, 'r', t, Hy, '--b', t, Hz, '-.m', 'LineWidth', 2);

axis ( [ 0 20 -3000 6500 ] )

title(' Question 1c ')

legend( 'Hx', 'Hy', 'Hz')

xlabel(' Time (s) ')

ylabel( 'Hi (kg*m2/s)' )

grid on

%d

subplot(212)

plot( Hx, Hy, 'r', 'LineWidth', 2)

title( ' Question 1d ' )

xlabel( ' Hx (kg*m2/s) ' )

ylabel( ' Hy (kg*m2/2) ' )

grid on

axis equal

%-----------------------------END OF CODE-------------------------------------

omega_x = state(:,1);

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