hw8_solutions_2023

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Jan 9, 2024

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Homework Assignment 8: SOLUTION ENAE 404: Space Flight Dynamics May 13, 2023 1. The orientation of a spacecraft is given in terms of the 3-2-1 Euler Angles (i.e., yaw-pitch-roll) as (30, 40, 10) degrees. (a) (6pts) Give the rotation matrix from the inertial to the body-fixed frame. R = 0 . 6634 0 . 3830 - 0 . 6428 - 0 . 3957 0 . 9087 0 . 1330 0 . 6350 0 . 1661 0 . 7544 (b) (6pts) Calculate the principal rotation axis ˆ e and the principal rotation angle φ . φ = 48 . 45 ˆ e = [ - 0 . 0221 , 0 . 8537 , 0 . 5203] (c) (6pts) Calculate the quarternions that describe this orientation of the spacecraft. ~ β = [0 . 9119 , - 0 . 0091 , 0 . 3503 , 0 . 2135] (d) (4pts) Assume that B ω = (0 . 1 , 0 . 2 , 0) radians/sec at the current time. What is the current quarternion rate of change (i.e., ˙ ~ β )? ˙ ~ β = [ - 0 . 0346 , 0 . 0242 , 0 . 1019 , - 0 . 0184] 2. Write a code to numerically integrate the angular velocity vector for a torque-free rigid body with principal body-fixed axes. The body has the following inertia matrix: [ I ] = 10 0 0 0 20 0 0 0 30 (1) (a) (4pts)Given ω = [10 , 0 , 30] deg/sec in the body-fixed frame, calculate the angular momentum magnitude and kinetic energy of the body. ANSWER: Angular momentum=15.8046kg m 2 /s, Kinetic energy=4.2646 kg m 2 /s 2 (b) (8pts) Given the above initial angular velocity, propagate the angular velocity for 100 seconds. On a single plot, plot the angular velocity components as a function of time. (c) (7pts) Plot the kinetic energy deviation and angular momentum magnitude deviation as a function of time. Discuss why you believe that your code is working. ANSWER: We know the code is working because the energy and angular momentum deviation are small. 1
0 20 40 60 80 100 Time (sec) -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 rad/sec w1 w2 w3 Figure 1: Answer 2b. (d) (5pts) Create the polhode plot for this system. The angular momentum sphere should be a single color and the kinetic energy ellipsoid should be a single color. On top of the polhode, plot (as points) the time-varying body-fixed angular momentum. (e) (5pts)Given ω = [1 , 15 , 0] deg/sec in the body-fixed frame, propagate the angular velocity for 100 seconds. On a single plot, plot the angular velocity components as a function of time. (f) (7pts) Create the polhode plot for this system. The angular momentum sphere should be a single color and the kinetic energy ellipsoid should be a single color. On top of the polhode, plot (as points) the time-varying body-fixed angular momentum. Discuss why plots in e and f look significantly different from the plots in part b and d. Compare the energy between the system in part a and the current system. ANSWER: The initial conditions in this problem are very close to the sepratrix for this system. The rotation is primarily about the intermediate axis, but also has an off-axis component. Thus, we see a very complex evolution of the angular velocity (i.e. ‘tumbling’). This tumbling occurs despite having a smaller energy and angular momentum than the previous system. 2
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