Question

A spaceship is preparing to dock with a space station. Throughout this problem the space station is motionless in the space frame (inertial frame). Prior to docking, some rotational maneuvers are necessary for the spaceship. At t = 0, the angular velocity of the spaceship is zero, and we have ei = x, e2 = ý and e3 = 2. A number of small rocket engines are attached to the exterior of the spaceship, and they can be switched either all the way on or off. They are used in pairs, so as to produce a torque about a principal axis but no net force. The magnitude of the torque is Imar. The captain will use one pair of rocket engines at a time. The principal moments of the spaceship are 11, 12 and 13. The plan is to first rotate the spaceship by /2 about 2. The angular velocity is to be zero at the conclusion of this maneuver. Next the spaceship will be rotated by 7/2 about x. Again the angular velocity is to be zero at the conclusion of this maneuver. The entire plan is to be carried out as quickly as possible. Your answers to the questions below will consist of four separate formulas for four consecutive time intervals. (a) The captain of the spaceship controls T1, T2 and 13 as functions of time, as explained above. What functions of time does she choose for the plan defined above? (b) Calculate the components of the unit vector e (t) with respect to the space frame axes. (c) Write out formulas for wilt), w2(t) and w3(t), the components of the angular velocity with respect to the body frame. (d) Use Euler's equations, Eq. (10.88), to find the components of the torque in the body frame. Do the results agree with your answer to part (a)?

A spaceship is preparing to dock with a space station. Throughout this problem the space station is motionless in the space frame (inertial frame). Prior to docking, some rotational maneuvers are necessary for the spaceship. At t = 0, the angular velocity of the spaceship is zero, and we have e_1 = x cap, e_2 = y cap and e_3 = z cap. A number of small rocket engines are attached to the exterior of the spaceship, and they can be switched either all the way on or off. They are used in pairs, so as to produce a torque about a principal axis but no net force. The magnitude of the torque is gamma_max. The captain will use one pair of rocket engines at a time. The principal moments of the spaceship are lambda_1, lambda_2 and lambda_3. The plan is to first rotate the spaceship by pi/2 about z cap. The angular velocity is to be zero at the conclusion of this maneuver. Next the spaceship will be rotated by pi/2 about x. Again the angular velocity is to be zero at the conclusion of this maneuver. The entire plan is to be carried out as quickly as possible. Your answers to the questions below will consist of four separate formulas for four consecutive time intervals. (a) The captain of the spaceship controls gamma_1, gamma_2 and gamma_3 as functions of time, as explained above. What functions of time does she choose for the plan defined above? (b) Calculate the components of the unit vector e_1 (t) with respect to the space frame axes. (c) Write out formulas for omega_1 (t), omega_2 (t) and omega_3 (t), the components of the angular velocity with respect to the body frame. (d) Use Euler's equations, Eq. (10.88), to find the components of the torque in the body frame. Do the results agree with your answer to part (a)?

Answer 1

Here we have assumed $\Gamma _{1},\Gamma _{2}\; and \; \Gamma _{3}$ as torque components about x,y and z axes in the fixed space station frame.
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