# HW16 - R to be inertial while B to be a body ±xed frame....

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MAE146: Astronautics – Homework Assigned: Wednesday, March 10 th , 2010 Due date: Friday , March 12 th , 2010, (at the beginning of the lecture) Please justify all of your answers. Write your answer cleanly with sentences and diagrams to explain. Don’t forget to write your name on your copy to receive credit. Problem 1 (20pts): On the inertia tensor Let R = (0 , ˆ e 1 , ˆ e 2 , ˆ e 3 ) abd B = (0 , ˆ b 1 , ˆ b 2 , ˆ b 3 ) be two reference frames centered at the center of mass of a rigid body. 1. Denoting the inertia tensor expressed in R and B as I 1 and I 2 , respectively (both are symmetric 3 × 3 matrices), show that these matrices are related by I 2 = A.I 1 .A T , where A denotes the rotation matrix from the R to the B frame. 2. Assume
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Unformatted text preview: R to be inertial while B to be a body ±xed frame. Compute the time derivative of the inertia tensor relative to R and B . Problem 2 (20pts): On integrals of motion Assume the angular velocity of a rigid body, vω , satis±es the rotational dynamics we have seen in class with no external torques applied. Show that the following quantities are integrals of motion of these rotational dynamics: 1. H = ∑ i ( ∑ j I i,j .ω j ) 2 2. T = 1 2 ∑ i,j I i,j ω i ω j where I i,j denotes the ( i, j ) component of the inertia matrix and ω i the coordinates of vω , both expressed in a body frame. 1...
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## This note was uploaded on 03/13/2010 for the course MAE 19100 taught by Professor Villac during the Winter '10 term at UC Irvine.

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