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# lecture21 - Attitude dynamics Ref Curtis Sec 9.4 Attitude...

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Attitude dynamics Ref.: Curtis Sec. 9.4 Attitude dynamics is the study of how space- craft orientation changes in time with or with- out external torques. The principal result we will obtain are the differential equations de- scribing how the angular velocity changes over time, called Euler’s equations. They are the equivalent in attitude dynamics of the orbital equation of motion arising from the law of gravitation. L. Healy – ENAE404 – Spring 2007 – Lecture 21 (Apr. 12) 1

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Coordinate of each point on spacecraft Since we will need to deal with the spacecraft as a body of finite extent, we define r , the vector from the center of earth to the c.m. of the spacecraft, as always. s , the vector of any point in the spacecraft from its c.m., so s r which will be rele- vant for gravity gradients. R = r + s , the vector from the center of the earth to the actual point on the satellite. L. Healy – ENAE404 – Spring 2007 – Lecture 21 (Apr. 12) 2
Rotating spacecraft For every point i on the spacecraft, these vec- tors are related by a sum, R i = r + s i , The velocity vector is the time derivative of each position vector, plus the angular motion about the center of mass ˙ R i = ˙ r + ˙ s i + ω i × s i . L. Healy – ENAE404 – Spring 2007 – Lecture 21 (Apr. 12) 3

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Angular momentum for each piece The angular momentum in the inertial frame about the satellite c.m. is H i = s i × m i ˙ R i = s i × m i r + ˙ s i + ω i × s i ) . Note that we’re talking about actual angular momentum, mass included, not specific angu- lar momentum, as we’ve been accustomed to doing in our orbit dynamics discussions. Also, this is not the orbital angular momentum h we’ve talked about in class up until now. For a rigid body ˙ s i = 0 and ω = ω i is the same for all pieces, H i = s i × m i r + ω × s i ) = m i s i × ˙ r + m i s i × ( ω × s i ) . To get the totals over the entire spacecraft, we sum over i for all the mass pieces in the spacecraft, or take the limit of small pieces, and turn the sum into an integral. L. Healy – ENAE404 – Spring 2007 – Lecture 21 (Apr. 12) 4
Angular momentum for the whole body Ref.: Curtis Sec. 9.5 The angular momentum of the entire body is the sum over all the mass units that make up the body, H = i m i s i × ˙ r + i m i s i × ω × s i , but the definition of the center of mass is i m i s i = 0, so the first term is zero, H = i m i s i × ( ω × s i ) . This double cross product can be rewritten us- ing the “BAC-CAB” rule s i × ( ω × s i ) = ω s 2 i - s i ( s i · ω ) . Now write out in IJK components, with the components of s s i = x i ˆ I + y i ˆ J + z i ˆ K . L. Healy – ENAE404 – Spring 2007 – Lecture 21 (Apr. 12) 5

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Angular momentum by component H = ˆ I ω x i m i ( y 2 i + z 2 i ) I x - ω y i m i x i y i I xy - ω z i m i x i z i I xz + ˆ J ω y i m i ( x 2 i + z 2 i ) I y - ω x i m i x i y i I yx - ω z i m i y i z i I yz + ˆ K ω z i m i ( x 2 i + y 2 i ) I z - ω x i m i x i z i I zx - ω y i m i y i z i I zy .
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