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**Unformatted text preview: **MAE146: Astronautics ‘ Homework #7 Assigned: Friday, March 6"", 2009 Due date: Friday , March 13””, 2009,
(at the beginning of the lecture) Note: “Textbook" refers to \Viesel’s book, “Spaceﬂight dynamics”. Please justify all of your answers. Problem 1 (20 pts): Sensitivity Analysis
Textbook, page, .314, # 5
5. Sensitivity analysis is an important part of the planning of any mission, since the ﬂight
hardware will never exactly achieve the required burnout velocity Taking differentials
of the energy equation for the departure hyperbola, 5 l 7 l 7 11 (ll 41}
:2: m vi 3 —— v“ .._. __. ./_
2 ix, 2 0 r0 - show that the speed error crossing the activity sphere, dugU is related to the error in
the burnout speed, duo, by dvxzm-dvo (lid?) In the heliocentric portion of the ﬂight, the Hohmann transfer ellipse has semimajor axis #6 Lo;
(1,: I: 6 It, (H44)
25 VI— ” hie/Re where V] : Va; + ex, Re, is the earth’s orbital radius, Va, is the velocity of the
earth, and [LG is the gravitational parameter of the sun. Write the aphelion distance as
R; 2 2a, — R9,, and by taking differentials show that the aphelion error ng is given by [KEV
(1122 = ”1.4..“ (1in (11.45)
(V172 _ Lie/Re) Problem 2 (30 pts): Euler Angles singularity 0 Show that the relation between the coordinate of the angular velocity in a body ﬁxed frame, (oz/‘1 , tag, gig) to the
Euler angle in a 3—1-3 relation (with angles 0. E). ii“) are given by the relations below 0 Textbook, page l21, # l I. The angular—velocity component expressions for the Euler angle rates (15, 9, and 11/,
an : (/5 sianinw +9 (3081/!
an : qb sin6cosw ——é sinw 1/} + (b c059 ll 503 are three linear equations in the three angular rates. Solve for these angular rates, and
show that difﬁculties occur for 6 ~—> O. Problem 3 (20 pts): Inertia matrix in inertial space Textbook, pages 121. #2 2. The expression H = 14.0“ is still correct when expressed in the unit vectors of an
inertial frame, although the moment-of-inertia tensor is no longer constant. Show that
the moment—of-inertia matrix obeys i . 0 "(1)3 a);
~— 1 = wb‘ x 1 = (1)3 0 Ma); {I} (4.98)
dt sz w] 0 and that the statement M = H becomes
M 2 (wb‘ x 1)wbi + 1w“ (4.99) in the inertial frame.
Problem £(30 pts): Gyrostat satellites o Textbook. page, 162. #3 3. A dual—spin satellite shown in Figure 5.22, sometimes referred to as a gyrostal, has
principal moments of inertia of A, B, and C about the axes b1. b3, and b3. It contains
a rotor with moment of inertia [,1 which spins on frictionless bearings with constant
angular rate to,” If the angular-velocity vector of the satellite is wbi : {am (03, any.
show that the total angular momentum is H3 = mu? + 133%? + (Cw3 + 1,119.)? (5.97) while the total kinetic energy is l T = 5 (A0)? + Brag + C0); + Leos) (5-98) and that both are constant. Show that both of these expressions describe ellipsoids, but
that one ellipsoid is no longer centered at the satellite’s center of mass. Argue that it
is still possible to deﬁne polhodes for a gyrostat as the intersection curves of the two
ellipsoids. I Use the code, provided on the, course web site to generate sample ellipsoids. polhorles (angular Velocity orbits)
and a sequence of frame for the rotational motion of a gyrostnt satellite. Choose .41. B. (7. II‘ to be, 1.0. 2.0. 3.0 and 1.0 Mint“). respectively and (an. egg. W13. my.) to be (0.1, i 0.1, 0.8., _, 0.3) rad/s. Then try with the
following angular veloeity (0.0. 0.9. 0.1). 0.1) rad/s. Can you interpret the results in terms of stability 2’ FIGURE 5.22
Dual~spin satellite. ' MN, [\5 ...

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