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Figure 1: Satellite in orbit around an astroid .
April 5th & 6th , 2013
University of Minnesota
Department of Aerospace Engineering & Mechanics
Ph.D. Written Preliminary Exam
Dynamics
Allotted Time: 2:30 Hours.
A cylindrical satellite of mass ms is shown in Figure 1 orbiting a large asteroid of mass ma . It orbits while
spinning at a constant rate ωs about the b̂3 axis. The b̂1 axis points in the direction of the velocity vector.
The b̂1 , b̂2 and b̂3 axes are principal and the principle moment of inertia tensor about the center of mass P
is given by:


I1 0 0
I =  0 I2 0 
0 0 I3
1. Assuming â1 , â2 and â3 are inertial, show that the translational equation of motion for the satellite is
the orbit equation given by:
r̈ + µ
r
=0
r3
where r = r2 −r1 , r =k r k and µ ≡ G(ma +ms ). Recall that the variable G is the gravitational constant
in Newton’s law of gravitation which describes the gravitational attraction force on an arbitrary mass
m1 imposed by another arbitrary mass m2 . This force is given by:
F = −G
m1 m2
ûd
d2
where d is the vector pointing from the mass m2 to the mass m1 and û is a unit vector along d.
2. If â1 , â2 and â3 are non-inertial, then show that the translational equation of motion for the satellite
relative to the center of mass of the asteroid/satellite system is given by
r̈2 + µ̃
r2
=0
r23
where r2 =k r2 k and
µ̃ =
ma
ma + ms
3
µ
3. Under what conditions on ma and ms will the equations of motion you derived in questions (1) and
(2) above become the same?
4. Determine the components of the absolute angular velocity vector of the satellite ω = ω1 b̂1 + ω2 b̂1 +
ω3 b̂1 in terms of quantities given in Figure 1 and your answers from questions (1) - (3).
5. Derive the rotational equations of motion of the satellite and determine an expression for the moment
M that has to be applied about the center of mass of the satellite continuously to ensure that the b̂3
axis of the spinning satellite is always pointing up along the line connecting the center of the asteroid
and the satellite.
6. Suppose the satellite is torque-free (i.e., M = 0) and, thus, b3 does not change directions in inertial
space. If we want the satellite to continue spinning about the b3 axis at a rate ωs , determine the
relationship between I3 and I1 as well as I3 and I2 that will ensure that small angular velocity perturbations δω1 , δω2 and δω3 remain bounded (i.e., do not grow continuously with time).
HINT: Write the rotational equations of motion in terms of ω1 + δω1 , ω2 + δω2 and ωs + δω3 . Then
determine what conditions on I1 , I2 and I3 will ensure that any non-zero perturbations δω1 , δω2 and
δω3 will decay with time.
7. The rotational kinetic energy of the satellite is TR = 21 ω · Iω. If ṪR < 0 (i.e., energy is being
dissipated) and the satellite is still torque-free, comment on how the relationships between I1 , I2 and
I3 will have to be modified to ensure angular velocity perturbations remain bounded. Justify your
answer mathematically or using a heuristic argument.
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