Notice that, even if
initially, the torques will drive nutation and precession anyway.
y
=
w
=
y
.
=
w
.
=
0
Equations (41)(44) are the final form for our symmetric, conically shielded spinning spacecraft.
They consist of terms describing forcefree motion (the terms containing
), with the addition of per
turbative terms due to pressures on the top of the spacecraft and on the protective conical shield.
These equations have been implemented in the numerical program, SymTop.
5 Precession
In this section, let us assume a fastspinning top, so that
. Further, assume the pressure
W
h
>>
W
w
,
W
y
terms are small. Take the last three equations of eqs. (41), differente them, substitute eqs. (41) for the
firstorder derivatives in the resulting equations. Finally, drop terms beyond first order in the small
quantities. We find the resulting secondorder system of ODEs,
(45)
sin
y d
2
dt
2
W
w
= 
(1

b)
2
W
h
2
W
w
sin
y
+
(1

b) W
h
K
2
(
a, b, h,
a , A
C
, A
T
,
w, y
)
d
2
dt
2
W
y
= 
(1

b)
2
W
h
2
W
y

(1

b) W
h
K
1
(
a, b, h,
a , A
C
, A
T
,
w, y
)
sin
y d
2
dt
2
W
h
=
(1

b)
2
W
h
2
W
w
sin
y

(1

b) W
h
K
2
(
a, b, h,
a , A
C
, A
T
,
w, y
)
cos
y
Notice that the value of
merely scales the time. Since
is large, we can assume it is slowly vary
W
h
ing compared to
and
. Hence, we may set
. There are two consequences of this
W
w
W
y
d
2
dt
2
W
h
l 0
from eqs. (45). First, the second equation implies simple harmonic mostion for
. Letting
W
y
, we have as solution
W
h
d const
(46)
W
y
l A cos((1

b) W
h
t)
+
B sin((1

b) W
h
t)

K
1
(
a,b,h,
a ,A
C
,A
T
,
w,y
)
(1

b) W
h
Notice the superimposed constant. This implies a small, monotonic drift in the inclination angle .
y
The second consequence is that we can solve for the precession rate. For the third equation of eqs.
(45) to hold, we require
(47)
W
w
l
K
2
(
a, b, h,
a , A
C
, A
T
,
w, y
)
(1

b) W
h
sin
y
If we further assume that the pressure is mainly along the fixedframe Z axis,
, then eq.
o
Z
>>
o
X
,
o
Y
(47) becomes
(48)
W
w
l
o
Z
(1

b) W
h
o
Z
cos
y

(
o
X
sin
w

o
Y
cos
w)
cos
2
y

sin
2
y
sin
y
G
(
a, b, h,
a , A
C
, A
T
)
This equation becomes more clear by further letting
and
.
o
X
=
o
Y
=
0,
o
Z
=
1, A
C
=
A
T
h A
a
=
o
2
Then we have
(49)
W
w
l
(1

A)
o b
2
h
(1

b) I
xy
W
h
P cos
y
Recall that
is the inclination of the symmetry axis to the fixedframe Z axis.
page 9 of 10