PROBLEM 18.158
The essential features of the gyrocompass are shown. The rotor
spins at the rate
about an axis mounted in a single gimbal,
which may rotate freely about the vertical axis AB. The angle
formed by the axis of the rotor and the plane of the meridian is
denoted by
,
and the latitude of the position on the earth is
denoted by
.
We note that line OC is parallel to the axis of the
earth, and we denote by
e
ω
the angular velocity of the earth about
its axis.
(a) Show that the equations of motion of the gyrocompass are
22
cos sin cos sin cos 0
ze e
II I
0
z
I
where
z
is the rectangular component of the total angular
velocity
ω
along the axis of the rotor, and I and I are the
moments of inertia of the rotor with respect to its axis of
symmetry and a transverse axis through O, respectively.
(b) Neglecting the term containing
2
,
e
show that for small values
of
, we have
cos 0
ze
I
I
and that the axis of the gyrocompass oscillates about the
north-south direction.
PROBLEM 18.158 (Continued)
The angular momentum
O
H of the rotor is
Oxz yy zz
II I
Hijk
We observe that the rotor is free to spin about the z axis and free to rotate about the y axis. Therefore,
the y and z components of O
M must be zero. It follows that the coefficients of j and k at the right-
ze e e e
Observing that the last two terms cancel out, we have
cos sin 0
ze
II
or cos sin 0
ze
I
I
(8)
PROBLEM 18.158 (Continued)
where the coefficient of sin
is a constant. The rotor, therefore, oscillates about the line NS as a
gyrocompass, therefore, cannot be used in the polar regions.