Environmental Engineering Reference
In-Depth Information
is parallel to the body-fixed
e
3
axis. As the gyroscope precesses with angular velocity
Let us now get rid of the approximation that
ω
ω
p
about the vertical
it gives rise to another contribution to the rotation about
e
3
, as measured in the
lab frame. At any time we can project the total
ω
on to the symmetry axis of the
gyroscope to obtain
=
+
ω
3
ω
t
ω
p
cos
θ,
(10.100)
where, as before,
θ
is the angle between the
e
3
axis and the vertical direction.
Previously, we made the approximation that
ω
3
≈
ω
t
which is valid only as long
as the top frequency is much higher than the precession frequency. Gyroscopes
are constructed so that there is very little friction in the rotation of the flywheel
so we shall treat
ω
t
as a constant. The fact that the torque is always perpendicular
to the
e
3
direction implies that the projection of the angular momentum onto the
symmetry axis of the gyroscope is a conserved quantity, i.e.
d
L
3
d
t
=
τ
3
=
0
,
(10.101)
where
L
3
=
I
3
ω
3
.
(10.102)
The angular momentum will also have a contribution from the precession in a
direction perpendicular to
e
3
whichwedenoteas
L
⊥
(see Figure 10.20). Assuming
the gyroscope to be a symmetric top we have
I
1
=
I
2
=
I
and can write
L
⊥
=
Iω
p
sin
θ.
(10.103)
The various contributions to
L
are shown in Figure 10.20 in the plane instanta-
neously containing the vertical and
e
3
. All of the vectors in Figure 10.20 are in a
common plane that is precessing about the vertical direction. As such, the compo-
nent of
L
in the horizontal plane describes a circle in the lab. Changes in direction
of this component are a result of the action of the gravitational torque. We can now
write an equation analogous to Eq. (10.98) that now includes the contribution of
the precession to
L
:
(L
3
sin
θ
−
L
⊥
cos
θ)δφ
=
τδt.
Vertical
L
3
⊗τ
L
⊥
q
L
3
sin q −
L
⊥
cos q
Figure 10.20
Components of
L
for a gyroscope. The torque is directed into the page.
