Environmental Engineering Reference
In-Depth Information
L
−
I
ω
=
Iω
1
e
1
+
Iω
2
e
2
+
I
3
ω
t
e
3
−
I
ω
,
=
(I
3
−
I)ω
t
e
3
.
(10.73)
Since
ω
t
is a constant, Eq. (10.73) describes a constant relationship between the
three vectors
L
,
ω
and
e
3
such that the three vectors lie in a plane. This certainly
does not mean that the plane formed by the three vectors is itself fixed in space;
although we do know that the direction of
L
is constant it is still possible for
ω
and
e
3
to rotate at the same rate about
L
. This is what happens, as we will
now show by examining the time dependence of
e
3
. To do this we transform the
time-derivative of
e
3
from the body-fixed frame to the lab. We have
d
e
3
d
t
body
+
ω
×
d
e
3
d
t
=
e
3
=
ω
×
e
3
,
(10.74)
since
e
3
is a constant vector in the body-fixed frame. Quite generally, we can
express
e
3
in terms of the basis vectors
i
,
j
,
k
in the lab as
e
3
=
cos
k
+
sin
(
cos
i
+
sin
j
),
(10.75)
where
represents the fixed angle between
e
3
and
L
,and
is the angle between
the projection of
e
3
into the
i
-
j
plane, and the
i
axis (see Figure 10.13). Now
Eq. (10.73) can be rearranged to give
ω
:
L
I
k
I
3
−
I
ω
=
−
ω
t
e
3
(10.76)
I
k
e
3
L
Θ
j
Φ
i
Figure 10.13
The
e
3
vector relative to the lab coordinate axes.
and we can use this in Eq. (10.74) to obtain
L
I
k
ω
t
e
3
d
e
3
d
t
I
3
−
I
L
I
k
=
−
×
e
3
=
×
e
3
.
(10.77)
I
