Environmental Engineering Reference
In-Depth Information
which allows us to write Euler's equations, Eq. (10.53) with
τ
=
0
as:
I
1
˙
ω
1
=
(I
2
−
I
3
)ω
2
ω
3
,
(10.85)
I
2
˙
ω
2
=
(I
3
−
I
1
)ω
1
ω
3
,
(10.86)
I
3
˙
ω
3
=
(I
1
−
I
2
)ω
1
ω
2
≈
0
.
(10.87)
Since Eq. (10.87) implies that
ω
3
is approximately constant we can set
ω
3
=
ω
t
where
ω
t
is a constant, and write
I
1
˙
ω
1
=
(I
2
−
I
3
)ω
2
ω
t
,
(10.88)
I
2
˙
ω
2
=
(I
3
−
I
1
)ω
1
ω
t
.
(10.89)
Differentiating Eq. (10.88) with respect to time and substituting for
ω
2
from
˙
Eq. (10.89) gives us
(I
2
−
I
3
)(I
3
−
I
1
)
ω
t
ω
1
.
ω
1
=−
¨
(10.90)
I
1
I
2
Eq. (10.90) is an equation describing a harmonic oscillation in
ω
1
with frequency
,where
(I
2
−
I
3
)(I
3
−
I
1
)
2
ω
t
.
=
(10.91)
I
1
I
2
Provided that
2
>
0, we can obtain a real frequency and the solution will be of
the form
ω
1
=
A
cos
(t
+
δ),
(10.92)
where
A
and
δ
are constants that are fixed by the orientation of the body at
t
0.
The oscillation will remain of small amplitude if it begins with small amplitude.
However if
2
<
0, then
is imaginary and there is no oscillation of
ω
1
about
zero. Rather, the general solution to Eq. (10.90) becomes
=
Ae
κt
Be
−
κt
,
ω
1
=
+
(10.93)
where
κ
=
i
is real and
A
and
B
are constants. This solution is unstable: even a
=
tiny
ω
1
at
t
0 will blow up as
t
increases. We conclude that for a stable rotation
we must therefore have
(I
2
−
I
3
)(I
3
−
I
1
)
2
ω
t
>
0
,
=
(10.94)
I
1
I
2
which occurs if
I
3
is the largest, or the smallest, of the three principal moments.
However, if
I
3
is the intermediate moment of inertia, i.e.
I
1
<I
3
<I
2
or
I
2
<I
3
<I
1
(10.95)
then
2
<
0 and the rotation is unstable.