Environmental Engineering Reference
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which allows us to write Euler's equations, Eq. (10.53) with
τ =
0 as:
I 1 ˙
ω 1 =
(I 2
I 3 2 ω 3 ,
(10.85)
I 2 ˙
ω 2 =
(I 3
I 1 1 ω 3 ,
(10.86)
I 3 ˙
ω 3 =
(I 1
I 2 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 2 ω t ,
(10.88)
I 2 ˙
ω 2 =
(I 3
I 1 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.
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