Environmental Engineering Reference
In-Depth Information
These are in general different from the principal moment of inertia I 3 . Thus the
third Euler equation gives
I 3 ˙
ω 3 =
0
(10.55)
from which we immediately conclude that
ω 3 =
constant
ω t .
(10.56)
We call ω t the top (or spin) frequency; it is the angular speed at which the top
spins about its symmetry axis. We now have the task of solving for ω 1 and ω 2 from
the first two of the Euler equations, which are now linear since ω t is constant:
I
ω 1 +
˙
(I 3
I)ω 2 ω t =
0 ,
(10.57)
I
ω 2 +
˙
(I
I 3 1 ω t =
0 .
(10.58)
Introducing the frequency
I 3
I
=
ω t
(10.59)
I
these equations can be rewritten as
ω 1 +
˙
ω 2 =
0 ,
(10.60)
ω 2
˙
ω 1 =
0 .
(10.61)
Differentiating Eq. (10.60) and substituting for
ω 2 using Eq. (10.61) leaves us with
˙
a second-order ordinary differential equation:
2 ω 1 =
ω 1 +
¨
0 .
(10.62)
In a similar fashion we can obtain the corresponding equation for ω 2 :
2 ω 2 =
ω 2 +
¨
0 .
(10.63)
Eq. (10.62) and Eq. (10.63) should be immediately recognisable as equations for
simple harmonic motion of frequency in each of the variables ω 1 and ω 2 . Hence,
the general solution for ω 1 is
ω 1 =
A cos (t
+
φ).
(10.64)
ω 2 is governed by a similar equation, except that the amplitude and phase of this
second equation are not independent of the constants A and φ . Rather, Eq. (10.60)
implies that
ω 2 =− ˙
ω 1
=
A sin (t
+
φ).
(10.65)
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