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
)ω
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)