Tumble
The wobble of a freely spinning body becomes unstable under a very special condition. The condition
is called the
tumble condition
. The tumble condition is developed below. Begin with Eq. (11.4 – 5).
Let
Σ
M
CX
=
M
CY
=
M
CZ
= 0 and assume that the body is initially spinning about the
Z
axis, almost,
so the angular velocity components of the body about the
X
and
Y
axes are initially small. In Eq. (11.4
– 5), the term
ϖ
X
Y
is “small squared” since the terms
X
and
Y
are each small. Neglecting the
X
Y
term in Eq. (11.4 – 5), it reduces to
(
a – c
)
Z
X
Z
Y
Z
X
Y
Z
Y
X
Y
Z
X
I
I
I
I
I
I
=

+
=

+
=
0
)
(
0
)
(
0
From (
c
),
Z
is a constant. Equations (
a
) and (
b
) are now linear, simultaneous equations in terms of
X
and
Y
. To decouple the equations, differentiate (
a
) with respect to time and substitute (
b
) into the
result eliminating
Y
. Similarly, differentiate (
b
) with respect to time and substitute (
a
) into the result
eliminating
X
. This yields
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 Spring '08
 Silverberg
 Angular Momentum, Moment Of Inertia, Rigid Body, MCX, angular velocity components, tumble condition

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