Page 505 - Dynamics of Mechanical Systems

0593_C14_fm Page 486 Tuesday, May 7, 2002 6:56 AM

486 Dynamics of Mechanical Systems

By inspection of Eqs. (14.4.1) and (14.4.2), we have:

a =−g k (14.4.4)

Thus, G moves as a projectile particle having a parabolic path (see Section 8.7). Note
further that points of B not lying on the central principal inertia axis of rotation will not
have a parabolic path. That is, as B rotates it rotates about the central inertia axis.
From an inspection of Eqs. (14.4.1) and (14.4.3) we have:

ω
ω
α
I⋅+αωω × ( I⋅ ) = 0 (14.4.5)
Let n , n , and n be mutually perpendicular unit vectors fixed in B and parallel to the
1
2
3
central principal inertia axes of B. Let ω and α be expressed in terms of n , n , and n as:
1
2
3
ωω = ω n + ω n + ω n = ω n
1 1 2 2 3 3 i i
(14.4.6)
αα = α n + α n + α n = α n
1 1 2 2 3 3 i i
Because the n (i = 1, 2, 3) are fixed in B the α are derivatives of the ω (see Eq. (4.4.6)). That is,
i
i
i
ω
α = ˙ (14.4.7)
i i
By substituting from Eqs. (14.4.6) and (14.4.7) into (14.4.5), we obtain:
−I ˙ ω + ω ω (I − ) = 0 (14.4.8)
I
11 1 2 3 22 33
−I ˙ ω + ω ω (I − ) = 0 (14.4.9)
I
22 2 3 1 33 11
−I ˙ ω + ω ω (I − ) = 0 (14.4.10)
I
33 3 1 2 11 22
where I , I , and I are the central principal moments of inertia.
33
22
11
Equations (14.4.8), (14.4.9), and (14.4.10) form a set of three coupled nonlinear ordinary
differential equations for the three ω (i = 1, 2, 3). To use these equations to determine the
i
stability of rotation of the body, let B be thrown into the air such that B is initially rotating
about the central principal inertia axis parallel to n . That is, let B be thrown into the air
1
such that its initial angular velocity components ω are:
i
ω =Ω (14.4.11)
1
ω = 0 (14.4.12)
3

ω = 0 (14.4.13)
2