Page 269 - Dynamics of Mechanical Systems

0593_C08_fm Page 250 Monday, May 6, 2002 2:45 PM

250 Dynamics of Mechanical Systems

*
passing through an arbitrary point (say G) together with a couple with torque T . Then,
*
*
F and T are:
N N
* ∑
F = F = m R a P i (8.6.3)
*
∑ i i
= i 1 = i 1
and
N N
T = r × F * = − ∑ m r × R a P i (8.6.4)
*
∑ i i ii
= i 1 = i 1
where N is the number of particles of B. Recall that we already examined the summation
in Eqs. (8.6.3) and (8.6.4) in Section 7.12. Specifically, by using the definitions of mass
*
center and inertia dyadic we found that F and T could be expressed as (see Eqs. (6.9.9),
*
(7.12.1), and (7.12.8)):
*
F =−M RG (8.6.5)
a
and
T =− I BG ⋅ −αωω × I ( BG ⋅ ) (8.6.6)
ωω
α
*
where M is the total mass of B.
Consider the form of the inertia torque: Suppose n , n , and n are mutually perpen-
3
2
1
dicular unit vectors parallel to central principal inertia axes of B. Then, the inertia dyadic
I B/G may be expressed as:
I BG = I n n + I n n + I n n (8.6.7)
11 1 1 22 2 2 33 3 3
Let the angular acceleration and angular velocity of B be expressed as:

αα = α n + α n 2 + α n 3 = α n i
1 1
i
3
2
(8.6.8)
and ωω = ω n + ω n 2 + ω n 3 = ω n i
i
3
2
1 1
*
Then, in terms of the α , ω , I , and the n (i = 1, 2, 3), the inertia torque T may be expressed
i
i
i
ii
as:
*
T = T n + T n + T n = T n (8.6.9)
1 1 2 2 3 3 i i
where the components T (i = 1, 2, 3) are:
i
2 (
T =−α I + ω ω I − ) (8.6.10)
I
1 1 11 3 22 33
3 (
T =−α I + ω ω I − ) (8.6.11)
I
2 2 22 1 33 11
1 (
T =−α I + ω ω I − ) (8.6.12)
I
3 3 33 2 11 22