Page 248 - Dynamics of Mechanical Systems

0593_C07_fm Page 229 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 229

B

G

R
FIGURE 7.12.1
A body B moving in an inertial frame R.

and

 N   N 
T =− ∑ m i r ×(αα × r ) − ω ×  ∑ m i r ×(ωω × r ) (7.12.2)
*
i
i
i
i
  = i 1     = i 1  
where a is the acceleration of G in R; ωω ωω and αα αα are the angular velocity and angular
G
acceleration, respectively, of B in R; and r is a position vector locating P relative to G.
i i
We are now in a position to further develop the terms of Eq. (7.12.2). Specifically, we
can conveniently express them in terms of the inertia dyadic of Section 7.4. Recall first
from Eq. (7.2.2) that the second moment of B for G, for a direction of a unit vector n , is:
a
N
a ∑ r)
BG
I = m ×(n a × i (7.12.3)
r
i i
i=1
(Observe the similarity of the sum of Eq. (7.12.3) with those of Eq. (7.12.2).) Then, recall
from Eq. (7.4.11) that the second moment may be expressed in terms of the inertia dyadic
of B for G as:
⋅
I BG = nI BG (7.12.4)
a a
Consider now the terms of Eq. (7.12.2). Let αα αα and ωω ωω be written in the forms:

ω
αα = αn and ωω = n ω (7.12.5)
α
where n and n are unit vectors in the directions of αα αα and ωω ωω at any instant, with α and
ω
α
ω then being three magnitudes of αα αα and ωω ωω. Then, the first term of Eq. (7.12.2) may be
expressed as:
N
∑ m r ×(αα × r ) = N m r ×(α n × )
i ∑
r
α
ii
ii
i
i=1 i=1
N
= αα ∑ m r ×( n × )
r
α
i
ii
i=1
= α I BG (7.12.6)
α
= α nI ⋅ BG
α
=⋅ I BG
αα