Page 226 - Dynamics of Mechanical Systems

0593_C07_fm Page 207 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 207

From Figure 7.6.1, we see that p , r , and p are related by:
i
i
G
p = p + r (7.6.3)
i G i
Then, by substituting into Eq. (7.6.2) we have:

N
a ∑ i ( { p + ) ×[ n ×( p + )]}
=
PO
I m G r i a G r i
i=1
N
G ∑
= ∑ m i p ×( n × ) + N m i p ×( n × )
r
p
a
i
G
a
G
i=1 i=1
N
G ∑
r ×(
+ ∑ m ii n × ) + N m ii r ×( n × )
p
r
i
a
a
i=1 i=1
O
 N    N 
= ∑ m p G ×(n a × ) +p G p G × n a × ∑ m i i 
r
i

i  =1   = i 1 
O
 N  N
×
+ ∑ m ii  (n a × ) + ∑ m i i ×(n a × ) r i
r
r
p
G
 i =1  = i 1
or
I SO = I GO + I SG (7.6.4)
a a a
where I GO is defined as:
a
GO D p ×( n × )
I = M p (7.6.5)
a G a G
where M is the total mass,
 N 
 ∑ m i ,
 i=1 
of the particles of S; and I GO is the second moment of a particle located at G with a mass
a
equal to the total mass M of S.
Eq. (7.6.4) is often called a parallel axis theorem. The reason for this name can be seen
from the analogous equation for the moments of inertia: that is, by examining the projec-
tion of the terms of Eq. (7.6.4) along n , we have:
a
I SO n ⋅ = I GO n ⋅ + I SG n ⋅
a a a a a a
or
I SO = I GO + I SG (7.6.6)
aa aa aa