Page 225 - Dynamics of Mechanical Systems

0593_C07_fm Page 206 Monday, May 6, 2002 2:42 PM

206 Dynamics of Mechanical Systems

Observe that the scalar components a and b of n and n of Eqs. (7.5.2) and (7.5.3) may
a
i
i
b
be identified with transformation matrix components as in Eq. (2.11.3). Specifically, let ˆ n j
(j = 1, 2, 3) be a set of mutually perpendicular unit vectors, distinct and noncollinear with
the n (i = 1, 2, 3). Let the transformation matrix components be defined as:
i
⋅
S = nn ˆ (7.5.6)
ij i j
Then, in terms of the S , Eq. (7.5.5) takes the form:
ij
ˆ PO = SS I PO (7.5.7)
I
k1 ik j1 ij

7.6 Parallel Axis Theorems
Consider once more the definition of the second moment vector of Eq. (7.2.1):

I PO = m p ×( n × p) (7.6.1)
a a
Observe that just as I PO is directly dependent upon the direction of n , it is also dependent
a a
upon the choice of the reference point O. The transformation rules discussed above enable
us to evaluate the second-moment vector and other inertia functions as n changes. The
a
parallel axis theorems discussed in this section will enable us to evaluate the second-
moment vector and other inertia functions as the reference point O changes.
To see this, consider a set S of particles P (i = 1,…, N) with masses m as in Figure 7.6.1.
i i
Let G be the mass center of S and let O be a reference point. Let pG locate G relative to
O, let p locate P relative to O, and let r locate P relative to G. Finally, let n be an arbitrary
i i i i a
unit vector. Then, from Eq. (7.2.2), the second moment of S for O for the direction of n is:
a
N
a ∑ p ×( n × )
SO
I = m i i a p i (7.6.2)
i=1

S
n a
P (m )
2 2
P (m )
1
1
P (m )
i
i
r
i
P (m )
N N G
p
p i
G
FIGURE 7.6.1
A set of particles with mass center G. O