Page 221 - Dynamics of Mechanical Systems

0593_C07_fm Page 202 Monday, May 6, 2002 2:42 PM

202 Dynamics of Mechanical Systems

Z
P(x,y,z)
n z

p
O
Y
n x
n y
FIGURE 7.3.3
A particle P in a Cartesian reference frame. X

and, similarly,

[
−
2
I PO = myx n +( x + z 2 n ) − yz n z] (7.3.7)
y x y
and
[
−
2
I PO = mzx n − zy n +( x + y 2 n ) z] (7.3.8)
z
y
x
Using the definitions of Eqs. (7.3.1) and (7.3.2), we see that the various moments and
products of inertia are then:
I PO = ( 2 y 2 I PO =− mxy, I PO =− mxz
mx + ),
xx xy xz
I PO =− myx, I PO = ( 2 y 2 I PO =− myz (7.3.9)
mx + ),
yx xy yz
my + )
I PO =− mzx, I PO =− mzy, I PO = ( 2 z 2
zx zy zz
Observe that the moments of inertia are always nonnegative or zero, whereas the products
of inertia may be positive, negative, or zero depending upon the position of P.
It is often convenient to normalize the moments and products of inertia by dividing by
the mass m. Then, the normalized moment of inertia may be interpreted as a length
squared, called the radius of gyration and defined as:

D PO m] 12
k = [ I (7.3.10)
a aa
(See also Eq. (7.3.3).)
Finally, observe that the moments and products of inertia of Eqs. (7.3.9) may be conve-
niently listed in the matrix form:

 ( y + ) − xy − xz 
2
2
z
 
2
2
x
I PO = m  − yx ( z + ) − yz  (7.3.11)
ab
 ( x + ) 
2
2
 − zx − zy y 
where a and b can be x, y, or z.