Page 224 - Dynamics of Mechanical Systems

0593_C07_fm Page 205 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 205

Finally, for systems of particles or for rigid bodies, the inertia dyadic is developed from
the contributions of the individual particles. That is, for a system S of N particles we have:

N
I SO = ∑ I PO (7.4.13)
i
i=1

7.5 Transformation Rules

Consider again the definition of the second moment vector of Eq. (7.2.1):

I PO = m p ×( n × p) (7.5.1)
a a

Observe again the direct dependency of I PO upon n . Suppose the unit vector n is
a a a
expressed in terms of mutually perpendicular unit vectors n (i = 1, 2, 3) as:
i
n = a n + a n + a n = a n (7.5.2)
a 1 1 2 2 3 3 i i

Then, by substitution from Eq. (7.5.1) into (7.5.2), we have:

I PO = m p ×( a i n × p) = a m p ×( n × p)
i
i
i
a
or
I PO = a I PO = a I PO + a I PO + a I PO (7.5.3)
a ii 1 1 22 33

Equation (7.5.3) shows that if we know the second moment vectors for each of three
mutually perpendicular directions, we can obtain the second moment vector for any
direction n .
a
Similarly, suppose n is a second unit vector expressed as:
b
n = b n + b n + b n = b n (7.5.4)
b 1 1 2 2 3 3 j j

Then, by forming the projection of I PO onto n we obtain the product of inertia I PO , which
a b ab
in view of Eq. (7.5.3) can be expressed as:

⋅
⋅
I PO = n I PO = ab n I PO
ab b a ij j i
or

I PO = ab I PO (7.5.5)
ab ij ij