Page 239 - Dynamics of Mechanical Systems

0593_C07_fm Page 220 Monday, May 6, 2002 2:42 PM

220 Dynamics of Mechanical Systems

Let P and P be a typical pair of particles taken from a set S of particles for which Π is
1
2
a plane of symmetry. Then, from Eq. (7.7.2) we see that the contributions of P and P to
2
1
the second moment of S relative to O for the direction of n are represented by:
×
×
I SO = m p ×( n p ) + m p ×( n p ) +K
n 1 1 2 2
+
= m( OP h n) ×[ n ×( OQ h n)]
+
(7.9.36)
−
−
+ m( OQ h n) ×[ n ×( OQ h n)] +K
2
= 2 m OQ n +K
where the last equality follows from expanding the triple products of the previous equality
and where the terms not written in Eq. (7.9.36) represent the contributions to I SO from the
a
other pairs of particles of S. However, by analyses identical to those shown in Eq. (7.9.36),
these contributions to I SO are also parallel to n. Therefore, I SO is parallel to n, and thus
a n
n is a principal unit vector of S for O.
A specialization of these ideas occurs if the system of particles all lie in the same plane,
as with a planar body. A unit vector normal to the plane is then a principal unit vector.
There are other ways of determining principal unit vectors without solving Eqs. (7.7.10)
and (7.7.12). For homogeneous bodies occupying common geometric shapes, we can
simply refer to a table of results as in Appendix II. We can also make use of a number of
theorems for principal unit vectors stated here, but without proof (see References 7.1 and
7.2 for additional information). If a line is parallel to a principal unit vector for a given
reference point, then that line is called a principal axis for that reference point. A principal
axis for the mass center as a reference point is called a central principal axis. Then, it can
be shown that:

1. If a principal axis for a reference point other than the mass center also passes
through the mass center, then it is also a central principal axis.
2. Mutually perpendicular principal axes that include a central principal axis for
any point on the central principal axis are parallel to mutually perpendicular
central principal axes.
3. A central principal axis is a principal axis for each of its points.

Finally, regarding the question as to whether the procedures of the example of the
previous section will always produce principal unit vectors and principal moments of
inertia, the answer is yes, but as seen above there may exist simpler procedures for any
given problem. That is, a principal unit vector may often be identified by inspection —
for example, as being normal to a plane of symmetry. In this case, the task of solving a
cubic equation may be reduced to that of solving a quadratic equation.
To illustrate the procedure, suppose that a principal unit vector found by inspection is
called n . Then, let n and n be unit vectors perpendicular to n and also perpendicular
2
3
3
1
to each other so that n , n , and n form a mutually perpendicular set. Then, because n 3
1
2
3
is a principal unit vector, the inertia dyadic I expressed in terms of n , n , and n has a
1
3
2
matrix of components in the form:
I  I 0 
 11 12 
I = I I 0 (7.9.37)
ij  21 22 
  0 0 I  
33