Page 247 - Dynamics of Mechanical Systems

0593_C07_fm Page 228 Monday, May 6, 2002 2:42 PM

228 Dynamics of Mechanical Systems

7.11 Inertia Ellipsoid
Geometrical interpretations are sometimes used to obtain insight into the nature of inertia
quantities. Of these, one of the most extensively used is the inertia ellipsoid, which may
be developed from Eq. (7.5.5). Specifically, for a given reference point O the moment of
inertia of a system for the direction of a unit vector n is:
a
I = a a I (7.11.1)
aa i j ij

where the a and the I are components of n and the inertia dyadic I relative to mutually
i ij a
perpendicular unit vectors n (i = 1, 2, 3). If the n are principal unit vectors, the products
i i
of inertia, I (i ≠ j), are zero. Eq. (7.11.1) then takes the simplified form:
ij
I = a I + a I + a I (7.11.2)
2
2
2
aa 1 11 2 22 3 33
Then, by dividing by I , we have:
aa
a 2 a 2 a 2
1 = 1 + 2 + 3 (7.11.3)
(I aa I 11) (I aa I 22) (I aa I 33)

Because the denominators of this expression are all positive, we may write it in the form:

a 2 a 2 a 2
1 = 1 + 2 + 3 (7.11.4)
a 2 b 2 c 2

where a, b, and c are defined by inspection in comparison with Eq. (7.11.3). Eq. (7.11.4) is
immediately seen to be the equation of an ellipsoid in the Cartesian coordinate space of
(a , a , a ). The distance from the ellipse center (at the origin O) to a point P on the surface
1 2 3
is proportional to the moment of inertia of a system for the direction of a unit vector n
a
parallel to OP. For example, if n is directed parallel to n such that (a , a , a ) is (1, 0, 0),
a 1 1 2 3
then I is I , a principal moment of inertia. This property has led the ellipsoid of Eq.
aa 11
(7.11.4) to be called the inertia ellipsoid.

7.12 Application: Inertia Torques
Consider a rigid body B moving in an inertial reference frame R (see Section 6.9) as
depicted in Figure 7.12.1. Let B be considered to be composed of a set of N particles P i
having masses m (i = 1,…, N). Let G be the mass center of B. Then, from Eqs. (6.9.9) and
i
(6.9.11), we recall that the system of inertia forces acting on B (through particles P ) is
i
equivalent to a single force F passing through G together with a couple having torque T *
*
*
*
where F and T are:
*
F =−M a (7.12.1)
G