Page 167 - The Mechatronics Handbook

0066-frame-C09 Page 41 Friday, January 18, 2002 11:01 AM

ω

i
G
z V A ρ
A
A
r A
O
y
FIGURE 9.34 Rigid body in general motion relative to
an inertial coordinate system, x, y, z. x x
and it relies on the specific location and orientation of coordinate axes in which it is defined. For a rigid
body, an origin and axes orientation can be found for which the inertia tensor becomes diagonalized, or

0 0
I x
I = 0 I y 0
0 0 I z

The orientation for which this is true defines the principal axes of inertia, and the principal moments
of inertia are now I x = I xx , I y = I yy , and I z = I zz (one should be a maximum and another a minimum of
the three). Sometimes this orientation can be determined by inspection. For example, if two of the three
orthogonal planes are planes of symmetry, then all of the products of inertia are zero, so this would
define principal axes of inertia.
The principal axes directions can be interpreted as an eigenvalue problem, and this allows you to find
the orientation that will lead to principal directions, as well as define (transform) the inertia tensor into
any orientation. For details on this method, see Crandall et al. [8].
Angular Momentum
For the rigid body shown in Fig. 9.34, conceptualized to be composed of particles, i, of mass, m i , the
angular momentum about the point A is defined as

( h A) i = ρ A × m i V i

where is the velocity measured relative to the inertial frame. Since V i = V A ω × ρ A , then
+
V i
( h A) i = ρ A × m i V i = m i ρ A × V A + m i ρ A × ( ω × ρ A)

Integrating over the mass of the body, the total angular momentum of the body is

h A = ( ∫ ρ Adm) × V A + ∫ ρ A × ( ω × ρ A) dm (9.24)
m m

This equation can be used to find the angular momentum about a point of interest by setting the
point A: (1) fixed, (2) at the center of mass, and (3) an arbitrary point on the mass. A general form arises
in cases 1 and 2 that take the form

h = ∫ ρ × ( ω × ρ)dm
m

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