Page 165 - The Mechatronics Handbook

0066-frame-C09 Page 39 Friday, January 18, 2002 11:00 AM

It is possible to find the quaternions and the elements of the direction cosine matrix independently by
integrating the angular rates about the principal axes of a body. Given the direction cosine matrix
elements, we can find the quaternions, and vice versa. For a more extended discussion and application,
the reader is referred to the listed references.

Dynamic Properties of a Rigid Body

Inertia Properties
The moments and products of inertia describe the distribution of mass for a body relative to a given
coordinate system. This description relies on the specific orientation and reference frame. It is presumed
that the reader is familiar with basic properties such as mass center, and the focus here is on those
properties essential in understanding the general motion of rigid bodies, and particularly the rotational
dynamics.
Moment of Inertia. For the rigid body shown in Fig. 9.33(a), the moment of inertia for a differential
element, dm, about any of the three coordinate axes is defined as the product of the mass of the differential
2
2
element and the square of the shortest distance from the axis to the element. As shown, r x = y + z ,
so the contribution to the moment of inertia about the x-axis, I xx , from dm is
2
2
2
dI xx = r x = ( y + z )dm
The total I xx , I yy , and I zz are found by integrating these expressions over the entire mass, m, of the body.
In summary, the three moments of inertia about the x, y, and z axes are

2
I xx = ∫ r x dm = ∫ ( y + z ) dm
2
2
m m
2
I yy = ∫ r y dm = ∫ ( x + z ) dm (9.22)
2
2
m m
I zz = ∫ r z dm = ∫ ( x + y ) dm
2
2
2
m m
Note that the moments of inertia, by virtue of their definition using squared distances and finite mass
elements, are always positive quantities.
z z a

dm
z G
O y
y a
r x
z G
x a
O
x
y
y G x G
x x
(a) (b)

FIGURE 9.33 Rigid body properties are defined by how mass is distributed throughout the body relative to a
specified coordinate system. (a) Rigid body used to describe moments and products of inertia. (b) Rigid body and
axes used to describe parallel-axis and parallel-plane theorem.

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