Page 166 - The Mechatronics Handbook

0066-frame-C09 Page 40 Friday, January 18, 2002 11:00 AM

Product of Inertia. The product of inertia for a differential element dm is defined with respect to a
set of two orthogonal planes as the product of the mass of the element and the perpendicular (or shortest)
distances from the planes to the element. So, with respect to the y − z and x − z planes (z common axis
to these planes), the contribution from the differential element to I xy is dI xy and is given by dI xy = xydm.
As for the moments of inertia, by integrating over the entire mass of the body for each combination
of planes, the products of inertia are

I xy = I yx = ∫ xy dm
m
I yz = I zy = ∫ yz dm (9.23)
m
I xz = I zx = ∫ xz dm
m

The product of inertia can be positive, negative, or zero, depending on the sign of the coordinates used
to define the quantity. If either one or both of the orthogonal planes are planes of symmetry for the
body, the product of inertia with respect to those planes will be zero. Basically, the mass elements would
appear as pairs on each side of these planes.
Parallel-Axis and Parallel-Plane Theorems. The parallel-axis theorem can be used to transfer the
moment of inertia of a body from an axis passing through its mass center to a parallel axis passing
through some other point (see also the section “Kinetic Energy Storage”). Often the moments of inertia
are known for axes fixed in the body, as shown in Fig. 9.33(b). If the center of gravity is defined by the
coordinates (x G , y G , z G ) in the x, y, z axes, the parallel-axis theorem can be used to find moments of
inertia relative to the x, y, z axes, given values based on the body-fixed axes. The relations are

I xx = ( I xx ) + my G + z G )
(
2
2
a
2
(
I yy = ( I yy ) + mx G + z G )
2
a
(
I zz = ( I zz ) + mx G + y G )
2
2
a
where, for example, (I xx ) a is the moment of inertia relative to the x a axis, which passes through the center
of gravity. Transferring the products of inertia requires use of the parallel-plane theorem, which provides
the relations
I xy = ( I xy ) +
a mx G y G
I yz = ( I yz ) + my G z G
a
I zx = ( I zx ) +
a mz G x G
Inertia Tensor. The rotational dynamics of a rigid body rely on knowledge of the inertial properties,
which are completely characterized by nine terms of an inertia tensor, six of which are independent. The
inertia tensor is

– I xy –
I xx I xz
I = –
I yx I yy – I yz
– I zx – I zy I zz

©2002 CRC Press LLC