Product of Inertia: Understand the Rule

[ian2012]

I am confused about the concept of product of inertia.

Wikipedia says: "Here Ixx denotes the moment of inertia around the x-axis when the objects are rotated around the x-axis, Ixy denotes the moment of inertia around the y-axis when the objects are rotated around the x-axis, and so on."

So, when do you get a product of inertia(what's the rule)? I am guessing only when the rotating (about the center of mass) object has mass in regions x, y, z > 0. If one of the coordinates are zero and the object is stuck rotating in a plane, then I am assuming it has stable rotation?

 

[Filip Larsen]

I would say that non-zero products of inertia means that rotation about the axis in question cannot be pure or, similarly, that the inertial body in question is not inertial symmetric around that axis.

If you look at the equation that really defines the meaning of the inertia tensor (in inertial space), namely L = I \omega you can see that if any of the product of inertia (the off-diagonal elements of I) is non-zero, the angular momentum vector will in general not point in the same direction as the rotation axis and any rotation will be a non-pure rotation (for pure rotation the angular momentum vector is parallel to the rotation axis).

For a body with fixed direction of rotation axis this means that there must be a resulting torque that will make the "tip" of the angular momentum vector move in circles around the the rotation axis, like what happens for dynamically unbalanced wheels, for instance. As the inertia tensor is usually defined in body coordinates, which rotates around the rotation axis, the inertia tensor in inertial space will thus always have off-diagonal elements that are non-zero.

On the other hand, for a torque free body (where the resulting torque is zero) the angular momentum vector stays fixed in inertial space and the rotation vector must then move around the angular momentum in some way, usually giving rise to body tumbling.

Also, as you probably know, it is possible to select a (principal) coordinate system for any rigid body such that the rotation around each of these principal axis is pure and in this coordinate system all the products of inertia is zero.