Moment of Inertia, Principal Moments of Inertia, Inertia Tensor:

The moment of inertia, denoted I, is the rotational equivalent of mass: it is proportional to the amount of torque required for a given angular acceleration. If an object is only capable of rotating around a single axis, then only one moment of inertia is required to describe its motion.

In many cases, however, an object is free to rotate about any axis. Because the moment of inertia depends of the distribution of mass of an object relative to its axis of rotation, there could be defined a moment of inertia about every possible rotational axis.

For the sake of convenience,principal moments of inertia are defined, which correspond to the moments of inertia about the x, y, and z axes in the body frame. Since the body frame remains fixed relative to the object, the principal moments of inertia are constant. This greatly simplifies many problems in rigid body dynamics.

An object called the inertia tensor is used to describe more completely an object’s rotational inertia. It is a 3x3 symmetric matrix with elements that characterize its moments of inertia from different axes of rotation. In situations where principal moments of inertia have been defined, the only nonzero components of the inertia tensor are the diagonal elements, which are Ixx,Iyy,and Izz.

One place where principal moments of inertia are so important is in the formulation of Euler’s equations for rigid body motion.

The torque on an object, a vector T, which is the rate of change angular momentum vector L can be represented by the time derivative of L.

Fully expanded, the above equation would have column vectors for T and omega and a 3x3 matrix for I. Because I depends on the distribution of the object’s mass in space, and thus the orientation of the body with respect to the reference frame, it will be a variable in the case of rotation (unless the object is symmetrical about the axis of rotation). To simplify the situation, the body frame of reference is adopted instead of an inertial frame, making the mass distribution in space fixed in that reference frame and thus making the inertia tensor constant, and reducing the above equation to: