Parallel Axes Rule - Moment of Inertia Matrix

Moment of Inertia Matrix

The inertia matrix of a rigid system of particles depends on the choice of the reference point. There is a useful relationship between the inertia matrix relative to the center of mass R and the inertia matrix relative to another point S. This relationship is called the parallel axis theorem.

Consider the inertia matrix obtained for a rigid system of particles measured relative to a reference point S, given by

where r_i defines the position of particle P_i, i=1, ..., n. Recall that is the skew-symmetric matrix that performs the cross product,

for an arbitrary vector y.

Let R be the center of mass of the rigid system, then

where d is the vector from the reference point S to the center of mass R. Use this equation to compute the inertia matrix,

Expand this equation to obtain

The first term is the inertia matrix relative to the center of mass. The second and third terms are zero by definition of the center of mass R,

And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix constructed from d.

The result is the parallel axis theorem,

where d is the vector from the reference point S to the center of mass R.