Parallel axis theorem

The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem,[1] named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes.

So, the equation becomes: The parallel axis theorem can be generalized to calculations involving the inertia tensor.

[2] Let Iij denote the inertia tensor of a body as calculated at the center of mass.

The generalized version of the parallel axis theorem can be expressed in the form of coordinate-free notation as where E3 is the 3 × 3 identity matrix and

Further generalization of the parallel axis theorem gives the inertia tensor about any set of orthogonal axes parallel to the reference set of axes x, y and z, associated with the reference inertia tensor, whether or not they pass through the center of mass.

The parallel axis theorem provides a convenient relationship between the moment of inertia IS around an arbitrary point S and the moment of inertia IR about the center of mass R. Recall that the center of mass R has the property where r is integrated over the volume V of the body.

The polar moment of inertia of a body undergoing planar movement can be computed relative to any reference point S, where S is constant and r is integrated over the volume V. In order to obtain the moment of inertia IS in terms of the moment of inertia IR, introduce the vector d from S to the center of mass R, The first term is the moment of inertia IR, the second term is zero by definition of the center of mass, and the last term is the total mass of the body times the square magnitude of the vector d. Thus, which is known as the parallel axis theorem.

[3] The inertia matrix of a rigid system of particles depends on the choice of the reference point.

Consider the inertia matrix [IS] obtained for a rigid system of particles measured relative to a reference point S, given by where ri defines the position of particle Pi, i = 1, ..., n. Recall that [ri − S] is the skew-symmetric matrix that performs the cross product, for an arbitrary vector y.

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 [d] 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.[4] In order to compare formulations of the parallel axis theorem using skew-symmetric matrices and the tensor formulation, the following identities are useful.

Also notice, that where tr denotes the sum of the diagonal elements of the outer product matrix, known as its trace.