3d translational transformation of inertia properties
The moment of inertia, or more precisely, the moment of inertia tensor, describes how mass is distributed within a body relative to a specific axis of rotation. The parallel axis theorem provides a method to determine the moment of inertia about an arbitrary axis, given the moment of inertia about a parallel axis passing through the center of mass.
Let’s consider a rigid body of mass m, with a center of mass at position \mathbf r_C. If a point mass is at a location \mathbf r, then we have \mathbf r' = \mathbf r - \mathbf r_C. The moment of inertia tensor I_{ij} with respect to a generic origin \mathbf O is given by
I_{ij} = \int \left(r^2\delta_{ij} - r_ir_j\right)\mathrm{d}m
where \delta_{ij} is the Kronecker delta. The moment of inertia with respect to the center of mass is:
I^C_{ij} = \int \left({r'}^2\delta_{ij} - r'_ir'_j\right)\mathrm{d}m
Through substitution and some algebra, the parallel axis theorem states that the moment of inertia tensor about any axis is:
I_{ij} = I^C_{ij} + m\left(r_C^2\delta_{ij} - r_{C,i}r_{C,j}\right)
In matrix form, the theorem can be expressed as:
\mathbf I = \mathbf I_{C} + m \left[\left\|\mathbf r_{C}\right\|^2\mathbf 1 - \mathbf r_{C}\mathbf r_{C}^T\right]
where \mathbf 1 is the identity matrix and \mathbf r_{C}\mathbf r_{C}^T represents the outer product.
To apply the theorem in Cartesian coordinates, consider a displacement vector \mathbf d = (d_x, d_y, d_z) from the center of mass to a new axis. The moment of inertia tensor \mathbf I becomes:
\mathbf I = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{xy} & I_{yy} & I_{yz} \\ I_{xz} & I_{yz} & I_{zz} \end{bmatrix} = \begin{bmatrix} I_{xx}^C & I_{xy}^C & I_{xz}^C \\ I_{xy}^C & I_{yy}^C & I_{yz}^C \\ I_{xz}^C & I_{yz}^C & I_{zz}^C \end{bmatrix} + m \begin{bmatrix} d_y^2+d_z^2 & -d_xd_y & -d_xd_z \\ -d_xd_y & d_x^2+d_z^2 & -d_yd_z \\ -d_xd_z & -d_yd_z & d_x^2+d_y^2 \end{bmatrix}
This expression provides a direct way to compute the moment of inertia tensor with respect to an axis displaced by \mathbf d.
For more insights into this topic, you can find the details here.