Moments of Inertia

We can express any tensor \(\mathbf{T}\) like so:

\begin{align} \mathbf{T} = \sum_{i=1}^3\sum_{j=1}^3 T_{ij} \mathbf{e}_i \otimes \mathbf{e}_j \notag \end{align}

Applied to a vector \(\mathbf{u} = u_1\mathbf{e}_1 + u_2\mathbf{e}_2 + u_3\mathbf{e}_3\), we can represent the tensor operation using matrix multiplication:

\begin{align} \begin{bmatrix}\mathbf{T}_{\mathbf{u}}\end{bmatrix} = \begin{bmatrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \notag \end{align}

We know that the inertia tensor is given by:

\begin{align} \mathbf{I}^C = \int_B \| \mathbf{r}-\mathbf{r}_C\|^2 - (\mathbf{r}-\mathbf{r}_C)\otimes(\mathbf{r}-\mathbf{r}_C)\text{ d}m \end{align}

Let \(\mathbf{r}-\mathbf{r}_C = x\mathbf{e}_x + y\mathbf{e}_y+z\mathbf{e}_z\), where \(\{\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z\}\) is a corotational basis of the rigid body. Then, we can express the parts inside the integral of \(\mathbf{I}^C\) as

\begin{align} (x^2+y^2+z^2)\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} - \begin{bmatrix}x^2 & xy & xz \\ xy & y^2 & yz \\ xz & yz & z^2 \end{bmatrix} = \begin{bmatrix} y^2+z^2 & -xy & -xz \\ -xy & x^2+z^2 & -yz \\ -xz & -yz & x^2+y^2 \end{bmatrix} \notag \end{align}

Then, the inertia tensor can be represented as

\begin{align} \boxed{\begin{bmatrix}\mathbf{I}^C\end{bmatrix} = \begin{bmatrix} \int_B y^2+z^2 \text{ d}m & -\int_B xy \text{ d}m & -\int_B xz \text{ d}m \\ -\int_B xy \text{ d}m & \int_B x^2+z^2 \text{ d}m & -\int_B yz \text{ d}m \\ -\int_B xz \text{ d}m & -\int_B yz \text{ d}m & \int_B x^2+y^2 \text{ d}m \end{bmatrix} = \begin{bmatrix} I_{xx}^C & I_{xy}^C & I_{xz}^C \\ & I_{yy}^C & I_{yz}^C \\ \text{sym.} & & I_{zz}^C \end{bmatrix}} \end{align}

The diagonal terms of the resulting symmetric matrix are called the moments of inertia, and the off-diagonal terms are the products of inertia. The moments of inertia relate to rotating about their respective axes.

We do not always need to calculate moments of inertia by hand. For common shapes, they are present in the form of tables.