Inertia matrix

The inertia matrix $\overline{I}$ is a 3X3 matrice that describes how a rigid body’s mass is distributed relative to a chosen origin.

It relates the body’s angular velocity $\vec{\omega}$ to its angular momentum $\vec{L}$:

$$ \vec{L} = \overline{I}\vec{\omega} $$

Definition in orthonormal basis

In a orthonormal basis (Cartesian coordinate system) $B$ with origin at point $O$, and position vector $\vec{r}_m=(x,y,z)$ of each mass element $dm$, the inertia matrix of an object or system $(S)$ is

$$\overline{I}(O,S)_B = \begin{pmatrix} I_{xx} & I_{xy} & I_{xz}\\ I_{yx} & I_{yy} & I_{yz}\\ I_{zx} & I_{zy} & I_{zz} \end{pmatrix}$$

where $\mathbf{I}(O,S)_B$ is the inertia tensor of object or system (S) according to cartesian coordinate system $B$ of origin $O$

with

$$ I_{xx}=\int_S (y^2+z^2)\,dm,\qquad I_{yy}=\int_S (x^2+z^2)\,dm,\qquad I_{zz}=\int_S (x^2+y^2)\,dm $$

and

$$ I_{xy}=I_{yx}=-\int_S xy\,dm,\qquad I_{xz}=I_{zx}=-\int_S xz\,dm,\qquad I_{yz}=I_{zy}=-\int_S yz\,dm $$

It can also be written as:

$$ \overline{I}(O,S)_B = \int_S \left[(\vec{r}_m\cdot\vec{r}_m)\,\mathbf{1}-\vec{r}_m\otimes\vec{r}_m\right]dm $$

where $\vec{r}_m=(x,y,z)$ is always the position vector of $dm$, the $\mathbf{1}$ is the 3x3 identity matrix, and $\otimes$ denotes the outer product.

Derivation

If the mass element $dm \in S$ at point $M$ rotates about $O$ with angular velocity $\vec{w}$, then its linear velocity

$$ \vec{v}(M)=\vec{w}\wedge\vec{r}. $$

Its angular momentum about $O$ $\vec{L}_O$ satisfies

$$ \vec{L}_O = \int_S \vec{r}\wedge\vec{v}(M)\,dm = \int_S \vec{r}\wedge(\vec{w}\wedge\vec{r})\,dm. $$

Since

$$ \vec{r}\wedge(\vec{w}\wedge\vec{r}) = (\vec{r}\cdot\vec{r})\vec{w} - (\vec{r}\cdot\vec{w})\vec{r} $$

Writing $\vec{r}=(x,y,z)$ and $\vec{w}=(w_x,w_y,w_z)$,

$$ \vec{r}\wedge(\vec{w}\wedge\vec{r}) = \begin{pmatrix} (y^2+z^2)u_x-xyu_y-xzu_z\\ -xyu_x+(x^2+z^2)u_y-yzu_z\\ -xzu_x-yzu_y+(x^2+y^2)u_z \end{pmatrix} $$

Integrating over the solid gives

$$ \int_S \vec{r}\wedge(\vec{w}\wedge\vec{r})\,dm = \begin{bmatrix} \int_S(y^2+z^2)\,dm & -\int_S xy\,dm & -\int_S xz\,dm\\ -\int_S xy\,dm & \int_S(x^2+z^2)\,dm & -\int_S yz\,dm\\ -\int_S xz\,dm & -\int_S yz\,dm & \int_S(x^2+y^2)\,dm \end{bmatrix} \begin{pmatrix} w_x\\ w_y\\ w_z \end{pmatrix} $$

Hence

$$ \overline{I}(O,S)_B\cdot\vec{w} = \begin{bmatrix} \int_S(y^2+z^2)\,dm & -\int_S xy\,dm & -\int_S xz\,dm\\ -\int_S xy\,dm & \int_S(x^2+z^2)\,dm & -\int_S yz\,dm\\ -\int_S xz\,dm & -\int_S yz\,dm & \int_S(x^2+y^2)\,dm \end{bmatrix} \vec{w} $$

By definition, the inertia matrix $\overline{I}(O,S)_B$ satisfies that

$$ \vec{L}_O=\overline{I}(O,S)_B\vec{w}. $$

Therefore,

$$ \overline{I}(O,S)_B = \begin{bmatrix} \int_S(y^2+z^2)\,dm & -\int_S xy\,dm & -\int_S xz\,dm\\ -\int_S xy\,dm & \int_S(x^2+z^2)\,dm & -\int_S yz\,dm\\ -\int_S xz\,dm & -\int_S yz\,dm & \int_S(x^2+y^2)\,dm \end{bmatrix} $$

Principal basis of inertia

$\forall O, \exists$ orthonormal basis $B \Rightarrow$

$$ \overline{I}_O(S) = \begin{pmatrix} I_{xx} & 0 & 0\\ 0 & I_{yy} & 0\\ 0 & 0 & I_{zz} \end{pmatrix}$ $$

$I_{xx},I_{yy},I_{zz}$ are then known as the principal moments of inertia for $(S)$ according to $O$.