Inertia matrix - Revision history

Rongzhou: /* Change of origin */

2026-10-07T13:16:05Z

Change of origin

← Older revision		Revision as of 13:16, 7 October 2026
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	Note $\int_S \overrightarrow{CM}\wedge(\vec{\omega}\wedge\overrightarrow{CM})\,dm=\vec{L}(C,S)_B = \overline{I}(C,S)_B\vec{\omega}$,		Note $\int_S \overrightarrow{CM}\wedge(\vec{\omega}\wedge\overrightarrow{CM})\,dm=\vec{L}(C,S)_B = \overline{I}(C,S)_B\vec{\omega}$,

	Because $C$ is the centre of ~~gravity~~,		Because $C$ is the centre of inertia,

	$$		$$

Rongzhou: /* Change of origin */

2026-10-07T13:15:16Z

Change of origin

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Rongzhou: /* Definition in orthonormal basis */

2026-10-07T12:19:41Z

Definition in orthonormal basis

← Older revision		Revision as of 12:19, 7 October 2026
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	== Definition in orthonormal basis ==		== Definition in orthonormal basis ==

	In a orthonormal basis (Cartesian coordinate system) 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		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

	{{#content:Q2052}}		{{#content:Q2052}}
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	$$		$$
	\overline{I(O,S)}		\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		\int_S \left[(\vec{r}_m\cdot\vec{r}_m)\,\mathbf{1}-\vec{r}_m\otimes\vec{r}_m\right]dm
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	$$		$$
	\overline{I}_O(S)\cdot\vec{w}		\overline{I}(O,S)_B\cdot\vec{w}
	=		=
	\begin{bmatrix}		\begin{bmatrix}
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	$$		$$

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

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

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	$$		$$
	\overline{I}_O(S)		\overline{I}(O,S)_B
	=		=
	\begin{bmatrix}		\begin{bmatrix}

Rongzhou: /* Principal basis of inertia */

2026-10-07T12:03:57Z

Principal basis of inertia

← Older revision		Revision as of 12:03, 7 October 2026
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	$$		$$

	=== Principal basis of inertia ===		== Principal basis of inertia ==

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

Rongzhou: Created page with "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) with origin at point $O$, and position vector $\vec{r}_m=(x,y,z)$ of each mass element $dm$, the inertia..."

2026-10-07T12:02:37Z

Created page with "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) with origin at point $O$, and position vector $\vec{r}_m=(x,y,z)$ of each mass element $dm$, the inertia..."

New page

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) 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

{{#content:Q2052}}

It can also be written as:

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

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

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

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

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

Therefore,

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

=== 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 $O$.