Inertia Notations

This reference will help you read the information given in the Measure Inertia dialog box for Inertia Matrix / G, Inertia Matrix / O, Inertia Matrix / P and Inertia Matrix / Axis System A.

This page discusses:

Moments and Products of 3D Inertia
Matrix of 3D Inertia
Moments and Principal Axes
Additional Notation Used in the Measure Inertia Command
Mass Radius of Gyration

Moments and Products of 3D Inertia

Notations for moments and products of 3D inertia.

Where M is the mass of the object (kg.m²).


$I_{o x}$	Moment of inertia of the object about the ox axis	$I_{o x} = \int (y^{2} + z^{2}) d M$
$I_{o y}$	Moment of inertia of the object about the oy axis	$I_{o y} = \int (x^{2} + z^{2}) d M$
$I_{o z}$	Moment of inertia of the object about the oz axis	$I_{o z} = \int (x^{2} + y^{2}) d M$
$P_{x y}$	Product of inertia of the object about axes ox and oy	$P_{x y} = \int (x . y) d M$
$P_{x z}$	Product of inertia of the object about axes ox and oz	$P_{x z} = \int (x . z) d M$
$P_{y z}$	Product of inertia of the object about axes oy and oz	$P_{y z} = \int (y . z) d M$

Matrix of 3D Inertia

Notations for the matrix of 3D inertia.

Where I is the matrix of inertia of the object with respect to orthonormal basis Oxyz.

I = [\begin{matrix} I_{o x} & - P_{x y} & - P_{x z} \\ - P_{x y} & I_{o y} & - P_{y z} \\ - P_{x z} & - P_{y z} & I_{o z} \end{matrix}]

Moments and Principal Axes

Notations for the moments and principal axes. The matrix of inertia being a real matrix (whose elements consist entirely of real numbers) and a symmetric matrix, there exists an orthonormal basis of vectors $(\vec{A_{1}}, \vec{A_{2}}, \vec{A_{3}})$ in this matrix of inertia.

PM = (\begin{matrix} M_{1} & 0 & 0 \\ 0 & M_{2} & 0 \\ 0 & 0 & M_{3} \end{matrix})

The principal axes are defined by vectors $(\vec{A_{1}}, \vec{A_{2}}, \vec{A_{3}})$ and inertia principal moments are expressed by $M_{1}, M_{2}, M_{3}$ .

$PM= T^{T} .I.T$ with $T = (\begin{matrix} A_{1 x} & A_{1 y} & A_{1 z} \\ A_{2 x} & A_{2 y} & A_{2 z} \\ A_{3 x} & A_{3 y} & A_{3 z} \end{matrix})$ and $T^{T}$ the transpose of $T$ .

Note:

(\vec{A_{1}}, \vec{A_{2}}, \vec{A_{3}})

is an orthonormal direct basis.

Expression in Any Axis System

I is the matrix of inertia with respect to orthonormal basis Oxyz.

Huygen's theorem is used to transform the matrix of inertia: O_xyz $\to$ P_xyz (parallel axis theorem).

Let I' be the matrix of inertia with respect to orthonormal basis Pxyz where $V = \bar{P O}$ , and $I^{}' = I + m . [\begin{matrix} (V_{y}^{2} + V_{z}^{2}) & - V_{x} . V_{y} & - V_{x} . V_{z} \\ - V_{x} . V_{y} & (V_{x}^{2} + V_{z}^{2}) & - V_{y} . V_{z} \\ - V_{x} . V_{z} & - V_{y} . V_{z} & (V_{x}^{2} + V_{y}^{2}) \end{matrix}]$

M = {u,v,w}: transformation matrix from basis (Pxyz) to basis (Puvw) TM is the transposed matrix of matrix M.

J is the matrix of inertia with respect to an orthonormal basis Puvw:

J = TM.I'.M

Additional Notation Used in the Measure Inertia Command

Additional notations.

Ixy = (-Pxy)

Ixz = (-Pxz)

Iyz = (-Pyz)

Note:

Since entries for the opposite of the product are symmetrical, they are given only once in the dialog box.


IoxG	Moment of inertia of the object about the ox axis with respect to the system Gxyz, where G is the center of gravity.
IoxO	Moment of inertia of the object about the ox axis with respect to the system Oxyz, where O is the origin of the document.
IoxP	Moment of inertia of the object about the ox axis with respect to the system Pxyz, where P is a selected point.
IoxA	Moment of inertia of the object about the ox axis with respect to the system Axyz, where A is a selected axis system.

Mass Radius of Gyration

Notations for the mass radius of gyration.


$k$	The radius of gyration about an axis parallel to the centroidal axis.
$k_{c}$	The radius of gyration about the centroidal axis.
$d$	The distance between the centroidal axis and the parallel axis.

Where

k^{2} = k_{c}^{2} + d^{2}


$k_{x}$	The radius of gyration about the x axis.
$I_{x}$	The moment of inertia about x axis.
$m$	The mass.

Where

I_{x} = k_{x}^{2} m