Notations Used for Inertia MatricesThis section 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. Moments and Products of 3D Inertia Moments and Products of 2D Inertia Matrix of Inertia Moments and Principal Axes Additional Notation Used in Measure Inertia Command

Moments and Products of 3D Inertia

IoxMoment of inertia of the object about the ox axis:

IoyMoment of inertia of the object about the oy axis:

IozMoment of inertia of the object about the oz axis:

PxyProduct of inertia of the object about axes ox and oy:

PxzProduct of inertia of the object about axes ox and oz:

PyzProduct of inertia of the object about axes oy and oz:

(where M is the mass of the object; units: kg.m2)

Moments and Products of 2D Inertia

IoxMoment of inertia of the surface about the ox axis:

IoyMoment of inertia of the surface about the oy axis:

PxyProduct of inertia of the surface about axes ox and oy:

(where A is the surface; units: m4)

Matrix of Inertia

3D Inertia:2D Inertia:

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

Moments and Principal AxesThe matrix of inertia being a real matrix (whose elements consist entirely of real numbers) and a symmetric matrix, there exists an orthonormal basis of vectorsin this matrix of inertia.

The principal axes are defined by vectorsand inertia principal moments are expressed by

Note: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:(parallel axis theorem).Let I' be the matrix of inertia with respect to orthonormal basis Pxyzwhere

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 Measure Inertia Command

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.

IoxGMoment of inertia of the object about the ox axis with respect to the system Gxyz, where G is the center of gravity.

IoxOMoment of inertia of the object about the ox axis with respect to the system Oxyz, where O is the origin of the document.

IoxPMoment of inertia of the object about the ox axis with respect to the system Pxyz, where P is a selected point.

IoxAMoment of inertia of the object about the ox axis with respect to the system Axyz, where A is a selected axis system.

etc.