Inertia Ellipsoid
The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body called the inertia ellipsoid. Let be the inertia matrix relative to the center of mass aligned with the principal axes, then the surface
or
defines an ellipsoid in the body frame. Write this equation in the form,
to see that the semi-principal diameters of this ellipsoid are given by
Let a point x on this ellipsoid be defined in terms of its magnitude and direction, x=|x|n, where n is a unit vector, then from the definition of the inertia matrix,
where In is the moment of inertia of the body around the axis in the direction n. Thus, the magnitude of a point x in the direction n on the inertia ellipsoid is
Read more about this topic: Inertia Tensor, Moment of Inertia Reference Frames
Famous quotes containing the word inertia:
“What is wrong with priests and popes is that instead of being apostles and saints, they are nothing but empirics who say “I know” instead of “I am learning,” and pray for credulity and inertia as wise men pray for scepticism and activity.”
—George Bernard Shaw (1856–1950)