PRINCIPAL MOMENTS OF INERTIA

The resulting moments of inertia at the principal axes are known as the principal moments of inertia. For more information about moments of inertia, see Moment of Momentum.

Using Cramer’s Rule to solve the following matrix derived in Principal Axes:

I = ||||||||| I^C_xx − I I^C_xy I^C_xz I^C_xy I^C_yy − I I^C_yz I^C_xz I^C_yz I^C_zz − I |||||||||

where I represents an eigenvalue for the principal moments of inertia. The eigenvector of this system of equations are Iωη. If the principal axes do not coincide with the axes of symmetry, this matrix must be solved three times, one for the x* axis, where I^C_xy and I^C_xz are zero, y* axis, where I^C_xy and I^C_yz are zero, and z* axis, where I^C_yz and I^C_xz are zero. Thus, if the principal axes are not also axes of symmetry, there will always be three real roots to this expression, along with their corresponding eigenvectors:

(1) (I^c_yy − I^c_zz)² + 4(I^c_yz)² ≥ 0

It is assumed that these calculations use principal axes originating from the mass center, C.