Editing Moment of inertia (section)

=== Alternate inertia convention ===
There are some CAD and CAE applications such as SolidWorks, Unigraphics NX/Siemens NX and MSC Adams that use an alternate convention for the products of inertia. According to this convention, the minus sign is removed from the product of inertia formulas and instead inserted in the inertia matrix:
<math display="block">\begin{align}
  I_{xy} = I_{yx} \ &\stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} x_{k} y_{k}, \\
  I_{xz} = I_{zx} \ &\stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} x_{k} z_{k}, \\
  I_{yz} = I_{zy} \ &\stackrel{\mathrm{def}}{=}\  \sum_{k=1}^{N} m_{k} y_{k} z_{k}, \\[3pt]
  \mathbf{I} = \begin{bmatrix}
     I_{11} & I_{12} & I_{13} \\[1.8ex]
     I_{21} & I_{22} & I_{23} \\[1.8ex]
     I_{31} & I_{32} & I_{33}
  \end{bmatrix} &= \begin{bmatrix}
     I_{xx} & -I_{xy} & -I_{xz} \\[1.8ex]
     -I_{yx} & I_{yy} & -I_{yz} \\[1.8ex]
      -I_{zx} & -I_{zy} & I_{zz}
  \end{bmatrix} \\[1ex]
  &=
  \sum_{k=1}^{N} \begin{bmatrix}
    m_k \left(y_k^2 + z_k^2\right) & - m_k x_k y_k & - m_k x_k z_k \\[1ex]
    - m_k x_k y_k & m_k \left(x_k^2 + z_k^2\right) & - m_k y_k z_k \\[1ex]
    - m_k x_k z_k & - m_k y_k z_k & m_k \left(x_k^2 + y_k^2\right)
  \end{bmatrix}.
\end{align}</math>

==== Determine inertia convention (principal axes method) ====
If one has the inertia data <math>(I_{xx}, I_{yy}, I_{zz}, I_{xy}, I_{xz}, I_{yz})</math> without knowing which inertia convention that has been used, it can be determined if one also has the [[#Principal axes|principal axes]]. With the principal axes method, one makes inertia matrices from the following two assumptions:

# The standard inertia convention has been used <math>(I_{12} = I_{xy}, I_{13} = I_{xz}, I_{23} = I_{yz})</math>.  
# The alternate inertia convention has been used <math>(I_{12} = -I_{xy}, I_{13} = -I_{xz}, I_{23} = -I_{yz})</math>. 
Next, one calculates the eigenvectors for the two matrices. The matrix whose eigenvectors are parallel to the principal axes corresponds to the inertia convention that has been used.