10.6: Parallel Axis Theorem

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Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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Often, the mass center of an body is at a different location than a more convenient measurement point, such as the geometric center of a vehicle. The parallel axis theorem allows one to translate the mass moments of inertia (MMOI) referenced to the mass center into another frame with parallel orientation, and vice versa. Sometimes a translation of coordinates to the mass center will make the cross-inertial terms \(I_{xy}, \, I_{yz}, \, I_{xz}\), small enough that they can be ignored; in this case \(\vec{r}_G = \vec{0}\) also, so that the equations of motion are significantly reduced, as in the spinning book example.

The formulas are:

\begin{align} I_{xx} \,\, &= \,\, \bar{I}_{xx} + m(\delta y^2 + \delta z^2) \\[4pt] I_{yy} \,\, &= \,\, \bar{I}_{yy} + m(\delta x^2 + \delta z^2) \\[4pt] I_{zz} \,\, &= \,\, \bar{I}_{zz} + m(\delta x^2 + \delta y^2) \\[4pt] I_{yz} \,\, &= \,\, \bar{I}_{yz} - m \delta y \delta z \\[4pt] I_{xz} \,\, &= \,\, \bar{I}_{xz} - m \delta x \delta z \\[4pt] I_{xy} \,\, &= \,\, \bar{I}_{xy} - m \delta x \delta y, \end{align}

where \(\bar{I}\) represents an MMOI in the axes of the mass center, and \(\delta x\), for example, is the translation of the \(x\)-axis to the new frame. Note that translation of MMOI using the parallel axis theorem must be either to or from a frame resting exactly at the center of gravity.