Parallel Axes Rule - in Classical Mechanics

In Classical Mechanics

In classical mechanics, the Parallel axis theorem (also known as Huygens-Steiner theorem) can be generalized to calculate a new inertia tensor Jij from an inertia tensor about a centre of mass Iij when the pivot point is a displacement a from the centre of mass:

where

is the displacement vector from the centre of mass to the new axis, and

is the Kronecker delta.

We can see that, for diagonal elements (when i = j), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.

This can also be formulated as follows. Once the moment of inertia tensor has been calculated for rotations about the center of mass of the rigid body, there is a useful labor-saving method to compute the tensor for rotations offset from the center of mass.

If the axis of rotation is displaced by a vector R from the center of mass, the new moment of inertia tensor equals


mathbf{I}^{mathrm{displaced}} = mathbf{I}^{mathrm{<a href=center}} + m left " src="http://upload.wikimedia.org/math/5/6/8/5685f05491ff00428551ec8ad91b5de0.png" />

where m is the total mass of the rigid body, E3 is the 3 × 3 identity matrix, and is the outer product.

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