Moment of Inertia Around An Arbitrary Axis
The moment of inertia of a body around an arbitrary axis in space is a scalar that is computed as the sum of the distance squared from the axis to each of the mass elements. This scalar can be computed from the moment inertia matrix of the body using the unit vector along the axis.
Let a rigid assembly of rigid system of N particles, Pi, i=1,...,N, have coordinates ri. Choose R as a reference point and compute the moment of inertia around an axis L defined by the unit vector S through the reference point R. The moment of inertia of the system around this line L=R+tS is computed by determining the perpendicular vector from this axis to the particle Pi given by
where is the identity matrix and is the outer product matrix formed from the unit vector S along the line L.
Introduce the skew-symmetric matrix such that y=S x y, then we have the identity
which relies on the fact that S is a unit vector.
The magnitude squared of the perpendicular vector is
The simplification of this equation uses the identity
where the dot and the cross products have been interchanged. Expand the cross products to compute
where is the skew symmetric matrix obtained from the vector ri-R.
Thus, the moment of inertia around the line L through R in the direction S is given by the scalar
or
where is the moment of inertia matrix of the system relative to the reference point R.
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