Moment of Inertia - Moment of Inertia Around An Arbitrary Axis

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.

Read more about this topic:  Moment Of Inertia

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Inertia Tensor - 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 ... 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 ...

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