moment of inertia of arigid body

Moment of Inertia of a Rigid Body

Based on [2], one can obtain the moment of inertia of a rigid by shown in Figure 2:

Figure 2
[3]

where r_i = the position of particle i, and n = the unit vector of the axis of rotation. Note here that the axis of rotation passes through the local reference frame, the OXYZ system. Let

[4]

and

[5]

where cosa, cosb & cosg = the three direction cosines of vector n to the XYZ system. Substituting [4] & [5] into [3] leads to

[6]

where

[7]

I_xx, I_yy & I_zz are called the moments of inertia while I_xy, I_yx, I_yz, I_zy, I_zx, & I_xz are the products of inertia. For a rigid body, the relative position of the particles do not change and one can write [7] as:

[8]

When the shape and the density distribution of the rigid body is precisely known, one can use [8] to compute the moments and products of inertia. (See BSP Equations for the MOI equations of the typical geometric shapes commonly used in human body modeling.) Otherwise, it is difficult to compute them through integration. Rather, the moment of inertia must be measured directly from the object. See Measuring MOI for the details.
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