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Expanding the triple vector product term yields:
H cg = ∫ {[(y 2 + z 2 x − xyω y - xzω z ] i +
[ −yxω x + (z 2 + x 2 y − yzω z ] j +
[−zxω x − zyω y + (x 2 + y 2 )ωz] k } dm
To simplify this equation, let's replace a few terms by letting:
I xx = ∫ (y 2 + z 2 ) dm
I yy = ∫ (z 2 + x 2 ) dm
I zz = ∫ (x 2 + y 2 ) dm
I xy = I yx = ∫ (xy) dm
I xz = I zx = ∫ (xz) dm
I yz = I zy = ∫ (yz) dm
Substituting these I variables, some of which should look familiar to you, back into the
expanded equation yields:
H cg = [I xx ω x − I xy ω y − I xz ω z ] i +
[−I yx ω x + I yy ω y − I yz ω z ] j +
[−I zx ω x − I zy ω y + I zz ω z ] k
Simplifying this a step further by letting I be a matrix:
I xx −I xy −I xz
I = −I yx I yy −I yz
−I zx −I zy I zz
yields the following equation:
H cg = I ω
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