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14.2.2 Derivation for τ = Iω in 3D<br />
Let A = Iω then using the rule<br />
( ) d<br />
τ =<br />
dt A ( ) d<br />
=<br />
dt A + ω × A<br />
resolved<br />
Then τ = Iω can be found for the general case<br />
⎡<br />
⎤<br />
A<br />
A<br />
{ ⎛ }} ⎞ ⎛ ⎞{<br />
⎛ ⎞ { ⎛ }} ⎞ ⎛ ⎞{<br />
τ = d I xx I xy I xz<br />
ω x<br />
ω x<br />
I xx I xy I xz<br />
ω x<br />
dt<br />
⎜I yx I yy I yz ⎟ ⎜ω y ⎟<br />
+<br />
⎝<br />
⎠ ⎝ ⎠<br />
⎜ω y ⎟<br />
⎝ ⎠ ×<br />
⎜I yx I yy I yz ⎟ ⎜ω y ⎟<br />
⎝<br />
⎠ ⎝ ⎠<br />
⎢<br />
⎣ I zx I yz I zz ω z<br />
⎥<br />
⎦ ω z I zx I yz I zz ω z<br />
⎛<br />
⎞ ⎛ ⎞ ⎛ ⎞ ⎛<br />
⎞<br />
I xx I xy I xz<br />
α x<br />
ω x<br />
I xx ω x + I xy ω y + I xz ω z<br />
=<br />
⎜I yx I yy I yz ⎟ ⎜α y ⎟<br />
⎝<br />
⎠ ⎝ ⎠ + ⎜ω y ⎟<br />
⎝ ⎠ × ⎜I yx ω x + I yy ω y + I yz ω z ⎟<br />
⎝<br />
⎠<br />
I zx I yz I zz α z ω z I zx ω x + I yz ω y + I zz ω z<br />
⎛<br />
⎞ ⎛ ⎞ ∣ ∣∣∣∣∣∣∣∣∣ I xx I xy I xz<br />
α x<br />
i j k<br />
=<br />
⎜I yx I yy I yz ⎟ ⎜α y ⎟<br />
⎝<br />
⎠ ⎝ ⎠ + det ω x ω y ω z<br />
I zx I yz I zz α z (I xx ω x + I xy ω y + I xz ω z ) (I yx ω x + I yy ω y + I yz ω z ) (I zx ω x + I yz ω y + I zz ω z ) ∣<br />
⎛<br />
⎞ ⎛ ⎞ ⎛<br />
⎞<br />
I xx I xy I xz<br />
α x<br />
ω y (I zx ω x + I yz ω y + I zz ω z ) − ω z (I yx ω x + I yy ω y + I yz ω z )<br />
=<br />
⎜I yx I yy I yz ⎟ ⎜α y ⎟<br />
⎝<br />
⎠ ⎝ ⎠ + ⎜ω x (I zx ω x + I yz ω y + I zz ω z ) − ω z (I xx ω x + I xy ω y + I xz ω z )<br />
⎟<br />
⎝<br />
⎠<br />
I zx I yz I zz α z ω x (I yx ω x + I yy ω y + I yz ω z ) − ω y (I xx ω x + I xy ω y + I xz ω z )<br />
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