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I am working on solving the equation of motion for a particular system. It has been a long time since I've worked with matrix equations and need help in simplifying the following:

$\frac{d}{dt}$$(I_G\omega)$ + $\omega \times(I_G\omega)$ + $My\times [(−\dot\omega\times y)$−$\omega \times (\omega \times y)]$ = 0

This equation came from substituting:

$a_G$ = $−\dot\omega \times y$ - $\omega \times (\omega \times y)$ into:

$\frac{d}{dt}(I_G\omega)$ + $\omega \times (I_G\omega)$ + $My \times a_G$ = 0

Any suggestions or references that would help me simplify this equation would be greatly appreciated! :)

$I_G$ is the inertia tensor at the center of gravity of the mean model. $M$ is the mass matrix. There are no external forces or moments. $\omega$ is the angular velocity vector and y is the distance vector $r_{GG}$ (an arbitrary point $P$ is coincident with $O$ and both are defined as $G$ bar from the mean model, but I have to call it $G$ since I can't find the over bar command). $a_G$ is the acceleration vector at the center of gravity of the mean model, therefore it can not be zero. The system is a block that is rotating and translating. The frame is body fixed.

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Any context, meaning of symbols or quantity to solve for would be greatly appreciated! :) – NikolajK Oct 23 '13 at 22:50
@NickKidman~ I'm sorry...should've thought about that! – Lanae Oct 24 '13 at 1:40
I figure you want to solve for $\omega$? Do any of the quantities $I_G, M_y, y$ depend on $t$ or $\omega$? Can you write the equation down in component form? – NikolajK Oct 24 '13 at 7:19
@NickKidman~I'm actually solving for the EOM and ultimately for $I_G$. I want to take the $I_G$ matrix and randomize it and get the response for ~1000 iterations. I've been doing this with thin curved beams (introducing randomness and getting the response of the system ie natural frequencies and mode shapes). The EOM is supposed to be in the form of $I_G$ = ..... – Lanae Oct 24 '13 at 23:46
@ja72~ is there any way to simplify that equation? You did an awesome job on the Newton-Euler derivation, that I thought maybe your spacial notation would work on this. But, I'm not sure..... – Lanae Oct 26 '13 at 23:22

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