I am trying to solve something of an inverse problem for inertia, and having a real tough time with it. Any suggestions would be helpful.(adsbygoogle = window.adsbygoogle || []).push({});

For a rectangular prism the mass and moments of inertia are:

m = rho*A*B*C

Ixx = m/12*(B^2 + C^2)

Iyy = m/12*(A^2 + C^2)

Izz = m/12*(A^2 + B^2)

Where m is mass, rho density, A width, B length, C height, and Ixx, Iyy, Izz are the principal moments of inertia. My problem involves a hollow rectangular prism, with wall thickness t. So the equations then become:

mOuter = rho*A*B*C

mInner = rho*(A-2*t)*(B-2*t)*(C-2*t)

m = mOuter - mInner

Ixx = mOuter/12*(B^2 + C^2) - mInner/12*((B-2*t)^2 + (C-2*t)^2)

Iyy = mOuter/12*(A^2 + C^2) - mInner/12*((A-2*t)^2 + (C-2*t)^2)

Izz = mOuter/12*(A^2 + B^2) - mInner/12*((A-2*t)^2 + (B-2*t)^2)

To see how these equations work imagine taking a rectangular prism and subtracting out a smaller rectangular prism from the inside. Given length (A), width (B), height(C), thickness (t), and density (rho) I can easily solve for the mass and moments of inertia. What I would like to do is given m, rho, Ixx, Iyy, and Izz solve for A,B,C, and t. I tried doing the hand-calc to solve the 4 equations for the 4 unknowns and quickly ran into a seemingly intractable 15th-degree polynomial in t with tons of unknown coefficients. If anyone has any ideas on how to solve this I would greatly appreciate it. It doesn't have to be a closed form solution (though that would be best), iterative or other methods would work as well.

Any ideas?

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# Solve for dimensions of rectangular prism given inertia

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