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Solve for dimensions of rectangular prism given inertia 
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#1
Dec2510, 09:42 PM

P: 2

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.
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*(A2*t)*(B2*t)*(C2*t) m = mOuter  mInner Ixx = mOuter/12*(B^2 + C^2)  mInner/12*((B2*t)^2 + (C2*t)^2) Iyy = mOuter/12*(A^2 + C^2)  mInner/12*((A2*t)^2 + (C2*t)^2) Izz = mOuter/12*(A^2 + B^2)  mInner/12*((A2*t)^2 + (B2*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 handcalc to solve the 4 equations for the 4 unknowns and quickly ran into a seemingly intractable 15thdegree 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? 


#2
Dec2610, 09:52 AM

PF Gold
P: 958

Welcome to PF!
If you want to pursue a numerical solution coded by yourself (as opposed to using a numerical solver in, say, Matlab or Mathematica), one of the standard tools in that toolbox would be to try solve it using NewtonRaphson [1] on your system of equations. If you want to try that, you should probably write it up as a system of equations of 4 unknowns and 4 knowns, as density is just a scale factor, and note that you can break symmetry in A, B and C by restricting the solution to [itex]A \geq B \geq C > 2t > 0[/itex] and make an initial guess that is asymmetric, like for instance [itex] B = A/2, C = A/4, t = A/16[/itex] which should give the initial guess [itex]A^3 = \frac{512}{43}\frac{m}{\rho}[/itex]. [1] http://en.wikipedia.org/wiki/Newton'...s_of_equations 


#3
Dec2710, 12:51 AM

P: 2

Thanks for the reply! I finally did try a newtonrhapson method coded in c++, and it seems to work pretty well. I've been using the eigenvalues of the inertia tensor as the initial guess, with some fraction of the smallest value as the thickness. I have noticed that it will periodically converge to a nonphysical value (i.e. A,B,C or t less than zero), but in this case taking the absolute value of the nonphysical answer and putting that back into the iterative solver seems to come up with a different (and realistic) solution. Not sure if there is a better way to ensure that the solution lands in the positive quadrant...perhaps a boundary function of some sort?



#4
Dec2710, 02:58 AM

PF Gold
P: 958

Solve for dimensions of rectangular prism given inertia
If the nonphysical solutions pop up after a large jump in state (due to a smallmagnitude Jacobian giving rise to a large state change) you could try limit the magnitude of the state change. If the nonphysical solution appears during a normal "slow" convergence sequence, then your initial guess must belongs to the attraction basin of that solution and a different initial guess is (as you say) obviously needed, probably one where the value for A, B and C are more equal in magnitude. If nonphysical solutions are a real problem a practical "workaround" may be to map out the attraction basins of different initial guesses to see if there is a class or pattern of initial guesses that always converge to the physical solution and then stick to those. If I recall correctly, there should also be a fair bit of convergence theory which may help you if you want to analyse convergence in a more theoretical way, although I'm not sure how well it will apply to your nonlinear system. 


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