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The problem I am facing is the reduction of vibrations of a rotating object. I have a series of vibration measurements taken at 5 different states with magnitude and phase components, and a set of coefficients corresponding to 4 adjustment types and each state. I'm trying to find the adjustments necessary to minimize the resulting vibration.

Basically, it's just a system of equations problem that I would like an automated solution for. For a single vibration and adjustment, the equation would be $\vec{v}_f=\vec{v}_i+a\vec{c}$, where $a$ is the scalar adjustment. I break the components into Cartesian coordinates from their polar form and solve the system $$ \left\{\begin{array}{1} v_{x_f}=v_{x_i}+aC_x\\ v_{y_f}=v_{y_i}+aC_y\\ \end{array}\right. $$

solving for $a$ such that $\sqrt{v_{x_f}^2+v_{y_f}^2}$ is minimized.

I've set up the vibrations, adjustments and coefficients into matrices, but I'm still a little foggy on what operations I need to perform to solve it. $$ \vec{V}_f = \begin{bmatrix} \vec{v}_{1_f} \\ \vec{v}_{2_f} \\ \vec{v}_{3_f} \\ \vec{v}_{4_f} \\ \vec{v}_{5_f} \\ \end{bmatrix} , \vec{V}_i = \begin{bmatrix} \vec{v}_{1_i} \\ \vec{v}_{2_i} \\ \vec{v}_{3_i} \\ \vec{v}_{4_i} \\ \vec{v}_{5_i} \\ \end{bmatrix} , \vec{C} = \begin{bmatrix} \vec{c}_{(1,a)} & \vec{c}_{(1,b)} & \vec{c}_{(1,c)} & \vec{c}_{(1,d)} \\ \vec{c}_{(2,a)} & \vec{c}_{(2,b)} & \vec{c}_{(2,c)} & \vec{c}_{(2,d)} \\ \vec{c}_{(3,a)} & \vec{c}_{(3,b)} & \vec{c}_{(3,c)} & \vec{c}_{(3,d)} \\ \vec{c}_{(4,a)} & \vec{c}_{(4,b)} & \vec{c}_{(4,c)} & \vec{c}_{(4,d)} \\ \vec{c}_{(5,a)} & \vec{c}_{(5,b)} & \vec{c}_{(5,c)} & \vec{c}_{(5,d)} \\ \end{bmatrix} , A = \begin{bmatrix} a_{a}\\ a_{b}\\ a_{c}\\ a_{d}\\ \end{bmatrix} $$ $$ \vec{V}_{f} = \vec{V}_i + \vec{C} A $$ Essentially I believe I need to find the values for $a$ such that the magnitudes of the vibrations are minimized. Is it possible to even solve using this equation using matrices, or am I completely off base?

I would really appreciate any discussion that might point me on the right track.

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    What's $A$, and where did $\vec{M}_f$ go in your final equation?2011-05-06
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    I made a change halfway through and forgot to change everything back to A..."Adjustment" sounded better than "Move". The most recent edit reflects the changes to 'A' from 'M'.2011-05-06
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    The dimensions of $\vec A$ and $\vec C$ don't seem to conform; you've four columns in $\vec A$ and five rows in $\vec C$...2011-05-06
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    On the other hand, if you meant $C\vec{A}$, then least squares is what'll save your bacon.2011-05-06
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    Basically, what I'm doing with the coefficient matrix is showing how much the vibration point will move at the particular state if adjusted by $a$.2011-05-06
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    Before you set up the matrices, you will want to write out each equation in full first. (I know it's excruciating, but if you haven't used matrices fully up until now, that's where you should start.) Make sure to line up corresponding variables in each equation, and then you can start building matrices from there.2011-05-06
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    Ah, now you can treat $\vec{C}A=\vec{V}_{f}-\vec{V}_i$ with least squares. What computing environment are you using?2011-05-06
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    I'll look into the Least Squares, thanks. The plan is to incorporate this into a mobile platform, so currently I'm using Java.2011-05-06
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    Since you're going to be implementing this... I would suggest that you use "QR decomposition". You'll encounter proposals to solve this by the method of "normal equations", and though it's more space-efficient than QR, it's more prone to roundoff error. [JAMA](http://math.nist.gov/javanumerics/jama/) seems to provide a QR routine, so you can use that.2011-05-06

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