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Math Help - Coupled mass to matrix problem

  1. #1
    Member Mollier's Avatar
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    Coupled mass to matrix problem

    Hi, I am new around here. Hope you don't mind me beginning my stay with a question.

    1. The Problem

    Suppose masses m_{1}, m_{2}, m_{3}, m_{4} are located at positions x_{1}, x_{2}, x_{3}, x_{4} in a line and connected by springs with constants k_{12}, k_{23}, k_{34} whose natural lengths of extension are l_{12}, l_{23}, l_{34}.
    Let f_{1}, f_{2}, f_{3}, f_{4} denote the rightward forces on the masses, e.g.,
    f_{1} = k_{12}(x_{2} - x_{1} - l_{12})

    a) Write the 4 X 4 matrix equation relating the column vectors  f and  x . Let  K denote the matrix in this equation.

    2. Attempt at solution
    I treat this as a equilibrium problem. I imagine the masses being pulled to the right and held in place. Then I find the forces neccessary to keep them in that new position. I'm not sure that this is what the problem asks for, but I don't want to just post the question

    f_{2} = k_{23}(x_{3} - x_{2} - l_{23}) - f_{1}
    f_{3} = k_{34}(x_{4} - x_{3} - l_{34}) - (f_{1} + f_{2})
    f_{4} = f_{1} + f_{2} + f_{3}

    I then expand and simplify each and expression.

    I then write these equations in the form:
    <br />
  \textbf{f} = \textbf{K}\textbf{x} + \textbf{c} <br />

    <br />
\left[\begin{array}[pos]{c}<br />
    f_{1} \\<br />
    f_{2} \\<br />
    f_{3} \\<br />
    f_{4} \\<br />
\end{array}\right]

    <br />
=<br />
\left[\begin{array}[pos]{cccc}<br />
    -k_{12} & k_{12}           & 0               & 0 \\<br />
    k_{12}  & (-k_{12}-k_{23}) & k_{23}          & 0 \\<br />
    0       & k_{23}           &(-k_{23}-k_{34}) & k_{34} \\<br />
    0       & 0                &-k_{34}          & k_{34}\\<br />
\end{array}\right]

    <br />
\left[\begin{array}[pos]{c}<br />
    x_{1} \\<br />
    x_{2} \\<br />
    x_{3} \\<br />
    x_{4} \\<br />
\end{array}\right]

    <br />
+<br />
\left[\begin{array}[pos]{c}<br />
    -k_{12}l_{12} \\<br />
    -k_{23}l_{23} + k_{12}l_{12} \\<br />
    -k_{34}l_{34} + k_{23}l_{23} \\<br />
    -k_{34}l_{34} \\<br />
\end{array}\right]<br /> <br />

    Sorry about the formatting of matrices, I get too long string errors when I tried writing them next to eachother..
    Somehow I feel that the masses should be included in the party as well...

    Any suggestions are appreciated, thanks.
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  2. #2
    MHF Contributor

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    That looks good to me!
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  3. #3
    Member Mollier's Avatar
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    Nothing better than that
    Thanks!
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