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Coupled mass problem

  1. Nov 29, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose masses [tex]m_{1}, m_{2}, m_{3}, m_{4}[/tex] are located at positions [tex]x_{1}, x_{2}, x_{3}, x_{4}[/tex] in a line and connected by springs with constants [tex]k_{12}, k_{23}, k_{34}[/tex] whose natural lengths of extension are [tex]l_{12}, l_{23}, l_{34}[/tex].
    Let [tex]f_{1}, f_{2}, f_{3}, f_{4}[/tex] denote the rightward forces on the masses, e.g.,
    [tex]f_{1} = k_{12}(x_{2} - x_{1} - l_{12})[/tex]

    a) Write the 4 X 4 matrix equation relating the column vectors [tex] f [/tex] and [tex] x [/tex]. Let [tex] K [/tex] denote the matrix in this equation.
    2. Relevant equations

    3. The attempt at a solution
    I'm trying to find the rightward force acting on every mass as the springs are stretched.

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

    It seems quite complicated to put this into matrix form, so I'm assuimg that I've done something wrong.
    Suggestions?
     
  2. jcsd
  3. Nov 30, 2009 #2
    By the way, here is a figure showing how I am visualizing this problem:
    I am assuming [tex]m_{1}[/tex] to be constrained. Just throwing out ideas :)

    Exercise1-2.jpg
     
    Last edited: Nov 30, 2009
  4. Nov 30, 2009 #3
    Here's my complete solution. I expand and simplify the equations given in my first post.
    Then I put together expressions for the x's and separate the constants.

    I'm sorry about the formatting.
    Does this look ok?

    [tex]

    \[ f_{1} = k_{12}(x_{2}-x_{1}-l_{12}) = k_{12}x_{2} - k_{12}x_{1} - k_{12}l_{12} \]
    \[ k_{12}l_{12} = const. \]

    \[ f_{2} = k_{23}(x_{3}-x_{2}-l_{23} - k_{12}(x_{2}-x_{1}-l_{12})) \]
    \[ \Rightarrow f_{2} = x_{2}(-k_{12}-k_{23}) + x_{1}k_{12} + x_{3}k_{23} - k_{23}k_{23} + k_{12}l_{12} \]
    \[ - k_{23}k_{23} + k_{12}l_{12} = const. \]

    \[ f_{3} = k_{34}(x_{4}-x_{3}-l_{34}) - k_{23}(x_{3}-x_{2}-l_{23}) \]
    \[ \Rightarrow f_{3} = k_{23}x_{2} + x_{3}(-k_{23}-k_{34}) + k_{34}x_{4} -k_{34}l_{34} + k_{23}l_{23} \]
    \[ -k_{34}l_{34} + k_{23}l_{23} = const. \]

    \[ f_{4} = k_{34}(x_{4}-x_{3}-l_{34}) = k_{34}x_{4} - k_{34}x_{3} - k_{34}l_{34} \]
    \[- k_{34}l_{34} = const. \]

    \[ \textbf{f} = \textbf{K}\textbf{x} + \textbf{c} \]

    \[
    \left[\begin{array}[pos]{c}
    f_{1} \\
    f_{2} \\
    f_{3} \\
    f_{4} \\
    \end{array}\right]=
    \left[\begin{array}[pos]{cccc}
    -k_{12} & k_{12} & 0 & 0 \\
    k_{12} & (-k_{12}-k_{23}) & k_{23} & 0 \\
    0 & k_{23} &(-k_{23}-k_{34}) & k_{34} \\
    0 & 0 &-k_{34} & k_{34}\\
    \end{array}\right]
    \left[\begin{array}[pos]{c}
    x_{1} \\
    x_{2} \\
    x_{3} \\
    x_{4} \\
    \end{array}\right]+
    \left[\begin{array}[pos]{c}
    -k_{12}l_{12} \\
    -k_{23}l_{23} + k_{12}l_{12} \\
    -k_{34}l_{34} + k_{23}l_{23} \\
    -k_{34}l_{34} \\
    \end{array}\right]
    \]


    [/tex]
     
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