Solving Coupled Mass Problem: 4x4 Matrix Equation

[Dafe]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I'm trying to find the rightward force acting on every mass as the springs are stretched.

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}

It seems quite complicated to put this into matrix form, so I'm assuimg that I've done something wrong.
Suggestions?

 

[Dafe]

By the way, here is a figure showing how I am visualizing this problem:
I am assuming m_{1} to be constrained. Just throwing out ideas :)

 

[Dafe]

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?

<br /> <br /> \[ f_{1} = k_{12}(x_{2}-x_{1}-l_{12}) = k_{12}x_{2} - k_{12}x_{1} - k_{12}l_{12} \]<br /> \[ k_{12}l_{12} = const. \]<br /> <br /> \[ f_{2} = k_{23}(x_{3}-x_{2}-l_{23} - k_{12}(x_{2}-x_{1}-l_{12})) \]<br /> \[ \Rightarrow f_{2} = x_{2}(-k_{12}-k_{23}) + x_{1}k_{12} + x_{3}k_{23} - k_{23}k_{23} + k_{12}l_{12} \]<br /> \[ - k_{23}k_{23} + k_{12}l_{12} = const. \]<br /> <br /> \[ f_{3} = k_{34}(x_{4}-x_{3}-l_{34}) - k_{23}(x_{3}-x_{2}-l_{23}) \]<br /> \[ \Rightarrow f_{3} = k_{23}x_{2} + x_{3}(-k_{23}-k_{34}) + k_{34}x_{4} -k_{34}l_{34} + k_{23}l_{23} \]<br /> \[ -k_{34}l_{34} + k_{23}l_{23} = const. \]<br /> <br /> \[ f_{4} = k_{34}(x_{4}-x_{3}-l_{34}) = k_{34}x_{4} - k_{34}x_{3} - k_{34}l_{34} \]<br /> \[- k_{34}l_{34} = const. \]<br /> <br /> \[ \textbf{f} = \textbf{K}\textbf{x} + \textbf{c} \]<br /> <br /> \[<br /> \left[\begin{array}[pos]{c}<br /> f_{1} \\<br /> f_{2} \\<br /> f_{3} \\<br /> f_{4} \\<br /> \end{array}\right]=<br /> \left[\begin{array}[pos]{cccc}<br /> -k_{12} &amp; k_{12} &amp; 0 &amp; 0 \\<br /> k_{12} &amp; (-k_{12}-k_{23}) &amp; k_{23} &amp; 0 \\<br /> 0 &amp; k_{23} &amp;(-k_{23}-k_{34}) &amp; k_{34} \\<br /> 0 &amp; 0 &amp;-k_{34} &amp; k_{34}\\<br /> \end{array}\right]<br /> \left[\begin{array}[pos]{c}<br /> x_{1} \\<br /> x_{2} \\<br /> x_{3} \\<br /> x_{4} \\<br /> \end{array}\right]+<br /> \left[\begin{array}[pos]{c}<br /> -k_{12}l_{12} \\<br /> -k_{23}l_{23} + k_{12}l_{12} \\<br /> -k_{34}l_{34} + k_{23}l_{23} \\<br /> -k_{34}l_{34} \\<br /> \end{array}\right]<br /> \]