# Thread: Coupled mass to matrix problem

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

a) Write the 4 X 4 matrix equation relating the column vectors $\displaystyle f$ and $\displaystyle x$. Let $\displaystyle K$ 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

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

I then expand and simplify each and expression.

I then write these equations in the form:
$\displaystyle \textbf{f} = \textbf{K}\textbf{x} + \textbf{c}$

$\displaystyle \left[\begin{array}[pos]{c} f_{1} \\ f_{2} \\ f_{3} \\ f_{4} \\ \end{array}\right]$

$\displaystyle = \left[\begin{array}[pos]{cccc} -k_{12} & k_{12} & 0 & 0 \\ k_{12} & (-k_{12}-k_{23}) & k_{23} & 0 \\ 0 & k_{23} &(-k_{23}-k_{34}) & k_{34} \\ 0 & 0 &-k_{34} & k_{34}\\ \end{array}\right]$

$\displaystyle \left[\begin{array}[pos]{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ \end{array}\right]$

$\displaystyle + \left[\begin{array}[pos]{c} -k_{12}l_{12} \\ -k_{23}l_{23} + k_{12}l_{12} \\ -k_{34}l_{34} + k_{23}l_{23} \\ -k_{34}l_{34} \\ \end{array}\right]$

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.

2. That looks good to me!

3. Nothing better than that
Thanks!