## Representing a Mechanical System in Matrix Form

Consider two masses connected between two fixed blocks by three springs. Initially the system is in equilibrium and all the springs are in tension. The masses are given small pushes and start to vibrate.

The net force on mass 1 is
$\mathbf{F}_1=--T_1+T_2$

The net force on mass 2 is
$\mathbf{F}_2=--T_2+T_3$

On the diagram above the net increase in length of the central spring is
$x_2-x_1$
so the tension in the central spring is
$T_2=k_2(x_2-x_1)$
.
Now use
$\mathbf{F}_i=m_i \mathbf{a}_i$
and
$\mathbf{T}_i=-k_- \mathbf{x}_i$
(note that
$x_3=x_2$
to get
$m_1 a_1=-k_1x_1+k_2(x_2-x_1)=-2k_1x_1+k_2x_2 \rightarrow a_1=-\frac{2k_1}{m_1}x_1 +\frac{k_2}{m_1} x_2$

$m_2 a_2=-k_2(x_2-x_1)-k_3x_2=k_2x_1-(k_2+k_3)x_2 \rightarrow a_1=\frac{k_2}{m_2}x_1 -\frac{k_2+k_3}{m_2}x_2$

We can write this in matrix form as
$\begin{pmatrix}a_1\\a_2\end{pmatrix}=\left( \begin{array}{cc} -\frac{2k_1}{m_1} & \frac{k_2}{m_1} \\ \frac{k_2}{m_2} & -\frac{k_2+k_3}{m_2} \end{array} \right) \begin{pmatrix}x_1\\x_2\end{pmatrix}$
.
Suppose that
$m_1=1, \: m_2=2, \: k_1=1, \: k_2=2, \: k_3=3$
. The system becomes
$\begin{pmatrix}a_1\\a_2\end{pmatrix}=\left( \begin{array}{cc} -2 & 2 \\ 1 & -5/2 \end{array} \right) \begin{pmatrix}x_1\\x_2\end{pmatrix}$
.
Find the eigenvalues of the matrix.
$det( \left( \begin{array}{cc} -2 & 2 \\ 1 & -5/2 \end{array} \right)- \lambda \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right))=\lambda^2+4.5 \lambda^2+3=0$
.
Then
$\lambda=\frac{-4.5 + \sqrt{8.25}}{2}$
.
Then mass 1 vibrates with a frequency
$2 \pi \omega_1= 2 \pi \sqrt{- \lambda_1} = 2 \pi \sqrt{\frac{4.5 + \sqrt{8.25}}{2}}$
, and mass 2 vibrates with a frequency
$2 \pi \omega_2= 2 \pi \sqrt{- \lambda_2} = 2 \pi \sqrt{\frac{4.5 - \sqrt{9.25}}{2}}$

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