## Modelling a Parallel Three Particle Mass Spring System Example

Suppose we have the mass spring system shown.The contact of the upper spring is halfway along the bar.

Applyin
$\mathbf{F}=m \mathbf{a}$
to each particle:
$m_1$
:
$m_1 \ddot{y}_1=-k_1 y_1 +\frac{1}{2}k_3(y_3+\frac{1}{2}(y_1+y_2))=(-k_1+\frac{1}{4}k_3)y_1+\frac{1}{4}k_3y_2+k_3y_3$

$m_2$
:
$m_2 \ddot{y}_2=-k_2 y_2 +\frac{1}{2}k_3(y_3+\frac{1}{2}(y_1+y_2))=\frac{1}{4}k_3y_1+(-k_2 +\frac{1}{4}k_3)y_2+k_3y_3$

$m_3$
:
$m_3 \ddot{y}_3=-k_3(y_3-\frac{1}{2}(y_1+y_2))=\frac{1}{2}k_3y_1+\frac{1}{2}k_3y_2-k_3y_3$

We can divide these equations by
$m_1, \: m_2, \: m_3$
respectively and writing in matrix form gives us the equation
$\begin{pmatrix}\ddot{y}_1 \\ \ddot{y}_2 \\ \ddot{y}_3 \end{pmatrix} = \left( \begin{array}{ccc} \frac{-k_1+k_3/4}{m_1} & \frac{k_3}{4m_1} & \frac{k_3}{4m_1} \\ \frac{k_3}{4m_2} & \frac{-k_2 +k_3/4}{4m_2} & \frac{k_3}{m_2} \\ \frac{k_3}{2m_3} & \frac{k_3}{2m_3} & \frac{k_3}{m_3} \end{array} \right) \begin{pmatrix}y_1 \\ y_2 \\ y_3 \end{pmatrix}$

Find the eigenvalues
$\lambda_i$
and the natural frequencues of the system are
$f=2 \pi \omega=2 \pi \sqrt{\lambda}$
.