## Modelling a Three Particle Spring System 0 Stiffness and Elasticity Matrices

Suppose we are to construct the elasticity and stiffness matrices for the system below, with masses
$m_1, \: m_2, \: m_3$
extended distances
$x_1, \: x_2, \: x_3$
from their equilibrium positions.

The force on each mass is as follows
$m_1$
:
$-k_1x_1+k_{12}(x_2-x_1)+k_{13}(x_3-x_1)=-(k_1+k_{12}+k_{13})x_1+k_{12}x_2+k_{13}x_3$

$m_2$
:
$-k_{12}(x_2-x_1)+k_{23}(x_3-x_2)=k_{12}x_1-(k_{12}+k_{23})x_2+k_{23}x_3$

$m_3$
:
$-k_{13}(x_3-x_1)-k_{23}(x_3-x_2)-k_3x_3=k_{13}x_1+k_{23}x_2-(k_{13}+k_{23}+k_3)x_3$

The stiffness matrix is then
$K= \left( \begin{array}{ccc} -(k_1+k_{12}+k_{13}) & k_{12} & k_{13} \\ k_{12} & -(k_{12}+k_{23}) & k_{23} \\ k_{13} & k_{23} & -(k_{13}+k_{23}+k_3) \end{array} \right)$

The elasticity matrix is nbsp;
$K^{-1}= \left( \begin{array}{ccc} -(k_1+k_{12}+k_{13}) & k_{12} & k_{13} \\ k_{12} & -(k_{12}+k_{23}) & k_{23} \\ k_{13} & k_{23} & -(k_{13}+k_{23}+k_3) \end{array} \right)^{-1}$
.