Image shows a two$-$mass translational mechanical system. The applied force $f_a(t)$ acts on mass $m_1.$ Displacements $z_1$ and $z_2$ are absolute positions of masses $m_1$ and $m_2,$ respectively, measured relative to fixed coordinates $($the static equilibrium positions with $f_a(t)=0).$ An oil film with viscous friction coefficient $b$ separates masses $m_1$ and $m_2.$ Derive the mathematical model of the mechanical system.

ANSWER $-$

$m_1{z}_1\text{'}\text{'}+b{z}_1\text{'}-b{z}_2\text{'}+k_1{z}_1=f_a(t)$

$m_2{z}_2\text{'}\text{'}+b{z}_2\text{'}-b{z}_1\text{'}+k_2{z}_2=0$