Damping and Damped Systems

A damper (dashpot) resists relative motion across it which appears in the ODE equation of motion as a fully linear ODE (x” + x’ + x)

MDOF Damped Systems

This system has multiple x’s (distances that the springs stretch with respect to)

You usually assume that the x’s further from the point of attachment is as follows
$x_2>x_1$
and the same goes for their derivatives.

For their Spring and damp forces we have as follows (if the mass is connected at the bottom to k2 and the top to k1)

At K1 $F_{k2}=k_2\left(x_2-x_1\right)$
and
$F_{c2}=c_2\left(x_2^{\prime }-x_1^{\prime }\right)$
Now you can write out the equations of motion for both masses using Newtons 2nd Law

For connected masses, motions are coupled