Formula 1

$z(t = y(t - x(t$: Relative displacement between the mass and base.

$$z(t = y(t - x(t$$

Formula 2

Absolute motion: $m\ddot{y}(t + c\dot{y}(t + ky(t = kx(t + c\dot{x}(t$

$$m\ddot{y}(t + c\dot{y}(t + ky(t = kx(t + c\dot{x}(t$$

Formula 3

Force equilibrium: $m\ddot{y}(t + c[\dot{y}(t - \dot{x}(t + k[y(t - x(t = 0$

$$m\ddot{y}(t + c[\dot{y}(t - \dot{x}(t + k[y(t - x(t = 0$$

Formula 4

Relative motion: $\ddot{z}(t + (c/m\dot{z}(t + (k/mz(t = -\ddot{x}(t$

$$\ddot{z}(t + (c/m\dot{z}(t + (k/mz(t = -\ddot{x}(t$$