State model of a mass-spring-damper system

The mass is pulled by $F(t)$, held back by the spring $K$ (left wall) and the damper $B$ (right wall). Both oppose the motion.

Differential equation

Newton's law (free-body diagram of $M$):

$$M\frac{d^2x}{dt^2} + B\frac{dx}{dt} + Kx = F(t)$$

State variables

Displacement and velocity of the mass (energy in the spring and the mass):

$$x_1 = x, \qquad x_2 = \dot{x}$$ $$\begin{aligned} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= -\frac{K}{M}x_1 - \frac{B}{M}x_2 + \frac{1}{M}F(t) \end{aligned}$$

State-space model

Taking the displacement $x$ as the output:

$$\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -\dfrac{K}{M} & -\dfrac{B}{M} \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \\ \dfrac{1}{M} \end{bmatrix}F(t)$$ $$y = \begin{bmatrix} 1 & 0 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$$

(If velocity is wanted as output, use $C = [0 \;\; 1]$.) The corresponding transfer function is $\dfrac{X(s)}{F(s)} = \dfrac{1}{Ms^2 + Bs + K}$.