How to Discretize?

The one-sided Laplace transform of a signal $x(t)$ is defined by

$$X(s) \buildrel$$$$\mbox{\tiny$\Delta$}$$$$\over={\cal L}_s\{x\} \buildrel$$$$\mbox{\tiny$\Delta$}$$$$\over=\int_0^\infty x(t) e^{-st}dt$$

where $t$ is real and $s=\sigma + j\omega$ is a complex variable.

The differentiation theorem for Laplace transforms states that

$$\frac{d}{dt}x(t) \leftrightarrow s X(s)$$

where $x(t)$ is any differentiable function that approaches zero as $t$ goes to infinity.

The transfer function of an ideal differentiator is $H(s)=s$, which can be viewed as the Laplace transform of the operator $d/dt$.

Given the equation of motion

$$m\ddot{x} + R\dot{x} + Kx = F(t),$$

the Laplace Transform is

$$s^2X(s) + 2\alpha sX(s) + \omega_0^2X(s) = F(s).$$