Damped Equation of Motion

The equation of motion for the damped system is obtained by adding the drag force into the equation of motion:

$$m\frac{d^2x}{dt^2} + R\frac{dx}{dt} + Kx = 0,$$

or alternatively

$$\frac{d^2x}{dt^2} + 2\alpha\frac{dx}{dt} + \omega_0^2x = 0,$$

where $\alpha=R/2m$ and $\omega_0^2 = K/m$.

The damping in a system is often measured by the quantity $\tau$, which is the time for the amplitude to decrease to $1/e$:

$$\tau = \frac{1}{\alpha} = \frac{2m}{R}.$$

When a simple oscillator is driven by an external force $F(t)$, the equation of motion becomes

$$m\frac{d^2x}{dt^2} + R\frac{dx}{dt} + Kx = F(t).$$