The Solution to the Damped Vibrator

The solution to the system equation

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

has the form

$$x = A(t)\cos(\omega_dt + \phi).$$

where $A(t)$ is the amplitude envelope

$$A(t) = e^{-t/\tau} = e^{-\alpha t},$$

and the natural frequency $\omega_d$

$$\omega_d = \sqrt{\omega_0^2-\alpha^2},$$

is lower than that of the ideal mass-spring system

$$\omega_0 = \sqrt{\frac{K}{m}}.$$

Peak $A$ and $\phi$ are determined by the initial displacement and velocity.