Damping

Nothing in life is perfect! Air friction, losses in a spring, and friction from surfaces conspire to remove energy from the spring and generally turn it into heat. Any real oscillator will damp. One useful model we can use to describe damping is to add an extra friction force proportional to velocity. The new force equation and Newton’s second law are: $$F = - k x – \beta v$$

Writing this in terms of Newton's second law, we have the following differential equation. $$\frac{d^2 x}{d t^2} = - 2 \Gamma \frac{d x}{d t} - \omega_0^2 x.$$

The new e.o.m. is damped as $$x_0 = x_0 e^{-\Gamma t} \sin (\omega t + \phi)$$

For weak damping (when $\Gamma \ll \omega$), the new frequency is slightly ``pulled'' to a lower value.