The Damped Oscillator.

Damping usually comes into play whenever we consider motion. We study the effect of damping on the spring-mass system. The damping force is assumed to be proportional to the velocity, acting to oppose the motion. The total force acting on the mass is

$$F=-k x - c \dot{x}$$ (2.1)

where in addition to the restoring force $- k x$ due to the spring we also have the damping force $- c \dot{x}$. The equation of motion for the damped spring mass system is

$$m \ddot{x} =-k x - c \dot{x}\,.$$ (2.2)

Recasting this in terms of more convenient coefficients, we have

$$\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x =0$$ (2.3)

This is a second order homogeneous equation with constant coefficients. Both $\omega_0$ and $\beta$ have dimensions ${\rm (time)}^{-1}$. Here $1/\omega_0$ is the time-scale of the oscillations that would occur if there was no damping, and $1/\beta$ is the time-scale required for damping to bring any motion to rest. It is clear that the nature of the motion depends on which time-scale $1/\omega_0$ or $1/\beta$ is larger.

We proceed to solve equation (2.4) by taking a trial solution

$$x(t)=A e^{\alpha t} \,.$$ (2.4)

Putting the trial solution into equation (2.4) gives us the quadratic equation

$$\alpha^2 + 2 \beta \alpha + \omega^2 =0$$ (2.5)

This has two solutions

$$\alpha_1=-\beta + \sqrt{\beta^2 - \omega^2}$$ (2.6)

and

$$\alpha_2=-\beta - \sqrt{\beta^2 - \omega^2}$$ (2.7)

The nature of the solution depends critically on the value of the damping coefficient $\beta$, and the behaviour is quite different depending on whether $\beta < \omega_0$, $\beta = \omega_0$ or $\beta > \omega_0$.