Damped Oscillations and Types of Damping

1The swing that stops

Pull a child's swing back and let go. In Unit 1 an ideal oscillator would keep swinging forever with the same amplitude. A real swing does not: each swing is a little smaller than the one before, and after a minute or so it hangs still. Air resistance and friction at the pivot take a little energy from every cycle. Engineers sometimes fight this loss, as in a clock, and sometimes add as much of it as they can, as in the shock absorbers of a car or minibus. This lesson describes the motion of an oscillator when a resistive force acts, and the three quite different ways it can come to rest.

2The damping force

Any force that opposes the motion and removes energy from an oscillator is called a damping force. For a body moving slowly through a fluid (air, water, oil), the damping force is close to proportional to the velocity:

where $bb$ is the damping constant, in $\mathrm{k}\mathrm{g}{\mathrm{s}}^{-1}\backslash mathrm\{kg\backslash ,s^\{-1\}\}$ (or $\mathrm{N}\mathrm{s}{\mathrm{m}}^{-1}\backslash mathrm\{N\backslash ,s\backslash ,m^\{-1\}\}$), and the minus sign means the force always opposes the velocity. Adding it to the restoring force of a spring, Newton's second law becomes

It is useful to write ${\omega }_0=\sqrt{\frac{k}{m}}\backslash omega\_0\; =\; \backslash sqrt\{\backslash frac\{k\}\{m\}\}$, the natural angular frequency the system would have without damping, and $\gamma =\frac{b}{2m}\backslash gamma\; =\; \backslash frac\{b\}\{2m\}$, which measures how strong the damping is. The equation then reads $\frac{d^{2}x}{d{t}^2}+2\gamma \frac{dx}{dt}+{\omega }_0^{2}x=0\backslash frac\{d^2x\}\{dt^2\}\; +\; 2\backslash gamma\backslash frac\{dx\}\{dt\}\; +\; \backslash omega\_0^2x\; =\; 0$.

3Light damping

When the damping is small (), the solution is an oscillation whose amplitude decays exponentially:

You can check this by substituting it into the equation. Two things happen. The amplitude $A_0{e}^{-\gamma t}A\_0e^\{-\backslash gamma\; t\}$ shrinks by the same factor in every equal interval of time, so the ratio of any two successive peaks is constant. The angular frequency ${\omega }_d\backslash omega\_d$ is a little smaller than ${\omega }_0\backslash omega\_0$, so the period is a little longer; for light damping the difference is tiny.

4Three kinds of damping

As the damping grows, the motion changes character. The deciding comparison is between $\gamma \backslash gamma$ and ${\omega }_0\backslash omega\_0$.

The critical damping constant $b_c=2\sqrt{mk}=2m{\omega }_{0}b\_c\; =\; 2\backslash sqrt\{mk\}\; =\; 2m\backslash omega\_0$ follows from setting $\gamma ={\omega }_0\backslash gamma\; =\; \backslash omega\_0$. Notice from the graph that heavy damping is not the quickest way to stop an oscillation: too much damping makes the return to equilibrium slower.

5Damping in a vehicle

Every car, minibus and motorcycle has springs to absorb bumps in the road and dampers, often called shock absorbers, to stop the body bouncing afterwards. Without dampers the body would bounce up and down for many cycles after each pothole; with far too much damping it would ride the bump stiffly and pass every jolt to the passengers.