So far, all the oscillators we've treated are *ideal*. There is no
friction or damping. In the real world, of course, things *always*
damp down. You have to keep pushing the kid on the swing or they slowly
come to rest. Your car doesn't *keep* bouncing after going through
a pothole in the road. Buildings and bridges, clocks and kids, real
oscillators all have damping.

Damping forces can be very complicated. There is kinetic friction,
which tends to be independent of speed. There are various fluid drag
forces, which tend to depend on speed, but in a sometimes complicated
way. There may be other forces that we haven't studied yet that
contribute to damping. So in order to get beyond a very qualitative
description of damping, we're going to have to specify a *form* for
the damping force (ideally one we can work with, i.e. integrate).

We'll pick the simplest possible one:

(84) |

We proceed with Newton's second law for a mass on a spring with
spring constant and a damping force :

(85) |

(86) |

Again, it looks like a function that is proportional to its own *first* derivative is called for (and in this case this excludes sine and
cosine as possibilities). We guess
as before,
substitute, cancel out the common and get the
characteristic:

(87) |

To solve for we have to use the dread *quadratic formula*:

(88) |

This isn't quite where we want it. We simplify the first term, factor a
out from under the radical (where it becomes , where
is the frequency of the *undamped*
oscillator with the same mass and spring constant) and get:

(89) |

(90) |

(91) |

Again, we can take the real part of their sum and get:

(92) |