What are Examples of Anharmonic Motion?

[James Starligh]

Dear all,

I've already known the basic forms and general equations which describes harmonic motion in case of description of simple oscillation system by means of Hooke laws. Could you provide me with some examples of anharmonic motions, with the vizual examples as well as math equations I've not found it in Wiki.


James

 

[sophiecentaur]

Dear all,

I've already known the basic forms and general equations which describes harmonic motion in case of description of simple oscillation system by means of Hooke laws. Could you provide me with some examples of anharmonic motions, with the vizual examples as well as math equations I've not found it in Wiki.
James

Simple Harmonic Motion is what you get when the restoring force is linearly related to displacement. There are plenty of systems in which the force is not linearly related. For instance, two springs, with different spring constants but with the stiffer one tightly coiled so that its coils only open when the oscillations have big displacements. Dampers on car suspension have valves which open when the wheel moves up and close when it moves down. That doesn't follow SHM, either. SHM is only a very special case and, not surprisingly, is the easiest one to analyse. That's why you get it first!

There are many electrical equivalents of oscillators with multiple components, including diodes and different capacitors. These are direct analogues of mass, springs, end stops and damping.

Complex multiple pendulums can also display Chaotic motion but that's another can of worms.

 

[MalachiK]

Is it just springs that you're interested in? It's possible to imagine lots of situations where non SHM oscillating motion occurs. For example...

I think there's a standard SHM example where you imagine a uniform density, spherical planet and then dig an imaginary tunnel though the middle. Jumping into the tunnel gets you SHM with the linear restoring force and parabolic potential energy curve. (You can prove that the g field strength falls off linearly as you go down the tunnel or just take my work for it.)

But, outside the sphere the force is non-linear (and the potential energy goes like 1/r). This means that a mass dropped into the tunnel from some distance outside the sphere will oscillate back and fore, but it won't be simple harmonic because a ≠ -ωr for all values of r between the amplitudes.

 

[AlephZero]

If you want to study a very simple system with surprisingly complicated behavior, google for "duffing oscillator" or "duffing equation".

Basically it is just a mass on a spring, except the force in the spring is $kx + ax^3$ not $kx$. The spring can get stiffer or more flexible as it stretches, depending on whether a is positive or negative.