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.

 

[PJC01]

, thank you for your question about anharmonic motion. Anharmonic motion refers to the motion of a system that does not follow the simple harmonic motion described by Hooke's law. This means that the restoring force is not directly proportional to the displacement, and the motion is not sinusoidal.

One example of anharmonic motion is the motion of a pendulum where the amplitude of the oscillations is large enough to cause the restoring force to deviate from Hooke's law. In this case, the motion is no longer sinusoidal and the period of the oscillations is longer than what would be predicted by a simple harmonic oscillator.

Another example is the motion of a mass on a spring with a nonlinear spring constant. This can be seen in systems where the spring is stretched or compressed beyond its linear range, causing the restoring force to deviate from Hooke's law.

Mathematically, anharmonic motion can be described using higher-order differential equations, such as the Duffing equation. This equation takes into account the nonlinearity of the system and can produce a wide range of complex motions, including chaotic behavior.

As for visual examples, you can find many demonstrations of anharmonic motion on YouTube or other educational websites. These can help you visualize the behavior of different systems and how they differ from simple harmonic motion.

I hope this helps to clarify anharmonic motion for you. Please let me know if you have any further questions. Keep exploring and learning about the fascinating world of oscillations and dynamics!