Relative periodicity of non-harmonic oscillators
 

Hello virtual friends.

You might enjoy this.

https://www.youtube.com/watch?v=o3Q7...A&index=3&t=1s

Each pendulum is set up to oscillate an integer number of times in a minute - the second longest manages one more than the longest, and so on and so on. The idea is sort of obvious, but the graphic result is perhaps unexpected until you see it.
cheers
Mark

Re: Relative periodicity of non-harmonic oscillators
 

Thanks for that, physics and maths in action shown visually, I think it's better than the Mona Lisa.

Re: Relative periodicity of non-harmonic oscillators
 

It must have taken a lot of patience to get the pendulum periods sufficiently accurate to recover the initial state so well. The maths is identical to the theory of mode locking in lasers to generate a train of ultra-short pulses.

Roger

Re: Relative periodicity of non-harmonic oscillators
 

Roger - I guess the short pulses occur when some things come in-phase - what about the brief times in the video where 1/2, or 1/3, or 1/4 of the oscillators are 'in-phase' and their addition would look momentarily not-incoherent?

(I'm lacking a language for lasers - I used to be in acoustics).

Re: Relative periodicity of non-harmonic oscillators
 

In a mode locked ultra-short pulse laser we're commonly synchronising a very large number of modes indeed, so except for the moment when they all come into synchronism the number that happen to add together never gets big enough to matter much. But in lasers which are oscillating on relatively few modes you can indeed see mode-beating effects which are similar to the behaviour of subsets of your pendulums.

(I should say the Roger and I both worked at the same laser institution for quite a while !)

Cheers,

GJ

Re: Relative periodicity of non-harmonic oscillators
 

My gut feeling is that as long as the periods are not long and you don't have too many then they come back into sync within a reasonable time (even if you don't pick 'nice' relationships between them).

As for 1/2, 1/3 , 1/4, of them, presumably that is because they have shorter times to come in sync, at worse the product of the periods.

Now I should properly think about this and decide if what I wrote above is true... :)

Re: Relative periodicity of non-harmonic oscillators
 

To get regular behaviour you do need the modes, or pendulum frequencies, to be evenly spaced, so in that sense you do need 'nice' relationships between them.

Cheers,

GJ

Re: Relative periodicity of non-harmonic oscillators
 

Yes, I can see that, otherwise they would just come together 'out of chaos' and the patterns would not be so nice.

Re: Relative periodicity of non-harmonic oscillators
 

I suppose if there were prime relationships between the periods of the oscillators - that is, their frequencies were all prime multiples of some fundamental bass 'sub-harmonic' - then they'd all come in-phase only at that (long) period, but would look incoherent elsewhere.

(Perhaps interestingly, in acoustics the ear can spot such a 'missing fundamental', and this forms part of the academic argument for how pitch perception works).

Re: Relative periodicity of non-harmonic oscillators
 

If you're listening to an SSB signal and adjusting the tuning, the apparent pitch of the audio slides. All audio components are offset by the same amount, so harmonic components are not in the right places compared to the fundamentals. This sounds wrong. Most people can detect the wrongness, but not put their finger on the reason why.

Twiddle the tuning and then at some point all the harmonics land on integer multiples of their fundamental's frequencies and it sounds 'right'.

For a bit of fun you can use an SSB receiver to demod one carrier of an AM station playing music. Frequency offset music sounds seriously weird!

In music and acoustics, frequency scaling by a multiplication factor is natural, and fits with our logarithmic perception of frequency. An additive/subtractive shift is very unnatural and can sound disquieting.

David

Re: Relative periodicity of non-harmonic oscillators
 

Yes - in acoustics the tendency to hear a bunch of harmonic modes as one fused tone with timbre set by the relative harmonic amplitudes, forms another part of the 'mechanisms of pitch perception' argument. That's one reason why bells sound weird - bell vibrations (like most bending waves in solids) are dispersive (the wave speed is a function of frequency) which means the modes are non-harmonic. The perceived pitch of a bell might be the mode which is loudest, not necc. the lowest one - and if you try to play harmonies on them the superposition of all the non-harmonic modes means all hell breaks loose! :)

(This is close to the argument that western music works the way it does, because once you decide to do harmony then spacing the superimposed (harmonic) modes of (most) musical instruments out, on the relevant membrane inside the inner ear, means you generate the set of musical intervals we are familiar with. In cultures which use other intervals, they tend not to be big on harmony).

Re: Relative periodicity of non-harmonic oscillators
 

Drifting onto the subject of vintage electronics (!) it's sometimes argued that 100Hz HT ripple in the output stages of push-pull valve amplifiers doesn't matter, because the effects of the resulting 100Hz current components through the valves are cancelled by the action of the output transformer, assuming they're equal in size. But there is another effect. The varying HT voltage modulates the stage gain and that imposes +/-100Hz sidebands on the audio. In an amp where the harmonic distortion has been effectively suppressed these two components are not unusually the largest ones present. And I wouldn't expect them to sound great ...

Cheers,

GJ

Re: Relative periodicity of non-harmonic oscillators
 

Conversely, when you tune to a "proper" SSB signal there is, quite sharply, a sweet spot where it sounds very good. Helped a lot these days by synthesized (from a crystal) transmitters and receivers.