3
$\begingroup$

This YouTube video shows a very interesting effect.

What is the underlying science?

I have recently started studying Fourier theory and DSP, and and trying to understand what is going on in terms of the material I've been learning. I would like to understand this from as many angles as possible.

For example, at 0:39 the signal forms two lines moving against one another, and shortly before that I can clearly see two sine waves in the signal.

This reminds me of the illustrations at http://www.dspguide.com/ch3/2.htm

PS not quite sure how to tag this

$\endgroup$
  • $\begingroup$ see: en.wikipedia.org/wiki/Pendulum_(mathematics). note that each pendulum have different "rod" size and thus different oscillation $\endgroup$ – Dov Mar 8 '12 at 10:48
  • $\begingroup$ Can someone edit the link in the above comment? The closing ) should be part of the link $\endgroup$ – P i Mar 8 '12 at 14:30
8
$\begingroup$

As was pointed out in a comment above, this is a simple consequence of the dynamics of a pendulum. There's nothing particularly signal-processing-related about this problem, just some simple physics and trigonometry.

For a pendulum that is displaced from its angular equilibrium point by an angle of $\theta_0$ with an angular velocity of $\omega_0 = 0$ at $t = 0$, for small initial angles, the motion of the pendulum can be approximated by simple harmonic motion:

$$ \theta(t) = \theta_0 \cos\left(\sqrt{\frac{g}{l}}t\right) $$

where $g$ is the magnitude of the force of gravity and $l$ is the length of the pendulum. In the video, a number of pendulums of varying lengths are given an approximately equal initial displacement $\theta_0$ and released simultaneously. Once they are released, each pendulum traces out a path whose displacement angle is a sinusoid with frequency related to the pendulum's length. The lengths were carefully chosen to give the interesting visual effect that you see in the video; the frequencies of the sinusoids are multiples of some fundamental frequency, so for particular values of $t$, they line up in interesting ways.

You'll notice toward the end, all of the balls "line up" again; this is because each of the sinusoids that describe their motion have moved through an integer number of periods, so the balls are back at a common angular displacement. If there were no effects of friction, air resistance, etc., the oscillation would continue, with the same patterns repeating indefinitely.

$\endgroup$
  • $\begingroup$ Thanks for the answer -- I kind of realised that each ball is like a sinusoidal oscillator of k*f_0, where k might run from 50 to 75 or something, which is sounding a lot like Fourier decomposition... $\endgroup$ – P i Mar 8 '12 at 14:33
  • $\begingroup$ Of course it's signal processing related. It's a discrete-time sine wave frequency sweep which bounces off the Nyquist frequency and aliases back to DC. :) $\endgroup$ – endolith Mar 8 '12 at 14:38
  • $\begingroup$ @endolith: I agree. It's just the physics that aren't really DSP-related. It's a really neat demo that can be used to teach a lot of signal processing concepts, though. Makes me want to try to build one with my kids. $\endgroup$ – Jason R Mar 8 '12 at 15:09
  • $\begingroup$ @endolith, RIGHT! Now we are getting somewhere! Would you be up for elaborating on that in a separate answer? Forget the physics, I'm interested in how this can teach signal processing concepts -- how much DSP can be learned through this demonstration? $\endgroup$ – P i Mar 8 '12 at 21:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.