Unique Pendulum Wave and Release Mechanism

10 Steps
This is the first half of a two-part instructable. This first half deals primarily with the math (algebra, trigonometry, geometry) involved in creating a pendulum wave and the release, deriving the pendulum wave design and dimensions. Part 2 will detail the process of constucting the pendulum wave. Although the math in this section is fairly easy, it can be very tedious at times. If you have a hard time following the math or even choose to skip the math altogether, you can still build the pendulum wave shown. I have all the necessary models and dimensioned drawings posted in steps 8 and 9, so no worries. :)

Below are a few pictures of the pendulum wave and release mechanism.

Pendulum wave at rest:

Release mechanism beginning to collect pendulums:

Just before releasing the wave:

Pendulum waves are simply a series of pendulums that are pulled back, then released at some angle(s). If the lengths and angles are just right, then each pendulum will cycle back and forth between its release position at a slightly different frequency than its neighbor. This results in some pretty neat alternating waveforms. Wave pendulums make great desktop toys for your rich dads, and have even been known to keep children quiet for up to 60 seconds. In addition, the can make nice props in classroom settings for illustrating physics principles such as potential and kinetic energy, air resistance, aliasing, and more.

About the Uniqueness of My Pendulum Wave

To my knowlege, the design for the pendulum wave in this instructable is unique for a number of reasons. First, the wave is intended to be viewed primarily from the top rather than a side (although it looks cool from the side, too). Consequently, it is important that the amplitude of each pendulum appears the same from above. While this requires a more complex release mechanism (instead of the typical flat board), it also leads to a very cool look. As far as I am concerned, a little extra coolness is worth a little extra effort, any day.

Release Mechanism

In order to release the pendulums at the varying release angles required for a top view, I designed a very original release mechanism. Each arm of the mechanism grabs its respective pendulum at a different time. This creates a wave effect as the mechanism curves up and around the pendulums. Once all pendulums have been selected and brought to their proper positions, the mechanism releases them all at the same time, creating the pendulum wave. The math for this is a little tedious, but the concept is pretty straightforward.

Here is a very crappy SolidWorks video demonstrating two arms of the release module in action:

Final Design

Here is an animation of the final design in action. Air resistance was neglected in this animation, which is why no damping occurs. However, since air resistance will not affect the period of the pendulum wave, that is fine.

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Step 1: Where the Waves Come From

As I touched on already, a wave pendulum is a series of pendulums with incremented frequencies. One way to think of the pendulum wave is as a series of points used to sample a wave of increasing frequency. This effect is shown in the video below, which I created using Matlab:

The Nyquist Sampling Theory (NST) states that to sample a wave of a given frequency, one needs to measure points at one-half cycle of that frequency. Or, for a given frequency f, one needs points spaced 1/(2*f) units of time apart (since T = 1/f). When the number of sampling points becomes less than the number required by the NST, aliasing occurs. This aliasing is the reason for the alternating waveforms present in a pendulum wave.
mgrunwel says: Dec 10, 2012. 8:09 AM
Very cool release mechanism. I must say though, all that math is making that sponge thing in my head throb.
Coxster (author) in reply to mgrunwelDec 12, 2012. 6:02 PM
Thanks for the feedback! Yes, I was a little worried about posting the math... In hindsight, it might not have been a bad idea to have the last page an appendix of sorts, dedicated solely to math for those interested. I know it's a lot to wade through.

Btw, you can always skip directly to the end for the bill of materials :)
Foxtrot70 says: Jun 9, 2012. 12:23 PM
And whos says science isn't fun!
locustboy says: Jun 7, 2012. 4:18 PM
Wow! That is amazing! You must be a genius or something with all of that math stuff, but I think I'll just follow your dimensions. This is a great instructable.
Coxster (author) in reply to locustboyJun 7, 2012. 5:57 PM
Ha, thanks! The dimensions should get you through. If you wait a couple months, I should have another instructable up detailing construction. However, go ahead and give it a try if you want. Feel free to ask if you have any questions!
ynze says: Jun 5, 2012. 2:14 AM
Beautiful! That's a great science-on-the-beach exhibit :-)
Coxster (author) in reply to ynzeJun 5, 2012. 12:32 PM
Why thank you!

Sand might result in a less-than-best viewing surface, however. ;)