The Helix is a simple mechanical device for finding the harmonic amplitudes and phases of periodic functions. It is a cylindrical rod with a number of thin bars through the rod, constructed in such a way that the thin bars spiral a full rotation across the rod's length (shaped like a DNA structure). The photo above shows the proof-of-concept model on the left side and a more accurate model on the right (thank you Tatjana J. van Vark for helping me construct this).
The Helix has no practical use but it helped me understand Fourier series and harmonic analysis.

Just like Lord Kelvin's Harmonic Analyzer, Michelson's Harmonic Analyzer and Henrici's Harmonic Analyzer, the Helix determines the Fourier coefficients of the Fourier series.

Usage : finding the first harmonic
  1. Print the graph of the function of interest onto a sheet of paper where the period of the function is the same width as the length of Helix's axis. It is assumed that the positive area of the function is equal to the negative area of the function, because the Helix is not suitable for finding the constant term of the Fourier series.

  2. The Helix has "slides" (the thin bars sticking out the axis) and "sliders" (the red or blue spherical weights which slide onto the slides). Start off with an empty Helix (all sliders removed).

  3. Position the Helix above the graph so that its axis runs along the graph's x-axis.

  4. While looking down from above the Helix, "record" the function as follows:
    For each of the Helix's slide, from left to right along the x-axis:
    - Twist the Helix's axis so that the slide is parallel to the graph's plane.
    - Find the position where the slider intersects the function.
    - Add a red slider to the slide and slide it to this position.
    The result will be a red slide on each slider, added to the Helix after turning a full period.

    The photo below shows the result after "recording" a square wave.
    Notice that the red sliders are all on one side of the Helix, with equal distances from the axis.



  5. Next, hold the Helix at each end and let it rotate freely, allowing gravity to do its work until the heavier side points down.

    Result - the phase: The angle between the resulting heavier bottom side and the bottom side of the starting position is the phase of the first harmonic (phase added to the sine).

    Note that the phase of the first harmonic of the square wave is zero, which is what we would expect because the first harmonic is a pure sine. If the phase were 90 degrees, then the harmonic would be a pure cosine.

  6. Now the tricky part:
    - Pick up the Helix, lift to eye-height and look down the Helix's axis, keeping the heavier side at the bottom.
    - Position a blue slider on each slide such that it forms a circle (when looking down the axis) on the top side (the lighter side), making sure the edge of the circle crosses the axis.
    - Resize the blue circle in such a way that it balances exactly the "rest-weight" of the red sliders on the other side, so that the Helix can rotate freely without weight-bias.

    Result - the amplitude: The diameter of the resulting circle is the amplitude of the first harmonic.

    The photo below shows the result after finding the first harmonic of a square wave.


    Note that the amplitude of the first harmonic (the diameter of the circle of blue sliders) is slightly larger than the amplitude of the square wave. This is as we would expect, because the theoretical ratio is 4 / π = 1.2732...

  7. Synthesis: Once you have found the balanced configuration as described above, while twisting the Helix above the x-axis (one full turn for the first harmonic), record the harmonic onto the graph paper by marking where the blue markers (from left to right) are above the graph when the slide is flat above the paper.
  8. Join the marked dots on the graph and there you have it: the first harmonic!
    The final result should look something like the situation in the left side of the above photo.

Usage : finding the second harmonic

Method 1
As described above, except in step 4: twist with twice the rate while crossing the graph, twisting 720 degrees for one period. Note that you will need twice as many sliders (red and blue), and that it may be difficult to position the sliders correctly.

Method 2
Use two Helix constructions, joined at the axes. Then proceed as described above, except in step 4: twist a full 720 degrees for one period. While doing so you will need to shift the Helix half a slide-distance towards the left after adding each slide.

Method 3
In step 6, use variable weights instead of blue slides to simply determine the amplitude and phase for each half of the period (separate measurements), and then combine the resulting weights for finding the resulting amplitude and phase. The latter alternative is useful if you just want to know the amplitude and phase, without the need for synthesis.

Method 4
Construct a more flexible Helix, where all the slides can be turned freely around the axis. For determining the second harmonic, simply evenly arrange all the sliders so that it turns a full 720 degrees over the axis. With this construction it is desired to have a large number of sliders, for accuracy. The advantage of this construction is that it is a more general construction, and can even be used to determine the constant term.


Note that this is a sort of discrete version of harmonic analysis. The device would be more accurate if it had a large number of slides. The design show on this page is not very inaccurate, especially when needing to determine the higher order harmonics. A more accurate and practical design would be a set of Helix constructions (one for each harmonic to analyze), where the axis of each Helix can be connected to the axis of another Helix.

An improved design: the Variable Helix Harmonic Analyzer
A better design would be one that has many freely rotating slides, in which case the recording process would be much easier (starting off with all sliders flat, like an abacus, no rotation necessary during recording) and only needs to be done just once for all harmonics. Then for each harmonic to analyse, apply an additional 360 degree rotation to the slides, evenly spread across the axis, and determine the rest-weight in each case.
This construction is a bit difficult to construct, so I am considering making a 3d animation.


I hope to find some time to post some full video instructions on how to use the Helix harmonic analyzer.



Tim Samshuijzen
2006