Summary

Models a 4-component, complex-valued square wave as a complex 1D Fourier transform. 3rd model of the Fourier Orrery collection.

Superseding the original square wave demo, this version uses all the learnings from the heart curve project to make a sturdier, more compact showpiece. A basic stand is provided, but wall mounting and faster spinning stands should be possible.

For the best print results and other info, check the in-depth guide.

Curve

            <div><p>f(t) = cos(t) + cos(3t)/3 + cos(5t)/5 + cos(7t)/7 + sin(t)i - sin(3t)/3i + sin(5t)/5i - sin(7t)/7i<br>Desmos input: <code>(cos(t) + cos(3t)/3 + cos(5t)/5 + cos(7t)/7, sin(t) - sin(3t)/3 + sin(5t)/5 - sin(7t)/7)</code></p><p>This version draws a square wave in both real and imaginary axes. The key insight is that flipping the phase and frequency of every other component draws the exact same square wave in the real axis (as can be clearly seen in cosine form), but now introduces a phase shifted symmetry in the imaginary axis. As expected, this curve will rapidly zip to each of the 4 corners and linger.</p><p>Ironically though, this actually makes an awful square curve due to the slow convergence and sharp overshoots. On a practical level, the large jerk and many negative frequency components makes it somewhat difficult to implement on scale. A mere 2-component complex triangle wave can draw a better square than a 500-component complex square wave. Nonetheless, this still remains an interesting shape with some mathematical significance.</p></div>                <p class="detail-title"><strong>Key Changes</strong></p>            <div><ul><li>Extra tolerances for 0.4mm printing, adding 0.25mm backlash to all gears on top of the previous 0.25mm arm length hack (at the cost of more backlash).</li><li>New stand uses significantly less filament at 60-70% infill.</li></ul></div>                <p class="detail-title"><strong>Quick Print Tips</strong></p>            <div><ul><li>60-70% infill recommended for shear/deformation resistance. Most of the print is already made of solid wall layers, so it's hardly any more filament/weight.</li><li>Z-seams and corner defects can cause fitting and teething problems, so print one piece at a time. Always place z-seams in the gear's teeth root and consider a 0.4mm wipe. Keep a small, round steel file and hobby knife on hand to smooth out defects.</li><li>Best results without a raft, but 90% support interface density is recommended.</li><li>No glue required, as pieces should fit very tightly. Parts should be detachable with some effort and care.</li></ul></div>                <p class="detail-title"><strong>Assembly (in Stack Order)</strong></p>            <div><p>The main 3mf file is organized with driving shafts on the left, driven arms on the right. Set 1 contains a double gear driver, a large "base" gear, and an arm of (center-to-center) length 12*7/5=16.8mm. The each row above make shaft/arm sets 2, 3, then 4.</p><p>The expected magnitude/phase coefficients for sets 1-4 are:</p><ul><li>F5 = 12*7/5mm, point right</li><li>F1 = 12*7mm, point right</li><li>F-3 = 12*7/3mm, point left</li><li>F-7 = 12mm, point left</li></ul><p>To assemble the stack, place arm 1 (F5) over the hex fit of the base gear and align. Pin it down through the hole using shaft 2, then mesh arm 2 (F1) with it right over the hex fit. Make sure that all components are still correctly oriented. Pin arm 2 down using shaft 3 and repeat. Seal off the last arm by pinning it with the shaft cap.</p><p>To assemble the stand, mesh shaft 1 with the aligned base and arm 1 gears, then place onto the stand. The base gear's round shaft should be seated in the middle to spin freely, while the long shaft 1 fits through the tube to the side. Feed the bevel gear with the shaft through the sideways hole from the inside and attach the handle on the other side. Finally mesh and connect up the other bevel gear with the bottom of shaft 1.</p><p>45 full, anti-clockwise rotations of the handle should now trace out the entire curve.</p></div>                <p class="detail-title"><strong>Measurements</strong></p>            <div><p>These are approximate specs when printed with 70% infill, cheap PLA at original size. Expect a few % variation.</p><ul><li>Weight: 66g total, stand parts account for ~30g.</li><li>Dimensions: 210mm x 140mm in "rest orientation" (see picture), extending out to 250mm. 129mm tall.</li></ul></div>

Fourier Orrery (Complex Square Wave)