Harmonograph Drawing Machine

Drawing with pendulums

A harmonograph is a drawing machine powered by pendulums. One pendulum moves a pen, another moves the card underneath it, and as they swing, the pen traces a curve that records both motions at once. Because the pendulums slowly lose energy, the lines spiral gently inward, and the finished drawing has the delicate, layered look that made harmonographs a favourite of Victorian parlours.

Ratios make the shape

Each pendulum swings with its own steady rhythm, set by its length. What matters for the drawing is the ratio of the rhythms. If the pen and the card swing at the same rate, a ratio of 1 : 1, the pen draws an ellipse or a circle. A ratio of 1 : 2 gives figure-of-eight shapes, and 2 : 3 or 3 : 4 give looping stars and knots. These closed curves are called Lissajous figures. The simple whole-number ratios are the musical ones too: as frequency ratios, 1 : 2 is an octave, 2 : 3 a perfect fifth and 3 : 4 a perfect fourth, so a harmonograph drawing is a picture of a musical interval.

Detune and damping

A perfect ratio would draw the same figure over and over. Real pendulums are never perfectly tuned, and that is where the beauty comes from. With a slight detune, the figure slowly rotates or morphs from one shape into another, filling the card with interwoven lines. Meanwhile friction makes every swing a little smaller than the last. The amplitude falls by the same fraction each second, a pattern called exponential decay, so the figure shrinks steadily toward the centre. Raise the damping in the Lab for a few bold loops, or lower it for a dense, lacy drawing.

The machine in this scene

This harmonograph has two pendulums, each hung on a gimbal so that it can swing in two directions at once: one carries the pen, the other carries the card. The pen circles one way, while the card swings in a line or round in an ellipse, set by the swing-shape slider, circling the same way or the opposite way. Every motion is a real damped pendulum swing, so the drawing evolves just as a wooden harmonograph’s would.

For the curious

Each pendulum angle obeys θ″ = −ω²·sin θ − 2γ·θ′, where ω sets its rhythm and γ its damping. For small swings the solution is close to θ(t) = θ₀·e^−γt·cos(ωt + φ). The pen’s position on the card is the pen pendulum’s displacement minus the table’s, so the drawing is approximately x = A·e^−γt·sin(ω₁t + φ₁) − B·e^−γt·sin(ω₂t + φ₂), with a matching formula for y. The scene integrates both pendulums, in both directions, with the fourth-order Runge–Kutta method 240 times per second, and lays down a ribbon of ink as the pen moves.