Harmonograph

The Equations of Motion

A harmonograph creates its drawings through the combined motion of multiple damped pendulums. In this simulation, we use four pendulums. Two pendulums drive motion along the X-axis, and two along the Y-axis. The exact position at any time $t$ is given by:

$$ \begin{aligned} x(t) &= A_1 \sin(\omega_1 t + \phi_1) e^{-d_1 t} + A_2 \sin(\omega_2 t + \phi_2) e^{-d_2 t} \\ y(t) &= A_3 \sin(\omega_3 t + \phi_3) e^{-d_3 t} + A_4 \sin(\omega_4 t + \phi_4) e^{-d_4 t} \end{aligned} $$

Where $A_i$ is the amplitude, $\omega_i$ is the angular frequency, $\phi_i$ is the phase, and $d_i$ is the damping coefficient. When damping is zero ($d_i = 0$), the figures are perfect, closed Lissajous curves (assuming rational frequency ratios). The damping term $e^{-d_i t}$ causes the pendulum's energy to dissipate, creating the characteristic spiral patterns of a harmonograph.

Harmony and Ratio

The most beautiful figures emerge when the frequencies $\omega_i$ are tuned to simple musical intervals. An Octave is a 2:1 ratio. A Perfect Fifth is a 3:2 ratio. Slight detuning (e.g., 2.01 instead of 2.0) causes the figure to slowly precess or rotate as the relative phases drift.

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