Om malependuler | Blackburn & Lissajous

Hvad er et malependul?

A painting pendulum—also called a Blackburn pendulum—is a mass suspended so it can swing freely in two horizontal directions. When the bob carries sand, ink, or paint, its path leaves a curve on paper or canvas. The Dutch name schilder pendel (“painting pendulum”) captures the idea: the swing itself becomes the brush.

Harmonisk svingning

Along one axis a free pendulum behaves like a simple harmonic oscillator: x(t) = A cos(ωt + φ). Amplitude A sets the swing size, angular frequency ω how fast it oscillates, and phase φ where the motion starts. Ideal SHM ignores friction, so the bob would swing forever at constant amplitude.

Sand som et svingende pendul har efterladt på papir

Y-ophæng og to frekvenser

Blackburn’s arrangement uses a Y-shaped (or V-shaped) suspension: the string splits so the effective lengths for east–west and north–south motion differ. Different lengths mean different restoring rates, so ωx ≠ ωy. The bob therefore combines two perpendicular oscillators at once.

Lissajous-figurer

The plane curve (x(t), y(t)) from two sinusoids is a Lissajous figure. When the frequency ratio ωx:ωy is a simple rational number, the path closes into a neat pattern. Irrational ratios never close and densely fill a region. Phase difference turns a 1:1 ratio from a line into an ellipse or circle.

Frekvensforhold i overblik

1:1 yields a line, ellipse, or circle depending on phase. 1:2 draws a figure-eight (lemniscate-like) family. 2:3 and 3:4 weave denser nets and mandala-like lattices. Those ratios are the classics of both classroom demos and sand pendulums.

En viktoriansk harmonograf, der tegner kurver med to penduler

Dæmpning og spiralmalen

Real pendulums lose energy. Multiplying each axis by e^(−kt) shrinks the amplitude over time, so loops spiral inward—the familiar “painting pendulum” look on paper. Larger k settles faster; tiny k lets many revolutions paint before the bob rests.

Prøv Painting Pendulum i PictorX

Open /painting-pendulum to paint damped Lissajous curves in your browser. Pick a ratio, tune amplitudes, phases, and damping, animate the trail, and save a PNG. Pair it with Spirograph or Fractal Generator for more geometric play.

Ofte stillede spørgsmål

What is a Blackburn pendulum?

A pendulum suspended so it can swing in two perpendicular directions with different effective lengths (often via a Y-shaped support). The combined motion traces Lissajous curves; with sand or paint it becomes a painting pendulum.

How do Lissajous figures relate to pendulums?

Each horizontal axis acts as a harmonic oscillator. Plotting both positions over time yields a Lissajous figure. Rational frequency ratios close; irrational ones never quite repeat.

Why does the paint spiral inward?

Damping (air drag, friction) reduces amplitude roughly like e^(−kt). Successive loops shrink toward the centre, creating the spiral paint effect.

What does a 1:2 or 3:4 ratio look like?

1:2 often looks like a figure-eight. 3:4 (and 2:3) produce woven, net-like patterns with more crossings—great for dense ornamental trails.

Is PictorX Painting Pendulum a physical device?

No. It is a browser simulation of the same mathematics: two damped oscillators forming Lissajous paint paths you can randomise, animate, and export.