Despre pendule de pictat | Blackburn și Lissajous

Ce este un pendul de pictat?

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.

Mașina oscilantă Lissajous (aparat cu pendul pictor)

Mișcare armonică simplă

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.

Nisip lăsat pe hârtie de un pendul în oscilație

Suspendare în Y și două frecvențe

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.

Figuri Lissajous

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.

Rapoarte de frecvență pe scurt

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.

Un harmonograf victorian care desenează curbe cu două pendule

Amortizare și efectul spirală

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.

Încearcă Painting Pendulum în 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.

Întrebări frecvente

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.