关于绘画摆 | 布莱克本与利萨如

什么是绘画摆？

绘画摆——又称布莱克本摆——是可在两个水平方向自由摆动的质量。摆锤带沙、墨或颜料时，轨迹会在纸或画布上留下曲线。荷兰语 schilderpendel（“绘画摆”）点出要点：摆动本身就是画笔。

简谐运动

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.

Colorful damped Lissajous spiral painted by a digital painting pendulum

Y 形悬挂与双频率

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.

利萨如图形

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.

频率比一览

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.

阻尼与螺旋涂料效果

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.

在 PictorX 中试用 Painting Pendulum

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. Nothing is uploaded. Pair it with Spirograph or Fractal Generator for more geometric play.

常见问题

什么是布莱克本摆？

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.

利萨如图形与摆有何关系？

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.

为什么涂料向内螺旋？

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

1:2 或 3:4 比例看起来如何？

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.