Fundamental Mathematics in Engineering II · Lecture 13 · demonstration
A shape that keeps its shape is an eigenvector
Pluck the chain by dragging it. Most shapes fall apart as time runs — but a few survive unchanged. Which ones, and what do they turn into as the number of masses grows?
Bars: how much of each mode is in the motion (the coefficients |ck|). Dashed curves: the continuum eigenfunctions sin(kπx). Drag anywhere on the chain to pluck it at that point.
How to read it
- Press mode 1, mode 2, mode 3: each shape oscillates without ever changing shape. Those shapes are the eigenvectors of the chain's stiffness matrix — its modes — and each has its own frequency.
- Pluck it anywhere else: the shape changes as it runs, because it is a mixture of modes ticking at different rates. The bars show the mixture.
- A center pluck excites only the odd modes, with weights 1, 1/9, 1/25, … — the same pattern that returns in week 10 as the Fourier sine series.
- Slide n to 64: the beaded chain becomes a string, and the modes become sin(kπx). Matter became continuous; the eigenvector became an eigenfunction.