Not Tension but Stiffness — Beam Vibration
A string was restored by its tension, but a diving board or the end of a ruler springs back and vibrates with no tension at all — a bent beam straightening under its own bending stiffness (EI). A beam is a continuous system with infinitely many modes too, but unlike a string its frequencies are not integer multiples of the fundamental (inharmonic).
Pull the tip of a cantilever (fixed at one end, free at the other) and let go. With no tension at all, the bent beam straightens under its own bending stiffness and vibrates. String: tension; beam: stiffness — the restoring force comes from a different place.
What sets a beam's frequency? f₁ ∝ √(EIρA)L². Move the stiffness EI and length L. Length entered to the first power for a string, but to the second power for a beam — double the length and the frequency drops to a quarter. That is why long structures are so floppy.
Pick a higher mode. A beam's mode shapes are not sines but a characteristic curve that swings more toward the free end. And the frequency ratios are 1 : 6.27 : 17.5 — not integer multiples, unlike a string's 1 : 2 : 3. This inharmonicity is why a struck bar gives a "clang," not a clear pitch.
How you hold a beam's ends changes everything. Toggle cantilever (fixed-free), simply-supported (pinned-pinned), and free-free. For the same beam the fundamental coefficient jumps 3.5 → 9.9 → 22.4. A free-free beam is exactly a tuning fork or a xylophone bar.
The same bending vibration is everywhere. Flip through the examples — a diving board (cantilever), an aircraft wing (cantilever bending, flutter danger), a tuning fork and a xylophone bar (free-free, with overtones tuned for music). Knowing a beam's modes tells you what shakes and when it turns dangerous.