beam vibration bending modes
A string is restored by a taut tension, but a diving board or the end of a ruler springs back and vibrates even with no tension at all. The bent beam straightens by its own bending stiffness (EI). A beam is also a continuous system with endless modes, yet unlike a string its frequencies are not whole-number multiples of the fundamental. That inharmonicity is what makes a beam sound different from a string.
Take a cantilever, a beam fixed at one end and free at the other, pull the free end, and let go. There is no tension at all, yet the bent beam straightens by its bending stiffness and vibrates. A string is restored by tension, and a beam is restored by stiffness. The source of the return is different. The shake can look the same while the force comes from somewhere else.
What sets a beam's frequency? f₁ ∝ √(EIρA)L². Move the stiffness EI and the length L. On a string the length entered to the power 1, but on a beam it is the power 2 of L, so making the length 2 times as long drops the frequency to 14. That is why a long structure is especially floppy. Even a small growth in length lowers the pitch a lot.
Choose a higher mode. A beam's mode shape is not a sine. It is a curve that ripples more toward the end. The frequency ratios are 1 : 6.27 : 17.5, not the string's 1 : 2 : 3, so they are not whole multiples. Because of that inharmonicity a struck rod does not give a pitched note. It gives a clack. The partials do not stack into a musical pitch.
How you hold the two ends changes everything. Toggle a cantilever (fixed and free), a simple support (pin and pin), and free and free, and the same beam's fundamental frequency coefficient jumps 3.5 → 9.9 → 22.4. The beam with both ends free is exactly a tuning fork and a xylophone bar. The way you hold it moves the pitch as a whole.
The same bending vibration shows up everywhere. Step through the examples and you meet a diving board (a cantilever), an airplane wing (cantilever bending, with a flutter risk), and a tuning fork or xylophone bar (free and free, with the harmonics trimmed to be musical). Know a beam's modes and you see what shakes and when it is dangerous. Change the boundary and the dangerous place changes too.