Is the natural frequency the highest frequency?

[Krampus]

I have a theoretical question. Does a structure's natural frequency have anything to do with the highest frequency of which it can vibrate? Or can the structure vibrate in any multiple of the natural frequency (until it brakes...)?

[FredGarvin]

A structure has, in theory, infinite numbers of modes. If you look at what is probably the most basic shape, a string held at two ends, the velocity in the string is derived to be
u'=2u_o\left(sin(\frac{n \pi x}{L})\right)\left(cos(2 \pi f t)\right)

Theoretically, the integer "n" can go to any number. In reality you can't do that.

[mathman]

I am not sure of what kind of structures you have in mind. However, for a simple string (like a violin or guitar string) the natural frequncy would correspond to the longest wavelength, i.e. double the string length. This would be the lowest possible frequency, not the highest.

[granpa]

perhaps the op is thinking of something like the plasma freq

[MikeyW]

Yes, it can oscillate in any integer multiple of the natural frequency, since these higher harmonics also satisfy the wave equation and its boundary conditions. And by superposition, (I think this is due to linearity in the wave equation), any sum of these harmonics will also satisfy the wave equation.

The fundamental is the LOWEST frequency solution, not the highest.