Undamped Natural Frequency

Every structure has a frequency it prefers, and this equation is where that preference comes from. Pull a mass on a spring aside and let go: the spring pulls it back, the mass overshoots because it has inertia, and the pair trade energy back and forth at a rate set by the competition between the two. Stiffness is what pulls, mass is what resists, and the ratio of those two is the entire physics. $f_n = \frac{1}{2\pi}\sqrt{k/m}$ says nothing more than that.

Notice what is not in the equation. The amplitude is absent, which is why a lightly plucked guitar string and a hard-struck one give the same note. Gravity is absent, which is why a spring-mass oscillator has the same natural frequency lying on its side as hanging vertically. And damping is absent, which is nearly true — a damped system rings slightly slower, and at the damping levels found in machinery the difference is under half a percent.

The trap on this page is the factor of 6.28. Every textbook derivation is written in $\omega_n = \sqrt{k/m}$, radians per second; every nameplate, analyser and specification is written in hertz, cycles per second. They differ by $2\pi$, and reading one as the other is the most common error in vibration arithmetic. If a result looks wrong by "about six", that is what happened. The other trap is subtler: the mass in the denominator is the mass actually carried by the spring, which on an installed machine means the equipment plus its base frame plus whatever liquid it is holding, not the shipping weight on the datasheet.

This same algebra runs the electrical world, and the correspondence is exact rather than poetic. An LC circuit resonates at $f_0 = 1/(2\pi\sqrt{LC})$: inductance plays the part of mass, because it resists changes in current the way mass resists changes in velocity, and the reciprocal of capacitance plays the part of stiffness. Anyone comfortable with one is a substitution away from the other, which is why the two fields borrowed each other's vocabulary — impedance, resonance, quality factor — and never gave it back.