physics//mechanics//vibration//natural frequency

2025-10-15

A natural frequency is the frequency at which a system oscillates on its own, without damping, once it is displaced and released, and it is the number an engineer compares with every excitation in the machine and with the speed a control loop is asked to have. For a mass on a spring it is

ωn=k/m,\omega_n=\sqrt{k/m},ωn=k/m,

in radians per second; dividing by 2π2\pi2π gives hertz. Stiffness raises it, mass lowers it, and those are the only two levers a designer has: stiffen a bracket or lighten it to push its frequency up, add mass to pull it down. A pendulum fits the same formula in disguise: gravity supplies a stiffness proportional to the mass and its frequency g/L\sqrt{g/L}g/L depends on the length alone (pendulum).

The same number appears in closed loops, where a controller supplies the stiffness. A PD controller on a drone's altitude acts as a spring of stiffness KpK_pKp on the drone's mass, so the loop has ωn=Kp/m\omega_n=\sqrt{K_p/m}ωn=Kp/m: 3.2 rad/s for Kp=12K_p=12Kp=12 N/m and 1.2 kg. Hang a payload under the drone and the same gains give a slower loop, by the square root of the mass ratio, which is why autopilots that carry varying loads scale their gains (gain scheduling).

With damping the oscillation that is actually seen runs a little slower, at the damped frequency ωn1−ζ2\omega_n\sqrt{1-\zeta^2}ωn1−ζ2, and the peak of a forced response sits slightly lower still; for the light damping of most structures the three are within a few percent, and for a loop tuned at ζ=0.7\zeta=0.7ζ=0.7 they are not (second-order system).

A structure has as many natural frequencies as it has degrees of freedom, one per mode, and the lowest few are the ones that matter in practice (modal analysis). What happens when an excitation lands near one of them is resonance.

In a linear system it is the modulus of a complex eigenvalue pair, ∣λ∣=ωn|\lambda|=\omega_n∣λ∣=ωn, with the real part setting the decay (modes).