formula.expert

Undamped Natural Frequency

Also known as natural frequency · resonant frequency of a spring-mass system · fn of a machine mount · single degree of freedom natural frequency · undamped natural frequency · omega n

fn=12πkmf_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}}

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Constant used — built into this formula, no need to enter
τ=6.283185307179586\tau = 6.283185307179586Tau (2π) · exact
See also
Natural Frequency from Static Deflection — The same frequency read off a deflection you can measure with a rule, when the spring rate is not on the drawing.Damped Natural Frequency — What the damping does to this number once the dashpot is connected.Force Transmissibility — The forcing frequency is compared against this one, and that ratio decides whether a mount isolates.Period of a Spring-Mass Oscillator — The same relation written as a period instead of a frequency — T = 1/f_n, and it is the physics-class form of this page.LC Resonant Frequency — The identical algebra in an electrical circuit: inductance plays the mass, and the reciprocal of capacitance plays the spring.

Learning zone

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. fn=12πk/mf_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 ωn=k/m\omega_n = \sqrt{k/m}, radians per second; every nameplate, analyser and specification is written in hertz, cycles per second. They differ by 2π2\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 f0=1/(2πLC)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.

Undamped Natural Frequency
fn=12πkmf_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}}
mkfn
Where
  • fnf_n= Natural frequency (Hz)
  • kk= Spring rate (N/m)
  • mm= Supported mass (kg)
Missing one of these? Work it out first, then come back