Natural Frequency (Spring–Mass)
f = (1/2π)√(k/m) — the fundamental note of any stiffness and mass.
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The engineering
Every structure is a spring-mass system wearing a disguise, and this is its phone number: excite it there and amplitudes multiply by the Q factor. The square root is the number that bites — to halve a natural frequency you need four times the mass or a quarter of the stiffness, which is why detuning a resonance problem is always more expensive than the first meeting assumed.
Isolation flips the formula into a design tool: soft mounts work by placing fn well *below* the excitation (√2 below is break-even; a factor of 3+ earns real attenuation). The classic field mistake is stiffening a machine mount to 'stop the shaking' and dragging fn up into the forcing frequency — turning an annoyance into a resonance.
Where this math comes from
Hooke's 1678 spring law contained the oscillator, and Newton's mechanics made √(k/m) inevitable, but vibration as an engineering discipline is really Lord Rayleigh's doing: his 1877 'Theory of Sound' organized modes, energy methods, and the approximation techniques engineers still reach for when the geometry stops being a textbook sketch.
The machine age made it urgent — unbalanced engines, whirling shafts, galloping bridges — and the field's working canon crystallized in the 1920s–30s with Timoshenko's 'Vibration Problems in Engineering' (1928) and Den Hartog's 'Mechanical Vibrations' (1934), the book whose tuned-absorber chapter still fixes real machines today.
- 1678Robert HookeThe linear spring — the oscillator's restoring force.
- 1877Lord Rayleigh'Theory of Sound' founds engineering vibration analysis.
- 1928Stepan Timoshenko'Vibration Problems in Engineering' — the working handbook.
- 1934Jacob den Hartog'Mechanical Vibrations' and the tuned damper canon.
See the full timeline of the math behind every calculator →
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