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Natural frequency

A mass–spring system disturbed from equilibrium oscillates at a particular frequency if no external force continues to act – the natural frequency. It depends only on the stiffness $k$ and the mass $m$ , not on how large the amplitude is.

\omega_n = \sqrt{k/m}

natural angular frequency

f_n = \dfrac{\omega_n}{2\pi}

natural frequency

Symbols

$\omega_n$	natural angular frequency	rad/s
$f_n$	natural frequency	Hz
$k$	spring stiffness	N/m
$m$	mass	kg

Example

A 5 kg machine on springs with $k = 10$ kN/m:

$f_n = \dfrac{1}{2\pi}\sqrt{10\,000/5} \approx 7.12$ Hz.

Quadrupling the mass halves the natural frequency – it follows

1/\sqrt m

, not

1/m

Practise free vibrations for free →

Springs in series and parallel →

Part of Machine Dynamics and Vibrations: Free vibrations.