VD-02Vibration & Structural Dynamics

Mass–Spring Natural Frequency

Natural frequency of a single-degree-of-freedom system from mass and stiffness or static deflection.

Inputs

Mass m

Stiffness k

N/m

200 N/mm

Natural frequency

10.07Hz

604 CPM

ωₙ

63.25rad/s

Period

99.35ms

Static deflection

2.452mm

Model

Critical damping c_c

6,325N·s/m

Weight m·g

490.3N

Isolation starts above

14.24Hz

forcing must exceed

\sqrt{2}\cdot f_n

Natural frequency vs static deflection

f_n = 15.76\,/\,\sqrt{\delta}

§ Theory of operation

The single-degree-of-freedom oscillator

A rigid mass m supported on a spring of stiffness k is the elementary building block of vibration analysis. Displaced and released, it oscillates at its undamped natural frequency:

\omega_n = \sqrt{k\,/\,m}\ \ \mathrm{rad/s} \qquad f_n = \dfrac{1}{2\pi}\sqrt{k\,/\,m}\ \ \mathrm{Hz}

The static deflection shortcut

Under gravity the spring compresses by δ = mg/k. Substituting k = mg/δ eliminates both mass and stiffness, the natural frequency depends only on how much the mount sags:

f_n = \dfrac{1}{2\pi}\sqrt{g\,/\,\delta} \approx \dfrac{15.76}{\sqrt{\delta\,[\mathrm{mm}]}}\ \ \mathrm{Hz}

This is why isolator catalogues are organised by deflection: 25 mm of static deflection gives about 3.2 Hz regardless of machine weight. (The relation strictly holds for linear springs; rubber mounts are stiffer dynamically than statically, so their real $f_n$ is typically 1.2–1.5× higher.)

Critical damping

c_c = 2\sqrt{k\,m} = 2\,m\,\omega_n \qquad \zeta = c\,/\,c_c

A damper with coefficient c equal to $c_c$ returns the mass to rest without overshoot. Most structures have ζ between 0.5 % and 5 %. See the SDOF Frequency Response calculator for the effect of damping.