Natural Frequency Calculator (fₙ from stiffness & mass or static deflection)

Understanding the natural frequency calculator — concept and principle

This calculator finds the natural frequency of a mass on a spring, either from stiffness and mass or from the static deflection under gravity. It gives the frequency in Hz, the angular frequency, the period and the equivalent speed in rpm. Engineers use it for machine mounts, vibration isolators, shelves, floors and simple structures.

Stiffer springs ring faster and heavier masses slower; avoid driving the system near 191 rpm.

How it is calculated

ω_n = √(k ÷ m); f_n = ω_n ÷ 2π; T = 1 ÷ f_n

Static deflection δ = m g ÷ k, so f_n = (1/2π) √(g ÷ δ)

Equivalent speed = f_n × 60 rpm

k in N/m, m in kg, δ in mm, g = 9.80665 m/s².

The principle

Pull a mass on a spring aside and let go: the spring pushes it back, it overshoots, and it oscillates. A stiffer spring pushes harder and makes it oscillate faster; a heavier mass responds more slowly. The balance gives ω_n = √(k/m). Because the sag under gravity also depends on k/m, measuring how much a mount compresses under its load tells you the natural frequency without knowing either value separately.

Worked example

Default inputs: stiffness 20,000 N/m, mass 50 kg.

ω_n = √(20,000 ÷ 50) = 20 rad/s.
f_n = 20 ÷ 2π = 3.183 Hz; T = 0.3142 s.
Equivalent speed = 3.183 × 60 = 191 rpm.
Static deflection = 50 × 9.80665 ÷ 20,000 = 24.52 mm.

Each point is the calculator re-run with only “Mass m (kg)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Mass m (kg)	Natural frequency fₙ (Hz)
12.5	6.366
25	4.502
37.5	3.676
50	3.183
62.5	2.847
75	2.599
100	2.251

How to read the result

A machine running near f_n × 60 rpm will resonate and vibrate strongly. Vibration isolation only starts when the forcing frequency is above about √2 × f_n, and works better the higher above it is. Soft mounts with large static deflection give low natural frequencies and good isolation for fast machines.

Common mistakes

Using weight in newtons as the mass.
Confusing Hz with rad/s.
Ignoring that several springs in parallel add their stiffness.
Expecting this single-mass model to describe complex structures with many modes.

Keep in mind: Real systems have damping, several vibration modes and non-linear mounts. Structural vibration design needs qualified engineering analysis.

Frequently asked questions

How do I find the natural frequency from deflection?

f_n ≈ 15.76 ÷ √δ(mm) Hz, from (1/2π)√(g/δ).

Does damping change the natural frequency?

Light damping lowers it only slightly.

What is resonance?

Large vibration when a forcing frequency matches the natural frequency.

Why do isolators need soft springs?

Isolation works only above √2 × f_n, so the natural frequency must be well below the machine's speed.

Sources: Standard undamped harmonic oscillator ω_n = √(k/m) with Hooke's law δ = mg/k; standard gravity 9.80665 m/s²

Checked 9 Nov 2026. Standard undamped harmonic-oscillator result ωₙ = √(k/m), fₙ = ωₙ/2π, combined with Hooke’s law for the static deflection δ = mg/k; standard gravity 9.80665 m/s².

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