Vibration Frequency Calculator

A vibration and noise calculator covers the four fundamental calculations in structural dynamics and industrial acoustics: undamped and damped natural frequency of a single-degree-of-freedom system, resonance transmissibility as a function of excitation frequency ratio, damping ratio from a known damping coefficient, and sound pressure level (dB SPL) with inverse-square-law attenuation over distance. These calculations are essential for machinery design, building acoustic assessments, vibration isolation mount selection, and compliance with workplace noise regulations.

Related tools include the reliability engineering calculator for fatigue life analysis (fatigue failures are often vibration-driven) and the mechanics calculator for the static loads that act alongside dynamic vibration loads.

1. Select the tab: Natural Frequency, Resonance, Damping Ratio, or Noise Level (dB).
2. For natural frequency, enter spring stiffness k (N/m) and mass m (kg) — the calculator gives ω_n (rad/s) and f_n (Hz).
3. For resonance, enter f_n, the excitation frequency f_e, and the damping ratio ζ — transmissibility T and Q factor are computed.
4. For noise, enter the sound pressure p (Pa) at a reference distance, then set d₁ and d₂ to estimate level attenuation over distance.

Vibration and acoustics formulas

Natural frequency — ω_n = √(k/m) rad/s; f_n = ω_n / 2π Hz

Damped natural frequency — f_d = f_n √(1 − ζ²); applies when ζ < 1 (underdamped)

Transmissibility — T = 1 / √((1 − r²)² + (2ζr)²); r = f_e/f_n; at r = 1, T = Q = 1/(2ζ)

Damping ratio — ζ = c / (2√(km)) = c / c_c; c_c (critical damping) = 2√(km)

Sound level — L = 20 log₁₀(p / 20 μPa) dB SPL; L₂ = L₁ − 20 log₁₀(d₂/d₁) (free-field point source)

Interpreting vibration and noise results

Resonance and isolation zones

The transmissibility plot divides into three zones: at r ≪ 1 (excitation well below natural frequency) T ≈ 1 — the mount does nothing but the system is safe. At r ≈ 1 (resonance) T = Q ≫ 1 — forces and displacements are amplified and structural damage is possible. At r > √2 the mount attenuates (T < 1) — this is the isolation zone. Isolation mounts are designed to keep r above 1.5 at minimum run speed, which requires a natural frequency at least 1.5× below the excitation frequency. Increasing damping ζ reduces the peak at resonance but also reduces isolation efficiency at high r.

Engineering tips and best practices

• The free-field inverse-square law (6 dB attenuation per doubling of distance) applies outdoors over hard ground. Indoors, room reflections (reverberant field) dominate beyond the critical distance — attenuation is much less than 6 dB per doubling.
• UK Control of Noise at Work Regulations 2005 set a lower action level at 80 dB(A) daily dose and an upper action level at 85 dB(A). Exposure limit is 87 dB(A). Use hearing protection above 80 dB(A).
• For rotating machinery, the primary excitation frequency is shaft speed in Hz (= rpm / 60). Resonance risk should be checked at all operating speeds including run-up and coast-down.
• See the reliability calculator to assess S-N curve fatigue life once you have a vibration stress amplitude — resonance dramatically shortens fatigue life.

Noise assessments for workplace compliance require calibrated sound level meters and must be carried out by a competent person. Vibration analysis for safety-critical structures (bridges, pressure vessels, aircraft) must be performed using validated simulation tools and reviewed by a chartered engineer. This calculator provides educational estimates only.

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