Vibration Analysis Calculator — Natural Frequency, Resonance & Isolation (ISO 10816)

Every mechanical system can be idealised as a single-degree-of-freedom (SDOF) mass–spring–damper at its first mode. The undamped natural frequency ωₙ = √(k/m) sets the resonant speed; the damping ratio ζ = c/c_cr (where c_cr = 2√(km)) determines whether the free response is underdamped (oscillatory decay), critically damped (fastest non-oscillatory return), or overdamped (sluggish return). The logarithmic decrement δ relates directly to ζ and lets the damping be measured from a free-decay trace.

Under harmonic forcing F₀·sin(ωt) the steady-state amplitude is scaled by the dynamic magnification factor (DMF). The DMF peaks near the resonance frequency ratio r = ω/ωₙ = 1, reaching 1/(2ζ) for an underdamped system. The quality factor Q = 1/(2ζ) is the amplification at resonance; the half-power bandwidth method gives ζ from measured FRF data.

ISO 10816-1 classifies rotating-machine health from the broad-band RMS vibration velocity measured on non-rotating parts. Four machine classes (I = small, IV = large flexible-mount) each carry four severity zones: Zone A (new-machine quality), B (acceptable long-term), C (alarm — short-term only), D (danger — immediate shutdown). The boundaries range from 0.71 mm/s RMS (Class I Zone A/B) up to 18 mm/s RMS (Class IV Zone C/D).

Vibration isolation works by making the system natural frequency much lower than the excitation frequency. The transmissibility TR is the ratio of force (or displacement) transmitted through the mounts to the source value. Isolation begins at r > √2 ≈ 1.414 where TR < 1; practical isolators target r ≥ 2–4. The isolator stiffness k is sized from the required isolation natural frequency and the supported mass, and the static deflection is checked against the isolator's rated deflection.

Rotor unbalance is modelled as a rotating eccentric mass m_u at radius e_u on a rotor of total mass M. The synchronous whirl amplitude X = (m_u·e_u/M)·r²/√((1−r²)²+(2ζr)²) at rotor speed ω, where r = ω/ωₙ. ISO 1940-1 grades the residual balance quality by G = (U/M)·ω_max in mm/s, where U = m_u·e_u is the unbalance; standard grades run from G0.4 (precision spindles) to G40 (agricultural equipment). The Campbell diagram plots each rotor natural frequency as a horizontal line and each engine order (1×, 2×, …) as a line through the origin; intersections are critical speeds that must be avoided in operation.

The shock response spectrum (SRS) characterises the peak response of a bank of SDOF oscillators to a base-acceleration pulse (half-sine, rectangular or terminal-peak sawtooth). It is computed here by RK4 numerical integration of the relative-displacement equation, and the maximax absolute acceleration is recorded at each natural frequency. The SRS envelope is used in MIL-STD-810 qualification to confirm that structural resonances do not amplify the shock beyond the design limit.