ISO 10816 Fundamentals Vibrations & Structural Dynamics

Damped Forced Harmonic Oscillator & Resonance Peak

Visualize mass-spring-damper resonance, phase lag φ(ω), and displacement amplification M(ω).

Solves the 2nd-order non-homogeneous differential equation of a forced mechanical oscillator. Witness the sharp amplitude spike at resonance frequency ω_n and the 90° phase shift between excitation force and displacement.

Governing Physical Law & Equations

m\ddot{x} + c\dot{x} + kx = F_0 \cos(\omega t), \quad M(\omega) = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}

Equations of motion and frequency-dependent magnification factor.

Law: Newton’s 2nd Law & Hookean Elastic Restoration | Standard Reference: ISO 10816-1 / ISO 1940-1 / ANSI S2.41 (Mechanical Vibration Evaluation)

Adjustable System Parameters

Parameter	Nominal Value	Dynamic Range	Physical Role
Oscillator Mass m (m)	5 kg	1 to 20 kg	Inertial mass vibrating on spring suspension
Spring Stiffness k (k)	350 N/m	50 to 1000 N/m	Linear elastic spring constant
Viscous Damper c (c)	8 N·s/m	1 to 40 N·s/m	Viscous fluid dashpot damping resistance
Harmonic Drive Freq (f_drive)	1.33 Hz	0.5 to 6 Hz	External sinusoidal excitation frequency

Analytical Proof & Derivation

Applying Newton’s 2nd Law to a mass-spring-damper: m ẍ + c ẋ + k x = F₀ cos(ω t). In canonical form: ẍ + 2ζω_n ẋ + ω_n² x = (F₀/m) cos(ω t), where natural frequency ω_n = √(k/m) and damping ratio ζ = c / (2√(k m)). Harmonic steady-state solution yields magnification factor M(r) = 1 / √[(1 - r²)² + (2ζ r)²] where r = ω / ω_n, with phase angle φ = arctan(2ζ r / (1 - r²)).

Verification Benchmark

ISO 10816 benchmark: m = 5 kg, k = 350 N/m, c = 8 N·s/m: natural frequency f_n = (1/2π)√(350/5) = 1.330 Hz. Critical damping c_c = 2√(350×5) = 83.66 N·s/m; ζ = 8 / 83.66 = 0.0956. At excitation f = 1.33 Hz (r = 1.0), magnification M = 1 / (2 × 0.0956) = 5.23x and phase lag φ = 90.0°.

Field Engineering Insights

ISO 10816-1 Zone A/B limits vibration severity velocity below 2.8 mm/s RMS for rigid machinery foundations. Operating within ±15% of the natural frequency resonance band causes fatigue failure in rotor shafts and bearing raceways.