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Free Vibration Simulator: Mass–Spring–Damper

As a first approximation, machines, foundations and supports behave like a mass, a spring and a damper. Give the system an initial displacement and velocity and watch how it oscillates and decays.

Natural frequency (fn) —Hz
Damping regime
—
Damped frequency (fd)
— Hz
ωn
— rad/s
Critical damping coefficient (cc)
— N·s/m
Damping coefficient (c)
— N·s/m
Logarithmic decrement (δ)
—
Quality factor (Q)
—
Settling time (2%)
—

Displacement x(t) and decay envelope

These tools are intended for preliminary assessment and training. Decisions on critical equipment should be based on on-site measurement and expert evaluation.

How to use

  1. Enter mass and stiffness; the natural frequency updates instantly.

  2. Move the damping slider: 0% is undamped, 100% is critical damping, above is overdamped.

  3. Change the initial conditions to mimic an impact or a pull-and-release test.

Theory & formulas

m·ẍ + c·ẋ + k·x = 0, ωn = √(k/m), fn = ωn/2π.

cc = 2√(k·m), ζ = c/cc, ωd = ωn√(1−ζ²).

Logarithmic decrement δ = 2πζ/√(1−ζ²) is the natural log of the ratio of two successive peaks and is used to find damping from an impact test.

Quality factor Q ≈ 1/(2ζ) approximates the dynamic amplification at resonance. Steel structures typically have ζ of 1–5%.

Frequently asked questions

Why does natural frequency matter?

When running speed or its harmonics approach a natural frequency, vibration is amplified by resonance. A machine at resonance vibrates strongly even when balance and alignment are good; the cure is a change of stiffness, mass or speed.

How is damping measured in the field?

In a bump test the ratio of successive peaks of the decaying signal gives the logarithmic decrement, and ζ ≈ δ/2π. The half-power bandwidth of the spectrum peak is another method.