Driven Resonance (2D) | mysimulator.uk

Mass–Spring–Damper

Stiffness k 10.0 N/m

Damping b 0.50 N·s/m

Driving Force

Force amplitude F₀ 5.0 N

Driving frequency ω 3.16 rad/s

ω/ω₀ = 1.00 — at resonance

Live Readout

Natural freq ω₀

3.16 rad/s

Quality factor Q

6.3

Amplitude A

0.00 m

Phase lag φ

0°

How it works

A unit mass on a spring (stiffness k) and damper (b) is pushed by a sinusoidal force F₀cos(ωt). The motion is integrated directly from Newton's second law with RK4 (not just plugged into the closed-form steady-state answer), so you see the transient build-up as well as the final steady oscillation:

m·x'' + b·x' + k·x = F0·cos(ωt)
ω0 = √(k/m)          natural frequency
A(ω) = (F0/m) / √((ω0²−ω²)² + (bω/m)²)
φ(ω) = atan2(bω/m, ω0²−ω²)

The resonance curve plot traces A(ω) for every driving frequency in range and marks where the current ω sits on it. For light damping the peak sits just below ω0, and the sharper that peak, the higher the quality factor Q = ω0/(b/m) — the same ratio that governs how narrowly a radio tunes a station or how violently a bridge deck can resonate with wind gusts.

Mass displacement x(t) Driving force F(t) Resonance curve A(ω)