Suspension Bridge Sway: Vortex-Shedding Resonance (2D)

Wind & Structure

Wind speed (V) 20 m/s

Deck width (D) 20 m

Torsional natural freq. (f_n) 0.22 Hz

Structural damping (ζ) 0.080

Live Readout

Elapsed time

0.0 s

Vortex freq. f_v

0.00 Hz

Frequency ratio r

0.00

Peak twist

0.0°

Structural status

SAFE

How it works

The deck is modeled as a single-degree-of-freedom torsional oscillator (the classic simplified account of the 1940 Tacoma Narrows collapse): a rotational mass-spring-damper driven by periodic vortex shedding off the bluff deck cross-section, integrated here with 4th-order Runge–Kutta at a fixed 5 ms sub-step:

θ'' = [ F0·sin(2π·fv·t) − c·θ' − k·θ ] / I
fv = St · V / D            (Strouhal shedding frequency, St ≈ 0.15)
k  = I·(2π·fn)²,  c = 2ζ·√(k·I)
F0 ∝ V² · D                 (aerodynamic force ~ dynamic pressure)

When the wind-driven shedding frequency f_v drifts close to the deck's own torsional natural frequency f_n (ratio r → 1), the forced response is amplified by roughly 1/(2ζ) at resonance — exactly the mechanism that tore the twist amplitude of the real bridge up to itself before it failed. Low damping and a wind speed that keeps r near 1 for a sustained stretch is what pushes the status readout from SAFE to WARNING to COLLAPSE RISK.

Bridge pane — towers, a Verlet-relaxed sagging main cable, and the deck rotating live by the twist angle θ.
Response curve — the analytic steady-state amplitude vs. frequency-ratio curve for this ζ, with a dot marking the live (r, amplitude) point.
Twist history — θ(t), so you can watch an amplitude envelope grow (or decay) as wind speed changes.

Deck twist θ(t) Resonance response curve Live (r, amplitude)