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The science
A cable-supported deck as a string with mass
For a suspension span the dominant restoring force on vertical motion is the horizontal cable tension H acting on the deck weight per length m, like a taut string. Its natural frequencies are fn = (n / 2L) √(H / m), n = 1, 2, 3…, and its shape in mode n is sin(nπx/L). The cable sags by s = m g L² / (8 H). A uniform wind force excites only the odd modes, because the even ones integrate to zero over the span.
Vortex shedding and resonance
Wind past a bluff deck sheds vortices at the Strouhal frequency fs = St U / D (St ≈ 0.2). The alternating pressure acts like a periodic force of amplitude ½ ρ U² D CL. Each mode is a damped oscillator, q'' + 2 ζ ωn q' + ωn² q = Fn / Mn sin ωst, whose steady amplitude is Fn / Mn / √((ωn² - ωs²)² + (2ζωnωs)²). It peaks sharply when fs = fn, which defines the critical wind speed Ucrit = fn D / St, and the peak height scales as 1/ζ.
What the model leaves out
Honest simplifications: real bridges also have bending stiffness, torsion and aerodynamic feedback (flutter, which destroyed the Tacoma Narrows Bridge, is a self-excited instability that this forced-resonance model does not contain). The lift coefficient is a fixed 0.1, the wind is steady and uniform, lock-in broadening is ignored, and the displacement is exaggerated on screen by the factor shown. Amplitudes are indicative orders of magnitude, not a design calculation.
What to try
- Sweep the wind speed and watch the amplitude spike at the red critical speeds, where the shedding frequency equals a mode frequency.
- Halve the damping ratio: the resonance peak doubles.
- Raise the cable tension: every mode frequency and critical wind speed rises with the square root of H.
- Use the exaggeration slider to see the mode shapes: mode 3 has three half-waves along the span.