Suspension Bridge Mode Lab

Question. Why can a steady breeze make a long suspension bridge sway, and why does the same wind do nothing at another speed? Because the span is a set of vibration modes, each with its own natural frequency, and the wind sheds vortices at a frequency set by its speed. When the two match, the mode builds up: resonance.

Modes of a stiffened suspension span. The deck of length L and mass μ per metre hangs from a cable with horizontal tension H set by the cable sag f (parabolic cable under its own weight: H = μ g L² / (8 f)) and has a bending stiffness EI. For small vertical motion y(x,t) with simply supported ends:

Every shape sin(nπx/L) is a mode, with the natural frequency

For a floppy girder (EI small) the frequencies are equally spaced like a guitar string, f_n = n f₁. A stiff girder bends the spectrum upward (f_n grows like n²). The deck is the sum y = Σ q_n(t) sin(nπx/L); this page keeps the first six modes.

Each mode is a damped oscillator. With modal mass M = μ L / 2, stiffness K = M (2πf_n)² and damping ratio ζ:

P_n is the load projected on the shape: P_n = ∫ p(x) sin(nπx/L) dx. A load that is uniform along the span (load 1) is symmetric, so it feeds only the odd modes 1, 3, 5 and exactly cancels the even ones; a gust on the left third (load 2) or a point load at L/4 (load 3) is not symmetric and excites all of them. The page integrates each mode exactly over every time step (the force is held constant within a step), so any damping and any time compression stays stable.

Vortex shedding. A bluff deck of depth D in a crosswind U sheds alternating vortices at the Strouhal frequency f_s = St U / D (St about 0.2), which pushes the deck up and down with an oscillating lift of amplitude p = ½ ρ U² D C_L per metre. Resonance occurs at the wind speed U_n = f_n D / St. At resonance the response is limited only by damping: the amplitude is P_n / (2 ζ K), a factor 1/(2ζ) (62 for ζ = 0.8 %) above the static deflection.

Three independent checks run in the page. (1) The fundamental frequency from a finite-difference discretisation of the equation (91 nodes, inverse iteration) must match the closed form above. (2) The static mid-span sag under a uniform load, computed by solving EI y'''' − H y'' = w with finite differences, must match the sum over modes; the page also shows how much the six-mode truncation misses. (3) Mode 1 driven by vortex shedding at its own frequency must reach the closed-form resonance amplitude P / (2 ζ K) when it is simulated for eight decay times.

Honest simplifications. Linear, small-amplitude vibration of one span with pinned ends; the cable is a parabola with hangers that do not stretch, so the symmetric modes of a real suspension bridge, which also stretch the cable (an extra stiffening term), are somewhat stiffer than this model gives; the side spans, flexible towers, torsion and twisting of the deck, aeroelastic flutter (the mechanism behind the 1940 Tacoma Narrows failure) and wind turbulence are not modelled; the vortex lift is a pure sine at f_s with no lock-in. The default numbers describe a generic long-span bridge, not a specific one. The deflection is drawn magnified (the real amplitude is shown in the readouts); the traffic and the ship are decoration only.

Press a preset and the deck is already built up to the resonance. Then try: (1) Same span, wind far off resonance against wind tuned to mode 1: the same wind, completely different swing. (2) Tune to mode 2: a uniform wind cannot excite an even mode, so the mid-span stays still. (3) Gust on the left third: now mode 2 moves. (4) Damping 4 %: the resonance is cut by a factor of five. (5) Press Tune load to selected mode after changing the span, sag, mass or girder stiffness. (6) Pluck shows free decay at the mode's own frequency.