Tacoma Narrows: Aeroelastic Flutter

Equivalent forms

\theta(t) = \theta_0\, e^{-\zeta_{\text{net}}\omega_n t}\cos\omega_d t

U_{\text{crit}}: \zeta_{\text{aero}}(U) = \zeta_{\text{struct}}

Not a forced resonance but a self-feeding one — the structure writes its own driving force out of the wind.

Symbol	Name	SI unit	Dimension	Range
$ζ_net$	Net damping ratiooutput Structural minus aerodynamic damping; negative means growth	dimensionless	1	-0.1 – 0.2
$U$	Wind speed Mean cross-wind speed over the deck	m/s	LT^-1	0 – 25
$ζ_s$	Structural damping Inherent mechanical damping ratio of the deck	dimensionless	1	0 – 0.2
$f_n$	Torsional frequency Natural torsional frequency of the span	Hz	T^-1	0.1 – 0.5

Unit systems

Symbol	SI	CGS	Imperial
$U$	m/s	cm/s	ft/s
$f_n$	Hz	Hz	Hz

Where it holds

Linearized flutter near the instability threshold; the exponential-growth model holds at small amplitude. Large amplitudes become nonlinear (limit cycles, geometric stiffening) before catastrophic failure.

Dimensional analysis

[T^{-2}]

(Î̈, with

\theta in rad)

[T^{-1}][T^{-1}] + [T^{-2}] = [T^{-2}]

Each term of the oscillator equation has units

rad/s^{2}

;

\zeta is

dimensionless.

\checkmark

Discovery

Theodore von Kármán / Robert Scanlan (flutter theory) · 1940

The Tacoma Narrows Bridge collapsed on 7 November 1940 in a 19 m/s wind. Early reports blamed resonance, but von Kármán and later Scanlan showed it was aeroelastic flutter: a self-excited, wind-driven torsional instability where aerodynamic forces supply negative damping. The footage reshaped bridge engineering.

Try this

A 40 mph wind — nothing extreme — tore apart a brand-new suspension bridge in 1940. The wind didn't push it down; it pumped it up.

Above a critical wind speed, the deck's own twisting sheds vortices in sync with its motion, feeding energy in faster than damping removes it. The amplitude grows exponentially until failure.

Research status: stable

Real-world applications

Flutter analysis of every modern suspension and cable-stayed bridge
Aircraft wing and tail flutter certification
Turbine blade and transmission-line galloping
Tuned mass dampers in skyscrapers

Common misconceptions

It was simple resonance with the wind's frequency — it was self-excited flutter; the wind was steady, not periodic
Marching-soldier / Karman-vortex resonance caused it — vortex shedding contributed but the killer was motion-induced flutter
Stronger materials would have saved it — the fix was aerodynamic shaping and damping, not brute strength

Experimental verification

Wind-tunnel section-model tests measure flutter derivatives and predict U_crit for every modern long-span bridge; the original 1940 film and Farquharson's eyewitness data match the torsional-flutter mechanism; the rebuilt 1950 bridge added open trusses and gaps to suppress it.

Derivation

Model the deck's torsional mode as Î̈ +

2\zeta _s\omega _n\theta

̇ +

\omega _n^{2}\theta = M

_aero

(\theta

,

\theta

̇,U).

The motion-induced aerodynamic moment contains a term proportional

to \theta

̇ (the flutter derivative A2*): M_aero

\approx \tfrac{1}{2}\rho U^{2}b^{2} A_{2}*(U) \theta

̇.

Folding it into the damping gives effective damping

\zeta

_

net = \zeta _s - \zeta

_aero(U).

When U rises past U_crit,

\zeta

_aero exceeds

\zeta _s

,

\zeta

_net goes negative,

and \theta (t) = \theta _0\,\mathrm{e}^{|\zeta

_net|

\omega _n t}

grows without bound — self-excited flutter, not external resonance.

Limiting cases

U < U_crit⟶

\zeta

_net > 0, decaying motionAerodynamic damping is small; any wobble dies out and the bridge is stable.

U = U

_crit⟶

\zeta

_

net = 0

, neutralEnergy in exactly balances energy out; sustained constant-amplitude oscillation — the flutter boundary.

U > U_crit⟶

\zeta

_net < 0, exponential growthAerodynamics pumps energy in; amplitude grows as

e^{|\zeta

_net|

\omega t}

until the structure fails.

What if…

What if the deck had open gratings (like the rebuilt bridge)?

Air bleeds through instead of forming coherent vortices, killing the negative damping — no flutter.

What if you add a tuned mass damper?

It raises effective structural damping $\zeta _s$ , pushing U_crit above any expected wind.

What if the wind were gusty instead of steady?

Flutter needs only that mean U exceed U_crit; steadiness isn't required — the instability is self-excited.

1

Crossing the flutter boundary

Given ·

U:: 19
ζ s:: 0.02

Find · stability

Steps

$\zeta$ _ $net = \zeta _s - \zeta$ _aero
$= 0.02 - 0.05 = -0.03$
Negative → amplitude grows exponentially: collapse

Result ·

If \zeta

_aero

(19\,\mathrm{m/s}) \approx 0.05

> 0.02, then

\zeta

_

net \approx -0.03

<

0 \to

unstable, growth

2

Growth time constant

Given ·

ζ net:: -0.03
f n:: 0.2

Find · amplitude e-folding time

Steps

$\omega _n = 2\pi f_n = 1.26\,\mathrm{rad/s}$
growth rate $= |\zeta$ _net| $\omega _n = 0.0377 /s$
$\tau \approx 26\,\mathrm{s}$ to double; minutes to destruction

Result ·

\tau = 1/(|\zeta

_net|

\omega _n) = 1/(0.03\times 2\pi \times 0.2) \approx 13\,\mathrm{s}

Simple Harmonic Motion Period→Simple Pendulum Period→Rotational Kinetic Energy→Torque→