Physics · Oscillations and waves · Grade 11

Bridge resonance

Why do soldiers break step when crossing a bridge?

The situation

A group of students is crossing a light pedestrian bridge over a city canal, in formation, in step. The step of one person is hardly noticeable, but after a few seconds the bridge begins to sway noticeably. It is not for nothing that soldiers are ordered to break step when crossing a bridge. How do small pushes, given at the right moment, produce a large oscillation, and how do engineers damp it?

Topic

Forced oscillations and resonance. Bridge resonance: why soldiers break step when crossing a bridge

Key ideas

Forced oscillations
Resonance
Dynamic damper

Formulas

\ddot{x} + 2 ζ ω_{0} \dot{x} + ω_{0}^{2} x = \frac{F_{0}}{M} \cos ω t

Equation of forced oscillations: the first bending mode of the bridge is a damped oscillator.

f_{0} = \frac{1}{2 π} \sqrt{\frac{k}{M}}

Natural frequency: $k$ is the modal stiffness, $M$ the modal mass.

A = \frac{F_{0} / M}{\sqrt{{(ω_{0}^{2} - ω^{2})}^{2} + {(2 ζ ω_{0} ω)}^{2}}}

Steady-state amplitude of oscillation (the amplitude–frequency response).

tg φ = \frac{2 ζ ω_{0} ω}{ω_{0}^{2} - ω^{2}}

Phase lag of the displacement behind the force: at resonance $φ = 90 °$ . The average power of the external force $⟨ P ⟩ = \frac{1}{2} F_{0} ω A \sin φ$ in the steady state equals the loss due to damping.

Q = \frac{1}{2 ζ} \approx \frac{f_{0}}{Δ f} = \frac{π}{δ}

Quality factor: $Δ f$ is the width of the curve at the level $A_{m a x} / \sqrt{2}$ , $δ = \ln (A_{k} / A_{k + 1})$ is the logarithmic decrement.

Theory

Natural and forced oscillation

A bridge is like a giant spring: the mass of the span and its bending stiffness determine its natural frequency. Each step presses the bridge down a little — a periodic force. First a transient is observed: the natural and forced oscillations add up and form “beats”. Once damping has removed the natural part, the bridge oscillates steadily at the stepping frequency. A constant external force — for example the weight of the people on the bridge — does not excite oscillation: it only shifts the equilibrium position by $x_{0} = F / k$ , and the oscillation takes place about this new equilibrium (the “constant deflection” on the panel; $x (t)$ is calculated from it).

Work done by the external force

In each period the stepping force does work $\oint F d x$ , with an average power $⟨ P ⟩ = ⟨ F v ⟩ = \frac{1}{2} F_{0} ω A \sin φ$ . In the steady state this equals the power that is turned into heat by damping, $\frac{1}{2} c ω^{2} A^{2}$ — it is exactly this energy balance that sets the amplitude. At resonance the force is in phase with the velocity ( $φ = 90 °$ ) — the work is greatest. At $f ≪ f_{0}$ or $f ≫ f_{0}$ the force is almost 90° out of phase with the velocity: for half a period it supplies energy and for half it takes it back, so the average work is close to zero. During the transient $⟨ F v ⟩$ is greater than the loss — the excess increases the oscillation energy. Compare the two powers on the panel.

Resonance

If the stepping frequency equals the natural frequency, the force always points in the same direction as the velocity of the bridge (the red and blue arrows) and leads the displacement by 90°. Energy is added in every period and only damping limits it: $A_{r e s} = Q F_{0} / k$ . In footbridges the damping is small (about $ζ \approx 0.3$ – $1.5$ %), so the oscillation is tens of times greater than the static deflection. When people step in unison the forces add up ( $\propto N$ ), and when they walk randomly they partly cancel (about $\sqrt{N}$ ).

Tuned mass damper (TMD)

A small mass $m_{d}$ hung on a spring is added to the bridge and its frequency is tuned to the natural frequency of the bridge. At resonance the damper mass oscillates in antiphase to the stepping force (and about 90° behind the bridge) with an amplitude several times greater than the bridge: it takes the stepping force on itself and converts the energy into heat in its own damper. Den Hartog's rule: $f_{d} = f_{0} / (1 + μ)$ , $μ = m_{d} / M$ ; in the model the internal damping of the damper is also chosen by this rule. A badly tuned TMD hardly helps at all.

Limits of the model

Only one vertical bending mode is taken into account. The walker's force is the first harmonic at the stepping frequency: $F \approx 0.4 G \approx 280$ N (a 70 kg person). The mass of the people and their adjusting their steps to the oscillation (as in the lateral sway of the Millennium Bridge) are not taken into account. The Tacoma Narrows Bridge, which collapsed in 1940, was destroyed not by simple resonance but by aeroelastic flutter with the wind — a self-amplifying oscillation.

Did you know?

The Millennium footbridge in London began to sway sideways under pedestrians on the very day it opened in June 2000, and was closed two days later. It reopened in 2002 after dampers had been installed.
A steel sphere of about 660 tonnes hangs in the upper storeys of the Taipei 101 skyscraper in Taipei — this giant tuned mass damper reduces oscillations caused by wind and earthquakes.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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The dutar string and sound — Grade 11
Waves and the Doppler effect — Grade 11
Coupled oscillations — Grade 11
Nonlinear oscillations — Year 2
Fourier: the spectrum of sound — Year 3