Problem — Footsteps on the bridge

Problem

An $80\;\text{m}$-long pedestrian bridge has a natural transverse (lateral) oscillation frequency $f_p \approx 0{,}9\;\text{Hz}$. A group of people walk across it in step. Estimate how many people are needed to bring the bridge to the dangerous amplitude of $\sim 5\;\text{cm}$, assuming each $70\;\text{kg}$ person transfers $\sim 30\;\text{J}$ of energy per step, and that mechanical damping dissipates $\sim 80\%$ of the energy per cycle. Bridge mass: $\sim 100$ tonnes.

Solution

Target oscillation energy, with $M = 10^5\;\text{kg}$, $\omega = 2\pi f_p \approx 5{,}65\;\text{rad/s}$, $A = 0{,}05\;\text{m}$:

$E = \tfrac{1}{2} M \omega^2 A^2 = \tfrac{1}{2}\cdot 10^5 \cdot 31{,}9 \cdot (0{,}05)^2 \approx 4\cdot 10^3\;\text{J}$

At steady state the energy input per cycle must balance the energy dissipated. If damping removes $80\%$ per cycle:

$E_\text{diss} \approx 0{,}8\, E \approx 3200\;\text{J/cycle}$

Each person gives $\sim 30\;\text{J}$ per step, and at $\sim f_p$ takes about one useful step per half-period: $\sim 30\text{–}60\;\text{J}$ per person per cycle. Number needed:

$N \approx \frac{3200}{60} \sim 50$

$\ev{N \sim 50 \text{ synchronised people}}$

Compare with the Millennium Bridge (London, 2000), closed a few days after opening due to lateral oscillations induced by the crowd.

Links

Topics: Oscillazioni e moto armonico Concepts: Risonanza Skills: Stime di Fermi