A pedestrian bridge deck is driven by footfall at a fixed angular frequency (a…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations · Simple, damped and driven oscillations

Mathematical Methods for Physics, Ch Second-order linear equations: damping, driving and resonance

Problem

A pedestrian bridge deck is driven by footfall at a fixed angular frequency

\omega = 12.57\,\mathrm{rad\,s^{-1}}

(a walking rate of

2.0\,\mathrm{Hz}

), with

F_0/m = 3.0\,\mathrm{m\,s^{-2}}

and damping rate

\gamma = 1.0\,\mathrm{s^{-1}}

fixed by the deck's construction. Engineers consider three stiffening designs giving natural angular frequencies

\omega_0 = 5.0\,\mathrm{rad\,s^{-1}}

\omega_0 = 10\,\mathrm{rad\,s^{-1}}

and

\omega_0 = 15\,\mathrm{rad\,s^{-1}}

. ((a)) Compute

|\tilde{A}|

and the phase lag

\phi

for each of the three designs. ((b)) State which design gives the largest response to the walking drive and explain the trend in one sentence.

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