Application: care crossing bridges

To avoid the possibility of driving bridges into large oscillations by the periodic impact of footsteps, the US Army infantry now never marches instep across bridges. In this problem, we will evaluate whether this concern is realistic by computing how closely timed the footsteps must be to match a bridge's natural frequency. To answer this question, we shall use a simple model of the bridge as a driven, damped oscillator.

Within this model, we take the march steps to correspond to a periodic driving force $F=F_o \cos(\omega t)$, where $\omega=\frac{2\pi}{T_{\mathrm{st}}}$ with $T_{\mathrm{st}}$ being the time between the footsteps of a unit marching in formation. To make the model realistic, we assume the following information:

• We take the bridge to have a mass $m$ and spring constant $k$ such that the bridge has a natural angular frequency $\omega_0=2\pi/T_o$, where $T_o \equiv 0.5$ sec is something close to the period of a march step (time between single steps).

• We assume that the natural damping of the bridge's motion is such that, if initially put in motion but then left undriven, the bridge will noticeably vibrate up and down about 20 times. Specifically, take the damping parameter $b$ to be such that the amplitude of undriven oscillations decays to $1/e$ of its original value in a time $t=10 T_o$

(a)

Use the above information to extract the numerical parameters ($\omega_o$, $b$) needed to make a computer or calculator plot of $A/(\frac{F_o}{m}$) versus the marching frequency $\omega$, where $A$ is the amplitude of the bridge's oscillation. Turn in a properly labeled sketch or a copy of the resulting plot.

(b)

Read off from your plot the range within which $T_{\mathrm{st}}$ must fall in order for the amplitude to be within 1/2 of its maximum value. (An approximate range read off from your plot is fine.)

(c)

Comment in a brief sentence or two on whether you feel the concern about driving such a bridge into resonance is realistic.