A suspension footbridge spanning a narrow valley has a measured natural frequency of vertical oscillation equal to 2.2Hz. A troop of soldiers, each step identical in rhythm, begins to cross the bridge while marching at a cadence of 120 steps per minute.
(a) Show that the marching can set up oscillations that are dangerously close to resonance for the bridge.
(b) Calculate the percentage difference between the soldiers’ step frequency and the bridge’s natural frequency.
(c) Explain why, even with this small mismatch, large amplitude vibrations may still develop if the soldiers continue marching in step for a sufficiently long time.
(d) Suggest two practical measures that the troop commander or bridge engineers could take to prevent resonance–induced damage in such situations and briefly explain the physics behind each measure.
(e) If engineers later add tuned mass dampers to increase the damping of the bridge, describe qualitatively how this will influence the maximum amplitude reached near resonance and the safety of the structure.