The Swinging Bridge Stabilizer · AI Generator #GEN-4DFEE7C8

Physics · mechanics

Problem

A pedestrian suspension bridge of length L=120  mL = 120\;\text{m} and mass M=8000  kgM = 8000\;\text{kg} is designed with a unique stabilization system. When the bridge oscillates laterally due to wind, a heavy pendulum bob of mass m=2000  kgm = 2000\;\text{kg} suspended by a cable of length ℓ=15  m\ell = 15\;\text{m} from the bridge center automatically swings to counteract the motion. During testing, engineers observe that when the bridge undergoes small lateral oscillations with angular frequency ωb=0.40  rad/s\omega_b = 0.40\;\text{rad/s}, the pendulum bob oscillates with maximum amplitude A=3.2  mA = 3.2\;\text{m} from its equilibrium position, perfectly synchronized to minimize the bridge's motion. Model the bridge as a rigid body that can only rotate about its center with moment of inertia I=112ML2I = \frac{1}{12}ML^2, and assume the pendulum cable remains straight during oscillations. (a) Find the angular frequency ωp\omega_p of the pendulum's oscillations. (b) Determine the maximum angular amplitude θmax\theta_{\text{max}} of the bridge's oscillations during this synchronized motion. (c) Calculate the total mechanical energy stored in this coupled oscillating system.
Intuitive explanation

This problem combines a damping mechanism with coupled oscillations—the key insight is that the pendulum acts as a tuned absorber, and you need to recognize when the pendulum and bridge oscillate at the same frequency for effective energy transfer. For part (a), use the small-angle approximation for a simple pendulum to find its natural frequency from the cable length alone. For part (b), relate the pendulum's linear amplitude to the bridge's angular motion by considering the geometry of how the bob swings relative to the bridge's rotation. For part (c), treat the two oscillators as a coupled system and sum their kinetic and potential energies at the moment when energy distribution is most revealing—typically when one object is at maximum displacement and the other passes through equilibrium.

Intuitive explanation

This problem combines a damping mechanism with coupled oscillations—the key insight is that the pendulum acts as a tuned absorber, and you need to recognize when the pendulum and bridge oscillate at the same frequency for effective energy transfer. For part (a), use the small-angle approximation for a simple pendulum to find its natural frequency from the cable length alone. For part (b), relate the pendulum's linear amplitude to the bridge's angular motion by considering the geometry of how the bob swings relative to the bridge's rotation. For part (c), treat the two oscillators as a coupled system and sum their kinetic and potential energies at the moment when energy distribution is most revealing—typically when one object is at maximum displacement and the other passes through equilibrium.

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