Analyzing Oscillations on a Bridge

Abstract

When people cross bridges, they create vibrations on these bridges because of the impulses they exert. In some cases, vertical oscillations have caused bridges to collapse due to a phenomenon called resonance. This paper utilizes simple harmonic motion to analyze the oscillations of bridges to create a mathematical model determining when certain bridges collapse. The paper first utilizes Hooke’s Law and Newton’s Second Law to create a second order differential equation of the motion of the bridge whose solution is a sine wave. Substituting in values for momentum for a singular impulse on the bridge, the paper then solves for the amplitude after the impulse. Adapting this model for impulses at different times and plotting the output graphs using Python, the bridge is shown to exhibit resonance and suggestions for damping are made.