Question

A bridge oscillates too much during high winds. The bridge is modeled as a spring with simple harmonic motion. The mass of the bridge is 5.0x106 kg and its spring constant is 4.9x10' N/m. The peak vertical displacement of the bridge is 1.0 m from its equilibrium position. In order to reduce this motion, dampers are added to the bridge with a damping coefficient of 3.13x10² kg/s. a) Derive, but do not solve, the equation of motion which describes the vertical displacement of the bridge as a function of time. Show all steps. b) Determine the period of oscillation of the system prior to the dampers being added. c) What type of damping does the bridge display once the dampers are added? Show all work. d) How long will it take for the amplitude of the oscillations to be reduced by 90% of its original value? e) If the frequency of the wind matches the natural frequency of the bridge what could occur? Explain

Answer 1

A bridge oscillates too much dering high words, 5*108 ug K= 4.9800 N/m b= 3.13 x 10 kg is moss M= 5 i egnation of motion F=-KR-by mdx dtz + bidx Tat +Kx=0 + v enem ) +x=0 b dtz my period of olsunh Ta 20 54100 4.98107 9x 215=2sec. lo 4 MK = 3.13 x177 - 4x5x18x 4.9x10 so The motion is inderdamped motion Now The amplitude varies as bt A, = Ao é amt 3.13x10 2x5x100 = e 7. t w(%) w 3.13810 2x5X10 matches the t=003367 sec. = 33.67 mitisec. The time taken = 33.67 milisec. when the fregnenly of the wind natural fregnely of the bridge then resonence and it will vibrates with Larger amplitude. mit becour