An engineer tasked with making sure that a suspension bridge does not sway insta

Question

An engineer tasked with making sure that a suspension bridge does not sway installs a series of devices to damp the swaying motion of the bridge. The devices used can have the amount of damping they provide adjusted. With the dampers the engineer determines that the equation of motion of the bridge is: x¨+3.8x+9.5x=0 where x is the amount of lateral displacement due to the sway and is the setting on the damping devices the engineer has installed. What value of (which is a dimensionless number) should the engineer set the devices to so that the sway is damped as rapidly as possible?

Explanation / Answer

for the bridge to come back to equilibrium position, the bridge has to be critically damped

given equation of motion of bridge
x" + 3.8alpha*x' + 9.5x = 0

so. for damped osscilations
c^2 = 4mk
here c= 3.8*alpha
m = 1
k = 9.5

so, 3.8^2*alpha^2 = 4*1*9.5
alpha = 1.6222

so alpha = 1.622 for the bridge to be critically dampe dso that it can come backl to its original position as soon as possible

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