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annihilator

RESONANCE

Soon after the mile-long Tacoma Narrows Bridge opened in July 1940,
motorists began to notice its tendency to vibrate frighteningly in even a
moderate wind. Nicknamed ?Galloping Gertie,? the bridge collapsed in a
steady 42-mile-per-hour wind on November 7 of the same year. The
following is an eyewitness report from a newspaper editor who found
himself on the bridge as the vibrations approached the breaking point.
?Just as I drove past the towers, the bridge began to sway violently from
side to side. Before I realized it, the tilt became so violent that I lost control
of the car... I jammed on the brakes and got out, only to be thrown onto
my face against the curb.
?Around me I could hear concrete cracking. I started to get my dog
Tubby, but was thrown again before I could reach the car. The car itself
began to slide from side to side of the roadway.
?On hands and knees most of the time, I crawled 500 yards or more to
the towers... My breath was coming in gasps; my knees were raw and
bleeding, my hands bruised and swollen from gripping the concrete curb...
Toward the last, I risked rising to my feet and running a few yards at a
time... Safely back at the toll plaza, I saw the bridge in its final collapse and saw my car plunge into the Narrows.?
The ruins of the bridge formed an artificial reef, one of the world?s
largest. It was not replaced for ten years. The reason for its collapse was not
substandard materials or construction, nor was the bridge underdesigned:
the piers were hundred-foot blocks of concrete, the girders massive and
made of carbon steel. The bridge was destroyed because of the physical
phenomenon of resonance, the same effect that allows an opera singer to
break a wine glass with her voice and that lets you tune in the radio station
you want. The replacement bridge, which has lasted half a century so far,
was built smarter, not stronger. The engineers learned their lesson and
simply included some slight modifications to avoid the resonance phenomenon
that spelled the doom of the first one.

ENERGY IN VIBRATIONS

One way of describing the collapse of the bridge is that the bridge kept
taking energy from the steadily blowing wind and building up more and
more energetic vibrations. In this section, we discuss the energy contained
in a vibration, and in the subsequent sections we will move on to the loss of
energy and the adding of energy to a vibrating system, all with the goal of
understanding the important phenomenon of resonance.
Going back to our standard example of a mass on a spring, we find that
there are two forms of energy involved: the potential energy stored in the
spring and the kinetic energy of the moving mass. We may start the system
in motion either by hitting the mass to put in kinetic energy by pulling it to
one side to put in potential energy. Either way, the subsequent behavior of
the system is identical. It trades energy back and forth between kinetic and
potential energy. (We are still assuming there is no friction, so that no
energy is converted to heat, and the system never runs down.)
The most important thing to understand about the energy content of
vibrations is that the total energy is proportional to the square of the
amplitude. Although the total energy is constant, it is instructive to consider
two specific moments in the motion of the mass on a spring as
examples. When the mass is all the way to one side, at rest and ready to
reverse directions, all its energy is potential. We have already seen that the
potential energy stored in a spring equals 1/2kx^2 , so the energy is proportional
to the square of the amplitude. Now consider the moment when the
mass is passing through the equilibrium point at x=0. At this point it has no
potential energy, but it does have kinetic energy. The velocity is proportional
to the amplitude of the motion, and the kinetic energy, 12
mv2 , is
proportional to the square of the velocity, so again we find that the energy is
proportional to the square of the amplitude. The reason for singling out
these two points is merely instructive; proving that energy is proportional to
A2 at any point would suffice to prove that energy is proportional to A2 in
general, since the energy is constant.
Are these conclusions restricted to the mass-on-a-spring example? No.
We have already seen that F=?kx is a valid approximation for any vibrating
object, as long as the amplitude is small. We are thus left with a very general
conclusion: the energy of any vibration is approximately proportional to the
square of the amplitude, provided that the amplitude is small.

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