To understand the resonance of buildings during earthquakes
we will look at other resonant systems. The natural frequency of a swing
makes the easy to maintain the motion using a little energy.
A prolonged wind swung the bridge on the right at its natural frequency
into it exceeded its structural integrity and tore apart.
The frequency of the wave refers to the number of waves that pass through a point
in one second.
A period is the amount of time that it takes one wave cycle
to pass a given point. A frequency of one cycle per second
has a one-second period. If the frequency doubles to two cycles per second
the period is only half of a second.
Resonance is the tendency of a system to oscillate with greater amplitude
at some frequencies than at others.
The resonant frequency of any given system
is the frequency at which the maximum-amplitude oscillation occurs.
For example the resonant frequency of the one-meter long pendulum
is one half of a hertz. If the string were one quarter the length,
it would vibrate at twice the frequency. Small amplitudes can be approximated by
this equation using length and the gravity constant.
Resonant systems can be used to generate
vibrations of a specific frequency
frequency to create music.
Strings under tension, such as with this cello, have resonant frequencies directly
related to the material of the string
as well as the mass, length, and tension of the string.
The mass of this low C string resonates at 264 hertz,
whereas the thinner A string of different material resonates at 440 hertz.
tNext we use a simplified analogy of boats at sea.
A series of waves, particularly when they hit broadside,
can initiate rolling behavior. The smaller boat is affected by
short-period low-amplitude waves that slap against the ocean liner with no effect.
Small boats, on the other hand,
will ride up and over long-period, high amplitude waves
as the larger boats begin a long slow rock.
Now let’s look at how buildings are affected by
earthquake-induced seismic waves. All buildings have a natural, period,
or resonance, which is the number of seconds it takes for the building to naturally
vibrate back and forth.
The ground also has a specific resonant frequency.
Hard bedrock has higher frequencies than softer sediments.
If the period of ground motion matches the natural resonance of a building,
it will undergo the largest oscillations possible and suffer the greatest damage.
Small buildings of one or two stories resonate naturally
at much less than one-second periods.
A one-second period will affect buildings of about 10 stories.
For example, a 30-story building resonates at a period of 3 seconds,
and a 50-story building at a period of 5 seconds.
During the 1985 Mexico City earthquake,
the ground beneath the city resonated with a two-second period for over a minute
thus medium-height buildings with similar natural periods suffered the most damage
while short old weak stone buildings and skyscrapers were relatively undamaged.
That proved to be what is called a resonance disaster,
which is how engineers describe the destruction of a building by seismic vibrations
at a system's resonant frequency.
It is because of prolonged energy input,
that the system swings more and more strongly, until its structural load limit is exceeded.
A key point here is that small buildings on hard rock
and large buildings on soft sediments may suffer more damaging
ground motion effects from an earthquake than small buildings on soft sediments and large buildings on hard rock.
Resonance is one factor that contributes to earthquake damage.
Of equal or greater importance are building design
By determining the resonant frequency of the ground beneath a building site,
the building design can be modified so that a resonance disaster doesn’t occur.
The following demonstration shows how to model resonance in the classroom.
the idea of the boss model is that different height buildings will respond
to different frequencies of horizontal ground motion.
A tall building will respond to a very slow or low-frequency
oscillations of the ground . So if I move the base slowly
back and forth I can get the tall building
to oscillate where the intermediate and short buildings are not moving.
If I move it more briskly
I can dial in the rate of horizontal oscillation
so that I move the center building, the
intermediate high building and the short and the tall ones are barely moving.
To get the short building to isolate
need to move the base very briskly back and forth and dial into the natural
frequency
of the short building and we can see that the intermediate and tall buildings are
barely moving.