May 5, 2011

Swaying high-rises and resonance frequencies

Posted by Austin Elliott

In a post a few weeks ago I linked to a humbling video of high-rise buildings in Tokyo swaying after the 9.0 quake. Well, those buildings were full of tens of thousands of people, plenty of whom had cameras. That video was only the beginning; below are many more that capture the dramatic oscillation of the towering steel edifices.

The swaying is especially clear in the following video, likely filmed during the 7.9 aftershock that happened shortly after the 9.0. The tsunami is already on the news, and smoke is billowing from the distance. It must have been alarming to start shaking again while watching the startling consequences of the first quake unfold.

The triplet of skyscrapers below is linked together by walkways high in the air. Clearly the walkways were engineered with earthquakes in mind, as they collapse and bend while the buildings swing differently.

From inside one of the swaying towers you can hear the surrounding building creak:

While high-rises do rattle and shake from earthquakes, the “gentle” swaying is a result of their resonant response to the low frequency waves unleashed by large earthquakes. Imagine an especially tall building being pushed to the side, from the bottom. The huge building has a lot of inertia, and it takes time for the force at the bottom to be transmitted up through the beams to the top. When the forces reach the top (a matter of less than a second, probably) the whole building will be in motion, moving to the side. If you suddenly stop the bottom, the top maintains its momentum and overshoots this position until the structure’s stiffness halts it and the elastic property that allowed it to flex forces it to recover that deformation and swing back the other way. This single impulse (pushing the building to the side a finite distance) results in an oscillation of the un-anchored end of the building (the top) as the force imparted at its base is gradually dampened or absorbed by flexing, heating, and creaking of the beams.

Now, if instead of stopping the bottom of the building you reverse it and push it back the other way, this would amplify the distance the top has to travel once it swings back, adding momentum to the return oscillation and enhancing the swaying. The same thing goes on when you push someone on a swing: you give them pushes just as they swing “forth” so that the energy you input into the system is added to the energy they already have being pulled forth by gravity. If you pushed them as they were coming at you, all of your energy would be expended resisting the force of their swing “back,” and the system (you and the swinger) would lose all of its energy. To make a tall building really sway, the seismic waves must drag its base back and forth at a frequency that matches that of the building’s natural oscillation–its resonant frequency. That is to say, if you “plucked” a building and let it wobble it would do so at a frequency that is determined by its material properties, geometry, and weight, among other things. If you then continue to shake it at that same frequency, you’ll accentuate the motion, just like the swing set, causing “resonance.”

Taller buildings have lower natural resonant frequencies than short buildings, meaning that if a broad spectrum of seismic wave frequencies is released, buildings with different resonant frequencies will sway differently. Small 2-story houses have extremely high resonant frequencies, and are thus more susceptible to the very sharp seismic waves experienced most strongly near the epicenter of a quake. Sky-scrapers may start swaying at huge distances from a quake, where high-frequency waves have died off and all that are left are the low-frequency seismic waves people can barely feel (we have even higher resonant frequencies than houses, if you want to think of it that way: it’s sort of why a bus braking hard makes us fall over whereas a train stopping for hundreds of yards leaves us upright).

In fact, there’s an impressive video from the May 2008 M8.3 Wenchuan earthquake in central China… filmed over 1,000 miles away in Taipei! At the top of Taipei101, currently the second tallest building in the world, is a “tuned mass damper”, essentially a giant dense metal pendulum designed to counteract the swaying of the building due to wind or earthquakes. On the afternoon of the Wenchuan quake, tourists looking at the orb witnessed it in action as it counterbalanced the passing seismic waves from the distant quake.

The video illustrates just how slow this swaying is. Without the pendulum for reference it is unlikely anyone would have noticed.

On the other hand, the magnitude of each oscillation in the Tokyo high-rises is almost certainly enough to have made plenty of their occupants nauseous. Fortunately the slow swaying keeps their contents from being thrown around too violently, making them among the safer places to be in a quake.