When 730 Tons is Beautiful

In our very first blog post, we mentioned that one of the goals of this blog was to highlight the art and craft of design in the built environment, with special emphasis on the arts and disciplines of architecture and structure. Often the art of architecture and the structure of building are perceived as being separate things.  It's easy to observe a superbly gifted athlete perform at the highest level of his or her chosen sport and admire the grace, fluidity, and elegance with which they execute their craft, without ever considering the structure [bone, muscle, tendon, ligament] that require being perfectly attuned in such a way as to allow the athlete to excel.  Something akin to this beauty/structure disconnect can occur when we look at, or personally experience works of architecture.  We risk forgetting the underlying substructure that allows the beauty to exist in the first place.  But there is something about the beauty of structure itself that is intrinsically thrilling and impressive, and when it occurs in unexpected places, in unexpected ways, and we are allowed to peak in, it can be downright fascinating.  Such was my reaction upon first seeing the picture of the tuned mass damper in an article titled Equation: Formula for Calculating a Skyscraper’s Sway in the December 2010 online issue of Wired Magazine.  The author explains it well.

Equation: Formula for Calculating a Skycraper’s Sway

• By Julie Rehmeyer
• November 29, 2010  | 
• 12:00 pm  | 
• Wired December 2010

Photo courtesy: Taipei 101

A skyscraper is a giant tuning fork. Give one a good knock — like with an earthquake or a heavy gust of wind — and it’ll start vibrating at its own natural resonance frequency (about seven octaves below the lowest notes on a piano). If you’re on the top floor of, say, the 1,667-foot-tall Taipei 101, you could find yourself swaying back and forth abruptly, a total of up to 2 feet within five seconds. Highly barfogenic. So Taipei 101’s designers hung a pendulum inside the building — in this case, they used an equation like the one below to determine that the megastructure needed a 730-ton weight with giant shock absorbers bolted to its bottom. It’s called a tuned mass damper, and when the tower starts to bend in the wind, the pendulum swings at the same frequency in the opposite direction, pulling the building upright and damping vibrations. It still sways, but subtly and smoothly. Here are the factors that come into play.

Damping (kg/s). This is the key variable. The goal is to halve acceleration, and more mass — building plus pendulum — means more damping.

Velocity of a swaying building at the top, in meters per second.

Acceleration (m/s²) at the top of a building — the main cause of nausea. It’s the derivative of velocity (), which is the derivative of distance (u).

The force applied to a structure by wind or an earthquake, measured in newtons (kg ⋅ m/s²).

Stiffness of a building, measured in newtons per meter (kg/s²) — the amount of force necessary to bend it a meter.

Displacement of a building at the top — that is, the distance it sways, in meters.

The dynamic mass of a building, in kilograms. The top of the structure will move more than the bottom, so the upper mass counts more.

END of ARTICLE.

What happens when structure and beauty merge into a supremely elegant solution?  A visceral satisfaction at witnessing architects, designers, and engineers performing at the very height of their craft in achieving a solution that simultaneously satisfies the highest aesthetic and functional requirements for their building. Having said that, there is one more vitally important element that cannot be overlooked.  It is the equation used to calculate the size of that pendulum.  Even without understanding all or any of the variables in play, one almost can't escape the fact that its almost impossible to determine which is more beautiful, the pendulum or the equation.  The reality is that without this particular equation and a whole host of many others, neither the aesthetic nor the functional and structural requirements of this building would have ever been realized.  Hence, such are the elements that allow beauty and structure to peacefully co-exist together in our built environment, and ultimately, keep us safe and bring us joy. 

Pendulum location at 87th floor in Taipei 101 Tower 

showing upper observation decks that permit direct 

views of pendulum.