It cracked on its first try-out in 1753. Recast by the Philadelphians John Pass and John Stow, it pealed the Declaration of Independence in 1776, and tolled significant doings thereafter until 1846, when another nasty fissure muzzled it for good. Today, thanks to Gary Koopmann, the Liberty Bell can be heard again—on the Internet.
Gary Koopmann's computerized Liberty Bell is divided into 450 squares, with an equation to describe the vibration of each one.
Koopmann, a distinguished professor of mechanical engineering at Penn State, is director of the University's Center for Acoustics and Vibrations. One of the classes he teaches is called "Designing Quiet Structures." The Bell project evolved after he got the idea to apply the principles of acoustically oriented design to one of his long-time side-interests. "I have always been fascinated with bells," he explains. "I'm a musician" —he plays piano and sings in a local chorale—"and so I'm interested in how you achieve harmonics in different instruments." When it came time to assign an appropriate semester project for his quiet-design students last spring, he asked them to recreate, through computer modeling, the most famous American bell of all.
"We started by defining the overall geometry," he recounts. Having established the Bell's dimensions, its weight (2,080 pounds), and its shape—a traditional English-bell profile—they created a structural model, dividing the surface of the bell into a grid made of 450 squares. "There's an equation describing the vibration of each square, and the computer can solve all the equations simultaneously," Koopmann says. Then they added an acoustical model, "so we can see a pattern of sound radiation at each frequency." He demonstrates the result on his lap-top computer: A keystroke sets the image of a bell swinging; the squares on the grid change from deep blue to green to red as the clapper strikes and the surface resonates.
The next step was to add sound. "A bell chord includes five tones," Koopmann explains. There's the dominant strike chord, and then there's a minor third, a major fifth, a nominal, and a hum tone." They also knew, from published records, what frequencies an English bell would include, and that the Liberty Bell itself had an E-flat strike note.
To tune the virtual bell properly, they had to match the spectrum of frequencies it would emit with what they knew should be present. "If the tones aren't properly distributed," Koopmann says, "we go back into the design loop." An optimization program fiddles with the relevant variable —in this case, the bell's thickness—until the frequencies come aright.
So much for recasting the unblemished original. "Then," says Koopmann, "we went back to see what happens when you add a crack." Remaking the surface grid, they found, "You lose symmetry in the vibration patterns. You get more of a mix of frequencies, less clarity. The bell has many more ways that it can vibrate." Indeed, on the laptop simulation, the color patterns on the swinging bell are dramatically altered.
Now for the moment of truth. Moving to a nearby tape recorder for its superior speakers, Koopmann pops in a cassette and presses a button to sound the Liberty Bell, here in 1999: first, the intact Declaration-era version, and then the damaged historical icon.
The first sound is clear and pronounced, a noise fit for ringing in a nation; the second is much lower, muddy, doleful. "I was on National Public Radio with this," Koopmann says, smiling, "and the host Liane Hansen remarked how sad the cracked bell sounds. People sent me letters saying we shouldn't anthropomorphize."
Gary H. Koopman, Ph.D., is distinguished professor of mechanical engineering and director of the Center for Acoustics and Vibrations in the College of Engineering, 157A Hammond Building, University Park, PA 16802; 814-865-2761, ghk@kirkof.psu.edu. Graduate students who worked on the virtual Liberty Bell project include Patricia Driesch, Jeremy Frank, Michael Yang, and Allison Robinson. Give the virtual Liberty Bell a ring at http://www.personal.psu.edu/staff/j/e/jef121/liberty.html.