 Udwadia and Kalaba |
A seminal equation formulated by two USC theoreticians lifts a 200-year-old roadblock in mechanics.
| “We have opened up the traditional boundaries of mechanics.” |
Last fall, two USC professors cracked it. It took Firdaus Udwadia and Robert Kalaba 15 years to build their mathematical tool to express the motions of constrained mechanical systems. They got no help from supercomputers. In fact, their research method strongly resembled that of 18th-century thinkers Jean le Rond d’Alembert and Joseph Louis Lagrange, who had formulated early theories describing motion.“Once or twice a week, we sat around and talked about mechanics,” says Udwadia, a professor of civil engineering, aerospace and mechanical engineering, mathematics, and information and operations management. “We hardly ever wrote anything down or even used a blackboard. We talked and talked until we started to understand. We developed a totally new way of thinking about the motion of mechanical systems. And so we discovered the remarkably simple and aesthetic way in which nature seems to operate.”PLANETS MOVING around the sun are examples of objects in unconstrained motion, while the movement of the tip of a fully articulated robotic arm is an example of constrained movement. Within constrained motion, the forces of constraint either work or they don’t.“The breakthrough here is that we show how to handle the ‘do’ case in a straightforward manner,” says Kalaba, a professor of biomedical engineering, electrical engineering and economics.The lack of a complete mathematical understanding of constrained motion hasn't hindered engineers historically.“We can build cars, although we can't write an accurate equation for a wheel rolling across a flat surface,” says Udwadia. “The Romans built arches and bridges without truly understanding them.”But accurate mathematical models are an important predictive tool. “For 200 years, all of mechanics has been based on an underlying assumption that isn't true for many situations,” Udwadia says.He and Kalaba aren't sure how others will use their newfound insights. “I can see applications in biomechanics and robotics where engineers need mathematical models that can accurately duplicate complex movements of the human body,” says Kalaba. “However, we can't predict where other researchers in other fields will go. We have opened up the traditional boundaries of mechanics.”– Bob Calverley
D’Alembert’s Blip
When writing equations to describe motion, mathematicians have long followed a principle named for the Frenchman who formulated it in 1743, Jean le Rond d’Alembert. It assumes that the forces of constraint do no work. When French physicist Joseph-Louis Lagrange published the classic Mécanique Analytique in 1788, it included d’Alembert’s principle. “Lagrange was a genius, and his simple assumption works so well most of the time,” says USC’s Firdaus Udwadia. “But it doesn’t work all the time.” Motions for which d’Alembert’s principle fails are those where the forces of constraint actually do work – for example, when sliding friction becomes important.
D’Alembert
While it isn’t clear what applications Udwadia and colleague Robert Kalaba’s new equation for constrained motion will have, there is inevitable value in simply having it. Consider this: Newton’s laws of motion weren’t essential to engineers of his day. Indeed, they almost went unpublished until someone decided they might be useful in military targeting. “People were aiming cannonballs, arrows and rocks long before Newton. But without Newton’s laws, we would not have been able to go to the moon,” Udwadia says.
 Collecting mud samples for lab analysis. |
“We immerse these young women in marine science,” says Judy Lemus, program specialist with USC Sea Grant and co-creator of the program, which first began in 1999. The idea is to expose girls to careers in science by letting them explore aquatic and terrestrial life in and around Catalina. All faculty and counselors are women, as are visiting speakers who range from research scientists and graduate students to park rangers. “It’s an empowerment message that women can do this,” says Lemus. In addition to studying plankton under a microscope, dissecting fish, exploring one of California’s few remaining mud flats, surveying life in intertidal zones, identifying seaweed, stroking a sea hare in a touch tank and listening to a crab’s heartbeat, there is time for snorkeling and kayaking. “One of the most exciting activities is when we go snorkeling at night. We see the bioluminescence from ostracods and dinoflagellates,” says Lynn Whitley, education coordinator for USC Sea Grant. – Bob Calverley
On Solid Ground
The Southern California Earthquake Center is now safer from the object of its research. Newly renovated and seismically retrofitted North Science Hall holds about 6,600 square feet of quake-safe space for the USC-headquartered SCEC, more than tripling the research center’s footprint on the University Park campus. “We were pretty spread out before,” says SCEC assistant director Mark Benthien. A consortium of more than 30 academic institutions, SCEC is one of the nation’s largest organized efforts for the study of earthquakes. USC is the lead institution for the center, which is funded by the National Science Foundation and the U.S. Geological Survey. SCEC’s new quarters bristle with custom electronics. Its media center and war room-style science lab brim with computer workstations and toiling students. Despite earthshaking changes on the inside, however, North Science Hall’s 1928 red-brick façade retains the pleasant, collegiate feel of its original design.
– Matthew Blakeslee
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Math in Motion
America may be anathema to the Muslim masses, but the same can't be said of American movies. Since 1985, the three top-grossing films in the Middle East are Titanic, The Mask of Zorro and Godzilla. Newer releases like America’s Sweethearts have continued to do well since September 11. The reason Yankee pop culture transcends politics, even war, says USC film expert Rick Jewell, is that Hollywood has perfected the craft of making crowd-pleasing mass entertainment. Hollywood filmmakers figured out early that there are “certain stories, certain kinds of mythic entertainment, that would speak to people no matter what type of culture they grew up in,” Jewell told the Los Angeles Times. “In terms of spectacle and technical polish and the quality of acting, nobody does it better.” |
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