Mathematics Created For Better Structures

Engineers say they have created a new mathematical method to design better structures, machines and computer-controlled robots.

Gordon Pennock, an associate professor of mechanical engineering at Purdue University, and Offer Shai, a civil engineer in the Department of Mechanics, Materials and Systems at Tel Aviv University in Israel, say they have created new theorems that improve the design process by combining the mathematics of both kinematics and statics.

"These new theorems represent a common language and provide an understanding of what we call the duality between kinematics and statics," Pennock said. "The practical result is that engineers can use this knowledge to design better structures and better machines."

The dual theorems could enable civil engineers to design structures that better withstand the forces and "moments," or torque, associated with motions such as those caused by earthquakes, perhaps at less expense than today's designs.

"Today, if you want to design a sturdy structure that does not become unstable, you have several choices," Pennock said. "You can use the highest quality material and add many supporting members.

"Having a dual language provides an alternative to the current approach by giving the designer a better understanding of the physics associated with the stability of the structure. This, in turn, should enable the designer to create a safer structure at or below the cost of current designs."

The mathematics associated with kinematics must factor in the effects of velocity and acceleration resulting from motion. Static structures, on the other hand, are inherently strong in one position but could become weak if that position changes due to unpredictable motion. Combining the mathematics of kinematics and statics provides the best of both worlds by enabling engineers to better design structures that can withstand the loads resulting from motion- related forces.

"Civil engineers understand the mathematics of forces and moments, and mechanical engineers understand the mathematics of velocity and acceleration,"

Pennock said. "We have shown that these concepts are, in fact, analogous. This duality was not fully appreciated until we presented several theorems and documented the proofs of these theorems."

Shai and Pennock received the A. T. Yang Memorial Award for their prize-winning paper, "The Duality between Kinematics and Statics," which was presented at the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, and the 29th ASME Mechanisms and Robotics Conference, all in September 2005.

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Key green features include a 4-story sky lit interior atrium with a translucent glass stair and walkway system, a raised access floor air distribution system, high-efficiency glazing, sophisticated lighting controls and rainwater collection system for landscape irrigation.

Completed in October 2005, the $30 million, 84,000-sq.-ft. facility houses laboratories, classrooms and offices for Northwestern's McCormick School of Engineering and Applied Science.

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Selection criteria include organizations that enhance the transportation industry through commitment to excellence and quality, have outstanding records of affirmative action in hiring and promoting at all levels, support continuing education and encourage women to enter the transportation field by providing internship opportunities.

The WTS is an international organization dedicated to enhancing professional development of women in transportation.

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