




Two of the most indispensable tools for solving geometry problems include a CAD system for generating geometry and a calculator for banging out sizing equations. The engineer provides an interface between the two. This arrangement works well as long as there are just a couple variables in the problem. But try juggling three or more variables and the challenge becomes a headache. Several software developers have recognized this shortcoming and responded by producing a branch of design called behavioral modeling or behavioral optimization.
The technology is a form of optimization although there is a difference between behavioral modeling and pure optimization. A design goal, for instance, might be to find the lowest weight beam to support a load with a maximumallowable stress. The height and width of the beam's cross section could be the variables of interest. A 2D shape optimizer could easily handle this job.
Behavioral modeling differs in that it would allow geometry modifications in 3D so the beam might also include arches. Furthermore, the cross section need not be uniform, and finiteelement analysis could be tied in along with a cost program to find the leastexpensive design.
Already there are at least three programs that include some form of behavioral modeling. The BMX extension from PTC in Needham, Mass., the most comprehensive of the three, comes from the developer of Pro/Engineer software. Behavioral Optimization comes from MechSoft.com, Austin, and Geomate Corp., in California, offers GrafiCalc, a 2D goalseeking program. Several examples illustrate the capability of the technology.
Behavioral modeling
Chad Jackson with PTC says there are three types of behavioral modeling in BMX, the company's offering. "One is internal analysis, in which the software works entirely within itself to reach the design goal. The method was used by a group within Ferrari's Formula One racing team that designs exhaust manifolds. On their V10 engine, engineers wanted equal lengths for the exhaust pipes to minimize pressure build up and not hurt engine performance. "Just through manual iteration, they could get to within a couple millimeters total difference," says Jackson.
Five pipes on each side of the engine are individually described by a trajectory or curve in space. The problem was set up in BMX so the curves passed through several different points defined by several dimensions and minimum bend radii. Exit and entry angles defined both ends of the trajectories, and upper and lower boundaries limited the range of each dimension. And the pipes could not cross. "It's a purely geometric problem and a design exploration," says Jackson. "It's not optimization because there is no gradientbased hunting or modification of an existing model."
Engineers then asked the software to calculate 2,000 variations on this design. By following up with an optimization routine within the software, they wound up with a design that had only 0.003mm difference between pipes.
To appreciate the size of such a problem consider that three different dimensions with 10 intervals each define a space of 10^{3 }or 1,000 variations. When there are four dimensions, there are 10^{4 }or 10,000 variations. The design space expands exponentially as dimensions are added. Ferrari engineers were looking at 100 million different possibilities. The problem becomes one of generating many equally spaced variations.
Another type of behavioral modeling involves software developed outside of PTC, explains Jackson. Team Prada, an America's Cup yacht racing team, used this variation to search a design space using a CFD code the team had developed. "The idea was to define a hull in Pro/E and let the CFD program calculate the design's drag. Prada engineers chose dimensions and parameters that could change and their boundaries. Then the CFD code calculated the drag and Pro/E used that value to reshape the hull. The software could go through 20 or 30 iterations in a day while manual methods limited them to two or three iterations per day. The automation eventually led them to a faster boat," says Jackson
The team enjoyed a good deal of success during qualifying runs. However, the crucible of racing uncovered other shortcomings in their boat design that were not originally considered. For instance, the mast proved underdesigned for such a fast boat and eventually became part of a more comprehensive study.
The point, says Jackson, is that virtual prototypes need not come from PTC software. "But the modeler provides a means to integrate them and do the improvement." This requires a little C programming to transfer needed information from one program to another. In Prada's case, engineers wrote a short loop to transfer information between Pro/E and the CFD software.
A custom analysis or userdefined analysis describes a third type of behavior modeling. Engineers at Cannondale Corp., the bicycle maker in Bethel, Conn., relied on it when designing the rear suspension on one of their racing bikes. The suspension uses a fourbar linkage and shock absorbers, so the entire back end of the bike "floats."
The problem required simultaneously tracking the instantaneous center of the rear wheel, a second phenomena called chain growth, and a leverage ratio. "Instantaneous centers come from having a fourbar link as the rear suspension," says Ron Litke, project engineer with Cannondale. "A simple swing arm pivots about a point. But using a fourbar linkage creates an instantaneous center about which the arm rotates. We try to place the pivot so that the suspension remains active and chain forces do not lock it up. If pedaling activates the suspension, the rider is putting energy into the bike instead of making the bike move. Experience gives us an idea of where the pivot works best as it reacts to bumps, for different pedaling loads, and making the bike feel efficient."
Chain growth, the second variable and a result of the moving suspension, resists pedaling effort as the suspension moves. "It feels like the bike is going over a bump when it's not," says Litke. "It kicks the petals a bit. But a little chain growth is OK because the bike stands up a bit under the suspension. Too much, however, wastes energy." And the leverage ratio, the third variable, characterizes motion between the wheel and shock.
"We set up a behavioralmodeling study with a range for each variable and it came back with a solution," says Litke. "We had gotten close by hand, but the software told us exactly what we wanted."
Ordinarily, it would take three days to go through six to 12 different designs. BMX software lets them examine 30 designs in a day. The payoff for Cannondale has been the Gemini, a design that won a recent series of downhill races. The Gemini was a limited production run, but its suspension will soon appear on other production bikes.
The advantage of behavioral modeling over spreadsheets or programming languages, is that the geometry is closely tied in, according to Litke. "The rest of the bike geometry can be involved as well. For instance, the shockabsorber location is limited, otherwise the suspension hits the seat tube. And there are tire clearances to maintain."
Litke also points out that the software tracks features that could be critical measures of an engineering quantity. "It's easy to write relations to other values. Tying these into behavioral modeling let you track and graph them as the software works through the design space," he adds. The analysis differs from mechanical dynamics in that the latter program calculates loads. Behavioral modeling generally does not.
Another advantage of behavioral modeling is its associativity to its geometry. This make it possible to change the length of a linkage, the size of a bearing, or the length of a shock, and have all the geometry automatically change to accommodate it.
Optimizing behavior
Another variation on behavioral modeling comes from MechSoft.com and its Behavioral Optimizer. It makes geometric and functional optimizations quick and easy tasks, say its developer. The technology, an optional module for the company's base product, lets users synchronize their optimizations with mechanical calculations within the developer's Calculations Library or with any customdefined calculation. The module works with the developer's Group Manager to automate design iterations.
In a design process, the Manager evaluates parts, calculations and relationships to determine mechanical feasibility. In revisions, the Manager identifies possible failures, makes recommendations based on standard mechanical laws and parts, and propagates userapproved changes throughout the design. Group Manager also handles the functions of an assembly, such as tracking links between parts, assembly calculations, and individual part functions. The Manager acts as a map, graphically representing a design's components and its functions.
Goal seeking in 2D
The developer of GrafiCalc calls its behavioral modeler generalpurpose software, but as the name suggests, it's more of a graphic calculator. The best way to sample its capability is to download and test it. An evaluation copy is available at no cost from www.graficalc.com. The download takes about 2 min over a dialup modem. The software includes several example models and tutorials. Unfortunately, testers are left to guess at the functions because an explanation of icons and other features is not included. Users can draw geometry in the software or import it by DXF from most any drawing or modeling program. One example involves the cross section of a beam or part with a rectangular hole. The problem is to adjust the width of the internal hole until you find one that provides a particular crosssectional modulus. In a Goal Seeking window, users need only identify the existing value of the required quantity (the current moment area), the target or goal value, and the characteristic that can be adjusted (the width). Calculations take only a fraction of a second. As input conditions change, the entire flexible model updates. The software includes over 100 geometryassociated functions for algebraic, trigonometric, and additional mathematical equations, according to the developer. 