A known mathematical modeling technique and recent advances in computing combine to solve complex mechanical models.

In comedy, they say timing is everything, but one engineer is discovering that timing is just as critical in becoming an entrepreneur. What does it take to move innovative university research to a marketplace success? Great intellectual property, good old fashioned blood and sweat, and if you’re lucky, favorable market conditions can offer a huge tailwind to get you moving.

John McPhee, at the Univ. of Waterloo’s Dept. of Systems Design Engineering, Ontario, Canada, is discovering how years of creativity and innovation in the lab can translate to commercial success. He has taken a 40-year-old mathematical modeling technique and adapted it to produce a commercial-grade software package called DynaFlexPro, available from Maplesoft, also of Waterloo.

DynaFlexPro allows the user to specify the general configuration and component parameters of complex mechanical and mechatronic models, and then using the symbolic capabilities of the Maple system, by Maplesoft, it automatically generates the required model equations. In essence, it automates the traditional modeling tasks that would require reams of paper and shelves full of reference books, letting the user develop models in a fraction of the time.

Underlying theory

DynaFlexPro and Maple combine to solve complex mechanical models including 3-D bipedal motion, which involves 12 degrees-of-freedom and nine joints. Image: Maplesoft Click to enlarge.

Although DynaFlexPro is a relatively new tool, the theory underneath is not. It uses a technique based on linear graph theory and the graph-theoretic method (GTM) for physical systems. Graph theory itself dates back to Euler in the 1700s and is one of the most important mathematical tools applied in the modern world. Nowadays many researchers are working in diverse fields, applying the theory, and extending the original concepts.

McPhee describes GTM as a simple, organized technique for creating mathematical models of discrete physical systems. GTM combines linear graph theory, which derives from topology (a branch of the mathematical field of combinatorics in how things are interconnected), with the physical characteristics of engineering components. The resulting technique lends itself easily to the creation and solution of mathematical models of physical systems. More specifically, GTM provides algorithms for automatically generating the differential-algebraic equations (DAEs) governing the dynamic behavior of a wide range of physical systems.

GTM is ideally suited for the modeling of multidimensional mechanical systems, otherwise known as multibody systems. By separating the system topological equations from the component constitutive equations, very efficient formulations and computer algorithms can be obtained. The explicit representation of topology provides physical insight into the structure of the DAEs of motions. Furthermore, there is no need to predefine a set of generalized coordinates, as is done in all conventional multibody formulations. In addition, both matrix and recursive formulations can be derived using GTM. As a result, GTM provides a general and unifying approach to the field of multibody dynamics.

The characteristic of GTM that distinguishes it from more traditional formulation methods is that the equations in the mathematical model describing the interconnection of system elements are derived separately from the equations describing their physical characteristics. This means that once a basic set of topological terms has been mastered, GTM can be very easily understood and applied to systems with widely different characteristics. In fact, GTM has been applied to a broad range of different types of systems: electrical circuits, fluid flow networks, mechanical and mechatronic systems, and even socio-economic systems.

GTM was pioneered in the 1960s and 1970s at the Univ. of Waterloo. However, it remained a theoretical novelty until technology caught up in the form of fast symbolic computation. The combination of DynaFlexPro and Maple is the first commercially viable implementation of this method. GTM is gaining popularity among the automotive, aerospace, and biomechanical sectors. Inside the DynaFlexPro package, the GTM technique has been relatively easily encoded into a sequence of Maple procedures due to its systematic nature. Because of Maple’s symbolic computation strengths, the velocity and acceleration equations can be directly obtained by symbolically differentiating the constraint equations with respect to time; this greatly reduces the amount of programming effort required. Although there are other significant advantages of a symbolic approach, a general solution to the nonlinear equations of motion can only be obtained using numerical methods. The ability of Maple to write these equations in either C++ or Fortran format facilitates their subsequent numerical solution.

Using conventional GTM, the system equations are automatically generated from the given system topology and for the tree selected by the user. A library of modeling components has been included that can easily be updated to include new components. The set of motion equations governing the kinematic or dynamic response of the given system is automatically assembled in a symbolic form that provides insight into their structure.

The end result of this combination of pure math, scientific creativity, and modern computing is generally the reduction of model development time from as long as months to days or less. Multibody dynamics researchers are relieved of the need to derive equations of motion by hand, a tedious and error-prone task. The models created by DynaFlexPro are comprised of a minimal set of ODEs (ordinary differential equations), meaning models are more efficient and manageable, and therefore easier and faster to work with. Constructing models in the DynaFlexPro environment is incredibly easy, and it is inevitable that more and more dynamics and control design will be impacted.

Practical applications

Companies starting to take notice include the major automobile companies, robotics companies, and even music companies. A major piano manufacturer collaborated with McPhee’s research team to develop a mathematical model of the piano’s “action”—the mechanism made of wood, leather, and felt that connects the pianist’s fingers to the hammer that strikes the strings. A detailed analysis of the kinematics and dynamics of this mechanism was undertaken using Maple and DynaFlexPro.

Piano manufacturers are constantly dealing with the question of how to improve the reliability of the piano action without radically changing the sound or feel of the piano, in keeping with the demands of conservative musician clients. Instead of the historic (and expensive) trial-and-error process of building and testing new pianos to answer their design questions, designers are starting to introduce and evaluate new innovations using virtual prototypes of the mechanism.

"With Maple, we obtain symbolic expressions for complex mechanical systems that often provide unique insight into the system's behavior,” says McPhee. “Furthermore, Maple’s code generation allows us to package and deliver the solution to our client so they can use it for their design work without getting into the mathematical detail."

Perhaps one of the most interesting applications of the DynaFlexPro software so far is the solution of a complex vehicle suspension dynamics modeling problem by a Maple user. The vehicle in question was a high-concept car from a high-end German automotive manufacturer, presented at the 1997 Frankfurt Motor Show. Designing such vehicles through from concept to modeling to prototyping, is very time-intensive and this had been no exception. The design and coding of the mathematical dynamics model of the suspension alone had taken four engineers nearly a year to complete using traditional methods.

In a mere 45 minutes, he was able to completely model the vehicle’s suspension dynamics. It is interesting to note that this was also the user’s first experience working with DynaFlexPro; no doubt his extensive background in this field was beneficial here, but he particularly commented on the software’s remarkable ease of use. He mentioned that when he presented his results to the team lead of the original dynamics and control development group, he found it hard to believe and was very impressed.

Researchers in the biomechanics field are also working with computer modeling programs to model kinematics and dynamics of natural events. At the 2007 Society of Automotive Engineers (SAE) Digital Human Modeling Conference this summer, Asif Mughal atthe Univ. of Arkansas, Fayetteville, presented his results on modeling 3-D human motion. Mughal’s research focuses on active leg prosthetics, and the complex bipedal mechanism he modeled involved 12 degrees-of-freedom and 9 joints. Mughal’s model employed the symbolic capabilities of the Maple system and DynaFlexPro for the model formulation.

Overall, McPhee is enjoying the emerging Math Renaissance that we’re experiencing in industry. Engineers from all sectors are starting to hit limits of traditional modeling techniques (even with powerful, expensive software tools) and are looking towards more fundamental, and often mathematical, approaches to modeling. McPhee’s work, among others, is making these goals a reality.

—Tom Lee, VP of Market Development, Maplesoft

Resources
• Maplesoft, Waterloo, Ontario, Canada, 800-267-6583, www.maplesoft.com
• Univ. of Waterloo, Ontario, Canada, 519-888-4567, www.uwaterloo.ca