Numerical-Analysis Software Constructs, Simplifies, and Solves Complex Equations

1/20/2009 by Bill Fane

In theory, almost any aspect of the physical world can be modeled and analyzed mathematically. In practice though, performing the mathematical manipulations required to perform an analysis can rapidly become difficult. For example, the mathematical theory underlying stresses and strains is simple, but the sheer volume of number-crunching involved meant that FEA wasn’t practical until computers came along.

Similar situations arise in other applications, such as evaluating transient responses. The basic formulas are relatively simple, but calculations can be quite tedious. Maple 12 software is a widely used mathematical-processing engine that lets users easily solve complex equations. In fact, the software includes many fill-in-the-blank equations that let users type-in the parameters, and away the engine goes.

A stepped force input is being applied to two spring-mass-damper systems in series. The graphs show the resultant forces and displacements over time as measured at the three probe locations, for two different sets of input parameters.

Back to the theory versus practice idea: Maple can easily solve equations, but as the system being analyzed gets more complex, even experienced users can sometimes lose their way. For instance, modeling a simple linear spring-massdamper system isn’t too difficult, but try applying the model to the suspension of a car. It’s still basically spring-mass-damper, except now the geometry changes as the suspension bounces and jounces.

A new product called MapleSim 1.0 works hand-in-hand with Maple 12 to solve this problem. Users simply model a system in schematic form using basic dragand- drop components and then apply the appropriate parameter values such as mass, spring rate, and damping coefficient. Maple- Sim constructs the appropriate equation or equations and solves them by passing them over to Maple 12.

MapleSim includes over 300 predefined components. These cover five major categories: Signal, Electrical, Mechanical, Multibody, and Thermal. Specific components include routers, bearings, gears, clutches, brakes, springs, dampers, resistors, capacitors, inductors, signal blocks, thermal conductors, and temperature sensors.

Users can mix-and-match elements so that, for instance, a model of a dc motor can include the inertia of the armature, friction in the bearings, and the resistance, capacitance, and inductance of the windings. Users can then feed the model a stepped voltage input and graph the transient torque and speed outputs.

Also, users can turn a collection of components into a named subsystem which can be used over and over again, with the same or different parameters. This makes it easy to model similar or complex multicomponent systems. Better yet, users can attach custom graphic images to subsystems so the symbol for an electric gearmotor in a complex machine looks like the actual electric gearmotor. Users can also add text to label the motor as a particular type or model. This capability makes it extremely easy to see and understand what the model represents.

When MapleSim is directed to run a simulation, the software gathers up all equations representing components and subsystems and simplifies them. In a trivial example, say a formula contains an expression such as 2x + 3x. MapleSim simplifies this to 5x.

In many cases, such simplifications slash the time needed to perform a simulation. In one case, for example, a major car manufacturer constructed a mathematical representation of the dynamics of an engine. The equation ran 10 pages, but MapleSim reduced this to one page.

Surprisingly, I got the same sort of reduction ratio when modeling a simple spring-mass-damper system. This helps explain why many engineers have not done a lot of numerical analysis in the past. We simply have not had the geologic time frames available for such calculations. We instead resorted to tweaking past techniques that worked and hoped for the best.

MapleSim 1.0 is intuitive and easy to learn. After working through a brief tutorial, I was able to construct a compound spring-mass-damper system, analyze it, and tinker with the parameters to optimize the system’s performance. The software also lets users mix units. For example, I applied a 1-lb force to a 1-kg mass without having to know, or even see, the underlying equations.

The default output is graphical. After users build a model they can insert virtual “probes” that measure speed, force, acceleration, voltage, current, and temperature. A series of output graphs shows the probed values. The graphs remain on-screen after each run, so users can change parameters, run the simulation again, and then easily compare results to previous trials. Users can also export graphs in several common graphic image formats for inclusion in other documents.

Users get parameter values in several ways. In some cases values come from manufacturers’ catalogs, through direct measurements, or by simple calculations. In addition, Maple 12 can obtain property values such as mass, moment of inertia, and center of gravity directly from an Inventor or SolidWorks part or assembly file. Everything is parametric, so, for instance, increasing the throw of a crankshaft in the 3D CAD updates the analysis in MapleSim.

All in all, MapleSim 1.0 helps users produce better designs much faster.

About the Author

Bill Fane
Professional engineer and retired instructor
British Columbia Institute of Technology

Article edited by Leslie Gordon, Sr Editor, Machine Design

Article reprinted by permission of Penton Media, publisher of Machine Design