 
NumericalAnalysis 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 numbercrunching 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
mathematicalprocessing engine that lets users easily solve complex
equations. In fact, the software includes many fillintheblank
equations that let users typein the parameters, and away the engine
goes.
A stepped force input is being applied to two springmassdamper 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 springmassdamper system isn’t too difficult, but try
applying the model to the suspension of a car. It’s still basically
springmassdamper, except now the geometry changes as the suspension
bounces and jounces.
A new product called MapleSim 1.0 works handinhand 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 mixandmatch 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 springmassdamper 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 springmassdamper
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 1lb force to a 1kg 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 onscreen 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 
