Spend Your Time Engineering, Not on Differential Equations
12/9/2008
by
Laurent Bernardin
Simulation software based on a symbolic computational approach bears
the brunt of the math.
Digital prototyping has become an essential tool to speed design
cycles. It lets designers replace expensive hardware prototypes with
virtual models to predict system behavior, providing insight into new
designs.
The concept of virtual modeling is not new: Designers create digital
models of physical engineering systems, such as spacecraft mechanisms
and hybridelectric vehicle drivelines, and then simulate their behavior
before physical prototypes are built. This lets engineers isolate and
solve problems early in the design cycle. Here, changes are much easier
and faster to implement than they would be after the hardware has
already been built. Depending on the project, virtual prototyping can
cut days, weeks, and even years off of the time needed to develop a
product. In fact, it can sometimes save a company millions of dollars.
Signalflow modeling
A downside to the signalflow modeling tools found in conventional
controlsystem design is they force users to spend a lot of time working
with the mathematics representing the system rather than helping users
develop, simulate, and evaluate designs. Thus physical modeling using
this scheme requires users to manually derive a set of equations before
implementing them as a block diagram for simulation. Such derivations
are time consuming, error prone, and require advanced mathematical
skills.
Even when working with something as simple as a springmassdamper
system, for example, users must draw the free body diagram and then
generate the governing equations between the physical components. Next,
users must derive the differential equations for the system, convert
them to integral form, and, finally, break the equations down to
represent blocks. The resulting block diagram looks nothing like the
original system representation. Several pages of derivations are
required just to translate a freebody diagram to a differential
equation. Imagine using this method to model an entire vehicle.
Obviously, it would be better to intuitively model and simulate a
system while letting the software bear the brunt of the math. In this
scenario, engineering designs would be described using components
representing their actual physical counterparts. The software would
represent a springmassdamper system, for example, with a compact,
intuitive component diagram. Similarly, electric circuits would be built
using resistors and inductors, and mechanical transmissions would be
built with gear sets and driveshafts. It would not be necessary to
derive or manually enter equations. Instead, the software stores and
manages the necessary relational, physical, and mathematical
information. Most importantly, the software generates and simplifies
model equations to produce concise models that permit highspeed
simulations of sophisticated systems.
Fostering multidomain simulation
An example of this kind of software is MapleSim, a new multidomain
simulation tool. The software provides a broad range of components
across several physical domains, including thermodynamics, multibody
mechanics, rotational and translational mechanics, and analog, digital,
and multiphase electric circuits. Each component contains information
about which physical laws it must obey. Two connected components
exchange information about which physical quantities, such as energy,
voltage, torque, and heat and mass flows, must be conserved.
MapleSim is built on a Maple math software foundation, so it contains
the same numerical and symbolic computation capabilities. Maple uses a
powerful computational engine to derive and solve complex sets of
equations, simplify large sets of equations, and develop advanced
mathematical models.
Unlike purely numeric computations, symbolic computations can
directly convert a physicalsystem representation to mathematical
equations. The upshot is MapleSim rapidly formulates the simulation
model from the model diagram without the errors associated with manual
derivations.
The simulation software generates equations symbolically, so it can
simplify complex models using sophisticated techniques, such as
differential elimination, before solving them numerically. Unlike purely
numeric solvers, these models do not rely on iterative numeric routines.
This boosts simulation speed without compromising fidelity in the
results. Users notice this when simulating large systems comprising
thousands of equations. Simplification reduces equations to a simpler
set that can be solved quickly.
Furthermore, the simulation model is fully parameterized. This gives
users the flexibility to choose variables for which to solve, and allows
operations such as parameter sweeps. Users can thus analyze and optimize
a system with minimal effort.
MapleSim has been used to model such complex systems as filters,
vehicle dynamics, microrobotics, biomechanical devices, spacecraft
mechanisms, hybridelectric vehicle powertrains and drivelines, and
mechatronic multidomain equipment. A recent example comes from a
semitrailing arm vehicle suspension, used in many passenger vehicles.
The independent suspension has one or more links, or “arms,” connecting
the axle and the chassis. The trailing arms are pivoted at inclined
angles.
This system is particularly difficult to model because it is moving
in three dimensions. Also, two large rubber bushings at the chassis
joints exhibit a dynamic response.
The software’s multibody modeling capabilities let engineers quickly
and easily define the geometric topology of the suspension mechanism,
from which the kinematic and dynamic behavior are computed. Accounting
for the dynamics introduced by the bushings is difficult in any modeling
software.
Toyota is
one of the first industrial companies to use virtual
prototyping based on a symbolic computational approach.
In fact, the automaker and the Maplesoft have partnered
to produce advanced physicalmodeling tools. The tools
are intended to help the automaker redesign its product
development cycle and take modelbased development to
the next level. This should cut costs, improve timetomarket,
and maintain highquality standards.
Anyone dealing with the design of 'real' systems,
such as automotive designers, power engineers, and
rocket scientists, will find the symbolic modeling tool
useful. It lets engineers sidestep lengthy mathematical
work and spend more time on design and analysis. In
other words, engineers can get their products out the
door faster, perhaps the most important task in today’s
marketplace.
About the Author
Laurent Bernardin
Vice President, Research and Development
Maplesoft
Article edited by Leslie Gordon,
Sr Editor, Machine Design

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