 An interesting example of how math really
matters in design optimization is in the manufacturing
of pianos. The mechanism that connects the pianist’s
fingers to the hammer that strikes the strings, known
as “the action”, has changed little since
its invention in the 17th century. Yet, manufacturers
have been struggling with trying to improve the reliability
of this complex mechanism, made of wood and leather
and felt, as well as its manufacture, while taking
care not to drastically change the feel and sound of
their pianos - musicians tend to be a conservative
group of people, and do not embrace radical changes
to the touch or tone of their instruments.
Specifically, designers are trying
to find out:
- How does varying certain parameters
affect the feel of the action?
- What effect would
different materials have on the behaviour of the
action?
- What are the magnitudes of the forces and
torques acting on various components?
- What are the
magnitudes of the speeds and accelerations of the
bodies?
- How much movement and rubbing is there between
different bodies?
- How fast is the hammer moving
for various intensities of finger blows on the
key?
For piano makers, finding the answers to these
questions is an expensive trial and error process
because the
only way to find out if a new design fulfills
its objectives is to build a new piano and test
it.
Therefore, there
has been increased interest in modeling and simulating
the mechanism in order to reduce development
costs and increase the likelihood of success.
However,
it seems that the piano action has only been
subject to
detailed analysis by a handful of engineers who
also happen to be musicians, most notably, the
composer
Rimsky-Korsakov in 1937.
One well-known piano company is collaborating with
a research team at the University of Waterloo, Ontario,
to develop a mathematical model of their piano action
so they can introduce and evaluate new innovations
using “virtual” prototypes of the mechanism.
To achieve this, a detailed analysis of the kinematics
and dynamics were undertaken using Maple with a specialized
tool, DynaFlexPro, developed in Maple by Dr John McPhee
and his research team.
" With Maple, we obtain symbolic expressions for complex mechanical systems
that often provide unique insight into the system's behavior.” Said Dr
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."
DynaFlexPro is an add-on development environment that incorporates specialized
functions into Maple and provides a visual programming interface that allows
the user to easily define the mechanism topology (2D and 3D) and component parameters
(dimensions, masses, etc). The equations of motion, including both kinematics
and dynamics, that represent the system are generated within Maple. The resulting
model can be further analyzed within the Maple environment and the solution converted
to program code (C, FORTRAN, Java, VB and Matlab supported) for deployment in
other tools.
You can find out more about this project at http://real.uwaterloo.ca/~morg/mandm.htm
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