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    Home : Company : User Case Studies : Commercial : How Math Matters in Design Optimization
User Case Studies - Commercial

How Math Matters in Design Optimization

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