June 22, 2008
June 22, 2008
June 25, 2008
13.1093.1 - 13.1093.15
Some Interesting Engineering Problems with Objects of Simple Geometry and Relatively Complex Mathematical Formulation
There are several interesting engineering problems related to objects of simple geometry that involve relatively complex mathematics. We consider three different problems in the area of Mechanics. These problems are discussed in our undergraduate classes without getting into the mathematical details. In the ET 1840 - Engineering Fundamentals class we discuss the “brachistochrone” (path of shortest time) which is a cycloid. This is the trajectory of a point on a disk that rolls without slipping along a straight line. The equation involves the radial distance, and sine and cosine functions. This problem analyzed by Bernoulli is considered to be the foundation of the calculus of variation. We discuss the theory briefly in the class and let the students get the details from the course website if they desire to do so. Students are required to work in teams, and build a cycloidal- and straight-path model and test it. In the ET 4830 - Vibration course, our students learn the theory of the Helmholtz resonator, build and test the device. The students learn to set up the equation of motion for the vibrating air mass in the resonator neck considering a small pressure difference. They derive an expression for the air- spring stiffness under adiabatic conditions. They obtain an expression for the natural frequency of the resonator in terms of the resonator body and neck geometry. The students are required to design and build the resonator for a particular natural frequency using AutoDesk Inventor, and rapid prototyping and CNC machines. They test their resonator using a microphone and a PC- based software, and compare the theory and experiment.
In the vibration course we introduce the vibration of continuous systems and teach the basics of partial differential equations. Instead of using a standard problem from the textbook the author gives an overview of a panel absorber type muffler that can be used to reduce the noise propagated in a pipe or duct. The author has worked on this muffler for several years and published articles. A panel absorber consists of a rectangular panel (mass) backed by an air gap (spring). The student get an overview of how the equation of motion, which is a fourth-order partial differential equation, is derived involving the panel deflection, geometric and material properties of the device, and the time and space variables. The students get a chance to see a full-size panel absorber model and some insertion loss results. They also learn the effect of the cavity geometry, panel material and thickness, and boundary (mounting) conditions on the muffler performance. We are in the process of setting up a vibration lab where students can run standard experiments as well as design, build and test panel absorbers.
Sridhara, B., & Prince, J. (2008, June), Some Interesting Engineering Problems With Objects Of Simple Geometry And Relatively Complex Mathematical Formulation Paper presented at 2008 Annual Conference & Exposition, Pittsburgh, Pennsylvania. 10.18260/1-2--3862
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