Using Sound to "Illustrate" Mechanical Vibrations

Objective:
When modeling the mechanical vibrations arising from springmass
systems, it is customary to illustrate the displacement of the mass as
it oscillates about the equilibrium position using graphs or animations.
This demo makes use of computer generated sound to
"illustrate" oscillatory solutions.
Level: A first course in ordinary differential equations. Prerequisites: Students should know how to solve homogeneous and nonhomogeneous second order linear ODEs and be familiar with the various forms of the solutions. Platform: This demo utilizes MATLAB (version 5.3) or Mathematica (version 2.2 or higher) software running on a PC platform. The accompanying MATLAB graphical user interface application has not been tested on Mac or Unix platforms. Instructor's Notes: A springmass system gives rise to a second order initial value problem of the form
where m, and k are positive constants, c is a nonnegative constant, y = y(t) is the displacement of the mass at time t, and f(t) is an applied force. When the parameters are such that , solutions are oscillatory. The nature of solutions depends on whether the system is damped (c > 0) or undamped (c = 0) and on whether the system is free (f(t) = 0 for all t in the relevant interval) or forced (f(t) is not identically zero in the interval). In the case where f(t) = 0 and c = 0, the resulting initial value problem can be written as . The natural frequency of the oscillation is . If an external force with frequency , , oscillations will become unbounded when . If the forcing frequency is close to the natural frequency, beats occur. If the system is damped with no forcing, solutions have the form
Audible waves originate from strings that vibrate within the frequency range 20 Hz to 20,000 Hz. This demo involves utilizing sound generating capabilities of MATLAB or Mathematica to illustrate the nature of the solutions to the differential equation when oscillations are sustained. To generate sound using MATLAB or Mathematica, sample code is given below.
Credits: This demo and MATLAB mfiles were submitted by Dr. Lila F. RobertsCollege of Information & Mathematical Sciences Clayton State University Morrow, GA 30260
and is included in Demos with Positive Impact with her permission. 

LFR 1/23/00. Last updated 9/15/2010 DRH
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