Using Sound to "Illustrate" Mechanical Vibrations
When modeling the mechanical vibrations arising from spring-mass
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.
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.
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.
Dr. Lila F. Roberts
College 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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