MATH 322 - MATLAB GUIs
One-dimensional Rectangular Elastic
Membrane
© GUI developed by Luis
GarcÃa-Naranjo, Spring
2007
Description
This MATLAB GUI illustrates graphically how the
vibrating modes of a rectangular membrane evolve in
time and interact with one another. The membrane is
clamped at its boundary and its deflection from the
horizontal, u, evolves according to the
two-dimensional wave equation,
utt =
c2(uxx +
uyy).
- The simplest vibrating motions correspond to
the periodic evolution of the so-called normal
modes, which have initial displacements of the
form umn(x, y) =
amn sin(m p x/a) sin(n
p y/b),
where m, n are positive integers,
and the membrane is the rectangle 0 £ x £ a and 0
£ y
£
b.
- The value of c is set as c =
(a + b)/2.
- The general motion of the membrane is
obtained by expressing the initial displacement
as a superposition of normal modes (using a
double Fourier sine series), and superimposing
their evolution.
- The Single Mode feature of this MATLAB
GUI allows the user to plot the first normal
modes (0 £
m, n £ 10) and see their
individual evolution in time.
- The Superposition of Modes feature
allows the user to plot a linear combination of
the first 4 normal modes (0 £ m, n
£ 2) and
see their evolution in time.
- The user is also allowed to modify the width
and the length of the membrane. The dynamics of
the superimposed modes can be seen to depend on
whether the membrane is a square or a
rectangle.
How to use the GUI
- Use the sliders in the top right panel to
adjust the dimensions of the membrane.
- Use the sliders in the Single Mode (enter
m, n) panel to choose m and n,
and to see the displacement corresponding to the
normal mode umn.
- Use the sliders in the panel labeled
Superposition of modes (enter weights) to
select the weights of the first four normal
modes, umn, with 1 £ m, n
£ 2, and to
see the displacement associated with the
corresponding linear combination.
- Click the Run button to see the
temporal evolution of the membrane.
- Click the Reset button to
reinitializes the GUI.
Download
Download (right click
on each link below) the following files into a
directory. Set the MATLAB path to that directory
and type
Rectangular_Membrane at the
MATLAB prompt.
Type
help Rectangular_Membrane at the
MATLAB prompt if you need to be reminded of how to
use the GUI.
