MATH 322 - MATLAB GUIs
One-dimensional Circular Elastic Membrane
© GUI developed by Luis
GarcÃa-Naranjo, Spring
2007 
Description
This MATLAB GUI illustrates graphically how the
vibrating modes of a circular 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 = Ñ2u.
- For the purpose of this problem, the
Laplacian is expressed in polar coordinates,
(r, q), as
Ñ2u
= urr + (1/r)
ur + (1/r2)
uqq.
- The simplest vibrating motions correspond to
the periodic evolution of the so-called normal
modes, which have initial displacements of the
form umn(r,q) = amn
Jn(kmn r)
cos(n q), where
m, n are positive integers,
Jn(z) is the
nth Bessel function of the
first kind, and kmn denotes the
mth non-negative zero of
Jn(z). (Note that
k0n = 0, so
u0n = 0).
- The general motion of the membrane is
obtained by expressing the initial displacement
as a superposition of normal modes (using a
Fourier-Bessel 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 £ 5) and see their
individual evolution in time.
- The Superposition of Modes feature
allows the user to plot a linear combination of
the first 9 normal modes (1 £ m £ 3, 0 £ n £ 2) and see their
evolution in time.
How to use the GUI
- 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 9 non-trivial
normal modes, umn, with 1
£ m
£ 3, 0
£ 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.
DownloadDownload (right click on each link below)
the following files into a directory. Set the
MATLAB path to that directory and type
Circular_Membrane at the MATLAB
prompt.
Type help
Circular_Membrane at the MATLAB prompt if you
need to be reminded of how to use the GUI.
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