MATLAB GUI: Circular Elastic Membrane

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, u_tt = Ѳu.

• For the purpose of this problem, the Laplacian is expressed in polar coordinates, (r, q), as Ѳu = u_rr + (1/r) u_r + (1/r²) u_qq.
• The simplest vibrating motions correspond to the periodic evolution of the so-called normal modes, which have initial displacements of the form u_mn(r,q) = a_mn J_n(k_mn r) cos(n q), where m, n are positive integers, J_n(z) is the n^th Bessel function of the first kind, and k_mn denotes the m^th non-negative zero of J_n(z). (Note that k_0n = 0, so u_0n = 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 u_mn.
• Use the sliders in the panel labeled Superposition of modes (enter weights) to select the weights of the first 9 non-trivial normal modes, u_mn, 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.

Download

Download (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.

• Circular_Membrane.m
• Circular_Membrane.fig

Type help Circular_Membrane at the MATLAB prompt if you need to be reminded of how to use the GUI.

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© GUI developed by Luis García-Naranjo, Spring 2007.