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Mathieu functions >> Mathieu functions > mathieu_membrane_mode

mathieu_membrane_mode

Calculate elliptical membrane mode for known semi-axes, mode numbers, mode type and boundary condition (Soft/Dirichlet or Hard/Neumann).

Calling Sequence

[X, Y, ZETA, is_even_or_odd, f, qs, ws, ka] = mathieu_membrane_mode(a, b, m, n, func_name, bc_type, N_xi, N_eta, mode_options)

Parameters

a

membrane semi-major axis (along x)

b

membrane semi-minor axis (along y)

m

angular variations (function order)

n

radial variations (number of q roots)

func_name

radial function name for mode (should be one of 'Mc1', 'Ms1', 'Ce', 'Se')

bc_type

boundary condition type (%t - Soft/Dirichlet, %f - Hard/Neumann)

N_xi

grid size on xi (ellipses) direction

N_eta

grid size on eta (hyperbolas) direction

mode_options

optional structure with options

q_start

initial mathieu_rootfinder guess for q (default 1e-3)

q_delta

mathieu_rootfinder step increment on search of q (default )

q_fun_tol

relative mathieu_rootfinder tolerance of zero value (fsolve function tolerance, default 1e-12)

q_maxiter

maximum number of mathieu_rootfinder iterations (default 1e4)

do_print

print mathieu_rootfinder infromation in console? (boolean)

do_plot_mode

plot membrane mode? (boolean, default %t)

font_size_axes

font size for the axes (default 3)

show_colorbar

show colorbar? (boolean, default %t)

X

N_xi*N_eta matrix with Cartesian x coordinate of mode surface

Y

N_xi*N_eta matrix with Cartesian y coordinate of mode surface

is_even_or_odd

mode type (boolean: %t - even, %f - odd)

f

ellipse foci

qs

array of found q values

ws

characteristic frequencies as in [2]

ka

dimensionless frequency parameter as in [1]

Description

Detailed theory on elliptic membrane is presented in N.W. McLachlan book [3, ch. 16].

Mathieu_membrane_mode calculates elliptical membrane mode for known semi-axes, mode numbers, mode type and boundary condition (Soft/Dirichlet or Hard/Neumann).

Let's assume that we have an ellipse with known semi-major axis a (along x) and semi-minor axis b (along y).

First of all function calculates , then it calculates elliptical coordinate     on membrane boundary.

Then mathieu_rootfinder finds corresponding q value for given function type, order m, root number n and boundary condition.

So we have two sets of q values:   for Soft/Dirichlet boundary condition and     for Hard/Neumann boundary condition.

The complete table with various function types and boundary conditions is presented below.

Func_name Bc_type Mode name Mathieu_rootfinder equation Mode expression
'Mc1' %t Even Soft/Dirichlet
'Mc1' %f Even Hard/Neumann
'Ms1' %t Odd Soft/Dirichlet
'Ms1' %f Odd Hard/Neumann
'Ce' %t Even Soft/Dirichlet
'Ce' %f Even Hard/Neumann
'Se' %t Odd Soft (Dirichlet)
'Se' %f Odd Hard/Neumann

On the last step mathieu_membrane_mode calculates mode surface in the points of N_xi*N_eta grid. It returns Cartesian coordinates (X, Y) of grid points and mode values (ZETA) in these points. Also for found qs this function calculates characteristic frequency (ws) and dimensionless frequency parameter (ka).

During unit-testing this function was tested by comparison with results from articles: [1, Tables 1-4] and [2, Tables II-V].

The results are very close.

Examples

// calculate two Soft modes of elliptic membrane - Even (m=0, n=1) and Odd (m=1, n=1)
a = 0.05;
b = 0.03;
N_xi = 101;
N_eta = 101;
options.font_size_axes = 4;

f_wh = get(0, 'screensize_px'); margin = 50;
f_h = figure('Background',-2);

subplot(1,2,1);
    mathieu_membrane_mode(a, b, 0, 1, 'Mc1', %t, N_xi, N_eta, options);
subplot(1,2,2);
    mathieu_membrane_mode(a, b, 1, 1, 'Ms1', %t, N_xi, N_eta, options);

f_h.figure_size = [f_wh(3) - 2*margin, f_wh(4) - 2*margin];
f_h.figure_position = [margin margin];

See Also

Authors

N. O. Strelkov, NRU MPEI

Bibliography

1. Wilson, Howard B., and Robert W. Scharstein. "Computing elliptic membrane high frequencies by Mathieu and Galerkin methods." Journal of Engineering Mathematics 57.1 (2007): 41-55. (online at http://scharstein.eng.ua.edu/ENGI1589.pdf or http://dx.doi.org/10.1007/s10665-006-9070-1 )

2. Gutiérrez-Vega, J., S. Chávez-Cerda, and Ramón Rodríguez-Dagnino. "Free oscillations in an elliptic membrane." Revista Mexicana de Fisica 45.6 (1999): 613-622. (online at http://optica.mty.itesm.mx/pmog/Papers/P001.pdf )

3. N.W. McLachlan, Theory and Application of Mathieu Functions, Oxford Univ. Press, 1947.


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