Vibrations of a Circular Membrane

This example computes the vibration modes, eigenvalues, and frequencies for a circular drum or membrane. The membrane is modeled by the unit circle and assumed to be attached to a rigid frame. The Poisson PDE equation is used with the Eigenvalue solver to compute the solution.

Tutorial

This model is available as an automated tutorial by selecting Model Examples and Tutorials... > Classic PDE > Vibrations of a Circular Membrane from the File menu. Or alternatively, follow the video tutorial or step-by-step instructions below.

1. To start a new model click the New Model toolbar button, or select New Model... from the File menu.
2. Select the Poisson Equation physics mode from the Select Physics drop-down menu.
3. Press OK to finish the physics mode selection.
4. To create a circle or ellipse, first click on the Create circle/ellipse Toolbar button. Then left click in the main plot axes window, and hold down the mouse button. Move the mouse pointer to draw the shape outline, and release the button to finalize the shape.
5. Select E1 in the geometry object Selection list box.
6. To modify and edit the selected ellipse, click on the Inspect/edit selected geometry object Toolbar button to open the Edit Geometry Object dialog box.
7. Enter 0 0 into the center edit field.
8. Enter 1 into the x_radius edit field.
9. Enter 1 into the y_radius edit field.
10. Press OK to finish and close the dialog box.
11. Switch to Grid mode by clicking on the corresponding Mode Toolbar button.
12. Enter 0.1 into the Grid Size edit field.
13. Press the Generate button to call the grid generation algorithm.
14. Switch to Equation mode by clicking on the corresponding Mode Toolbar button.
15. Press OK to finish the equation and subdomain settings specification.
16. Switch to Boundary mode by clicking on the corresponding Mode Toolbar button.
17. Select all boundaries (1-4) in the Boundaries list box.
18. Select Dirichlet boundary condition from the Poisson Equation drop-down menu.
19. Enter 0 into the Dirichlet coefficient edit field.
20. Press OK to finish the boundary condition specification.
21. Switch to Solve mode by clicking on the corresponding Mode Toolbar button.

Open the Solver Settings dialog box, and select the Eigenvalue solver which per default will try to find the six smallest eigenvalues and corresponding eigenvectors.

1. Press the Settings Toolbar button.
2. Select Eigenvalue from the Solution and solver type drop-down menu.

3. Press the Solve button.

   

Plot the first, second, and fourth modes and confirm that they feature a corresponding number of peaks and troughs.

1. Press the Plot Options Toolbar button.
2. Select the Height Expression check box.
3. Select the fourth solution (25.9617 (0.810936 Hz)) from the Available solutions/eigenvalues (frequencies) drop-down menu.
4. Press OK to plot and visualize the selected postprocessing options.

From the analytical solution, the frequency of the second and third mode to the first one should be a factor 1.59 times higher, fourth and fifth 2.14, and sixth 2.30.

The vibrations of a circular membrane classic pde model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB m-script text file, or GUI script (.fes) file.

Video