Vibration Modes of a Hollow Cylinder

This model studies the vibration modes of a free and hollow cylinder using an axisymmetric approximation. No loads or constrains are applied as the free modes are sought. The cylinder is 10 m long, has a center diameter of 2 m, and is 0.4 m thick. Moreover, the material of the cylinder is considered to be steel with E = 2·10¹¹ Pa, density 8000 km/m³, and a Poisson's ratio of 0.3. Target free vibration frequencies are given in [1].

Tutorial

This model is available as an automated tutorial by selecting Model Examples and Tutorials... > Structural Mechanics > Vibration Modes of a Hollow Cylinder from the File menu. Or alternatively, follow the 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 Axisymmetry radio button.
3. Select the Axisymmetric Stress-Strain physics mode from the Select Physics drop-down menu.
4. Press OK to finish the physics mode selection.
5. To create a rectangle, first click on the Create square/rectangle 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.
6. Select R1 in the geometry object Selection list box.
7. To modify and edit the selected rectangle, click on the Inspect/edit selected geometry object Toolbar button to open the Edit Geometry Object dialog box.
8. Enter 1.8 into the x_min edit field.
9. Enter 2.2 into the x_max edit field.
10. Enter 0 into the y_min edit field.
11. Enter 10 into the y_max edit field.
12. Press OK to finish and close the dialog box.
13. Switch to Grid mode by clicking on the corresponding Mode Toolbar button.
14. Enter 0.2 into the Grid Size edit field.
15. Press the Generate button to call the grid generation algorithm.
16. Switch to Equation mode by clicking on the corresponding Mode Toolbar button.
17. Enter 0.3 into the Poisson's ratio edit field.
18. Enter 2e11 into the Modulus of elasticity edit field.
19. Enter 8000 into the Density edit field.
20. Press OK to finish the equation and subdomain settings specification.
21. Switch to Boundary mode by clicking on the corresponding Mode Toolbar button.
22. Select 1, 2, 3, and 4 in the Boundaries list box.
23. Press OK to finish the boundary condition specification.
24. Switch to Solve mode by clicking on the corresponding Mode Toolbar button.
25. Press the Settings Toolbar button.
26. Select Eigenvalue from the Solution and solver type drop-down menu.
27. Press the Solve button.

Open the postprocessing settings dialog box, look at the Solution eigenvalue/frequency drop-menu box, and verify that they correspond to the target values 0, 243.8, 378.5, 394, 397.5, 405 Hz. Select the second mode and plot its Total displacement as well as Deformation plot with a scale factor of 1.

1. Press the Plot Options Toolbar button.
2. Select Total displacement from the Predefined surface plot expressions drop-down menu.
3. Select the Deformation Plot check box.
4. Enter 0.2 into the Deformation scale factor edit field.
5. Select 2.34132e+06 (243.529 Hz) from the Available solutions/eigenvalues (frequencies) drop-down menu.
6. Press OK to plot and visualize the selected postprocessing options.

The vibration modes of a hollow cylinder structural mechanics 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 (available as the example ex_axistressstrain1 script file), or GUI script (.fes) file.

CLI Postprocessing

To visualize the full 3D solution from the axisymmetic model, the data can be exported and processed on the MATLAB command line interface (CLI) console with the Export Model Data Struct to MATLAB option from the File menu. The postrevolve and postplot functions can then be applied to revolve and visualize the data, for example

fea_revolved = postrevolve( fea, 36, 1 );

% Note that the radial coordinate is replaced by
% "x" and "y" in the revolved 3D fea data struct
postplot( fea_revolved, 'surfexpr', 'sqrt((sqrt(x^2+y^2)*u)^2+w^2)', ...
          'deformexpr', {'x*u', 'y*u', '0'}, 'deformscale', -0.2, ...
          'solnum', 4, 'parent', figure, 'axis', 'off', 'colorbar', 'off' )
view(3)

Reference

[1] Abbassian F, Dawswell DJ, Knowles NC. Free Vibration Benchmarks. NAFEMS, Test 41, 1987.