FEATool Multiphysics
v1.10 Finite Element Analysis Toolbox |

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 = 2·10 ^{11} Pa*, density

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.

- To start a new model click the
**New Model**toolbar button, or select*New Model...*from the*File*menu. - Select the
**Axi 2D**radio button. - Select the
**Axisymmetric Stress-Strain**physics mode from the*Select Physics*drop-down menu. - Press
**OK**to finish the physics mode selection. - 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. - Select
**R1**in the geometry object*Selection*list box. - 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. - Enter
`1.8`

into the*x*edit field._{min} - Enter
`2.2`

into the*x*edit field._{max} - Enter
`0`

into the*y*edit field._{min} - Enter
`10`

into the*y*edit field._{max} - Press
**OK**to finish and close the dialog box. - Switch to
**Grid**mode by clicking on the corresponding*Mode Toolbar*button. - Enter
`0.2`

into the*Grid Size*edit field. - Press the
**Generate**button to call the grid generation algorithm. - Switch to
**Equation**mode by clicking on the corresponding*Mode Toolbar*button. - Enter
`0.3`

into the*Poisson's ratio*edit field. - Enter
`2e11`

into the*Modulus of elasticity*edit field. - Enter
`8000`

into the*Density*edit field. - Press
**OK**to finish the equation and subdomain settings specification. - Switch to
**Boundary**mode by clicking on the corresponding*Mode Toolbar*button. - Select
**1**,**2**,**3**, and**4**in the*Boundaries*list box. - Press
**OK**to finish the boundary condition specification. - Switch to
**Solve**mode by clicking on the corresponding*Mode Toolbar*button. - Press the
**Settings***Toolbar*button. - Select
**Eigenvalue**from the*Solution and solver type*drop-down menu. - 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*.

- Press the
**Plot Options***Toolbar*button. - Select
**Total displacement**from the*Predefined surface plot expressions*drop-down menu. - Select the
**Deformation Plot**check box. - Enter
`1`

into the*Deformation scale factor*edit field. - Select
**2.34132e+06 (243.529 Hz)**from the*Available solutions/eigenvalues (frequencies)*drop-down menu. - 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, or GUI script (.fes) file.

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