# Cylinder Under Rotational Force¶

## Overview¶

The aim of this test case is to validate the following functions:

- centrifugal force

The simulation results of SimScale were compared to the numerical results presented in [HPLA100]. The mesh used in (A) and (C) was created with the automatic-tetrahedralization-tool on the SimScale platform. The mesh used in (B) und (D) was created locally.

## Geometry¶

A | B | C | D | |
---|---|---|---|---|

x [m] | 0.0195 | 0.0205 | 0.0205 | 0.0195 |

y [m] | 0 | 0 | 0 | 0 |

z [m] | 0.01 | 0.01 | 0 | 0 |

To obtain the solid body the face ABCD is rotated 45° around the z-axis. Because of the symmetry of the cylinder only one quarter was modelled.

## Analysis type and Domain¶

**Tool Type** : Code_Aster

**Analysis Type** : Static

**Mesh and Element types** :

Case | Mesh type | Number of nodes | Element type |
---|---|---|---|

(A) | quadratic tetrahedral | 2893 | 3D isoparametric |

(B) | quadratic hexahedral | 2583 | 3D isoparametric |

(C) | quadratic tetrahedral | 17460 | 3D isoparametric |

(D) | quadratic hexahedral | 9285 | 3D isoparametric |

## Simulation Setup¶

Material:

- isotropic: E = 200 GPa, \(\nu\) = 0.3, \(\rho\) = 8000 kg/m^3

Constraints:

- Face AA’BB’ and face CC’DD’ zero z-displacement
- Symmetry boundary condition on face ABCD and A’B’C’D’

Loads:

- Centrifugal force with a rotational speed \(\omega\) = 1 rad/s around the z-axis applied on the whole body

## Reference Solution¶

All stated equations used to solve the problem are derived in [HPLA100]. The parameters \(h\) and \(R\) in \(\eqref{ref5}\) are the thickness of the cross section and the radius of the middle surface of the cylinder respectively.

## Results¶

Important

The values for the comparison of the displacement \(u_{r}\) and the stresses \(\sigma_{zz}\) are averaged over on edge and an area respectively

Comparison of the displacement \(u_{r}\) and the stresses \(\sigma_{zz}\) of the inner and the outter face obtained with SimScale and the results derived from [HPLA100].

Case | Quantity | [HPLA100] | SimScale | Error |
---|---|---|---|---|

(A) | \(u_{r}(r=0.0195 m)\) [m] | 2.9424E-013 | 2.83491E-013 | 3.653% |

(A) | \(u_{r}(r=0.0205 m)\) [m] | 2.8801E-013 | 2.8972E-013 | -0.594% |

(A) | \(\sigma_{zz}(r=0.0195 m)\) [N/m^2] | 0.99488 | 0.992259 | 0.263% |

(A) | \(\sigma_{zz}(r=0.0205 m)\) [N/m^2] | 0.92631 | 0.929711 | -0.367% |

(B) | \(u_{r}(r=0.0195 m)\) [m] | 2.9424E-013 | 2.94195E-013 | 0.015% |

(B) | \(u_{r}(r=0.0205 m)\) [m] | 2.8801E-013 | 2.87966E-013 | 0.015% |

(B) | \(\sigma_{zz}(r=0.0195 m)\) [N/m^2] | 0.99488 | 1.00893 | -1.412% |

(B) | \(\sigma_{zz}(r=0.0205 m)\) [N/m^2] | 0.92631 | 0.914645 | 1.259% |

(C) | \(u_{r}(r=0.0195 m)\) [m] | 2.9424E-013 | 2.94234E-013 | 0.002% |

(C) | \(u_{r}(r=0.0205 m)\) [m] | 2.8801E-013 | 2.88003E-013 | 0.002% |

(C) | \(\sigma_{zz}(r=0.0195 m)\) [N/m^2] | 0.99488 | 0.995217 | -0.034% |

(C) | \(\sigma_{zz}(r=0.0205 m)\) [N/m^2] | 0.92631 | 0.926616 | -0.033% |

(D) | \(u_{r}(r=0.0195 m)\) [m] | 2.9424E-013 | 2.94237E-013 | 0.001% |

(D) | \(u_{r}(r=0.0205 m)\) [m] | 2.8801E-013 | 2.88007E-013 | 0.001% |

(D) | \(\sigma_{zz}(r=0.0195 m)\) [N/m^2] | 0.99488 | 0.994995 | -0.012% |

(D) | \(\sigma_{zz}(r=0.0205 m)\) [N/m^2] | 0.92631 | 0.926415 | -0.011% |