# Free vibrations on elastic support¶

## Overview¶

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

- Elastic support

The simulation results of SimScale were compared to the analytical results derived from [SCHAUM]. The mesh used was created using first order tetrahedralization meshing algorithm on the SimScale platform.

## Geometry¶

The square box mass has a length, width and height of \(1 m\) with upper face partitioned in to half.

## Analysis type and Domain¶

**Tool Type** : Code_Aster

**Analysis Type** : Linear static and dynamic

**Mesh and Element types** :

Case | Mesh type | Number of nodes | Number of 3D elements | Element type | Analysis type | Elastic support type - face EIGJ | Elastic support type - face IFJH | Elastic support type - combined (face EFGH) |
---|---|---|---|---|---|---|---|---|

(A-1) | linear tetrahedrals | 21 | 26 | 3D isoparametric | Static | isotropic total | ||

(A-2) | linear tetrahedrals | 21 | 26 | 3D isoparametric | Dynamic | isotropic total | ||

(B-1) | linear tetrahedrals | 33 | 61 | 3D isoparametric | Static | isotropic total | orthotropic total | |

(B-2) | linear tetrahedrals | 33 | 61 | 3D isoparametric | Static | isotropic distributed | orthotropic distributed | |

(B-3) | linear tetrahedrals | 33 | 61 | 3D isoparametric | Static | isotropic total and distributed | orthotropic total and distributed | isotropic total |

## Simulation Setup¶

Material:

- isotropic: \(E\) = 205 GPa, \(\nu\) = 0.28, \(\rho\) = 10 kg/m³

Constraints:

*Case A-1/A-2:*

- total isotropic spring stiffness of \(K\) = 9810 N/m on face EFGH

*Case B-1:*

- total isotropic spring stiffness of \(K\) = 4905 N/m on face EIGJ
- total orthotropic spring stiffness of \(K_x\), \(K_y\), \(K_z\) = 4905 N/m on face IFJH

*Case B-2:*

- distributed isotropic spring stiffness of \(\frac {K}{A}\) = 9180 N/m³ on face EIGJ
- distributed orthotropic spring stiffness of \(\frac {K_x}{A}\), \(\frac {K_y}{A}\), \(\frac {K_z}{A}\) = 9180 N/m³ on face IFJH

*Case B-3:*

- total isotropic spring stiffness of \(K\) = 1962 N/m on face EIGJ
- total orthotropic spring stiffness of \(\frac {K_x}{A}\), \(\frac {K_y}{A}\), \(\frac {K_z}{A}\) = 1962 N/m on face IFJH
- distributed isotropic spring stiffness of \(\frac {K}{A}\) = 3924 N/m³ on face EIGJ
- distributed orthotropic spring stiffness of \(\frac {K_x}{A}\), \(\frac {K_y}{A}\), \(\frac {K_z}{A}\) = 3924 N/m³ on face IFJH
- total isotropic spring stiffness of \(K\) = 1962 N/m on face EFGH

## Reference Solution¶

*Case A-1/B-1/B-2/B-3:*

*Case A-2:*

where,

angular frequency, \(\omega\) = \(\sqrt \frac {k}{m}\) = \(\sqrt \frac {9810}{10}\) = 31.32 rad/s

initial velocity, \(v_o\) = -0.01 m/s

position of initial release, \(x_o\) = -0.01 m

time, 2s <= \(t\) <= 4s

The equation (1), (2) used to solve the problem is derived in [SCHAUM].

## Results¶

Comparison of the displacement \(d_z\) computed on face ABCD from SimScale case A-1/B-1/B-2/B-3 with [SCHAUM] equation (1).

Quantity | [SCHAUM] | Case A-1 | Error | Case B-1 | Error | Case B-2 | Error | Case B-3 | Error |
---|---|---|---|---|---|---|---|---|---|

x | 0.01 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 | 0.01 | 0 |

Comparison of the oscillations \(d_z\) computed on face ABCD from SimScale case A-2 with [SCHAUM] equation (2).