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.
Import validation project into workspace
Geometry
The square box mass has a length, width and height of 1m 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, ν = 0.28, ρ = 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 Kx, Ky, Kz = 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:
(1)
$$x = \frac {mg}{k} = \frac {10.(9.81)}{9810} = 0.01 m$$
Case A-2:
(2)
$$x = \frac {v_o}{\omega} sin \omega t + x_o cos \omega t$$
where,
angular frequency, \(\omega\) = \(\sqrt \frac {k}{m}\) = \(\sqrt \frac {9810}{10}\) = 31.32 rad/s.
initial velocity, Vo = -0.01 m/s
position of initial release, Xo = -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 dz 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 dz computed on face ABCD from SimScale case A-2 with [SCHAUM] equation (2).
References
| [SCHAUM] | (1, 2, 3, 4, 5) (2011)”McGraw-Hill Schaum’s outlines, Engineering Mechanics: Dynamics”, pg 271-273, N. W. Nelson, C. L. Best, W. J. McLean, Merle C. Potter |