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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.

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Geometry

FreeVibrationsOnElasticSupport-geometry
Geometry of the square box mass

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
FreeVibrationsOnElasticSupport-mesh-caseA
Mesh used for the SimScale case A-1/A-2
FreeVibrationsOnElasticSupport-mesh-caseB
Mesh used for the SimScale case B-1/B-2/B-3

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).

Comparison of the displacement [m]
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
FreeVibrationsOnElasticSupport-result
Comparison of the oscillations computed on face ABCD for case A-2

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

References

[SCHAUM] (12345) (2011)”McGraw-Hill Schaum’s outlines, Engineering Mechanics: Dynamics”, pg 271-273, N. W. Nelson, C. L. Best, W. J. McLean, Merle C. Potter
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