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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.

Import validation project into workspace

Geometry

FreeVibrationsOnElasticSupport-geometry

Geometry of the square box mass

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
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, \(\nu\) = 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:

(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, \(v_o\) = -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).

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

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

FreeVibrationsOnElasticSupport-result

Comparison of the oscillations computed on face ABCD for case A-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