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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

Geometry of the square box mass

The square box mass has a length, width and height of 1m

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
Mesh used for the SimScale case A-1/A-2
Mesh used for the SimScale case B-1/B-2/B-3

Simulation Setup

Material:

  • isotropic: E
    E

     

    = 205 GPa, ν

    ν

     

    = 0.28, ρ

    ρ

     

    = 10 kg/m³

Constraints:

Case A-1/A-2:

  • total isotropic spring stiffness of K
    K

     

    = 9810 N/m on face EFGH

Case B-1:

  • total isotropic spring stiffness of K
    K

     

    = 4905 N/m on face EIGJ

  • total orthotropic spring stiffness of Kx
    Kx

     

    Ky

    Ky

     

    Kz

    Kz

     

    = 4905 N/m on face IFJH

Case B-2:

  • distributed isotropic spring stiffness of KA
    KA

     

    = 9180 N/m³ on face EIGJ

  • distributed orthotropic spring stiffness of KxA
    KxA

     

    KyA

    KyA

     

    KzA

    KzA

     

    = 9180 N/m³ on face IFJH

Case B-3:

  • total isotropic spring stiffness of K
    K

     

    = 1962 N/m on face EIGJ

  • total orthotropic spring stiffness of KxA
    KxA

     

    KyA

    KyA

     

    KzA

    KzA

     

    = 1962 N/m on face IFJH

  • distributed isotropic spring stiffness of KA
    KA

     

    = 3924 N/m³ on face EIGJ

  • distributed orthotropic spring stiffness of KxA
    KxA

     

    KyA

    KyA

     

    KzA

    KzA

     

    = 3924 N/m³ on face IFJH

  • total isotropic spring stiffness of K
    K

     

    = 1962 N/m on face EFGH

Reference Solution

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

(1)

x=mgk=10.(9.81)9810=0.01m

x=mgk=10.(9.81)9810=0.01m

 

Case A-2:

(2)

x=voωsinωt+xocosωt

x=voωsinωt+xocosωt

 

where,

angular frequency, ω

ω

 = km−−√

km

 = 981010−−−−√

981010

 = 31.32 rad/s

initial velocity, vo

vo

 = -0.01 m/s

position of initial release, xo

xo

 = -0.01 m

time, 2s <= t

t

 <= 4s

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

Results

Comparison of the displacement dz

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

Comparison of the oscillations dz

dz

 computed on face ABCD from SimScale case A-2 with [SCHAUM] equation (2).

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