# 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 $$1 m$$ 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

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: $$E$$ = 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).

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