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 :

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 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).

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

Last updated: January 29th, 2019

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