Transient Response of a Spring-mass System with Imposed Acceleration

This test lets you check analysis results for a spring-mass system, in the context of a transient dynamic response case. You will use a 1D mesh.

This test is used to validate the following attributes:

  • 1D beam element

  • Transient dynamic response solve algorithms.

Reference:

Guide de validation des prologiciels de calcul de structures, AFNOR Technique, SFM, pp182-183

 

Specifications

Geometry Specifications

Mass:
m = 1 kg

Stiffness:
k = 1000 N_m 

Length:
L=1 m

Section:
area = 1.10-5 m2  

Analysis Specifications

Young Modulus (material):
E = 100 MPa

Mesh Specifications:
Size = 187.5 mm

Restraints:

  • Clamp on A

  • Ty = Tz = Rx = Ry = Rz = 0 for each beam

Loads:

  • 0 s < t < 0.1 s

  • Damping:
    d = 0 %

 

Results

Analytical Solution

The natural frequencies are the roots of the following equation:

The response at point D is given by:

with imposed acceleration coefficient:

and modal participated factor for a translational motion with u = 1.


Frequency Analysis

Mode

Frequencies [Hz]

Error [%]

Analytic solution

Values

1

2.239

2.2399

0.038

2

6.275

6.2760

0.015

3

9.069

9.0690

0


Transient Dynamic Analysis:

Time [s]

Displacement [mm]

Error [%]

 Analytic solution 

Values

0.02

2.7

2.69

0.25

0.04

42.6

42.77

0.40

0.05

104.1

104.32

0.21

0.06

215.8

216.18

0.18

0.08

681.3

682.16

0.13

0.1

1658

1659.71

0.10

To Perform the Test:

The Transient_response_spring_mass_system.CATAnalysis document presents a complete analysis of this case.

Proceed as follow:

  1. Open the CATAnalysis document.

  2. Compute the case and generate an image called Deformed mesh.