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TEST CASE 8.2 DYNAMIC PROBLEM UNDER ACTION OF HARMONIC DISTURBING FORCE

Reference:

T. Thomson, Vibration Theory and Applications, 2nd Printing, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965, pg. 99, article 4.1.

 

Problem description:

A rack with a concentrated mass M at its free end is loaded with a transverse harmonic force F(t) (see a graph of the change in force intensity over time in the figure below). Determine the transverse displacement u at the moment T = 0.1 sec, T = 0.2 sec, T = 0.3 sec.

 

Problem sketch:

Type of created problem:

Flat frame (X, Z, UY).

 

Geometric characteristics:

L = 2.54 m.

 

Material properties:

EIy = 7028.62 t·m2.

 

Boundary conditions:

Node 1: X = Z = UY = 0;

Node 2: Z = 0.

 

Loads:

Concentrated mass: М = 1.785 t·sec 2/m;

Harmonic influence: F = 45.36·sin(θt) t;

Circular frequency: θ = 30 rad/sec.

 

Model description:

The system is modeled by one bar element (FE 10 type). Perform calculation of dynamics in time (integration time 0.3 sec, integration step 0.001 sec).

 

Analytical solution:

Calculation results:

Target value

Analytical solution

LIRA 10

Deviation, %

u, m; T = 0.1 sec

0.04093

0.04089

0.106

u, m; T = 0.2 sec

-0.08145

-0.0812

0.307

u, m; T = 0.3 sec

0.12045

0.12023

0.180

 

Moving along X at different times