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:

EI_y = 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:

Moving along X at different times