Reference:

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

Problem description:

A spring with a mass m at its end (point 2) is stretched by a longitudinal force F (graph in the figure) for 0.0025 seconds. Determine the displacement w of point 2 with the damping parameter ξ = 0.00 at time T = 0.0025 sec, T = 0.08 sec, T = 0.1 sec.

Problem sketch:

Type of created problem:

Flat frame (X, Z, UY).

Geometric characteristics:

Bar: L = 1 m.

Material properties:

Spring stiffness: K = EA/L = 200 tf/m.

Boundary conditions:

Node 1: X = Z = 0;

Node 2: X = 0.

Loads:

Concentrated mass: m = 0.5 t·sec ²/m;

Rectangular impulse: F = 4000 tf.

Model description:

The system is modeled by one bar element (type FE 10).

Analytical solution:

Calculation results:

Moving in Z at different times