Reference:

Analytical solution.

Problem description:

A rack with a concentrated mass M is exposed to accelerograms in three directions. There is no damping. Determine the displacements u_x1, u_x2, u_x3, u_y1, u_y2, u_y3, u_z1, u_z2, u_z3 at the moment T = 0.1 sec, T = 0.2 sec, Т = 0.3 sec (subscripts x, y, z denote the direction, 1, 2, 3 — time points 0.1 sec, 0.2 sec, 0.3 sec, respectively).

Problem sketch:

Type of created problem:

Spatial design (X, Y, Z, UX, UY, UZ).

Geometric characteristics:

L = 2.54 m;

A = 0.0025 m²;

I_y = 5.2083·10^-7 m⁴;

I_z = 5.2083·10^-7 m⁴.

Material properties:

E = 3·10⁶ tf/m².

Boundary conditions:

Node 1: X = Y = Z = UX = UY = UZ = 0.

Loads:

Concentrated mass: M = 2 tf;

Laws of harmonic action change:

a_x = 5·sin(θt) m/sec²,

a_y = 6·sin(θt) m/sec²,

a_z = 10·sin(θt) m/sec²;

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 is 0.3 sec, integration step is 0.0001 sec).

Analytical solution:

Calculation results:

Moving at Т = 0.1 sec

Moving at Т = 0.2 sec

Moving at Т = 0.3 sec