2.6.2 Linear behavior of spring and dashpot elements

Product: ABAQUS/Standard  

Problem description

Analytical solutions

The analytical solutions for each of the analyses are presented here.

Grounded tests

Force balance on the system yields the second-order linear differential equation for a single degree of freedom damped oscillator. Let x be the position of the mass node along the x-axis. We obtain the equation

which is solved subject to and to yield

where the constants have the following values:

Node-to-node tests

The analytical solution to this problem is obtained by writing out the force balance equation for both masses. This yields a coupled system of two ordinary differential equations. Let and refer to the position of the middle and end mass, respectively. The following equations are obtained:

subject to , , , and .

Taking the Laplace transform and solving for and yields

By using partial fraction expansion and taking the inverse transform of each of the terms, the following closed-form solution is obtained:

where the constants have the following values:

Results and discussion

Input files

Tables

Table 2.6.2–1 Nodal results at end of dynamic step: grounded tests.

Node
3–1.2019 × 10–36.905–29.89
13–1.2019 × 10–36.905–29.89
103–1.2019 × 10–36.905–29.89

Table 2.6.2–2 Element results at end of dynamic step: grounded tests.

Element S11E11
1–3.6056 ×10–2–1.2019 × 10–3
3–3.6056 × 10–2–1.2019 × 10–3
5–3.6056 × 10–2–1.2019 × 10–3
20.00.0
40.00.0
60.8286–1.2019 × 10–3
70.8286–1.2019 × 10–3
80.8286–1.2019 × 10–3

Table 2.6.2–3 Nodal results at end of dynamic step: node-to-node tests.

Node
12–0.2105–6.531151.8
102–0.2105–6.531151.8
13–0.2698–11.5591.23
103–0.2698–11.5591.23

Table 2.6.2–4 Element results at end of dynamic step: node-to-node tests.

ElementS11E11
2–6.315–0.2105
4–6.315–0.2105
3–1.778–5.9276 × 10–2
5–1.778–5.9276 × 10–2
7–0.6022–5.9276 × 10–2
8–0.6022–5.9276 × 10–2


Figures

Figure 2.6.2–1 Three independent spring/dashpot/mass systems.

Figure 2.6.2–2 Free node displacement history: grounded tests.

Figure 2.6.2–3 Free node displacement history: node-to-node tests.