[★★☆☆☆] Mass-Spring-Dashpot System

This example is taken from the ABAQUS benchmark manual, see section 2.6.1. The original problem is also reported in the research paper, see 10.1002/nme.1620170902.

The model can be downloaded.

The System

The configuration of the model is shown below.

Numerical Model

Nodes

There are two DoFs in the system, two masses are connected to fixed points via nonlinear elastic springs. In order to do so, we define four nodes.

node 1 0 0
node 2 1 0
node 3 2 0
node 4 3 0

Here we use unit distance between two adjacent nodes. If the elements used are based on strain and strain rate, the unit distance is the only correct choice. If the elements used are based on displacement and velocity, the unit distance is not a must.

Materials

The left spring uses a \(\tanh\) function. To model it, we use Tanh1D material.

material Tanh1D 1 1000

The right spring uses a \(\sinh\) function. To model it, we use Sinh1D material.

material Sinh1D 3 1000

The middle spring is a linear spring, we simply use Elastic1D material.

material Elastic1D 2 100

The dashpot is linear. We use \(\alpha=1\) in Viscosity01 material. The viscosity coefficient is \(5\).

material Viscosity01 4 1 5

Elements

The springs can be modelled by using either T2D2, which uses strain and strain rate as the basic quantities, or Spring01, which uses displacement and velocity as the basic quantities.

element Spring01 1 1 2 1
element Spring01 2 2 3 2
element Spring01 3 3 4 3

For a linear dashpot, we use Damper01.

element Damper01 4 2 3 4

In addition to the above, it is necessary to define two mass elements.

mass 5 2 1 1
mass 6 3 1 1

IBC

The vertical DoFs of all nodes shall be fixed. The horizontal DoFs of the first and last nodes shall be fixed.

fix2 1 1 1 4
fix2 2 2 1 2 3 4

The initial condition can be applied to node 2 via

initial velocity 100 1 2

Load

To apply a step load, one shall use tabular amplitude to define the load curve. The following table can be stored in file h.

0 0
0.499999999999 0
0.5 1
0.55 1
0.550000000001 0
100 0

Then the load can be applied such that

amplitude Tabular 1 h

cload 1 1 3000 1 3

Response

To record response, we define two recorders, one for displacement, one for velocity.

hdf5recorder 1 Node U1 2 3
hdf5recorder 2 Node V1 2 3

Step and Analysis

The remaining settings are pretty standard. We use a dynamic step with a default integration scheme (Newmark). If one wishes, other integration schemes can be used.

step dynamic 1 1
set ini_step_size 1E-3
set fixed_step_size 1

converger RelIncreDisp 1 1E-11 10 1

analyze

save recorder 1 2

exit

Results

One could compare the results with the original results in the paper.