Reference problem
=====================

Device description
-------------------------

The viscous damper is represented by the rheological model below.


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    :height: 2.4098in


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**Figure** 1.1-a: Rheological model of the viscous damper.

The values of the various stiffness :math:`{E}_{1}`, :math:`{E}_{2}`, :math:`{E}_{3}` and the characteristics of the non-linear viscous part :math:`{C}_{3}`, :math:`{\alpha }_{3}` are derived from tests. The equations governing the behavior are [:ref:`R5.03.17 <R5.03.17>`]:

:math:`{\dot{F}}_{1}\left(\frac{1}{{E}_{1}}+\frac{1}{{E}_{3}}+\frac{{E}_{2}}{\left({E}_{1}.{E}_{3}\right)}\right)=\left({\dot{U}}_{2}-{\dot{U}}_{1}\right)\left(1+\frac{{E}_{2}}{{E}_{3}}\right)-{\langle \langle \frac{{F}_{1}}{{C}_{3}}\left(1+\frac{{E}_{2}}{{E}_{1}}\right)-\frac{{E}_{2}}{{C}_{3}}\left({U}_{2}-{U}_{1}\right)\rangle \rangle }^{1/{\alpha }_{3}}`

with :math:`\begin{array}{cc}{\langle \langle x\rangle \rangle }^{a}={x}^{a}& \mathit{si}x\ge 0\\ {\langle \langle x\rangle \rangle }^{a}=-{∣x∣}^{a}& \mathit{si}x\le 0\end{array}`

The dissipation increment is:

:math:`\Delta D={C}_{3}.{∣\frac{{F}_{1}}{{C}_{3}}\left(1+\frac{{E}_{2}}{{E}_{1}}\right)-\frac{{E}_{2}}{{C}_{3}}\left({U}_{2}-{U}_{1}\right)∣}^{1+1/{\alpha }_{3}}`


Modellations
-------------

Note: The units of the parameters must agree with the unit of effort, the unit of length, and the unit of time of the problem [:ref:`R5.03.17 <R5.03.17>`]. For all models the units are homogeneous to [N], [m], [s].


Modeling A
~~~~~~~~~~~~~~~

This modeling compares a system of discretes (stiffness and linear damper) assembled in series and in parallel to a discrete affected by the law of behavior DIS_VISC, with :math:`{\alpha }_{3}=1`, and a point mass :math:`m=1.0\mathit{kg}`. This comparison is carried out in transient linear dynamics by simulating a release test.


B, C, D, E models
~~~~~~~~~~~~~~~~~~~~~~~~~~

This modeling is carried out in non-linear dynamics and simulates the loading due to an earthquake. The acceleration is sinusoidal for 4 periods, then zero acceleration. The CALC_CHAR_SEISME command allows you to define the solicitation.


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    :height: 1.339in


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**Figure** 1.2.2-a **: Model system.**

Modeling B is carried out with the operator DYNA_NON_LINE. The C, D, E models are carried out with the operator DYNA_VIBRA (TYPE_CALCUL =' TRAN ', ='', BASE_CALCUL =' GENE ') equivalent to the command DYNA_TRAN_MODAL, with a time diagram of the type Euler, Runge-Kutta of order 5 and Runge-Kutta of order 5 and Runge-Kutta of order 3, respectively.


Material properties
--------------------

Modeling A
~~~~~~~~~~~~~~~

The characteristics of the shock absorber are:

:math:`\mathit{K1}=120.0`, :math:`\mathit{K2}=10.0`, :math:`\mathit{K3}=60.0`, :math:`C=1.7`, :math:`\mathit{PUIS}\text{\_}\mathit{ALPHA}=1.0`

The mass value is :math:`m=1.0\mathit{kg}`.


B modeling
~~~~~~~~~~~~~~~

The characteristics of the shock absorber are:

:math:`\mathit{K1}=120.0`, :math:`\mathit{K2}=10.0`, :math:`\mathit{K3}=60.0`, :math:`C=1.7`, :math:`\mathit{PUIS}\text{\_}\mathit{ALPHA}=0.50`

The spring stiffness is :math:`k=1.0{\mathit{N.m}}^{-1}`, the mass is :math:`m=1.0\mathit{kg}`.


C, D, E modeling
~~~~~~~~~~~~~~~~~~~~~~

The characteristics of the shock absorber are given under the keyword DIS_VISC of the DYNA_VIBRA or DYNA_TRAN_MODAL command.

:math:`\mathit{K1}=120.0`, :math:`\mathit{K2}=10.0`, :math:`\mathit{K3}=60.0`, :math:`C=1.7`, :math:`\mathit{PUIS}\text{\_}\mathit{ALPHA}=0.50`

The spring stiffness is :math:`k=1.0{\mathit{N.m}}^{-1}`, the mass is :math:`m=1.0\mathit{kg}`.


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Boundary conditions and loads
-------------------------------------

When the discrete is a SEG2, one of the nodes is blocked, on the other a displacement condition is imposed. When the discrete is a POI1 the displacement condition is imposed on this node.

In modeling A, the displacement is imposed and remains constant: :math:`{U}_{0}=0.1`

The on-the-go condition is a function of time for B, C, D, and E models:

:math:`{U}_{0}.\mathrm{sin}(2\pi .\mathit{f.t})` with :math:`f=5\mathit{Hz}` for 4 periods, then :math:`0`.