Reference solutions
======================

Calculation method used for reference solutions
----------------------------------------------------------

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

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


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The equations of the differential system, which is of 3rd order, describing the analytical solution are:

:math:`\{\begin{array}{c}{F}_{1}=m\ddot{{U}_{1}}\\ {\dot{F}}_{1}.\left(\frac{1}{{E}_{1}}+\frac{1}{{E}_{3}}+\frac{{E}_{2}}{{E}_{1}.{E}_{3}}\right)=\left({\dot{U}}_{2}-{\dot{U}}_{1}\right)\left(1+\frac{{E}_{2}}{{E}_{3}}\right)-\frac{1}{{C}_{3}}.\left({F}_{1}.\left(1+\frac{{E}_{2}}{{E}_{1}}\right)-{E}_{2}.\left({U}_{2}-{U}_{1}\right)\right)\end{array}` [:ref:`éq2.1.1-1 <éq2.1.1-1>`]

The initial conditions are, with :math:`{U}_{0}=\mathrm{0,1}m`:

:math:`{\ddot{U}}_{1}(t=0)=0` :math:`{\dot{U}}_{1}(t=0)=0` :math:`{U}_{1}(t=0)=0` :math:`{U}_{2}(t)={U}_{0}.H(t)\mathit{avec}H\mathit{la}\mathit{fonction}\mathit{de}\mathit{Heaviside}`

The numerical integration of this differential system is obtained with the Laplace transform technique:

:math:`{U}_{1}(t)={A}_{s}\mathrm{exp}(-{\lambda }_{\mathit{sc}}t)\mathrm{sin}({\omega }_{t}t)-{A}_{c}\mathrm{exp}(-{\lambda }_{\mathit{sc}}t)\mathrm{cos}({\omega }_{t}t)+{A}_{e}\mathrm{exp}(-{\lambda }_{e}t)+\frac{1}{10}` [:ref:`éq2.1.1-2 <éq2.1.1-2>`]

:math:`{F}_{1}(t)=-{B}_{s}\mathrm{exp}(-{\lambda }_{\mathit{sc}}t)\mathrm{sin}({\omega }_{t}t)+{B}_{c}\mathrm{exp}(-{\lambda }_{\mathit{sc}}t)\mathrm{cos}({\omega }_{t}t)+{B}_{e}\mathrm{exp}(-{\lambda }_{e}t)` [:ref:`éq2.1.1-3 <éq2.1.1-3>`]

With

:math:`\begin{array}{ccc}{\omega }_{t}=\frac{14593}{4792}{s}^{-1}& & \\ {A}_{s}=\frac{5516}{214807}& {\lambda }_{\mathit{sc}}=\frac{1573}{2072}{s}^{-1}& {B}_{s}=\frac{5625}{7831}\\ {A}_{c}=\frac{3137}{29305}& & {B}_{c}=\frac{9170}{11289}\\ {A}_{e}=\frac{413}{58610}& {\lambda }_{e}=\frac{38132}{1685}{s}^{-1}& {B}_{e}=\frac{12692}{3517}\end{array}`


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**Figure** 2.1.1-a **: Evolution of effort as a function of time, modeling A.**


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**Figure** 2.1.1-b **: Evolution of effort as a function of displacement** :math:`{U}_{1}` **, modeling A.**


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

The equations governing behavior are nonlinear differential equations. To validate the answer obtained with *Code_Aster*, an integration by a Runge-Kutta method is carried out with a tool external to *Code_Aster*.

Comparisons are made on the displacement and on the effort.


Uncertainty about the solution
---------------------------

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

*For the effort response, displacement:*

The reference solution is obtained by numerical integration of a differential system, using the Laplace transform technique. There is no uncertainty, the solution is analytical.


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

*For the effort response, displacement:*

The reference solution is obtained by numerical integration of a differential system.