Modeling B — Direct Transient Analysis
====


Problem and sketch of the calculation
----

We now want to calculate the movement of the plate on its vibrating table. We start with the most natural calculation, the direct calculation on a physical basis. Two calculations are carried out: without and with Rayleigh damping type.

An essential question lies in modeling the loading on the plate. Here, we choose to use a force of gravity, intensity and direction corresponding to the settings of the vibrating table.

The main steps of the command file are:


*   start:* DEBUT *


*   reading the mesh:* LIRE_MAILLAGE *


*   model construction:* AFFE_MODELE *


*   definition of materials:* DEFI_MATERIAU *


*   assignment of materials to the different groups of the mesh:* AFFE_MATERIAU *


*   boundary conditions:* AFFE_CHAR_MECA *


*   load definition:* AFFE_CHAR_MECA (PESANTEUR =_F (GRAVITE =300. , DIRECTION =( -1,0.1)) *


*   construction of the mass and stiffness matrices and the second member vector:* ASSEMBLAGE *


*   definition of time modulation (sine of period :math:`15\times 2\mathrm{\pi }` seconds):* FORMULE *


*   transitory analysis:* DYNA_VIBRA (TYPE_CALCUL =' TRAN ', BASE_CALCUL =' PHYS') *


*   We study the response of the structure up to :math:`0.5s` with a time step of :math:`0.002s` (to be entered under the keyword* INCREMENT *)


*   recovery of the movement at a point:* RECU_FONCTION *


*   curve printing:* IMPR_FONCTION *


Results
----

Let's see the results curve:


.. figure:: images/deplacement_P_DZ_transitoire_direct_sans_amor.png
    :width: 6.3071in
    :height: 4.4563in
    :name: deplacement-P-DZ-transitoire-direct-sans-amor


    Vertical displacement (DZ) in P



A sinusoidal response to the excitation frequency is observed. However, this sinusoidal response is strongly modulated by the response of the first natural mode of the plate.


Taking depreciation into account
----

It is now proposed to take depreciation into account in the context of the calculation carried out previously. To do this, we consider the Rayleigh proportional damping model, with the following parameters:

* :math:`\alpha = 0.00011575 s`: value to be entered in* AMOR_ALPHA *in* DEFI_MATERIAU *;
* :math:`\beta = 1.142397 {s}^{-1}`: value to be entered in* AMOR_BETA *in* DEFI_MATERIAU *.

These parameters correspond to a target reduced damping of 2\%, over the frequency range [5; 50] Hz.

Once these parameters have been entered in the *DEFI_MATERIAU* operator, it is necessary to assemble the damping matrix via the *ASSEMBLAGE* macro command (option *AMOR_MECA*). The transitory calculation with amortization can then be carried out via the *DYNA_VIBRA* operator, by entering the amortization matrix assembled via the *MATR_AMOR* option.

The movements obtained are then compared with those obtained without damping. The temporal evolution of the DZ component of displacements with and without Rayleigh damping is shown below:

.. figure:: images/Compa-depl_P_DZ_transitoire_direct_avec_et_sans_amor.png
    :width: 6.3071in
    :height: 4.4563in
    :name: Compa-depl-P-DZ-transitoire-direct-avec-et-sans-amor


    Comparison of vertical displacement between two calculations: without and with Rayleigh damping