Modeling A
==============


Characteristics of modeling
-----------------------------------

We use the Euler Bernouilli POU_D_E beam element


.. image:: images/1000113E000019F1000011286EA9EF6A07FE2E5D.svg
    :width: 334
    :height: 221


.. _RefImage_1000113E000019F1000011286EA9EF6A07FE2E5D.svg:


.. csv-table::

    "3 beams:", ":math:`\mathrm{ABC}`, :math:`\text{DEF}`, :math:`\mathrm{HGI}` each cut into 10 meshes SEG2
    Nodes :math:`(B,H)` and :math:`(E,I)` have the same coordinates."



+-------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Boundary conditions:                                         |                                                                                                                                                                                                       |
+-------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|beams :math:`\mathrm{ABC}` and :math:`\text{DEF}`            |                                                                                                                                                                                                       |
+-------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|DDL_IMPO:                                                    | .. code::                                                                                                                                                                                             |
+                                                             +                                                                                                                                                                                                       +
| beam :math:`\mathrm{HGI}` knots ends                        |    ( GROUP_NO: (PABC, PDEF)    DX: 0., DY: 0., DRY: 0. )                                                                                                                                              |
+                                                             +                                                                                                                                                                                                       +
|                                                             |    ( GROUP_NO: (PHGI)            DX: 0., DY: 0., DRX: 0. )                                                                                                                                            |
+                                                             +                                                                                                                                                                                                       +
|                                                             |    ( GROUP_NO: (NACDF)        DZ: 0. )                                                                                                                                                                |
+-------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                             |                                                                                                                                                                                                       |
+-------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|DDL_link:                                                    |:math:`{\mathrm{DZ}}_{B}–{\mathrm{DZ}}_{H}=0.`                                                                                                                                                         |
+                                                             + and                                                                                                                                                                                                   +
| nodal_force:                                                | :math:`{\mathrm{DZ}}_{E}–{\mathrm{DZ}}_{I}=0.`                                                                                                                                                        |
+                                                             +                                                                                                                                                                                                       +
|                                                             | Node: :math:`G` :math:`\mathrm{Fz}`: :math:`–1.E5`                                                                                                                                                    |
+                                                             +                                                                                                                                                                                                       +
|                                                             |                                                                                                                                                                                                       |
+-------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+



.. csv-table::

    "Node names:", ":math:`A=\mathrm{N1}` "," :math:`B=\mathrm{N6}` "," :math:`C=\mathrm{N11}`"
    "", ":math:`D=\mathrm{N21}` "," :math:`E=\mathrm{N26}` "," :math:`F=\mathrm{N31}`"
    "", ":math:`H=\mathrm{N41}` "," :math:`G=\mathrm{N46}` "," :math:`I=\mathrm{N51}`"



Characteristics of the mesh
----------------------------


.. csv-table::

    "Number of knots:", "33"
    "Number of meshes and types:", "3*10 = 30 SEG2"



notes
---------

The blocking of the degrees of freedom :math:`\mathrm{DX}` and :math:`\mathrm{DY}` at all the nodes makes it possible to select only the transverse flexure modes (in the "vertical" plane).


Tested sizes and results
------------------------------

**Frequency (** :math:`\mathrm{Hz}` **)**


.. csv-table::

    "**Clean mode order**", "**Reference**", "**Aster**", "**% difference**"
    "1
    2", "16.456
    38.165", "16.4190
    38.0468", "—0.22
    —0.31"


**Clean mode: value of** :math:`{W}_{B}/{W}_{G}`


.. csv-table::

    "**Symmetric eigenmode order**", "**Reference**", "**Aster***", "**% difference**"
    "1
    2", "1.213
    —0.412", "1.213
    —0.412", "0.
    0."



.. csv-table::

    "*", ":math:`{W}_{B}=\mathrm{DZ}` in :math:`B` (:math:`\mathrm{N6}`)", ":math:`{W}_{G}+{W}_{B}=\mathrm{DZ}` in :math:`G` (:math:`\mathrm{N46}`)"
    "mode 1:", ":math:`{W}_{B}=0.5480` "," :math:`{W}_{G}+{W}_{B}=1.`"
    "mode 2:", ":math:`{W}_{B}=–0.6698` "," :math:`{W}_{G}+{W}_{B}=0.9559`"


**Harmonic response:**


.. csv-table::

    "**Point**", "**Value type**
    **(** :math:`m` **)**", "**Reference**", "**Aster**", "**% difference**"
    ":math:`B,E`
    :math:`G`
    :math:`G` "," :math:`{W}_{B}\mathrm{max}`
    :math:`{W}_{G}\mathrm{max}\ast`
    :math:`{W}_{B}+{W}_{G}\mathrm{max}` ", "—0.098
    —0.125
    —0.227", "—0.1003
    —0.1271
    —0.2274", "2.45
    1.60
    0.18"



notes
---------

Calculations made by:


.. code-block:: text

    CALC_MODES
OPTION = 'PLUS_PETITE'
CALC_FREQ =_F (NMAX_FREQ = 3)
SOLVEUR_MODAL =_F (METHODE = 'TRI_DIAG')


An antisymmetric mode is obtained for a frequency :math:`f=22.5676\mathrm{Hz}`. This natural frequency depends on the torsional constant provided; this is not defined in the reference data.

The :math:`{W}_{B}/{W}_{G}` values are not verified in the test but are obtained manually from :math:`{W}_{B}` and :math:`{W}_{G}+{W}_{B}`.

The (WG) max value is not checked in the test. We only have access to :math:`{W}_{B}\mathrm{max}` and :math:`({W}_{B}+{W}_{G})\mathrm{max}`. :math:`{W}_{G}\mathrm{max}` is obtained manually by difference.

**Contents of the results file:**

First 3 natural frequencies, displacement of the :math:`B,E,G` nodes in harmonic response.