D modeling
==============


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

Transposition of the reference test to the case of degrees of freedom of rotation (torsion+inertia spring) using the discrete element of translation/rotation stiffness.


.. image:: images/10000F1400001AAA000008FEEBA8DA432A18FD92.svg
    :width: 344
    :height: 115


.. _RefImage_10000F1400001AAA000008FEEBA8DA432A18FD92.svg:

Characteristics of the elements

.. csv-table::

    "ORIENTATION:", "in all the nodes", "with an angle :math:`\alpha \mathrm{=}53.130102°` ","
    "DISCRET:", "", "", ""
    "with nodal masses", "", "", ""
    "in all nodes", "M_ TR_D_N ", "in local coordinate system", ":math:`(m\mathrm{=}10.)`"
    "stiffness matrices", "", ""
    "in every mesh", "K_ TR_D_L ", "in local coordinate system", ":math:`({\mathit{KR}}_{x}\mathrm{=}{1.10}^{5})`"
    "at the end nodes", "K_ TR_D_N ", "in local coordinate system", ":math:`({\mathit{KR}}_{x}\mathrm{=}{1.10}^{5})`"
    "damping matrices", "", "", ""
    "in every mesh", "A_ TR_D_L ", "in local coordinate system", ":math:`({\mathit{CR}}_{x}\mathrm{=}50.)`"
    "at the initial node", "A_ TR_D_N ", "in local coordinate system", ":math:`({\mathit{CR}}_{x}\mathrm{=}250.)`"
    "to the final node", "A_ TR_D_N ", "in local coordinate system", ":math:`({\mathit{CR}}_{x}\mathrm{=}25.)`"


Boundary conditions:

+------------+-------------------------------------------------------------------------------------------------+
|DDL_IMPO:   | .. code::                                                                                       |
+            +                                                                                                 +
|            |    ( TOUT:’OUI’    DX: 0., DY: 0., DZ: 0., DRZ: 0. )                                            |
+------------+-------------------------------------------------------------------------------------------------+
|LIAISON_DDL:| .. code::                                                                                       |
+            +                                                                                                 +
|            |    (telle que :math:`\mathrm{3DRy}\mathrm{=}\mathrm{4DRx}`en tous les nœuds)                    |
+------------+-------------------------------------------------------------------------------------------------+

Node names: :math:`{P}_{\mathrm{1,}}{P}_{\mathrm{2,}}\mathrm{....},{P}_{8}`


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


.. csv-table::

    "Number of knots:", "8"
    "Number of meshes and types:", "7 SEG2"


Nodes :math:`{P}_{1}` and :math:`{P}_{8}` are connected to a fixed fictional node by nodal springs (K_ TR_N, A_ TR_N).


Contents of the results file
----------------------------

Results obtained with:

.. code-block:: text

    CALC_FREQ: (LIST_FREQ: (6., 10., 15., 15., 15., 15., 15., 19., 19., 19., 24., 29., 31.))

    CALC_MODE: (NMAX_MODE: 75)


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


.. csv-table::

    "**Frequency**", "**Reference**"
    "Clean mode order 1", "5.53"
    "Clean mode order 2", "10.90"
    "Clean mode order 3", "15.93"
    "Clean mode order 4", "20.45"
    "Clean mode order 5", "24.34"
    "Clean mode order 6", "27.49"
    "Clean mode order 7", "29.84"
    "Clean mode order 8", "31.29"



.. csv-table::

    "**Depreciation**", "**Reference**"
    "Clean mode order 1", "1.521e—2"
    "Clean mode order 2", "2.877e—2"
    "Clean mode order 3", "3.960e—2"
    "Clean mode order 4", "4.709e—2"
    "Clean mode order 5", "5.098e—2"
    "Clean mode order 6", "5.183e—2"
    "Clean mode order 7", "5.115e—2"
    "Clean mode order 8", "5.036e—2"



.. csv-table::

    "**Nature of clean mode**", "**Point**", "**Clean mode Reference in 10** **—3**
    **Real part**
    **Imaginary part**"
    "Rotation 1
    (:math:`\mathit{DRx}`)
    :math:`{\Phi }_{1}` ", "P1
    P2
    P3
    P4
    P5
    P6
    P7
    P8", "—2.442, 2.736
    —4.782, 4.968
    —6.54, 6.6
    —7.5, 7.5
    —7.5, 7.4
    —6.66, 6.54
    —4.944, 4.824
    —2.646, 2.55"
    "Rotation 8
    (:math:`\mathit{DRx}`)
    :math:`{\Phi }_{8}` ", "P1
    P2
    P3
    P4
    P5
    P6
    P7
    P8", "—1.338, 0.684
    —2.226, 1.788
    —2.85, 2.646
    —3.15, 3.162
    —3.084, 3.258
    —2.664, 2.928
    —1.938, 2.214
    —0.996, 1.206"


Eigenmode standardized to the unit modal mass: :math:`{\phi }_{i}^{t}C{\phi }_{i}+2{\lambda }_{i}{\phi }_{i}^{t}M{\phi }_{i}\mathrm{=}1`

:math:`\lambda` is the eigenvalue associated with damping and natural frequency.


Contents of the results file
----------------------------

The 8 depreciations and natural frequencies, as well as the associated natural vectors.