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


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

The system is modelled by:


* 3 discrete elements K_T_D_L,


* 2 discrete M_T_D_N elements.


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

The mesh consists of 3 SEG2 meshes.


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


Natural frequencies
~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**MODE**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`1` "," :math:`1.000E+00` "," :math:`0.1`"
    ":math:`2` "," :math:`2.236E+00` "," :math:`0.1`"



Static training modes
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Fashion :math:`\mathrm{1 }`: absolute movements :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "Tolerance :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`0.4E+00` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`0.6E+00` "," :math:`0.1`"


Fashion :math:`\mathrm{2 }`: absolute movements :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`0.4E+00` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`0.6E+00` "," :math:`0.1`"



Static modes for static correction
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Fashion :math:`\mathrm{1 }`: absolute movements :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`1.317E-02` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`1.216E-02` "," :math:`0.1`"


Fashion :math:`\mathrm{2 }`: absolute movements :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`1.216E-02` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`1.317E-02` "," :math:`0.1`"



Global response on a complete modal basis (uncorrelated multi-support calculation)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The :math:`1` and :math:`2` modes are taken into account.


* **calculation** :math:`n°1`

COMB_MODE =' SRSS '

             For each active degree of freedom :math:`2` and :math:`\mathrm{3 }`:


* response from support :math:`j=1` (node :math:`\mathrm{NO1}`): :math:`{R}_{1}=\sqrt{{\mathrm{Rm}}_{11}^{2}+{\mathrm{Rm}}_{21}^{2}}` (cumulative on modes :math:`1` and :math:`2`)


* response from support :math:`j=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}=\sqrt{{\mathrm{Rm}}_{12}^{2}+{\mathrm{Rm}}_{22}^{2}}` (cumulative on modes :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{1}^{2}+{R}_{2}^{2}}` (cumulative support)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`5.65E-03` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`5.65E-03` "," :math:`0.1`"



* **calculation** :math:`n°2`

COMB_MODE =' ABS '


* response from support :math:`j=1` (node :math:`\mathrm{NO1}`): :math:`{R}_{1}=∣{\mathrm{Rm}}_{11}∣+∣{\mathrm{Rm}}_{21}∣` (cumulative on modes :math:`1` and :math:`2`)


* response from support :math:`j=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}=∣{\mathrm{Rm}}_{12}∣+∣{\mathrm{Rm}}_{22}∣` (cumulative on modes :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{1}^{2}+{R}_{2}^{2}}` (cumulative support)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`6.476E-03` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`6.476E-03` "," :math:`0.1`"



* **calculation** :math:`n°3`

COMB_MODE =' DPC '


* response from support :math:`j=1` (node :math:`\mathrm{NO1}`): :math:`{R}_{1}=\sqrt{{\mathit{Rm}}_{11}^{2}+{\mathit{Rm}}_{21}^{2}}` (cumulative on modes :math:`1` and :math:`2`)


* response from support :math:`j=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}=\sqrt{{\mathrm{Rm}}_{12}^{2}+{\mathrm{Rm}}_{22}^{2}}` (cumulative on modes :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{1}^{2}+{R}_{2}^{2}}` (cumulative support)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`5.65E-03` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`5.65E-03` "," :math:`0.1`"



* **calculation** :math:`n°4`

COMB_MODE =' CQC '

modal depreciation = :math:`0.05`


* response from support :math:`j=1` (node :math:`\mathrm{NO1}`): :math:`{R}_{1}=\sqrt{{\rho }_{12}{\mathrm{Rm}}_{11}{\mathrm{Rm}}_{21}}` (cumulative on modes :math:`1` and :math:`2`)


* response from support :math:`j=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}=\sqrt{{\rho }_{12}{\mathrm{Rm}}_{12}{\mathrm{Rm}}_{22}}` (cumulative on modes :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{1}^{2}+{R}_{2}^{2}}` (cumulative support)

absolute displacements: :math:`\mathit{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathit{NO2}` "," :math:`5.65E-03` "," :math:`0.1`"
    ":math:`\mathit{NO3}` "," :math:`5.65157E-03` "," :math:`0.1`"



* **calculation** :math:`n°5`

COMB_MODE =' DSC '

modal depreciation = :math:`0.05`

duration: 15 s


* response from support :math:`j=1` (node :math:`\mathrm{NO1}`): :math:`{R}_{1}=\sqrt{{\rho }_{12}{\mathrm{Rm}}_{11}{\mathrm{Rm}}_{21}}` (cumulative on modes :math:`1` and :math:`2`)


* response from support :math:`j=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}=\sqrt{{\rho }_{12}{\mathrm{Rm}}_{12}{\mathrm{Rm}}_{22}}` (cumulative on modes :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{1}^{2}+{R}_{2}^{2}}` (cumulative support)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`5.649E-03` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`5.6521E-03` "," :math:`0.1`"



Global response on a complete modal basis (single-support calculation with :math:`{\mathit{SRO}}_{\mathit{NO}1}`)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~

The :math:`1` and :math:`2` modes are taken into account.


* **calculation** :math:`n°1`

COMB_MODE =' SRSS '

             For each active :math:`\mathrm{ddl}` :math:`2` and :math:`\mathrm{3 }`:


* :math:`\mathrm{1 }` mode response: :math:`{R}_{1}={\mathrm{Rm}}_{11}+{\mathrm{Rm}}_{12}`


* :math:`\mathrm{2 }` mode response: :math:`{R}_{2}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}`


* global response: :math:`R=\sqrt{{R}_{1}^{2}+{R}_{2}^{2}}` (cumulative mode)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`1.01321E-02` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`1.01321E-02` "," :math:`0.1`"



* **calculation** :math:`n°2`

COMB_MODE =' ABS '


* :math:`\mathrm{1 }` mode response: :math:`{R}_{1}={\mathrm{Rm}}_{11}+{\mathrm{Rm}}_{12}`


* :math:`\mathrm{2 }` mode response: :math:`{R}_{2}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}` r


* global response: :math:`R=∣{R}_{1}∣+∣{R}_{2}∣` (cumulative mode)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`0.01013` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`0.01013` "," :math:`0.1`"



* **calculation** :math:`n°3`

COMB_MODE =' DPC '


* :math:`\mathrm{1 }` mode response: :math:`{R}_{1}={\mathrm{Rm}}_{11}+{\mathrm{Rm}}_{12}`


* :math:`\mathrm{2 }` mode response: :math:`{R}_{2}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}`


* global response: :math:`R=\sqrt{{R}_{1}^{2}+{R}_{2}^{2}}` (cumulative mode)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathrm{NO2}` "," :math:`0.01013` "," :math:`0.1`"
    ":math:`\mathrm{NO3}` "," :math:`0.01013` "," :math:`0.1`"



* **calculation** :math:`n°4`

COMB_MODE =' CQC '

modal depreciation = :math:`0.05`


* :math:`\mathrm{1 }` mode response: :math:`{R}_{1}={\mathrm{Rm}}_{11}+{\mathrm{Rm}}_{12}`


* :math:`\mathrm{2 }` mode response: :math:`{R}_{2}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}`


* global response: :math:`R=\sqrt{{\rho }_{12}{R}_{1}{R}_{2}}` (cumulative mode)

absolute displacements: :math:`\mathit{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathit{NO2}` "," :math:`0.01013` "," :math:`0.1`"
    ":math:`\mathit{NO}3` "," :math:`0.01013` "," :math:`0.1`"



* **calculation** :math:`n°5`

COMB_MODE =' DSC '

modal depreciation = :math:`0.05`

duration: 15 seconds


* :math:`1` mode response: :math:`{R}_{1}={\mathit{Rm}}_{11}+{\mathit{Rm}}_{12}`


* :math:`2` mode response: :math:`{R}_{2}={\mathit{Rm}}_{21}+{\mathit{Rm}}_{22}`


* global response: :math:`R=\sqrt{{\rho }_{12}{R}_{1}{R}_{2}}` (cumulative mode)

absolute displacements: :math:`\mathit{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance** :math:`(\text{\%})`"
    ":math:`\mathit{NO}2` "," :math:`0.01013` "," :math:`0.1`"
    ":math:`\mathit{NO}3` "," :math:`0.01013` "," :math:`0.1`"



Overall response on an incomplete modal basis (single-support calculation with static correction)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

     Modal base consisting of mode 2 only.


* **calculation** :math:`n°1`

COMB_MODE =' ABS '

             For each active degree of freedom :math:`2` and :math:`3`:


* response from mode :math:`i=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}` (cumulative support :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{2}^{2}+{U}^{2}}` (cumulative modal response and static correction)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance**"
    ":math:`\mathrm{NO2}` ", "0.02302302705"," :math:`0.001`"
    ":math:`\mathrm{NO3}` ", "0.02302302705"," :math:`0.001`"



* **calculation** :math:`n°2`

COMB_MODE =' SRSS '

             For each active degree of freedom :math:`2` and :math:`3`:


* response from mode :math:`i=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}` (cumulative support :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{2}^{2}+{U}^{2}}` (cumulative modal response and static correction)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance**"
    ":math:`\mathrm{NO2}` ", "0.02302302705"," :math:`0.001`"
    ":math:`\mathrm{NO3}` ", "0.02302302705"," :math:`0.001`"



* **calculation** :math:`n°3`

COMB_MODE =' DPC '

             For each active degree of freedom :math:`2` and :math:`3`:


* response from mode :math:`i=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}` (cumulative support :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{2}^{2}+{U}^{2}}` (cumulative modal response and static correction)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance**"
    ":math:`\mathrm{NO2}` ", "0.02302302705"," :math:`0.001`"
    ":math:`\mathrm{NO3}` ", "0.02302302705"," :math:`0.001`"



* **calculation** :math:`n°4`

COMB_MODE =' CQC '

modal depreciation = :math:`0.05`

             For each active degree of freedom :math:`2` and :math:`3`:


* response from mode :math:`i=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}` (cumulative support :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{2}^{2}+{U}^{2}}` (cumulative modal response and static correction)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance**"
    ":math:`\mathrm{NO2}` ", "0.02302302705"," :math:`0.001`"
    ":math:`\mathrm{NO3}` ", "0.02302302705"," :math:`0.001`"



* **calculation** :math:`n°5`

COMB_MODE =' DSC '

             For each active degree of freedom :math:`2` and :math:`3`:


* response from mode :math:`i=2` (node :math:`\mathrm{NO4}`): :math:`{R}_{2}={\mathrm{Rm}}_{21}+{\mathrm{Rm}}_{22}` (cumulative support :math:`1` and :math:`2`)


* global response: :math:`R=\sqrt{{R}_{2}^{2}+{U}^{2}}` (cumulative modal response and static correction)

absolute displacements: :math:`\mathrm{DEPL}`


.. csv-table::

    "**NOEUD**", "**Reference**", "**Tolerance**"
    ":math:`\mathrm{NO2}` ", "0.02302302705"," :math:`0.001`"
    ":math:`\mathrm{NO3}` ", "0.02302302705"," :math:`0.001`"