B modeling
==============


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

Discrete elements of stiffness, damping and mass.


.. image:: images/Object_10.svg
    :width: 300
    :height: 109


.. _RefImage_Object_10.svg:

Characteristics of the elements:


.. csv-table::

    "DISCRET:", "nodal mass", "M_T_D_N"
    "", "linear stiffness", "K_T_D_L (:math:`{k}_{\mathrm{N1N2}}=\mathrm{10k}`, :math:`{k}_{\mathrm{N2N3}}=k/10`)"
    "", "linear amortization", "A_T_D_L"


Boundary conditions: at node :math:`\mathrm{N1}` DDL_IMPO :math:`\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=0`.

Node names: :math:`A=\mathrm{N1}`, :math:`C=\mathrm{N2}`, :math:`B=\mathrm{N3}`.

Calculation methods:


* Integration into physical space with Newmark (:math:`\alpha =\mathrm{0,25}`, :math:`\delta =\mathrm{0,5}`)

No time :math:`\Delta t={10}^{-3}s`


* Full modal integration with Euler

No time :math:`\Delta t={10}^{-3}s` then modal recombination


* Full modal integration with 2nd order adaptive :math:`\Delta t`

No initial time :math:`\Delta t={10}^{-3}s` then modal recombination


* Integration on a complete modal basis with :math:`\Delta t` adaptive by the Runge-Kutta order method (32). Relative error tolerance is :math:`{10}^{-5}`.


* Full modal integration with :math:`\Delta t` adaptive using the Runge-Kutta order method (54). Relative error tolerance is :math:`{10}^{-6}`.

Observation time: :math:`\mathrm{2,5}s`.


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

Number of knots: 3

Number of meshes and type: 2 meshes SEG2


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


* Move (:math:`m`) from point :math:`B`


.. csv-table::

    "", ""
    "**Time**", "**Reference**"
    "(:math:`s`)", ""
    "0.19", "2.9334 E-3"
    "0.38", "1.0959 E-3"
    "0.57", "2.2468 E-3"
    "0.76", "1.5260 E-3"
    "0.95", "1.9773 E-3"
    "1.19", "-1.2107 E-3"
    "1.38", "7.5880 E-4"
    "1.57", "-4.7553 E-4"
    "1.76", "2.9796 E-4"
    "1.95", "-1.8668 E-4"
    "2.14", "1.1694 E-4"
    "2.33", "-7.3246 E-5"



* Speed (:math:`{\mathrm{m.s}}^{-1}`) of point :math:`B`


.. csv-table::

    "", ""
    "**Time**", "**Reference**"
    "(:math:`s`)", ""
    "0.09", "2.4261 E-2"
    "0.28", "-1.5210 E-2"
    "0.47", "9.5332 E-3"
    "0.66", "-5.9745 E-3"
    "0.85", "3.7438 E-3"
    "1.08", "-2.6037 E-2"
    "1.27", "1.6302 E-2"
    "1.46", "-1.0204 E-2"
    "1.66", "6.387-E-3"
    "1.85", "-4.0059 E-3"
    "2.04", "2.5114 E-3"
    "2.23", "-1.5743 E-3"
    "2.42", "9.8676 E-4"



notes
---------

The results are tested at the level of the respective peaks of movement and speed where the values are the most significant.