Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

Direct modal calculation (without dynamic substructuring)

**Calculation of natural modes in air:**

With the option 'BANDE' of the operator CALC_MODES the first 4 natural frequencies of the air system (mass-spring system) are calculated:


.. csv-table::

    "mode 1:", "vibration of the two cylinders in phase according to :math:`\mathrm{Ox}`"
    "mode 2:", "vibration of cylinder No. 2 according to :math:`\mathrm{Oy}` (on the right)"
    "mode 3:", "vibration of the two cylinders in phase opposition according to :math:`\mathrm{Ox}`"
    "mode 4:", "vibration of cylinder No. 1 according to :math:`\mathrm{Oy}` (on the left)"


These modes can be determined analytically [:ref:`bib1 <bib1>`].

The Code_Aster calculation provides for natural frequencies in air:


.. csv-table::

    "mode 1:", ":math:`{f}_{1}=17.3555\mathrm{Hz}` ", "mode 2:", ":math:`{f}_{2}=18.2034\mathrm{Hz}`"
    "mode 3:", ":math:`{f}_{3}=42.6760\mathrm{Hz}` ", "mode 4:", ":math:`{f}_{4}=57.5418\mathrm{Hz}`"


**Calculation of the mass matrix added on a modal basis:**

On this modal basis, we calculate the 4th order added mass matrix with the operator CALC_MATR_AJOU [:external:ref:`U4.55.10 <U4.55.10>`] option 'MASS_AJOU' keyword MODE_MECA (lower triangular terms):


.. csv-table::

    ":math:`\mathrm{m11}=300.67\mathrm{kg}/m` "," :math:`\mathrm{m12}=0.001\mathrm{kg}/m`"
    ":math:`\mathrm{m13}=269.98\mathrm{kg}/m` "," :math:`\mathrm{m14}=0.009\mathrm{kg}/m`"
    ":math:`\mathrm{m22}=269.98\mathrm{kg}/m` "," :math:`\mathrm{m23}=0.009\mathrm{kg}/m`"
    ":math:`\mathrm{m24}=31.05\mathrm{kg}/m` "," :math:`\mathrm{m33}=301.71\mathrm{kg}/m`"
    ":math:`\mathrm{m34}=–0.011\mathrm{kg}/m` "," :math:`\mathrm{m44}=269.86\mathrm{kg}/m`"


**Adding this matrix to the generalized mass matrix:**

The matrix thus determined is added to the generalized mass matrix (operator COMB_MATR_ASSE [:ref:`U4.53.01 <U4.53.01>`]) and then the natural frequencies of the immersed structure are calculated with the operator CALC_MODES option 'PLUS_PETITE' [:ref:`U4.52.02 <U4.52.02>`].

The calculation finds the following natural frequencies:


.. csv-table::

    "mode 1:", ":math:`f{\text{'}}_{1}=15.8782\mathrm{Hz}` ", "mode 2:", ":math:`f{\text{'}}_{2}=16.7811\mathrm{Hz}`"
    "mode 3:", ":math:`f{\text{'}}_{3}=39.0389\mathrm{Hz}` ", "mode 4:", ":math:`f{\text{'}}_{4}=53.0488\mathrm{Hz}`"



Benchmark results
----------------------

Natural frequencies determined by Code_Aster in a direct calculation.


Bibliographical references
---------------------------


1. R. J GIBERT - Vibrations of Structures. Interactions with fluids. Eyrolles (1988).