Reference problem
=====================


Geometry
---------

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|                                                                                                                   |Straight pipe with a hollow circular cross section:Outer tube diameter: :math:`\mathrm{15,90 }\mathrm{mm}` Inner tube diameter: 1 :math:`\mathrm{2,72 }\mathrm{mm}`                                                                                                                           |
+ .. image:: images/1000035C000069D5000041D6D770488074B08F73.svg                                                    +                                                                                                                                                                                                                                                                                              +
|    :width: 227                                                                                                    |                                                                                                                                                                                                                                                                                              |
+    :height: 141                                                                                                   +                                                                                                                                                                                                                                                                                              +
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|                                                                                                                   |Total Tube Length: :math:`\mathrm{1,350 }m` Part Length :math:`\mathrm{AB}`: :math:`\mathrm{0,305 }m` Part Length :math:`\mathrm{BC}`: Part Length: :math:`\mathrm{0,300}m` Part Length :math:`\mathrm{CD}`: :math:`\mathrm{0,305 }m` Part Length :math:`\mathrm{DE}`: :math:`\mathrm{0,440}m`|
+ .. image:: images/1000F498000069D500002D7A9208FED44367B886.svg                                                    +                                                                                                                                                                                                                                                                                              +
|    :width: 227                                                                                                    |                                                                                                                                                                                                                                                                                              |
+    :height: 141                                                                                                   +                                                                                                                                                                                                                                                                                              +
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Material properties
------------------------

The values of the physical quantities characteristic of the elements of the structure are:

**Brass tube:**


.. csv-table::

    ":math:`E=\mathrm{1,1}{10}^{11}N/{m}^{2}` "," :math:`\nu =\mathrm{0,3}` "," :math:`\rho =8330\mathrm{kg}/{m}^{3}`"


**Internal fluid**: air

:math:`{\rho }_{i}=\mathrm{1,3}\mathrm{kg}/{m}^{3}`


.. csv-table::

    "**External fluid**:", "tube immersed in water over the part of its length between the two supports,
    tube immersed in air over the cantilever part of its length,
    transverse flow incident on part :math:`\mathrm{BC}` of the tube."


:math:`{\rho }_{e}=\mathrm{1000,0}\mathrm{kg}/{m}^{3}` or :math:`\mathrm{1,3}\mathrm{kg}/{m}^{3}` depending on the external zone

An equivalent density is attributed to the structure when calculating its modal fluid base at rest; this equivalent density includes the density of the internal fluid, that of the structure and that of the external fluid; the inertial effect of the latter is evaluated by means of an added mass coefficient.


Boundary conditions and loads
-------------------------------------

The beam is simply supported at points :math:`A` and :math:`D`. A distributed random load, transverse to the tube, is imposed on section :math:`\mathrm{BC}`. This loading is defined, on the one hand using a speed profile along the excited zone, and on the other hand using a dimensionless excitation spectrum.