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


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

The reference solution is an approximate analytical solution. The approximate analytical fluctuating potentials to calculate the added stiffness are written as [:ref:`bib1 <bib1>`]:

:math:`\mathrm{\{}\begin{array}{ccc}{\varphi }_{1}(r,\theta ,y)\mathrm{=}& \frac{{R}_{e}^{2}}{{R}_{e}^{2}\mathrm{-}{R}_{i}^{2}}(r+\frac{{R}_{i}^{2}}{r})& \mathrm{sin}\theta \mathrm{sin}\frac{\pi (y+l\mathrm{/}2)}{l}\\ {\varphi }_{2}(r,\theta ,y)\mathrm{=}& \frac{{R}_{e}^{2}}{{R}_{e}^{2}\mathrm{-}{R}_{i}^{2}}(r+\frac{{R}_{i}^{2}}{r})& \mathrm{sin}\theta \mathrm{cos}\frac{\pi (y+l\mathrm{/}2)}{l}\end{array}`

The stiffness added to the first mode of flexure of the outer cylinder considered as a swivel and swivel beam is written [:ref:`bib1 <bib1>`]:

:math:`{K}_{A}\mathrm{=}\mathrm{-}\frac{\rho }{2}\frac{{V}_{0}^{2}{\pi }^{3}{R}_{e}^{3}}{({R}_{e}^{2}\mathrm{-}{R}_{i}^{2})l}({R}_{e}+\frac{{R}_{i}^{2}}{{R}_{e}})`

This stiffness, calculated on a cylindrical geometry, is then assigned to a model with an equivalent degree of freedom.

The system with an equivalent degree of freedom is an equivalent mass-spring system to which is assigned a mass equal to the mass of the system increased by the mass added by the fluid and a stiffness equal to the stiffness of the system increased by the stiffness added by the flow for various speeds.

The mass of the system in air is:

:math:`M\mathrm{=}10292\mathit{kg}`

for an external cylindrical shell with a thickness:

:math:`C\mathrm{=}{2.10}^{\mathrm{-}3}m`

For equivalent stiffness in air of the "outer shell" system, we take the stiffness of a beam subjected to a force distributed over its entire length:


.. image:: images/10000D66000027610000052B117D0A01DB11CF31.svg
    :width: 508
    :height: 66


.. _RefImage_10000D66000027610000052B117D0A01DB11CF31.svg:

So :math:`K\mathrm{=}2.533{10}^{4}N\mathrm{/}m`

The equivalent system coupled to the flow is represented by the following diagram:


.. image:: images/1000027200000B0F0000052BB5A3FD7B4A90F759.svg
    :width: 508
    :height: 66


.. _RefImage_1000027200000B0F0000052BB5A3FD7B4A90F759.svg:

with :math:`m\mathrm{=}M+{M}_{A}` :math:`k\mathrm{=}K+{K}_{A}`

The natural pulsation of the coupled system evolves according to the square of the flow speed. If we call :math:`{V}_{{0}_{C}}` the critical flow speed for which the stiffness :math:`k` is cancelled out:

:math:`\mathrm{\exists }{V}_{{0}_{C}},K+{K}_{A}({V}_{{0}_{C}})\mathrm{=}0` with :math:`{V}_{{0}_{C}}^{2}\mathrm{=}\frac{2({R}_{e}^{2}\mathrm{-}{R}_{i}^{2})lK}{({R}_{e}+\frac{{R}_{i}^{2}}{{R}_{e}})\rho {\pi }^{3}{R}_{e}^{3}}`

so we show that:

:math:`\omega ({V}_{0})\mathrm{=}{\omega }_{e}(0)\sqrt{1\mathrm{-}{x}^{2}}`

where we put:

:math:`{\omega }_{e}(0)\mathrm{=}\sqrt{\frac{K}{M+{M}_{A}}}` (natural pulsation of the system in fluid at rest)

:math:`x\mathrm{=}\frac{{V}_{0}}{{V}_{{0}_{C}}}` (reduced flow speed)

The pulsation of the fluid at rest is equal to: :math:`\omega \mathrm{=}0.085\mathit{rad}\mathrm{/}s`.


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

The natural frequency of the system is calculated for different flow velocities.


.. csv-table::

    ":math:`{V}_{0}(m\mathrm{/}s)` ", "0.5", "1.5", "2. ", "2.2", "2.688"
    ":math:`{M}_{A}(\mathit{kg})` ", "3.486E6", "3.486E6", "3.486E6", "3.486E6", "3.486E6", "3.486E6"
    ":math:`{K}_{A}(N/m)` ", "—876.5", "—7888.50", "—7888.50", "—14023.95", "—16968.98", "—25330"
    ":math:`{M}_{\mathrm{total}}(\mathrm{kg})` ", "3.491E+6", "=", "=", "=", "=", "=", "="
    ":math:`{K}_{\mathit{total}}(N\mathrm{/}m)` ", "24453.5", "17441.5", "17441.5", "11306.05", "8361.00", "0."
    ":math:`f({V}_{0})\mathrm{\times }{10}^{\mathrm{-}2}(\mathit{Hz})` ", "1.318", "1.112", "1.112", "0.896", "0.772", "0."



Uncertainty about the solution
---------------------------

Semi-analytical solution.


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


1. ROUSSEAU G., LUU H.T. - Mass, damping and stiffness added for a vibrating structure placed in a potential flow - Internal note EDF/DER, HP-61/95/064/A (1995).