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


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

For the calculation of the added coefficients:

we show [:ref:`bib1 <bib1>`] that the added mass and damping coefficients depend on the permanent potential of the fluid velocities :math:`\stackrel{ˉ}{\phi }` as well as on two fluctuating potentials :math:`{\phi }_{1}` and :math:`{\phi }_{2}`: these potentials can be written in the case of the flexural movement of the plate [:ref:`bib1 <bib1>`]:

For the first mode:

.. image:: images/Object_14.svg
    :width: 178
    :height: 141


.. _RefImage_Object_14.svg:

For the second mode:

.. image:: images/Object_15.svg
    :width: 178
    :height: 141


.. _RefImage_Object_15.svg:

However, the added modal coefficients projected onto these flexure modes can be written as:


.. image:: images/Object_16.svg
    :width: 178
    :height: 141


.. _RefImage_Object_16.svg:

either:


.. image:: images/Object_17.svg
    :width: 178
    :height: 141


.. _RefImage_Object_17.svg:


* Digital applications:

An added damping calculation was made which corresponds, for the given speed, to a damped vibratory behavior of the structure:

Speed :math:`{V}_{0}` to :math:`4{\mathrm{m.s}}^{-1}`

The values of the mechanical system are:


.. image:: images/Object_19.svg
    :width: 178
    :height: 141


.. _RefImage_Object_19.svg:

The mass added by the flow is equal to:


.. image:: images/Object_20.svg
    :width: 178
    :height: 141


.. _RefImage_Object_20.svg:

The depreciation added is equal with :math:`{V}_{0}=4{\mathrm{m.s}}^{-1}`:


.. image:: images/Object_22.svg
    :width: 178
    :height: 141


.. _RefImage_Object_22.svg:

The added stiffness applies with :math:`{V}_{0}=4{\mathrm{m.s}}^{-1}`:


.. image:: images/Object_24.svg
    :width: 178
    :height: 141


.. _RefImage_Object_24.svg:


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

Analytical result.


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


*ROUSSEAU G., LUU H.T.: Mass, damping and stiffness added for a vibrating structure placed in a potential flow - Bibliography and implementation in the*Code_Aster* - HP-61/95/064


* BLEVINS R.D: Formulas for natural frequency and mode shape. Ed. Krieger 1984