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


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

To calculate the added coefficients:

we show [:ref:`bib1 <bib1>`] that the added mass and damping coefficients depend on the permanent potential of the fluid speeds :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 rotational movement of the external cylinder around the pivot C [:ref:`bib1 <bib1>`]:


.. image:: images/Object_12.svg
    :width: 382
    :height: 166


.. _RefImage_Object_12.svg:

However, the added modal coefficients projected onto this mode of rotation can be written as:


.. image:: images/Object_13.svg
    :width: 382
    :height: 166


.. _RefImage_Object_13.svg:

either:


.. image:: images/Object_14.svg
    :width: 382
    :height: 166


.. _RefImage_Object_14.svg:

For the system with an equivalent degree of freedom:

It is a mass-spring-shock absorber system representing the rotational movement of the cylinder around the downstream pivot :math:`C`.


.. image:: images/1000032C00000CEB000008450E8D38D607DC8481.svg
    :width: 382
    :height: 166


.. _RefImage_1000032C00000CEB000008450E8D38D607DC8481.svg:


* The inertia of the mechanical system subjected to the flow is written as:

    .. image: images/Object_15.svg
        :width: 382
        :height: 166


    .. _RefImage_Object_15.svg:

where

.. image:: images/Object_16.svg
    :width: 382
    :height: 166


.. _RefImage_Object_16.svg:

is the inertia of the pivoting external cylinder with respect to the axis :math:`\mathrm{Cx}` (see figure below) in air.

We show [:ref:`bib2 <bib2>`] that this inertia is equal to:


.. image:: images/Object_17.svg
    :width: 382
    :height: 166


.. _RefImage_Object_17.svg:

where

.. image:: images/Object_18.svg
    :width: 382
    :height: 166


.. _RefImage_Object_18.svg:

is the mass of the cylinder:


.. image:: images/Object_19.svg
    :width: 382
    :height: 166


.. _RefImage_Object_19.svg:

where

.. image:: images/Object_20.svg
    :width: 382
    :height: 166


.. _RefImage_Object_20.svg:

is the thickness of the cylinder,

.. image:: images/Object_21.svg
    :width: 382
    :height: 166


.. _RefImage_Object_21.svg:

its total length.


.. image:: images/Object_22.svg
    :width: 382
    :height: 166


.. _RefImage_Object_22.svg:

is the density of the cylinder.


.. image:: images/10002806000024CB000009180AB86993A0037FC4.svg
    :width: 382
    :height: 166


.. _RefImage_10002806000024CB000009180AB86993A0037FC4.svg:

thus

.. image:: images/Object_23.svg
    :width: 382
    :height: 166


.. _RefImage_Object_23.svg:


* the damping of the mechanical system subjected to the flow is written as:

    .. image: images/Object_24.svg
        :width: 382
        :height: 166


    .. _RefImage_Object_24.svg:

where

.. image:: images/Object_25.svg
    :width: 382
    :height: 166


.. _RefImage_Object_25.svg:

is the damping of the mechanical system in air. Usually,

.. image:: images/Object_26.svg
    :width: 382
    :height: 166


.. _RefImage_Object_26.svg:

is equal to a few% of the critical damping of the system:

.. image:: images/Object_27.svg
    :width: 382
    :height: 166


.. _RefImage_Object_27.svg:

.

where

.. image:: images/Object_28.svg
    :width: 382
    :height: 166


.. _RefImage_Object_28.svg:

is the inertia of the air cylinder calculated above and

.. image:: images/Object_29.svg
    :width: 382
    :height: 166


.. _RefImage_Object_29.svg:

spring stiffness at the pivot point

.. image:: images/Object_30.svg
    :width: 382
    :height: 166


.. _RefImage_Object_30.svg:

. We take the reduced depreciation

.. image:: images/Object_31.svg
    :width: 382
    :height: 166


.. _RefImage_Object_31.svg:

equal to 1%.

Thus, the total damping of the system under flow is written as:


.. image:: images/Object_32.svg
    :width: 382
    :height: 166


.. _RefImage_Object_32.svg:


* the stiffness of the mechanical system subjected to flow is written as:

    .. image: images/Object_33.svg
        :width: 382
        :height: 166


    .. _RefImage_Object_33.svg:

where

.. image:: images/Object_34.svg
    :width: 382
    :height: 166


.. _RefImage_Object_34.svg:

is the stiffness of the spring in air.

.. image:: images/Object_35.svg
    :width: 382
    :height: 166


.. _RefImage_Object_35.svg:

is the stiffness added by the flow. We show [:ref:`bib1 <bib1>`] that it is zero in this rotation mode.


.. image:: images/Object_36.svg
    :width: 382
    :height: 166


.. _RefImage_Object_36.svg:

Thus the total rigidity of the system is independent of the flow speed.


.. image:: images/Object_37.svg
    :width: 382
    :height: 166


.. _RefImage_Object_37.svg:


* We then calculate the complex modes of this mechanical system under flow (damped free vibrations):


.. image:: images/Object_38.svg
    :width: 382
    :height: 166


.. _RefImage_Object_38.svg:

The complex natural frequencies of this system are written as [:ref:`bib3 <bib3>`]:


.. image:: images/Object_39.svg
    :width: 382
    :height: 166


.. _RefImage_Object_39.svg:

with

.. image:: images/Object_40.svg
    :width: 382
    :height: 166


.. _RefImage_Object_40.svg:


.. image:: images/Object_41.svg
    :width: 382
    :height: 166


.. _RefImage_Object_41.svg:

: reduced system damping


.. image:: images/Object_42.svg
    :width: 382
    :height: 166


.. _RefImage_Object_42.svg:

: natural pulsation.


* Digital applications:

Three calculations of added damping were made corresponding to three flow velocities that cause three vibratory behaviors of the structure:

Speed to :math:`4m/s`

Speed to :math:`4.24m/s`

Speed to :math:`6m/s`

The values of the mechanical system are:


.. image:: images/Object_43.svg
    :width: 382
    :height: 166


.. _RefImage_Object_43.svg:

The added weight and damping provided by the flow are equal to:


.. image:: images/Object_44.svg
    :width: 382
    :height: 166


.. _RefImage_Object_44.svg:

(independent of the flow speed value)

Depending on the fluid entry speed, we have:


.. image:: images/Object_45.svg
    :width: 382
    :height: 166


.. _RefImage_Object_45.svg:

The amortization of the fluid/structure system is written as:


* to

    .. image: images/Object_46.svg
        :width: 382
        :height: 166


    .. _RefImage_Object_46.svg:

The flow does not amplify the vibrations. The internal structural damping is significant enough to dissipate the energy provided by the flow to the structure. The system is still depreciated.


* to

    .. image: images/Object_47.svg
        :width: 382
        :height: 166


    .. _RefImage_Object_47.svg:

The depreciation of the system is cancelled out.


* to

    .. image: images/Object_48.svg
        :width: 382
        :height: 166


    .. _RefImage_Object_48.svg:

The damping of the system at this last speed is negative: the system then enters **floating instability**.

The corresponding reduced amortization is written as:


.. image:: images/Object_49.svg
    :width: 382
    :height: 166


.. _RefImage_Object_49.svg:

The natural pulsation remains unchanged:

.. image:: images/Object_50.svg
    :width: 382
    :height: 166


.. _RefImage_Object_50.svg:

.


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

Analytical result.


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


1. 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


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


3. SELIGMANN D, MICHEL R: Solving Algorithms for the Quadratic Problem [:ref:`R5.01.02 <R5.01.02>`], *Aster* Reference Manual.