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

The natural modes of the system are calculated after having verified those of each substructure. The mass added to the air modes is then evaluated.


Breakdown into substructures
--------------------------------

**First substructure: intermediate cylinder**

The first substructure consists of the intermediate cylinder and four springs of :math:`{k}_{\mathrm{2 }}={10}^{7}{\mathrm{N.m}}^{-1}` stiffness. These springs are embedded at the interface with the external cylinder which constitutes the second substructure (interface type CRAIG - BAMPTON).


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Mass of cylinder 1: :math:`{m}_{1}=2.041{10}^{6}\mathrm{kg}`

Since the cylinder is rigid, its movement can be modelled by a mass-spring system with one degree of freedom:


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At each end, the cylinder is connected to two springs in parallel: the equivalent stiffness of each is :math:`k’=2{k}_{2}`

The natural frequency is then equal to:

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, that is:

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**Second substructure: outer cylinder**

The second substructure is the external cylinder connected on the one hand to the interface by the same springs, on the other hand to a fixed frame:


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Mass of cylinder 2: :math:`{m}_{2}=\mathrm{3,674}{10}^{6}\mathrm{kg}`

Since the equivalent stiffness of an attachment of this cylinder by the system of springs in series :math:`{k}_{1}` and :math:`{k}_{2}` is equal to :math:`\mathrm{9,9}{10}^{6}{\mathrm{N.m}}^{-1}` (four attachments of the same type connect the cylinder in parallel to an installation), the natural frequency is given by:


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, that is:

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**N.B.**: the third cylinder (inner cylinder) was not modeled in our case because it is a fixed cylinder. It therefore constitutes a fixed wall of the fluid domain.

**Air modes of the complete structure (intermediate cylinder and external cylinder)**

It is a system with two degrees of freedom:


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The natural frequencies of this system are given by the exact formula [:ref:`bib2 <bib2>`]:


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,

either

:math:`{f}_{1}=0.497\mathrm{Hz}` and :math:`{f}_{2}=5.263\mathrm{Hz}`.

The two eigenmodes admit, for numerical value:


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.


Calculating the added mass matrix
-------------------------------------

**Fluid potentials**

Repeating [:ref:`bib1 <bib1>`], we establish that:


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The shape of the mass matrix added, in this configuration, is:


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With:


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The inertial coupling coefficient

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is considered to be negligible against the self-mass added coefficients

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and

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. As a first approximation, the natural frequencies of the system depend only on these last two coefficients.


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