2. 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.

2.1. Breakdown into substructures

First substructure: intermediate cylinder

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

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

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

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

The natural frequency is then equal to:

, that is:

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:

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

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

, that is:

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:

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

,

either

\({f}_{1}=0.497\mathrm{Hz}\) and \({f}_{2}=5.263\mathrm{Hz}\).

The two eigenmodes admit, for numerical value:

.

2.2. Calculating the added mass matrix

Fluid potentials

Repeating [bib1], we establish that:

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

With:

The inertial coupling coefficient

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

and

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

2.3. Benchmark results

Analytical result.

2.4. 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