2. Reference solution

2.1. Calculation method used for the reference solution

Analytical calculation:

We are going to assume that the movements of the cylinders and the fluid are essentially plane. Longitudinal effects will be overlooked in the face of transversal effects. The problem is two-dimensional. Given the symmetry, the coordinate system used is a cylindrical coordinate system \((r,\theta )\) linked to the central cylinder (see figure above). In this coordinate system and with this particular geometry, the normal derivative \(\frac{\partial \mathrm{.}}{\partial n}\) is equal to the derivative \(\frac{\partial \mathrm{.}}{\partial r}\) with respect to \(r\).

Throughout this part, the variable \(p\) designates the hydrodynamic pressure field in the fluid created by the natural vibrations of the structures, \({X}_{1/2}\) designates the natural modes of the cylinder \(1\) or \(2\) respectively.

The natural modes of border shells \(({\Gamma }_{1})\) and \(({\Gamma }_{2})\) in the absence of fluid are of the form (\(n\) refers to the order of the mode):

\({X}_{\mathrm{1n}}(r)=\{\begin{array}{}\mathrm{cos}n\theta \text{ou}\mathrm{sin}n\theta \\ 0\end{array}\) and \({X}_{\mathrm{2n}}(r)=\{\begin{array}{}0\\ \mathrm{cos}n\theta \text{ou}\mathrm{sin}n\theta \end{array}\)

\(\theta\) is the azimuthal angle. These modes are, of course, decoupled. The first component corresponds to the normal movement of the inner shell, the second to that of the outer shell. In the fluid volume, we therefore have two problems to solve:

\(\Delta {p}_{\mathrm{1n}}=0\) \({(\frac{\partial {p}_{\mathrm{1n}}}{\partial n})}_{{\Gamma }_{1}}=-{\rho }_{f}\{\begin{array}{}\mathrm{cos}n\theta \\ \text{ou}\\ \mathrm{sin}n\theta \end{array}\) \({(\frac{\partial {p}_{\mathrm{1n}}}{\partial n})}_{{\Gamma }_{2}}=0\) eq 2.1-1

and:

\(\Delta {p}_{\mathrm{2n}}=0\) \({(\frac{\partial {p}_{\mathrm{2n}}}{\partial n})}_{{\Gamma }_{1}}=0\) \({(\frac{\partial {p}_{\mathrm{2n}}}{\partial n})}_{{\Gamma }_{2}}=-{\rho }_{f}\{\begin{array}{}\mathrm{cos}n\theta \\ \text{ou}\\ \mathrm{sin}n\theta \end{array}\) eq 2.1-2

Field \({p}_{\mathrm{1n}}\) corresponds to the pressure field generated in the fluid if the central shell \({\Gamma }_{1}\) vibrates alone, field \({p}_{\mathrm{2n}}\) is that created by the outer shell \({\Gamma }_{2}\) if it vibrates alone. The linearity of the Laplace equation makes it possible to solve each problem independently and then to superimpose them to find the total pressure field.

The solution to the problem [éq2.1-1] is, in polar coordinates, of the type [bib1]:

\({p}_{\mathrm{1n}}(r,\theta )=\left\{A{r}^{n}+B{(\frac{1}{r})}^{n}\right\}\left\{\begin{array}{}\mathrm{cos}n\theta \\ \text{ou}\\ \mathrm{sin}n\theta \end{array}\right\}\)

We must have \(n\ne 0\), because otherwise we have the non-conservation of the volume of the fluid.

The constants \(A\) and \(B\) are determined by the boundary conditions:

\({(\frac{\partial {p}_{\mathrm{1n}}}{\partial n})}_{{R}_{1}}=-{\rho }_{f}\left\{\begin{array}{}\mathrm{cos}n\theta \\ \text{ou}\\ \mathrm{sin}n\theta \end{array}\right\}\) and \({(\frac{\partial {p}_{\mathrm{1n}}}{\partial n})}_{{R}_{2}}=0\)

We then find that the pressure field for each of the two problems is written as:

\({p}_{\mathrm{1n}}(r,\theta )=\frac{{\rho }_{f}{R}_{1}}{n}\frac{{(r/{R}_{1})}^{n}+{({R}_{2}/{R}_{1})}^{n}{({R}_{2}/r)}^{n}}{{({R}_{2}/{R}_{1})}^{\mathrm{2n}}-1}\left\{\begin{array}{}\mathrm{cos}n\theta \\ \text{ou}\\ \mathrm{sin}n\theta \end{array}\right\}\)

and:

\({p}_{\mathrm{2n}}(r,\theta )=\frac{{\rho }_{f}{R}_{2}}{n}\frac{{({R}_{2}/{R}_{1})}^{n}{(r/{R}_{1})}^{n}+{({R}_{2}/r)}^{n}}{{({R}_{2}/{R}_{1})}^{\mathrm{2n}}-1}\left\{\begin{array}{}\mathrm{cos}n\theta \\ \text{ou}\\ \mathrm{sin}n\theta \end{array}\right\}\)

The modal added mass coefficients \({m}_{\mathrm{ijnm}}^{A}\) are calculated using the following formula [R4.07.03] if \(i=1\) or \(2\), \(j=1\) or \(2\), \((n,m)\) belongs to \({i}^{2}\).

\({m}_{\mathrm{ijnm}}^{A}={\int }_{{\Gamma }_{j}}{p}_{\mathrm{jn}}{X}_{\text{im}}(r)\mathrm{.}n({\Gamma }_{j})d{\Gamma }_{j}\)

The indexing here is a bit more complex than in the formula presented in [R4.07.03]: the indices i and j refer to shells

and

, and the indices \(m\) and \(n\) are associated with shell modes. It is noted that there is a coupling of the modes of the various shells, external and internal.

We note, on the one hand, that the fluid does not couple the modes with different indices \(n\) because the integrals \({\int }_{\Gamma }\mathrm{cos}n\theta \mathrm{cos}m\theta d\Gamma\) cancel each other out; on the other hand, the fluid does not couple the modes \(\mathrm{cos}n\theta\) and \(\mathrm{sin}n\theta\) either because \({\int }_{(\Gamma )}\mathrm{cos}n\theta \mathrm{sin}n\theta d\Gamma =0\). The only existing coupling is a coupling between the two shells for modes of the same nature.

Each \(n\) mode is associated with a symmetric fourth-order matrix. A sub-matrix corresponding to the projection on the \(n\) mode is written:

\({M}_{1}^{A}=(\begin{array}{cc}{m}_{\mathrm{11nn}}^{A}& {m}_{\mathrm{12nn}}^{A}\\ {m}_{\mathrm{21nn}}^{A}& {m}_{\mathrm{22nn}}^{A}\end{array})\)

The global matrix is written as: \(\left[\begin{array}{cc}{M}_{1}^{A}& 0\\ 0& {M}_{2}^{A}\end{array}\right]\) with \({M}_{1}^{A}={M}_{2}^{A}\)

with \({m}_{\mathrm{11nn}}^{A}=L{R}_{1}\underset{0}{\overset{2\pi }{\int }}{p}_{1}({R}_{\mathrm{1,}}\theta )\left\{\begin{array}{}\mathrm{cos}n\theta \\ \text{ou}\\ \mathrm{sin}n\theta \end{array}\right\}d\theta\)

Either:

\({m}_{\mathrm{11nn}}^{A}=\frac{\pi }{n}{\rho }_{f}{R}_{1}^{2}L\frac{{({R}_{2}/{R}_{1})}^{\mathrm{2n}}+1}{{({R}_{2}/{R}_{1})}^{\mathrm{2n}}-1}\) eq 2.1-3

we will get:

\({m}_{\mathrm{22nn}}^{A}=\frac{\pi }{n}{\rho }_{f}{R}_{2}^{2}L\frac{{({R}_{2}/{R}_{1})}^{\mathrm{2n}}+1}{{({R}_{2}/{R}_{1})}^{\mathrm{2n}}-1}\) eq 2.1-4

and:

\({m}_{\mathrm{21nn}}^{A}={m}_{\mathrm{12nn}}^{A}=-\frac{\pi }{n}{\rho }_{f}{R}_{1}{R}_{2}L\frac{2({R}_{2}/{R}_{1})}{{({R}_{2}/{R}_{1})}^{\mathrm{2n}}-1}\) eq 2.1-5

\(L\) here refers to the height of the cylinder shells in the longitudinal direction.

In our case, we only consider the modes of order \(n=1\) of the shells: they correspond respectively to the modes of translation of each of the shells along an axis passing through the center of the central tube: we take those corresponding to the axis \(\mathrm{Ox}\) arbitrarily: the linear added mass coefficients are written as:

\({m}_{11}^{A}=\pi {\rho }_{f}{R}_{1}^{2}\frac{{({R}_{2}/{R}_{1})}^{2}+1}{{({R}_{2}/{R}_{1})}^{2}-1}\)

\({m}_{22}^{A}=\pi {\rho }_{f}{R}_{2}^{2}\frac{{({R}_{2}/{R}_{1})}^{2}+1}{{({R}_{2}/{R}_{1})}^{2}-1}\)

\({m}_{21}^{A}={m}_{12}^{A}=-\pi {\rho }_{f}{R}_{1}{R}_{2}\frac{2({R}_{2}/{R}_{1})}{{({R}_{2}/{R}_{1})}^{2}-1}\)

The generalized equation of motion of the two coupled shells is written as:

\((\begin{array}{cc}{m}_{1}& 0\\ 0& {m}_{2}\end{array})(\begin{array}{}{\ddot{x}}_{1}\\ {\ddot{x}}_{2}\end{array})+(\begin{array}{cc}{k}_{1}& 0\\ 0& {k}_{2}\end{array})(\begin{array}{}{x}_{1}\\ {x}_{2}\end{array})=-(\begin{array}{cc}{m}_{11}& {m}_{12}\\ {m}_{12}& {m}_{22}\end{array})(\begin{array}{}{\ddot{x}}_{1}\\ {\ddot{x}}_{2}\end{array})\)

The natural pulsations of the coupled system are given by the equation of degree 4:

\(\mathrm{det}\left[(\begin{array}{cc}{m}_{1}+{m}_{11}& {m}_{12}\\ {m}_{12}& {m}_{2}+{m}_{22}\end{array}){\Omega }^{2}-(\begin{array}{cc}{k}_{1}& 0\\ 0& {k}_{2}\end{array})\right]=0\)

Digital application:

\({K}_{1}={10}^{7}N/m\) \({K}_{2}={10}^{7}N/m\)

\({m}_{11}=33060\mathrm{kg}/m\)

\({m}_{22}=40004\mathrm{kg}/m\)

\({m}_{12}=–36200\mathrm{kg}/m\)

Two natural frequencies are obtained:

\({f}_{1}=1.696\mathrm{Hz}\) \({f}_{2}=4.128\mathrm{Hz}\)

2.2. Benchmark results

Analytics

2.3. Bibliographical references

R.J GIBERT. Vibrations of Structures. Interactions with fluids. Eyrolles (1988).