2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference solution is an approximate analytical solution. The approximate analytical fluctuating potentials to calculate the added stiffness are written as [bib1]:
\(\mathrm{\{}\begin{array}{ccc}{\varphi }_{1}(r,\theta ,y)\mathrm{=}& \frac{{R}_{e}^{2}}{{R}_{e}^{2}\mathrm{-}{R}_{i}^{2}}(r+\frac{{R}_{i}^{2}}{r})& \mathrm{sin}\theta \mathrm{sin}\frac{\pi (y+l\mathrm{/}2)}{l}\\ {\varphi }_{2}(r,\theta ,y)\mathrm{=}& \frac{{R}_{e}^{2}}{{R}_{e}^{2}\mathrm{-}{R}_{i}^{2}}(r+\frac{{R}_{i}^{2}}{r})& \mathrm{sin}\theta \mathrm{cos}\frac{\pi (y+l\mathrm{/}2)}{l}\end{array}\)
The stiffness added to the first mode of flexure of the outer cylinder considered as a swivel and swivel beam is written [bib1]:
\({K}_{A}\mathrm{=}\mathrm{-}\frac{\rho }{2}\frac{{V}_{0}^{2}{\pi }^{3}{R}_{e}^{3}}{({R}_{e}^{2}\mathrm{-}{R}_{i}^{2})l}({R}_{e}+\frac{{R}_{i}^{2}}{{R}_{e}})\)
This stiffness, calculated on a cylindrical geometry, is then assigned to a model with an equivalent degree of freedom.
The system with an equivalent degree of freedom is an equivalent mass-spring system to which is assigned a mass equal to the mass of the system increased by the mass added by the fluid and a stiffness equal to the stiffness of the system increased by the stiffness added by the flow for various speeds.
The mass of the system in air is:
\(M\mathrm{=}10292\mathit{kg}\)
for an external cylindrical shell with a thickness:
\(C\mathrm{=}{2.10}^{\mathrm{-}3}m\)
For equivalent stiffness in air of the « outer shell » system, we take the stiffness of a beam subjected to a force distributed over its entire length:
So \(K\mathrm{=}2.533{10}^{4}N\mathrm{/}m\)
The equivalent system coupled to the flow is represented by the following diagram:
with \(m\mathrm{=}M+{M}_{A}\) \(k\mathrm{=}K+{K}_{A}\)
The natural pulsation of the coupled system evolves according to the square of the flow speed. If we call \({V}_{{0}_{C}}\) the critical flow speed for which the stiffness \(k\) is cancelled out:
\(\mathrm{\exists }{V}_{{0}_{C}},K+{K}_{A}({V}_{{0}_{C}})\mathrm{=}0\) with \({V}_{{0}_{C}}^{2}\mathrm{=}\frac{2({R}_{e}^{2}\mathrm{-}{R}_{i}^{2})lK}{({R}_{e}+\frac{{R}_{i}^{2}}{{R}_{e}})\rho {\pi }^{3}{R}_{e}^{3}}\)
so we show that:
\(\omega ({V}_{0})\mathrm{=}{\omega }_{e}(0)\sqrt{1\mathrm{-}{x}^{2}}\)
where we put:
\({\omega }_{e}(0)\mathrm{=}\sqrt{\frac{K}{M+{M}_{A}}}\) (natural pulsation of the system in fluid at rest)
\(x\mathrm{=}\frac{{V}_{0}}{{V}_{{0}_{C}}}\) (reduced flow speed)
The pulsation of the fluid at rest is equal to: \(\omega \mathrm{=}0.085\mathit{rad}\mathrm{/}s\).
2.2. Benchmark results#
The natural frequency of the system is calculated for different flow velocities.
\({V}_{0}(m\mathrm{/}s)\) |
0.5 |
1.5 |
2.2 |
2.688 |
|||
\({M}_{A}(\mathit{kg})\) |
3.486E6 |
3.486E6 |
3.486E6 |
3.486E6 |
3.486E6 |
3.486E6 |
|
\({K}_{A}(N/m)\) |
—876.5 |
—7888.50 |
—7888.50 |
—14023.95 |
—16968.98 |
—25330 |
|
\({M}_{\mathrm{total}}(\mathrm{kg})\) |
3.491E+6 |
= |
= |
= |
= |
= |
= |
\({K}_{\mathit{total}}(N\mathrm{/}m)\) |
24453.5 |
17441.5 |
17441.5 |
11306.05 |
8361.00 |
||
\(f({V}_{0})\mathrm{\times }{10}^{\mathrm{-}2}(\mathit{Hz})\) |
1.318 |
1.112 |
1.112 |
0.896 |
0.772 |
2.3. Uncertainty about the solution#
Semi-analytical solution.
2.4. Bibliographical references#
ROUSSEAU G., LUU H.T. - Mass, damping and stiffness added for a vibrating structure placed in a potential flow - Internal note EDF/DER, HP-61/95/064/A (1995).