2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The advantage of the test here is to calculate and test the modal effort autospectrum obtained from a pressure spectrum characteristic of established turbulent flows.
For this case, the mode of travel is imposed. It is therefore not necessary to perform a modal calculation for the structure. The modal force autospectrum, calculated from the static deformation, is written as:
\(\mathit{DSP}(\omega )\mathrm{=}{S}_{p}(\omega ){J}_{{{A}_{\text{ij}}}_{}}^{2}(w)\)
The spectrum chosen here is constant and then zero based on a cutoff frequency:
The pressure spectrum is:
\({S}_{{p}_{}}(\omega )\mathrm{=}{K}^{2}({\mathit{\rho U}}^{2}{)}^{2}{d}^{3}\) for \(\mathrm{0,1}<\frac{\omega d}{2\pi U}<10\)
With the settings selected, \({S}_{{p}_{}}(\omega )\mathrm{\approx }1.0\)
The coherence function chosen in the case of this circular cylinder subjected to parallel flow comes from a AU_YANG model:
\({r}^{(s)}(x\mathrm{-}x\text{',}\omega )\mathrm{=}{e}^{\mathrm{-}(x\mathrm{-}x\text{'})\mathrm{/}\lambda }\text{cos}(\omega (x\mathrm{-}x\text{'})\mathrm{/}{U}_{c})\)
\({r}^{(s)}(\theta \mathrm{-}\theta \text{',}\omega )\mathrm{=}{e}^{\mathrm{-}R(\theta \mathrm{-}\theta \text{'})\mathrm{/}\lambda \text{'}}\)
The parameters \(\lambda\) and \(\lambda \text{'}\) are the correlation lengths along the axis and the orthoradial direction respectively.
\({U}_{c}\) is the axial convective speed of vortices: it is equal to the product of the axial speed coefficient by the fluid speed.
The acceptance function is defined by:
\({J}_{{{A}_{\text{ij}}}_{}}^{2}(\omega )\mathrm{=}\underset{A}{\mathrm{\int }}\underset{A}{\mathrm{\int }}r(x\mathrm{-}x\text{',}\omega ){f}_{{i}_{\alpha }}(x){f}_{{j}_{\alpha \text{'}}}(x\text{'}){n}_{\alpha }(x){n}_{\alpha \text{'}}(x\text{'})\text{dA}\text{dA}\text{'}\)
In our case it applies to:
\({J}_{{{A}_{\text{nm}}}_{}}^{2}(\omega )\mathrm{=}\underset{A}{\mathrm{\int }}\underset{A}{\mathrm{\int }}{e}^{\mathrm{-}R\mathrm{\mid }\theta \mathrm{-}\theta \text{'}\mathrm{\mid }\mathrm{/}\lambda \text{'}}{e}^{\mathrm{-}R\mathrm{\mid }x\mathrm{-}x\text{'}\mathrm{\mid }\mathrm{/}\lambda }\text{cos}(\frac{\omega (x\mathrm{-}x\text{'})}{{U}_{c}})x\text{cos}\theta x\text{'}\text{cos}\theta \text{'}\mathit{dxRd}\theta \mathit{dx}\text{'}\mathit{Rd}\theta \text{'}\)
\({J}_{{{A}_{\text{nm}}}_{}}^{2}(\omega )\mathrm{=}\underset{0}{\overset{2\pi }{\mathrm{\int }}}\underset{0}{\overset{2\pi }{\mathrm{\int }}}{e}^{\mathrm{-}R\mathrm{\mid }\theta \mathrm{-}\theta \text{'}\mathrm{\mid }\mathrm{/}\lambda \text{'}}\text{cos}\theta \text{cos}\theta \text{'}{R}^{2}d\theta d\theta \text{'}\underset{0}{\overset{H}{\mathrm{\int }}}\underset{0}{\overset{H}{\mathrm{\int }}}{e}^{\mathrm{-}R\mathrm{\mid }x\mathrm{-}x\text{'}\mathrm{\mid }\mathrm{/}\lambda }\text{cos}(\frac{\omega (x\mathrm{-}x\text{'})}{{U}_{c}})xx\text{'}\text{dxdx}\text{'}\)
In the following table, values of this integral are given using Python:
\(\omega (\mathrm{rad}/s)\) |
|
0.06283 |
252.701 |
0.6283 |
249.663 |
2.2. Benchmark results#
Analytical result.
2.3. 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 1984.
ROUSSEAU G. Specification of the acceptance calculation in the*Code_Aster*. Spectral response of structures to random turbulent excitation HP51 /97/027/A