2. Benchmark solution#
2.1. Calculation method used for the reference solution#
For the calculation of the added coefficients, we refer to test case FDLV109.
A calculation of damping added to the flow speed was done (\({V}_{0}=4{\mathrm{m.s}}^{-1}\)).
The mass added by the flow is equal to:
\(\begin{array}{}{M}_{\text{11}}^{a}=0\text{.}\text{625}{\text{10}}^{5}\text{kg},\\ {M}_{\text{22}}^{a}=0\text{.}\text{625}{\text{10}}^{5}\text{kg},\\ {M}_{\text{12}}^{a}=0\text{.}\end{array}\)
The depreciation added is equal with \({V}_{0}=4{\mathrm{m.s}}^{-1}\):
\(\begin{array}{}{C}_{\text{11}}^{a}=\mathrm{0,}\\ {C}_{\text{22}}^{a}=\mathrm{0,}\\ {C}_{\text{12}}^{a}=0\text{.}\text{266}{\text{10}}^{5}N\text{.}{m}^{-1}\text{.}\end{array}\)
The added stiffness applies with \({V}_{0}=4{\mathrm{m.s}}^{-1}\):
\(\begin{array}{}{K}_{\text{11}}^{a}\text{=-}0\text{.}\text{3943}{\text{10}}^{4}N\text{.}{m}^{-1}\text{.}{\text{rad}}^{2},\\ {K}_{\text{22}}^{a}\text{=-}0\text{.}\text{1577}{\text{10}}^{5}N\text{.}{m}^{-1}\text{.}{\text{rad}}^{2},\\ {K}_{\text{12}}^{a}=0\text{.}\end{array}\)
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.
The spectrum chosen here is constant and then zero based on a cutoff frequency:
Our DSP pressure is:
\({S}_{{p}_{}}(\omega )\mathrm{=}{K}^{2}({\mathit{\rho U}}^{2}{)}^{2}{d}^{3}\) for \(\mathrm{0,1}<\frac{\omega d}{\mathrm{2pU}}<10\)
The coherence function chosen in the case of this plate subjected to parallel flow comes from a Corcos model:
\({r}^{(s)}(x\mathrm{-}x\text{',}\omega )\mathrm{=}{e}^{\mathrm{-}{k}_{L}(x\mathrm{-}x\text{'})}{e}^{\mathrm{-}{k}_{T}(y\mathrm{-}y\text{'})}\text{cos}(\omega (x\mathrm{-}x\text{'})\mathrm{/}{U}_{c})\text{.}\)
The parameters \({k}_{T}\) and \({k}_{L}\) are called Bakewell parameters and are valid according to the pulse:
\({k}_{L}\mathrm{=}0.1\frac{\omega }{{U}_{c}}\) and \({k}_{T}\mathrm{=}0\text{.}\text{55}\frac{\omega }{{U}_{c}}\)
The acceptance function, generally 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:
\(\begin{array}{c}{J}_{{{A}_{\text{nm}}}_{}}^{2}(\omega )\mathrm{=}\underset{A}{\mathrm{\int }}\underset{A}{\mathrm{\int }}{e}^{\mathrm{-}{k}_{T}\mathrm{\mid }y\mathrm{-}y\text{'}\mathrm{\mid }}{e}^{\mathrm{-}{k}_{L}\mathrm{\mid }x\mathrm{-}x\text{'}\mathrm{\mid }}\text{cos}(\frac{\omega (x\mathrm{-}x\text{'})}{{U}_{c}})\text{sin}(\frac{{k}_{n}x}{L})\text{sin}(\frac{{k}_{m}x\text{'}}{L})\text{dxdy}\text{dx}\text{'}\text{dy}\text{'}\\ \mathrm{=}\underset{\mathrm{-}l\mathrm{/}2}{\overset{l\mathrm{/}2}{\mathrm{\int }}}\underset{\mathrm{-}l\mathrm{/}2}{\overset{l\mathrm{/}2}{\mathrm{\int }}}{e}^{\mathrm{-}{k}_{T}\mathrm{\mid }y\mathrm{-}y\text{'}\mathrm{\mid }}\text{dydy}\text{'}´\underset{0}{\overset{L}{\mathrm{\int }}}\underset{0}{\overset{L}{\mathrm{\int }}}{e}^{\mathrm{-}{k}_{L}\mathrm{\mid }x\mathrm{-}x\text{'}\mathrm{\mid }}\text{cos}(\frac{\omega (x\mathrm{-}x\text{'})}{{U}_{c}})\text{sin}(\frac{{k}_{n}x}{L})\text{sin}(\frac{{k}_{m}x\text{'}}{L})\text{dxdx}\text{'}\text{.}\\ \end{array}\)
The first factor integral has an analytic expression and is equal to:
\(\underset{-l/2}{\overset{l/2}{\int }}\underset{-l/2}{\overset{l/2}{\int }}{e}^{-{k}_{T}\mid y-y\text{'}\mid }\text{dydy}\text{'}=\frac{\mathrm{2l}}{{k}_{T}}-2(\frac{1-{e}^{-{k}_{T}l}}{{k}_{{}_{T}}^{2}})\text{.}\)
In the following table, values of this integral are given:
\(\omega (\mathrm{rad}/s)\) |
|
0.01 |
24.9121 |
0.1 |
24.1414 |
18.0988 |
|
13.8102 |
|
4.2803 |
The other factor is more complex to assess. We have therefore calculated this integral numerically using the Maple V.5 software:
\(\omega (\mathrm{rad}/s)\) |
|
0.01 |
1006.601 |
0.1 |
815.3964 |
14.319 |
|
6.5836 |
|
1.288 |
Thus, for pulsations \(0.01\mathit{rad}\mathrm{/}s\) and \(1\mathit{rad}\mathrm{/}s\), the DSP modal effort value is respectively:
\(\omega (\mathit{rad}\mathrm{/}s)\) |
|
|
0.01 |
7.28848E8 |
|
7.53237E6 |
||
For the case of modeling A |
the definition of a turbulence spectrum is also tested using any frequency function |
using the keyword SPEC_CORR_CONV_2. The function chosen is: |
\({S}_{{p}_{}}(\omega )\mathrm{=}{10}^{10}{e}^{\mathrm{-}(\omega \mathrm{/}0.1{)}^{2}}\)
Thus, for pulsations \(0.01\mathrm{rad}/s\) and \(\mathrm{0,1}\mathit{rad}\mathrm{/}s\), the DSP modal effort value is respectively:
\(\omega (\mathrm{rad}/s)\) |
|
0.01 |
24,82703E13 |
0.1 |
7,24164E13 |
2.2. Benchmark results#
Analytical result.
2.3. Bibliographical references#
ROUSSEAU G., LUU H.T. : Masse, amortissement et raideur ajoutés pour une structure vibrante placée dans un écoulement potentiel - Bibliographie et implantation dans le Code_Aster - HP-61/95/064.
BLEVINS R.D : Formulas for natural frequency and mode shape. Ed. Krieger 1984.
ROUSSEAU G. Specification of acceptance calculation in Code_Aster. Spectral response of structures to random turbulent excitation HP51 /97/027/A