Benchmark solution ===================== 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 (:math:`{V}_{0}=4{\mathrm{m.s}}^{-1}`). The mass added by the flow is equal to: :math:`\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 :math:`{V}_{0}=4{\mathrm{m.s}}^{-1}`: :math:`\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 :math:`{V}_{0}=4{\mathrm{m.s}}^{-1}`: :math:`\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: .. image:: images/10000200000006A9000005092E78E7DE07EF2274.png :width: 6.3591in :height: 4.8075in .. _RefImage_10000200000006A9000005092E78E7DE07EF2274.png: Our DSP pressure is: :math:`{S}_{{p}_{}}(\omega )\mathrm{=}{K}^{2}({\mathit{\rho U}}^{2}{)}^{2}{d}^{3}` for :math:`\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: :math:`{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 :math:`{k}_{T}` and :math:`{k}_{L}` are called Bakewell parameters and are valid according to the pulse: :math:`{k}_{L}\mathrm{=}0.1\frac{\omega }{{U}_{c}}` and :math:`{k}_{T}\mathrm{=}0\text{.}\text{55}\frac{\omega }{{U}_{c}}` The acceptance function, generally defined by :math:`{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: :math:`\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: :math:`\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: .. csv-table:: ":math:`\omega (\mathrm{rad}/s)` "," :math:`{I}_{T}(\omega )`" "0.01", "24.9121" "0.1", "24.1414" "1. ", "18.0988" "2. ", "13.8102" "10. ", "4.2803" The other factor is more complex to assess. We have therefore calculated this integral numerically using the Maple V.5 software: .. csv-table:: ":math:`\omega (\mathrm{rad}/s)` "," :math:`I(\omega )`" "0.01", "1006.601" "0.1", "815.3964" "1. ", "14.319" "2. ", "6.5836" "10. ", "1.288" Thus, for pulsations :math:`0.01\mathit{rad}\mathrm{/}s` and :math:`1\mathit{rad}\mathrm{/}s`, the DSP modal effort value is respectively: .. csv-table:: ":math:`\omega (\mathit{rad}\mathrm{/}s)` "," :math:`\mathit{DSP}(\omega )`" "0.01", "7.28848E8" "1. ", "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: :math:`{S}_{{p}_{}}(\omega )\mathrm{=}{10}^{10}{e}^{\mathrm{-}(\omega \mathrm{/}0.1{)}^{2}}` Thus, for pulsations :math:`0.01\mathrm{rad}/s` and :math:`\mathrm{0,1}\mathit{rad}\mathrm{/}s`, the DSP modal effort value is respectively: .. csv-table:: ":math:`\omega (\mathrm{rad}/s)` "," :math:`\mathrm{DSP}(\omega )`" "0.01", "24,82703E13" "0.1", "7,24164E13" Benchmark results ---------------------- Analytical result. Bibliographical references -------------------------- .. [bib1] 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. .. [bib2] BLEVINS R.D : Formulas for natural frequency and mode shape. Ed. Krieger 1984. .. [bib3] ROUSSEAU G. Specification of acceptance calculation in *Code_Aster*. Spectral response of structures to random turbulent excitation HP51 /97/027/A