.. include:: /replace.txt .. _section-airfoil: Airfoil (MH-104) ================ Problem description ------------------- .. figure:: /figures/examplemh1040.png :name: fig_mh104_draw :width: 5.5in :align: center Sketch of a cross section for a typical wind turbine blade [CHEN2010]_. This example demonstrates the capability of building a cross section having an airfoil shape, which is commonly seen on wind turbine blades or helicopter rotor blades. This example is also studied in [CHEN2010]_. A sketch of a cross section for a typical wind turbine blade is shown in :numref:Fig. %s . The airfoil is MH 104 (http://m-selig.ae.illinois.edu/ads/coord_database.html#M). In this example, the chord length :math:CL=1.9 m. The origin O is set to the point at 1/4 of the chord. Twist angle :math:\theta is :math:0^\circ. There are two webs, whose right boundaries are at the 20% and 50% location of the chord, respectively. Both low pressure and high pressure surfaces have four segments. The dividing points between segments are listed in :numref:Table %s . Materials are given in :numref:Table %s  and layups are given in :numref:Table %s . A complete :math:6\times 6 stiffness matrix is given in :numref:Table %s . Complete input files can be found in examples\ex_airfoil\, including mh104.xml, basepoints.dat, baselines.xml, materials.xml, and layups.xml. .. csv-table:: Dividing points :name: table_div_pts :header-rows: 2 :align: center "Between segments", "Low pressure surface", "High pressure surface" , ":math:(x, y)", ":math:(x, y)" "1 and 2", "(0.004053940, 0.011734800)", "(0.006824530, -0.009881650)" "2 and 3", "(0.114739930, 0.074571970)", "(0.126956710, -0.047620490)" "3 and 4", "(0.536615950, 0.070226120)", "(0.542952100, -0.044437080)" .. figure:: /figures/examplemh1041.png :name: fig_mh1041 :width: 6.5in :align: center *Base point*\ s of the tube cross section. .. figure:: /figures/examplemh1042.png :name: fig_mh1042 :width: 6.5in :align: center *Base line*\ s of the tube cross section. .. figure:: /figures/examplemh1043.png :name: fig_mh1043 :width: 6.5in :align: center *Segment*\ s of the tube cross section. .. figure:: /figures/examplemh1044.png :name: fig_mh104 :width: 6.5in :align: center Meshed cross section viewed in Gmsh. .. csv-table:: Material properties :name: table_mh104_materials :header-rows: 2 :align: center "Name", "Type", "Density", |e1|, |e2|, |e3|, |g12|, |g13|, |g23|, |nu12|, |nu13|, |nu23| , , |den_si_k|, |mod_si_g|, |mod_si_g|, |mod_si_g|, |mod_si_g|, |mod_si_g|, |mod_si_g|, , , "Uni-directional FRP", "orthotropic", 1.86, 37.00, 9.00, 9.00, 4.00, 4.00, 4.00, 0.28, 0.28, 0.28 "Double-bias FRP", "orthotropic", 1.83, 10.30, 10.30, 10.30, 8.00, 8.00, 8.00, 0.30, 0.30, 0.30 "Gelcoat", "orthotropic", 1.83, 1e-8, 1e-8, 1e-8, 1e-9, 1e-9, 1e-9, 0.30, 0.30, 0.30 "Nexus", "orthotropic", 1.664, 10.30, 10.30, 10.30, 8.00, 8.00, 8.00, 0.30, 0.30, 0.30 "Balsa", "orthotropic", 0.128, 0.01, 0.01, 0.01, 2e-4, 2e-4, 2e-4, 0.30, 0.30, 0.30 .. csv-table:: Layups :name: table_mh104_layups :header-rows: 2 :align: center "Name", "Layer", "Material", "Ply thickness", "Orientation", "Number of plies" , , , |len_si|, :math:\circ, "layup_1", 1, "Gelcoat", 0.000381, 0, 1 , 2, "Nexus", 0.00051, 0, 1 , 3, "Double-bias FRP", 0.00053, 20, 18 "layup_2", 1, "Gelcoat", 0.000381, 0, 1 , 2, "Nexus", 0.00051, 0, 1 , 3, "Double-bias FRP", 0.00053, 20, 33 "layup_3", 1, "Gelcoat", 0.000381, 0, 1 , 2, "Nexus", 0.00051, 0, 1 , 3, "Double-bias FRP", 0.00053, 20, 17 , 4, "Uni-directional FRP", 0.00053, 30, 38 , 5, "Balsa", 0.003125, 0, 1 , 6, "Uni-directional FRP", 0.00053, 30, 37 , 7, "Double-bias FRP", 0.00053, 20, 16 "layup_4", 1, "Gelcoat", 0.000381, 0, 1 , 2, "Nexus", 0.00051, 0, 1 , 3, "Double-bias FRP", 0.00053, 20, 17 , 4, "Balsa", 0.003125, 0, 1 , 5, "Double-bias FRP", 0.00053, 0, 16 "layup_web", 1, "Uni-directional FRP", 0.00053, 0, 38 , 2, "Balsa", 0.003125, 0, 1 , 3, "Uni-directional FRP", 0.00053, 0, 38 Result ------ .. table:: Effective Timoshenko stiffness matrix :name: table_airfoil_result =================================== =================================== =================================== =================================== =================================== =================================== :math:\phantom{-}2.395\times 10^9 :math:\phantom{-}1.588\times 10^6 :math:\phantom{-}7.215\times 10^6 :math:-3.358\times 10^7 :math:\phantom{-}6.993\times 10^7 :math:-5.556\times 10^8 :math:\phantom{-}1.588\times 10^6 :math:\phantom{-}4.307\times 10^8 :math:-3.609\times 10^6 :math:-1.777\times 10^7 :math:\phantom{-}1.507\times 10^7 :math:\phantom{-}2.652\times 10^5 :math:\phantom{-}7.215\times 10^6 :math:-3.609\times 10^6 :math:\phantom{-}2.828\times 10^7 :math:\phantom{-}8.440\times 10^5 :math:\phantom{-}2.983\times 10^5 :math:-5.260\times 10^6 :math:-3.358\times 10^7 :math:-1.777\times 10^7 :math:\phantom{-}8.440\times 10^5 :math:\phantom{-}2.236\times 10^7 :math:-2.024\times 10^6 :math:\phantom{-}2.202\times 10^6 :math:\phantom{-}6.993\times 10^7 :math:\phantom{-}1.507\times 10^7 :math:\phantom{-}2.983\times 10^5 :math:-2.024\times 10^6 :math:\phantom{-}2.144\times 10^7 :math:-9.137\times 10^6 :math:-5.556\times 10^8 :math:\phantom{-}2.652\times 10^5 :math:-5.260\times 10^6 :math:\phantom{-}2.202\times 10^6 :math:-9.137\times 10^6 :math:\phantom{-}4.823\times 10^8 =================================== =================================== =================================== =================================== =================================== =================================== .. table:: Results from reference [CHEN2010]_ :name: table_airfoil_result_ref =================================== =================================== =================================== =================================== =================================== =================================== :math:\phantom{-}2.389\times 10^9 :math:\phantom{-}1.524\times 10^6 :math:\phantom{-}6.734\times 10^6 :math:-3.382\times 10^7 :math:-2.627\times 10^7 :math:-4.736\times 10^8 :math:\phantom{-}1.524\times 10^6 :math:\phantom{-}4.334\times 10^8 :math:-3.741\times 10^6 :math:-2.935\times 10^5 :math:\phantom{-}1.527\times 10^7 :math:\phantom{-}3.835\times 10^5 :math:\phantom{-}6.734\times 10^6 :math:-3.741\times 10^6 :math:\phantom{-}2.743\times 10^7 :math:-4.592\times 10^4 :math:-6.869\times 10^2 :math:-4.742\times 10^6 :math:-3.382\times 10^7 :math:-2.935\times 10^5 :math:-4.592\times 10^4 :math:\phantom{-}2.167\times 10^7 :math:-6.279\times 10^4 :math:\phantom{-}1.430\times 10^6 :math:-2.627\times 10^7 :math:\phantom{-}1.527\times 10^7 :math:-6.869\times 10^2 :math:-6.279\times 10^4 :math:\phantom{-}1.970\times 10^7 :math:\phantom{-}1.209\times 10^7 :math:-4.736\times 10^8 :math:\phantom{-}3.835\times 10^5 :math:-4.742\times 10^6 :math:\phantom{-}1.430\times 10^6 :math:\phantom{-}1.209\times 10^7 :math:\phantom{-}4.406\times 10^8 =================================== =================================== =================================== =================================== =================================== =================================== .. note:: The errors between the result and the reference are caused by the difference of modeling of the trailing edge. If reduce the trailing edge skin to a single thin layer, then the difference between the trailing edge shapes is minimized, and the two resulting stiffness matrices are basically the same, as shown in :numref:Fig. %s . .. figure:: /figures/examplemh104_comparison.png :name: fig_mh104_comparison :width: 6in :align: center Comparison of stiffness matrices after modifying the trailing edge.