Airfoil

_images/examplemh1040.png

Figure 37 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 Fig. 37. The airfoil is MH 104 (http://m-selig.ae.illinois.edu/ads/coord_database.html#M). In this example, the chord length \(CL=1.9\) m. The origin O is set to the point at 1/4 of the chord. Twist angle \(\theta\) is \(0^\circ\). 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 Table 24. Materials are given in Table 25 and layups are given in Table 26. A complete \(6\times 6\) stiffness matrix is given in Table 27. Complete input files can be found in examples\ex_airfoil\, including mh104.xml, basepoints.dat, baselines.xml, materials.xml, and layups.xml.

Table 24 Dividing points
Between segments Low pressure surface High pressure surface
  \((x, y)\) \((x, y)\)
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)
_images/examplemh1041.png

Figure 38 Base points of the tube cross section.

_images/examplemh1042.png

Figure 39 Base lines of the tube cross section.

_images/examplemh1043.png

Figure 40 Segments of the tube cross section.

_images/examplemh1044.png

Figure 41 Meshed cross section viewed in Gmsh.

Table 25 Material properties
Name Type Density \(E_{1}\) \(E_{2}\) \(E_{3}\) \(G_{12}\) \(G_{13}\) \(G_{23}\) \(\nu_{12}\) \(\nu_{13}\) \(\nu_{23}\)
    \(10^3\ \mathrm{kg/m^3}\) \(10^9\ \mathrm{Pa}\) \(10^9\ \mathrm{Pa}\) \(10^9\ \mathrm{Pa}\) \(10^9\ \mathrm{Pa}\) \(10^9\ \mathrm{Pa}\) \(10^9\ \mathrm{Pa}\)      
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
Table 26 Layups
Name Layer Material Ply thickness Orientation Number of plies
      \(\mathrm{m}\) \(\circ\)  
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
Table 27 Effective Timoshenko stiffness matrix
\(\phantom{-}2.395\times 10^9\) \(\phantom{-}1.588\times 10^6\) \(\phantom{-}7.215\times 10^6\) \(-3.358\times 10^7\) \(\phantom{-}6.993\times 10^7\) \(-5.556\times 10^8\)
\(\phantom{-}1.588\times 10^6\) \(\phantom{-}4.307\times 10^8\) \(-3.609\times 10^6\) \(-1.777\times 10^7\) \(\phantom{-}1.507\times 10^7\) \(\phantom{-}2.652\times 10^5\)
\(\phantom{-}7.215\times 10^6\) \(-3.609\times 10^6\) \(\phantom{-}2.828\times 10^7\) \(\phantom{-}8.440\times 10^5\) \(\phantom{-}2.983\times 10^5\) \(-5.260\times 10^6\)
\(-3.358\times 10^7\) \(-1.777\times 10^7\) \(\phantom{-}8.440\times 10^5\) \(\phantom{-}2.236\times 10^7\) \(-2.024\times 10^6\) \(\phantom{-}2.202\times 10^6\)
\(\phantom{-}6.993\times 10^7\) \(\phantom{-}1.507\times 10^7\) \(\phantom{-}2.983\times 10^5\) \(-2.024\times 10^6\) \(\phantom{-}2.144\times 10^7\) \(-9.137\times 10^6\)
\(-5.556\times 10^8\) \(\phantom{-}2.652\times 10^5\) \(-5.260\times 10^6\) \(\phantom{-}2.202\times 10^6\) \(-9.137\times 10^6\) \(\phantom{-}4.823\times 10^8\)
Table 28 Results from reference [CHEN2010]
\(\phantom{-}2.389\times 10^9\) \(\phantom{-}1.524\times 10^6\) \(\phantom{-}6.734\times 10^6\) \(-3.382\times 10^7\) \(-2.627\times 10^7\) \(-4.736\times 10^8\)
\(\phantom{-}1.524\times 10^6\) \(\phantom{-}4.334\times 10^8\) \(-3.741\times 10^6\) \(-2.935\times 10^5\) \(\phantom{-}1.527\times 10^7\) \(\phantom{-}3.835\times 10^5\)
\(\phantom{-}6.734\times 10^6\) \(-3.741\times 10^6\) \(\phantom{-}2.743\times 10^7\) \(-4.592\times 10^4\) \(-6.869\times 10^2\) \(-4.742\times 10^6\)
\(-3.382\times 10^7\) \(-2.935\times 10^5\) \(-4.592\times 10^4\) \(\phantom{-}2.167\times 10^7\) \(-6.279\times 10^4\) \(\phantom{-}1.430\times 10^6\)
\(-2.627\times 10^7\) \(\phantom{-}1.527\times 10^7\) \(-6.869\times 10^2\) \(-6.279\times 10^4\) \(\phantom{-}1.970\times 10^7\) \(\phantom{-}1.209\times 10^7\)
\(-4.736\times 10^8\) \(\phantom{-}3.835\times 10^5\) \(-4.742\times 10^6\) \(\phantom{-}1.430\times 10^6\) \(\phantom{-}1.209\times 10^7\) \(\phantom{-}4.406\times 10^8\)

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 Fig. 42.

_images/examplemh104_comparison.png

Figure 42 Comparison of stiffness matrices after modifying the trailing edge.