Airfoil (MH-104)

Baseline MH-104 airfoil example.

Overview

Figure 31 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 Figure 31. The airfoil is MH 104. 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 16.1% and 51.1% of the chord, respectively. Both low pressure and high pressure surfaces have four segments. The dividing points between segments are listed in Dividing points. Materials are given in Material properties and layups are given in Layups.

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)

Table 25 Material properties

Name	Type	Density	\(E_{1}\)	\(E_{2}\)	\(E_{3}\)	\(G_{12}\)	\(G_{13}\)	\(G_{23}\)	\(\nu_{12}\)	\(\nu_{13}\)	\(\nu_{23}\)
		\(10^3\ \mathrm{kg/m^3}\)	\(\mathrm{GPa}\)	\(\mathrm{GPa}\)	\(\mathrm{GPa}\)	\(\mathrm{GPa}\)	\(\mathrm{GPa}\)	\(\mathrm{GPa}\)
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

Figure 32 Base points of the tube cross-section.

Figure 33 Base lines of the tube cross-section.

Figure 34 Segments of the tube cross-section.

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Input

• mh104.xml

• mh104.dat

• materials.xml

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Run the example

prevabs -i mh104.xml --hm

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Output

Figure 35 Cross-section viewed in gmsh.

• mh104.png

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Analysis Result

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 Figure 36.

Figure 36 Comparison of stiffness matrices after modifying the trailing edge.