ahmed h. bayoumy a continuum based three-dimensional...

Ahmed H. BayoumyGraduate Student

Mechanical Design & Production,

Engineering Department,

Faculty of Engineering,

Cairo University,

Giza, 12613, Egypt

e-mail: [email protected]

Ayman A. Nada1,2

Assistant Professor

Mechanical Engineering Department,

Benha Institute of Technology,

Benha University,

Benha, 13512, Egypt


Said M. MegahedProfessor

Mechanical Design & Production,

Engineering Department,

Faculty of Engineering,

Cairo University,

Giza, 12613, Egypt


A Continuum BasedThree-Dimensional Modelingof Wind Turbine BladesAccurate modeling of large wind turbine blades is an extremely challenging problem.This is due to their tremendous geometric complexity and the turbulent and unpredictableconditions in which they operate. In this paper, a continuum based three dimensional fi-nite element model of an elastic wind turbine blade is derived using the absolute nodalcoordinates formulation (ANCF). This formulation is very suitable for modeling of large-deformation, large-rotation structures like wind turbine blades. An efficient model of sixthin plate elements is proposed for such blades with non-uniform, and twisted nature.Furthermore, a mapping procedure to construct the ANCF model of NACA (National Ad-visory Committee for Aeronautics) wind turbine blades airfoils is established to mesh thegeometry of a real turbine blade. The complex shape of such blades is approximatedusing an absolute nodal coordinate thin plate element, to take the blades tapering andtwist into account. Three numerical examples are presented to show the transientresponse of the wind turbine blades due to gravitational/aerodynamics forces. The simu-lation results are compared with those obtained using ANSYS code with a good agree-ment. [DOI: 10.1115/1.4007798]

Keywords: ANCF, wind turbine blade, thin plate element

1 Introduction

In recent years the aerodynamic performance of wind turbineblades has been considerably improved. This has contributed to anoverall reduction in the cost of wind power produced electricity.The energy capture is approximately proportional to the square ofthe blade length while the blade weight is approximately propor-tional to the cube of its length [1]. To counteract the weightincrease, the development of blades goes towards long and rela-tively flexible structures [1].

The most important aeroelastic components of a wind turbineare the blades. The purpose of the blades is to extract aerodynamicforces from the passing airflow; therefore, they are highly affectedby aerodynamic forces. The development of larger wind turbineshas resulted in long slender blades with high flexibility. The ideaof modeling such flexible multibody systems is to introduce amoving frame of reference to each substructure [2]. Relative tothe moving frame, the elastic displacements are relatively smalland rendering linear analysis possible [3]. Hence, nonlinearitiesare confined to the description of the moving frame. The standardformulation of this method presumes that the moving frame isfixed to the rigid body motion of the substructure. The coordinatesdefining the position and orientation of the moving frame becomea part of the degrees of freedom of the multibody system. How-ever, the use of a mixed set of referential and elastic coordinatesleads to highly nonlinear inertial couplings between the rigid bodymotion and elastic deformation. In the case of rotating machinery,the problem of geometric stiffness arises and wrong results shouldbe obtained if the rotating speed reaches the basic natural fre-quency of the flexible blade [4]. To overcome the geometric stiff-ness effect, the internal elastic coupling between different formsof motion should be taken into consideration. Furthermore, realwind turbine blades are made of composite materials, making

them anisotropic which increase the internal elastic couplingeffect of blade motion. This cannot be described by the movingframe of reference, especially with high rotating speeds. The mod-eling computation problem increases as the rotor blade diameterincreases. For instance, the Enercon E-126 is the largest wind tur-bine model built to date, manufactured by the German wind tur-bine producer Enercon. With a hub height of 135 mð Þ, rotordiameter of 126 mð Þ and a total height of 198 mð Þ, this large modelcan generate up to 7.58 Megawatts of power per turbine [5].

The recently developed ANCF had been used in the analysis oflarge deformation of flexible multibody systems including beltdrives [6–8], rotor blade [9], large deformation piezo-electric lami-nated plates [10], flexible robotic manipulators [4], and cable appli-cations [11]. The important advantage of using this formulation inmultibody computer simulations is the constant mass matrix thatcan be obtained for fully nonlinear dynamic problems. Therefore,this nonlinear finite element formulation can be implemented usingnonincremental solution procedures in a general framework of mul-tibody computer algorithms. The elastic forces; in contrast, are cal-culated using a general continuum mechanics approach. Thisallows for describing the cross-section deformation modes as wellas the deformation modes that appear in the existing beam theories(bending, torsion, longitudinal, and shear deformations). In a gen-eral continuum mechanics theory, the deformation and rotationfields within an infinitesimal volume can be uniquely defined usingnine components of the displacement gradients in the three-dimensional applications. For this reason, the nine independent gra-dient coordinates are used in the ANCF to describe the large rota-tional motion defined using the three independent rotationalparameters as well as the deformation defined using the six straincomponents [3,12]. Using such gradient coordinates leads to sim-pler expressions of the generalized inertia forces and the exact mod-eling of the rigid body motion [12–14].

Recent advances in the ANCF, involving the method of calcu-lating the strain energy [15], illuminating high frequency modes[16,17] and development of stiff integrators [18,19]; help inreducing the calculation time and enhance the sensitivity of thesystem equations. Also the formulation of 3D joint constraints iswell established and verified [20], which enable constructing the

1Corresponding author.2Present address: College of Engineering, Jazan University, Jazan P.O.706, KSA.Contributed by the Design Engineering Division of ASME for publication in

the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript receivedDecember 5, 2011; final manuscript received August 18, 2012; published onlineOctober 30, 2012. Assoc. Editor: Khaled E. Zaazaa.

Journal of Computational and Nonlinear Dynamics JULY 2013, Vol. 8 / 031004-1Copyright VC 2013 by ASME

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model of complete wind turbine blades structure. Further advan-ces in the ANCF were carried out including the modeling of inter-nal damping [21] and nonlinear viscoelasticity [22].

In this paper, a mapping procedure to construct ANCF modelof NACA airfoils of large wind turbine blades is established. Thevariation of the cross sectional layouts across the blade are manip-ulated and the corresponding mapping equations are derived.Because of the special cross-sectional geometric features/parame-ters of wind turbine blades; the use of ANCF and the computa-tional tools that provide fast and accurate results, opensopportunities to improve the design of flexible blades. Theobtained results are necessary for improving the dynamics fordesign (DFD) process concerning large-rotation, large-deforma-tion wind turbine blades.

2 Airfoil Shape Parameters

Wind turbines on the propeller blades possess various bladeprofiles such as NACA, LS and LM profiles standards [23]. Inhorizontal axis wind turbines NACA profiles standards ofNational Advisory Committee for Aeronautics is applied in Ref.[24]. The NACA 44 series profiles were used on older wind tur-bines (up to 95 KW models). This profile was developed duringthe 1930s, and has good all-round properties, giving a good powercurve and a good stall. The shape of the airfoil can be chosen inthe famous NACA 4 digits library [24]. This simple library isinteresting because the shape is analytically expressed as a func-tion of only three parameters, which control the maximum cam-ber, maximum camber location, and maximum thickness of theairfoil, see Fig. 1. A wide variety of airfoils can be obtained byvarying these three parameters, as shown in Fig. 2. A numberingsystem is used to define NACA wing sections by 4-digits: the firstdigit indicates the maximum value of the mean-line ordinate inpercent of the chord. The second digit indicates the distance fromthe leading edge to the location of the maximum camber in tens ofthe chord. The last two digits indicate the section thickness as apercentage of the chord. Thus, the NACA 4512 has 4% camberlocated at 50% of the chord from the leading edge, and is 12%thick. It is noted that these digits do not really have to be integers.

By extension of the notation, a NACA 4.23 2.21 17.2 would have4.23% camber located at 22.1% of the chord from the leadingedge, and be 17.2% thick.

One of the significant factors of a blade profile is the chordlength which is the distance between leading edge and trailingedge. Chord length may have various values at the square and endpoints on the blade.

The camber line can be expressed in the xz plane, as:

zc

c¼ m

p22pn� n2� �

; � � � n � p (1)

¼ m

ð1� pÞ2ð1� 2pþ 2pn� n2Þ; n � p (2)

In these expressions, c is the airfoil chord length, m is the maxi-mum camber, p is the maximum camber location, and n ¼ x=cð Þis the parametric position. The thickness distribution for theNACA 4-digits sections is given by:

zth ¼ 5ch�0:2969n1=2 � 0:1260n� 0:3516n2

þ 0:2843n3 � 0:1015n4�

(3)

where h is the maximum thickness expressed as a fraction of thechord.

Fig. 1 Airfoil shape parameters

Fig. 2 NACA airfoils

031004-2 / Vol. 8, JULY 2013 Transactions of the ASME


The wing section is obtained by combining the camber line andthe thickness distribution as shown on the Fig. 2,

xu ¼ x� zth sinðhÞ (4)

zu ¼ zc þ zth cosðhÞ (5)

xl ¼ xþ zth sinðhÞ (6)

zl ¼ zc � zth cosðhÞ (7)

where xu; zuð Þ and xl; zlð Þ are the coordinates of the upper andlower airfoil surfaces, respectively. Here, h is the slope of thecamber line, and can be calculated using the following equation:

dzc

dx¼ d zc=cð Þ

dn¼ 2m

p2p� nð Þ; � � � n � p (8)

¼ dðzc=cÞdn

¼ 2m

ð1� pÞ2ðp� nÞ; n � p (9)

3 Absolute Nodal Coordinates Formulation (ANCF)

In the absolute nodal coordinate formulation, the nodal coordi-nates of the elements are defined in a fixed inertial coordinate sys-tem, and consequently no coordinate transformation is required.The element nodal coordinates represent global displacements andslopes, and no infinitesimal or finite rotations are used as nodalcoordinates. Furthermore, no assumptions on the magnitude of theelement rotations are made. In the absolute nodal coordinate for-mulation, the global position vector, of an arbitrary point on theplate is defined in terms of the nodal coordinates and the elementshape function as [12]:

r ¼ r1 r2 r3½ �T¼ S x; y; zð Þe (10)

where S is the element shape function matrix, x; y, and z are thelocal coordinates of the element defined in the element coordinatesystem and e is the vector of element nodal coordinates. InEq. (10), the plate is described as a continuous volume, making itpossible to relax the assumption of rigid cross sections [12].

For a 4-noded plate element ð1; 2; 3; 4Þ, as shown in Fig. 3, 12nodal coordinates can be used for each node. The coordinates ofnode 1, e1, can be written as:

e1 ¼ r1T @r1T

@x

@r1T

@y

@r1T

@z

� �T

(11)

¼ ½ r1T r1Tx r1T

y r1Tz �T (12)

where vector r1 defines the global position of node 1 and the threevectors @r1=@x, @r1=@y, and @r1=@z, define the position vector

gradients (slopes) at node 1. The nodal coordinates of one elementcan then be given by the vector e:

e ¼ e1T e2T e3T e4T� �T

(13)

3.1 Thin Plate Element. For thin plates, the deformation ofthe element along the thickness direction can be neglected. Thisleads to reduced set of deformation modes since the displacementfield of the element becomes dependent on the spatial coordinatesx and y only. In this case, the position vector gradients obtainedby differentiation with respect to z are not considered as nodalcoordinates, leading to a reduced order element with 36 degreesof freedom. The normal vector n of the mid surface of the platecan always be defined using cross product of the vectors rx and ry,with subscript x and y refer to partial derivatives with respect tothese coordinates; see Fig. 4. The shape functions of the thin plateelement can be directly obtained from the shape function matrixof the plate element by omitting the components that depend onthe z coordinate [6]. For the reduced order element, the elementnodal coordinate vector at node ð1Þ is defined as follows:

e1 ¼ r1T r1Tx r1T

y

� �T(14)

r ¼ Sðx; yÞe (15)

3.2 ANCF of Airfoil Geometry. The blade profile is a hol-low profile usually formed by two (shell) structures glued to-gether, one upper shell on the suction side, and one lower shell onthe pressure side. It is required to construct the wind turbine bladewith specific NACA code, and therefore, the NACA profile equa-tions should be used to estimate the node position and gradients.In this section, two ANCF thin plate element should be used toconstruct the wind turbine blade and the resulting error should bementioned. One plate on the upper and the other on the lower sideof the wind turbine blade.

The thickness distribution of the airfoil can be described by theslope in xz direction of the first and last nodes, i.e., nodes number(1,4), which have the same slope in x-direction. For the upper ele-ment, the gradient can be estimated as:

rkx ¼

dxu

dx0

dzu

dx

� �T

; � � � k ¼ 1; 2; 3; 4 (16)

rky ¼ 0 1 0½ �T ; � � � k ¼ 1; 2; 3; 4 (17)

where rkx is the gradient at node k, with respect to x. For nodes

number (1) and number (4), Eq. (8) should be used at start pointx 6¼ 0, and Eq. (9) is used for nodes (2) and (3) at the x ¼ c.

For the lower element, the gradient can be estimated as:

rkx ¼ �

dxu

dx0

dzu

dx

� �T

; � � � k ¼ 5; 8 (18)Fig. 3 Four nodes plate element

Fig. 4 Normal vector

Journal of Computational and Nonlinear Dynamics JULY 2013, Vol. 8 / 031004-3


rkx ¼

dxl

dx0

dzl

dx

� �T

; � � � k ¼ 6; 7 (19)

For nodes (5) and (8), use the start point at which x 6¼ 0, and fornodes (6) and (7), the equation is calculated at the x ¼ c. Figure 5shows the elements, nodes and the corresponding gradients. It isfound that, using only two elements for the upper and lower surfa-ces, results a large deviation between the resulting blade and theNACA profile; see Fig. 6.

3.3 Efficient Six-Element ANCF Model. In order toenhance the obtained ANCF model, further elements should beadded to construct the blade model to coincide with the NACAcode. In this section, six ANCF thin plate element should be usedto construct the wind turbine blade, and the NACA thickness dis-tribution (see Eqs. (4)–(7)) should be used to estimate the nodeposition and gradients along the chord length. Three plate ele-ments are used for the upper surface and three others for the lowerside of the wind turbine blade.

The mapping procedure is tabulated in Tables 1 and 2. Thetables show the nodal coordinates and gradients of the upper andlower surfaces, respectively. Figure 7 shows the elements, nodes

and the corresponding gradients, as described in Tables 1 and 2.Figure 8 shows the resulting blade constructed using six plate ele-ments using the ANCF. It shows three different blades constructedusing the ANCF, the first is for NACA 2412, the second forNACA 8412 and the last for inverted NACA 4412. It shown thatthe deviation is minimized between the resulting blade andNACA airfoil blades.

4 Blade Tapering

It is important that the blade sections near the hub are able toresist forces and stresses from the rest of the blade. Therefore, theblade profile near the root is both thick and wide. Further, along theblade, the blade profile becomes thinner so as to obtain acceptableaerodynamic properties. Therefore, the effect of tapering the bladeis obvious; it tends to decrease the stresses. Also, as the blade speedincreases towards the tip, the lift force will increase towards the tip.Decreasing the chord width towards the tip will contribute to coun-teract this effect. From aerodynamical point of view, it improvesthe wind rotor performance, although it increases the manufacturingcost. In other words, the blade tapers from a point somewhere nearthe root towards the tip. In general, the blade profile constitutes acompromise between the desire for strength and the desire for goodaerodynamic properties. At the root, the blade profile is usually nar-rower and tubular to fit the hub. In this section, a method of con-structing the ANCF model of tapered (nonuniform) blade isintroduced using the lofted surface geometry [25].

4.1 Lofted Surfaces. Lofted surfaces are defined as thosesurfaces that through every point on it, there is a straight line thatlies completely on the surface. In the previous section it wasshown the possibility of modeling a uniform wind turbine bladewith the ANCF, with its derived shape functions, S [12]. In thecase of obtaining the global position vector for the nonuniformwind turbine blade, the global position, r, should be linearly inter-polated between the blade starting chord and the ending chord.This bounded curves can be denoted by r n; 0ð Þ and r n; 1ð Þ and bytwo straight segments r 0; gð Þ and r 1; gð Þ connecting them. Surfacelines in g direction are therefore straight, i.e., lofted surfaces,whereas each line in the n direction is a blend of r n; 0ð Þ andr n; 1ð Þ this blend constitutes the following surface expression:

r n; gð Þ ¼ 1� gð Þr n; 0ð Þ þ gr n; 1ð Þ (20)

where g and n are parametric domains such that n; g 2 ½0; 1� and canbe estimated as n ¼ x=a; g ¼ y=b; with a, and b are the plate ele-ment length and width, respectively. It should be mentioned herethat this kind of surface is fully defined by specifying the two bound-ary curves. The four corner points of the plate elements are implicitin these curves. These surfaces are sometimes called “ruled,”because straight lines are an important part of their description.

4.2 Tapering of Blade. Tapering is connecting two crosssections along the span length, Ls, within the blade surface, with

Fig. 5 Constructing airfoil using ANCF (2 elements)

Fig. 6 NACA and ANCF airfoils (2 elements)



an angle called, taper angle, a; see Fig. 9 to achieve the requiredstrength properties for the wind turbine blade. The change in thechord length, Dc; can be obtained as:

Dc ¼ Ls tan a (21)

c2 ¼ c1 � Dc (22)

where c is the chord length, and c1; c2 are the chord length at the“start” and “end” cross sections, respectively. Depending on c1

and c2, the boundary curves can be obtained as:

r n; 0ð Þ ¼ ri ¼ S x; 0ð Þe � � � ; x 2 ½0; c1� (23)

where e is the nodal coordinates of the starting cross section i; seeFig. 9. Thus, the ending cross section can be concluded as:

r n; 1ð Þ ¼ rj ¼ 0 Ls 0½ �Tþ

c2

c1

0 0

0 1 0

0 0c2

c1

266664

377775r n; 0ð Þ (24)

By substituting Eqs. (23) and (24) into Eq. (20) gives the curvesof cross sections between ri and rj. Then, Eq. (20) can be solvedfor the nodal coordinates e of the ending cross section. i.e., thenodal coordinates of nodes number 3; 4; 6; and 14, on the uppersurface and the nodal coordinates of nodes number 9; 10; 12; and16 on the lower surface; see Sec. 3.3. If the upper and lower surfa-ces should be divided into more than one element along the spanlength, Eq. (20) should be used to calculate the correspondingnodal coordinates.

Figure 10 shows a tapered wind turbine blade according toNACA 4412 with tapering angle a ¼ 5 deg, span length Ls ¼ 5 m,and plate thickness t ¼ 0:006 m; while Fig. 11 shows a taperedwind turbine blade according to NACA 8812 with tapering anglea ¼ 8 deg with the same span length and plate thickness.

4.3 Blade Twist. The blade is twisted along its axis so as toenable it to follow the change in the direction of the resultingwind along the blade, which the blade will experience when itrotates. Hence, the pitch will vary along the blade. The pitch is the

Table 1 Mapping equations of upper surface

k x rk rkx rk

y

1 0:1c xu 0 zu½ �T dxu=dx 0 dzu=dx½ �T 0 1 0½ �T

2 p xu 0 zu½ �T dxu=dx 0 dzu=dx½ �T 0 1 0½ �T

3 p xu Ls zu½ �T dxu=dx 0 dzu=dx½ �T 0 1 0½ �T

4 0:1c xu Ls zu½ �T dxu=dx 0 dzu=dx½ �T 0 1 0½ �T

5 c xu 0 zu½ �T dxu=dx 0 dzu=dx½ �T 0 1 0½ �T

6 c xu Ls zu½ �T dxu=dx 0 dzu=dx½ �T 0 1 0½ �T

13 0 xu 0 zu½ �T 0 0 1½ �T 0 1 0½ �T

14 0 xu Ls zu½ �T 0 0 1½ �T 0 1 0½ �T

Table 2 Mapping equations of lower surface

k x rk rkx rk

y

7 0:055c xl 0 zl½ �T dxl=dx 0 dzl=dx½ �T 0 1 0½ �T

8 p xl 0 zl½ �T dxl=dx 0 dzl=dx½ �T 0 1 0½ �T

9 p xl Ls zl½ �T dxl=dx 0 dzl=dx½ �T 0 1 0½ �T

10 0:055c xl Ls zl½ �T dxl=dx 0 dzl=dx½ �T 0 1 0½ �T

11 c xl 0 zl½ �T dxl=dx 0 dzl=dx½ �T 0 1 0½ �T

12 c xl Ls zl½ �T dxl=dx 0 dzl=dx½ �T 0 1 0½ �T

15 0 xl 0 zl½ �T 0 0 �1½ �T 0 1 0½ �T

16 0 xl Ls zl½ �T 0 0 �1½ �T 0 1 0½ �T

Fig. 7 Constructing airfoil using ANCF (6 elements)



angle between the chord of the blade profile and the rotor plane.In order to construct the wind turbine blade with certain twistangle b. The lofted surface described in the previous section ismodified such that the end cross section of the wind turbine bladeis twisted by the angle b; see Fig. 12. This can be done by multi-plying the global position vector matrix of the end section, rj, i.e.,

r n; 1ð Þ, by a rotation matrix around y-axis with the twist angle b.Thus, the global position vector equations of the end cross sectioncan be calculated as:

r n; 1ð Þ ¼cos bð Þ 0 � sin bð Þ

0 1 0

sin bð Þ 0 cos bð Þ

264

375r n; 0ð Þ (25)

where r n; 0ð Þ can be estimated using Eq. (23), the lofted surface,Eq. (20) can be used in the case of dividing the wind turbine bladeinto more than one element along the span length.

Figures 13 and 14 show a tapered/twisted wind turbine bladeaccording to NACA 4412 with tapering angle a ¼ 5 deg, andtwist angle of b ¼ 30 deg, span length Ls ¼ 5 m, and thicknessh ¼ 0:006 m. While Figs. 15 and 16 show a tapered/twisted windturbine blade according to NACA 8812 with tapering anglea ¼ 8 deg twist angle of b ¼ 30 deg, with the same span lengthand thickness.

Twisting the wind turbine blade is vital key for achieving theacceptable aerodynamic properties. The twisting of the blade tipincreases the strength of the blade as the stiffness increases. Also,it is found that by increasing the twist angle and decreasing theeigenvalue, this results in twisting decreases the probability ofreaching the resonant frequency of the blade [26].

5 Equations of Motion

The general dynamic equations of the flexible multibody sys-tems can be written as:

M€e ¼ Q (26)

where M is the mass matrix associated with the ANCF [27], andthe vector Q includes the generalized forces associated with nodalcoordinates, which can be written as:

Q ¼ Qk þQe (27)

Qk is the vector of elastic forces, and Qe is the vector of the exter-nal forces (gravity and aerodynamic forces).

5.1 Mass Matrix. The total kinetic energy T is described by:

T ¼ 1

2

ðV0

q _rT _rdV0 ¼1

2q _eT

ðV0

STSdV0

� �_e (28)

Fig. 8 ANCF models of NACA airfoils, red lines representNACA code profile

Fig. 9 Nonuniform wind turbine blade

Fig. 10 WTB NACA 4412, a 5 5 (deg)



where _r is the velocity vector of an arbitrary point on the plate, qis the material density and V0 is the element initial volume. In thecase of wind turbine blade model, the plate elements are initiallycurved; otherwise, the integration should be estimated withrespect to the straight reference configuration. The relationbetween volumes in the un-curved reference and initially curvedconfiguration can be defined using the gradient transformationmatrix J, and can be expressed as follows [6,28]:

V0 ¼ Jj jV (29)

where V0 is the volume of the element in initial curved configura-tion and V is volume in the un-curved configuration. Therefore,the mass matrix can be written as:

M ¼ð

V

qSTS Jj jdV (30)

5.2 Elastic Forces. The elastic force vector Qk is obtainedby differentiating the strain energy U with respect to the nodalcoordinate vector e, as:

Qk ¼ �@U

@e

� T

(31)

The thin-plate element is based on Kirchhoff’s plate theory.Accordingly, the strain energy can be written as the sum of twoterms: one term is due to membrane and shear deformations at theplate mid-surface, whereas the other term is due to the plate

Fig. 11 WTB NACA 8612, a 5 8 (deg)

Fig. 12 Twisted wind turbine blade

Fig. 13 WTB: twisted NACA 4412, b 5 30 (deg)



bending and twist. The strain energy can then be written for a thinplate as follows [6,12]:

Ue ¼ Ume þ Uj

e

¼ 1

2

ðV0

emT

cEem Jj jdV þ 1

2

ðV0

ejT

cEej Jj jdV(32)

Where cE is the elastic coefficient matrix for homogeneous iso-tropic material [10]. The membrane and shear strain vector em isdefined as:

em ¼ exx eyy 2exy½ �T (33)

where exx and eyy are the normal strain components in x and ydirection and exy is the shear strain. The curvature strain vector ej

is given by:

ej ¼ zj (34)

The curvature vector j ¼ jxx jyy 2jxy½ �T can be expressed interms of the gradient vectors as the following relations [10]:

jxx ¼nTrxx

nk k3; jyy ¼

nTryy

nk k3; jxy ¼

nTrxy

nk k3(35)

where n is the normal to the element mid surface such that:n ¼ rx � ry, and its norm is defined as nk k ¼

ffiffiffiffiffiffiffiffinTnp

.It is important to note that the reference configuration defined

by Jj j remains constant in the ANCF since nonincremental proce-dures are used, which do not require updating reference configura-tion. Using the gradient transformation, the strains of the curvedconfiguration are expressed with respect to those defined in theelement coordinate system. The effect of strains at the initiallycurved configuration is eliminated using Almansi strain [29].

Fig. 14 WTB: front view: twisted NACA 4412, b 5 30 (deg)

Fig. 15 WTB: twisted NACA 8612, b 5 30 (deg)



5.3 Gravitational and Aerodynamic Forces. The general-ized forces of an external force vector F can be expressed as:

Qe ¼ð

V0

STFdV0 (36)

The gravity force per unit volume of plate can be expressed as qg,where q is the density of the material g ¼ 0 0 g½ �T is the grav-ity acceleration vector. Hence, the generalized gravity forces canbe written as:

Qg ¼ qð

V0

STgdV0 (37)

For free stream wind velocity V1, the aerodynamic loads areexpressed in the following forms [30,31]:

FL ¼1

2qa � V2

1 � c � CL (38)

FD ¼1

2qa � V2

1 � c � CD (39)

where FL and FD are the lift and drag forces, qa is density of air.The aerodynamics characteristics of the blade are determinedthrough the lift and drag coefficients, CL, CD, respectively. Thecoefficients mainly depend on the flow angle of attack, /; seeFig. 17. The angle of flow is measured from the inlet tangent tothe mean camber line of the airfoil. The generalized forces due to

aerodynamics load can be established by substituting the resultantforce FR into Eq. (36). FR are distributed over a mid-line of theblade.

6 Dynamic Simulation

The dynamic simulation of a fixed cantilevered wind turbineblade is carried out using multibody system computer code, system-atic analysis of multibody systems (SAMS2000) [32]. Three exam-ples are carried out in order to demonstrate the use of the developedprocedure for modeling large-size wind turbine blades. The numeri-cal integration is carried out by using the sparse HHT integrator[19], with the following parameters: relative error: 1� 10�6, abso-lute error: 1� 10�6, and constraint tolerance: 1�10�8.

The first example concerns fixed cantilevered blade with thefollowing data: NACA 4412, chord length of c1 ¼ 2 m, spanlength Ls ¼ 10 m, taper angle a ¼ 0 deg, twist angle b ¼ 25 deg,plate thickness t ¼ 15 mm. The Polyethylene which is isopara-metric plastic material is chosen with modulus of elasticityE ¼ 1 GPa, Poisson ratio � ¼ 0:4; and density q ¼ 1200 Kg=m

3.

The external force is considered to be the standard gravitationalforces of the blade. In order to compare the results with otherFEM codes, the ANCF model is converted to 2D sections andthen to 3D parasolid models. This 3D model is used to constructthe FEM model using the ANSYS code [33]; see Fig. 18. TheFEM model of the blade is constructed using tetrahedrons mesh-ing and consists of 4208 elements, 8359 nodes. It is observed thatthe numerical results of the ANCF model and ANSYS-FEMmodel are in good agreement; see Fig. 19. The difference betweentransverse deflection curves is due to the oscillatory nature of theANCF model. This nature is the direct effect of the coupled defor-mation modes of the ANCF model [17] comparing with thesmooth results of the ANSYS solution. It is found that, in the caseof a very flexible structure, as in the case of large-size wind tur-bine blade, the inclusion of the ANCF-coupled deformationmodes becomes necessary to obtain an accurate solution. There-fore, in this case, the use of the general continuum mechanicsapproach leads to an efficient solution algorithm and to moreaccurate numerical results.

The second example concerns fixed cantilevered tapered-bladewith the following data: NACA 4412, chord length of c1 ¼ 3 m,span length Ls ¼ 10 m, taper angle a ¼ 3 deg, twist angleb ¼ 25 deg with the same material used in the first example. TheANSYS-FEM model is shown in Fig. 20. The transient solutionFig. 17 Aerodynamic forces

Fig. 16 WTB: front view: twisted NACA 8612, b 5 30 (deg)



Fig. 18 FEM blade model with ANSYS (nontapered blade)

Fig. 19 Transverse deflection of tip edge point

Fig. 20 FEM blade model with ANSYS (tapered blade)



due to gravity force is illustrated in Fig 21. It is observed that,tapering the blade increases the difference between the transversedeflection curves of the two models, not only in the amplitude ofthe deflection but also in the periodic time (frequency) of the twomodels. To obtain good results, the authors believe that the non-conforming plate elements [34] produced due to the tapering prop-erty of the blade should be manipulated carefully. A modifiedmass matrix and elastic force vector should be estimated to takeinto account the nonconforming structure of the reference configu-ration of the plate elements. Other suggestion is increasing thenumber of elements along the span length of the tapered blade,which can decrease the error significantly. The results obtained inthis section, encourage the authors to extend the work to formalizeoptimal design process for large-size wind turbine blades.

In the third example, the blade of the second example is sub-jected to air stream with velocity of 3 m=s. It is shown by Ref.[31] that the aerodynamics characteristics of NACA 4412 are asfollows: mid-blade lift coefficient is CL ¼ 1:35 at flow angle of/ ¼ 12:5 deg, and drag coefficient of CD ¼ 0:03. Other values ofthe coefficients along the mid-line of the blade are obtained fromRefs. 35 and 36. In order to compare the ANCF results withANSYS, advanced coupled numerical method of computationalfluid dynamics (CFD) module and computational flexible multi-body dynamics (CFMBD) module has been developed in order toinvestigate the aero elastic response of this example. The meshingdomains of both the blade wall and fluid are shown in Fig. 22,while the ANSYS solution is shown in Fig. 23. The comparisonbetween the transient responses of the ANCF and ANSYS-FEMmodels is shown in Fig. 24, the numerical results are in good

Fig. 21 Transverse deflection of tip edge point

Fig. 22 ANSYS meshing domains of blade-wall and fluid

Fig. 23 Transient response of aerodynamic forces with ANSYS



agreement with the same noticeable errors due to the nonconform-ing structure of the reference configuration of the plate elements.

7 Comments and Discussion

The numerical examples in the last section are addressed forsix-elements blade model, i.e., only one section is consideredalong the span length of the wind turbine blade. In order toimprove the dynamic simulation results, it is suggested to increasethe number of elements along span length. Figure 25 shows differ-ent blade models with 1, 2 and 4 sections along the span lengthwithin 6, 12 and 24 elements, respectively. It is found that, as thenumber of degree of freedoms increases, the calculation timeincreases dramatically.

In the first numerical example, a reduced order dynamic modelis obtained by excluding some gradients from the integration pro-cess. Several attempts of excluding gradients are carried out tomaintain the correct results of the model with the remaining gra-dients at the nodal points. It is identified that the gradients of@r1=@x; @r2=@x; @r1=@y, and @r2=@y, introduce very high frequen-cies to the blade motion. If those frequencies are excluded, the nu-merical results expected to coincide with ANSYS simulationresults. This is because the position vector r1; r2; r3; and the gra-dients of @r3=@x, and @r3=@y are “only” included in the numericalintegration.

The dynamic simulation of the blade model of first example, inwhich, c ¼ 2 m, Ls ¼ 10 m, b ¼ 25 deg and a ¼ 0 deg is shownin Fig. 26. It shown that the transverse deflection of the 2- and

Fig. 24 Comparison of aerodynamics responses

Fig. 25 WTB model with multisections along the span length

Fig. 26 Comparison of different models along the span length



4-sections reduced order models are in a good agreement with the1-section full order model. Also, as expected, the results of the2- and 4-sections, reduced order models are coincide with ANSYSsimulation results.

8 Summary and Conclusions

In this paper, an efficient procedure is developed for mappingNACA airfoil wind turbine blades into absolute nodal coordinateformulation (ANCF) models. The procedure concerned the windturbine blade with the nonuniform, twisted nature. The variationsof the cross sectional layouts across the blade is manipulated andthe corresponding mapping equations are derived. Several exam-ples of mapping various NACA airfoils are illustrated over the pa-per with good agreements.

The ANCF models of large-size wind turbine blades are solvedusing the HHT integrator implemented in SAMS2000 code, whichis used for dynamic simulation of the developed blade models.The ANCF blade model is converted to 3D parasolid, which canbe invoked by ANSYS code for comparisons. Regardless, the os-cillatory nature of the ANCF response, the numerical results showa very good agreement in transient solution of straight (Nonta-pered) blade due to gravitational forces. Noticeable difference isfound in the case of tapered blades due to the nonconformingstructure of the plate element in the reference configuration. It isfound that, in the case of a very flexible structure, as in the case oflarge-size wind turbine blade, the inclusion of the ANCF-coupleddeformation modes becomes necessary in order to obtain an accu-rate solution. In conclusion, the simulation results show a numeri-cal convergence and good accuracy. Since the ANCF is suited forlarge-deformation, large-rotation problems, which is the case oflarge blades, i.e., the use of ANCF opens opportunities to improvethe design process of such blades. The obtained results are neces-sary for improving the dynamics for design (DFD) process con-cerning large-rotation, large-deformation wind turbine blades.

This work will be extended in later papers to modify the massmatrix and elastic force vector in order to take into account the non-conforming structure of the reference configuration of the plate ele-ments. Modeling slope discontinuities, and the optimum selectionof the number of elements across the span length of the blade.

Acknowledgment

We are grateful to Dynamic Simulation Lab, University of Illi-nois at Chicago, for generous support of license agreement ofSAMS2000 software package and technical support.

Nomenclature

a ¼ length of plate in initial configuration (m)a ¼ taper angle of WTB (rad)b ¼ width of plate in initial configuration (m)

B1 ¼ connectivity matrixB2 ¼ boundary condition matrixb ¼ twist angle of WTB (rad)c ¼ chord length (parametric)

cE ¼ matrix of the elastic constantsCD ¼ drag force coefficientCL ¼ left force coefficient

e ¼ vector of absolute nodal coordinatesE ¼ Young’s modulus of elasticitye ¼ strain vectorF ¼ external force vectorg ¼ gravity acceleration constantg ¼ gravity acceleration vectorh ¼ maximum thickness of airfoil (parametric)g ¼ parametric position along y-axisJ ¼ initial position gradient matrixj ¼ curvature vector

Ls ¼ span length of wind turbine blade (m)

m ¼ maximum camber (parametric)M ¼ mass matrixn ¼ normal vector of the midplane of the platet ¼ Poission ratiop ¼ maximum camber location (parametric)q ¼ mass densityQ ¼ generalized force vector

Qe ¼ generalized external force vectorQg ¼ generalized gravity force vectorQk ¼ generalized elastic force vector

r ¼ position vector in the global coordinate systemrx ¼ longitudinal gradient vectorry ¼ transverse gradient vectorS ¼ shape function matrixt ¼ thickness of plate element (m)

T ¼ kinetic energyU ¼ potential energyV ¼ volume of plate elementx ¼ vector of local coordinates x; y; zð Þ

zc ¼ camber line position coordinates z-axis (parametric)zth ¼ thickness distribution of the NACA airfoil (parametric)/ ¼ flow anglen ¼ parametric position along x-axis

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