Engineering Computations

Engineering Computations Emerald Article: Investigating a Flexible Wind Turbine Using Consistent Time-Stepping Schemes Denis Anders, Stefan Uhlar, Melanie Krüger, Michael Groß, Kerstin Weinberg

Article information: This is an EarlyCite pre-publication article: Denis Anders, Stefan Uhlar, Melanie Krüger, Michael Groß, Kerstin Weinberg, (2012),"Investigating a Flexible Wind Turbine Using Consistent Time-Stepping Schemes", Engineering Computations, Vol. 29 Iss: 7 (Date online 5/9/2012) Downloaded on: 20-06-2012 To copy this document: [email protected]

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Article Title Page Investigating a Flexible Wind Turbine Using Consistent Time-Stepping Schemes Author Details Denis Anders Department of Mechanical Engineering, University of Siegen, Siegen, Germany Stefan Uhlar Basic Development, Voith Hydro Holding GmbH & Co. KG, Heidenheim, Germany Melanie Krüger Department of Mechanical Engineering, University of Siegen, Siegen, Germany Michael Groß Department of Mechanical Engineering, University of Siegen, Siegen, Germany Kerstin Weinberg Department of Mechanical Engineering, University of Siegen, Siegen, Germany

Corresponding author: Denis Anders Corresponding Author’s Email: [email protected] Structured Abstract: Purpose - Wind turbines are of growing importance for the production of renewable energy. The kinetic energy of the blowing air induces a rotary motion and is thus converted into electricity. From the mechanical point of view the complex dynamics of wind turbines become a matter of interest for structural optimization and optimal control in order to improve stability and energy efficiency. Design/Methodology/Approach - In our contribution we present a mechanical model based upon a rotationless formulation of rigid body dynamics coupled with flexible components. The resulting set of differential-algebraic equations will be solved byusing energyconsistent time-stepping schemes. Rigid and orthotropic-elastic body models of a wind turbine show the robustness and accuracy of these schemes for the relevant problem. Findings - Numerical studies prove that physically consistent time-stepping schemes provide reliable results, especially for hybrid wind turbine models. Originality/value - The application of energy-consistent methods for time discretization intends to provide computational robustness and to reduce the computational costs of the dynamical wind turbine systems. Our model is aimed to give a first access into the investigation of fluid-structure interaction for wind turbines. Keywords: wind turbine; conserving time integration; constrained mechanical systems; flexible multibody dynamics; differentialalgebraic equations Article Classification: Research paper

For internal production use only Running Heads:

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Engineering Computations: International Journal for Computer-Aided Engineering and Software c Emerald Group Publishing Limited

INVESTIGATING A FLEXIBLE WIND TURBINE USING CONSISTENT TIME-STEPPING SCHEMES

Denis Anders Chair of Solid Mechanics, Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Str. 9-11 57076 Siegen, Germany [email protected] http://www.uni-siegen.de/fb11/fkm Stefan Uhlar Voith Hydro Holding GmbH & Co. KG, Basic Development 89518 Heidenheim, Germany Melanie Kr¨ uger Chair of Computational Mechanics, Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Str. 9-11 57076 Siegen, Germany [email protected] http://www.uni-siegen.de/fb11/nm Michael Groß Chair of Computational Mechanics, Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Str. 9-11 57076 Siegen, Germany [email protected] http://www.uni-siegen.de/fb11/nm Kerstin Weinberg Chair of Solid Mechanics, Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Str. 9-11 57076 Siegen, Germany [email protected] http://www.uni-siegen.de/fb11/fkm Received (– – 2011) Revised (– – 2011) Purpose - Wind turbines are of growing importance for the production of renewable energy. The kinetic energy of the blowing air induces a rotary motion and is thus converted into electricity. From the mechanical point of view the complex dynamics of wind turbines become a matter of interest for structural optimization and optimal control in order to improve stability and energy efficiency. 1

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2 Design/Methodology/Approach - In our contribution we present a mechanical model based upon a rotationless formulation of rigid body dynamics coupled with flexible components. The resulting set of differential-algebraic equations will be solved by using energy-consistent time-stepping schemes. Rigid and orthotropic-elastic body models of a wind turbine show the robustness and accuracy of these schemes for the relevant problem. Findings - Numerical studies prove that physically consistent time-stepping schemes provide reliable results, especially for hybrid wind turbine models. Originality/value - The application of energy-consistent methods for time discretization intends to provide computational robustness and to reduce the computational costs of the dynamical wind turbine systems. Our model is aimed to give a first access into the investigation of fluid-structure interaction for wind turbines. Keywords: wind turbine; conserving time integration; constrained mechanical systems; flexible multibody dynamics; differential-algebraic equations Paper type Research paper

1. Introduction As the world faces running dry of fossil fuels wind power becomes a prominent source of energy that can be relied on in the long-term future. Turbines converting wind into electrical energy are typically designed with three propeller-like blades (rotor) which are attached to a nacelle containing the electrical generator, see Fig. 1. The wind turbine is revolvable set on a tower to adaptively point into the wind. Because winds are stronger on higher distances from the ground the towers are on average 80 m high, some of them even exceed 200 m. Figure 1 here. The process of energy conversion in wind turbines is simple. The wind causes the blades to rotate and, when the rotor spins, the turbine generates electricity out of its kinetic energy. However, the mechanics behind wind turbine design is rather challenging. Aside of structural optimization of single components, cf. [Molenaar and Dijkstra (1999)], the multibody modeling of the whole structure has gained attention in recent years, see, e.g., [Larsen and Nielsen (2006); Holm-Jorgensen and Nielsen (2009); Baumjohann et al. (2002); Zhaoa et al. (2007)]. In this context it became evident that robust and energy conserving numerical schemes for time integration are prerequisite to capture the motion of rotating wind turbines in the long run. Therefore, this contribution presents a mechanical model of a 3-blade wind turbine with a momentum and energy conserving time integration of the system. Energyconsistent time integration schemes have been introduced in general e.g., in [Uhlar (2009); Bauchau and Bottasso (1999); Bottasso et al. (2001); Bauchau (1998); Bauchau et al. (2003); Betsch and Steinmann (2002a); Uhlar and Betsch (2008b); Uhlar and Betsch (2009)]. Here we follow these ideas and exploit them for modeling the relevant system of wind turbines. To this end we study a rigid multibody model of a 3-blade turbine as well as a flexible multibody model with orthotropic-elastic blades. Both systems result in sets of differential-algebraic equations of second order and are efficiently solved by means of a basic energy-momentum and a basic hybrid

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3

energy-momentum scheme, respectively. By means of comparison the effect of large elastic deformations on the system’s energy and stress distribution is shown. The remaining of the paper is organized as follows: In Section 2 we summarize the governing equations of motion. Furthermore, the essential mathematical background of multibody systems is presented and we introduce a model consisting exclusively of rigid bodies. The rigid-body system is, of course, a strong simplification of the real structure. Nonetheless, in case of negligible deformations this model provides sufficiently good results. In Section 3 the model is extended to a hybrid model by means of flexible components able to map large elastic deformations of the blades. Numerical simulations of the rigid and the hybrid wind turbine model including academic model problems as well as technical relevant scenarios are presented in Section 4. 2. Multibody systems In this section we describe rigid body systems and introduce the corresponding equations of motion arising from the kinematic assumptions and external constraints due to the linkage of rigid bodies. This leads to a system of differential algebraic equations. The formulations are based on works of Betsch, Uhlar and Leyendecker [Uhlar (2009); Uhlar and Betsch (2007); Uhlar and Betsch (2008a); Betsch and Uhlar (2007); Betsch and Steinmann (2001); Betsch and Leyendecker (2006)]. 2.1. Rigid body kinematics and dynamics The configuration of a rigid body B in the three dimensional Euclidean space relative to an inertial Cartesian basis {e1 , e2 , e3 } can be characterized by the location of its center of mass ϕ(t) ∈ R3 and a right-handed body frame {di (t)} , di ∈ R3 (i = 1, 2, 3). Here the system {di (t)} specifies the orientation of the body, see Fig. 2. Figure 2 here. Let X = Xi ei a be an arbitrary material point in the reference configuration B0 ⊂ R3 . The spatial position x ∈ B at time t can be characterized by x (X, t) = ϕ(t) + Xi di (t).

(1)

For simplicity we assume that the axes of the body frame coincide with the principal axes of the rigid body. Then the kinetic energy of the rigid body can be written as 3

T =

1X 1 2 2 Mϕ kvϕ k + Ei kvi k , 2 2 i=1

˙ vi = d˙ i and where vϕ = ϕ, Z Mϕ = ̺ (X) dV, B

a In

Ei =

Z

2

(Xi ) ̺ (X) dV. B

this paper the summation convention applies to related lower case Roman indices.

(2)

(3)

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In this context ̺ (X) is the mass density at X ∈ B. Mϕ denotes the total mass of the body and Ei are the principal values of the Euler tensor with respect to the center of mass. Note that the spectral decomposition of the current Euler tensor E with respect to the center of mass is given by E=

3 X i=1

Ei di ⊗ di ,

(4)

where the operation ⊗ denotes the dyadic product. The Euler tensor is symmetric positive definite and can be linked to the customary inertia tensor J via the relationship J = (trE) I − E.

(5)

Apparently, the configuration of a rigid body can be described employing a rotationless formulation (see also [Betsch and Steinmann (2001); Betsch and Uhlar (2007); Betsch and Leyendecker (2006); Shabana (1998)]) by the vector of redundant coordinates q ∈ R12 with iT h (6) q = ϕ d1 d2 d3 . By means of Eq.(6) the kinetic energy expression (2) can be reformulated in a more compact manner introducing the constant mass matrix   Mϕ I3×3 03×3 03×3 03×3    03×3 E1 I3×3 03×3 03×3  , M= (7)  0   3×3 03×3 E2 I3×3 03×3  03×3 03×3 03×3 E3 I3×3 where I3×3 and 03×3 are the 3 × 3 identity and zero matrices. Then it holds T =

1 1 q˙ · Mq˙ = v · Mv. 2 2

(8)

According to the assumption of rigidity the body frame {di (t)} has to form for all times an orthonormal system fulfilling di · dj = δij . In this framework there are m = 6 independent internal constraints condensed in an associated constraint function Φint : R12 7−→ R6 with   d1 · d1 − 1    d2 · d2 − 1    d · d − 1  3 3  Φint (q) =  (9) .  d1 · d2       d1 · d3  d2 · d3

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The internal constraint function Φint gives rise to the corresponding Jacobian Gint (q) ∈ R6×12 given by   0T 2dT1 0T 0T  T T   0 0 2dT2 0T    T T T T  ∂Φint (q)   0 0 0 2d3  Gint (q) = = T T (10) ,  0 d2 dT1 0T  ∂q    T T   0 d3 0T dT1  0T 0T dT3 dT2

where the symbol 0T denotes a suitable zero vector. 2.2. Incorporation of joints

In this subsection we present the concept of coordinate augmentation as a technique to incorporate rotational variables into a multibody system [Betsch and Uhlar (2007); Betsch and Leyendecker (2006); Uhlar and Betsch (2008a); Uhlar and Betsch (2007)]. Here we consider a representative multibody system consisting of 2 rigid bodies which are connected by a joint. Accordingly, each rigid body can be characterized by a configuration vector. To distinguish between the different bodies we number the configuration vector of rigid body A = 1, 2 as qA . A ∈ R6 are numIn this way the individual functions of internal constraints ΦA int q A A 6×12 bered as well as the corresponding matrices Gint q ∈ R and MA ∈ R12×12 . For the representation of the global system a global configuration vector iT h q = q1 q2 is introduced. In this notation the global function of internal constraints Φint , the corresponding Jacobian Gint and the mass matrix M are given by " # " # " # Φ1int G1int 06×12 M1 012×12 Φint = , Gint = , M= . (11) Φ2int 06×12 G2int 012×12 M2

The interconnection between rigid bodies in a multibody system is referred to external constraints. For a revolute joint that allows the rotation of one part with respect to another part about a common axis (illustrated in Fig. 3), the vector valued constraint function is given by   ϕ2 − ϕ1 + η 2 − η 1   , (12) Φext (q) =  d12 · d21   d12 · d23

where the vector

η A = ηiA dA i

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specifies the position of the joint on body A with respect to the body frame dA i . Figure 3 here. Therefore the corresponding Jacobian Gext has the following representation   −I −η11 I −η21 I −η31 I I η12 I η22 I η32 I  T T T ∂Φext (q)  Gext = . (13) = 0 0T  0T 0T 0T d12 0T d21  ∂q  T 2 T T T T T T 1 T 0 d3 0 0 0 0 0 d2

Then the constraint function Φ (q) and the constraint Jacobian are given by " # " # Φint (q) Gint (q) Φ (q) = , G (q) = . (14) Φext (q) Gext (q) Now we come to the introduction of an additional rotational variable Θ specifying the rotation of body 1 relative to body 2 about a common axis. First we extend the configuration vector h iT q = q1 q2 Θ . (15) The new coordinate Θ is now related to the original set of coordinates by an additional constraint function Φaug (q) = d21 · d11 + sin Θ + d21 · d13 − cos Θ = Φ1aug (qori ) + Φ2aug (Θ) .

(16)

For simplicity we split additively Φaug in a part depending on the original configuration vector iT h qori = q1 q2 and the rotational variable Θ, where

Φ1aug (qori ) = d21 · d11 + d21 · d13 ,

Φ2aug (Θ) = sin Θ − cos Θ.

The Jacobian of (16) is then given by ∂Φaug (q) ∂q i h T T T = 0T d21 0T d21 0T d11 + d13 0T 0T (sin Θ + cos Θ) h i = G1aug (qori ) G2aug (Θ) .

Gaug (q) =

(17)

Finally, the global constraint function Φ, the corresponding Jacobian G and the mass matrix M can be stated as # " # " # " Φori (qori ) Mori 024×1 Gori (qori ) 015×1 . Φ (q) = ,M= , G (q) = 01×24 0 G1aug (qori ) G2aug (Θ) Φaug (q) (18) Actually we are able to consider other types of joints by means of augmented constraints. In the scope of this manuscript we will employ exclusively revolute joints to

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link structural components of our wind turbine system. For a more detailed insight into modeling techniques of a multifaceted class of joints in multibody systems we refer to [Betsch and Leyendecker (2006)]. 2.3. Equations of motion in the rotationless framework In the present subsection we introduce the basic set of equations of motion providing a uniform framework for the rotationless formulation of multibody dynamics subjected to internal and external constraints. Internal constraints arise from the rigidity of the bodies whereas external constraints follow from the presence of joints [Betsch and Steinmann (2001); Betsch and Uhlar (2007); Betsch and Leyendecker (2006); Uhlar and Betsch (2007)]. This manner of formulation circumvents the requirement to introduce terms of angular velocities and accelerations in the subsequent time discretization schemes. Conveniently, we focus here on discrete mechanical systems which are addressed by the following form of equations of motion T

q˙ − v = 0

Mv˙ − f (q) + G (q) λ = 0

(19)

Φ (q) = 0.

Here q (t) ∈ Rn characterizes the configuration of the mechanical system at time t. Therefore in the present work q will be referred to as configuration vector and v (t) ∈ Rn denotes the velocity vector. Furthermore, vector f (q) ∈ Rn summarizes all loads exerted on the system and it may be additively decomposed into f = Q − ∇V (q). In this context V (q) ∈ R is a potential energy function and Q ∈ Rn signifies all loads which cannot be derived from a potential. Vector Φ (q) ∈ Rm characterizes the function of geometric (holonomic) constraints and G = Dq Φ(q) ∈ Rm×n is the corresponding constraint Jacobian. λ ∈ Rm is a vector containing Lagrange multipliers related to the relative magnitude of constraint forces. Since we study a constrained mechanical system the configuration space for (19) is given by Q = {q (t) ∈ Rn | Φ (q) = 0} .

(20)

2.4. Discretization of the equations of motion The system (19) constitutes a set of differential-algebraic equations (DAEs). For the temporal discretization we consider a representative time interval T which is divided into nt subintervals In = [tn , tn+1 ] according to T =

n[ t −1 n=0

In .

(21)

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The first order time derivative in (19)1 is approximated by finite differences with (equidistant) time-step ∆t = tn+1 − tn . For time integration we employ the socalled basic energy-momentum (BEM) scheme providing algorithmic conservation properties [Betsch and Uhlar (2007); Uhlar (2009)]. The discretized version of (19) reads: ∆t qn+1 − qn = (vn + vn+1 ) 2 (22) ¯=0 ¯ T (qn , qn+1 ) λ M (vn+1 − vn ) − ∆t f (qn , qn+1 ) + ∆t G Φ (qn+1 ) = 0

¯ (qn , qn+1 ). Here the overline denotes the with f (qn , qn+1 ) = Q (qn , qn+1 ) − ∇V discrete gradient of the function below. If this function is at most quadratic in terms of the configuration vector q the discrete gradient coincides with the standard midpoint evaluation. The discrete version of Eq. (18)2 reads   015×1 Gori (qori )n+ 1 2 . (23) G (qn , qn+1 ) =  G1aug (qori )n+ 1 G2aug (Θn , Θn+1 ) 2

Φ1aug

Since the constraint functions Φori and are at most quadratic polynomials with respect to the configuration vector, the discrete gradient is replaced by the midpoint evaluation of the respective continuous constraint Jacobians. Unfortunately a midpoint evaluation for Φ2aug (Θ) will be inexact because Φ2aug is a highly nonlinear function. In this case we choose  2 ) − Φ2aug (Θn ) Φ (Θ   aug n+1 if Θn+1 6= Θn Θn+1 − Θn . (24) G2aug (Θn , Θn+1 ) =  ′  2 if Θn+1 = Θn Φaug (Θn )

2.5. Rigid wind turbine system

For the rigid multibody system representing a 3-blade wind turbine we choose a model consisting of five rigid bodies. The blades are allocated uniformly around the nave so they enclose an angle of 120◦ among themselves. The geometry of our wind turbine model with the respective body frames is presented in Fig. 4 and Fig. 5. The tower is regarded to be a part of the environment. The nacelle is connected to the tower by a revolute joint allowing a rotation around the vertical axis of the tower in order to point the rotor into the wind. Θ1 is the rotational variable to specify this motion. The nave is connected with the nacelle by a revolute joint as well to depict the motion of the rotor by Θ2 . All three blades share a revolute joint with the nave. The variables Θ3 , Θ4 , Θ5 can be used to adapt the blade settings to changing wind profiles and thus influence the motion of the rotor. This kind of bearing takes the load of all system components during turbulent wind and storms. Otherwise system components may be damaged by strong oscillations.

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Figure 4 here. Now we embed this model in the framework of the BEM. The global configuration vector for the wind turbine model is given by iT h (25) q = q 1 q 2 q 3 q 4 q 5 Θ1 Θ2 Θ3 Θ4 Θ5 .

The internal constraints due to the rigidity of the system are straightforward; they coincide with relation (9) for each body.   di1 · di1 − 1  i   d2 · di2 − 1     di · di − 1    3 3 Φiint (q) =  for i = 1, 2, 3, 4, 5. (26)   di1 · di2       di1 · di3  di2 · di3 The external constraints arising from the revolute joint bearing between the rigid bodies can be stated as       d1 + d2 2 1 3 3 3 1 ϕ + d c − ϕ ϕ + d ϕ 3 3 2   2     rev    3 2   1 rev   ◦  , Φ = , Φ = Φrev = 2 1    d · d + sin (60 )  d ·e 2 1  3  d3 · d1   2 1  1 2   3 2 1 d · e d13 · e2 3 d3 · d2 2 (27)



   Φrev = 4  

4

−ϕ +

d42

L2 + L4 2

d42

·

d21

+ϕ

2





ϕ5 + d52 c5 − ϕ3



     5 2 , Φrev ◦  5 =  d2 · d1 − cos (30 )  ,     5 d2 · e3

d42 · d23

(28)

b2 Li + . The constraints due to the coordinate augmentation ◦ 2 cos (30 ) 2 technique are given by

with ci =

Φ1aug = d13 · e1 + d13 · e3 + sin Θ1 − cos Θ1 ,

Φ2aug = d22 · d11 + d22 · d12 + sin Θ2 − cos Θ2 ,

Φ3aug = d33 · d21 + d33 · d23 + cos (60◦ ) sin Θ3 − cos Θ3 ,

Φ4aug Φ5aug

=

=

d43 d53

·

·

d21 d21

+

+

d43 d53

·

·

d23 d23

(29)

+ sin Θ4 − cos Θ4 ,

+ cos (60◦ ) sin Θ5 − cos Θ5 .

As initial configuration we choose the following settings, cf. Fig. 5, ϕ1 = 0,

d11

= e1 ,

d12 = e2 ,

d13

= e3 ,

(30)

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ϕ2 = 0 0 − d21 = e1 , d22 d23

d1 + d2 2

,

= e2 , = e3 ,

ϕ4 = 0

L2 + L4 d 1 + d 2 − 2 2

d41 = e1 ,

d + d2 ϕ3 = c3 sin (60◦ ) −c3 cos (60◦ ) − 1 , 2 h i d31 = cos (60◦ ) sin (60◦ ) 0 , h i d32 = − sin (60◦ ) cos (60◦ ) 0 ,

(31)

d33 = e3 ,

d42 = e2 , d43 = e3 ,

d + d2 ϕ = −c5 cos (30 ) −c5 sin (30 ) − 1 , 2 h i 5 d1 = cos (60◦ ) − sin (60◦ ) 0 , h i d52 = cos (30◦ ) sin (30◦ ) 0 , 5

◦

◦

,

d53 = e3 .

(32) Figure 5 here. 3. Hybrid multibody systems In the following we will present a mathematical concept of the coupling between rigid and flexible bodies according to [Bottasso et al. (2001); Betsch and Steinmann (2002a); Betsch and Sänger (2009); Betsch and Steinmann (2002b); Uhlar (2009); Betsch et al. (2010); Bauchau and Bottasso (1999); Bauchau (1998); Bauchau et al. (2003); Géradin and Cardona (2001)]. In many technical applications system components have to meet contrasting requirements. On the one hand, these components need to be extremely stiff in order to ensure the stability of the whole structure, on the other hand, they have to be as light as possible for reason of energy efficiency. As a consequence, modern mechanical structures often combine very stiff and heavy components with light elastic parts to achieve an optimal performance. Numerical methods have to take this fact into account and, therefore, we will present here an energy-consistent computational approach for structures composed of rigid and flexible parts according to [Uhlar (2009); Uhlar and Betsch (2009); Uhlar and Betsch (2008b)]. 3.1. Flexible body dynamics We consider the motion of a flexible body B in the 3-dimensional Euclidean space during a time interval T = [0, te ]. The motion starts at time t = 0 with the reference configuration B0 . Any position x of the body for t > 0 arises from the reference configuration B0 by means of a deformation mapping ϕ(X, t). Thereby, X denotes the position of a material point of the body in the reference configuration B0 . The approximation of this mapping owing to the displacement-based finite element

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concept (see [Hughes (2000)]) is given by x = ϕ(X, t) =

nX node

NA (X) qA (t),

(33)

A=1

where the index A = 1, . . . , nnode indicates the spatial finite element node, and NA and qA denote the corresponding global shape function and position vector, respectively. The material velocity vector is then v=

nX nX node node ∂ϕ NA v A NA q˙ A = = ∂t

(34)

A=1

A=1

with the nodal velocity vectors vA . By the deformation mapping (33), we obtain the deformation gradient F = ∇X ϕ =

nX node A=1

qA ⊗ ∇X NA

(35)

where symbol ∇X denotes here the gradient operator with respect to X. As deformation metric, we use the right Cauchy-Green tensor C = FT F =

nX node

qA · qB ∇X NA ⊗ ∇X NB .

(36)

A,B=1

Considering nonlinear elastodynamics with respect to hyperelastic materials, the symmetric second Piola-Kirchhoff stress tensor S emanates from the scalar-valued (C) . The potential energies strain energy function W (C) by the relation S = 2 ∂W∂C Vint and Vext associated with the conservative internal and external forces, respectively, are given by Z Z Z Vint = W (C) dV and Vext = − ̺0 b · ϕ dV − t · ϕ dA. (37) B0

B0

∂B0

Here, b denotes a constant volume force vector, t a constant surface force vector and ̺0 the mass density in the reference configuration. The deformation mapping (33) then leads to nX node q A · FA (38) Vext = − ext = −q · Fext A=1

with the external nodal forces Z Z FA = N ̺ b dV + A 0 ext B0

NA t dA

(39)

∂B0

as well as the configuration vector qT = [q1 . . . qnnode ] and the global force vector node ]. The kinetic energy T of the body follows from the material FText = [F1ext . . . Fnext velocity vector (34) by the relation Z nnode 1 1 X 1 MAB vA · vB = v · M v (40) ̺0 v · v dV = T = 2 B0 2 2 A,B=1

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with the global mass matrix    M=  

M11 I3×3 .. .

...

M1,nnode I3×3 .. .

Mnnode ,1 I3×3 . . . Mnnode ,nnode I3×3

corresponding to the components MAB =

Z

     

̺0 NA NB dV

(41)

(42)

B0

and the configuration velocity vector vT = [v1 . . . vnnode ]. By means of Hamilton’s principle associated to the semi-discrete Lagrange function L = T − Vint − Vext , we obtain the first order semi-discrete equations of motion q˙ − v = 0

(43)

M v˙ + K q − Fext = 0 describing the motion of the spatial finite element nodes. Thereby, the matrix K takes the form   K I . . . K I 11 3×3 1,nnode 3×3     .. ..   (44) K= . .    Knnode ,1 I3×3 . . . Knnode ,nnode I3×3

with the components

KAB =

Z

SAB dV, B0

(SAB = S : ∇NA ⊗ ∇NB ) .

(45)

The term semi-discrete refers to a formulation involving a system discretized only in one component either in time or space. Here we consider a formulation where the spatial discretization is performed at first. This system of first order ordinary differential equations leads to the conservation of the total energy H = T + Vint + Vext . This can be shown by employing the semi-discrete equations of motion in the time derivative H˙ = v · M v˙ + [K q − Fext ] · q˙ − q · F˙ ext

(46)

which leads to H˙ = 0 in view of constant external forces. The total angular momentum L of the spatial finite element discretization with respect to the reference frame is defined by Z nX node L= ̺0 ϕ × v dV = ̺0 qA × vA . (47) B0

A=1

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Differentiation of the total angular momentum with respect to time leads to L˙ =

nX node A=1

̺0 q˙ A × vA + qA × v˙ A .

(48)

Employing the semi-discrete equations of motion and taking the symmetry of the components MAB and KAB into account, the skew-symmetry of the cross product renders L˙ = T with the external total torque T=

nX node A=1

q A × FA ext .

(49)

Hence, we arrive at a constant total angular momentum L if no external forces act on the body. The total linear momentum P of the spatial finite element discretization reads Z nX node P= ̺0 vA (50) ̺0 v dV = B0

A=1

since the global shape function NA satisfies the completeness condition (partition Pnnode of unity) A=1 NA = 1 (see [Hughes (2000)]). After employing the semi-discrete equations of motion, the time derivative of the total linear momentum is given by ! Z nX nX nX node node node B ˙ =N− FA P ∇X NB dV (51) ∇X NA · S int = N − q A=1

B0

A=1

B=1

with the acting total force N=

nX node

FA ext .

(52)

A=1

Taking the completeness condition of the sum vanishes and we obtain the relation mentum is conserved if no external forces

global shape functions into account, the ˙ = N. Therefore, the total linear moP act on the body.

3.2. Coupling of rigid and flexible components At this point we will outline the basic ideas of the coupling between rigid body components with flexible structures. This combination will be accomplished by introducing so-called coupling constraints which employ a closed loop of vectors according to Fig. 6 connecting the rigid body B RB with a representative node on the contact surface taken from the finite element mesh characterizing the flexible component B flex [Uhlar (2009); Uhlar and Betsch (2008b); Betsch and Sänger (2009)]. Consequently, the coupling constraint which depends linearly on the configuration is given by Φicoup = ϕRB + ϕRF − ϕiN .

(53)

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The number of coupling constraints equals the number of the nodes located at the contact surface between rigid and flexible part. Vector ϕRF remains constant in length for all times and can be obtained by means of a rotation matrix R and the corresponding vector ϕRF from initial configuration by the following relation 0 h i ϕRF = RϕRF , with R = d d d 0 1 2 3 .

(54)

Figure 6 here.

3.3. Basic hybrid energy momentum scheme (BHEM) Here we present the basic set of equations to depict the dynamics of the coupled system. First of all we introduce a global configuration vector containing all unknowns of our system h iT q = qRB λint λcoup λext λaug qflex .

(55)

Using this notation the description of the system’s dynamics can be condensed to the following set of continuous DAEs q˙ = v M

RB

v˙

RB

+f

RB

+

GTint λint

+

GText λext

+

GTcoup λcoup

+ GTaug λaug Φint qRB RB flex

Φcoup q

M

flex flex

v˙

+f

flex

+

,q

Φext q

RB

Φaug q

RB

=0 =0 =0

=0

GTcoupF λcoupF

= 0,

(56)

=0

with Gcoup =

∂Φcoup ∂qRB

and

GcoupF =

∂Φcoup . ∂qflex

(57)

Although system (56) may look rather complicated at the first glance, it simply involves the equations of motion for the rigid bodies subjected to internal, external, coupling and augmented constraints (56)2 . Eq. (56)7 describes the dynamics of the flexible system components opposed to coupling constraints to the rigid bodies. The temporal discretization of system (56) follows Eq. (21) and the numerical integration is performed by the basic hybrid energy momentum scheme (BHEM), see

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[Uhlar (2009)].

2MRB RB RB RB T + ∆t GTint qRB q qn+1 − qRB λ + ∆t G 1 1 int n − ∆t vn ext n+ 2 n+ 2 λext ∆t ¯ T λaug + ∆t ¯f RB = 0 =Rrigid +∆t GTcoup λcoup + ∆t G aug n+1,n Φint qRB n+1 = 0 =Rint flex Φcoup qRB n+1 , qn+1 = 0 =Rcoup Φext qRB n+1 = 0 =Rext Φaug qRB n+1 = 0 =Raug

2Mflex flex flex − 2Mflex vnflex + ∆t GTcoupF λcoupF + ∆t ¯fn+1,n = 0 =Rflex qn+1 − qflex n ∆t (58) In this context

h iT R = Rrigid Rint Rcoup Rext Raug Rflex

(59)

denotes the global residual vector.

4. Numerical studies In the last section we present the numerical results obtained for our rigid and hybrid multibody models of one representative wind turbine. All simulations are performed in the presence of gravity within a representative time interval of 12 s and with a time-step size of ∆t = 0.02 s. The nacelle and the nave are always modeled as rigid bodies. In our hybrid wind turbine framework we treat the blades as flexible structures involving a St. Venant-Kirchhoff material model. The blades have a length of 54 m and a maximum diameter of 3 m. The mesh for the blade geometry was generated in Abaqus-Cae using 144 cubic three-dimensional trilinear finite elements. We regard the system to be initially at rest, which is expressed by v0 = 0. Since all revolute joints in the system are extended with augmented rotational variables Θi it is possible to trigger these joints with associated torques Fi (t). In this work both systems are loaded by torques Fi (t) with the shape of a hat function during a loading period of 3 s for i = 1, 2, 3, 4, 5, see Fig. 7.

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For the employed Fi (t) it holds  f   it if 0 ≤ t ≤ t1 Fi (t) = t1 f i   (t − t2 ) if t1 ≤ t ≤ t2 t1 − t2 (60) We take the following set of parameters during all simulations for Fi (t): t1 = 1.5

t2 = 3

f1 = 0

f2 = 3.6 · 104 kNm

f3 = f4 = f5 = 0

Figure 7 here.

The set of nonlinear DAEs is numerically solved by the application of a NewtonRaphson method with a given tolerance of ε = 10−7 . This, actually, will be the tolerance to qualify the conservation of energy and angular momentum. As quantities to test the consistency in energy and angular momentum we introduce a ∆-Hamiltonian and ∆-functions of angular momentum separated into the three spatial components. In general we define a ∆-(•) function as the difference of the considered physical quantity at time-step n and n + 1. Both, the ∆-Hamiltonian and ∆-angular momentum are defined by ∆-Hamiltonian: ∆-angular momentum:

∆H = Hn+1 − Hn

(61)

∆L = Ln+1 − Ln

All computations are performed using dimensionless parameters. For the rigid model the scaled mass m, the principal values of the Euler tensor and the geometrical dimensions are summarized in Tab. 1. body

m [t]

1 2 3 4 5

200 61 16.5 16.5 16.5

E1 [t m2 ] 600 102.94 183.5 183.5 183.5

E2 [t m2 ] 600 102.94 1449.5 1449.5 1449.5

E3 [t m2 ] 600 81.33 18.9 18.9 18.9

length [m]

diam. [m]

width [m]

6 4.5 54 54 54

6 4 1.8 1.8 1.8

6 4.5 3 3 3

Table 1: Parameters for the rigid body model.

The chosen dimensions coincide with the geometrical data of contemporary wind turbines. Let us remark that the principal values of the Euler tensor are obtained from the principal moments of inertia from the blade geometry of the flexible part to keep both models comparable.

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4.1. Simulation results for the rigid wind turbine model The simulations for the rigid wind turbine model show a very robust numerical behavior. After the loading period of 3 s the system’s energy and angular momentum reach a certain fixed value and remain constant within the tolerance of the Newton iteration. Figure 8 and 9. In the left part of Fig. 8 we see that our numerical scheme is energy consistent. The ∆-Hamiltonian is far below the given tolerance. The consistency in angular momentum is illustrated in Fig. 9, where we split the angular momentum in spatial components L1 , L2 and L3 . The left image of Fig. 9 again indicates the algorithmic conservation of angular momentum in our rigid wind turbine model. The different configurations during simulation are presented in Fig. 10 and Fig. 11b . To illustrate the rotatory motion in snapshots one blade is partly dyed in light green. Figure 10 and 11 here. 4.2. Simulation results for the anisotropic and isotropic hybrid wind turbine model In order to present the simulation results obtained for our hybrid wind turbine model we follow two strategies. At first we choose a realistic orthotropic material. Since our hybrid model is based on the St. Venant-Kirchhoff material model in the context of linear elasticity, the stress-strain relation can be expressed by means of stiffness matrix D in the usual Voigt notation with   C11 C12 C13 0 0 0 C C C 0 0 0   12 22 23   (62) D = C013 C023 C033 C0 00 00  . 44   0 0 0 0 C55 0 0 0 0 0 0 C66 The strain-stress relation for orthotropic linear elastic material behavior can then be written by means of the compliance tensor D−1 with  1 −ν −ν  12 13 0 0 0 E1 E2 E3  −ν12 1 −ν23 0 0 0   E1 E2 E3   −ν31 −ν32 1  0 0 0   (63) D−1 =  E1 E2 E3 1 , 0 0 G12 0 0   0   1 0 0 0 G23 0   0 0 0 0 0 0 G131

where Ei denotes the Young’s modulus along axis i, Gij is the shear modulus in direction j on the plane whose normal points in direction i and νij is the Poisson’s ratio that corresponds to a contraction in direction j when an extension is applied b The

visualization was accomplished by means of ray-tracing program Pov-Ray in order to depict the wind turbine model in high visual quality.

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in direction i. Assuming an idealized homogeneous distribution of fibers, we reduce the orthotropic material symmetry class to the subclass of transversely isotropic materials. In this case the demand of an isotropic plane mandates E 2 = E3 ,

G31 = G21 ,

ν31 = ν21 ,

G32 =

E2 . 2 (1 + ν31 )

(64)

Due to the symmetry of the stiffness matrix in Eq. (62) it holds ν13 ν31 = , E1 E3

ν21 ν12 = , E1 E2

ν32 ν23 = . E2 E3

(65)

As a result we have to specify five material parameters • E|| = E1 = 45000 N/mm2 : Young’s modulus parallel to the fiber direction • E⊥ = E2 = 15000 N/mm2 : Young’s modulus perpendicular to the fiber direction • G||⊥ = G31 = G12 = 5400 N/mm2 : shear moduli • G⊥⊥ = G23 = 5400 N/mm2 : shear modulus • ν⊥|| = ν12 = ν13 = ν23 = 0.3: Poisson’s ratio All material values have to be regarded as averaged material data for carbon reinforced material with a fiber volume fraction of 60%. The average density is ρ = 1.5 · 103 kg/m3 . The given set of material parameters accounts for a light and very stiff material adumbrating the fiber reinforced composite material as it is used for the design of wind turbine blades, see [Sch¨ urmann (2005), p. 202]. In our model the strain energy density is given by W (C) =

1 E:D:E 2

with

E=

1 (C − I) . 2

(66)

Here E denotes the Green-Lagrange strain tensor and D is the fourth-rank elasticity tensor. In the second example we consider an extremely soft isotropic material. In this case the elasticity tensor is comprised of the two material parameters namely the so-called Lamé parameters Λ and µ by D = ΛI ⊗ I + 2µI.

(67)

The corresponding stiffness matrix in Voigt notation then consequently is given by   2µ + Λ Λ Λ 0 0 0 0 0 0  Λ 2µ + Λ Λ  Λ 2µ + Λ 0 0 0  . (68) D= Λ 0 0 µ 0 0   0 0 0 0 0µ0 0 0 0 0 0µ

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In order to characterize the stress state in both examples we use the Frobenius norm of the Cauchy stress tensor σ, given by

kσkF :=

√

v u 3 uX σ:σ=t σij σij .

(69)

i,j=1

The Cauchy stress can easily be obtained from the second Piola-Kirchhoff stress tensor S by means of Piola transformation as σ = det F−1 FSFT .

(70)

Figure 12 and 13 here. The simulation results for the fiber reinforced blade-material show that our hybrid wind turbine model complies with the physical conservation properties in energy and angular momentum, cf. Fig. 12 and Fig. 13. After the loading period the system undergoes continuous decreasing oscillations which can be seen in terms of the strain energy in the left part of Fig. 12 and the rotation angle of the nacelle in Fig. 14. In Fig. 15 and Fig. 16 we see that high stress values (reddish parts of the mesh) are obtained at the coupling domains between the rigid hub and the flexible blades. This mirrors the realistic stress distribution, where high stresses are present at the bearings. The observed deformations are rather small taking into account the very stiff material. Figure 14, 15 and 16 here. In order to illustrate the robustness of our hybrid wind turbine model we now simulate an extreme physical configuration. As material parameters we choose Λ = 39 · 102 N/mm2 and µ = 9.75 · 102 N/mm2 which corresponds to about 1/15 of the stiffness chosen above. These parameters describe a rather soft material. Therefore we have to “switch off” gravity otherwise the structure will collapse under its own weight. Additionally we induce a rotation of the nacelle in order to replicate a pointing of the rotor into the wind. The loading period is still 3 s. As shown in Fig. 17 and Fig. 18, even during this extreme simulation energy and one component of angular momentum are conserved. Note that this example serves here as an academic model problem to show the numerical possibilities of our solution scheme. Figure 17 and 18 here. The effect of the soft material property is twofold. On the one hand we observe large deformations during simulation, see Fig. 20 and Fig. 21. On the other hand the system is much more susceptible to oscillations. Due to the high deformations during the loading period the strain energy reaches very high values. After that loading period the system transitions into a state of uniform oscillations. This can be seen in terms of strain energy and the oscillating rotation angles of the hub and nacelle, cf. Fig. 19. Figure 19, 20 and 21 here.

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5. Conclusion The presented work illustrates the incorporation of two mechanical models of a wind turbine into the framework of an energy-consistent scheme for multibody dynamics. In the case of small deformations and low wind speeds the rigid wind turbine model has shown to be a robust and very efficient model for the dynamics of the real structure. However, more insight in the deformation of a real wind turbine provides the hybrid model of coupled orthotropic-elastic and rigid parts. Here the deformation and the stress state of the blades in the presence of turbulent wind profiles modeled by typical applied moments is observable. The computational cost of the hybrid model, however, is higher than the cost of a rigid body model. In summary it can be stated, that both models are relatively simple, readily identifiable and energy-consistent. Their differential-algebraic structure allows for additional constraints, e.g. for purpose of control, by simple addition into the original set of equations. Therefore our model may serve as a first step towards investigations of fluid-structure interactions of wind turbines. To capture fluid-structure interactions a back-coupling between the moving structure (displayed by our model) and the fluid medium has to be considered. References Bauchau, O.A. and Bottasso, C.L. (1999). On the design of energy preserving and decaying schemes for flexible, nonlinear multi-body systems. Computer Methods in Applied Mechanics and Engineering, Vol. 169, pp. 61–79. Bauchau, O.A. (1998). Computational schemes for flexible nonlinear multi-body systems. Multibody System Dynamics, Vol. 17 No. 2, pp. 169–225. Bauchau, O.A., Bottasso, C.L. and Trainelli, L. (2003). Robust integration schemes for flexible multibody systems. Computer Methods in Applied Mechanics and Engineering, Vol. 192, pp. 295–420. Baumjohann, F., Hermanski, M. and Diekmann, R. (2002). 3D-Multi Body Simulation of Wind Turbines with Flexible Components. Proceedings of the German Wind Energy Conference DEWEK 2002, Wilhelmshaven, Germany. Betsch P., Hesch, C., Snger, N. and Uhlar, S. (2010). Variational integrators and energymomentum schemes for flexible multibody dynamics. ASME Journal of Computational and Nonlinear Dynamics , Vol. 5 No. 3, pp. 031001. Betsch, P. and Leyendecker, S. (2006). The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: Multibody dynamics. International Journal for Numerical Methods in Engineering, Vol. 67 No. 4, pp.499–552. Betsch, P. and S¨ anger, N. (2009). A nonlinear finite element framework for flexible multibody dynamics: Rotationless formulation and energy-momentum conserving discretization. Multibody Dynamics: Computational Methods and Applications, Vol. 12, pp. 119–141. Betsch, P. and Steinmann, P. (2002). A DAE approach to flexible multibody dynamics. Multibody System Dynamics, Vol. 8, pp. 367–391. Betsch, P. and Steinmann, P. (2002). Conservation properties of a time finite element method. Part III: Mechanical systems with holonomic constraints. International Journal for Numerical Methods in Engineering Vol. 53, pp. 2271–2304. Betsch, P. and Steinmann, P. (2001). Constrained Integration of Rigid Body Dynamics. Computer Methods in Applied Mechanics and Engineering, Vol. 191, pp. 467–488.

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Betsch, P. and Uhlar, S. (2007). Energy-momentum conserving integration of multibody dynamics. Multibody System Dynamics, Vol. 17, No. 4, pp. 243–289. Bottasso, C.L., Borri, M. and Trainelli, L. (2001). Integration of elastic multibody systems by invariant conserving/dissipating algorithms. II. Numerical schemes and applications. Computer Methods in Applied Mechanics and Engineering, Vol. 190, pp. 3701–3733. Géradin, M. and Cardona, A. (2001). Flexible multibody dynamics A finite element approach, John Wiley & Sons: New York, 2001. Holm-Jorgensen, K. and Nielsen, S.R.K. (2009). System reduction in multibody dynamics of wind turbines. Multibody System Dynamics, Vol. 21, pp. 147–165. Hughes, T.J.R. (2000). The Finite Element Method, Dover, Mineola. Larsen, J.W. and Nielsen, S.R.K. (2006). Non-linear dynamics of wind turbine wings. International Journal of Non-Linear Mechanics, Vol. 41 No. 5, pp. 629–643. Molenaar, D. P. and Dijkstra, S. (1999). State of the Art of Wind Turbine Design Codes: main features overview for cost-effective generation. Wind Engineering, Vol. 23, No. 5, pp. 295–311. Sch¨ urmann H. (2005). Konstruieren mit Faser-Kunststoff-Verbunden. Springer. Shabana, A. A. (1998). Dynamics of multibody systems (2nd edn). John Wiley & Sons. Uhlar, S. (2009). Energy Consistent Time-Integration of Hybrid Multibody Systems, Ph.D. thesis, University of Siegen. Uhlar, S. and Betsch, P. (2007). On the rotationless formulation of multibody dynamics and its conserving numerical integration. In Bottasso, C.L., Masarati, P. and Trainelli, L., editors: Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics, Politecnico di Milano, Italy, 25–28 June. Uhlar, S. and Betsch, P. (2008). Conserving integrators for parallel manipulators. In Ryu, J. H., editor: Parallel Manipulators I-Tech Education and Publishing, www.books.itechonline.com, Vol. 5, pp. 75–108. Uhlar, S. and Betsch, P. (2008). A uniform discretization approach to flexible multibody systems. Proceedings in Applied Mathematics and Mechanics, Vol. 8, No. 1, pp. 10147– 10148. Uhlar, S. and Betsch, P. (2009). A unified modeling approach for hybrid multibody systems applying energy-momentum consistent time integration. In Wojtyra, M., Arczewski, K. and Fraczek, J., editors: Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics, Warsaw University of Technology,Poland, 29 June–2 July. Zhaoa, X., Maißer, P. and Wu, J. (2007). A new multibody modelling methodology for wind turbine structures using a cardanic joint beam element. Renewable Energy, Vol. 32, No. 3; pp. 532–546.

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Figures

Fig. 1: Sketch of a classical three-blade windturbine.

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d3

B

d2 ϕ

e3

d1 e2

e1

Fig. 2: Configuration of a spatial rigid body in rotationless framework.

d12 body 1

d11 d13

ϕ1

η1

e2 e1 e3

Θ ϕ2 η2 d22 d21 d23 body 2

23

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Θ1

L2

d22 d21

d23 d2

d12

L1 d13

d11 d1

Fig. 4: Illustration of the nave and nacelle.

Θ2

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b4 Θ4

d42 d41

L4 d43

d22 d23

d52

d53

d21 d32

d31 d33

d51

Θ3

Θ5 Fig. 5: Configuration of body frames of the rotor components.

d2

d1 d3

ϕRB

ϕRF

e2

e1

ϕiN

e3 Fig. 6: Graphical interpretation of the coupling constraints.

25

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Fi (t)

t t1

0

t2

Fig. 7: Magnitude of the torques during the load period.

-8

x 10

3500 10 Hamiltonian

2500 2000 1500

energy [kJ]

energy [kJ]

3000 ∆H tolerance

8 6 4

1000

2

500

0

0 0

2

4

6

time

8

10

12

3

4

5

6

7

8

9

10

11

12

time

Fig. 8: Shape of the Hamiltonian and ∆-Hamiltonian for the rigid wind turbine model.

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1

x 10

-10

4

x 10 2

0 -1

angular momentum

angular momentum

27

L1 L2 L3

-2 -3 -4 -5 -6

1 0 -1 -2 -3

-7 -4

-8 0

2

4

6

time

8

10

12

3

4

5

6

7

8

9

10

11

12

time

Fig. 9: Shape of angular momentum [kNm] split in spatial components and plot of ∆L3 -function.

Fig. 10: Visualization of the rigid multibody model for t = 0 s and t = 5 s.

Fig. 11: Visualization of the rigid multibody model for t = 9 s and t = 12 s.

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28 -7 x 10

6000

∆H tolerance

2.5

energy [kJ]

energy [kJ]

Hamiltonian strain energy

5000

4000

3000

2000

2 1.5 1 0.5 0 -0.5

1000

-1 0

0

2

4

6

8

10

12

3

4

5

6

7

8

9

10

11

12

time

time

Fig. 12: Shape of the Hamiltonian, strain energy function and ∆-Hamiltonian in the hybrid model with stiff orthotropic material parameters.

1

-8 x 10

4 x 10 10

-1

L1 L2 L3

-2

angular momentum

angular momentum

0

-3 -4 -5 -6

∆L2 tolerance

8 6 4 2 0

-7 -2 -8 0

2

4

6

time

8

10

12

2

4

6

8

10

12

time

Fig. 13: Shape of the angular momentum [kNm] and plot of ∆L2 -function in the hybrid model with stiff orthotropic material parameters.

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INVESTIGATING A FLEXIBLE WIND TURBINE MODEL -4 x 10

0

Θ1

angle of rotation in [rad]


2

29

1.5 1 0.5 0 -0.5 -1 -1.5 -2

-0.2

Θ2

-0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6

0

2

4

6

time

8

10

12

0

2

4

6

8

10

12

time

Fig. 14: Rotation angle of nacelle (left) and hub (right).

Fig. 15: Visualization of the hybrid model with stiff material parameters for t = 0 s and t = 5 s.

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Fig. 16: Visualization of the hybrid model with stiff material parameters for t = 9 s and t = 12 s.

x 10

8000

Hamiltonian strain energy

∆H tolerance

1.5

6000

energy [kJ]

energy [kJ]

7000

-7

2

5000 4000 3000

1 0.5 0 -0.5

2000

-1

1000

-1.5

0 0

2

4

6

time

8

10

12

4

5

6

7

8

9

10

11

12

time

Fig. 17: Shape of the Hamiltonian, strain energy function and ∆-Hamiltonian in the hybrid model with soft isotropic material parameters.

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1

x 10

-9 x 10

4 10

0

L1 L2 L3

-1

angular momentum

angular momentum

31

-2 -3 -4 -5 -6

∆L2 ∆L3 tolerance 5

0

-7 -5 -8 0

2

4

6

8

10

12

3

4

5

6

7

8

9

10

11

12

time

time

Fig. 18: Shape of the angular momentum [kNm] in the hybrid model with soft isotropic material parameters.

-9 x 10

0

2



4

Θ1

0 -2 -4 -6 -8 -10 -12 -14 -16

Θ2

-0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6

0

2

4

6

time

8

10

12

0

2

4

6

time

Fig. 19: Rotation angle of nacelle (left) and hub (right).

8

10

12

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Fig. 20: Visualization of the hybrid model with smooth material parameters for t = 0 s and t = 5 s.

Fig. 21: Visualization of the hybrid model with smooth material parameters for t = 9 s and t = 12 s.

1135x1975mm (600 x 600 DPI)

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