A Finite Element Approach for Aeroelastic Instability Prediction of Wind Turbines P.A Castillo Capponi

A Finite Element Approach for Aeroelastic Instability Prediction of Wind Turbines

Thesis dissertation, submitted in partial fulﬁllment of the requirements for the degree of the Master Program of Aerodynamic

Pablo A. Castillo Capponi

October, 2010

Graduation committee

Prof.dr.ir. G.J.W van Bussel Ir. T. Ashuri Dr.ir. J. Holierhoek Delft University of Technology Faculty of Aerospace Engineering Wind Energy Research Group

Contents

Nomenclature 1

1 Introduction 5 1.1 WindEnergy ...... 5 1.2 Motivation ...... 6 1.3 Goalofthethesis...... 9 1.4 Outline ...... 9 2 Literature review 13 2.1 LiteratureReview ...... 13 2.1.1 Aeroelastic Instability in Airplanes ...... 14 2.1.1.1 Historic Approaches to Predict Aeroelastic Insta- bilities in Airplanes ...... 14 2.1.1.2 Instability Prediction for Airplanes ...... 16 2.1.2 Aeroelasticity Instabilities in Wind Turbines ...... 18 2.1.3 State of Art - Aeroelastic Codes for Wind Turbines Insta- bilities ...... 22 2.2 Multibody and Finite Element Method ...... 24 3 The Finite Element Approach for Aeroelastic Instability Prediction 29 3.1 FormulationoftheStructuralModel ...... 30 3.1.1 The Finite Element Method for Structural Components (FEM)...... 30 3.1.2 Generalized Body Forces in a Non-Inertial Reference Frame 31 3.1.3 Mass, Damping and Stiﬀness Matrices for Forces due Ac- celerationsinaNon-InertialFrame ...... 34 3.1.4 Coupling the Generalized Body Forces to the FEM Method 36 3.2 FormulationoftheAerodynamicModel ...... 39

i ii CONTENTS

3.2.1 AerodynamicModel ...... 39 3.2.1.1 Basis of the Aerodynamic Model: The Theodor- sen Solution for a Flat Plate ...... 39 3.2.1.2 Drag Model for the Theodorsen Solution . . . . . 41 3.2.1.3 The Aerodynamic Model used in this Thesis . . . 41 3.2.2 The Finite Element Method applied to the Aerodynamic Model ...... 45 3.2.2.1 The Basis idea: Minimization of the Aerodynamic EnergyFunctional ...... 45 3.2.2.2 Variables Definition of the Aerodynamic Element 49 3.2.2.3 Deduction of the Aerodynamic Element Matrices . 50 3.2.2.4 Assembling of Global Aerodynamic Matrices . . . 66 3.2.2.5 A Non True Finite Element Matrices for the Ae- rodynamicModel ...... 68 3.3 Integration of the Structural & Aerodynamic Models ...... 69 3.3.1 Complete Formulation of the Aeroelastic Method ...... 69 3.3.2 Methodology to find the First Unstable Operational Point . 70 3.4 ImplementationoftheMethod ...... 78 4 Verification of the Method 81 4.1 Finite element model for the 5MW Reference Wind Turbine . . . . 82 4.1.1 Tower ...... 82 4.1.2 HubandNacelleModel ...... 83 4.1.3 Blades...... 84 4.1.4 Boundary Conditions and Connection between the Blades, Nacelle,HubandTower ...... 84 4.1.5 Pitch, Rotational and Wind Speed Control Curves . . . . . 86 4.1.6 ParametricModelinPATRAN ...... 86 4.1.7 Modelsummary ...... 88 4.1.8 Simple Model for Stability Analysis ...... 89 4.2 Unstable Operational Points for the 5MW Wind Turbine . . . . . 92 4.2.1 Static Unstable Points ...... 92 4.2.2 Dynamic Unstable Points ...... 93 5 Application: Analysis of a 20MW Wind Turbine 95 5.1 Upscaling process of the 5 MW NREL wind turbine to an optimum 20MW...... 95 5.2 Finite Element Model for the 20MW Reference Wind Turbine . . . 96 5.2.1 Tower ...... 96 5.2.2 HubandNacelleModel ...... 97 5.2.3 Blades...... 98 5.2.4 Boundary conditions and connection between the blades, nacelle,hubandtower...... 98 CONTENTS iii

5.2.5 Pitch, Rotational and Wind Speed Control Curves . . . . . 101 5.2.6 ParametricModelinPATRAN ...... 101 5.2.7 Modelsummary ...... 102 5.3 Unstable Operational Scenarios for the 20MW Wind Turbine . . . 104 5.3.1 Static Unstable Points ...... 104 5.3.2 Dynamic Unstable Points ...... 105 6 Conclusions and Recommendations 107 6.1 Conclusions ...... 107 6.2 Recommendations ...... 109 A Appendix I: Blade layout for the 5MW Wind Turbine 111 Bibliography 115

Nomenclature

List of Symbols

A Sectional blade area [m2] a Acceleration [m/s2] b Diameter ratio between the upscaled and the reference blade [-] C Theodorsen function or tensor notation [ ] − Cl Aerodynamic lift coefficient [ ] − Cm Aerodynamic moment coefficient [ ] D Sectional drag [−N/m] d1 Distance from the elastic axis to 1/4 of the airfoil chord [m] d2 Distance from the elastic axis to 3/4 of the airfoil chord [m] Cd Aerodynamic drag coefficient [ ] − Da Aerodynamic damping matrix * Dr Damping matrix due the rotational frame * Ds Structural damping matrix * e1 Unitary vector which defines the direction of the h DOF * e Lift direction without rotational speed of the blade [ ] 10 − e2 Unitary vector which defines the direction of the s DOF * e3 Unitary vector which defines the direction of the α DOF * e30 Drag direction without rotational speed of the blade [ ] F Generalized force *−

Fpkm Force acting on the node k in the direction m [N] Fr Generalized force due the rotational frame * h DOF deﬁned on the lift direction [m] H Hankel function [ ] − h0 Value around h is linearized or plunging amplitude [m] i Imaginary number unit [ ] k Reduced frequency or index [−] − Ka Aerodynamic stiﬀness matrix *

1 2 CONTENTS 0.0

Kr Stiﬀness matrix due the rotational frame * Ks Structural stiﬀness matrix * L Sectional lift [N/m] m Mass [kg] M Sectional moment [N/m]

Mpkm Moment acting on the node k in the direction m [Nm] Ma Aerodynamic mass matrix * Mri Equivalent mass matrix due the rotational frame * Ms Structural mass matrix * n Direction cosines [Pa] p Position vector [m] R Position vector [m] R1 Lift functional [Nm] R2 Moment functional [Nm] R3 Drag functional [Nm] s DOF deﬁned on the drag direction [m] S Surface [m2] s0 Value around s is linearized [m] t Time or surface traction [s], [pa] u Generalized displacements * uk3m Displacement on the node number k in the direction m [m] uk3θm Angular rotation on the node k in the direction m [rad] V Volume domain [m3] x Coordinate x in space [m] X External forces [N/m3] x1 First basis component of the reference coordinate system [m] x2 First basis component of the second coordinate system [m] x3 ﬁrst basis component of the global coordinate system [m] xs x axis parallel to the wind speed [ ] y Coordinate y in space [m−] y1 Second basis component of the reference coordinate system [m] y2 Second basis component of the second coordinate system [m] y3 Second basis component of the global coordinate system [m] ys y axis parallel to the rotational wind speed [ ] z Coordinate z in space [m−] z3 third basis component of the global coordinate system [m]

Greek symbols

α Angle of attack [rad] α0 Value around α is linearized or pitching amplitude [m] Ω Angular velocity [rad/s] 0.0 CONTENTS 3

ω Frequency or weight function [rad/s], [ ] ρ Density [kg/m3] − ε Mechanical deformation [ ] σ Mechanical stress [Pa− ]

Subscripts

0 Conﬁguration initial a Refers to aerodynamic blade Complete blade domain e Element matrix i Element index, direction index j Counter index, direction index n Identiﬁcation number or coordinate component of a vector ns Number of sections on the blade p Pitch of the blade section r Angle of attack of the blade section section Refers to a section of the blade t Twist of the blade section up Upper side of the blade us Lower side of the blade V Wind speed

Superscripts

(1) Hankel function of ﬁrst order (2) Hankel function of second order

Notations bold Vector or matrix ()n Respect to the n coordinate system DOF Degree of freddom DOFs Degree of freddoms Determinant <| · |, > Inner product · ·

The symbol * means the vector or matrix has the unit of its contained equations.

Chapter 1

Introduction

1.1 WindEnergy

The development of the human society is inﬂuenced by the use of energy. The energy helps the society to manage the natural resources doing easier the adaptation to a new environments. This is the reason why managing the energy is inevitable in any society. The development of energy resources is essential for transportation, agriculture, waste collection and communications, which play an important role in a developed society. The energy consumption has been increasing since the industrial revolution and this brought with it a number of serious problems. These problems are related to a critical damage of natural environments. One example is the global warming which present potentially grave risks to the world. Today the consumption per capita is 115 times higher than the energy consumption for an primitive human (See ﬁgure 1.1).

The energy becomes every day an important subject. Therefore, diﬀerent types of renewable energy are in development and under investigation, with wind energy as one of those. Wind energy promises to be one of the aﬀordable green alternative energies.

The mechanism of energy generation of wind turbines is the conversion of the kinetic energy of the free streaming air to a mechanical power, which in turn can be used to rotate a generator to produce electricity. The increment use of wind energy to obtain electricity is presented in ﬁgure 1.2. At the end of 2009, the energy generated by wind was 2% of worldwide electricity usage. However, still the electricity produced by other technologies is cheaper than the electricity produced by wind. Therefore, many scientist and engineers are attempting every day to developed technologies which decreases the costs of

5 6 INTRODUCTION 1.2

Estimated Daily Consumption of Energy per Capita at Diﬀerent Historical Points 250 Transportation Industry and Agriculture Home and Commerce 200 Food

150 Kcal

1000 100

0 1 2 3 4 5 6

Figure 1.1: Estimated Daily Consumption of Energy per Capita at Diﬀerent Historical Points Adapted from: E. Cook, “The Flow of Energy in an Industrial Society” Scientiﬁc American, 1971 p. 135. The legend is shown in table 1.1 .

electricity produced by wind. A promissing way to achieve that is to build bigger wind turbines which decreases the costs [1], see figure 1.3. However, the idea of increasing the size of the wind turbines is only possible with the application of new materials, new technologies and better design methodologies. The new approach to wind Turbine designs should be improved with more emphasis on integrated design. This gives the possibility to do make larger wind turbine that are lighter and more flexible. Today one of the biggest project to find solutions for very large wind turbine designs is the UpWind European project. This project is funded under the EU’s Sixth Framework Programme (FP6) and it looks towards the wind power of to- morrow, searching for new design of very large wind turbines between 8 to 20MW for onshore and offshore.

1.2 Motivation

The conventional approach to study aeroelastic instabilities in Wind Tur- bines is to use a Multibody formulation of the Wind Turbine. A Multibody model represents the dynamic of a body (mechanical part) with only few degrees 1.2 MOTIVATION 7

1 Technological Man 2 Industrial Man 3 Advanced Agricultural Man 4 Primitive Agricultural Man 5 Hunting Man 6 Primitive Man

Table 1.1: Legend of ﬁgure 1.1

Figure 1.2: World total installed wind energy capacity. The market is growing with an exponential rate [1].

of freedom and it uses the global structural properties of it (such as mass, moment of inertia for example). Another method to analyze the dynamics of a wind turbine is Finite Element Method, which uses many degrees of freedom and is based on local properties of the structure. The Multibody method has the advantage to simulate the dynamics of a body with less degrees of freedom in comparison with Finite Element Method. When a model has less degrees of freedom, it means that the number of equations involved on the are smaller and it requires less time to solve it. On the other hand the Finite Element Method gives much accurate results in comparison with the Multibody model. The designers of Wind Turbines usually starts with a Multibody model of the Wind Turbine, because that model is fast to simulate and it is only based on the global properties of the Wind Turbine. That idea is very convenient when the designer does not know too much details about wind turbine and thus the method 8 INTRODUCTION 1.2

Electricity cost comparison for diﬀerent Wind Turbines size. 12 Coastal site Inland site 10

6 cEUR/kWh

0 95 150 225 300 500 600 1000 kW Turbine size

Figure 1.3: Total Cost of Wind Power (cEUR/kWh, Constant 2001 Prices) by Turbine Size

is proper for conceptual and preliminary designs. When the designer is finished with the Multibody design of the Wind Turbine (he knows the global properties of its Wind Turbine) he needs to change his strategy. Now he must use an accurate simulation of the Wind Turbine based on the local properties and he should do a Finite Element Model for the wind turbine. The detailed designs of the Wind Turbine (The Finite Element Model) must be in match with the Multibody model, otherwise the Finite Element Model of the Wind Turbine will not be representative of the Wind Turbine. Thus it can be seen that the designer should use that in detail design level, where he already has a good understanding of the global properties of the turbine. The entire process is time consuming because in many iterations the Finite Element Model has to be remeshed at least in the modified areas of each iteration. Other disadvantage is the designer has to be very intuitive to change the right part of the Finite Element Model to achieve a desired global properties value. The explained task turns to be more difficult when new designs are studied. One of the problems in the new designs of Wind Turbines are the instabilities. Tools to simulate instabilities in time domain and frequency domain exists for a Multibody model of the Wind Turbine. The motivation of this thesis is to create a 1.4 GOAL OF THE THESIS 9 tool to find instabilities for a Finite Element Model of the Wind Turbine without the necessity to pass throw a Multibody simulation and increasing the accuracy.

Normal approach for instabilities study

Real Model Finite Element Model

Global properties Solver work to find instabilities

Aerodynamic Model MultiBody Model Good representation for the dynamicsfor Goodthe representation Instabilities

Figure 1.4: Common approach to study instabilities on the industries.

1.3 Goalofthethesis

This thesis presents the development of a new method to find aeroelastic instabilities of Wind Turbines based on the Finite Element Method. This new method is programmed in NASTRAN and integrated as a new feature with NAS- TRAN. The method is automatized and it gives a capability for a user to find the instabilities of the Wind Turbine without passing through many details. The proposed method helps the designers to save time in his design process to find instabilities and gives the capability to get a quick and accurate results (See figure 1.5). This method can help the designer to analyze new innovative blade designs, which associate the risk of increasing instability. For example designs that rely on aeroelastic tailoring (the blade twists as it bends under the action of aerodynamic loads to shed load resulting from wind turbulence) increases the possibility of making the blade unstable [2].

1.4 Outline

Figure 1.6 shows the structure of this dissertation. As shown in this ﬁgure, chap- ters 1 and 2 deal with the state of art and the objective of this thesis. The development of the methodology is explained in the chapter 3. Chapter 4 shows the validation of the methodology using the 5MW NREL wind turbine and chapter 5 10 INTRODUCTION 1.4

Thesis proposed approach for Instabilities study

Solver work to find instabilities

Aerodynamic Model in Frequency Domain

Real Model Finite Element Model

Solver for Instabilities

Instabilities

Figure 1.5: Approach proposed in this thesis for instabilities studies.

analyze the stability of a new design of a 20MW wind turbine. The conclusions and recommendations of this presentation are showed in chapter 6.

1.4 OUTLINE 11

1. Introduction

2. Literature Review

3. The Finite Element Approach for Aeroelastic Instability Prediction

3.1 Formulation of the Structural Model considering Non-Inertial Frame

3.2 Formulation of the Aerodynamic Model

3.3 Integration of the Structural & Aerodynamic Models

3.4 Implementation of the Method

4. Verification of the Method using the 5MW NREL Wind Turbine

5. Application: Analysis of a 20MW Wind Turbine

RESULTS PROGRAMME BACKGROUND 6. Conclusion and Recommendation

Figure 1.6: The structure of this dissertation.

Chapter 2

Literature review

2.1 Literature Review

The introduction on the first chapter describes the motivation and the goal of create a tool to find instabilities for Wind Turbines using the Finite Element Model of the Wind Turbine. This methodology gives the capability to jump one step on the design process of a Wind Turbine and it eliminates the tedious and difficult iteration process to match the properties between the Finite Element Model and the Multibody model doing the design faster and more accurate. To- day accuracy on the designs of Wind Turbines is really important because that permits to build bigger Wind Turbines and decrease the cost (see figure 1.3).

This chapter is divided into two sections. The ﬁrst section explain how the study of the aeroelastic stability begins in airplanes and how in the past the aeroelastic instabilities was not considering in Wind Turbine designs until the designs become lighter and stronger. This section shown new approaches and studies in aeroelasticity of the Wind Turbines in chronological order. The ﬁnal part shown the state of art of the codes for aeroelastic stability analysis. On the literature is not found an idea with the same objective and methodology of the approach for the instabilities given in this thesis. The second section describes, explains and compares the Mutibody method and the Finite Element Method from a theorist and practical point of views. The Mul- tibody method is the base of the methodology for the approach which is using today for the Aeroelastic Analysis and the Finite Element Method is the base of the method proposed in this thesis.

A paper from Sandia Laboratories is important for this thesis [3]. This paper shows a method between the normal Multibody approach to study Aeroelastic instabilities with some characteristics of a Finite Element Method. Although this

13 14 LITERATURE REVIEW 2.1 paper does not present a validation of the results, the results are on the right order of magnitude with comparison of similar works. The references presented on the chapter three are used to formulate the base of the Aerodynamic model for the work presented in this thesis. The Aerodynamic model presents some modiﬁcations to include drag and it uses the real airfoil characteristics for the lift and moment coeﬃcients.

2.1.1 Aeroelastic Instability in Airplanes 2.1.1.1 Historic Approaches to Predict Aeroelastic Instabilities in Airplanes

The people who designed the first airplanes on the World War I did not rea- lize the aeroelastic instabilities on their aircraft because at that time were flying at low speed and the aircraft structures were rigid. The phenomena appeared early in the control surfaces of the airplanes. The first recorded flutter incident was on a Handley Page O/400 twin engine biplane bomber in 1916. The flutter mechanism was a coupling of the fuselage torsion mode with an antisymmetric ele- vator rotation mode. The elevators were independently actuated on this airplane and the solution for this problem was to connect the elevators with a torque tube [4]. During the first world war the control surface flutter began to appear. The flutter for the interaction wing-aileron was encountered in many airplanes during this time [5]. Von Baumhauer and Koning were the first people who tried to solve this flutter mechanism, the solution was based on flying experience. They sug- gested the use of a mass balance about the control surface hinge line to avoiding this type of flutter. Sometimes a flutter on the control surface were encountered afterward and the solution for this flutter was increasing the mass balance of the control surface. After World War I, higher airspeeds are reached and a shift from external wire-braced biplanes to an aircraft with cantilevered have been done.

The first formal flutter test was carried out by Von Schlippe in Germany in 1935 [6]. His approach was to vibrate the aircraft at resonant frequencies at progressively higher speeds and plot the amplitude response as a function of the wind velocity. When the amplitude response is big means that point could be a point with reduced aerodynamic damping and the flutter speed could occurring at asymptote of the infinite amplitude as shown in figure 2.1. The idea was applied successfully to many German aircraft during World War II until a Junkers JU90 fluttered and crashed during flight tests in the year 1938. When Von Schlippe in Germany was carried its experimental method to find aeroelastic instabilities, Theodore Theodorsen was developing a mathematic theory in United States. He was working at the NACA facility. Theodore Theodorsen in 1935 publish a paper named “General theory of aerodynamic instability and the mechanism of flutter” [7]. This paper is based on the potential flow equation and it shown a closed 2.1 LITERATURE REVIEW 15 Maximum responseamplitude

Airspeed Vflutter

Figure 2.1: Von Schlippes ﬂight ﬂutter test method

solution for the unsteady lift and moment for a flat plate. This mathematical model was the first unsteady aerodynamic model capable to predict unsteady forces. Theodorsen explains in the paper basic applications on how to coupled this unsteady forces to an structure and find the unstable velocities for the structure. This is the first known mathematical aeroelastic model. The application of Theodorsen model to a real aircraft carry three main kinds of problems at this moment. Firstly without the use of computers is hard to obtained the structural properties of a wing. Although is possible to use a experimental way to obtain the structural properties the wing most of the wings are tapered. The Theodorsen aerodynamic model applied for tapered wings gives different reduced frequencies for each section of the wing and the mathematical problem becomes very difficult to solve for that time. Secondly for each velocity the aerodynamic model changes and is needed to solve a new problem, that means many iteration for different velocities are needed to find the instability point and without the computers help that was difficult. Thirdly the model neglect drag and it is for a flat plate. The structural requirements for heavy airplanes force the airplanes to have a big camber and the idea of a flat plate is not longer true for those airplanes. The results of the method was not accurate and most of the people prefer the experimental approach to find instabilities at that moment.

In the late 1950s the common approach to ﬁnd instabilities was evaluate the response of the system using excitation systems consisted of inertia shakers, manual control surface pulses, and thrusters (bonkers). The instrumentation was improved and the response signals started to be telemetered to a ground sta- tion for further anlisis. Today in some airplanes some programs still displayed signals of response on oscillographs in the airplane. Many people who started to do experiments realized the real importance of adequate structural excitation for obtaining a high signal-to-noise ratio to decrease the error associated to the experiment. The people started to use oscillating vanes to excite the airplane during this time. 16 LITERATURE REVIEW 2.1

From the 1950s until the 1970s, many aircrafts were equipped with excitation systems. Frequency sweeps were made to identify resonances on the structure. These sweeps were followed by a frequency quick stop at each resonant frequency to identify possible instabilities. The analysis in ﬂight was usually limited due to the low computation capability in ﬂight and the analysis was to study the log decrement of the accelerometer to determine damping.

The traditional method of testing the flutter margins of aircraft using flight testing only is risky due to the inherently unreliable nature of the damping. This method has two main disadvantage, firstly each test flight needs large amount of time to prepare it and secondly the test flights are expensive, also the expansion of the flight envelope occurs very slowly because the flight test conditions must be changed in very small increments to avoid unpredictable sudden changes in instabilities which can produce a flight accident.

Thus since the 1970s, the introduction of digital computers to predict aeroelastic instabilities have significantly affected flight flutter testing techniques. The Fast Fourier Transformation (FFT) doing by the computers from the experimental data is fast and accurate. With the use of FFT is easier to analyze the experimental data and predict the flutter speed. Today the development of more sophisticated data processing algorithms are adapted useful for analysis of response data from either steady state or transient excitation. Nowadays the frequency and damping are estimated with parameter identification techniques and is also possible to do in a real-time manner.

2.1.1.2 Instability Prediction for Airplanes Six different main group of methods to predict and analyze aeroelastic instability for airplanes are developed: Theodorsen’s method • The µ-method; a technology patented by NASA • The K-method • The PK-method • The P Method • State Space Methods • The first 5 methods works in frequency domain, the State Space methods works in time domain. The next lines explain briefly each method. For more information please go to the references of the method. 2.1 LITERATURE REVIEW 17

Theodorsen’s method The Thedorsen method is based on the simultaneous solution of the real and imaginary parts for the 2D system of equation for the wing motion (Torsion and Bending). A quasi-steady Aerodynamic model is used, with the modiﬁcation only on the lift forces by the use of unsteady 2-D aerodynamics [8].

µ-method The µ-method is a method for Robust Flutter Prediction in Expanding a Safe Flight Envelope for an Aircraft Model Under Flight Test. It is a technology Pa- tented by NASA under the Patent: US6,216,063 [9]. The NASA’s method of flutter margin prediction uses a computer model of the aircraft structure. A structured singular value, µ is defined using the particular plant characteristic (in this case the structural model of the airplane), thus using the the singular value µ a robust flutter speed margins are computed.

The K-method The k-method is also known as the V-g method or the American method of ﬂut- ter solution to determine the aeroelastic stability of a system. Many aerodynamic formulations lead to aerodynamic matrices which are only valid for harmonic motion without amplitude changing on time. Using these simple harmonic loads, and introducing an artiﬁcial structural damping factor, complex roots are obtained from the equations. The value of zero for the real part of the complex roots means the system converged to an unstable point [10].

The PK-method The p-k method attempts to improve upon the k-method by allowing the reduced frequency to be complex. The equations of motion are written in a form indi- cating that the aerodynamic matrix is available only for harmonic motion (only function of the frequency). The eigenvalues of this approximate system can be solved, producing complex roots. The real part of the roots gives the unstable or stable behavior of the system [11].

P Method The p-method is the simplest method to understand, but perhaps the most dif- ﬁcult to apply. Utilizing the p-method means simply solving for the complex eigenvalues of the governing equations. The governing equations usually has many degrees of freedom and solve the systems involves large amount of CPU time and computing power. This approach is not normally used directly for big structures due to the large requirement of computer power to solve the system for the eigenvalues, but it is the basis model for model reduction techniques [11]. 18 LITERATURE REVIEW 2.1

State Space Methods This is the only method presented on this thesis based on time domain analysis. The basic idea of the State Space methods is to linearize the system of ordinary differential equations and integrate on time using the discrete system of equation for the continuous system of equations. When the system is integrated on time and the solution is known a post-processing techniques to find instabilities are applied. The common approach is to use the Fast Fourier Transform (FFT) or a model identification technique to find the unstable points. The solution on time domain has the advantage to give the possibility of an easy physical interpretation of the results but difficulties to find the instability points in comparison with a solution in the frequency domain.

Those six methods were the precursors of the aeroelastic methods to ﬁnd instabilities for Wind Turbines. Today modiﬁcations on the Theodorsen’s method, the P method and State Space Methods are the basis of the methods to compute instabilities for Wind Turbines. The aerodynamic and the structural model is adapted to a Wind Turbine models in order to simulate Wind Turbines instead of airplanes.

2.1.2 Aeroelasticity Instabilities in Wind Turbines The Aeroelastic instabilities historically has not been driving a issue in wind turbines design. The Aeroelastic phenomena is rarely addressed in the past for wind turbines. However, nowadays the problem started to become serious on the design of wind turbine. A wind turbine with higher speed, lighter and softer blades increases the chances to be unstable for diﬀerent operational scenarios.

The ﬁrst known electricity generating windmill operated, was a battery char- ging machine installed in 1887 by James Blyth in Scotland. The ﬁrst windmill for electricity production in the United States was built in Cleveland, Ohio by Charles F Brush in 1888, and in 1908 there were 72 wind-driven electric generators from 5 kW to 25 kW. The largest machines were on 24 m (79 ft) towers with four-bladed 23 m (75 ft) diameter rotors. Around the time of World War I, American windmill makers were producing 100,000 farm windmills each year, mostly for water-pumping. By the 1930s, windmills for electricity were common on farms, mostly in the United States where distribution systems had not yet been installed. In this period, high-tensile steel was cheap, and windmills were placed atop prefabricated open steel lattice towers. This wind turbines never reach theirs unstable boundaries.

The ﬁrst theory to compute unsteady lift and moment on a ﬂat plate was proposed by Theodore Theodorsen in 1935 [7]. This theory is based on potential 2.1 LITERATURE REVIEW 19

flow and a oscillatory motion without damping of a flat plate. He did the first aeroelastic study for instabilities when he coupled this aerodynamic model to the structural model and he found the first operational unstable points for airplanes.

In 1981 David C. Janetzke and Krishna.V. Kaza at the NASA Lewis Research Center in Cleveland publish the first paper directly related to aeroelasticity in wind turbines [12]. They explore the possibility of whirl flutter and search the effect of pitch-flap-coupling on teetering motion of a 2 blade wind turbine. The wind turbine had 3.5 meters of diameter and they determined a unstable point on this wind turbine at a wind speed of 77.1 m/s and angular velocity of 320 RPM.

M.E Bechly and P.D. Clausen in 1995 publish a paper called “Structural design of a composite wind turbine blade using Finite Element Analysis” [13]. Although this paper does not treat the instability problem, the paper shows the coupling between the steady aerodynamic loads obtained from a panel code and a structural ﬁnite element model of a composite blade.

On the next years the idea to automatic adjust the pitch of a blade by doing a pitch-bending coupling as a passive power control is developed. The studies in this area takes into account the steady aeroelastic phenomena but the instabilities are hardly touched. In 1998 Sandia Laboratories & National Renewable Energy Laboratory publish the article “Aeroelastic Tailoring in Wind Turbine Blade applications” [2]. They used the Aeroelastic properties of the blades to increase the cost-eﬀective, passive means to shape the power curve and reduce loads. They analyze the aeroelastic stability of the wind turbine using the potential energy gains as a function of twist coupling on the blade. Other similar studies are shown in the papers “Compliant blades for wind turbines”[14] and “Compliant blades for passive power control of wind turbines” [15] where the articles shown investigations about the capability of a constant speed wind turbine to automatically shed power in gust by feathering the blades.

A mathematical model of an unsteady separated flow around an oscillating airfoil is published by O.Yu.Korotkov and G.M.Shumskii on 2000 [16]. This model uses a viscid-inviscid approach. The points of separation and the intensity of vor- ticity displaced to the external flow is determined using boundary layer equations in an integral form and the mechanism of antidamping for instability is discover. One year later is published a paper which shown the optimization of wind turbine blades based on the maximum frequency design criterion [17]. This paper analyzed also the Aeroelastic instabilities boundaries for the wind turbine using the Floquet’s transition matrix theory. This paper shows how the instabilities boundaries of a wind turbine becomes an important issue when it is optimized. From this time to now on publications related to find aeroelastic instabilities for wind turbines are common. 20 LITERATURE REVIEW 2.1

Yoshimasa Tomonari extended the Theodorsen function to assuming a arbi- trary and constant convection velocity for wake vortices in 2002 [8]. Some correc- tions of the Theodorsen model are based in this extension later on. The Energy research Centre of the Netherlands ECN published in 2003 an article which showed the importance to study the aeroelastic boundaries in the next generation of wind turbines [18]. The article presents the ECN wind energy research projects on the aero-elastic stability of rotor blade vibrations. They showed three diﬀerent investigations projects the STABTOOL-3, the DAMPBLADE and the STABCON.

Three articles were publish on the same year about instabilities on Wind Tur- bines. The first is an article published by Hans Ganander, it refers to the use of a code-generating system for the derivation of the equations for Wind Turbine Designs [19]. This code has the capability to tuning the eigenfrequency of the model with the measured eigenfrequencies of the real wind turbine, that gives the possibility to match the measured static eigenfrequencies to the model eigenfrequencies, the Multi-Body approach is used in this paper. The second is an article of the VISCEL proyect [20]. This article shows the study of a 2D airfoil stability using two different structural and aerodynamic models, one is the classical flutter test case and the other is the Stall-induced Flap-Lag flutter model. The third article called “The present status of the Aeroelasticity of Wind Turbines” shows the available aerodynamic 3D models at this time and the three aeroelastic modeling possibilities for the structural dynamics [21]. The three possibilities were a Finite Element Model, a Multibody Model or a Modal representation of the Wind Turbine. This article shown and compares the direction of vibration for the first two modes of the LM 19.1 m blade using a Multibody model and a Finite Element Model for the blade. The results for the direction of the vibration shows a large difference between the models (More than 50%) for r/R <0.36.

Publications of the year 2004 shown interest to find instabilities of the wind turbines based on the eigenvalues approach. Three papers published on this year show the stability analysis of wind turbines based on the eigenvalue approach and they uses the Multi-Body approach for the structural model of the wind turbine [22], [23] and [24]. The implemented aerodynamic model were a linearized aerodynamic model in [22], a linearized structural-aerodynamic coupling equations in [23] and the Theodorsen solution for an Oscillating Flat Plate in [24]. The article form Risoe presents a code called HAWCStab and the results of a simulation for the instabilities of a 600kW Wind Turbine [23]. It shows a comparison between the computed eigenfrequencies and a measured frequency using a developed experimental tool to estimate the aeroelastic damping. The results between the measurement and the predicted values for the eigenfrequencies are close to each other. This is one of the first tool for find instabilities on wind turbines which is validate in comparison with the measurement and it is obtained good agrement. 2.1 LITERATURE REVIEW 21

Don W. Lobitz on 2005 published an article called “Parameter Sensitivities Affecting the Flutter Speed of a MW-Sized Blade” [3]. This paper studies the sensitivity of two parameters on the value of flutter speed of a 35 meters wind turbine. The parameters are the chordwise location of the center of mass and the ratio of the flapwise natural frequency to the torsional natural frequency. The paper shows how different values for the parameters can highly conditioning the flutter speed. The same year The Technical University of Denmark publish a paper where a blade of a wind turbine in steady state is simulated [25]. A Finite Element Model for the structure of the blade is used and the distribution of the pressures over the blade are obtained from XFOIL. This paper did not do a stability analysis but it shown how the results of the stresses computed on the blade changes for different pressure distributions.

The mainly research objectives on the next years were to find news structural and aerodynamic models to find instabilities for large wind turbines. The people realized the problem of instabilities will be a critic factor for large wind turbines. This models should have the capability to simulate large deformations (non-linear models) with high accuracy. A review of the wind turbine aeroelasticity is publish in 2007 [26]. This review shown examples of simulation of wind turbines in time domain and how extract the unstable frequency by applying a FFT. An article called “Aeroelastic Stability of Wind Turbine Blade Section” [27] and “Investigation into the possibility of flap- lag flutter”[28] were published. The article “Aeroelastic Stability of Wind Turbine Blade Section” presents an stability analysis of a large wind turbine based on the Beddoes-Leishman model. The article “Investigation into the possibility of flap- lag flutter” concluded the flap-lag-stall instability is not likely to happen but the negative damping of the edgewise mode is an instability that can show up in blades. M.H. Hansen published an article in 2008 which deals with the aeroelastic instabilities that have occurred and may still occur for modern commercial turbines [29]. The treated instabilities are stall-induced vibrations for stall wind turbines and classical flutter for pitch regulated turbines. The same year the aeroelastic instability is studied for the new Floating Wind Turbine Concept [30].

Today new approaches and applications of the aeroelasticity for wind turbines are in use to create more efficient designs of wind turbines. One example is the “Smart” wind turbine rotor for load alleviation at Delft University of Technology. The proof of this concept is shown in the paper [31]. The paper shown a proof of the concept using two trailing edge flaps to suppress higher loads on the blades due to vibrations. A aeroelastic model for a non-rotating blade is studied and a control system for the trailing edge is tested. The results proof the feasibility of this new concept in aeroelasticity for wind turbines. Other projects in currently developing are the idea of search for aeroelastic instabilities using reduced order system identification based on flexible Multi-Body 22 LITERATURE REVIEW 2.1

[32], a new non-linear aeroelastic models for stall induced vibration [33], models for unsteady aerodynamic forces on small wings [34] and stability analysis for high angles of attack on parked wind turbine blades [35].

2.1.3 State of Art - Aeroelastic Codes for Wind Turbines Instabilities The following softwares are available for wind turbines aeroelastic analysis. ADAMS/WT (Automatic Dynamic analysis of Mechanical Systems - Wind Turbine) ADAMS/WT is programmed an application and as a specific add-on to Adams solver and ADAMS/View. ADAMS (Automatic Dynamic Analysis of Mechanical Systems) is developed by MSC Software and it is a general multipurpose multibody body dynamics code with unlimited degrees of freedom. This code is used also to model robots, satellites, and cars [36]. BLADED The software is developed by Garrad Hassan and Partners, Ltd. BLADED is an integrated simulation package for wind turbine design and analysis [37]. The solver is in time domain. FAST (Fatigue, Aerodynamics, Structures, and Turbulence) The FAST code is being developed through a subcontract between National Re- newable Energy Laboratory (NREL) and Oregon State University. NREL has modified FAST to use the AeroDyn subroutine package developed at the Univer- sity of Utah to generate aerodynamic forces along the blade [38]. YawDyn YawDyn is developed at the Mechanical Engineering Department, University of Utah, US with support of the National Renewable Energy Laboratory (NREL), National Wind Technology Center. YawDyn simulates the yaw motions or loads of a horizontal axis wind turbine, with a rigid or teetering hub [39]. FLEX4 The code is developed at the Fluid Mechanics Department at the Technical Uni- versity of Denmark. The program simulates turbines with one to three blades; fixed or variable speed generators pitch or stall power regulation. The code uses a Multi-Body formulation for the structures [40]. GAST (General Aerodynamic and Structural Prediction Tool for Wind Turbines) GAST is developed at the fluid section, of the National Technical University of Athens. The program includes a simulator of turbulent wind fields, time-domain aeroelastic analysis of the full wind turbine configuration and postprocessing of loads for fatigue analysis [41]. S4WT (SAMCEF for Wind Turbines) SAMTECH S.A is a European specialist in computer aided engineering software for finite element analysis and multi-disciplinary optimization. SAMCEF for wind turbines provides access to detailed linear or non-linear analyses of all relevant wind turbine components such as gearbox, blade and generator. This program gives to the user the capability to use Multi-Body and a Finite Element models for the simulations. The analysis is on time domain [42]. 2.2 LITERATURE REVIEW 23

PHATAS (Program for Horizontal Axis Wind Turbine Analysis Simulation) The PHATAS programs are developed at ECN Wind Energy of the Netherlands. The program is developed for the design and analysis of onshore and oﬀshore horizontal axis wind turbines. This program is based on a time domain simulation [43].

GAROS (General Analysis of Rotating Structures) The program is developed by Aerodyn Energiesysteme, GmbH and it is a general purpose design code for the dynamic analysis of coupled elastic rotating and non rotating structures with special attention to horizontal axis wind turbines. This software is based in time domain simulations [44].

FOCUS FOCUS (Fatigue Optimization Code Using Simulations) is an integrated wind turbine design tool, developed by Knowledge Centre Wind turbine Materials and Constructions (WMC), and includes modules developed by ECN. FOCUS consists of four main modules, SWING (stochastic wind generation), FLEXLAST (calculation load time cycles), FAROB (structural blade modeling) and Graph (output handling). A special module to compute the eigenfrequencies and eigenmodes for the structure of the wind turbine is available [45].

HAWC2 and HAWCStab HAWC2 and HAWCStab are developed at Riso in Denmark. The model is based on the Multi-Body method approach. The code HAWC2 predicts the response of horizontal axis two-or three bladed machines in time domain. HAWCStab predict the unstable operational points of a wind turbine. It is uses the eigenvalue approach to ﬁnd instabilities directly based on the dynamics of the system [46].

DHAT DHAT (Dynamic analysis of Horizontal Axis Turbines) originates from Germani- scher Lloyd (GL) WindEnergie GmbH, which is a certifying Multi-Body for wind turbines [47]. It is a HAWT-speciﬁc code with a similar design philosophy to FAST.

Mostly all the codes are based in time domain solution. The only code from the shown codes which is based on frequency domain is HAWCStab. A simulation in time domain always needs a post-processing technique to ﬁnd the unstable operational points (like FFT or parameter identiﬁcation techniques), instead from a frequency domain simulation is possible to known directly when the system is unstable. 24 LITERATURE REVIEW 2.2

2.2 Multibody and Finite Element Method

A general mechanical system is described by a system of partial differential equation (PDE) with its boundaries conditions. The system is discretized because is impossible to find close solutions for all the possible PDE systems in the conti- nuum space and time with different boundaries conditions. The discretization of the problem results in a system of equations which is possible to solve using a computer. The discretization of a PDE could be done using different methods. There is 4 main groups of methods: - The Finite Difference Method. - The Finite Volume Method. - The Finite Element Method. - The Multi-Body Method.

The Finite Element Method and the Multi-Body method are normally used to analyze mechanical systems because the mechanical systems are usually described using the Lagrange approach and the last two methods work well on PDE which are based on Lagrange formulations.

Multibody The Multi-Body method can be formulated using different ways. A motion can be represented by superimposing a rigid body motion and a relative flexible motion in multibody systems. If additionally the relative flexible motion is given in a body fixed frame (non-inertial frame), this is the classical flexible multibody formulation, see [48] and [49]. In the classical formulation, there exist the rigid body variables for each flexible body as unknown variables. The classical formulation can be characterized by the superimposed motion with the rigid body variables and a relative displacement vector given in a non-inertial body fixed frame. The classical formulation comes from rigid Multi-Body mechanics by adding flexibility to the bodies. Exist others variations for Multi-Body formulation but the basic idea is the same. The mechanical parts or “bodies” could be linear or non-linear and they are formulated with the idea to use a few nodes on the body and represent the behavior of the complete body. The boundary nodes are used to connect the body to another body or impose boundary conditions on the body. The middle nodes are used for the formulation of the body element. When a mechanical part has larger displacements is a good idea to use more than one element per body to increase the accuracy. This method does not require a mesh for the body, it only required the global mechanical properties for each body (for example mass, moment of inertia). The mechanical structure is conform by given the position and the connection between the boundary nodes for each of the bodies on the mechanism. The Multibody me- 2.2 MULTIBODY AND FINITE ELEMENT METHOD 25 thod is commonly used in dynamic analysis because it well represents the global dynamics of the bodies with a few degrees of freedom. The use of few degrees of freedom decreases the size of the problem doing it solution faster. The capability of a fast solution for different body configurations and properties do the Multi- body approach the appropriate method on the early stages of the Wind Turbine design where a lot of iterations with different configurations has to be done. Usually the Multi-Body formulation of a body is validate by a comparison to a Finite Element Model of the same body (See figure 2.2).

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Figure 2.2: Multibody validation diagram.

Finite Element Method The Finite Element Method (FEM) is a numerical technique to solve partial differential equations (PDE) by minimizing an integral error on the domain. The solution approach is based on eliminating the differential equation completely for steady state problems or change the PDE into an approximating system of ordinary differential equations, which can be integrated numerically using standard techniques such as Euler’s method or Runge-Kutta for example. The idea of this method is to minimize a functional which it has the spatial model and time as a domain. When this method is applied to a solid structure the Energy becomes the functional that the FEM minimizes. That means the method searches for the displacements on the nodes of the body mesh which equals the work done by the internal displacements on the body and the work done by the applied forces, the displacements must fulfilled the boundary conditions contrains. 26 LITERATURE REVIEW 2.2

The FEM is used for detailed analysis and for each body is required a mesh. Create the mesh of a body is not straight forward process although exist automatic meshers. The automatic meshers usually are designed to create tetrahedral mesh on the bodies and they do not have the capability to create quad meshes. A quad mesh gives higher accuracy on the FEM analysis, the disadvantage is the user has to do the mesh manually and this process is time consuming. A parametric mesh is created when the user has to simulate different modifications of a mechanism or optimize a mechanism which is an iterative process. A parametric mesh decreases the time to create the mesh because the model is meshing automatically. The first stages of a mechanism design do not require a detailed analysis but many simulations of different mechanism configuration, that is the reason why this method is commonly used only on the final and detailed analysis of the mechanism. Many methods are developed based on the FEM with the objective to find an efficient way to use the FEM method in early stages of the design. The intuitive approach is to use only the degrees of freedom of some nodes instead of all the degrees of freedom on the mesh. That concept is the same as model the mechanical part with a coarse mesh, the problem is the method has large errors for coarse meshes. This error decreases when the right nodes on the mesh are selected to represent the mechanism. The selection of the nodes are based on which dynamic behavior the model represents for the specific prescribed boundaries conditions. Three important methods were developed to reduce the size of the model. The modal reduction techniques, the static condensation and the dynamic substructuring.

The modal reduction technique changes the problem to a frequency domain and decoupled the system of ODE’s. The result is a system of equation in which every equation is orthogonal to the others. The system is solved by adding the contribution of the solution for each equation separately. The idea of this reduction technique is to use only the equations which are important to describe the system and neglect the small contributions to the solution of the other equations. The equations are usually selected based on two criteria, the high value for the projection of the spectral content of the excitation on the eigenmode of the decoupled equation or the excitation frequency is closer to the eigenfrequency of the decoupled equation (Natural frequency of the model). The idea is to solve only the important decoupled equations for the dynamic of the system doing the solution procedure faster.

The static condensation reduces the model by using an approximate displacement for some nodes. The diﬀerent variants of this method diﬀers only in how the method approximate the displacement in some nodes. The most common approach of this method is the Guyan’s static condensation. The Guyan’s static condensation method uses the static solution of the problem 2.2 MULTIBODY AND FINITE ELEMENT METHOD 27

(same as neglect the mass and damping on the system) and computes the displacement for the nodes. When is known the static displacement for the nodes the user selects on which nodes he want to impose this displacement and then the system is solved for the left degree of freedom. This algorithm is convenient because the error on the solution is only because the system do not consider the dynamical part of the solution of some nodes, from the static point of view the system is solved exactly. The dynamic substructuring is a method to split huges structures into smaller ones. The idea is to express the behavior of the body based only on a few degrees of freedom. The most common method is the Craig Bampton which uses the static solution for the boundary nodes plus the internal vibration modes for the structure as a basis for the displacements. This method is exact for static response of the interface nodes (Nodes to coupling diﬀerent parts). The ﬁgure 2.3 shows an schema for the common methods used to study mechanical systems.

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Figure 2.3: Diﬀerent methods to model a general mechanism.

The connection between the literature review and the work presented in this thesis

As the literature shows, there are two methods to study instabilities of wind turbines, the multibody and FEM. The main advantage of the multibody method 28 LITERATURE REVIEW 2.2 is lesser computational time when compared with a FEM, which for a dynamic response problem is of a great importance. However, for an instability problem, the designer is only interested to know the safe envelope of operating points and thus a frequency domain analysis is preferred. In addition, when the comparison comes to accuracy, the FEM promises better results. The reason for that is the usage of many degrees of freedom to discretize the domain under study, which results in a better capture of local properties. However, this does not mean that the designer can not use a multibody formulation in time domain to ﬁnd instabilities. It is indeed rather complicated than a FEM formulation in frequency domain. The main complexities are:

- Adding the numerical damping to the system because of time integration, if not impossible is very diﬃcult to do.

- An algorithm is needed to detect the unstable operational points. The numerical algorithms to detect instabilities in time domain are commonly based on FFT or model identiﬁcation which is an extra time consuming activity.

Now a question arises: Why do not create a methodology in which the designer can directly solve the problem with a high accuracy? The method developed in this work uses the FEM to analyze the stability of the wind turbine in frequency domain, solving the problem of both accuracy and time together. The method developed in this thesis has the capability to predict directly the stability of the system based on its eigenvalues. This approach is more appropriate to predict instabilities, than the multibody time domain formulation. Chapter 3

The Finite Element Approach for Aeroelastic Instability Prediction

The literature review gives the base of the work done in this thesis. This chapter explains how is developed the new methodology to compute the unstable operational points of a wind turbine based on its finite element model. This chapter is divided into four sections with specific subjects. The first section is the formulation of the structural model where is explained the FEM formulation and how it is coupled to a generalized body forces in a non-inertial reference frame, that gives the capability to include the rotational blade stiffness on the analysis. The second section is the formulation of the aerodynamic model. It is developed from the Theodorsen solution for an oscillating flat plate (potential flow) by doing some modifications on the lift and moment coefficients and adding the drag capability to the model. After that is explained how “convert” this model to an equivalent complex aerodynamic “mass”, “damping” and “stiffness” matrices. The third section explains the integration of the complex aerodynamic “mass”, “damping” and “stiffness” matrices to the common finite element model matrices. The integration of the aerodynamic and structural models conform the aeroelastic formulation of the proposed method. This section also explains the solution method to find the first unstable eigenvalues for different operational scenarios of the wind turbine. The fourth section relates the implementation of this method in a solver. The method is implemented in a solver for the NASTRAN finite element software. The result is a tool which predict the unstable operational points of a wind turbine design. The solver is used in the next chapter to verify the proposed method. The chapter fourth presents results for the 5MW baseline wind turbine for the EU funded UPWIND project and comparison with other method to find instabilities.

29 30 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.1

3.1 Formulation of the Structural Model 3.1.1 The Finite Element Method for Structural Components (FEM) The dynamic behavior of a mechanical part or body is represented using the elastodynamics equations. The elastodynamics equations are composed by a system of partial diﬀerential equations and its boundary conditions. The elastodynamics PDE for structural components and its boundary conditions is shown in the equation 3.1.

∂σji + Xj ρ0u¨j =0 in V0 ∂xi − tj = niσij = tj on Sσ σ = σ ij ji (3.1) σij = Cijklεkl ε = 1 ( ∂uj + ∂ui ) 2 ∂xi ∂xj uj = uj on Su Where: V (t): volume in the deformed configuration, mass density ρ. V0: volume in the undeformed configuration, mass density ρ0 (Lagrangian des- cripton). xi = [x1, x2, x3]: the cartesian coordinates. ui(xj ,t) = [u1,u2,u3]: the displacement field. S = Su + Sσ: the total surface of undeformed body, Su: area where the displacement are imposed ui = ui, Sσ: area where the surface traction are imposed: ti = ti. Xi: the applied body forces. ti: the surface traction imposed on Sσ. ni: the direction cosines. ui: the displacements imposed on Su.

A diagram of the PDE variables in shown in the ﬁgure 3.1. When the Finite Element Method is used to solve the system 3.1 the result is a ordinary system of diﬀerential equation which read as:

d2u(t) du(t) [Ms] + [Ds] + [Ks]u(t)= F(t) (3.2) dt2 dt

Where: Ms: structural mass matrix. Ds: structural damping matrix. 3.1 FORMULATION OF THE STRUCTURAL MODEL 31

ui ti

Su Sσ

V 0 Xi x3

V (t) x2

Figure 3.1: Elastodynamics of a continuous system.

Ks: structural stiﬀness matrix. u: vector which contains all the nodes displacement in function of time.

The system of equation 3.2 represents the dynamics of the system considering its structural properties and geometry. This thesis is not going deep on how obtained the matrices shown in equation 3.2. Many books explains deeply how apply the ﬁnite element method to structures and obtained the system of equation 3.2 as a result, for example the reference [50] explains clearly all the procedures to get the system of equation as shown in 3.2.

The work presented here uses the structural matrices Ms, Ds and Ks obtained from a ﬁnite element software. This matrices contains the structural information of the wind turbine which is used for the method to compute the instabilities.

3.1.2 Generalized Body Forces in a Non-Inertial Reference Frame

This section treats the derivation of the acceleration in an inertial reference frame. A general movement for a differential of mass is assumed to derive the equations for the velocity and acceleration. The non-inertial forces are obtained by applying the non-inertial acceleration to 32 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.1 a differential of mass following the Second Newton’s law. The forces are written in terms of the displacement with respect to the non-inertial frame. The figure 3.2 shows a diagram for the notation used in this section. Time derivative The time derivative for a vector in a non-inertial reference frame also takes into account the non-inertial reference frame movement. The time derivative for a position vector p in a non-inertial reference frame is given by:

dp(x,y,z) dp(x,y,z) = + (Ω) p(x,y,z) (3.3) dt dt 2 × 1 2 Where: ()1 is the derivative with respect to the inertial reference frame 1. ()2 is the derivative with respect to the non-inertial reference frame 2. Ω = (Ωx, Ωy, Ωz) is the angular velocity for the non-inertial reference frame respect to the inertial reference frame. Velocity, Acceleration and Forces The position for the point p2 showed in ﬁgure 3.2 is given by (respect to the inertial reference frame):

p2(x,y,z)= R1(x,y,z)+ R2(x,y,z) (3.4) Where: R1(x,y,z) is the position for the non-inertial reference frame. R2(x,y,z) is the position of the point p2 with respect to the non-inertial frame.

The velocity of p2 is obtained by derived the position of p2 with respect to time. Thus applying equation 3.3:

dp2(x,y,z) dp1(x,y,z) dp2(x,y,z) = + Ω (p2(x,y,z)) + (3.5) dt dt × 2 dt 1 1 2 The acceleration of p2 is obtained by deriving the velocity of the point p2 with respect to time. Applying the equation 3.3 to the equation 3.5 yields:

2 2 d p2(x,y,z) d p1(x,y,z) d dp2(x,y,z) = + Ω p2(x,y,z)+ dt2 dt2 dt × dt 1 1 1 (3.6) Applying equation 3.3 to 3.6:

2 2 d p2(x,y,z) d p1(x,y,z) dp2(x,y,z) dt2 = dt2 + Ω Ω p2(x,y,z)+ dt + 1 1 × × 2 d dp2(x,y,z) dt Ω p2(x,y,z)+ dt (3.7) × 2 3.1 FORMULATION OF THE STRUCTURAL MODEL 33

The last expression is rewritten as:

2 2 d p2(x,y,z) d p1(x,y,z) dΩ = dt2 + Ω Ω (p2(x,y,z))2 + dt (p2(x,y,z))2 + dt2 1 × × × 1 2 dp2 d p2(x,y,z) 2Ω dt (x,y,z) + dt2 (3.8) × 2 2 The equivalent force acting a body of mass m due to the acceleration is obtained by applying the second Newton’s law:

d2p F = m (3.9) · dt

d2p Where m is the mass and dt is the acceleration. Replacing equation 3.8 on equation 3.9 is obtained an expression for the force:

2 d p1(x,y,z) dΩ F = + Ω Ω (p2(x,y,z)) + (p2(x,y,z)) + dt2 × × 2 dt × 2 ZV 1 2 dp2(x,y,z) d p2(x,y,z) 2 Ω dt + dt2 dm (3.10) · × 2 2 Where V is the volume domain of the mass. 34 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.1

Inertial frame 1 z1 Non inertial frame 2 dm z2 ap2 R2

p2(x2,y2,z2)

y1 p1(x1,y1,z1)

Figure 3.2: General acceleration on point p2 in a rotating and translational reference frame 2 (Non-inertial frame) respect to the inertial reference frame 1.

3.1.3 Mass, Damping and Stiffness Matrices for Forces due Accelerations in a Non-Inertial Frame

The interest on this thesis is apply the non-inertial forces to the finite element matrices. The finite element method consider the deformation of the body on the displacement variables u. Thus, applying the definitions of the subsection 3.1.2 to the displacements, the position of the point p2 for a deformed body is given by:

p2(x,y,z)= p1(x,y,z)+ R2(x,y,z)+ u(x,y,z) (3.11)

Where: R2(x,y,z) is the position vector for the point p2 in the non-inertial reference frame and undeformed state. u(x,y,z) is the displacement due to the forces in the undeformed point p2.

From this part p1(x,y,z) and R2(x,y,z) is deﬁned constant over time. Thus 3.1 FORMULATION OF THE STRUCTURAL MODEL 35

the velocity and acceleration of the point p2 is given by:

dp (x,y,z) du(x,y,z) 2 = (3.12) dt dt

d2p (x,y,z) d2u(x,y,z) 2 = (3.13) dt2 dt2

The expressions for the position (equation 3.11), velocity (equation 3.12) and acceleration (equation 3.13) for the point p2 are replaced in equation 3.10 using the following notations for the vector components:

Fx Fe = Fy (3.14)   Fz e   Ωx Ωe = Ωy (3.15)   Ωz e   R2,x R = R ,y (3.16) 2e  2  R2,z e   ux ue = uy (3.17)   uz e   The resultant expression for the force is expressed in terms of the element matrices Mse, Dre, Kre, Mrie and vector Fre :

d2u du dΩ F = [Ms] + [Dr] + [Kr] u + [Mri] + Fr (3.18) e e dt2 e dt e e dt e

Where:

V dm 0 0 [Ms]e = 0 V dm 0 (3.19)  R 0 0 dm  R V e   Me is the mass matrix of the element. R

0 2 Ωzdm 2 Ωydm V − · V · [Dr]e = 2 Ωzdm 0 2 Ωxdm (3.20)  V · R RV − ·  2 Ωydm 2 Ωxdm 0 RV − · V · R e   R R 36 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.1

Dr is the damping element matrix associated to the rotation of the frame.

[Kr]e = C1 C2 C3 e (3.21) 2 2 V Ωydm V Ωzdm − dΩz − C1 = V dt dm + V ΩxΩydm (3.22)  − R dΩy R  dm + ΩxΩzdm − RV dt RV e  dΩz  RV dt dm + RV ΩyΩxdm − 2 2 C2 = V Ωzdm V Ωxdm (3.23)  −R dΩx −R  dm + ΩyΩzdm − VR dt RV e  dΩy  RV dt dm + RV ΩzΩxdm − dΩ C3 = x dm + Ω Ω dm (3.24)  RV dt RV z y  − Ω2 dm Ω2dm −R V x −R V y e   Kre is the element stiﬀness matrixR associatedR to the rotation of the frame.

0 V Rzdm V Rydm [Mri]e = Rzdm 0 Rxdm (3.25)  − V R RV  Rydm Rxdm 0 VR − V R e   The matrix Mrie is the elementR matrixR for body forces due to angular acceleration of the non-inertial reference frame.

2 2 V ΩyΩxRydm V ΩyRxdm V ΩzRxdm + V ΩzΩxRzdm − 2 − 2 [Fr]e = V ΩxΩyRxdm V ΩxRydm V ΩzRydm + V ΩzΩyRzdm  R − R 2 − R 2 R  ΩxΩzRxdm Ω Rzdm Ω Rzdm + ΩyΩzRydm RV − RV x − RV y RV e  (3.26) R R R R The vector Fre contains the constant body forces due to the rotation of the non-inertial frame.

Assembling global Mass, Damping and Stiffness matrix for forces due accelerations in a non-inertial frame

The element matrices [Mri]e,[Dr]e and [Kr]e are assembled to a global matrices [Mri], [Dr], [Kr] and [Fr]. The global matrices relates the forces for all the elements on the ﬁnite element model to a global force vector called F. The assemble procedure is shown in ﬁgure 3.15.

3.1.4 Coupling the Generalized Body Forces to the FEM Method The subsection 3.1 explains how get the structural ﬁnite element model from the elastodynamics PDE. The resultant equation which describes the structure is given by (See section 3.1): 3.1 FORMULATION OF THE STRUCTURAL MODEL 37

d2u du [Ms] + [Ds] + [Ks]u = F(t) (3.27) dt2 dt

This subsection explains how modify the equation 3.27 to include the body forces due to the non-inertial frame shown on the equation 3.18. Equation 3.18 is rewritten as:

du dΩ d2u [Dr] [Kr]u [Mri] Fr = [Ms] F (3.28) − dt − − dt − dt2 −

Following the Second’s Newton law is possible to interpret the right hand side of the equation 3.28 as a external forces applied to the mass in a inertial reference frame. The forces on the equation 3.29 are changed to a internal forces of the body. The reaction to the external forces determined the equation of motion. Thus, the equation 3.29 reads: du dΩ d2u F + [Dr] + [Kr]u + [Mri] + Fr = [Ms] (3.29) dt dt dt2

Where the right hand side contains all the internal forces for a movement of the body. The ﬁnite element model is described by the equation 3.27 is rewritten as: du d2u F(t) [Ds] [Ks]u = [Ms] (3.30) − dt − dt2

d2u The term [Ms] dt2 represent the force due to the mass acceleration. The variable u represent the displacement due to the forces produces by the non-inertial reference frame and the forces given from the ﬁnite element model equation. Thus, the sum up of the equation 3.29 and 3.30 results in a equation which couples the forces:

du dΩ du d2u F(t) + [Dr] + [Kr]u + [Mri] + Fr [Ds] [Ks]u = [Ms] (3.31) dt dt − dt − dt2

Equation 3.31 is factorized and reorganized similarly to the ﬁnite element model equation 3.27:

d2u du dΩ [Ms] + [Ds Dr] + [Ks Kr]u = F(t) + [Mri] + Fr (3.32) dt2 − dt − dt 38 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.1

The system of equation has the same structure of the Finite Element Method equations. The implementation of this model in a finite element code is easy and straightforward. The only task is modify the structural matrices of the initial finite element model and add the right equivalent forces. The new finite element model reads: d2u du [Msr] + [Dsr] + [Ksr]u = F(t)sr (3.33) dt2 dt

Where: Msr = Ms Dsr = Ds Dr − Ksr = Ks Kr − dΩ F(t)sr = F(t)+ Mri dt + Fr

This thesis uses the ﬁnite element model to compute the vibration of the model. The linear model for vibration reads : d2u du [Msr] + [Dsr] + [Ksr]u = 0 (3.34) dt2 dt where no external forces are applied to the model. In this presentation is assumed a constant rotational speed of the non-inertial reference frame, thus the matrix Kre is simpliﬁed to:

2 2 V (Ωy +Ωz)dm V ΩyΩxdm V ΩzΩxdm − 2 2 [Kr]e = V ΩxΩydm V (Ωz +Ωx)dm V ΩzΩydm  R − R R 2 2  ΩxΩzdm ΩyΩzdm (Ω +Ω )dm RV R V − RV x y e  (3.35) R R R The constant angular velocities are moved outside the integral terms of Dr and Kr: 0 2 Ωz 2 Ωy − · · [Dr]e = 2 Ωz 0 2 Ωx me (3.36)  · − ·  2 Ωy 2 Ωx 0 − · · 2 2  (Ωy +Ωz) ΩyΩx ΩzΩx − 2 2 [Kr]e = ΩxΩy (Ωz +Ωx) ΩzΩy me (3.37)  − 2 2  ΩxΩz ΩyΩz (Ω +Ω ) − x y   where me = V dm is the element mass. R This way of coupling do not required any information of the nodes position on the mesh, this is an advantage. Furthermore to implement the method only the matrices of the structural ﬁnite element model are needed because the matrix Ms contains the mass of the elements and the angular velocities are known, thus the 3.2 FORMULATION OF THE AERODYNAMIC MODEL 39

matrices Dre and Kre are easily computed. The eﬀect of include the forces due to the non-inertial reference frame produces changes in the stiﬀness and damping of the structure.

3.2 Formulation of the Aerodynamic Model

3.2.1 Aerodynamic Model 3.2.1.1 Basis of the Aerodynamic Model: The Theodorsen Solution for a Flat Plate

The aerodynamic model is based on the unsteady lift an moment solution for a 2D flat plate. The solution assumed potential flow and it is publish by Theodorsen in [7]. A scheme for the solution is shown in figure 3.3.

L y α, M U x ba d1 2b z,h

Figure 3.3: Flat Plate schematic for Theodorsen solution.

The Theodorsen solution is applicable for a ﬂat plate which simultaneously pitching and plunging in an oscillatory fashion as described below:

iωt iωt h = h0e α = α0e (3.38)

Where h0 and α0 are complex constants. The solution for the lift and the moment are:

iωC(k) iωb ω2b ω2b2a L =2πρU 2b[ h + C(k)α + [1 + C(k)(1 2a)] α h + α ] U 0 0 − 2U 0 − 2U 2 0 2U 2 0 (3.39) 2 iωC(k) iωb 2πρU b[d1[ h0 + C(k)α0 + [1 + C(k)(1 2a)] α0]+ U 2 2 2U 2 3 M = iωb ω ab − 1 2 ω b d α 2 h + ( + a ) 2 α ] 2 2U 0 − 2U 0 8 2U 0 ! (3.40) ωb Where C(k) is the Theodorsen function and k = U is the reduced frequency. The Theodorsen function is a complex valued function of the reduced frequency, 40 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2 it is given by the expression: (2) H1 (k) C(k)= (2) (2) (3.41) H1 (k)+ iH0 (k) Where H denotes the Hankel function. The real and imaginary part of C are displayed graphically in ﬁgure 3.4.

1.2 0

−0.02 1.1

−0.04

1 −0.06

−0.08 ) ) 0.9 C C ( ( −0.1

0.8 −0.12 Real Imag

−0.14 0.7

−0.16

0.6 −0.18

0.5 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ωb ωb k = U k = U

Figure 3.4: The Theodorsen function.

When a harmonic movement of the ﬂat plate is assumed 3.38 is possible to derive the velocities and acceleration of the ﬂat plate, Thus: h = h eiωt h˙ = h eiωtiω h¨ = h eiωtω2 0 0 0 (3.42) α = α eiωt α˙ = α eiωtiω α¨ = −α eiωtω2 0 0 − 0 Replacing expressions 3.42 into equations 3.39 and 3.40 is obtained the following expressions for the lift and moment:

C(k) b b b2a L =2πρU 2b h˙ C(k)α+[1+C(k)(1 2a)] α˙ h¨ + α¨ (3.43) − U − − 2U − 2U 2 2U 2 h i 2 C(k) ˙ b 2πρU b d1 U h + C(k)α + [1 + C(k)(1 2a)] 2U α˙ + M = − − 2 3 − (3.44) b ab 1 2 b  h dh α˙ 2 h¨ + ( + a ) 2 α¨ i  − 2 2U − 2U 8 2U The expression of the equations 3.43 and 3.2.1.3 arei the base 2D mode l used in this thesis for the lift and the moment prediction. 3.2 FORMULATION OF THE AERODYNAMIC MODEL 41

3.2.1.2 Drag Model for the Theodorsen Solution The lift and the moment expressions obtained in the section 3.2.1.1 are based on potential flow (incompressible and inviscid). This section shows how is done the incorporation of the drag to the aerodynamic model of the 2D section. Since is assumed the separation on the airfoil does not happened and the flow regime is laminar, a linear relation between the drag coefficient Cd and the angle ∂Cd of attack α is assumed. The slope ∂α is computed by fitting a linear curve to the Cd versus alpha curve of the airfoil using the points before the separation angle of attack. The drag coefficient is computed using the equation 3.45.

∂Cd Cd = αunsteady (3.45) ∂α ·

The unsteady angle of attack αunsteady is obtained from the Theodorsen solution of a Flat Plate. Equation 3.43 is rewritten with the objective to obtained the unsteady angle of attack expression from the Theodorsen solution:

L C(k) b b b2a αunsteady = =2 h˙ C(k)α+[1+C(k)(1 2a)] α˙ h¨+ α¨ πbρU 2 − U − − 2U − 2U 2 2U 2 h (3.46)i Replacing the unsteady angle of attack from equation 3.46 into 3.45:

2 ∂Cd C(k) b b b a Cd = 2 h˙ C(k)α+[1+C(k)(1 2a)] α˙ h¨ + α¨ (3.47) ∂α · − U − − 2U − 2U 2 2U 2 h i Thus the sectional drag is:

∂C C(k) b b b2a D = ρU 2b d h˙ C(k)α+[1+C(k)(1 2a)] α˙ h¨+ α¨ (3.48) ∂α · − U − − 2U − 2U 2 2U 2 h i This is the expression for the drag model for a 2D airfoil used in this thesis.

3.2.1.3 The Aerodynamic Model used in this Thesis The aerodynamic model used in this thesis is a modiﬁcation of the Theodorsen solution 3.2.1.1 for a Flat Plate and the drag model explained in section 3.2.1.2. Those 2D models are assembled over the radius of the wind turbine blade conforming the 2 dimensional model for the 3 dimensional blade.

2D model The lift expression for the Theodorsen solution showed in the equation 3.43 ∂Cl uses a value of 2π for the ∂α slope, this value is right for a potential ﬂow solution. ∂Cl In the model used in this thesis the 2π value is replaced by the real value of ∂α 42 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

∂Cl for each airfoil. The ∂α slope is obtained by ﬁt a linear curve to the points on the Cl versus alpha curve of the airfoil until the separation angle of attack. The resultant lift expression is:

∂C C(k) b b b2a L = l ρU 2b h˙ C(k)α+[1+C(k)(1 2a)] α˙ h¨+ α¨ (3.49) ∂α − U − − 2U − 2U 2 2U 2 h i The expression of the moment shown in the equation is also modiﬁed by re- ∂Cl placing 2π with the real slope of the airfoil ∂α , thus:

∂Cl 2 C(k) ˙ b ∂α ρU b d1 U h + C(k)α + [1 + C(k)(1 2a)] 2U α˙ + M = − − 2 3 − (3.50) b ab 1 2 b  h dh α˙ 2 h¨ + ( + a ) 2 α¨ i  − 2 2U − 2U 8 2U  i  The drag model is the same presented in section 3.2.1.2

∂C C(k) b b b2a D = ρU 2b d h˙ C(k)α+[1+C(k)(1 2a)] α˙ h¨+ α¨ (3.51) ∂α · − U − − 2U − 2U 2 2U 2 h i The last three expressions for sectional lift, moment and drag constitutes the aerodynamic model. A schema for the 2D aerodynamic model is shown in ﬁgure 3.5.

L z,h

U α, M y x d1 ba

b 2b

Figure 3.5: Schema of the Aerodynamic model. 3.2 FORMULATION OF THE AERODYNAMIC MODEL 43

Blade aerodynamic model based on the 2D model The blade of the wind turbine is thinking as a conglomerate of airfoil sections along the radius as shown the picture 3.6

Figure 3.6: Conglomerate of 2D sections conform the blade.

The lift, moment and drag for each section are:

∂Cl 2 C(k) ˙ b ∂α ρU b U h C(k)α + [1 + C(k)(1 2a)] 2U α˙ + Lsection = − − 2 − (3.52) b b a h 2 h¨ + 2 α¨ Asection − 2U 2U · i

∂Cl 2 C(k) ˙ b ∂α ρU b d1 U h + C(k)α + [1 + C(k)(1 2a)] 2U α˙ + Msection = − − 2 3 − b ab 1 2 b hd α˙h 2 h¨ + ( + a ) 2 α¨ Asection i − 2 2U − 2U 8 2U · i (3.53)

2 ∂Cd C(k) ˙ b ρU b ∂α U h C(k)α + [1 + C(k)(1 2a)] 2U α˙ + Dsection = · − − 2 − (3.54) b b a h 2 h¨ + 2 α¨ Asection − 2U 2U · i Where: Asection: the area of a part of the blade. The chord is represented using a constant 44 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2 value for each area.

Each blade section has defined their own direction for the flapwise, torsional and edgewise degree of freedom given by the unitary vectors e1, e2 and e3. This definition is given in the vector basis of the global coordinate system. (See figure 3.7).

e1, h

e2, α

Global system of coordinates e3, s z

Figure 3.7: Direction vectors for the ﬂapwise, torsional and edgewise degree of freedom.

Where: h: ﬂapwise degree of freedom. α: torsional degree of freedom. s: edgewise degree of freedom. e1: unitary vector which gives the direction of the ﬂapwise degree of freedom. e2: unitary vector which gives the direction of the torsional degree of freedom. e3: unitary vector which gives the direction of the edgewise degree of freedom.

The complete generalized aerodynamic forces (lift, drag and moment) acting on a part of the blade is computed by sum the generalized forces acting on each blade section which are on this part of the blade. Thus the total lift, drag and moment acting on the blade is:

ns ∂Cl 2 C(k) ˙ b j=1 ∂α ρU b U h C(k)α + [1 + C(k)(1 2a)] 2U α˙ + L = − − 2 − blade b ¨ b a P h h 2 h + 2 α¨ Asection − 2U 2U · j i i (3.55) 3.2 FORMULATION OF THE AERODYNAMIC MODEL 45

ns ∂Cl 2 C(k) ˙ b j=1 ∂α ρU b d1 U h + C(k)α + [1 + C(k)(1 2a)] 2U α˙ + M = − − 2 3 − blade b ab ¨ 1 2 b P h d2 hα˙ h2 h + ( + a ) 2 α¨ Asection i − 2U − 2U 8 2U · j i i (3.56)

ns 2 ∂Cd C(k) ˙ b j=1 ρU b ∂α U h C(k)α + [1 + C(k)(1 2a)] 2U α˙ + D = · − −2 − blade h h b ¨ b a P U 2 h + U 2 α¨ Asection − 2 2 · j i i (3.57) Where: ns: total number of sections on the blade. j: number of section.

3.2.2 The Finite Element Method applied to the Aerodynamic Model The ﬁnite element method is applied to the Aerodynamic model represented by equations 3.55, 3.56 and 3.57.

The purpose to apply the ﬁnite element method to the aerodynamic model is obtain the lift, drag and moment written as the system of equation 3.58:

d2u du [Ma] + [Da] + [Ka]u = Fa(t) (3.58) dt2 dt

Where: Ma: complex aerodynamic matrix. Da: complex damping matrix. Ka: complex stiﬀness matrix. Fa(t): vector of generalized aerodynamic forces (lift, drag and moment) applied to the nodes of the blade.

This subsection explains how obtained the system of equation 3.58 from the basic aerodynamic model given by the equations 3.55, 3.56 and 3.57.

3.2.2.1 The Basis idea: Minimization of the Aerodynamic Energy Functional The ﬁnite element method is used in this section to ﬁnd an approximation for a force by minimizing the integral, weighted by the displacement of this force, with respect to the displacements. The idea is shown in equation 3.59. 46 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

F min < F(u), u > (3.59) ≈ h i Where: F: force vector. u: displacement vector. < , >: inner product. · · The inner product for continuous function is the integral over the whole domain. Equation 3.59 reads:

F min F(u) du (3.60) ≈ · ZV The use of the right hand on the equation 3.60 to express the force has two main advantages. Firstly the possibility to divide the whole domain V into small do- n mains V = j=1 Vj and using the property of the integral (the integral of the complete domain is the same as the sum of the integral for all the sub-domains) is possible to integrateP the equation using only small domains. Secondly is possible to use diﬀerent displacement approximation for the displacement vector u in each of the sub-domains. This do the minimization process easier with the disadvantage to add a small error.

Lift, Drag and Moment functionals The lift, moment and drag expression given in equations 3.55, 3.56 and 3.57 are replaced into equation 3.60 to obtained an approximated expression for the lift, moment and drag. The integral part of the right hand side of the equation is called the functional and in this case represent the work done by the displacement due to the generalized force F.

ns ∂Cl 2 C(k) Lblade =min ρU b h˙ C(k)α ∂α − U − V j=1 Z X h h (3.61) b b b2a + [1 + C(k)(1 2a)] α˙ h¨ + α¨ Asection du − 2U − 2U 2 2U 2 · j · i i

ns ∂Cl 2 C(k) Mblade =min ρU b d h˙ + C(k)α + [1 + C(k)(1 2a)] − ∂α − 1 U − V j=1 Z X h h h 2 3 b b ab 1 2 b α˙ d2 α˙ h¨ + ( + a ) α¨ Asection du 2U − 2U − 2U 2 8 2U 2 · j · i i i (3.62) 3.2 FORMULATION OF THE AERODYNAMIC MODEL 47

ns 2 ∂Cd C(k) Dblade =min ρU b h˙ C(k)α ∂α · − U − V j=1 Z X h h (3.63) b b b2a + [1 + C(k)(1 2a)] α˙ h¨ + α¨ Asection du − 2U − 2U 2 2U 2 · j · i i The integral is split for the diﬀerent sections:

ns ∂Cl 2 C(k) Lblade =min ρU b h˙ C(k)α ∂α − U − j=1 Vj X Z h h (3.64) b b b2a + [1 + C(k)(1 2a)] α˙ h¨ + α¨ Asection du − 2U − 2U 2 2U 2 · · j i i

ns ∂Cl 2 C(k) Mblade =min ρU b d h˙ + C(k)α + [1 + C(k)(1 2a)] − ∂α − 1 U − j=1 Vj X Z h h h 2 3 b b ab 1 2 b α˙ d2 α˙ h¨ + ( + a ) α¨ Asection du 2U − 2U − 2U 2 8 2U 2 · · j i i i (3.65)

ns 2 ∂Cd C(k) Dblade =min ρU b h˙ C(k)α ∂α · − U − j=1 Vj X Z h h (3.66) b b b2a + [1 + C(k)(1 2a)] α˙ h¨ + α¨ Asection du − 2U − 2U 2 2U 2 · · j i i

Functionals considering the upper and the lower side of the blade.

The aerodynamic discretization of the blade is done using 2D quad element on the skin of the blade (See figure 3.11). The aerodynamic model given in equations 3.64, 3.65 and 3.66 is defined in the sectional area Asection of the blade, not in the skin of the blade as it needed. The proposed solution for this issue is to imagine the work done by the lift, moment and drag independently, due to the movement of the upper skin of the blade, is a half of the total work done by the sectional area Asection. The work done for the lower skin is also a half of the total work produce by the lift, moment and drag independently. The result is the work produced by the lower and the upper skin is the same as the total work. Thus the lift, moment 48 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2 and drag are possible to write in terms of the skin displacements. The functionals are defined as:

ns 1 ∂Cl 2 C(k) Lblade = min ρU b h˙ C(k)α 2 ∂α − U − j=1 Vj X Z h h b b b2a + [1 + C(k)(1 2a)] α˙ h¨ + α¨ Asection du + − 2U − 2U 2 2U 2 · · j us ns i i 1 ∂C C(k) min l ρU 2b h˙ C(k)α 2 ∂α − U − j=1 Vj X Z h h b b b2a + [1 + C(k)(1 2a)] α˙ h¨ + α¨ Asection du − 2U − 2U 2 2U 2 · · j us i i (3.67)

ns 1 ∂Cl 2 C(k) Mblade = min ρU b d h˙ + C(k)α + [1 + C(k)(1 2a)] 2 − ∂α − 1 U − j=1 Vj X Z h h h 2 3 b b ab 1 2 b α˙ d2 α˙ h¨ + ( + a ) α¨ Asection du + 2U − 2U − 2U 2 8 2U 2 · · j us i ns i i 1 ∂C C(k) min l ρU 2b d h˙ + C(k)α + [1 + C(k)(1 2a)] 2 − ∂α − 1 U − j=1 Vj X Z h h h 2 3 b b ab 1 2 b α˙ d2 α˙ h¨ + ( + a ) α¨ Asection du 2U − 2U − 2U 2 8 2U 2 · · j ls i i i (3.68)

ns 1 2 ∂Cd C(k) Dblade = min ρU b h˙ C(k)α 2 ∂α · − U − j=1 Vj X Z h h b b b2a + [1 + C(k)(1 2a)] α˙ h¨ + α¨ Asection du + − 2U − 2U 2 2U 2 · · j us ns i i 1 ∂C C(k) min l ρU 2b d h˙ + C(k)α + [1 + C(k)(1 2a)] 2 − ∂α − 1 U − j=1 Vj X Z h h h 2 3 b b ab 1 2 b α˙ d2 α˙ h¨ + ( + a ) α¨ Asection du 2U − 2U − 2U 2 8 2U 2 · · j ls i i i (3.69)

Where the subscripts us is the upper skin and ls is the lower skin. 3.2 FORMULATION OF THE AERODYNAMIC MODEL 49

The aerodynamic model formulated as equations 3.67, 3.68 and 3.69 has two main problems. Firstly the integration is not possible because the displacement vector u is not known, these displacements are the unknowns. Secondly under the supposition the displacements u are known is not possible to find a close solution for all the different domains. This difficulties are solved when a a displacement function for the sub-domains which has a closed form solution is assumed. Thus instead of a continuous body is used a discrete model of the body, where the body geometry is represented using sub-domains called elements. The next parts of this section explains how obtained the expression for the lift, moment and drag given in equation 3.67, 3.68 and 3.69 using the discrete body geometry.

3.2.2.2 Variables Deﬁnition of the Aerodynamic Element

The following variables are used to deduce the ﬁnite element model from the aerodynamic model, all the variables can have a subindex j which represents the section number and the subindex i which represents the element number on the section.

Geometric variables p13, p23 , p33 and p43 are the names of the nodes.

Degree of freedoms variables uk3m : Displacement on the node number k in the direction m. uk3θm : Angular rotation on the node k in the direction m.

Generalized forces variables

Fpkm : Force acting on the node k in the direction m.

Mpkm : Moment acting on the node k in the direction m.

General use variables x3, y3 and z3: the basis vector of the global coordinate system. R1: Lift functional. R2: Moment functional. R3: Drag functional.

The variables deﬁnition on the Quad element are shown in ﬁgure 3.8. 50 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

u33z , u33θz

Fp3z , Mp3z u43z , u43θz F , M p4z p4z p33(x3,y3,z3)

u33y , u33θy p43(x3,y3,z3) Fp3 , Mp3 u43y , u43θy y y

Fp4y , Mp4y u33x , u33θx u23z , u23θz u43x , u43θ x Fp3x , Mp3x Fp2z , Mp2z Fp4x , Mp4x

u1 , u1 u23 , u23 Global coordinate system 3z 3θz y θy Fp1z , Mp1z Fp2y , Mp2y z3 u2 , u2 3x 3θx p23(x3,y3, x3)

Fp2x , Mp2x u13y , u13θy

y3 Fp1y , Mp1y p13(x3,y3,z3) u13x , u13θx x3 Fp1x , Mp1x

Figure 3.8: Variables deﬁnition on the quad element.

3.2.2.3 Deduction of the Aerodynamic Element Matrices This subsection shows the deduction of the aerodynamic element matrices. The obtained expressions are very large and it is not possible to show them (could take an entire book to show the expressions). The idea to divide the deduction in nine steps is the reader can follows the deduction of the matrices having in mind the expressions for the next steps are defined in the anterior step and so on, a scheme for the steps in presented in figure 3.9. The different steps are programmed in MATLAB using its symbolic toolbox. The expressions for the equivalent mass, damping and stiffness aerodynamic matrices are big (around 100 pages of symbolic expression) but the evaluation is almost instantaneously when they are compiled in MATLAB. 3.2 FORMULATION OF THE AERODYNAMIC MODEL 51

u xy,, u xy ,, u xy , x3( 11) y 3( 11) z 3 ( 11 ) e1 ,e 2 ,e 3 θxy,, θ xy ,, θ xy , x3( 11 )( y 3 11 )( z 3 11 )

hhxy( 011,,,) hxy( 11 ,,) hxy( 11 , )

αα(011,,,xy )( α xy 11 ,, )( α xy 11 , )

" # ,,,,, ue ,,,,, ue R1 =OO O( Lhhh( ααα) d3 ⋅= 1 ) dAR3 2 OO O ( Mhhh( ααα ) d3 ⋅ 2) dA 3 A3 u3⋅ e 1 A3 u3⋅ e 2 ,,, , , u e R3 =OO O ( Dhhh(α α α ) d3 ⋅ 3 ) dA 3 A3 u3⋅ e 3

+ $

CCdA S SCCdA S S ,,, , , 3 ,,, , , 3 R1 = D Lhhh(α α α ) JD T dAd1 TDu3⋅ e 1 R 2 = Mhhh(α α α ) JD T dAd1 T u3 ⋅ e 2 O OODdA TD O OO dA T u3⋅ e 1 A1 EE1 U UEu3 ⋅ e 2A 1 E1 U U CCdA S S ,,, , , 3 R3 = D Dhhh(α α α ) JD T dAd1 T u3 ⋅ e 3 O OO DdA T u3⋅ e 3 A1 EE1 U U

* !

, " % &

( '

)

F= Mu + Du + Ku [ ]ee[ a] ( )e[ a] e( ) e[ a ] e ( ) e

Figure 3.9: Steps to deduce the element matrices for the aerodynamic model. 52 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

1. Definition of the section orientation The definition of the section orientation is given by the unitary vectors e1, e2 and e3 which defines flapwise, torsion and edgewise degree of freedom. The vectors are shown in the figure 3.7. Those vectors also defines the lift, moment and drag direction respectively. The definition of those vector are used for all the elements along the chordwise direction of the blade section. This vectors defines also the direction in which the aerodynamic work due by the lift, moment and drag are taken into account.

The vector e1, e2 and e3 are deﬁned by the user for each of the sections along the blade.

e3 βt

βp e30 βr

βV U k k xs e2 Ω r k × k

e10

Figure 3.10: Change on the direction of e1, e3 vectors and angle of attack due to the angular velocity Ω of the blade. The wind velocities are represented by shadow lines.

Where: xs: x axis parallel to the wind speed. ys: y axis parallel to the rotational wind speed. e10 : Geometrical e1 direction. e30 : Geometrical e3 direction. βV : wind speed angle. βp: pitch angle of attack. 3.2 FORMULATION OF THE AERODYNAMIC MODEL 53

βt: twist angle of attack. βr: angle of attack of the blade section.

The angle of attack βr is given by the equation 3.70.

π Ω r βr = βt βp arctan k × k (3.70) 2 − − − U k k The direction of the vector e2 is based on the position of the blade section, the direction does not change in function of the wind speed and the rotational speed of the blade. Vectors e1 and e3 are deﬁned in function of the wind speed and angular speed of the blade. The expression for those vectors are given in equations 3.71 and 3.72 and a schema is presented in ﬁgure 3.10.

(e10 (e10 e2)e2) cos(βr + βp) + (e30 (e30 e2)e2) sin(βr + βp)+ e2 e1 = − · · − · · (e10 (e10 e2)e2) cos(βr + βp) + (e30 (e30 e2)e2) sin(βr + βp)+ e2 k − · · − · · (3.71)k

(e30 (e30 e2)e2) cos(βr + βp) (e10 (e10 e2)e2) sin(βr + βp)+ e2 e3 = − · · − − · · (e30 (e30 e2)e2) cos(βr + βp) (e10 (e10 e2)e2) sin(βr + βp)+ e2 k − · · − − · · (3.72)k Where: e10 : Lift direction without rotational speed of the blade. e30 : Drag direction without rotational speed of the blade.

2. Local interpolation for displacement and element geometry to the 2D square reference frame The 3D model geometry is discretize using 2D quad elements for the aerodynamic model. The structural model could use any element topology for the mesh, the only requirement is the nodes of the Quad elements for the aerodynamic model must be coincident with the external nodes of the structural model. This means is possible to use different meshes for the aerodynamic and structural models since the external nodes of the structural model are coincident with the aerodynamic mesh. In this thesis the external geometry of the blade is meshed with 2D quad elements to applied the aerodynamic model. The blade is discretize by dividing it in radial sections and meshing each of the external resultant areas with the 2D quad elements as shown in figure 3.11. The 2D quad elements are defined in the 3D space. The quad elements defined in the 3D space are transform firstly to a local element 2D coordinate system and secondly to a reference coordinate system. This gives the capability to do all the mathematics operations independent of the geometry of the body and the reference integration domain is always constant in shape and area (see figure 3.12). 54 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

From 3D coordinate system to a 2D coordinate system The blade sections are discretize using 2D quad elements. The quad elements on the discretize blade are in the global 3D coordinate system for the aerodynamic mesh (the system is called system of coordinate 3). All the elements are deﬁned by those four nodes positions in the coordinate system 3. The quad elements are transform from the 3D coordinate system to the 2D system of coordinates called system of coordinates 2 (see ﬁgure 3.11) applying the following formula for a node p3 (subindex means the number of coordinate system).

p2 = [(p3 (p1 + p2 + p3 + p4 )) x32 , − 3 3 3 3 · (3.73) (p (p1 + p2 + p3 + p4 )) y ] 3 − 3 3 3 3 · 32

Where x32 and y32 are the basis vectors of the coordinate system 2 written in terms of the basis vectors of the coordinate system 3. Those vectors are deﬁned as follows:

(y32 (e2 y32 )) x 2 = × × (3.74) 3 (y (e y )) k 32 × 2 × 32 k

(p43 p13) y 2 = − (3.75) 3 p4 p1 k 3 − 3k Where e2 is defined in section 3.2.1.3, see figure 3.7. Applying equation 3.74 and 3.75 to the nodes is possible to rewrite the elements from the global 3D coordinate system (coordinate system number three) to a local 2D coordinate system for each element (coordinate system number 2) as shown in the diagram on the figure 3.11. 3.2 FORMULATION OF THE AERODYNAMIC MODEL 55

3D discrete blade model

Quad element in 3D mesh

p43 p33

p13 p23

System of coordinate 3 z3

p43(x,y,z) p33(x,y,z)

p2 (x,y,z) p13(x,y,z) 3

System of coordinate 2 y2 p3 (x, y) p42(x, y) 2

p2 p12(x, y) 2(x, y)

Figure 3.11: From the 3D blade geometry to the 2D Quad element scheme. 56 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

From 2D coordinate system to the reference coordinate system The quad elements are transformed from the 2D coordinate system (called coordinate system number 2) to the reference coordinate system (called coordinate system number 1) using the bilinear interpolation functions between the corners values of the quad elements. Thus the geometry are transformed using expressions 3.76. A scheme for the variables names and the transformation are shown in the ﬁgure 3.12.

p2(x1,y1)=p1 w1(x1,y1)+ p2 w2(x1,y1) 2 · 2 · (3.76) + p3 w (x ,y )+ p4 w (x ,y ) 2 · 3 1 1 2 · 4 1 1 The expression on equation 3.76 contains 4 shape bilinear functions: w (x ,y )= y 1 x 1 1 1 1 1 − 2 · 1 − 2 w (x ,y )= y 1 x +1 2 1 1 − 1 − 2 · 1 2 w (x ,y )= y + 1 x + 1 3 1 1 1 2 · 1 2 w (x ,y )= y +1 x 1 4 1 1 − 1 2 · 1 − 2 Those functions are the basis functions for the interpolation between the second coordinate system to the reference coordinate system. Those functions are linear independent to each others and when they are weighted by the values of the nodes conform the local interpolation functions for values at the interior of the elements. The shape functions are plotted in the ﬁgure 3.13. 3.2 FORMULATION OF THE AERODYNAMIC MODEL 57

Quad-bilinear ﬁnite element representation on the reference coordinate system

Element 2D system of coordinates Reference system of coordinates

y2 y1

p32 ( 0.5, 0.5) (0.5, 0.5) − p31 p42 equation 3.76 p41

x2 x1

p22 p11 p21 p12 ( 0.5, 0.5) (0.5, 0.5) − − −

Figure 3.12: Quad-bilinear ﬁnite element representation for the 2D geometry on the reference coordinate system. 58 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

Function for node 1 Function for node 2

1 1

0.5 0.5

0 0 0.5 0.5 0.5 0.5 0 0 0 0 y −0.5 −0.5 x y −0.5 −0.5 x Function for node 3 Function for node 4

1 1

0.5 0.5

0 0 0.5 0.5 0.5 0.5 0 0 0 0 y −0.5 −0.5 x y −0.5 −0.5 x

Figure 3.13: Shape functions for the bilinear quad element in the reference system of coordinates.

3D displacements representation on the reference coordinate system using bilinear interpolation The displacements for the 6 degrees of freedom of each node are interpolate using the bilinear basis functions presented in 3.76 and plot in ﬁgure 3.13. The interpolation expression reads:

1 1 1 1 u3(x1,y1)=u13 y1 x1 u23 y1 x1 + · − 2 · − 2 − · − 2 · 2 (3.77) 1 1 1 1 + u3 y + x + u4 y + x 3 · 1 2 · 1 2 − 3 · 1 2 · 1 − 2 3.2 FORMULATION OF THE AERODYNAMIC MODEL 59

Displacement representation using the reference coordinate system domain

Global system of coordinates Reference system of coordinates

u33 z3 y1 ( 0.5, 0.5) (0.5, 0.5) − u3 u43 u41 1 Equation 3.77

y3 x1

u23 u1 x3 u21 1 ( 0.5, 0.5) (0.5, 0.5) − − − u13

Figure 3.14: Quad-bilinear ﬁnite element representation for 3D displacements on the reference coordinate system as a domain.

3. Aerodynamic variables definition in terms of the nodes displacement The aerodynamic generalized forces are dependant on the variables: h, h˙ , h¨, α,α ˙ andα ¨. Those variables are written in function of the nodes displacements using the interpolation function for displacements 3.77 and the direction vectors used in the definition of the section orientation figure 3.10. The obtained expressions are:

h(x ,y )= h + [u (x ,y ), u (x ,y ), u (x ,y )] u (3.78) 1 1 0 31 1 1 32 1 1 33 1 1 · 1 h˙ (x ,y ) = [ ˙u (x ,y ), ˙u (x ,y ), ˙u (x ,y )] u (3.79) 1 1 31 1 1 32 1 1 33 1 1 · 1 h¨(x ,y ) = [ü (x ,y ), ü (x ,y ), ü (x ,y )] u (3.80) 1 1 31 1 1 32 1 1 33 1 1 · 1 α(x ,y )= α + [u (x ,y ), u (x ,y ), u (x ,y )] u (3.81) 1 1 0 3θ1 1 1 3θ2 1 1 3θ3 1 1 · 2 α˙ (x ,y ) = [ ˙u (x ,y ), ˙u (x ,y ), ˙u (x ,y )] u (3.82) 1 1 3θ1 1 1 3θ2 1 1 3θ3 1 1 · 2 α¨(x ,y ) = [ü (x ,y ), ü (x ,y ), ü (x ,y )] u (3.83) 1 1 3θ1 1 1 3θ2 1 1 3θ3 1 1 · 2 Where: h0: initial distance in the ys direction from the center of the blade section to the center of the element. This term disappeared when the functional is minimized.

α0: the steady angle of attack. This angle is deﬁned in the deﬁnition of the 60 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

section orientation and its called βr. The angle is shown in the ﬁgure 3.10.

4. Functionals definition The aerodynamic model results in 3 functionals (lift, moment and drag). The functionals are defined for blade sections because the aerodynamic forces are defined for blade section as well. The functionals reads :

R1 = L(h, h,˙ h,¨ α, α,˙ α¨)du3 e1 dA3 (3.84) A3 u3 e1 · Z Z Z ·

R2 = M(h, h,˙ h,¨ α, α,˙ α¨)du3 e2 dA3 (3.85) A3 u3 e2 · Z Z Z ·

R3 = D(h, h,˙ h,¨ α, α,˙ α¨)du3 e3 dA3 (3.86) A3 u3 e3 · Z Z Z · Where: R1: functional for the lift force. R2: functional for the torsional moment. R3: functional for the drag. A3: domain of integration which is the area of the element in the coordinate system number 3 or global coordinate system. The expressions for h, h˙ , h¨, α,α ˙ α¨ are replaced into the functional expressions. Those functional expressions are function of the elements displacements .

5. Changing the functional domain to the 2D square reference system The integration domain of the functionals given by the expressions 3.84, 3.85 and 3.86 are in the coordinate system 3 or global coordinate system. The explained geometry transformation from the 3D geometry to the 2D geometry (equation 3.73) is used to change the domain of integration from the 3D to a 2D domain of integration. The 2D domain of integration in the coordinate system number 2 is transformed to the reference domain of integration in the system of coordinates number 1 using equation 3.76 with the idea to have the same domain of integration for all the elements. The result are expressions for the functionals deﬁned in the reference integration domain A1. The functionals reads :

dA2 R1 = L(h, h,˙ h,¨ α, α,˙ α¨)du3 e1 J dA1 (3.87) A1 u3 e1 · · dA1 Z Z Z · dA2 R2 = M(h, h,˙ h,¨ α, α,˙ α¨)du3 e2 J dA1 (3.88) A1 u3 e2 · dA1 Z Z Z · 3.2 FORMULATION OF THE AERODYNAMIC MODEL 61

dA2 R3 = D(h, h,˙ h,¨ α, α,˙ α¨)du3 e3 J dA1 (3.89) A1 u3 e3 · dA1 Z Z Z · Where: A1: reference area of integration in the coordinate system 1. The area is shown in the pictures 3.14 and 3.12. dA2 J dA1 : Jacobian of the coordinate transformation from the domain in the coordinate system number 2 to the domain in the coordinate system number 1 or reference coordinate system. The jacobian of the transformation is given by the expression:

∂x2 ∂x2 dA2 ∂x1 ∂y1 J = ∂y2 ∂y2 (3.90) dA1 ∂x1 ∂y1

Where: is the symbol for a determinant of a matrix. | · |

6. Symbolic integration over the displacements The functional expressions 3.87, 3.88 and 3.89 are integrate over the displacements. The integration is done exactly using the basis interpolation functions presented in 3.76. The following variables are deﬁned to integrate expression 3.87:

a1= w1 e1x b1= w1 e1y c1= w1 e1z a2= w · e b2= w · e c2= w · e 2 · 1x 2 · 1y 2 · 1z a3= w3 e1x b3= w3 e1y c3= w3 e1z a4= w · e b4= w · e c4= w · e 4 · 1x 4 · 1y 4 · 1z The integration reads:

dA2 R1 = g1 J dA1 (3.91) A1 dA1 Z Z Where:

g = L a du1 + L a du2 + L a du3 + L a du4 1 · 1 · 31 · 2 · 31 · 3 · 31 · 4 · 31 Z Z Z Z + L b du1 + L b du2 + L b du3 + L b du4 · 1 · 32 · 2 · 32 · 3 · 32 · 4 · 32 Z Z Z Z + L c du1 + L c du2 + L c du3 + L c du4 · 1 · 33 · 2 · 33 · 3 · 33 · 4 · 33 Z Z Z Z (3.92)

The following variables are deﬁned to integrate expression 3.88: 62 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

a1= w1 e2x b1= w1 e2y c1= w1 e2z a2= w · e b2= w · e c2= w · e 2 · 2x 2 · 2y 2 · 2z a3= w3 e2x b3= w3 e2y c3= w3 e2z a4= w · e b4= w · e c4= w · e 4 · 2x 4 · 2y 4 · 2z The integration reads:

dA2 R2 = g2 J dA1 (3.93) A1 dA1 Z Z Where: g = L a du1 + L a du2 + L a du3 + L a du4 2 · 1 · 3θ1 · 2 · 3θ1 · 3 · 3θ1 · 4 · 3θ1 Z Z Z Z + L b du1 + L b du2 + L b du3 + L b du4 · 1 · 3θ2 · 2 · 3θ2 · 3 · 3θ2 · 4 · 3θ2 Z Z Z Z + L c du1 + L c du2 + L c du3 + L c du4 · 1 · 3θ3 · 2 · 3θ3 · 3 · 3θ3 · 4 · 3θ3 Z Z Z Z (3.94)

The following variables are deﬁned to integrate expression 3.89:

a1= w1 e3x b1= w1 e3y c1= w1 e3z a2= w · e b2= w · e c2= w · e 2 · 3x 2 · 3y 2 · 3z a3= w3 e3x b3= w3 e3y c3= w3 e3z a4= w · e b4= w · e c4= w · e 4 · 3x 4 · 3y 4 · 3z The integration reads:

dA2 R3 = g3 J dA1 (3.95) A1 dA1 Z Z Where:

g = L a du1 + L a du2 + L a du3 + L a du4 3 · 1 · 31 · 2 · 31 · 3 · 31 · 4 · 31 Z Z Z Z + L b du1 + L b du2 + L b du3 + L b du4 · 1 · 32 · 2 · 32 · 3 · 32 · 4 · 32 Z Z Z Z + L c du1 + L c du2 + L c du3 + L c du4 · 1 · 33 · 2 · 33 · 3 · 33 · 4 · 33 Z Z Z Z (3.96)

7. Functional integration using Gauss-Legendre 2D integration quadrature The expressions 3.91, 3.93 and 3.95 are diﬃcult (sometimes impossible) to integrate exactly since the jacobian expression for the transformation is not linear. The functionals are integrated over the reference domain area using the two points 3.2 FORMULATION OF THE AERODYNAMIC MODEL 63

Gauss-Legendre integration quadrature. The Gauss-Legendre integration quadrature is an approximate integration method described on detail in [51]. This method represent the integral of a function f(x) in the domain [ 1, 1] as a sum of the weighted values which takes the functions f for diﬀerent points− on the domain. The idea is presented in equation 3.97:

1 n f(x)dx ωi f(xi) (3.97) ≈ · 1 i=1 Z− X Where: n: number of evaluation points. ωi: weighting value. xi: evaluation point.

The error on this method is decreasing by increasing the number of evaluated points and choosing the right weighting values. The variants of this method are based on the number of evaluation points and weighting values [51]. This paper presents many variants of this method with the evaluating points and the weighting values on the interval [ 1, 1]. − Gauss-Legendre Two-Point Rule: The method used to integrate the functional over the area is called Gauss-Legendre Two-Point Rule. This method uses two points and two weighting values presented in the table 3.1. The reference integration domain is deﬁned between [ 1/2, 1/2] −

Points number ωi xi 1 1.0 1 √3 1 2 1.0 − √3

Table 3.1: Gauss-Legendre Two-Point Rule. in x1 and y1. For this domain the evaluation points and weighting functions are presented in the table 3.2. The application of the Gauss-Legendre Two-Point rule

Points number ωi xi 1 2.0 1 2√3 1 2 2.0 − 2√3

Table 3.2: Gauss-Legendre Two-Point Rule in domain [−1/2, 1/2]. 64 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2 to a function f(x, y) deﬁned in the reference domain is:

1 1 2 2 1 1 1 1 1 1 f(x, y)dxdy 4 f( − , − )+ f( − , )+ f( , − )+ 1 1 − − ≈ 2√3 2√3 2√3 2√3 2√3 2√3 Z 2 Z 2 1 1 f( , ) 2√3 2√3 (3.98)

The integral expression for the functionals using the equation 3.98 to integrate over the area are:

1 1 1 1 1 1 1 1 R1 =4 g1 ( − , − ) +g1 ( − , ) +g1 ( , − ) +g1 ( , ) (3.99) · | 2√3 2√3 | 2√3 2√3 | 2√3 2√3 | 2√3 2√3

1 1 1 1 1 1 1 1 R2 =4 g2 ( − , − ) +g2 ( − , ) +g2 ( , − ) +g2 ( , ) (3.100) · | 2√3 2√3 | 2√3 2√3 | 2√3 2√3 | 2√3 2√3 1 1 1 1 1 1 1 1 R3 =4 g3 ( − , − ) +g3 ( − , ) +g3 ( , − ) +g3 ( , ) (3.101) · | 2√3 2√3 | 2√3 2√3 | 2√3 2√3 | 2√3 2√3 Where g1, g2 and g3 are deﬁned in equations 3.92, 3.94 and 3.96 respectively.

8. Minimization of the functionals The functionals have to be minimized to get the generalized forces expressions. The functionals R1, R2 and R3 are function of the four nodes displacements of the quad elements. The functionals are deriving with respect to the degrees of freedom and is obtained an expression for the generalized forces acting on the nodes. The procedure is shown in the equations 3.102, and 3.103.

∂(R1 + R3) Fpkm = (3.102) ∂uk3m

∂R2 Mpkm = (3.103) ∂uk3θm Where k=1,2,3,4 is the node number and m=1,2,3 is the degree of freedom direction. The expression for the generalized forces are written in the vector [F] in the following organization:

[F]= Fp11 Fp12 Fp13 Mp11 Mp12 Mp12 Fp21 Fp22 Fp23 Mp21 Mp22 Mp22 h t (3.104) Fp31 Fp32 Fp33 Mp31 Mp32 Mp32 Fp41 Fp42 Fp43 Mp41 Mp42 Mp42 e i The expression for [F] is dependant on the generalized displacements, velocities and acceleration of the nodes. 3.2 FORMULATION OF THE AERODYNAMIC MODEL 65

9. Element matrices for the Aerodynamic Model The result for the expression 3.104 deﬁnes the complete generalized aerodynamic forces acting on a quad element. This expression for [F] 3.104 is organized in a matrices called element matrices [MA]e,[DA]e and [KA]e. Those matrices relates the complete set of degrees of freedom of a quad element with respect to the aerodynamic forces acting on its nodes [F]e. Since the model has many elements, many times this matrices has to be evaluated. The expression for the element matrices is shown in equation 3.105 (the subindex e means element). The vectors has dimension [1X24] and the matrices [24X24].

[F]e = [MA]e[¨u]e + [DA]e[ ˙u]e + [KA]e[u]e (3.105)

Where: [F]e: the generalized forces vector acting on the nodes of the quad element. The vector is shown in equation 3.106. [u]e: vector which contains all the displacements degree of freedom of the nodes on the element. The vector is shown in equation 3.107. [ ˙u]e: vector which contains all the generalized velocities degree of freedom of the nodes on the element. The vector is shown in equation 3.108. [¨u]e: vector which contains all the generalized acceleration degree of freedom of the nodes on the element. The vector is shown in equation 3.109. [MA]e: element matrix for the generalizes aerodynamic forces due the generalized acceleration degree of freedom. This matrix contains the terms of expression 3.104 factorized by the vector [¨u]e. [DA]e: element matrix for the generalizes aerodynamic forces due the generalized velocity degrees of freedom. This matrix contains the terms of expression 3.104 factorized by the vector [ ˙u]e. [KA]e: element matrix for the generalizes aerodynamic forces due the generalized displacement degrees of freedom. This matrix contains the terms of expression 3.104 factorized by the vector [u]e.

[F]e = Fp11 Fp12 Fp13 Mp11 Mp12 Mp12 Fp21 Fp22 Fp23 Mp21 Mp22 Mp22 h t (3.106) Fp31 Fp32 Fp33 Mp31 Mp32 Mp32 Fp41 Fp42 Fp43 Mp41 Mp42 Mp42 e i

[u]e = u131 u132 u133 u13θ1 u13θ2 u13θ3 u231 u232 u233 u23θ1 u23θ2 u23θ3 h t (3.107) u331 u332 u333 u33θ u33θ u33θ u431 u432 u433 u43θ u43θ u43θ 1 2 3 1 2 3 e i

[ ˙u]e = u˙131 u˙132 u˙133 u˙13θ1 u˙13θ2 u˙13θ3 u˙231 u˙232 u˙233 u˙23θ1 u˙23θ2 u˙23θ3 h t (3.108) u˙331 u˙332 u˙333 u˙33θ u˙33θ u˙33θ u˙431 u˙432 u˙433 u˙43θ u˙43θ u˙43θ 1 2 3 1 2 3 e i 66 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

[¨u]e = u¨131 u¨132 u¨133 u¨13θ1 u¨13θ2 u¨13θ3 u¨231 u¨232 u¨233 u¨23θ1 u¨23θ2 u¨23θ3 h t (3.109) u¨331 u¨332 u¨333 u¨33θ u¨33θ u¨33θ u¨431 u¨432 u¨433 u¨43θ u¨43θ u¨43θ 1 2 3 1 2 3 e i 3.2.2.4 Assembling of Global Aerodynamic Matrices The element aerodynamic matrices for each of the quad elements are assembled in a global matrices. This global aerodynamic matrices represent the aerodynamic forces over the nodes for the complete blade on the global coordinate system. The number of the element in the global system is deﬁned with the subindex l. Thus for an element l the equation 3.105 is written:

[F]l = [MA]l[ü]l + [DA]l[ ˙u]l + [KA]l[u]l (3.110) The global force vector is defined in the global system as: t [F]= ..., [F]l 1, [F]l, [F]l+1, ... (3.111) − i The global vectors which contains the degree of freedom are defined in equations 3.112, 3.113 and 3.114. t [u]= ..., [u]l 1, [u]l, [u]l+1, ... (3.112) − h it [ ˙u]= ..., [ ˙u]l 1, [ ˙ul], [ ˙u]l+1, ... (3.113) − h it [ü]= ..., [ü]l 1, [ü]l, [ü]l+1, ... (3.114) − The matrices [MA], [DA] andh [KA] are defined to satisfiedi equation 3.115. They are conform from the element matrices [MA]l,[DA]l and [KA]l respectively.

[F] = [MA][¨u] + [DA][ ˙u] + [KA][u] (3.115)

The process to conform the global matrices matrices [MA], [DA] and [KA] from the element matrices [MA]l,[DA]l and [KA]l is called assembling.

The local element matrix [MA]l,[DA]l and [KA]l for each element is divided in 16 parts, the column position of those sub-matrices in the global matrix are given by the position on the degree of freedom vector in the global matrix and the row position is given by the position of the generalized force vector for the element on the global matrix. The assembling procedure is repeated three times for all the elements of the FEM model to build the sparse “mass”, “damping” and “stiffness” aerodynamic matrices. The assembling procedure is showed in figure 3.15, the figure shows how the equation form the element matrix is positioning in the global matrices to conform the global matrix. 3.2 FORMULATION OF THE AERODYNAMIC MODEL 67

Element matrix

Generalized element Local Element force element DOF vector vector matrix

Forces node 1 DOF node 1 Forces node 2 DOF node 2 Forces node 3 DOF node 3 Forces node 4 DOF node 4

Equation for generalize forces acting on second node

Forces acting on second node Equation for generalize forces acting on second node

Generalized Global matrix DOF vector forces vector

Figure 3.15: Global matrices assembling process. 68 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.2

The Equivalent Complex “Mass”, “Damping” and “Stiffness” Matrices of the Aerodynamic Model The resultant aerodynamic model from the assembling process is:

[Fa] = [MA][¨u] + [DA][ ˙u] + [KA][u] (3.116)

Interpretation of the equation 3.116: The equation 3.116 can be interpreted as a mechanical system with a mass matrix [MA], damping matrix [DA] and stiffness matrix [KA]. The differences with respect to the structural matrices is the aerodynamic matrices are complex and the mass, damping and stiffness are not always positive definite matrices. The complex values of the matrices introduces forces which are not in phase with the generalized displacements on the nodes, this means the force response is not in phase with the movement of the nodes. The negatives or positive values of the eigenvalues of the aerodynamic matrices gives the possibility to have an unstable system, this is different in comparison with the structural matrices for a mechanical system the matrices for

3.2.2.5 A Non True Finite Element Matrices for the Aerodynamic Model One of the advantage of the ﬁnite element method for structural systems is the mass, damping and stiﬀness matrix for the elements are only dependant on the local properties of the element, that means the element matrices needs only properties of the local element to be evaluated. The aerodynamic matrices do not have this property. The aerodynamic matrices for the elements [MA]e,[DA]e and [KA]e developed in this thesis depends on aerodynamic and structural properties of the complete blade section. Five depen- dencies are distinguished:

1. Positional: The elements matrix are dependant on the radial position of the blade section on the blade. This position relates the angular velocity of the blade.

2. Dimension: The total chord of the blade 2b inﬂuences the elements on each of the sections along the blade.

3. Steady angle of attack (αi): The element aerodynamic matrices depends of the steady angle of attack of the complete blade section. All the elements which are in the same blade section has the same initial angle of attack. This angle is computed taken into account the structural pitch angle, the wind speed and the angular velocity of the blade.

4. Position of the elastic axis on the blade section (a): The element matrices changes for diﬀerent distance between the middle chord point and the elastic axis. This value is the same for all the elements which are on the same blade 3.3 INTEGRATION OF THE STRUCTURAL & AERODYNAMIC MODELS 69 section.

ω b 5. Reduced frequency (k): The reduced frequency is given by k = U . Al- though the vibration frequency ω is the same for the blades, the reduced frequency changes for diﬀerent blades section in function of the value for the half chord of the blade section b and the total velocity acting on the blade section U (composed by the wind speed and the velocity due to the blade rotation).

The physical meaning of why the local element depends on global and local properties is because the aerodynamics (represented by the aerodynamic model) depends on global properties and local properties. On the other hand the equation 3.1 for the structural model depends only in local properties and boundary conditions, that is the reason why the elements matrices for the structural part only depends on local properties.

3.3 Integration of the Structural & Aerodynamic Models

The deduced aerodynamic model, the structural model (including the rotational forces) are coupled in this section. The obtained system of equations represents the complete dynamic behavior of the wind turbine.

3.3.1 Complete Formulation of the Aeroelastic Method The coupling between the structural model of the wind turbine and the body forces in a rotational reference frame leads to the equation 3.33 which is :

d2u du [Msr] + [Dsr] + [Ksr]u = F(t)∗ (3.117) dt2 dt

The aerodynamic model leads to the system of equation 3.116 which is:

[Fa] = [MA][¨u] + [DA][ ˙u] + [KA][u] (3.118)

The external forces F(t)∗ on the system of equation 3.117 are the aerodynamic forces [Fa] of the system 3.118. The equation 3.118 is replaced into 3.117:

d2u du [Msr] + [Dsr] + [Ksr]u = [MA][¨u] + [DA][ ˙u] + [KA][u] (3.119) dt2 dt

The term of equation 3.119 are reorganized: 70 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.3

d2u du [Msr MA] + [Dsr DA] + [Ksr KA]u = 0 (3.120) − dt2 − dt −

The equation 3.120 is the system of equation which represents the complete behavior of the wind turbine (aerodynamic, rotational and structural models). The equation 3.120 is rewritten as: d2u du [Mt] + [Dt] + [Kt]u = 0 (3.121) dt2 dt

Where: [Mt]: the complex mass matrix of the complete model [Msr MA]. − [Dt]: the complex damping matrix of the complete model [Dsr DA]. − [Kt]: the complex stiﬀness matrix of the complete model [Ksr KA]. −

3.3.2 Methodology to find the First Unstable Operational Point Solve for the eigenvalues The complex system 3.121 is solved for the eigenvalues using the the Arnoldi algorithm with spectral transformation [52]. This algorithm computes the eigenvalues and the eigenvectors for the complex pencil shown in the equation 3.122: ([A] λ[B])X = 0 (3.122) − Where [A] and [B] are the complex matrices, λ are the eigenvalues and X are the eigenvectors. The second order ODE system of equation 3.121 must be transform to a system of equation like 3.122 to compute the eigenvalues using the Arnoldi algorithm. The transformation is done in two steps. The first step when the second order of ODE system given by the equation 3.121 is transform to a first order ODE and the second step when a general solution for an ODE linear system is replaced into the first order system of equation.

The chosen transformation of the second order ODE system 3.121 to a ﬁrst order ODE results in the system of equation 3.122: dv [A]V [B] = 0 (3.123) − dt

Where:

du V = dt (3.124) u 3.3 INTEGRATION OF THE STRUCTURAL & AERODYNAMIC MODELS 71

I 0 [A]= (3.125) 0 Kt 0 I [B]= (3.126) Mt Dt − − Where 0 is a null matrix and I is an identity matrix. The general solution for a linear ODE system u = C Xeλt is replaced into equation 3.123, where C is a constant, X are the eigenvectors· and λ are the eigenfrequencies. The resultant expression reads:

([A] λ [B])X = 0 (3.127) −

Equation 3.127 is in the form of equation 3.122 and it is solved using the Arnoldi algorithm with spectral transformation.

Iteration Procedure to ﬁnd the First Unstable Operational Points

Static Instabilities

The algorithm searches for the smaller rotor speed at which an static instability (divergence) occurs for a given wind speed. The ﬁrst unstable operational points are computed using an iterative algorithm. The algorithm has the following steps:

1. The user selects a wind speed and a rotor speed which he knows do not occur instabilities.

2. A ﬁnite element model of the wind turbine is given for the selected wind speed. This ﬁnite element model consider the pitch of the blade for the selected wind speed.

3. The user selects diﬀerent values for the reduced frequency k (that he wants to test for instabilities). This thesis used eight values for the reduced frequency k [0.001, 0.1429, 0.2857, 0.4286, 0.5714, 0.7143, 0.8571, 1.0000].

4. A set of frequencies ω are computed using the formula 3.128

k V ω = · (3.128) b Where: k: Reduced frequency. 72 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.3

V : Total velocity (velocity due rotational speed and wind speed) at the 75% of the blade length. b: Half of the chord at the section on the 75% of the blade.

5. The algorithm computes and assembles all the required matrices, for the given wind and rotor speed, to obtain the eigenvalue problem show on equation 3.127. For each of the selected frequencies ω the aerodynamic matrices changes.

6. The eigenvalue problem on step 5 is solved for all the selected frequencies ω. The eigenvalue problem is solved for the ﬁrst 20 eigenvalues with smaller modulus for each of the ω. The eigenvalue with smaller value of the imaginary part (absolute value) is saved for each of the ω.

7. The eigenvalue with smaller value of the imaginary part (absolute value) is selected from the saved values. The imaginary part of this eigenvalue is compared with the the imaginary part of the previous eigenvalue calculated for a lower rotational speed. If the imaginary values have the same sign the algorithm come back to step 3 with a bigger rotational speed.

8. The imaginary part of the last obtained eigenvalue has diﬀerent sign. This means the eigenvalue pass a point in which the eigenfrequency is zero. The rotational speed where the eigenfrequency is zero is computed using a linear interpolation. This point could be an unstable operational because the mode is in phase.

9. The real value for the possible unstable rotational speed is computed using a linear interpolation. A positive value means for this rotational speed the system is unstable because the mode is in phase and the amplitude increases in time, a negative value means only the mode is in phase and the instability do not occur, in this case the algorithm comes back to step 3.

10. The user come back to step 1 and start the algorithm again to ﬁnd the unstable rotational speed for a diﬀerent wind speed.

Note: Sometimes the sign does not change and the computed unstable speed is higher than the reality. This behavior is recognized when closer wind speed have very different unstable rotational speeds. When this happens the user can see the obtained eigenfrequencies in the step 7 are very close to zero at least in two iteration for different rotational speed. In this case, a second order polynomial fit is applied to find the angular speed for which a zero value of the eigenfrequency is obtained. The polynomial fit is done using the point with smaller eigenfrequency and the two points closer to it (see figure 3.17).

A scheme for the iteration procedure is shown in the ﬁgure 3.16. 3.3 INTEGRATION OF THE STRUCTURAL & AERODYNAMIC MODELS 73

ω !"# k⋅ V ω = b

d2u d u M+ D + u = 0 tdt2 t dt t

(A−λ B) X = 0

Figure 3.16: Iteration to ﬁnd the ﬁrst unstable operational points (static). 74 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.3

! "

Figure 3.17: Conditions for static unstable operational point. 3.3 INTEGRATION OF THE STRUCTURAL & AERODYNAMIC MODELS 75

Dynamic Instabilities

The algorithm searches for the smaller rotor speed at which a dynamic instability (ﬂutter) occurs for a given wind speed. The ﬁrst unstable operational points are computed using an iterative algorithm. The algorithm is the same as the algorithm to solve for static instabilities until the step number 6.

The next steps are new for dynamic instabilities:

7. The sign of the real part of the eigenvalue with smaller eigenfrequency is analyzed. A positive or a zero values for the real part is interpreted as instability. If the real part is negative, that mens the response of the system is decaying in amplitude and the system is stable, the used increment the angular speed and come back to step 3.

8. The user found a dynamic unstable point and he come back to step 1 and start the algorithm again to ﬁnd the unstable rotational speed for a diﬀerent wind speed.

When the author of this thesis test the method, he selected a small amount of different rotational speed to find the unstable points. This rotational speeds are based in experience and the objective was to do faster the process to find the different unstable points.

A scheme for the iteration is showed in figure 3.18 and the different possibilities of stability are showed in the figure 3.19 based on the eigenvalues of the system. 76 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.3

ω !"# k⋅ V ω = b

d2u d u M+ D + u = 0 tdt2 t dt t

(A−λ B) X = 0

Figure 3.18: Iteration to ﬁnd the ﬁrst unstable operational points (dynamic). 3.3 INTEGRATION OF THE STRUCTURAL & AERODYNAMIC MODELS 77

" !

# $

Figure 3.19: Possibilities for unstable operational point based on the eigenvalues. 78 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY PREDICTION 3.4

3.4 Implementation of the Method

The pre-processing is done using the PCL code written for the parametric model of the wind turbine. The solver is implemented in NASTRAN using the DMAP environment with an interface to a functions written in MATLAB. The post-processing part is done using the GUI of PATRAN.

The implementation of the method is divided in three main steps: pre-processing, processing and post-processing. Those three main steps are divided into ten substeps. The steps are showed in the ﬂux diagram of ﬁgure 3.20.

Pre-processing The pre-processing step is the substep number 1 in the figure 3.20. A code based in a parametric model of a wind turbine is written in PCL. This code creates the finite element model of the wind turbine efficiently. It is programmed due to the high difficulty to mesh a wind turbine. The programmed code has the capability to draw a wind turbine, create a complete mesh of the wind turbine using only quad elements, create and assign the properties for the different materials or elements. The time spent to create the finite element model of the wind turbine is reduced up to 97%. The code has the capability to change the pitch and the blade position easily. The connection between the tower, nacelle, rotor and blades is rigid. The nacelle and rotor are modeled as a disks with equivalent inertia and mass properties.

Processing The processing step is divided into eight substeps. The substeps are numbering from two to nine in the ﬁgure 3.20.

The substep two is the modification of the solver 107 for eigenvalues of NAS- TRAN. This solver computes the eigenfrequencies and eigenmodes of a structure using the direct approach (solve the eigenvalue of the complete system). A DMAP subroutine is compiled for this solver with the objective to modify the solver and create an access to all the finite element properties that is required to implement the method to include the aerodynamic and rotational forces due to the non inertial frame. The data of the finite element model is printed in the substep three.

The substep four until the substep eight are programmed in MATLAB. The substep number four read the data written by the DMAP subroutine in the NAS- 3.4 IMPLEMENTATION OF THE METHOD 79

TRAN solver and its convert in MATLAB readable files. The substeps five and six computes the aerodynamic matrices and the matrices to include the forces due to the non-inertial reference frame. The substep seven assemble and coupled the aerodynamic and the structural matrices considering the forces on the non-inertial reference frame. Thus it is conforming a system of equation which is solved for the eigenvalues and eigenmodes using the Arnoldi algorithm with spectral transformation [52]. This step uses the algorithm described in the section 3.3.2 to find the unstable operational points of the wind turbine. The results for the instabilities are sent to NASTRAN and they are printed in the step nine.

Post-processing The post-processing is the substep number ten in the figure 3.20. The operational unstable conditions are sent to the PATRAN GUI interface of NASTRAN. Al- though the complex eigenmode associated to the instability is computed, it is not possible to see using the PATRAN GUI. PATRAN is designed for structural analysis where the modes are real and it only has the capability to show real modes. The modes associated to the instability are complex due to the complexity of the aerodynamic matrices, then the implementation to see the complex modes were not possible in PATRAN. The flux diagram of the solver is shown in figure 3.20 80 THE FINITE ELEMENT APPROACH FOR AEROELASTIC INSTABILITY

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Figure 3.20: Implementation of the method using MATLAB, NASTRAN and PA- TRAN. Chapter 4

Veriﬁcation of the Method

The third chapter explains the theory behind the proposed approach, the development of the method and also how the method is programmed to build an aeroelastic solver. The result is a method implemented in NASTRAN which gives the capability to find unstable operational points for wind turbines. This chapter shows the verification of this method using the programmed solver in NASTRAN. The 5MW baseline wind turbine for the EU project Upwind is the selected wind turbine. The 5MW wind turbine is selected because the available data and research studies for it. The idea is to compare the instability boundaries predicted by the proposed method in this thesis, the instabilities boundaries obtained from the paper [32] and a simple model which is created due to scarce information of the unstable operational points in function of the wind speed. A parametric finite element model of the 5MW turbine is programmed in NASTRAN. The model is composed by the meshes of the tower and the blade. The Nacelle and hub are modeled as a disk at the center of gravity of each component and the drive train is simulated rigid. These disks have the same mass and inertias of each of the components. The tower and the blade are modeled using the real geometry with all the layers of the different composites materials and considering their different directions. The structural tower and blade properties for the finite element model matches very close to the global properties of the turbine. The blade layout is obtained from the report “UPWIND WMC5MW laminate data” [53]. This report tries to match the global properties of the layout for the 5MW Turbine to the global properties, although it is a serious report, the blade of the finite element model does not match completely with the multibody model. The report says “It should be noted, that the section properties of the URT are not 100% interconsistent. There are some differences in mass and (especially torsional) stiffness distribution” [53]. The next chapter shows an application of the solver, the instability boundaries for an optimum 20MW wind turbine.

81 82 VERIFICATION OF THE METHOD 4.1

4.1 Finite element model for the 5MWReferenceWindTurbine

The reference “NREL oﬀshore 5-MW baseline wind turbine” is the wind turbine model used to veriﬁed the method. The properties of this wind turbine are shown in [54]. This wind turbine is a three-bladed upwind variable-speed variable blade- pitch-to-feather-controlled turbine. This model was created by using the design information from the published documents of turbine manufacturers, with especial focus on the REpower 5M machine. Although the detailed data for the REpower 5M machine wind turbines was unavailable, it uses the public available properties from the conceptual models in the WindPACT, RECOFF, and DOWEC projects.

The wind turbine model for this thesis is divided into 5 diﬀerent subcomponents: The tower, nacelle, drive train, hub and blades. Those components are parametric modeling using the PCL language of PATRAN. The general properties of the 5MW wind turbine are shown in table 4.1.

Wind Regime IEC Class 1B / Class 6 winds Rotor Orientation Clockwise rotation - Upwind Control Variable Speed Collective Pitch Cut in wind speed 4 m/s Cut out wind speed 25 m/s Rated power 5 MW Number of blades 3 Rotor Diameter 126.0 m Hub Diameter 3.0 m Hub Height 90.0 m Maximum Rotor Speed 12.1 rpm Maximum Generator Speed 1,173.7 rpm Gearbox Ratio 97.0 Maximum Tip Speed 80.0 m/s Hub Overhang 5.0 m Shaft Tilt Angle 5.0 Rotor Precone Angle -2.5 Rotor Delta3 (sweep) Angle 0.0

Table 4.1: NREL 5MW wind turbine speciﬁcations.

4.1.1 Tower The tower is the deep support and it is modeled using the ﬁnite element method. The geometry is shown in ﬁgure 4.1.The material is isotropic with properties given 4.1 FINITE ELEMENT MODEL FOR THE 5MW REFERENCE WIND TURBINE 83 in table 4.2.

3.87m 0.0247m Tower Top

87.6m

38.23m

0.0351m Tower bottom 6.0m Variable Mudline

Figure 4.1: Deep tower. The boundary condition is applied at the tower bottom.

Young’s Elasticity Modulus 2.10E+11 N/m2 Shear Modulus 8.08E+10 N/m2 Eﬀective Density 8500 kg/m3

Table 4.2: Structural properties of the tower material. .

The tower is meshed using quad plate elements.

4.1.2 Hub and Nacelle Model

The Nacelle and the Hub are modeled as a disk with equivalent mass and inertia properties in the center of mass of each component. The diagram 4.2 shows the position of the hub and nacelle centers of mass. The consider properties for the Hub and Nacelle are in the table 4.3. 84 VERIFICATION OF THE METHOD 4.1

Yaw axis Nacelle

1.9m

0.21m Shaft axis

5.01m Hub

89.56mg Tower Top

Tower bottom

Figure 4.2: Nacelle and Hub center of mass.

4.1.3 Blades The aerodynamic properties for the blade section are given in table 4.4, the airfoils names are shown in table 4.5. The geometry, composites layout and materials are obtained from the UpWind report “WMC5MW laminate lay-out of reference blade for WP”[53]. The blade is modeling using the quad plate composites elements available in NASTRAN. The ﬁnite element model takes the real layout into account considering all the details and materials on the layouts shown in the Appendix I.

4.1.4 Boundary Conditions and Connection between the Blades, Nacelle, Hub and Tower The boundary condition is imposed by ﬁxing the nodes on the bottom of the tower. That means the displacements and the rotation degree of freedom of the bottom nodes are zero. The blades, nacelle,hub and tower are connected by imposing a condition on the 4.1 FINITE ELEMENT MODEL FOR THE 5MW REFERENCE WIND TURBINE 85

Hub Mass 56.780 Kg Inertia about the shaft axis 115.926 Kgm2 Nacelle Mass 240.000 Kg Inertia about the yaw axis 2.607.890 Kgm2

Table 4.3: Hub and Nacelle properties. .

Figure 4.3: A picture of the ﬁnite element model. The rigid connectors are represented by pink lines.

nodes which are the top of the tower, the center node of the disk which represents the nacelle, the center node of the disk which represents the hub and the nodes which are on the border of the blade close to the shaft. The condition is all the nodes must have the same displacements for all the degrees of freedom, that means have equals displacements and rotations. The result is a connection which is rigid between the nodes. A scheme for this rigid connection is show in the ﬁgure 4.3. The large displacements on a wind turbine are on the blades and the tower, thus the assumption to impose a rigid connection between the blades and the tower considering a rigid hub and nacelle is possible. 86 VERIFICATION OF THE METHOD 4.1

Radial position Twist Radial distance Chord Airfoil number [m] [deg] [m] [m] [ ] − 0.000 13.308 2.733 3.542 1 4.100 13.308 2.733 3.854 1 6.833 13.308 2.733 4.167 2 10.250 13.308 4.100 4.557 3 14.350 11.480 4.100 4.652 4 18.450 10.162 4.100 4.458 4 22.550 9.011 4.100 4.249 5 26.650 7.795 4.100 4.007 6 30.750 6.544 4.100 3.748 6 34.850 5.361 4.100 3.502 7 38.950 4.188 4.100 3.256 7 43.050 3.125 4.100 3.010 8 47.150 2.319 4.100 2.764 8 51.250 1.526 4.100 2.518 8 54.667 0.863 2.733 2.313 8 57.400 0.370 2.733 2.086 8 61.500 0.106 2.733 1.419 8

Table 4.4: Distributed blade aerodynamic properties. .

4.1.5 Pitch, Rotational and Wind Speed Control Curves The control of the blade pitch and the rotational speed in function of the wind speed is obtained from [54]. The figure 4.4 shown the curves for the blade pitch and rotational speed of the rotor in function of the wind speed. The values shown in figure 4.4 are the curves used to define the operational points of the wind turbine.

4.1.6 Parametric Model in PATRAN The wind turbine is parameterized and a code is programmed using the PCL (PA- TRAN) language to automatize the drawing and the mesh of the wind turbine. One of the big advantages of the programmed parametrization is for any wind turbine geometry based on the 5MW wind turbine is possible to create automatically a quad mesh for all the parts. A diagram for the parametric code is shown in the ﬁgure 4.5.

Mesh Requirements The only requirement to apply this method is the mesh of the aerodynamic surfaces must be meshed with quad elements. This wind turbine is 4.1 FINITE ELEMENT MODEL FOR THE 5MW REFERENCE WIND TURBINE 87

NFoil Airfoil name Normalized thickness [t/c] 1 cylinder1 100 2 cylinder2 90 3 DU40 A17 40 4 DU35 A17 35 5 DU30 A17 30 6 DU25 A17 25 7 DU21 A17 21 8 NA64 A17 17

Table 4.5: Airfoils for the 5MW wind turbine. .

Blade Pitch [◦] Rotor RPM 20

5 10 15 20 25 m Wind Speed s h i Figure 4.4: 5MW Wind turbine control curve.

meshed in such way that the structural model also is used to deﬁned the aerodynamic model using the discretization of the external skin of the blade. The tower and the blades are meshed with only quad elements. This is chosen because the 88 VERIFICATION OF THE METHOD 4.1

Input Parameters PCL Code Operation Point

- 2D Airfoils geometry - Blade Pitch - Angular position of - Airfoil position Draw Geometry - Chord and twist distribution the blades around shaft -Spars , web and shell axis dimension -Tower dimensions

- Composite material direcction -Composite materials properties and plies -Number of elements per section of the blade Mesh geometry - Mass and inertia of the (Quad elements) Nacelle and Hub -Number of elements for the tower

Rigid conection between nacelle, hub, tower and blades

Impose boundary conditions on the tower

Structural Wind Turbine Finite Element Model

Figure 4.5: Diagram for the parametric wind turbine model in PATRAN

quad elements gives better results since the element integration on the reference frame has less error although do a mesh with only quad elements is harder. The geometry and the mesh of quad elements is parameterized.

4.1.7 Model summary

The summary of the ﬁnite element model for the 5MW wind turbine are show in the table 4.6: 4.1 FINITE ELEMENT MODEL FOR THE 5MW REFERENCE WIND TURBINE 89

Material properties (Plies laminate and directions) 53 Number of element (CQUAD4) 4650 Number of nodes 4455 Number degrees of freedom (s-set) 26136

Table 4.6: Summary of the 5MW ﬁnite element model. .

4.1.8 Simple Model for Stability Analysis A simple model is proposed to study the instabilities and compare the order of magnitude for the instabilities boundaries. The model is two dimensional and it is located in a plan which cut the blade perpendicular at 75% of its radius. The model has two degree of freedom, h and α and it is used the structural properties of the blade at 75% of its radius. The aerodynamic model is the Theodorsen solution for a ﬂate plate. A diagram of the model is presented in the ﬁgure 4.6.

y2 βn Kh V Ω r + V k k k × k h,L

x z Ω r βs Kα k × k

βp βa M,α βn

Figure 4.6: Simple model for stability analysis at 75% of the blade radius.

Where: βa: Wind angle of attack. βs: Structural angle of the blade at 75% of the radius. 90 VERIFICATION OF THE METHOD 4.1

βp: Pitch angle of attack. Kh: Blade flapwise stiffness. Kα: Torsional stiffness of the blade. h: Plunge degree of freedom. α: Rotational degree of freedom.

When the second Newton law is applied to the elastic axis of the blade section (axis z) is obtained the equation of motion for the model: ¨ Fy2 = m h (4.1) · X Mz = Iθ θ (4.2) · The equations are rewritten as: X

m h¨ L cos(βn)+ Kl h =0 · − · · (4.3) Iθθ¨ L d cos(βn) Mθ + Kαα =0 − · · − The lift and the moment are given by: C(k) b b b2a L =2πρu2b − h˙ + C(k)α +(1+ C(k)(1 2a)) α˙ h¨ α¨ (4.4) u − 2u − 2u − 2u2 2 C(k) ˙ b M =2πρu b d1( u h + C(k)α +(1+ C(k)(1 2a)) 2u α˙ + − 2 3 − (4.5) b ab 1 2 b d α˙ 2 h¨ ( + a ) 2 α¨ 2 2u − 2u − 8 2u The equation of motion are reorganized as: MX¨ + DX˙ + KX = 0 (4.6) Where the vector X and the matrices are deﬁne by:

X = [h α]t (4.7)

m b ab2 2πρu2 b + 2u2 cos(βn) 2u2 cos(βn)

M  2 3  = ab b Iθ 1 2 b (4.8) 2u2 dcos(βn)( 2u2 cos(βn)) 2πρu2b + ( 8 + a ) 2u2 +  − ab2   d cos(βn)( 2 cos(βn))   − · 2u   

C(k) b u cos(βn) (1+ C(k)(1 + 2a)) 2u cos(βn)   D = C(k) b b u d1+ d2 2u (1 + C(k)(1 2a)) 2u d1+  C(k) − − − b   d cos(βn)( cos(βn)) d cos(βn)((1 + C(k)(1 + 2a)) cos(βn))   − · u − · 2u   (4.9) 4.1 FINITE ELEMENT MODEL FOR THE 5MW REFERENCE WIND TURBINE 91

Kl 2πρu2b C(k)cos(βn) K = Kθ − (4.10) 0 2 C(k)d + d cos(βn)C(k)cos(βn) " 2πρu b − 1 · #

This method is implemented in the 5MW and 20MW wind turbines assuming the structural frequency to compute the reduced frequency is constant for all the diﬀerent wind speed. The values assumed are 0.20 Hz for the 5MW wind turbine and 0.11 Hz for the 20MW wind turbine. Those values are the eigenfrequency obtained from the ﬁnite element model of the wind turbines with 4% of structural damping.

Sectional property 5MW 20MW Chord [m] 2.7 8.0 Mass [kg/m] 138.9 709.8 Momentum of Inertia [kgm] 27.7 149.8 Flapwise stiﬀness [N/m2] 2.7E3 8.5E3 Torsional Stiﬀness [N] 3.5E5 2.4E6 Radial position [m] 47.7 109.5

The wind turbines are pitch regulated and their pitch control curves are shown in the figures 4.4 and 5.5. The first unstable angular velocity of the blade for a range of wind speed is showed in the figure 4.7 The unstable speeds are highly dependant on the reduced frequency value. Although that means this model is not accurate, it is in the right order of magnitude [32]. This model is used as an estimation of the instabilities for different wind speed and obtain a guess for the instabilities of the 20MW wind turbine. 92 VERIFICATION OF THE METHOD 4.2

Instabilities for the simple model of 5MW Instabilities for the simple model of 20MW 36.6 28.5

28 36.55 27.5

27 36.5

26.5

36.45 26 Rotational speed [RPM] Rotational speed [RPM] 25.5 36.4 25

36.35 24.5 0 10 20 30 0 10 20 30 Wind speed [m/s] Wind speed [m/s]

Figure 4.7: Instabilities for the simple model.

4.2 Unstable Operational Points for the 5MW Wind Turbine

4.2.1 Static Unstable Points

The static unstable operational points (divergence) for the 5MW are plotted in the figure 4.8. The plot shows two types of solutions, the “potential flow solution” and the “viscous solution”. The potential flow solution is the solution obtained with the implementation of the methodology explained in this thesis using 2π value for the lift coefficient without including drag. The “viscous solution” is the solution obtained when it is included the real aerodynamic characteristics of the airfoils taken into account the lift coefficients, the drag and the moment coefficient for the different airfoils along the blade.

The “viscous” model predicts higher rotational speed as it is expected. This makes sense because the drag dissipates energy from the wind turbine. 4.2 UNSTABLE OPERATIONAL POINTS FOR THE 5MW WIND TURBINE 93

Predicted unstable operational points for the 5MW wind turbine 25 Potential flow Viscous flow

10 Rotational speed [RPM]

0 0 5 10 15 20 25 Wind speed [m/s]

Figure 4.8: Static instabilities for the 5MW wind turbine (divergence).

The predicted rotational speed is higher than the rated rotor speed for the 5MW wind turbine. Thus divergence instabilities are not expected for the wind turbine.

4.2.2 Dynamic Unstable Points The static unstable operational points (flutter) for the 5MW are plotted in the figure 4.9. The plot shows four types of solutions. The solution for the simple model, the solution using the “potential flow”, the “viscous solution” and the solution from the paper “AIAA 2008-1302” which is the unstable point showed in the paper [2]. The unstable operational point for the “viscous” and the “potential flow” solution at the rated wind speed (10m/s) surrounds the solution from the paper “AIAA 2008-1302”. It is possible to deduce (at least for this wind speed) the methodology proposed in this thesis to predict the instabilities is accurate.

The unstable rotational speed of the rotor predicted by the “potential” solution is always smaller than the speed predicted by the “viscous” solution for all the points. This behavior matches with the physics behind the aerodynamic model 94 VERIFICATION OF THE METHOD 4.2

Predicted unstable operational points for the 5MW wind turbine

35 Simple model Potential flow 30 Viscous flow AIAA 2008−1302 25

15 Rotational speed [RPM] 10

0 5 10 15 20 25 Wind speed [m/s]

Figure 4.9: Dynamic instabilities for the 5MW wind turbine (ﬂutter).

because from the energy point of view, the energy of the system should increase when the system are in the unstable point, the lift add energy to the system and the drag take out energy of the system, this mean when drag is added to the system the wind turbine could rotate faster because the drag is dissipating energy.

The lower unstable rotor speed is at lower wind speed , that agrees with the lower aerodynamic damping predicted at lower wind speed in [55].

The simple model overestimate the maximum rotational speed of the wind turbine.

The predicted rotational speed for ﬂutter is higher than the rated rotor speed for the 5MW wind turbine. Flutter instabilities are not expected for this wind turbine. Chapter 5

Application: Analysis of a 20MW Wind Turbine

The fourth chapter shows the veriﬁcation of the method by testing the solver in the 5MW baseline wind turbine for the Upwind EU project. The Upwind project searches for new design and solutions for bigger wind turbines. This chapter shows the analysis of the aeroelastic instabilities for an optimum 20MW wind turbine developed using the 5MW baseline wind turbine. The parametric model for wind turbines programmed in NASTRAN is used to create the mesh for the tower and the blade of the 20MW turbine. The drive train is modeled as a rigid connection between the blades and the tower, the nacelle and hub are modeled as a equivalent disk with equivalent mass and inertia following the same idea as the chapter 3 and 4. The analysis of this wind turbine presents a real application of the methodology and the solver.

5.1 Upscaling process of the 5 MW NREL wind turbine to an optimum 20MW

Comparing to traditional design methods, design search and optimization has many advantages. It forces the design team to set up an organized approach to solve the problem. Immediate interdisciplinary interaction instead of a sequential interaction that goes slowly through each discipline for each design iteration is the main advantage of multidisciplinary design optimization (MDO). This makes the end results of a MDO globally optimum among all the disciplines, rather than the local optimum of each discipline that does not necessarily guarantee the global optimality of the system.

Although the MDO methodology has some signiﬁcant advantages over the tra-

95 96 APPLICATION: ANALYSIS OF A 20MW WIND TURBINE 5.2 ditional design methodologies, there are also some disadvantages. The design solution of a MDO methodology can be sensitive to the robustness of the analysis codes, since an optimizer will quickly exploit any weakness in an analysis code. Therefore, the design team should ensure that the analysis codes are valid for the entire design space. MDO methodology also should not be viewed as a substi- tute for expert knowledge and engineering judgement. It is simply another tool in the designer’s toolbox, which allows more eﬃcient exploration of the design space.

Considering all the advantages and disadvantages of the MDO, it is conside- red as the best alternative to design the 20 MW wind turbine, since it enables the design of the wind turbine as a complex system, which its physics involves couplings between various interacting disciplines/phenomena. However, still the linear upscaling approach is used to get the initial design variables of the 20 MW wind turbine from the 5 MW NREL wind turbine.

Explaining the design optimization process of the 20 MW wind turbine is out of the scope of this thesis, and only the ﬁnal results of the multidisciplinary design optimization of the 20 MW wind turbine are used here. The interested reader can refer to the PhD dissertation of Mr. Turaj Ashuri, Delft University of Tech- nology. However, the usage of the optimum 20 MW wind turbine data guarantees that the model in which the aroelastic instabilities are checked for is a realistic model. This increases the conﬁdence on addressing the aeroelastic instabilities issues of larger scale wind turbines, since the results of any aeroelastic instability study is model dependent.

5.2 Finite Element Model for the 20MW Reference Wind Tur- bine

The general properties of the the 20MW wind turbine are show in the table 5.1:

5.2.1 Tower

A scheme for the geometry of the tower of the 20MW wind turbine is showed in ﬁgure 5.1. The material is isotropic with properties given in table 5.2.

The mesh of the tower for the 20 MW wind turbine is also meshed using quad elements. 5.2 FINITE ELEMENT MODEL FOR THE 20MW REFERENCE WIND TURBINE 97

Table 5.1: 20MW wind turbine speciﬁcations.

Young’s Elasticity Modulus 2.10E+11 N/m2 Shear Modulus 8.08E+10 N/m2 Eﬀective Density 8500 kg/m3

Table 5.2: Material properties of the tower for the 20MW wind turbine. .

5.2.2 Hub and Nacelle Model

The Nacelle and the Hub are modeled as a disk with its center in the center of mass of each component. Those point have the equivalent mass and inertia properties of the components. The diagram 5.2 shows the position of the hub and nacelle center of mass. The consider properties for the Hub and Nacelle are in the table 5.3:

Hub Mass 264.371 Kg Inertia about the shaft axis 2.160.000 Kgm2 Nacelle Mass 936.800 Kg Inertia about the yaw axis 82.860.699 Kgm2

Table 5.3: Hub and Nacelle properties for the 20MW wind turbine. . 98 APPLICATION: ANALYSIS OF A 20MW WIND TURBINE 5.2

7.96m 0.0528m Tower Top

157.1m

0.0986m Tower bottom 16.0m Variable Mudline

Figure 5.1: Tower for the 20MW wind turbine. The boundary condition is applied at the bottom of the tower.

5.2.3 Blades The blade is modeled using the parameterized model written in PCL. The exact composite layout of the wind turbine is unknown, then it is necessary to do a model of the wind turbine with some assumptions to came up with a finite element model of the wind turbine. The blades were model with the structure shown in the figure 5.3, using the “equivalent” materials properties given in table 5.6 is possible to compute and “equivalent” thickness of the layers showed in the table 5.5. This properties are obtained form the phd work....(). Each layer is modeled with quad plate elements using the equivalent thickness and material properties. The values between the defined radial positions on the table 5.4 are linear interpolated, this task is done using the PATRAN software.

5.2.4 Boundary conditions and connection between the blades, nacelle, hub and tower A rigid connection between the blades, nacelle, hub and tower are used. The rigid connection is the same used for the 5MW wind turbine and it is explained in subsection 4.1.4. The boundary condition is imposed on the nodes at the bottom of the tower. Those nodes have zero displacements in all the degrees of freedom. 5.2 FINITE ELEMENT MODEL FOR THE 20MW REFERENCE WIND TURBINE 99

Radial position Chord Shell thickness Web thickness Spar thick [m] [m] [m] [m] [m] 0.000 9.000 0.080 0.000 0.000 2.771 9.000 0.080 0.000 0.000 7.390 9.000 0.080 0.010 0.010 12.009 9.312 0.070 0.010 0.018 16.628 9.999 0.051 0.010 0.024 21.247 10.686 0.041 0.010 0.027 25.880 11.000 0.041 0.010 0.027 30.499 10.992 0.041 0.010 0.027 35.118 10.969 0.042 0.011 0.026 42.053 10.903 0.042 0.011 0.026 55.924 10.659 0.043 0.013 0.024 69.781 10.256 0.045 0.015 0.022 83.651 9.685 0.047 0.018 0.019 97.508 8.936 0.048 0.020 0.017 111.379 8.000 0.050 0.021 0.014 125.250 3.028 0.050 0.022 0.012 132.171 0.525 0.041 0.019 0.011 135.639 0.120 0.032 0.016 0.011 139.106 0.120 0.023 0.013 0.010 142.119 0.120 0.015 0.010 0.010

Table 5.4: Structural properties of the 20MW wind turbine blades. . 100 APPLICATION: ANALYSIS OF A 20MW WIND TURBINE 5.2

Radial position Twist Distance(1) Airfoil [m] [deg] [m] [ ] − 0.000 13.308 0.500 Cylinder1 2.771 13.308 0.500 Cylinder1 7.390 13.308 0.500 Cylinder1 12.009 13.308 0.460 Cylinder2 16.628 13.308 0.420 Cylinder2 21.247 13.308 0.390 XDU00W401 25.880 12.265 0.375 XDU00W401 30.499 10.252 0.375 XDU00W401 35.118 8.919 0.375 XDU00W350 42.053 8.335 0.375 XDU00W350 55.924 7.559 0.375 XDU97W300 69.781 7.116 0.375 DU91W2250 83.651 6.773 0.375 XDU93W210 97.508 6.300 0.375 NACA64618 111.379 5.464 0.375 NACA64618 125.250 3.542 0.375 NACA64618 132.171 2.157 0.375 NACA64618 135.639 1.412 0.375 NACA64618 139.106 0.655 0.375 NACA64618 142.119 0.000 0.375 NACA64618

Table 5.5: Aerodynamic properties of the 20MW wind turbine blades. Distance (1) is the distance from the leading edge to middle of the airfoil normalized by the chord. .

E11 E22 G12 nu12 Density Kg [GP a] [GP a] [GP a] [ ] [ 3 ] − m Shell 10.37 10.37 4.5 0.3 643.3 Web 10.37 10.37 4.5 0.3 676.7 Spar 27.1 27.1 4.5 0.3 1700

Table 5.6: Properties for the materials of the 20MW wind turbine blades. . 5.2FINITE ELEMENT MODEL FOR THE 20MW REFERENCE WIND TURBINE 101

Yaw axis Nacelle

3.8m

3.5m Shaft axis

10.0m Hub

161.9mg Tower Top

Tower bottom

Figure 5.2: Nacelle and Hub center of mass for the 20MW wind turbine.

5.2.5 Pitch, Rotational and Wind Speed Control Curves

The control of the blade pitch and the rotational speed in function of the wind speed is shown in the figure 5.5. The values shown in figure 5.5 are the curves used to define the operational points of the wind turbine when is search for unstable operational points.

5.2.6 Parametric Model in PATRAN

The model for the properties explained in section 5.2.3 for the blades is used. All the other components are modeled in the same way as the 5MW wind turbine with diﬀerent values for the distances and properties. The wind turbine is meshed with the same PCL program as the 5MW wind turbine. 102 APPLICATION: ANALYSIS OF A 20MW WIND TURBINE 5.2

Web Spar Shell

0.15 c 0.5 c · ·

Figure 5.3: Structural diagram for the sections of the 20MW wind turbine.

Figure 5.4: Picture of the ﬁnite element blade for the 20MW wind turbine, the colors represents diﬀerent properties on the PCL program.

5.2.7 Model summary The ﬁnite element model for the wind turbine model used in this thesis is done by the code written in PCL. The model summary of the model 20MW Wind turbine ﬁnite element model are summarize in the table 5.7 5.2FINITE ELEMENT MODEL FOR THE 20MW REFERENCE WIND TURBINE 103

18 Blade Pitch [◦] Rotor RPM 16

5 10 15 20 25 m Wind Speed s h i Figure 5.5: 20MW Wind turbine control curve.

Material properties (Plies laminate and directions) 4 Number of element (CQUAD4) 4650 Number of nodes 4455 Number degrees of freedom (s-set) 26136

Table 5.7: Summary of the 20MW Finite Element Model . 104 APPLICATION: ANALYSIS OF A 20MW WIND TURBINE 5.3

5.3 Unstable Operational Scenarios for the 20MW Wind Tur- bine 5.3.1 Static Unstable Points The unstable operational points for divergence of the 20MW wind turbine are shown in ﬁgure 5.6.

Predicted unstable operational points for the 20MW wind turbine

14 Potential flow Viscous flow 12

Rotational speed [RPM] 6

5 10 15 20 25 Wind speed [m/s]

Figure 5.6: Static Instabilities for the 20MW wind turbine (divergence).

The plot two types of solution, the “potential flow” solution and the “viscous solution”. The potential flow solution is the solution obtained with the implementation of the methodology explained in this thesis using 2π value for the lift coefficient without including drag. The “viscous solution” is the solution obtained when it is included the real aerodynamic characteristics of the airfoils taken into account the lift coefficients, the drag and the moment coefficient for the different airfoils along the blade.

The predicted values of the unstable rotor speed are higher than the rotational speed of the wind turbine, this means instabilities on the wind turbine are 5.3UNSTABLE OPERATIONAL SCENARIOS FOR THE 20MW WIND TURBINE 105 not expected.

The maximum rotational speed predicted for the “viscous” solution is higher than the values predicted using the “potential ﬂow” solution, this is because the drag take out energy of the wind turbine.

The unstable rotor speed is lower in comparison with the 5MW wind turbine. A lower rotor speed is expected because the natural frequencies of the wind turbine decreases linearly with respect to the linear upscale factor.

5.3.2 Dynamic Unstable Points The unstable operational points for ﬂutter of the 20MW wind turbine are shown in ﬁgure 5.7.

Predicted unstable operational points for the 20MW wind turbine 30

Simple model Potential flow 20 Viscous flow

10 Rotational speed [RPM]

0 5 10 15 20 25 Wind speed [m/s]

Figure 5.7: Dynamic Instabilities for the 20MW wind turbine (ﬂutter).

The plot showed two types of solution, the solution for the simple model, the 106 APPLICATION: ANALYSIS OF A 20MW WIND TURBINE 5.3

“potential ﬂow” solution and the “viscous” solution. The simple model overestimate the unstable angular velocity of the rotor.

The worst scenario for the wind turbine is for lower wind speed, this conclusion is the same for the 5MW wind turbine, this agrees with the lower aerodynamic damping predicted at lower wind speed in [55]. The “viscous” solution predicts higher rotational speed than the “potential” solution . Flutter instabilities are not expected on the wind turbine at the rated rotor speed. Chapter 6

Conclusions and Recommendations

6.1 Conclusions

Parametric Model. A complete detailed ﬁnite element model for the structure of the wind turbine is done. The parametric model works correctly in PATRAN and it reduced considerably the time of drawing and meshing the wind turbine. It meshes a three blade wind turbine with only quad element and it has the capability to include orthotropic composites materials in diﬀerent directions. The time using the parametric model is reduced from one month of meshing task to a couple of days (for the 5MW wind turbine).

Methodology. A method to couple the structural finite element model of a wind turbine to an aerodynamic model to find instabilities using the eigenvalues approach is deduced. One of the advantages of this method in comparison with a time domain method is this method does not required post-processing tools to find the unstable points. The aerodynamic model is represented by complex aerodynamic matrices equivalent to a “mass”, “damping” and “stiffness” properties. The aerodynamic matrices are band matrices with almost the same band profile as the structural matrices. This means the complexity to solve this problem should not increment when the aerodynamic model is coupled to the structural model. On the other hand there is an issue, the aerodynamic matrices are complex and the structural matrices are real, this increases the difficulty to solve the problem and the time spent by the computer to find the eigenvalues increases considerably as well. The aerodynamic matrices are function of local and global properties, this is a difference in comparison with the structural finite element method where the structural matrices only depends on local properties. The eigenmodes obtained from the coupled system of structural and aerodynamic

107 108 CONCLUSIONS AND RECOMMENDATIONS 6.1 matrices are complex. This means the shape of the modes are not constant in time.

Implementation. The method is implemented in DMAP/MATLAB works stable. Although the part of the program in DMAP was difficult to implement due to the low language level it is runs fast. The methodology takes into account the changes in the stiffness and damping of the wind turbine blades due to the rotational speed based on the mass, damping and stiffness matrices of the finite element model of the wind turbine. This is implemented using only the structural model of the wind turbine doing the algorithm fast. The equivalent “mass”, “damping” and “stiffness” matrices for the aerodynamic model are complex, non-symmetric, sparse and band. The computer do not require large amount of memory to save them. The complex character of the matrices when the aerodynamic and structural model are coupled increases the difficulty to solve the eigenvalue problem. Although the matrices were sparse and band the computer spent long time to solve the system of eigenvalues using the Arnoldi method with spectral transformation. This part of the algorithm is the part in which the computer spent more time to process.

Results. The obtained flutter solutions are accurate. The predicted unstable rotational speed have fluctuations because the algorithm to find the unstable points. The algorithm do not test for all the possibilities of reduced frequencies, in this thesis is used eight values between zero and one for the reduced frequency. The aerodynamic damping predicted by the model based on the Theodorsen solution for a flat plate is highly dependant on the reduced frequency. Therefore when there is a small change in the reduced frequency the Theodorsen function changes considerably its complex value. These changes have a big effect on the damping which is performed by the aerodynamic model of the complete model of the wind turbine (aerodynamic and structural model) and when not all the possibilities of values for the reduced frequencies are tested, the small fluctuations for the predicted unstable points appears. Other authors in the academic world have also observed this high dependency of the flutter speed which is in its turn depended on the reduced frequency. Work performed by the Sandia Laboratories [24] has showed this high dependence when they computed the flutter speed of a 1.5 MW wind turbine. The results of this flutter speed at 10 m/s decreased almost 45% when on the solution of the Theo- dorsen function the imaginary part is neglected.

The predicted unstable rotational speed for the 5MW and 20MW is higher when the drag model is implemented in the aerodynamic model. This behavior match with the reality in the sense the drag take out energy form the wind turbine and it becomes unstable at higher rotational speed.

The unstable angular velocity for the 5MW wind turbine predicted in the pa- 6.2 RECOMMENDATIONS 109 per [32] is 19.1 RPM. This rotational speed its bounded by the values of 19 RPM and 21 RPM predicted by the “potential” and “viscous” model respectively for the unstable operational points at the same wind speed. This is the only unstable operational point found in the literature and it match close to the values predicted by the method deduced in this thesis.

The predicted unstable rotational speed for the 20MW wind turbine is lower than the rotational speed predicted for the 5MW wind turbine. This match with the expected behavior where the frequency is reduced linearly respect to the upscale value in the linear upscaling. The 5MW and the 20MW wind turbine have unstable rotational speed higher than the operational rotational speed, thus aeroelastic instabilities are not expected for these wind turbines.

Although the accuracy of the static solutions is unknown they are in the right order of magnitude. In general the rotational speed predicted for the divergence is slightly higher than the speed predicted for ﬂutter. This result depends on the position of the center of mass of the blade respect to the elastic axis.

6.2 Recommendations

Improve the estimation of the reduce frequency. The same methodology showed in this thesis could be used with a more accurate aerodynamic model to predict instabilities. The value of the reduced frequency highly affects the results and this turns the solution slightly oscillating in a narrow band. The value of the reduced frequency could be determined using other approaches different than test a possible values for it. An algorithm which has the capability to catch the right eigenvalue of a desired mode could be implemented. This algorithm is not easy to develop due to the complex eigenmodes of the solution for a complex system of ordinary differential equations. The complex eigenmodes does not have the physical interpretation of a constant shape like the real eigenmodes and that do the problem more difficult. Interpretations for the complex eigenvalues of the mode shapes could be developed to have more control in how to interpret the unstable solution.

Including axial and rotational induction factors. The aerodynamic model used in this thesis neglects the rotational and axial induced factors of the wind turbine. Those factors have an inﬂuence on the angle of attack and in the wind speed acting on the blade sections. Those factors change the reduced frequency as well.

Numerical algorithm to find eigenvalues of a band complex system. A better numerical algorithm could be developed to find the eigenvalues for the particular complex system of ordinary differential equations generated by the pro- 110 CONCLUSIONS AND RECOMMENDATIONS 6.2 posed method. The Arnoldi algorithm with spectral transformation takes long time to find the eigenvalues and when one applies this method to complex matrices and the accuracy can be low.

Solve directly for the unstable points. Another different method to find the unstable speed can be implemented. The method used in this thesis is not a “true” solver for the instabilities, that means it is only for the eigenvalues. In fact different values for the reduced frequencies are tested. A new idea could be developed in the sense to create an algorithm to iterate and find the unstable operational scenarios without testing different scenarios. This idea should increase the accuracy of the method and improve the computation time.

Extend the method for systems not marginally stable. The aerodynamic model assumes the system is marginally stable. This means the aerodynamic model considers only the oscillatory part and neglects the damping of the modes. A new aerodynamic model could be developed considering the damping terms, thus the real part of the solution for the eigenvalues can show better the stability of the wind turbine.

Possibility of model reduction. The time spent by the solver to find the unstable point could be smaller if a reduction technique is applied to the coupled structural and aerodynamic model. Although this idea is difficult to perform because the matrices are complex and many of the reduction techniques for the finite element models are based in physical behaviors (not complex), this can decrease the time spent by the solver and it can be the base to develop fast algorithms to find stability boundaries for a wind turbine based on the eigenvalues of the system.

Changes in the wake velocity. The Theodorsen solution used in this thesis assumed the wake shedding by the airfoils of the blade has the same velocity of the free stream velocity [7] . In the reality this is not true and the difference on the wake velocity could be very important especially on the tips of the blade where the freestream velocity and the wake velocity have big differences. This effect can be included according to the paper [8]. This paper explains how to add a term to the Theodorsen solution and modify this velocity. Although the implementation of this idea is simple and straightforward, it is difficult to determine the real wake velocity. This is the reason why this modification is usually not used until the velocity of the wake is known. Appendix A

Appendix I: Blade layout for the 5MW Wind Turbine

This appendix showed the layout properties for the 5MW wind turbine used in this thesis. All the information is from the upwind report “WMC5MW laminate lay-out of reference blade for WP” [53] which deﬁnes the layout for the wind turbine as a reference for further studies.

UTS UCS E11 E22 G12 nu12 density (mean) (mean) [MPa] [MPa] [MPa] [ ] [kg/m3] [MPa] [MPa] − OD OB 38887 9000 3600 0.249 1869 810 507 UD45R 24800 11500 4861 0.416 1826 436 349 R4545 11700 11700 9770 0.501 1782 180 144 SKINFOAM 256 256 22 0.3 200 WEBPS 25 25 12 0.3 45

Table A.1: Material properties.

111 112 APPENDIX I: BLADE LAYOUT FOR THE 5MW WIND TURBINE A.0

Layer ID Material Label Radius Layers Layer Total name number thickness thickness [m] [ ] [mm] [mm] − TRIAX-1 UD45R 2 3 0.94 2.82 TRIAX-1 UD45R 63.5 3 0.94 2.82 TRIAX-2 UD45R A 2 106 0.94 100 TRIAX-2 UD45R B 2.3 106 0.94 100 TRIAX-2 UD45R C 2.8 28 0.94 26 TRIAX-2 UD45R D 3.6 28 0.94 26 TRIAX-2 UD45R E 11 1 0.94 0.94 UD UD OB F 3.5 1 0.47 0.47 UD UD OB G 9 105 0.47 49.26 UD UD OB H 14 205 0.47 96.33 UD UD OB I 19 200 0.47 94.05 UD UD OB J 24 186 0.47 87.37 UD UD OB K 29 172 0.47 80.94 UD UD OB L 34 153 0.47 71.77 UD UD OB M 39 124 0.47 58.37 UD UD OB N 44 99 0.47 46.45 UD UD OB O 49 55 0.47 25.81 UD UD OB P 54 21 0.47 9.99 UD UD OB Q 60 1 0.47 0.47 UD T E UD OB R 3.5 1 0.47 0.47 UD T E UD OB S 7.5 32 0.47 15 UD T E UD OB T 11 45 0.47 21 UD T E UD OB U 14 29 0.47 13.6 UD T E UD OB V 47 1 0.47 0.47 SKINFOAM SKINFOAM W 3.5 1 SKINFOAM SKINFOAM X 7.5 40 SKINFOAM SKINFOAM Y 43 40 SKINFOAM SKINFOAM Z 57 6 SKINFOAM SKINFOAM AA 60 6 TRIAX-3 UD45R 2 3 0.94 2.82 TRIAX-3 UD45R 63.5 3 0.94 2.82

Table A.2: Layout description. A.0 113

TRIAX−1 A−D E

TRIAX−2

F G HIJKLM N O P Q

UD RTUS V

TE_UD

W XYZ AA

SKINFO AM

TRIAX−3

. 114 APPENDIX I: BLADE LAYOUT FOR THE 5MW WIND TURBINE A.0

trailing edge 3 * UD45R n * UD trailing edge UD 3 * UD45R 3 * UD45R trailing edge sandw ich panel SKINFO AM 3 * UD45R

chord

sandw ich shear webs

2 * R4545 50 mm WEBPS 2 * R4545 3 * UD45R UD spar cap n * UD 3 * UD45R up wind side dow n wind side

3 * UD45R leading edge sandw ich panel SKINFO AM 3 * UD45R

leading edge

Radius= 2001 m m Radius= 3501 m m Radius= 9001 m m

Radius= 14001 m m Radius= 19001 m m Radius= 24001 m m Radius= 29001 m m

R adius= 34001 m m R adius= 39001 m m Radius= 39001 m m Radius= 44001 m m

R adius= 49001 m m R adius= 54001 m m R adius= 60001 m m Bibliography

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