Structural Dynamic Modeling of Wind Turbine Blades

Faculty of Postgraduate Studies and Scientific Research German University in Cairo

Structural Dynamic Modeling of Wind Turbine Blades

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechatronics Engineering By Mary Victor Bastawrous Supervised by Prof. Ayman A. El-Badawy

July, 2012

Faculty of Postgraduate Studies and Scientific Research German University in Cairo

Structural Dynamic Modeling of Wind Turbine Blades

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechatronics Engineering By Mary Victor Bastawrous Supervised by Prof. Ayman A. El-Badawy

July, 2012

Approval Sheet This thesis has been approved in partial fulfillment for the degree of:

Master of Science in Mechatronics Engineering by the Faculty of Postgraduate Studies and Scientific Research at the German University in Cairo (G.U.C) on ................................ (July 2012)

Declaration This is to certify that:

(i) the thesis comprises only my original work toward the Master Degree (ii) due acknowledgement has been made in the text to all other material used

Mary Victor Bastawrous

Acknowledgements The academic and scientific style of this work may mislead some to miss what is within its lines. The fact is that this work is the fruit of the effort, time and support of many people, without whom it would not have made it to existence. It will not be really complete before acknowledging their role and expressing my feelings of gratitude towards them. I would like to thank my supervisor Dr. Ayman for his mentoring, guidance and friendship. I can not give his generous character nor his guidance their due right in these lines. I specially appreciate his patience in letting me realize myself, follow my intuition and learn things the hard way, without thinking its a waste of time. I would also like to acknowledge the financial support from the Science and Technology Development Fund (STDF), Cairo, Egypt under grant number 1495. I thank each and everyone of my colleagues for their help and support. There has not been a single advice not too significant nor a single word of support that missed the point or the right moment. I feel deeply grateful to all my friends in the GUC or outside its walls. I am indebted to your friendship, support and encouragement, literally. The hardest part comes when I have to thank my family: my parents and my sisters, and my close and loved ones. How can one thank his rock, shelter and strength in this life?

Mary Victor Bastawrous

Abstract A study is developed to investigate the effect of geometry, material stiffness and the rotational motion on the coupled flapwise bending and torsional vibration modes of a wind turbine blade. The assumed modes method is used to discretize the derived kinetic and potential energy terms. Lagrange’s equations are used to derive the modal equations from the discretized terms, which are solved for the vibration frequencies. The parametric study utilizes dimensional analysis techniques to study the collective influence of the investigated parameters by combining them into few non-dimensional parameters, thus providing deeper insight to the physics of the dynamic response. Results would be useful in providing rules and guidelines to be used in blade design.

Contents

List of Figures

XI

List of Tables

XIII

1 Introduction and Literature Review

1

1.1

Wind Energy: History and Economic Importance . . . . . . . . . . .

1

1.2

Aerodynamics of Wind Turbine Blades . . . . . . . . . . . . . . . . .

2

1.3

Wind Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Blade Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.5

Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.6

Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Uniform Blade Structural Dynamic Model

12

2.1

Kinematic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

Fixed Uniform Blade Model . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1

Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . 16

VII

2.3

2.4

2.2.2

Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.3

Hamilton’s principal . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.4

Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 20

Spatial Reduction of the Governing Equations . . . . . . . . . . . . . 24 2.3.1

Exact solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2

Assumed Modes Method: Fixed Uniform Blade Case . . . . . 31

Fixed Uniform Blade Case Study . . . . . . . . . . . . . . . . . . . . 34 2.4.1

Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.2

Approximate Solution by Assumed Modes . . . . . . . . . . . 37 2.4.2.1

Trial functions selection . . . . . . . . . . . . . . . . 38

2.4.2.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5

A Study on the Effect of the Blade Properties on Coupled BendingTorsion Vibrations of Cantilever Uniform Blades . . . . . . . . . . . . 39

2.6

Extending the Governing Equations to Rotating Blades . . . . . . . . 44

2.7

2.6.1


2.6.2

Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.6.3

Governing Partial Differential Equations by Hamilton’s principal 46

Spatial Reduction of the Governing Equations . . . . . . . . . . . . . 46 2.7.1

Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.7.2

Assumed Modes Method . . . . . . . . . . . . . . . . . . . . . 47

2.7.3

Uniform Rotating Blade Case Assumed Modes Discretization . 49 VIII

2.8

Rotating Blade Case Study 2.8.1

2.9

. . . . . . . . . . . . . . . . . . . . . . . 51

Assumed Modes Approximate Solution . . . . . . . . . . . . . 51 2.8.1.1

Trial functions selection . . . . . . . . . . . . . . . . 52

2.8.1.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . 52

A Study on the Effect of the Blade Properties on Coupled BendingTorsion Vibrations of Rotating Uniform Blades . . . . . . . . . . . . 53

3 Non-Uniform Blade Structural Dynamic Model 3.1

3.2

Non-Uniform Rotating Blade Model . . . . . . . . . . . . . . . . . . . 56 3.1.1


3.1.2

Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.3

Governing Partial Differential Equations by Hamilton’s principal 58

Assumed Modes Method . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2.1

3.3

3.4

56

Rotating Non-Uniform blade Case Assumed Modes Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Controls Advanced Research Turbine (CART) Blade Case Study . . . 61 3.3.1

Trial functions selection . . . . . . . . . . . . . . . . . . . . . 61

3.3.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A Study on the Effect of the Blade Properties on Coupled BendingTorsion Vibrations of Rotating Non-Uniform Blades . . . . . . . . . . 65

4 Conclusion and Recommendations

IX

69

Bibliography

72

Appendices

76

A Dimensional Analysis

76

B Partial Differential Equations with Variations

80

C Partial Differential Equations Derivation Code

86

D Coupled Vibrations Exact Analytical Solution Code

94

E Assumed Modes Discretization Code

108

F Controls Advanced Research Turbine (CART) Blade Data

117

X

List of Figures

1.1

A horizontal axis wind turbine blade (HAWT). From:ec.europa.eu . .

3

1.2

Geometrical properties of airfoils. From: Fundamentals of Aerodynamics by John D. Anderson, McGraw-Hill, 1984. . . . . . . . . . . .

3

Lift and drag forces acting on an airfoil cross-section. From: Aerodynamics of Wind Turbines by Martin O. L. Hansen, second edition, 2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Geometrical properties of airfoils. . . . . . . . . . . . . . . . . . . . .

8

2.1

An airfoil cross-section blade. . . . . . . . . . . . . . . . . . . . . . . 13

2.2

The blade cross-section twisted by the torsional angle β about the y2 axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3

Coupled Bending-Torsional modes of a uniform cantilever blade. . . . 37

2.4

The parameter λb =

2.5

ω The parameter λb = µL for rotating uniform blades with different EIx2 GJy2 L 2 values of χ = EIx2 ( r ) and f = Ix2m Ω2 . . . . . . . . . . . . . . . . . 55

3.1

λb -χ plot for both uniform and non-uniform rotating blades, where 4 ω2 y2 L 2 and χ = GJ ( ) and f = Ix2m Ω2 . . . . . . . . . . . . . 67 λb = µL EIx2 EIx2 r

1.3

µ ω 2 L4 E Ix2 4

for blades with different χ =

λb λt

=

G Jy2 E Ix2

L 2 . r

41

2

XI

3.2

λ-χ plot for a non-uniform rotating blade showing the first two cou4 ω2 y2 L 2 and χ = GJ ( ) . . . . . . . . . . . . 67 pled modes, where λb = µL EIx2 EIx2 r

3.3

λ-χ plot for a non-uniform rotating blade showing the first coupled 4 ω2 y2 L 2 mode, where λb = µL and χ = GJ ( ) . . . . . . . . . . . . . . . 68 EIx2 EIx2 r

3.4

λ-χ plot for a non-uniform rotating blade showing the second coupled 4 ω2 y2 L 2 mode, where λb = µL and χ = GJ ( ) . . . . . . . . . . . . . . . 68 EIx2 EIx2 r

F.1 Distribution of bending stiffness EIx2 along the blade length and the exponential fit EIx2 (y) = 1.719E + 8e−0.2214 y . . . . . . . . . . . . . . 118 F.2 The torsional stiffness distribution along the blade length and the exponential fit utilized GJy2 (y) = 4.423E + 7e−0.1995 y . . . . . . . . . . 119 F.3 Distribution of mass per unit length µ along the blade length and the exponential fit µ(y) = 232.6e0.06363 y − 0.6628y 2 − 27.95y. . . . . . . . 119 F.4 Distribution of mass moment of inertia Ix2m along the blade length and the exponential fit Ix2m (y) = 12.88e−0.2294 y . . . . . . . . . . . . . 120 F.5 Distribution of shear center offset rx along the blade length and the fit utilized is rx = 0.08718 + 0.005621 sin(0.5485y) + 0.02088 sin(0.2574y) .120 F.6 Distribution of the polar radius of gyration of the blade cross-section about the centroidal axis along the blade length and the fit utilized r(y) = 0.3616 + 0.07763 sin(0.2905 y) . . . . . . . . . . . . . . . . . . 121

XII

List of Tables

2.1

Computed exact natural frequencies using exact solution approach . . 36

2.2

Computed approximate natural frequencies using assumed modes approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3

Computed approximate natural frequencies for a rotating blade using assumed modes approach . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1

Computed approximate natural frequencies for a rotating blade using assumed modes approach . . . . . . . . . . . . . . . . . . . . . . . . . 65

F.1 Distributed CART blade characteristics. . . . . . . . . . . . . . . . . 117

XIII

List of Abreviations p pressure. rc radius of curvature. Vr flow speed relative to a point on the airfoil. ρair air density. r radius of curvature. l lift force per unit length. d drag force per unit length. CL lift coefficient. CD drag coefficient. c chord length. P power output. CP power coefficient. A rotor area. U wind speed. [xyz]0 Inertial frame fixed at the shear center of the blade root. O0 Origin of [xyz]0 . Ω Rotation speed. [xyz]1 Rotating frame at the shear center of the blade root. O1 Origin of [xyz]1 . P A point on the blade. 1 h Position of point P relative [xyz]1 . xp yp zp Position coordinates of point P relative [xyz]1 . [xyz]2 Frame fixed at the shear center at a distance y away from the blade root. O2 Origin of [xyz]2 . w(y, t) Bending deflection of point P . [xyz]3 Frame rotated by angle β about y2 . XIV

O3 Origin of [xyz]3 . β(y, t) torsional angle about the y2 axis. L Blade length. 1 T2 Transformation matrix between frames 1 and 2. 2 T3 Transformation matrix between frames 2 and 3. 1 A2 Link Transformation matrix between frames 1 and 1. 1 E2 Elastic Transformation matrix between frames 1 and 1. (0 ) differential with respect to the blade length. (˙) differential with respect to time. V0 elastic potential energy per unit volume. σ elastic stress of the blade. elastic strain of the blade. E Young’s elastic modulus. G Shear rigidity modulus. u displacement of point P due to blade deflection. rx shear center offset away from the blade cross-section centroid. [˜ xy˜z˜]2 Frame fixed at the centroid of a cross section at a distance y away from the blade root. Ix2 area moment of inertia of the blade cross section about the x2 axis. Jy2 area polar moment of inertia of the blade cross section about the y2 axis. J¯y2 area polar moment of inertia of the blade cross section about the y˜2 axis. I¯x2 area moment of inertia of the blade cross section about the x˜2 axis. T blade kinetic energy. ρ blade density per unit volume. h˙ velocity of a point P on the blade. Ix2m mass moment of inertia of the blade cross section about the x2 axis. Jy2m mass polar moment of inertia of the blade cross section about the y2 axis. J¯y2m mass polar moment of inertia of the blade cross section about the y˜2 axis. I¯x2m mass moment of inertia of the blade cross section about the x˜2 axis. µ mass per unit length. W (y) mode shape of the bending deflection. q(t) time function of the coupled vibrations. B(y) mode shape of the torsional deflection. ω coupled vibrations frequency. s coefficient of trigonometric functions in the spatial mode shape. wˆ amplitude of the bending mode shape. XV

ˆb amplitude of the torsional mode shape. r mass radius of gyration of the blade cross section about the y2 axis. 2 4 λb A dimensionless parameter equal to λb = µEωIx2L . Jy2 L 2 χ A dimensionless parameter equal to χ = λλbt = G ( ). E Ix2 r qi (t) discretized generalized coordinate. φwi (t) discretized spatial mode shape for bending deflections. φβi (t) discretized spatial mode shape for torsional deflections. M Inertia matrix in the assumed modes discretization. K Stiffness matrix in the assumed modes discretization. mij Inertia matrix element. kij Stiffness matrix element. g gravitational acceleration vector. T0 terms in the kinetic energy of the rotating blade that are coefficients of the square of the rotational speed. T1 terms in the kinetic energy of the rotating blade that are coefficients of the rotational speed. T2 terms in the kinetic energy of the rotating blade that are no coefficients of the rotational speed. Jz1m blade mass moment of inertia about the rotation axis. G Gyroscopic matrix in the assumed modes discretization. gij elements of the gyroscopic terms. f a factor that is the product of the mass moment of inertia of the blade cross-section about the airfoil axis and the square of the rotational speed used to study the effect of the rotational motion.

XVI

Chapter 1 Introduction and Literature Review 1.1

Wind Energy: History and Economic Importance

In a world ruled by economy and limited resources, wind has inspired people to make use of it in daily life for several centuries. Wind mills have been used to harvest wind energy and convert it into mechanical energy that was used to grind grains or to irrigate lands. The rotational motion of the wind mills was used to power pumps used in irrigation and to move water to higher grounds. Their designs have undergone several variations according to the nature of the installment place and application. With the advent of electricity, engineers and scientists started to look for ways to use wind for electric power generation. Thanks to electric generators, wind mills became wind turbines. The first attempts at using wind energy to generate electricity can be traced back to the 12-KW DC windmill generator constructed by Brush in the USA and LaCour’s research work in Denmark [1]. Though several wind turbines have been built in the period starting the 1930s till the 1960s in the USA and Europe, wind power generation remained costly and inefficient, resulting in the take-over of fossil fuels as a power source. The turning point was the oil crisis in 1973 when the oil prices increased dramatically, which drew attention to the strategic importance of energy and the importance of its availability and control. That crisis, accompanied by a growing awareness of the 1

CHAPTER 1. INTRODUCTION AND LITERATURE REVIEW

2

limitedness of fossil fuels, directed the political interest at the time to fund and invest in renewable energy resources; resources that cannot be exhausted. Naturally, wind energy was a very strong candidate due to its availability and the fact that, though yet to be developed, wind power technology actually existed. Today, an additional main motive behind promoting and investing in wind power is to limit and slow down the climate change as wind energy is a clean energy source and wind turbines production cycle involves low CO2 emission [1]. According to the World Wind Energy Association (WWEA), worldwide capacity of installed wind energy reached 196630 MW, out of which about 19% are installed in 2010. Also, by the end of 2010, the installed wind turbines worldwide was to be sufficient to provide for 2.5% of the global electricity demand [2]. Also in Egypt, renewable energy got its share of attention as soon as the political environment was ready. A national energy plan had been developed in the early 1980s. It aimed at increasing the renewable energy share of the Egyptian energy market up to 20% by 2020, of which 12% were planned to be produced by wind power[3]. Thus, the New and Renewable Energy Authority (NREA) was established in 1986 to promote renewable energy technology in Egypt. Investments have also been made to manufacture some wind turbine components in Egypt in 2010 [4]. Egypt was ranked number 24 in the world for the year 2010 according to the total wind power capacity which reached 5500 MW, of which 22% had been installed in 2010 [2].

1.2

Aerodynamics of Wind Turbine Blades

Now that the importance of wind energy is highlighted, some of the basic principals and concepts of wind turbines are presented. In this work, we are mainly interested in Horizontal Axis Wind Turbines (HAWTs). In the most common designs, a Horizontal Axis Wind Turbine (HAWT) consists of a horizontal axis rotor and a nacelle which are supported by a tower Fig. 1.1. As the name implies, the rotor part consists of rotating components, namely the blades and the hub. HAWTs normally have two or three blades. Blades are long and slender structures that start at the hub and extend radially. Blade properties like the cross-section area and the twist angle vary along the blade length. In this section, some of the basic aerodynamic principals of blades are quickly reviewed. The terms ”spanwise” and ”streamwise”


3

will be used to denote the radial direction along the blade and the perpendicular to the plane of rotation, respectively.

Figure 1.1: A horizontal axis wind turbine blade (HAWT). From:ec.europa.eu Wind turbine blades have an airfoil cross section. An airfoil is characterized by certain properties as shown in Fig. 1.2; the mean camber line is the locus lying midway between the upper and lower surfaces; the foremost and the last points on the mean camber line are the leading and trailing edges, respectively; the chord line is the straight line joining the leading and trailing edges, and it’s length is called the chord c of the airfoil; the camber of the airfoil is the maximum distance between the mean camber line and the chord line. If an airfoil is symmetric about the chord line, then the upper and lower surfaces are mirrored about the chord line and the airfoil has zero camber [5].

Figure 1.2: Geometrical properties of airfoils. From: Fundamentals of Aerodynamics by John D. Anderson, McGraw-Hill, 1984. When the wind flows past a wind turbine, the span-wise velocity component along the blade is much less than the stream-wise component. Therefore, the flow


4

can be simplified to a two dimensional flow where the wind flows past the blade cross section as shown in Fig. 1.3. Since the blade airfoil cross-section forces the flow streamlines passing through them to deviate and flow around the airfoil shape, the flow in turn exerts forces on the blade. The nature of these forces depends on the model employed to describe the aerodynamic flow. In a viscid flow, for example, in which viscosity effects are considered, the forces exerted on the airfoil originate from two sources: normal pressure distribution over the airfoil and tangential shear stresses due to viscous effects.

Figure 1.3: Lift and drag forces acting on an airfoil cross-section. From: Aerodynamics of Wind Turbines by Martin O. L. Hansen, second edition, 2008. In an inviscid flow, only the forces due to the normal pressure distribution on the airfoil surface are included because the fluid viscosity is neglected. The normal pressure distribution on the airfoil surface is because of the curvature induced in the streamlines by the airfoil shape that necessitates the presence of a pressure gradient to act like a centripetal force [5]. V2 ∂p = ρair r ∂rc rc

(1.1)

where p is the air pressure, Vr is the flow speed relative to a point on the airfoil, ρ is the air density, and rc is the radius of curvature. Thus, the pressure at the airfoil surface has to be lower than the atmospheric pressure in order to have a pressure gradient that forces the streamlines to curve around the airfoil. As seen in Eq. 1.1, the pressure gradient is directly proportional to the curvature 1r , i.e., the pressure at the airfoil surface is lower at higher curvatures in order to attain higher pressure


5

differentials. So, if the upper and lower surfaces are symmetric, then the pressure distribution will be the same and net forces will cancel each other. However, if the airfoil is cambered, it will have a higher curvature at the upper surface resulting in an upward directed net force called the lift force. On the other hand, the net force resulting from the integration of the shear stresses over the airfoil surface is called the drag force. Lift and drag forces are directed along the wind speed and perpendicular to it, respectively. They can be calculated by 1 l = CL ρVr2 c 2 1 d = CD ρVr2 c 2

(1.2) (1.3)

where l and d are the lift and drag forces per unit length, respectively, c is the chord length, and CL and CD are the lift and drag coefficients, respectively. CL and CD are functions of, among other variables, the angle between the wind direction and the airfoil chord, which is called the angle of attack α. The coefficient CL increases linearly as α increases till a certain critical limit, called the stall angle, after which CL decreases drastically. CD , on the other hand, remains almost constant till the stall angle, after which it increases. The net forces acting on the blade can be found by integrating the lift and drag forces along the blade length. They can be resolved to in-plane and out-of-plane forces [6]. Owing to the aerodynamic forces, wind turbine blades can rotate to generate electricity. Naturally, these forces cause the blades to deflect in bending and torsion.

1.3

Wind Power

The power output of a wind turbine is calculated by [6] P =

1 Cp ρair A U 3 2

(1.4)

where P is the power output, Cp is the power coefficient which describes the fraction of wind power that can be converted to mechanical power by a certain wind turbine, ρ is the air density, A is the rotor swept area and U is the wind speed. The air density ρ is about 1.225 kg/m3 . The coefficient Cp has a maximum theoretical value of 0.63 which is known as the Betz limit, but real life turbines have smaller power coefficients. From Eq. 1.4, it is evident that wind speed and rotor swept area are


6

the major factors that can effectively influence a wind turbine power output. Wind speed can be increased by placing wind turbines in places where wind speeds are higher or by increasing the tower height to make use of the increased wind speed at high altitudes due to reduced friction with the terrain. The rotor swept area can be increased by increasing the blade length. Manufacturers have been trying to optimize rotor size and height depending on the power output, manufacturing cost and operation. However, blade sizes have been continuously increasing with the advantage of lighter blade materials. The increasing demand for optimizing wind turbine power production, wind turbine weight and costs including material and manufacturing costs has lead the blade design process to be a crucial process in the wind turbine industry. An important aspect in designing blades is to account for blade vibration characteristics, which can tell about how they will respond to various excitiations and the possible causes of failure. Blade modeling and simulation tools help aid the design process in the primary stages as they can predict the blade vibration characteristics and propose possible optimization solutions. It is evident that a successful modeling stage contributes to minimizing the blade cost, and thus the wind turbine cost.

1.4

Blade Structural Model

It was previously mentioned how aerodynamic loads originate from the pressure difference between the upper and lower surfaces of an airfoil, resulting in a lifting force that causes the wind turbine blade to rotate. Since these forces are aerodynamic in nature, they have the periodic nature of wind [7]. It is necessary to study the vibration characteristics of the blade and how it responds to such excitations. Therefore, a blade model is needed to describe how it interacts with the surroundings. Apart from foreseeing the blade vibrational behavior, it can also be used to explain it based on the blade properties, thus optimize the blade characteristics so that the desired performance is attained. In the following sections, some principal structural concepts that will be used throughout this thesis are presented, as well as a justification for the choice of some of the utilized methods and approaches in this work. In the beginning, the employed blade model is introduced. Then, coupled vibrations and why they happen in the case of a wind turbine blade is discussed. Finally, some parametric studies in the


7

literature that attempted to understand and predict how blades vibrate and interact with external loadings are reviewed. Generally, a wind turbine blade may stretch axially in the direction of the elastic axis, bend in the flap-wise direction, the edge-wise direction and be twisted about the elastic axis. Wind turbine blades were usually modeled as beams because they can be seen as slender structures with one dimension much larger than the other two [6, 8, 9]. The Euler-Bernoulli beam model accounts for the axial strain due to bending as well as the shear strain in the plane perpendicular to the elastic axis. That plane is assumed to stay perpendicular to the elastic axis after deformation. As rotor geometry became more complicated, models grew more complicated as attention has been directed to describe the shear stresses in situations where it cannot be neglected as in composite materials, beams with low slenderness ratio, or high torsional deformation with the consequent shear stresses and warping. However, the beam model is still used in case of preliminary analysis even in case of low slenderness ratio rotors [10]. Also, beam models are preferred when a simple blade model is desired to describe the underlying physical principals of the blade deformation. An elastic beam is a continuous structure that has infinitely many degrees of freedom. Since the distance between two different particles is infinitesimal and the displacement field must be continuous, the elastic deformation of an Euler-Bernoulli beam can be sufficiently described by a finite number of displacement variables such as the deflection of a reference point in a given cross-section in bending and the torsional angle of rotation about this point, given that these variables are functions of the spatial coordinates as well as of time. Thus, Partial Differential Equations (PDEs) are used to model the continuous beam behavior [8]. Also, boundary and initial conditions are used to solve the equations. The PDEs can be solved in an exact sense, if possible, or using approximate numerical techniques. With the development of computational technology, approximate solving techniques like the finite elements analysis and the assumed modes method have become the usual approach to solve vibrations problems, especially in complicated problems for which closed form solutions are not established. Vibration mode shapes and natural frequencies have been calculated in classic vibrations books for bi-axially symmetric beams, in which the elastic axis coincide with the centroidal axis [11]. However, real life cantilever beam problems can be more complicated as the cross-sections are not bi-axially symmetric. This results in an offset of the beam’s elastic axis from its centroidal axis. The significance of


8

such offset becomes more evident as we review the definition of the elastic axis of a beam. The flexural center of a beam cross section can be defined as a point on the cross-section at which a shear force can be applied without causing the rotation of the cross-section in its own plane. The center of twist, on the other hand, is a point that remains stationary when a twisting torque is applied. If the twisting center and the flexural center coincide, their loci along the beam length form the elastic axis [7]. The response of a beam in which the elastic and centroidal axes do not coincide is always coupled. By the term coupled, we mean that the torsional and flexural oscillations are dependent on each other and take place simultaneously at the same natural frequency. The geometrical shape of the airfoil cross-section of a wind turbine blade is such that there is an offset between the elastic and centroidal axes of the blade, which leads to coupling multiple degrees of freedom in the blade vibration modes.

Shear Center Chord Trailing Edge

Leading Edge Centroid

Figure 1.4: Geometrical properties of airfoils.

Dokumaci and Rao have explored linear models of beams with small deformations [12, 13] while Da Silva and Hodges modeled beams with moderate deflections [14, 15]. Dokumaci explored the exact solution for coupled uni-axial bending and torsional linear oscillations in elastic uniform beams having single axis cross-section symmetry. In non-linear analyses, coupling phenomena between different deflections, torsion and bending for instance, is considered by including higher order terms in the deformation and strain expressions [14]. In a linear analysis though, it has to be integrated in the equations of motion by other means. In literature, the coupling was integrated in the governing equations by explicitly including an offset between the cross-section’s centroid and shear center [12, 13].


1.5

9

Parametric Studies

As it was previously mentioned, the sizes of wind turbine blades have increased immensely over the past decade, driven by the need to maximize the power output, presenting more challenges to the design process. Therefore, understanding the dynamic behavior of blades is crucial. Parametric studies provide deep insight to how certain material and geometric parameters, individually and collectively, affect the vibrational behavior. Such studies contribute to reaching general guidelines that can aid the preliminary design process. Their importance grows even more central if they can be employed to inversely determine the structural material and geometric properties required to attain, or to avoid, certain vibration characteristics. Countless works aimed at studying the coupled vibrations of blades, which were usually modeled as beams [16, 15, 17, 12, 18, 19, 20, 21, 22, 23, 24]. Timoshenko developed the equations of motion of a beam in coupled linear bending-torsional vibrations in [16], while Hodges et al. developed the equations for non-linear moderate coupled bending-bending-torsional vibrations in [15]. K. B. Subrahmanyam et el. solved for coupled vibration frequencies and mode shapes for rotating uniform blades using the Reissner method and the classical potential energy method [17]. The effect of shear center offset on the vibration frequencies as well as the rotation effect was discussed. It was noticed that rotational motion caused the bending frequencies to increase, especially in the first mode. That was justified by the increased stiffness due to rotation. Coriolis forces were found to have little effect on the frequencies magnitude. It was mentioned that the bending coupled frequencies decrease in a nonlinear manner with increasing offset. It was Dokumaci who first developed an exact closed form solution for the coupled PDEs describing the coupled bending and torsional vibrations of an Euler-Bernoulli beam, thus contributing significantly to the understanding of coupled vibrations at that time [12]. He used the advantage of his closed form solution to investigate the parameters affecting the dynamic behavior of blades. He demonstrated how the natural frequencies of coupled bending-torsion vibrations change with the torsional to bending stiffness ratio and the slenderness ratio. Bercin and Tanaka studied the coupled bending-torsion vibrations of a Timoshenko beam, in which they included shear deformations and warping effects [18]. It was concluded that such effects grow more pronounced as the mode order increases,


10

which makes modeling a blade as an Euler-Bernoulli beam a reasonable approximation if the lower modes are in question. Atma R. Sahu found that the vibration of blades of non-uniform exponential cross-section is affected by the rotation speed and the range of change of the cross-sectional area as a function of the blade length [19]. Bazoune aimed at deriving explicit expressions for the mass and stiffness matrices for double-tapered beams in terms of the taper ratio by the finite elements method [21]. He noted that both the taper ratio and the rotational speed affected the vibration frequencies and tried to find physical trends governing that influence. Hsu used the Euler-Bernoulli beam model to describe the response of a wind turbine blade [22]. The effect of rotation on the blade vibration characteristics was investigated and the rotation speed was found to have the highest influence on the first mode of vibration. Kaya et al. developed the EOMs of coupled bending-torsion vibrations of rotating Timoshenko beams using Hamilton’s principal and solved for the vibration frequencies [24]. In his results, he found that the rotation effect is most obvious in the fundamental bending mode, decreases as the frequency order increases and it was almost negligible in the torsional modes, which agrees with the results in [17]. The effect of the shear forces, which characterizes the Timoshenko beam and makes it different from the Euler-Bernoulli beam, were found to decrease the bending frequencies, especially with increasing mode order, and have no effect on the torsional frequencies, which is consistent with the results in [18]. Coriolis terms were neglected as they had an almost negligible effect on the vibration frequencies. That work was extended by Ozgumus et al. to double-tapered Timoshenko beams [20]. It was found that the taper ratio of the beam affects the vibration frequency. Change trends were drawn out of the results but no physical explanation could be drawn out, which is consistent with Bazoune’s result in [21]. These findings, besides Sahu’s findings in [19], lead to the conclusion that each specific distribution of masses and material properties in a beam has to be studied separately to investigate how it affects the beam dynamic behavior. Finite differences method was used by Altintas to solve PDEs modeling coupled chord-wise, flap-wise bending and torsion vibrations of thin-walled non-uniform Euler-Bernoulli beams [23]. He found that the ratio of the moduli expressed in Poisson’s ratio, rather than the moduli themselves, is important in coupled vibrations. It can alter the frequencies, as well as change the dominant type of vibrations in the fundamental mode.


1.6

11

Objective

In this work, Lagrange’s equations are used to derive the modal equations of the rotating non-uniform tapered blade problem discretized by the assumed modes method. A parametric study is done to investigate the effects of the rotational motion, as well as the blade structural properties on the vibration frequencies. Dimensional analysis techniques, as well as the problem physics, are employed to extract parameters comprising these variables upon which blades can be considered similar in terms of their vibration characteristics.

Chapter 2 Uniform Blade Structural Dynamic Model In this chapter, the structural model of a wind turbine blade of uniform cross-section and physical properties along the blade length is constructed. As it was mentioned earlier in the literature review, the Euler-Bernoulli beam model is used to describe the blade. This model accounts for the axial stress due to bending and the torsional shear stresses in the plane perpendicular to the elastic axis. First, the kinematic analysis of the blade is conducted, specifying the rotating and fixed frames. Then, the potential and kinetic energies of the cantilever fixed blade are derived. The energy expressions are used to derive the governing equations by Hamilton’s principal. The uniform fixed cantilever blade PDEs are solved analytically. To prove the sanity of the results, the same problem is solved using the assumed modes method and the approximate results are compared with the exact ones obtained by the analytical solution. Then, the rotational rigid body motion is added to the structural model and the uniform rotating blade problem is discretized using the assumed modes method. A parametric study aiming to find out the influence of the blade physical properties on its vibrational behavior is presented after each blade model.

2.1

Kinematic Analysis

Consider the inertial fixed frame [xyz]0 with an origin O0 that is located at the shear center of a blade root as shown in Fig. 2.1. The blade is rotating about the 12

CHAPTER 2. UNIFORM BLADE STRUCTURAL DYNAMIC MODEL

13

z0 axis with a constant angular speed Ω. Another frame [xyz]1 is fixed at the shear center of the blade root such that the z1 axis extends along the z0 axis. The frame [xyz]1 is fixed to the blade and is rotating with it with the angular speed Ω. The homogeneous transformation matrix between frames 0 and 1 can be expressed as  0

  T1 =   

cos(Ωt) − sin(Ωt) sin(Ωt) cos(Ωt) 0 0 0 0

0 0 1 0

0 0 0 1

     

(2.1)

Now, let P be a point on the blade whose position is described by 1 h = [xp yp zp 1]T

Figure 2.1: An airfoil cross-section blade. relative to frame [xyz]1 . Also, let the coordinate system [xyz]2 be displaced by a distance yp away from O1 along the y1 axis. The origin point O2 is attached to the shear center at that cross section even after deformation and the y2 axis extends along the deformed elastic axis at that cross section. Using the notation x for xp , y for yp and z for zp for the sake of generality and convenience, the position of point P is described as 2 h = [x 0 z 1] relative to [xyz]2 in the undeformed state of the blade. Also, let the coordinate system [xyz]3 be fixed such that O2 coincides with O3 and the y2 axis is collinear with the y3 axis. The x3 axis is rotated by an angle β from the x2 axis about the y2 axis as shown in Fig. 2.2. Uni-axial bending and torsional deflections are considered to be the blade degrees of freedom in this work. The geometry of an airfoil cross-section causes it to be stiffer in chord-wise bending than in flapwise bending as the area moment of inertia


14

z2

z3

β x2

β

x3

Figure 2.2: The blade cross-section twisted by the torsional angle β about the y2 axis. will be less in the latter. Therefore, flapwise bending and torsional deflections are considered in this model. The blade can deflect in the z2 axis direction and can be twisted in torsion about the y2 axis. So, the origin O2 is translated by [0 0 w] in the x2 , y2 and z2 directions, respectively. The homogeneous transformation matrix from frame 1 to frame 2 can be expressed as [25]  1

  T2 = A2 E2 =    1

1

1 0 0 0

0 1 0 0

0 0 1 0

0 y 0 1

     

1 0 0 0 0 0 1 −w (y, t) 0 0 0 w (y, t) 1 w(y, t) 0 0 0 1

     

(2.2)

. The homogeneous transformation matrix where w0 (y, t) is used to express ∂w(y,t) ∂y 1 A2 represents the displacement due to the link between the two frames and 1 E2 is due to the elastic deformations. Similarly, the transformation matrix 2 T3 can be expressed as   1 0 β(y, t) 0     0 1 0 0 2  T3 =  (2.3)  −β(y, t) 0  1 0   0 0 0 1 Assuming no warping occurs in torsion so that 3 h, after torsional deformation, stays the same as 2 h before torsion, the position of P relative to the rotating frame


15

after deformation can be expressed as 1

h = 1 T3 3 h = 1 T2 2 T3 3 h

 1

  h=  

x + zβ y − zw0 + xβ w0 z + w − xβ 1

(2.4)

     

(2.5)

Assuming that the blade is undergoing small deformations, the product β w0 is neglected since its value is of a much smaller order of magnitude compared to the rest of the expression. So, 1 h is simplified to  1

  h=  

x + zβ y − z w0 z + w − xβ 1

     

(2.6)

and the velocity of P can be written as    h=  

1˙

z β˙ −z w˙ 0 w˙ − x β˙ 0

     

(2.7)

Referring to equations 2.6 and 2.1, the position of point P can be described relative to the inertial frame as   x cos(Ωt) − y sin(Ωt) + z cos(Ωt)β + z sin(Ωt)w0    y cos(Ωt) + x sin(Ωt) + z sin(Ωt)β − z cos(Ωt)w0  0 0 1  h = T1 h =  (2.8)   z + w − xβ   1


16

and the velocity as    h=  

0˙

−yΩ cos(Ωt) − xΩ sin(Ωt) − zΩ sin(Ωt)β + z cos(Ωt)β˙ + zΩ cos(Ωt)w0 + z sin(Ωt)w˙ 0 xΩ cos(Ωt) − yΩ sin(Ωt) + zΩ cos(Ωt)β + z sin(Ωt)β˙ + zΩ sin(Ωt)w0 − z cos(Ωt)w˙ 0 w˙ − xβ˙ 0 (2.9)

2.2

Fixed Uniform Blade Model

2.2.1

Potential energy

Both the gravitational and elastic energies contribute to the fixed cantilever blade potential energy. As the deflections are assumed to be small, the changes in the gravitational energy are ignored. The strain energy of the deflections can be calculated by [26] ˆ

σ dε

V0 =

(2.10)

0

where V0 is the strain energy, σ is the stress and is the strain. Since the deflections are assumed to be small, it can be safely assumed that the blade deflections are linearly elastic. Thus, equation 2.10 becomes ˆ V0 = ˆ

0 yz

yz dyz 0

ˆ

yy

E yy dyy + 2 G ˆ zy ˆ + zy dzy + 0

0

ˆ

xy

0 zx

xy dxy + ˆ zx dzx +

yx

0 xz

yx dyx + xz dxz

(2.11)

0

where E and G are the blade’s tensile elastic modulus and shear elastic modulus, respectively, yy is the axial strain in the direction of the elastic axis, and the notation ij is used to denote the shear strain perpendicular to the axis i and parallel to the axis j. Letting u be the displacement of the point P in deflection, it can be expressed as shown   zβ   u =  −z w0  (2.12) w − xβ

     


17

The elastic strain is defined as [8] 1 ij = 2

∂uj ∂ui + ∂i ∂j

i,j = x, y and z.

(2.13)

For example, yx

1 = 2

∂ux ∂uy + ∂y ∂x

=

1 (zβ 0 (y, t)) 2

Therefore, the elastic strain is expressed as 

  1 0  yx zβ (y, t) 2      yy  =  −zw00 (y, t)  1 yz xβ 0 (y, t) 2

(2.14)

The strain yz was calculated to be zero. The relations yx = xy yz = zy zx = xz were used. Substituting Eq. 2.14 into Eq. 2.11, the elastic potential energy per unit volume becomes 1 1 (2.15) V0 = E z 2 w002 + G β 02 (z 2 + x2 ) 2 2 The potential energy is integrated with respect to the cross-sectional area to get the elastic potential energy per unit length of the blade. To simplify the integrations, they are carried with respect to the cartesian coordinate system [˜ xy˜z˜]2 in which x˜2 , y˜2 and z˜2 axes are parallel to the x2 , y2 and z2 axes respectively. However, the only difference is that the y˜2 axis extends along the centroidal axis instead of the elastic axis, which simplifies the area integrations as they will vanish about the x˜2 and z˜2 axes. Though the blade is not symmetric about the x2 axis, the offset between the elastic and the centroidal axis in the z2 direction rz is ignored and only the offset in the x2 direction rx is considered. In spite of the fact that it is actually the offset rz that is responsible for developing the pressure difference between the upper and lower airfoil surfaces and thus the lifting force, that offset can be neglected without much loss of accuracy in the model for the following reasons. First, that offset is responsible for coupling the chordwise bending deflections with the flapwise bending and torsional deflections. Since chordwise deflections are not included in


18

the model, then there is no need to take the offset rz into consideration. Secondly, offsets normal to the airfoil chord are difficult to measure and their effect on the vibrational properties is negligible [27]. Throughout this text, the y notation is used instead of y˜ as the independent variable for the bending and torsion deflections for ease. However, the difference between them is underlined in the treatment of the moments of inertia. The two coordinate systems are related by Eq. 2.16. x˜2 + rx = x2

y˜2 = y2

z˜2 = z2

(2.16)

Integrating equation 2.15 with respect to the blade’s cross-section area d A = d˜ x2 d˜ z2 to get the elastic potential energy per unit length yields ˆ

ˆ V0 d V = 0

L

ˆ ˆ L ˆ 1 02 1 002 2 2 2 Ew z dA dy + Gβ (˜ x + rx ) + z dA dy 2 2 0 ˆ L 1 1 002 02 = E Ix2 w + GJy2 β dy (2.17) 2 2 0

Ix2 and Jy2 are the blade area moment of inertia about the x2 axis and the polar area moment of inertia about the y2 axis, respectively. The product GJy2 indicates the blade stiffness in torsion and the product EIx2 indicates the blade’s stiffness in bending about the x2 -axis. The polar moment of inertia Jy2 is related to the polar moment of inertia about the y˜2 axis J¯y2 by Jy2 = J¯y2 + A rx2

(2.18)

where A indicates the cross-sectional area of the blade.

2.2.2

Kinetic energy

The blade kinetic energy can be calculated by [26] ˆ T =

1 ~˙ T ~˙ ρ h . h dV 2

(2.19)

where ρ is the density per unit volume and ~h˙ is the velocity of a point P relative to the rotating frame in Eq. 2.7. Substituting Eq. 2.7 in Eq. 2.19, the kinetic energy


19

becomes ˆ T =

1 2 ρ w˙ + z 2 w˙ 2 − 2x w˙ β˙ + (x2 + z 2 ) β˙ 2 dV 2

(2.20)

To find the kinetic energy of the blade per unit length, equation 2.20 is integrated with respect to the blade cross-sectional area. Note that the centroidal axes are used here as well. So, we express the kinetic energy as ˆ T =

1 2 2 2 2 2 ˙2 ˙ ρ w˙ + z˜ w˙ − 2(˜ x + rx )w˙ β + ((˜ x + rx ) + z ) β dV 2

(2.21)

The kinetic energy per unit length is ˆ T = 0

L

1 2 2 2 2 0 2 ˙ y , t) + µ(r + r )β(˜ ˙ y , t) + Ix2m w˙ (˜ µ w(˜ ˙ y , t) − 2rx w(˜ ˙ y , t)β(˜ y , t) dy x 2 (2.22)

´ where µ is the material’s density per unit length calculated by µ = A ρ dA = ρ A, Ix2m is the mass moment of inertia per unit length about the x2 axis and Jy2m is the polar mass moment of inertia per unit length about the y-axis. In the above integrations, the parallel axis theorem was used to relate the moments of inertia about the elastic and centroidal coordinates. Jy2m = J¯y2m + µrx2

Ix2m = I¯x2m

(2.23)

where J¯y2m is the polar mass moment of inertia about the y˜2 axis and I¯x2m is the mass moment of inertia about the x˜2 axis.

2.2.3

Hamilton’s principal

In this work, we are using analytical mechanics techniques to derive the equations of motion. In analytical mechanics, the degrees of freedom of a certain system are represented by generalized coordinates, which are not unique. They provide a more general and abstract approach to model a problem, as compared to Newtonian mechanics [11]. In analytical mechanics, the concept of virtual displacements is used. Virtual displacements can be defined as slight variations in the generalized coordinates that are consistent with the system constraints and happen instantaneously, which means that the variation of time occurring in parallel with the virtual displacements is zero, δt = 0. Virtual displacements are assumed to be completely

20


arbitrary and, being infinitesimal, differential calculus can be applied to them as well. In this work, the governing equations for the blade are derived by Hamilton’s principal, which states that ˆ

t2

t1

(δT − δV ) dt = 0

(2.24)

where T and V are the kinetic and potential energies, respectively. Hamilton’s principal is a variational principal, i.e., it deals with variations and virtual displacements, which reduces a mechanics problem to a problem of investigation of a scalar integral. The principal states that The actual path in the configuration space renders the values of the ´t definite integral I = t12 (T −V )d t stationary with respect to all arbitrary variations of the path between two instants t1 and t2 provided that the path variations vanish at these two end points [28]. To find the minimal path, the variations of the displacements are taken and equated to zero. Accordingly, Hamilton’s principal is known to be a principal of least action where the integrals of the variations of the energy terms over a time interval from t1 to t2 have to be minimal. The conditions that yield the integral stationary eventually lead to the equations of motion. The variations of the kinetic and potential energies can be found by deriving the energy expression with respect to time and replacing the time derivative with the variation sign [28]. Though the energy terms are integrated with respect to time, no such operations are needed to derive the governing equations. The independence, and hence the arbitrariness, of the generalized coordinates used in Hamilton’s principal is in fact the key to deriving the governing equations. Invoking the arbitrary nature of the virtual displacements, the governing equations can be extracted from the coefficients of the independent arbitrary virtual displacements without applying the time integrals in Eq. 2.24.

2.2.4

Equations of Motion

To apply Hamilton’s principal, the variation of the length integral of the potential energy term is calculated. ˆ

ˆ

t2

t2

ˆ

δV dt = t1

t1

0

L

(GJy2 β 0 δβ 0 + EIx2 w00 δw00 ) d y d t

(2.25)

21


The energy terms must be coefficients of δw and δβ as only those variations are assumed to be arbitrary. So, the variations in Eq. 2.25 can be integrated by parts to obtain variations of w and β.

ˆ

t2

ˆ

ˆ

L

0

0

ˆ

t2

(GJy2 β

GJy2 β δβ dydt = t1

t2

ˆ

ˆ

L

00

ˆ

0 t2

EIx2 w

= t1

t1 t2

ˆ EIx2 w00 δw0 |L0 dt −

δβ) |L0 dt−

t2

00

δw0 |L0

dt −

(EIx2 w00 )0 δw|L0 dt +

t1

ˆ

t1

L

(GJy2 β 0 )0 δβdydt (2.26)

0

ˆ

t2

00

EIx2 w δw dydt = t1

0

0

t1

0

ˆ

ˆ

L

t2

ˆ

t1

ˆ

L

(EIx2 w00 )0 δw0 dy dt

0

t2

t1

ˆ

L

(EIx2 w00 )00 δwdydt

0

(2.27) The variation of the potential energy in Hamilton’s principal is ˆ

ˆ

t2

t2

ˆ

L

((EIx2 w00 )00 δw − (GJy2 β 0 )0 δβ) dy dt+

δV dt = t1

GJy2 β

0

t1

0

δβ|L0

+ EIx2 w00 δw0 |L0 − (EIx2 w00 )0 δw|L0

(2.28)

Repeating the same procedures with the blade’s kinetic energy, the variation of the length integral of the kinetic energy term is calculated and then integrated with respect to time. ˆ

ˆ

t2

t2

ˆ

δT d t = t1

t1

0

L

˙ w˙ − rx wδ ˙ β˙ + Ix2m w˙ 0 δ w˙ 0 d y d t µwδ ˙ w˙ − µrx βδ ˙ β˙ + J0m βδ (2.29)

Integrating Eq. 2.29 by parts in order to get all the variations in terms of w and β only, the terms in Eq. 2.29 can be divided into two classes, ˆ

t2

ˆ

t1

and

ˆ

t2

t1

L

˙ ˙ F (y)δ G(y, t)K(y, t)dydt

0

ˆ 0

L

F (y)δ G˙ 0 (y, t)K˙ 0 (y, t)dydt

22


For the sake of an example, one term of each class is integrated in Eq. 2.30 and Eq. 2.31. ˆ

L

ˆ

ˆ

t2

L

µ(y)δ w˙ wdt ˙ dy = 0

0

t1

ˆ t2 µwδw| ˙ t1 dy

−

ˆ

L

t2

µwδwdtdy ¨ 0

(2.30)

t1

In Eq. 2.30 the integrations are assumed to be interchangeable and the variation of w between t1 and t2 vanishes because the varied path of δw is assumed to have the same start and end point as the actual dynamic path [28]. The same assumptions apply to the second example. ˆ

t2

t1 t2

ˆ =

t1

ˆ

L

ˆ Ix2m δ w˙ 0 w˙ 0 d y d t =

0

ˆ

Ix2m w˙ 0 δ w| ˙ L0 dt −

0

t2

t1 L

ˆ Ix2m w˙ 0 δ w| ˙ L0 dt −

(Ix2m w˙ 0 )0 δw|tt21 d y +

ˆ

t2

t1

t2

L

(Ix2m w˙ 0 )0 δ wdy ˙ dt

0

t1

ˆ

ˆ

L

(Ix2m w¨0 )0 δwdy dt

(2.31)

0

All integrations can be performed in the same manner. The variations and derivations of the equations of motion are detailed in Appendix B. Eventually, the time integral of the variation of kinetic energy in equation Eq. 2.29 with respect to time is ˆ

ˆ

t2

t2

ˆ

L

δT = t1

t1

0

−µw¨ + µrx β¨ + (Ix2m w¨0 )0 δw + µrx w¨ − Jy2m β¨ δβdy dt ˆ t2 − Ix2m w¨0 δw|L0 dt (2.32) t1

Equation 2.32 and Eq. 2.28 are substituted in Hamilton’s principal. ˆ

t2

t1

ˆ

t2

+ t1

ˆ 0

L

−µw¨ + µrx β¨ + (Ix2m w¨0 )0 − (EIx2 w00 )00 δw+ 0 0 ¨ µrx w¨ − Jy2m β + (GJy2 β ) δβdy dt

−Ix2m w¨0 δw|L0 − GJy2 β 0 δβ|L0 − EIx2 w00 δw0 |L0 + (EIx2 w00 )0 δw|L0 dt

(2.33)

To satisfy Eq. 2.33, either the variations δw and δβ are equal to zero, which is the trivial solution, or their coefficients should be equal to zero so that the equation can hold for all values of δw and δβ. Taking the coefficient of each variation and equating it to zero yields the blade governing equations. As the model is dealing with small deflections and is not interested in the higher


23

modes of vibration that may cause wrinkling of the blade, the rotary inertia terms Ix2m w¨0 are ignored. The governing PDEs of the fixed cantilever blade become −µw¨ + µrx β¨ − (EIx2 w00 )00 = 0 µrx w¨ − Jy2m β¨ + (GJy2 β 0 )0 = 0

(2.34)

The boundary conditions are GJy2 β 0 δβ|L0 = 0

(2.35)

EIx2 w00 δw0 |L0 = 0

(2.36)

(EIx2 w00 )0 δw|L0 = 0

(2.37)

The boundary condition in Eq. 2.35 addresses the torsional angle and the applied torque, represented by the first derivative of the torsional angle β at the root and the tip of the blade. The second boundary condition in Eq. 2.36 deals with the slope of the blade, represented by the first derivative of the bending deflection w and the bending moment EIx2 w00 at the blade tip and root. Finally, the boundary condition in Eq. 2.37 imposes conditions on the bending deflection and the shear force (EIx2 w00 )0 at the blade tip and root. For the case of a blade fixed at y = 0, w(0, t) = 0

(2.38)

β(0, t) = 0

(2.39)

w0 (0, t) = 0

(2.40)

GJy2 β 0 (L, t) = 0

(2.41)

EIx2 w00 (L, t) = 0

(2.42)

(EIx2 w00 )0 (L, t) = 0

(2.43)

The first three boundary conditions mean that the bending and torsional deflections at y = 0, besides the derivative of the bending deflection with respect to the blade length is equal to zero. These conditions are concluded by applying the boundary conditions obtained by Hamilton’s principal in Eq. 2.35, Eq. 2.36 and Eq. 2.37 at y = 0. The last three boundary conditions in equations 2.38, 2.39 and 2.40 imply that the blade is in free response and that the applied torsional torque, bending moment and transverse shear force at the free end of the blade are equal to zero,


24

respectively. Since we have a system of homogeneous boundary conditions of the fixed blade at free response, the matrix formed by these equations will naturally be rank deficient.

2.3

Spatial Reduction of the Governing Equations

In this part of the chapter, the problem of the uniform fixed cantilever blade is solved by the exact solution and by the assumed modes approximate method. The advantage of using an exact method is that the closed form solution is more accurate and computationally more efficient. Unfortunately, more complicated problems cannot be solved in a similar manner, which is why numerical approximate techniques are used in such cases. The assumed modes method is the approximate numerical technique that is used in this work. The approximate results calculated from the characteristic equation produced by the assumed modes method will be verified against the exact results calculated for the case of the uniform fixed cantilever blade.

2.3.1

Exact solution

In this section, the vibration characteristics of a fixed cantilever uniform blade are solved for using exact analytical techniques of solving PDEs. The separation of variables method is used to separate the displacements w(y, t) and β(y, t) to distinct functions of space and time. A sixth order characteristic equation is obtained and is solved for the frequencies, which are then substituted in the mode shapes of the blade. First, the partial differential equations in Eq. 2.34 are reduced to ordinary differential equations with respect to the variable y using the separation of variables method. w(y, t) = W (y) q(t)

(2.44)

β(y, t) = B(y) q(t)

(2.45)

where W (y) and B(y) are functions of the spatial coordinate y and are used to describe the vibration spatial mode shapes of the bending and torsional vibrations respectively, and q(t) is the generalized coordinate used to describe the frequency


25

of the blade vibration with the given mode shapes. Note that the time generalized coordinate q(t) is the same in both types of vibration due to the coupled nature of the vibrations which causes the two types to be present in the same mode at the same frequency. The generalized coordinate q(t) is a sinusoidal oscillation function with frequency ω expressed as u eiωt . Substituting equations 2.44 and 2.45 in Eq. 2.34 and dividing by eiωt yields the ordinary differential equations µω 2 W (y) − µrx ω 2 B(y) − (EIx2 W 00 (y))00 = 0

−µrx ω 2 W (y) + Jy2m ω 2 B(y) + (GJy2 B 0 (y))0 = 0

(2.46)

A possible solution for the ordinary equations in Eq. 2.46 is in the form sy

(2.47)

W (y) = wˆ exp L sy B(y) = ˆb exp L

(2.48)

where wˆ and ˆb are the amplitudes of the mode shapes. Thus, Eq. 2.46 can be rewritten as, !(

4

ω 2 µ − E Ix2 Ls 4 −µ rx ω 2 −ω 2 µ rx ω 2 (J¯y2m + µrx2 ) +

s2 G Jy2 L2

wˆ ˆb

) =0

(2.49)

Now, let s

J¯my2 µ

(2.50)

µ ω 2 L4 λb = E Ix2

(2.51)

µ r2 ω 2 L2 λt = G Jy2

(2.52)

r=

=1+

rx2 r2

(2.53)

The coefficient matrix in Eq. 2.49 becomes λb − s4 −rx λb −λt rrx2 λt + s2

! (2.54)

where r is the polar mass radius of gyration of the blade about the centroidal axis y˜2 ,


26

λb and λt are dimensionless parameters that can be obtained by dimensionless analysis techniques [29], and is a dimensionless quantity measuring the offset between the centroidal and elastic axes, where = 1 indicates the case of zero offset. The physical significance of λb and λt becomes evident when the roots of the determinant of the uncoupled coefficient matrix of wˆ and ˆb are solved for. This determinant constitutes the uncoupled characteristic equation of the ordinary differential equation with respect to y. In the uncoupled ODEs (Ordinary Differential Equations), the value of rx is equal to zero and the characteristic equation becomes (s4 −λb )(s2 +λt ) = 0. This means that λb and λt are related to the spatial frequencies for the mode shapes of uncoupled bending and torsional vibrations. Back to the coupled vibrations coefficient matrix, equating the determinant of the coefficient matrix in Eq. 2.54 to zero results in the characteristic equation of the coupled ODEs of an asymmetric blade with an offset rx between the centroidal and elastic axes. s6 + s4 λt − s2 λb − λb λt = 0 (2.55) Equation 2.55 is a sixth order symbolic polynomial of the variable s, in which all the terms are even-powered. This equation cannot be solved numerically as the coefficients include the variable ω, the vibration frequency. Thus, the equation includes two unknowns, s and ω. In Uspensky’s theory of equations, instructions on how to solve even powered sixth-order polynomials are detailed [30]. Letting r = s2 , it becomes r3 + r2 λt − r λb − λb λt = 0 (2.56) Setting r =a+k

(2.57)

and substituting Eq. 2.57 in the third order polynomial in Eq. 2.56 yields a3 + a2 (3k − λt ) + a(3k 2 − λb − 2kλt ) + k 3 − k λb − k 2 λt − λb λt = 0 Setting k to

λt 3

(2.58)

such that a2 vanishes, equation 2.58 becomes a3 + P a + Q = 0

(2.59)


27

where λt a = s2 + 3 2 λ2t P = −λb 1 + 3λb 2 3 λ2t Q = −λb λt 1 − − 3 27λb

(2.60)

(2.61)

Equation 2.59 can be solved by Cardan’s formula if the discriminant ∆ = 4 P 3 + 27 Q2

(2.62)

is negative [30]. Substituting the values of P and Q from Eq. 2.61 in the discriminant in Eq. 2.62 and rearranging the term yields ∆=

−λ3b

2 λ2t 2 3 λt 2 4 + ( + 18 − 27) + 4 ( ) λb λb

(2.63)

In this paragraph, it is proved that the discriminant in Eq. 2.63 is negative. Notice λ2 the term between brackets in the discriminant is a quadratic polynomial in λtb . The discriminant in Eq. 2.63 is negative if the term 2 + 18 − 27 is positive. Solving for the roots of that term, it is found that it has two roots at = ±1.4. Since the physics of the problem implies that must be at least one, only the positive root needs to considered. The sign of the discriminant is investigated in the intervals 1 ≤ ≤ 1.4 and ≥ 1.4. The positive root is at approximately 1.4, which marks a change of sign in the term (2 + 18 − 27) at that value of . To find the direction of the sign change, the global minima and maxima of the polynomial are calculated. It is found that a global minimum lies at = -9 before the root, concluding that the value of that term must be increasing after the global minimum at -9. Thus, the term (2 + 18 − 27) must be negative in the interval 1 ≤ ≤ 1.4 and positive when ≥ 1.4, which makes the discriminant δ negative in the interval ≥ 1.4. Next, the roots of the whole discriminant, the quadratic polynomial of λ2t /λb , are solved for to check if the discriminant is negative in the


28

interval 1 ≤ ≤ 1.4 as well.

√ 27 − (−9 + )3/2 −1 + − 18 − 2 = 82 √ 27 + (−9 + )3/2 −1 + − 18 − 2 λ2t /λb = 82

λ2t /λb

(2.64)

To find out the nature of the roots at the desired interval, the discriminant √ ∆0 = (−9 + )3/2 −1 + λ2

that can be found in the roots of the λtb polynomial in equations 2.64 is investigated in the desired interval of . Equation 2.64 has complex roots in the interval 1 ≤ ≤ 1.4 because the discriminant ∆0 is imaginary in that interval. Therefore, the polynomial of λ2t /λb in Eq. 2.63 does not change sign and stays positive in the interval 1 ≤ ≤ 1.4. This proves that the value of ∆ in Eq. 2.63 is negative in the interval ≥ 1. Hence the roots are all real and can be solved for by Cardan’s formula yielding r

−P φ a1 = 2 cos( ) 3 3 r −P φ 2π cos( + ) a2 = 2 3 3 3 r φ 4π −P a3 = 2 cos( + ) 3 3 3 where

(2.65) (2.66) (2.67)

√ 27 Q √ cos(φ) = 2 P −P r s i = ± ai −

λt 3

(2.68)

i = 1, 2 and 3

(2.69)

Note that the angle in cos(φ) is expressed in radians. The general solution for the ordinary differential equations is f (y) = Σi=1,2,3 ai (exp

si y L

+ exp−

si y L

)

(2.70)

where si are the roots of the characteristic equation calculated by Cardan’s formula. Using Descartes’ rule of signs 1 , it is found that there is one variation of sign 1

The number of positive real roots of an equation with real coefficients f (x) = a0 xn + a1 xn−1 +


29

in the characteristic equation in Eq. 2.55. Also, substituting s by −s in the same equation yields only one variation of sign. Hence, there are only two real roots for Eq. 2.55, one positive and the other negative. The remaining four roots are pure imaginary roots as the roots of the cubic equation in Eq. 2.59 are all real as proved earlier. Thus, the six roots of the characteristic equation in Eq. 2.55 are s1 , −s1 , i s2 , −i s2 , i s3 and −i s3 . Then, using euler’s formula 2 , the general solution for W(y) and B(y) for the case of an asymmetric blade is rewritten in the form s1 y s2 y s2 y s1 y ) + A2 sinh( ) + A3 cos( ) + A4 sin( )+ L L L L s3 y s3 y ) + A6 sin( ) A5 cos( L L s1 y s1 y s2 y s2 y B(y) = B1 cosh( ) + B2 sinh( ) + B3 cos( ) + B4 sin( )+ L L L L s3 y s3 y B5 cos( ) + B6 sin( ) (2.71) L L

W (y) = A1 cosh(

To find the values constants A1 , A2 , ... and A6 , the general solution in equations 2.71 is substituted in the ODEs in Eq. 2.46. This yields twelve equations from the coefficients of the trigonometric functions of s1 , s2 and s3 in both equations. The system of equations is over-determined as only six out of the twelve equations can be used. For example, if the coefficients of cosh( s1Ly) in both governing equations are considered, it is found that the value of B1 can be either B1 =

−A1 (s4 − λb ) rx λb

(2.72)

A1 rx λt (s2 + λt )

(2.73)

or B1 =

r2

As the value of rx approaches zero, the value of decreases at an even faster rate because of the rx2 term, thus the values of the roots of the coupled characteristic equation tend to those of the uncoupled roots. Taking this into consideration, as rx tends to zero, the value of B1 in Eq. 2.72 tends to zero, while the value of B1 in Eq. 2.73 tends to infinity. Hence, the value of B1 in Eq. 2.72 is used here as it is more general and consistent with the literature results for symmetric blade vibrations. Accordingly, the general solution of W (y) and B(y) can be rewritten in .. + an = 0 is never greater than the number of variations in the sequence of its coefficients a0 , a1 , ... and an , if less, then always by an even number. 2 expı θ = cos(θ) + ı sin(θ)

30

CHAPTER 2. UNIFORM BLADE STRUCTURAL DYNAMIC MODEL terms of the constants A1 , A2 , A3 , A4 , B5 , and B6 as

s y s y s y s y 1 2 2 1 + A2 sinh + A3 cos + A4 sin + L L L L s y s y r2 (s23 + λt ) 3 3 B5 cos + B6 sin rx λt L L 4 −(s1 − λb ) s1 y s1 y −(s41 − λb ) B(y) = A1 cosh + A2 sinh + r λ L L r λ s y s y s y x b s yx b 2 3 3 2 + A4 sin + B5 cos + B6 sin (2.74) (A3 cos L L L L

W (y) = A1 cosh

Substituting Eq. 2.74 in the boundary conditions in Eq. 2.38-Eq. 2.43, Ax = 0

(2.75)

where A is the coefficient matrix of the constants constructed from the six boundary conditions of the fixed cantilever blade, x is a vector consisting of the constants A1 , A2 , A3 , A4 , B5 and B6 .                   

1 0

0

s21 cosh(s1 ) L2

s1 L s21 sinh(s1 ) L2

s31 sinh(s1 ) L3

s31 cosh(s1 ) L3

− r1x +

s41 rx λb

1) − s1 sinh(s + Lrx

2

s2 L 2 s sin(s ) − 2 L2 2

−

s32 cos(s2 ) L3

0 2) − s2 cos(s + Lrx

1) − s1 cosh(s + Lrx

− rrx − rx +

0

s52 cos(s2 ) Lrx λb

s51 cosh(s1 ) Lrx λb

−

+

r2 s33 sin(s3 ) L3 rx

rx s23 cos(s3 ) L2

−

s22 cos(s2 ) L2

s32 sin(s2 ) L3

s2 sin(s2 ) Lrx

−

s42 rx λb s52 sin(s2 ) Lrx λb

r2 s23 rx λt

−

rx s33 sin(s3 ) L3



0 2

0

r2 s23 cos(s3 ) L2 rx

−

− r1x +

0

s51 sinh(s1 ) Lrx λb

1 0

r2 s43 cos(s3 ) L2 rx λt

+

r2 s53 sin(s3 ) L3 rx λt

r2 s33 rx s3 + L Lrx λt rx s23 sin(s3 ) r2 s43 sin(s3 ) − L2 L2 rx λt

− rLrsx3 −

r2 s23 sin(s3 ) L2 rx

+

r2 s33 cos(s3 ) L3 rx

+

rx s33 cos(s3 ) L3

1

0

3) − s3 sin(s L

s3 cos(s3 ) L

−

r2 s53 cos(s3 ) L3 rx λt

                 


31

Since the system of equations is rank deficient, the determinant of A must be equal to zero in order to get a non-trivial solution for the vector x, which constitutes the characteristic equation of the problem. After substituting the values of s1 , s2 and s3 calculated by Cardan’s formula, A can be considered as essentially a function of ω, the blade vibration frequency. Solving the characteristic equation yields the natural frequencies of the blade which, given that s is a trigonometric function of ω, are of infinite number as the blade is a continuous structure with infinite degrees of freedom. Due to the nature of the transcendental characteristic polynomial, numerical techniques should be employed to solve for the roots. In this work, Newton’s method was employed to solve for the roots of the characteristic polynomial and thus the free response frequencies of the fixed cantilever blade in coupled uni-axial and torsional vibrations. This was executed via the FindRoot command in the software Mathematica . The values for the constants A2 , A3 , A4 , B5 and B6 can be calculated in terms of A1 . The code lines used to solve the PDEs analytically can be found in appendix D.

2.3.2

Assumed Modes Method: Fixed Uniform Blade Case

In this section, we discuss spatial discretization using the assumed modes method. The assumed modes method is an approach for discretizing distributed parameter systems that is closely related to the Rayleigh-Ritz method [11]. It is basically a series discretization technique where the solution is assumed to be a linear combination of a set of N trial functions. It is chosen in this work for its simplicity and the conservative nature of the system. The objective is to produce a finite degree of freedom system that best approximates the conservative distributed parameter system at hand. The assumed modes method aims at developing the discretized modal equations of motion by first discretizing the energy and virtual work expressions. Then, Lagrange’s equations are employed to produce the modal equations of motion from the discretized energy expressions. The problem is to model the coupled bending-torsional vibrations of a homogeneous uniform wind turbine blade with an asymmetric cross-section. The EulerBernoulli beam model is employed due to the slender geometry of the blade. The vibrational displacements are expressed as the product of two functions in space and


32

time as w(y, t) = β(y, t) =

N X i=1 N X

φwi (y) qi (t) φβi (y) qi (t)

(2.76)

i=1

where φwi (y) and φβi (y) are the trial functions used to approximate the spatial mode shapes, N is the number of employed trial functions and i is a counter that takes values from i to N , qi (t) is the generalized coordinate used to describe the time response of the blade vibrations. It is expressed as qi (t) = ai eλi t

(2.77)

where λi is the time frequency of vibration of the trial function i and ai determines the contribution of the trial function i to the total solution. It should be noted that though φwi (y) and φβi (y) are the trial functions used to represent the bending and torsional vibrations, yet they are multiplied by the same generalized coordinate qi (t) due to the coupling effect which makes them both effectively a single coupled bending-torsional mode, i.e. bending and torsional vibrations occur simultaneously in coupled modes. It is important that the utilized trial functions in the assumed modes approach satisfy the problem boundary conditions. Boundary conditions can be classified into geometrical boundary conditions, which satisfy the geometrical constraints imposed on the spatial functions, or natural boundary conditions which satisfy constraints involving spatial derivatives of the dependent variables, namely the blade displacement at a certain point and the slope. Trial functions in the assumed modes method determine how close a certain discretized system to the original distributed system is. They can be admissible or comparison functions. Comparison functions satisfy both geometric and natural boundary conditions of the problem. They should be as many times differentiable as the order of the spatial ordinary differential equation describing the problem. Admissible functions, on the other hand, are only half times as differentiable as the comparison functions and they satisfy the geometric boundary conditions only. Comparison functions yield better results than admissible functions as they better approximate the problem. In this section, the fixed cantilever blade problem is solved using the assumed

33


modes method. In the previous sections, the elastic potential energy per unit length of a fixed cantilever uniform blade was expressed as ˆ

L

Ve = 0

1 1 002 02 dy E Ix2 w + GJy2 β 2 2

(2.17)

while the kinetic energy per unit length was expressed as ˆ

L

1 2 2 2 ˙ 2 0 2 ˙ T = µ w(˜ ˙ y , t) − 2rx w(˜ ˙ y , t)β(˜ y , t) + µ(r + rx )β(˜ y , t) + Ix2m w˙ (˜ y , t) dy 2 0 (2.22) Substituting the discretized displacements in Eq. 2.76 into the expressions of the energies per unit length inside the integral in Eq. 2.17 and Eq. 2.22, the discretized energies are T =

N X N X 1 2

i=1 j=1

µ φwi q˙i φwj q˙j − 2 rx µ φwi q˙i φβj q˙j + 1 + Ix2m φ0w i qi φ0w j qj 2

Ve =

N X N X 1 i=1 j=1

2

EIx2 φ00wi qi

φ00wj

1 µ(r2 + rx2 ) φβi q˙i φβj q˙j 2

1 qj + GJy2 φ0βi qi φ0βj qj 2

(2.78)

(2.79)

The discretized energy expressions are integrated with respect to the blade length and are substituted in Lagrange’s equations d dt

δ Tl δ q˙k

−

δTl δVl + = Qk δqk δqk k = 1 , 2 , ... , N

(2.80)

where Vl and Tl are the potential and kinetic energies integrated along the beam length. Equation 2.80 is used to write N modal equations of motion. Lagrange’s equations are more suitable to work with in case of discretized systems compared to Hamilton’s principal, which is more suitable for continuous systems. The governing modal equations can be written as ¨ + Kq = Qk Mq

(2.81)

where M is the inertia matrix with dimensions N xN . From the energy expressions

34


in Eq. 2.79 and Eq. 2.78, the elements in the inertia matrix are the coefficients of the second derivatives of the modal generalized coordinates qk . ˆ mij = 0

L

µ φwi φwj − µ rx φwi φβj − µ rx φwj φβi + µ (r2 + rx2 ) φβi φβj dy

(2.82) Similarly, K is the stiffness matrix with dimensions N xN where the matrix elements are the coefficients of the generalized modal coordinates qk in the energy expressions in Eq. 2.79 and Eq. 2.78. ˆ kij = 0

L

GJy2 φ0βi φ0βj + EIx2 φ00wi φ00wj dy

(2.83)

Also, in Eq. 2.81, q is the vector of generalized coordinates and Qk is the vector of generalized forces. Since our system is conservative, the generalized forces vector is reduced to zero. Qk = 0 (2.84) Substituting qi (t) = ai eλi t

(2.85)

in Eq. 2.81, we obtain the algebraic eigen-value problem (λ2 M + K)a = 0

(2.86)

where a is the vector of the constants ai determining the contribution of each trial function in the approximate mode shapes. The system of equations in Eq. 2.86 is rank deficient. Therefore, the determinant of the matrix λ2 M + K = 0 must be equal to zero which yields the characteristic equation of the system, a polynomial of λ of degree N 2 . Solving the characteristic equation for λ yields the vibration frequencies for the first N modes in the assumed modes discretization of the system.

2.4

Fixed Uniform Blade Case Study

In the previous sections, the problem of a uniform wind turbine blade vibrating in bending and torsion was solved using two approaches. The first is the exact analytical approach which used Partial Differential Equations solving techniques to


35

solve for the vibration frequencies and mode shapes. The advantage of the fact that the characteristic equation was a sixth-order even-powered polynomial was used to solve for the roots of the characteristic polynomial by Cardan’s formula which is a method used for solving third order polynomial equations. Then, the approximate assumed modes discretization method was introduced. The principal of this method is to discretize the spatial mode shapes of the blade by a finite number of spatial trial functions that satisfy the blade boundary conditions. The accuracy of the assumed modes method depends on how closely the employed trial functions describe the blade actual mode shapes. Usually, the employed trial functions are taken to be the exact mode shapes of a simpler problem. In this section, a case study is conducted so that the two methods, the exact solution technique method and the assumed modes method, are applied to solve for the actual mode shapes and vibration frequencies of a wind turbine blade in free response. The structural properties of the wind turbine blade in the case study are in fact the average properties of the CART wind turbine blade [31]. The CART wind turbine blade is discussed in detail in the next chapter. The averaged properties of the blade are shown in equations 2.87. µ rx GJy2 EIx2 r L

2.4.1

= = = = = =

102.097 0.09 1.09E7 3.8441E7 0.3632 19.955

kg/m m N.m2 N.m2 m m

(2.87)

Exact Solution

Following the previously mentioned approach in subsection 2.3.1, the mode shapes and vibration frequencies of a uniform wind turbine blade with the structural properties shown in equations 2.87 are solved for in this section. The blade properties are substituted in the coefficient matrix to solve for the roots of the characteristic equation. The coupled bending-torsional vibration frequencies are tabulated in Table 2.1. The form of the general solution of the blade coupled bending-torsional vibrations can be found in equations 2.74. The relative values of the constants A1 , A2 , A3 , A4 , B5 and B6 can be found after substituting the values of the vibration frequencies in the coefficient matrix A and solving for the vector x in Eq. 2.75. Ax = 0

(2.75)

36

CHAPTER 2. UNIFORM BLADE STRUCTURAL DYNAMIC MODEL Table 2.1: Computed exact natural frequencies using exact solution approach Mode Exact Frequency(rad/s) 1 2 3 4

5.41696 33.9079 94.8126 185.181

The resulting coupled bending-torsional mode shapes for the wind turbine blade are presented in equations 2.88-2.95 and plotted in Fig. 2.3. W1 (y) = −0.0003855 cos(0.006039y) − 0.9996 cos(0.09396y) + cosh(0.0939561y) −0.00002836 sin(0.006039y) + 0.7339 sin(0.09396y) − 0.734 sinh(0.09395y) (2.88) W2 (y) = −0.00241481 cos(0.0378124y) − 0.997585 cos(0.235187y) + cosh(0.235045y) −0.00281346 sin(0.0378124y) + 1.01838 sin(0.235187y) − 1.01854 sinh(0.235045y) (2.89)

W3 (y) = −0.00504467 cos(0.0787436y) − 0.994955 cos(0.339522y) + cosh(0.339095y) +11.8222 sin(0.0787436y) − 1.74068 sin(0.339522y) − 1.00246 sinh(0.339095y) (2.90)

W4 (y) = −0.00680078 cos(0.105753y) − 0.993199 cos(0.393573y) + cosh(0.392906y) +0.00780018 sin(0.105753y) + 0.995424 sin(0.393573y) − 0.999214 sinh(0.392906y) (2.91)

B1 (y) = 0.00491 cos(0.006039y) − 0.002467 cos(0.09396y) − 0.002447 cosh(0.09395y) +0.0003615 sin(0.006039y) + 0.001811 sin(0.09396y) + 0.001796 sinh(0.09395y) (2.92)

B2 (y) = 0.03075 cos(0.03781y) − 0.01576 cos(0.2351y) − 0.01499 cosh(0.235y) +0.03583 sin(0.03781y) + 0.01609 sin(0.2351y) + 0.01526 sinh(0.235y)

(2.93)

B3 (y) = 0.06411 cos(0.07874y) − 0.03374 cos(0.3395y) − 0.03037 cosh(0.339y)− 150.249 sin(0.07874y) − 0.05903 sin(0.3395y) + 0.03044 sinh(0.339y)

(2.94)

B4 (y) = 0.0862 cos(0.1057y) − 0.04615 cos(0.3935y) − 0.04006 cosh(0.3929y) −0.0989 sin(0.1057y) + 0.04626 sin(0.3935y) + 0.04003 sinh(0.3929y)

(2.95)

37


First Bending Mode

−3

1

4

0.5

2

0

0

5

10 15 Blade Length (m)

20

0

x 10

0

Second Bending Mode 0.04

0

0.02 5


20

0

0

Third Bending Mode


20

5


20

Third Torsional Mode

1

0

0.5 0

5

Second Torsional Mode

1

−1 0

First Torsional Mode

−10 0

5


20

−20 0

Fourth Bending Mode 0.1

0

0 5



20

Fourth Torsional Mode

1

−1 0

5

20

−0.1 0

5


20

Figure 2.3: Coupled Bending-Torsional modes of a uniform cantilever blade.

2.4.2

Approximate Solution by Assumed Modes

In this section, the wind turbine blade in the case study is solved using the assumed modes method procedures as detailed in subsection 2.3.2. The blade properties in Eq. 2.87 are substituted into Eq. 2.86. (λ2 M + K)a = 0

(2.86)

38


where M and K are the N -dimensional inertia and stiffness matrices, the elements of which are shown in Eq. 2.82 and Eq. 2.83. ˆ mij = 0

L

µ φwi φwj − µ rx φwi φβj − µ rx φwj φβi + µ (r2 + rx2 ) φβi φβj dy (2.82) ˆ kij = 0

2.4.2.1

L

GJy2 φ0βi φ0βj + EIx2 φ00wi φ00wj dy

(2.83)

Trial functions selection

The selection of the employed trial functions determines how close the discretized system is to the original distributed system. The types of trial functions that can be used in the assumed modes method and how they affect accuracy was mentioned earlier in subsection 2.3.2. For the problem at hand, the trial functions are chosen to be the first four mode shapes for the coupled torsional and bending vibrations computed for a uniform cantilever blade with the same physical properties using the exact solution approach shown in equations 2.95. The employed trial functions in this problem are classified as comparison functions. The code lines used to solve the case study at hand by the assumed modes method is shown in Appendix E.

2.4.2.2

Results

The characteristic equation resulting from the determinant of the coefficient matrix in Eq. 2.86 is solved for the approximate vibration frequencies λ. The resulting inertia and stiffness matrices are displayed in Eq. 2.96 and Eq. 2.97 

M

   K=  

 1977.94 27.9849 40.4206 24.1582    27.9849 1960.28 −52.7732 −11.7377    =  40.4206 −52.7732 1989.9 −39.7103   24.1582 −11.7377 −39.7103 1961.65  58460.1 −20291.2 32594.5 −45482.7  −20291.2 2.29853E6 −412698. 526400.   32594.5 −412698. 1.82742E7 65.1989   −45482.7 526400. 65.1989 7.05322E7

(2.96)

(2.97)

The resulting frequencies are listed in Table 2.2. It is noted that the approximate


39

Table 2.2: Computed approximate natural frequencies using assumed modes approach Mode Exact Frequency(rad/s) Approximate Frequency(rad/s) % 1 2 3 4

5.41696 33.9079 94.8126 185.181

5.42421 34.1622 95.8411 189.708

natural frequencies are always larger than the exact ones. This can be justified in light of Rayleigh Ritz approach as follows in this paragraph. It is known that the computed frequencies from Rayleigh-Ritz represent a local minimum, a value at which Rayleigh’s quotient is stationery. This value can get approach the exact one as close as the choice of the trial function is. Thus, the approximate value can be equal to or more than the exact value. This applies to the assumed modes method as well since it is closely related to Rayleigh-Ritz method as mentioned earlier. Regarding the solution accuracy, it can be seen that the highest error is in the fourth mode frequency and is less than 1% which is still acceptable.

2.5

A Study on the Effect of the Blade Properties on Coupled Bending-Torsion Vibrations of Cantilever Uniform Blades

In this section, a study is carried to investigate the effect of the blade properties on its vibration characteristics. Some of the previous literature works documenting similar efforts are mentioned in the introduction. In this thesis, it is attempted to use dimensionless parameters instead of individual properties and study their effect on the blade response. The importance of using dimensionless parameters instead of individual blade characteristics, e.g., stiffness, is highlighted at the end of this study. A guess at which blade properties possibly affect the blade behavior would leave us with the distributed mass µ, the bending stiffness EIx2 , the torsional stiffness GJy2 , the blade length L and the cross-section mass radius of gyration r. Performing Buckingham Pi theory on these parameters would yield some significant parameters y2 like the slenderness ratio Lr , the stiffness ratio GJ , λb which is the uncoupled EIx2

CHAPTER 2. UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 2

40

4

, andλt which is the uncoupled bending mode shape spatial frequency λb = ωEIµL x2 ω 2 µL2 r2 torsional mode shape spatial frequency λt = GJy2 . The details of the dimensional analysis procedures can be found in Appendix A. Taking Dokumaci’s study as a guideline, λb is plotted on the vertical axis versus a parameter χ on the horizontal Jy2 L 2 axis in Fig. 2.4, where χ = λλbt = G is a dimensionless parameter constituted E Ix2 r from two other dimensionless parameters; the stiffness ratio and the slenderness ratio of the blade. The log scale is chosen in order to be able to deal with a wide range of the blade properties. The parameter λb includes the blade coupled bending-torsional vibration frequency ω. The blade vibration frequency is solved for by means of the exact solution technique mentioned earlier. In order to understand the physics of the blade coupled vibrations, the uncoupled bending and torsion frequencies of the blade were also solved for and plotted by means of the parameter λb in Fig. 2.4. Therefore, the vibration frequencies in the λb lines in the figure are not only for coupled blade bending-torsion vibrations, but also for their uncoupled bending and torsion counterparts occurring in supposedly symmetric blades in which the shear center offset rx is set to zero. The uncoupled λb lines are intended to help comprehend the differences between coupled and uncoupled vibrations and how they vary with changing the blade dimensionless parameter χ including the slenderness and the stiffness ratios. Starting with the uncoupled λb lines in Fig. 2.4, it is found that the uncoupled bending λb lines for the first four uncoupled bending modes are zero-slope straight lines at different values of λb ascensdingly according to the mode order. The uncoupled torsional modes lines, on the other hand, are non-zero-slope straight lines. The slopes of the uncoupled torsional λb lines increase with the mode order. Moving on to coupled bending-vibrations λb lines, it is found that the lines have no constant trends for different values of χ. On the contrary, the coupled lines keep varying from constant-slope to zero-slope lines that seem to take the uncoupled mode lines as an asymptote or a guideline. It is important to add the reminder that the coupled bending-torsion vibration lines are neither pure bending nor pure torsional vibrations, but rather coupled. Taking a closer look at the coupled modes λb - χ plot, it is noticed that the intersection between the uncoupled torsional and bending λb always marks the change in the bahavior of the coupled λb lines. It is found that the first coupled mode line always sticks to the lowest uncoupled λb for a given value of χ. For example, the first uncoupled torsion mode λb line has the lowest uncoupled λb value for χ less than approximately 5 before λb for the first

λb

6

10

−2

10

0

10

2

10

4

10

0

10

3

for blades with different χ =

2

µ ω 2 L4 E Ix2

Figure 2.4: The parameter λb =

10 10 χ = (GJy2/EIx2)(L/r)2

Third Coupled Mode

1

10

First Coupled Mode

Second Coupled Mode

Fourth Coupled Mode

Fourth Torsional Third Mode Torsional Second Mode Torsional Mode

λb λt

=

4

10

G Jy2 E Ix2

L 2 . r

5

10

First Bending Mode

Second Bending Mode

Third Bending Mode

First Torsional Fourth Bending Mode Mode

CHAPTER 2. UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 41


42

bending mode becomes the lowest value. Thus, the first coupled mode is asymptotic to the first uncoupled torsional mode for χ less than 5 and then becomes asymptotic to the first uncoupled bending mode. The second coupled mode is asymptotic to the second lowest uncoupled λb , which makes it asymptotic to the first bending mode λb at χ less than 5 and to the first torsional mode after χ exceeds 5. The first torsional mode λb stays the second lowest value till χ ≈ 300 then the second uncoupled bending mode λb becomes the second lowest λb drawing the second coupled mode vibration frequency closer to the second uncoupled bending frequency instead of the first torsional mode frequency at values of χ more than 300. Hence, it is concluded that for the nth mode, the curve representing the nth coupled mode is asymptotic to the uncoupled mode of the nth lowest λb for a given value of χ. The plot covers a wide range of χ starting at χ less than one till 100000, which makes the results applicable to a wide range of applications employing the beam model. The results detailed earlier can lead to important conclusions about the nature of blade uncoupled and coupled vibrations. Starting with uncoupled bending vibrations represented by the uncoupled bending λb lines drawn as zero-slope lines, it is noticed that the λb value for uncoupled bending vibrations stays the same regardless of the blade slenderness and stiffness ratios. Fixing the product µL4 for a certain blade, it is found that the bending vibrations frequency increases parabolically with increasing the bending stiffness EIx2 , resulting in the zero-slope line. Taking the other parameters in λb into consideration, it is found that the blade bending vibration frequency is also directly proportional with the blade slenderness ratio and inversely proportional with the distributed mass µ. Fixing the bending stiffness EIx2 of the blade and changing the torsional stiffness GJy2 would not alter the uncoupled bending frequencies, hence the zero-slope line is also drawn. This can be understood in light of the problem physics discussed in the previous chapters because the offset is set to zero in case of uncoupled bending and torsional vibrations, thus decoupling the partial differential equations. The same applies to the uncoupled torsional vibrations frequencies which are represented by the constant non-zero-slope lines. Fixing µL4 and varying the torsional stiffness GJy2 results in a constant-slope the product EI x2 straight line that reveals a parabolic relationship between the uncoupled torsional frequency and stiffness. Moving to coupled vibrations of blades, it was mentioned earlier that the coupled λb line of the nth mode follows no certain trend or slope but is asymptotic to the uncoupled λb line with the nth lowest value for a certain value of χ. Apparently, this


43

predicts the frequency of the coupled fundamental vibration mode and higher order modes as well. It implies that the coupled vibration frequencies of a certain blade stray not very far away from the uncoupled frequencies of a blade with the same structural parameters and that the lowest uncoupled frequency at a certain value of χ is the closest to the coupled frequency value, which is pretty logical in light of known physics rules. Here, it is important to add a remark about the coupled nature of the blade vibrations that the two types of vibrations, bending and torsion, occur in the same mode at the same frequency, regardless of the value of this frequency and where its corresponding λb lies on the plot. However, its not contradictory to the aforementioned fact to expect the dominance of a certain type of vibrations in a given mode shape. The dominant type of vibrations in a certain coupled mode can be determined by the magnitude of its frequency, and whether its closer to the uncoupled torsional or bending vibration frequency. Taking the first coupled mode for example, the bending vibrations in the first coupled mode are dominant after χ = 5, which marks the intersection of the uncoupled torsional and bending vibration lines at which the coupled λb begins to be asymptotic to the uncoupled bending line instead of the uncoupled torsion one. Before χ = 5, the first coupled mode was torsional dominant. Analyzing this, one can draw the conclusion that the uncoupled vibration type with the nth lowest uncoupled λb for a given χ is the dominant type of vibration in the nth coupled mode. In light of the previous discussion, one can draw a link between coupled and uncoupled bending and torsional vibrations to comprehend the physical principals upon which coupled vibrations can be understood. It seems that coupled vibrations follow the most energy-effective route that can be expected based on their uncoupled counterparts regarding the vibration frequency and dominance. As important as this is, though seemingly simple and expected, the true significance of this plot truly lies in the dimensionless parameters utilized in this study. As mentioned previously, the fact that the blade individual material and geometric properties are combined together in a few dimensionless parameters enables one to predict the blade behavior based on the collective effect of these parameters. It is also important to point out that since the parameters in the study are dimensionless, this study is generally applicable to any uniform blade.


2.6

44

Extending the Governing Equations to Rotating Blades

Earlier in this chapter, the structural model for a uniform cantilever blade in coupled bending-torsional vibrations was developed based on the Euler-Bernoulli beam model. The governing partial differential equations for a blade in coupled bendingtorsion vibrations were derived using Hamilton’s principal. The PDEs were solved using exact analytical techniques and the assumed modes discretization method. A study was then conducted utilizing the blade model to investigate the effect of the blade material and geometric properties on the coupled bending-torsional vibrations. In the following sections, the model is extended to include the rotational motion in the case of a rotating uniform wind turbine blade. The same procedures are followed to derive the governing differential equations. In the end of the chapter, a similar parametric study is conducted to find the effect of the rotating uniform blade structural parameters on the coupled vibration frequencies, in addition to the effect of the rotational motion on the blade frequencies.

2.6.1

Potential energy

Proceeding from the kinematic analysis in Sec. 2.1, it is found that in the case of a rotating blade, the potential energy originates from the elastic deformations and earth gravity. Elastic potential energy remains the same as in the case of the non-rotating blade. 1 1 (2.98) Ve = EIx2 w002 + GJy2 β 02 2 2 Assuming the origin point lies at the nacelle of the wind turbine, the gravitational potential can be calculated by Vg = µh.g (2.99) where g is the gravity vector [0 g 0]T and h is the position of a certain point P on the blade relative to the inertial fixed frame. Since this work deals with linear deflections, the changes in the gravitational potential energy due to deflections are ignored and only those changes due to the rotational motion are included. Multiplying with the dot product in Eq. 2.99, the gravitational energy is expressed as shown in Eq. 2.100


45

Vg = g y ρ cos(Ω t) − g x ρ sin(Ω t) − g z ρ sin(Ω t) βy, t − g z ρ cos(Ω t) w0 (y, t) (2.100) Performing the area integrations with respect to the centroidal axes, the gravitational potential energy can be finally written as Vg = g y µ cos(Ω t) − g rx µ sin(Ω t)

(2.101)

where the angle Ω t is assumed to be zero when the blade is in the upward vertical position. The gravitational potential energy will be ignored when deriving the governing PDEs as it does not include any of the generalized coordinates so the variation of the gravitational potential energy is taken to be zero.

2.6.2

Kinetic Energy

Using the inertial velocity vector derived in Eq. 2.9 and following the same procedures in the non-rotating blade case, the kinetic energy per unit length for the rotating blade is

1 = Ω2 2

T = T0 + T1 + T2 1 ˙ 0 − 2Ix2m β w˙ 0 + Jz1m + Ix2m β 2 + Ix2m w02 + Ω 2Ix2m βw 2 1 2 µw˙ − 2rx µw˙ β˙ + Jy2m β˙ 2 + Ix2m w˙ 02 (2.102) 2

where

1 T0 = Ω2 Jz1m + Ix2m β 2 + Ix2m w02 2 and is called the centrifugal kinetic energy, 1 ˙ 0 − 2Ix2m β w˙ 0 T1 = Ω 2Ix2m βw 2 which is the term that gives rise to the gyroscopic forces, and T2 =

1 2 µw˙ − 2rx µw˙ β˙ + Jy2m β˙ 2 + Ix2m w˙ 02 2

which is the kinetic energy term that arises due to the beam vibrations without any influence of the rotational motion.


2.6.3

46

Governing Partial Differential Equations by Hamilton’s principal

Taking the variations of equations 2.102, 2.98 and 2.101, it is found that the gravitational potential energy in Eq. 2.101 does not contribute to the governing equations. This is because the blade degrees of freedom w(y, t) and β(y, t) are not present in the gravitational potential energy term. Substituting the variations in Hamilton’s equation yields the governing equations. The code utilized to derive the equations of motion for the rotating blade is shown in Appendix C. −µw¨ + rx µ β¨ − 2Ix2m Ωβ˙ 0 − Ix2m Ω2 w00 + Ix2m w¨ 00 − EIx2 w0000 = 0

(2.103)

Ix2m Ω2 β + rx µw¨ − Jy2m β¨ − 2Ix2m Ωw˙ 0 + GJy2 β 00 = 0

(2.104)

Ignoring the rotary inertia terms, the governing equations are simplified to −µw¨ + rx µ β¨ − 2Ix2m Ωβ˙ 0 − Ix2m Ω2 w00 − EIx2 w0000 = 0 Ix2m Ω2 β + rx µw¨ − Jy2m β¨ − 2Ix2m Ωw˙ 0 + GJy2 β 00 = 0

2.7

(2.105)

Spatial Reduction of the Governing Equations

In this section, it is attempted to solve the governing PDEs of the uniform rotating wind turbine blade for the vibration characteristics of the blade. First, the exact solution is approached, which proves unfit for the case of the rotating blade. Then, the assumed modes discretization technique is used.

2.7.1

Exact Solution

The separation of variables technique is used to convert the PDEs in Eq. 2.105 to ODEs. w(y, t) = W (y) eλ t

β(y, t) = B(y) eλ t

(2.106)

47


ˆ e Ls y and Be ˆ Ls y , respectively, Dividing by eλt and substituting W (y) and B(y) by W 4 2 2 2 2 2I s λ Ω El I s I s Ω Ix2m s Ω x2m x2 x2m 2 2 ˆ rx µ λ − ˆ − B +W + − µλ − L L4 L2 L2 = 0 G Jy2 s2 2Ix2m s λ Ω 2 2 2 2 2 ˆ ˆ ¯ +B = 0 W rx µ λ − − Jy2m λ − rx µ λ + Ix2m Ω L L2 (2.107) The system of algebraic equations in Eq. 2.107 can be rewritten in matrix form as 4

2 2 Ix2m s2 λ2 − µλ2 − Ix2mLs2 Ω L2 rx µ λ2 − 2Ix2mL s λΩ

− EILx24 s +

G Jy2 s2 L2

rx µ λ2 −

− µr2 λ2 −

=0

2Ix2m s λ Ω L 2 2 µ rx λ +

! Ix2m Ω2 (2.108)

Thus, the characteristic equation will be the determinant of the coefficient matrix in Eq. 2.108 E Ix2 G Jy2 E Ix2 , µ(r2 + rx2 )λ2 Ix2m G Jy2 λ2 4 s − + − + s L6 L4 L4 2 (E Ix2 + G Jy2 ) Ix2m Ω2 3Ix2m λ2 Ω2 −Ix2m µ(r2 + rx2 ) λ4 G Jy2 λ2 µ 2 − − + s L4 L2 L2 L2 2 Ω4 Ix2m Ix2m µ(r2 + rx2 )λ2 Ω2 4 Ix2m rx λ3 µ Ω − + − s L2 L2 L 6

− Ix2m λ2 µ Ω2 + µ2 r2 λ4 µ = 0

(2.109)

Equation 2.109 is a sixth order polynomial as in the case of the cantilever blade characteristic equation. There is a difference between the two cases, however, which is the s term in Eq. 2.109. Due to that term, the polynomial is not even-powered anymore and therefore, it cannot be converted to a third order equation that can be solved by Cardan’s formula as in the case of the cantilever blade case. Hence, only approximate numerical techniques like the assumed modes discretization method are used to solve the rotating blade problem.

2.7.2

Assumed Modes Method

In the previous section, the characteristic equation of the uniform blade undergoing coupled uni-axial bending and torsional vibrations problem from the determinant

ˆ W ˆ B

!


48

of the boundary conditions matrix was derived and it was proved that it cannot be solved analytically. In this section, the rotating blade problem is discretized using the assumed modes method. As it was mentioned earlier, the assumed modes method is a spatial discretization technique that is closely related to the RayleighRitz method. The continuous spatial displacements were approximated as the linear combination of a set of N trial functions. Though the system involves rotational motion, it is still conservative, which makes the assumed modes method a suitable technique of discretization. As in the case of the cantilever blade, our objective is to produce a finite degree of freedom system that best approximates a continuous rotating wind turbine blade. The problem is to model the coupled bending-torsional vibrations of a uniform homogeneous rotating wind turbine blade with asymmetric cross-section. Due to the slender geometry of the wind turbine blade, Euler-Bernoulli beam model is used to model the blade. It is worth notice that the frequencies in the case of a rotating blade should be higher than those in the case of the cantilever blade. This is due to the stiffening of the blade caused by the centrifugal force effect which is apparent in the added stiffness terms due to the Ω2 contribution. The vibrational displacements are expressed as the sum of N functions in space and time w(y, t) =

N X

φwi (y) qi (t)

β(y, t) =

i=1

N X

φβi (y) qi (t)

(2.110)

i=1

where φwi (y) and φβi (y) are the trial functions used to approximate the spatial mode shapes, N is the number of employed trial functions and i is a counter that takes values from i to N , qi (t) is the generalized coordinate used to describe the time response of the blade vibrations. It is expressed as qi (t) = ai eλi t

(2.111)

where λi is the time frequency of vibration and ai determines the contribution of the trial functions i to the total solution. It should be noted that though φwi (y) and φβi (y) are the trial functions used to represent the bending and torsional vibrations, yet they are multiplied by the same generalized coordinate qi (t) due to the coupling effect which makes them both effectively a single coupled bending-torsional mode. In the following section, the assumed modes discretization is applied to the case of a rotating blade.


2.7.3

49

Uniform Rotating Blade Case Assumed Modes Discretization

Starting with recalling the energy expressions for the case of a uniform rotating blade, it was previously mentioned that the potential energy for a rotating blade originates from the elastic deflections and gravity, as mentioned before. The elastic potential energy can be written as 1 1 Ve = EIx2 w002 + GJy2 β 02 2 2

(3.1)

Assuming the origin point lies at the nacelle of the wind turbine, the gravitational potential can be calculated by Vg = µh.g where g is the gravity vector [0 g 0]T and h is the distance between a certain point P on the blade and the turbine nacelle. Vg = g y µ cos(Ω t) − g rx µ sin(Ω t)

(3.3)

The gravitational potential energy will be ignored when deriving the modal equations as it does not include any of the generalized coordinates. Therefore, substituting the gravitational potential energy in Lagrange’s equations yields nothing. Also, the kinetic energy can be written as 1 1 ˙ 0 − 2Ix2m β w˙ 0 + T = Ω2 Jz1m + Ix2m β 2 + Ix2m w02 + Ω 2Ix2m βw 2 2 1 2 µw˙ − 2rx µw˙ β˙ + Jy2m β˙ 2 + Ix2m w˙ 02 2

(3.4)

Substituting the discretized dependent variables in the energy expressions, the potential energy can be rewritten as Ve =

N X N X 1 i=1 j=1

2

EIx2 φ00wi qi

φ00wj

1 qj + GJy2 φ0βi qi φ0βj qj 2

(2.112)

50

CHAPTER 2. UNIFORM BLADE STRUCTURAL DYNAMIC MODEL and the kinetic energy can be rewritten as T =

N X N X 1

2

i=1 j=1

Ω2 µ ro2 + Ix2m φβi qi φβj qj + Ix2m φ0wi qi φ0wj qj

1 + Ω 2 Ix2m φβi q˙i φ0wj qj − 2 Ix2m φβi qi φ0wj q˙j 2 1 +µ φwi q˙i φwj q˙j − 2 rx µ φwi q˙i φβj q˙j + µ(r2 + rx2 ) φβi q˙i φβj q˙j 2 1 + Ix2m φ0w i qi φ0w j qj 2

(2.113)

Substituting the discretized energy expressions in Lagrange’s equations d dt

δ Tl δ q˙k

− :


(2.114) (2.80)

where Vl and Tl are the potential and kinetic energies integrated along the beam length. Equation 2.80 can be used to write N equations of motion. The governing modal equations can be written as Mq ¨ + Gq˙ + Kq = 0

(2.115)

where M is an N by N inertia matrix constructed from the coefficients of q¨i in which each element can be expressed as ˆ

L

mij = 0

µ φwi φwj − µ rx φwi φβj − µ rx φwj φβi + µ (r2 + rx2 ) φβi φβj dy

(2.116) G is an N by N gyroscopic matrix constructed from the coefficients of q˙i in which element can be written as ˆ gij = 0

L

Ix2m Ω φ0wi φβj − Ix2m Ω φ0wj φβi dy

(2.117)

and K is an N by N stiffness matrix constructed from the coefficients of qi in which each element can be expressed as ˆ kij = 0

L

Ix2m Ω2 (φβi φβj + φ0wi φ0wj ) + GJy2 φ0βi φ0βj + EIx2 φ00wi φ00wj dy (2.118)

51

CHAPTER 2. UNIFORM BLADE STRUCTURAL DYNAMIC MODEL Substituting the general solution for the generalized coordinates qi (t) = ai eλi t

(2.85)

Equation 2.85 changes the differential equation 2.115 to the algebraic equation (λ2 M + λG + K)a = 0

(2.119)

The matrix (λ2 M + λG + K) cannot be inverted in order to have a non-trivial solution, i.e., it has to be singular which means it is rank deficient. Thus, the values of λ can be solved for by finding the roots of the determinant of (λ2 M + λG + K), which constitutes the characteristic equation of the assumed modes discretized rotating blade problem. The mode shapes can be found by solving the underdetermined system of equations for each value of λ.

2.8

Rotating Blade Case Study

In this section, the rotating uniform blade problem is discretized by the assumed modes method. The rotational speed is assumed to be uniform. The blade properties are the integrated average of the structural properties of the CART wind turbine blade [31]. µ rx GJy2 EIx2 r Ix2m L

2.8.1

= = = = = = =

102.097 0.09 1.09E7 3.844E7 0.363152 2.7847 19.955

kg/m m N.m2 N.m2 m kg.m m

(2.120)

Assumed Modes Approximate Solution

The blade properties are substituted in the eigen-value problem in Eq. 2.119. The inertia, gyroscopic and stiffness matrices elements are calculated as shown in equations 2.116, 2.117 and 2.118.

CHAPTER 2. UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 2.8.1.1

52


To approximate the modes of a rotating uniform blade, the exact mode shapes of a cantilever blade with the same properties were used as trial functions. The mode shapes were derived earlier in this chapter and are shown in equations 2.88-2.95.

2.8.1.2

Results

Taking an angular speed of 3 rad/sec for example, the inertia, gyroscopic and stiffness matrices are shown as follows   1977.94 27.9849 40.4206 24.1582    27.9849 1960.28 −52.7732 −11.7377   M = (2.121)  40.4206 −52.7732 1989.9 −39.7103    24.1582 −11.7377 −39.7103 1961.65   0. 1.08938 −1.57943 −0.304428    −1.08938  0. 5.14569 7.88767  G= (2.122)  1.57943 −5.14569  0. 0.41833   0.304428 −7.88767 −0.41833 0.   58466. −20300.6 32599.7 −45491.2    −20300.6 2.29857E6 −412730.  526419.  K= (2.123)  32599.7 −412730. 1.82743E7 28.1941    −45491.2 526419. 28.1941 7.05323E7 The characteristic equation obtained from the determinant of the algebraic eigenvalue problem in Eq. 2.119 yields the coupled vibration frequencies. The resulting frequencies are listed in Table 2.3. It is noted that the approximate natural frequencies computed by assumed modes method for the rotating blade are always higher than those for the cantilever blade, though the effect is not very pronounced. This is due to the stiffening effect discussed earlier, which is induced by the centrifugal forces.


53

Table 2.3: Computed approximate natural frequencies for a rotating blade using assumed modes approach Mode

Fixed Blade Frequency rad/s

Rotating Blade Frequency rad/s

Rotating Speed rad/s

5.42421

5.42424 5.42433 5.42448

1 2 3

2

34.1622

34.1622 34.1623 34.1625

1 2 3

3

95.8411

95.8411 95.8412 95.8414

1 2 3

189.708

189.708 189.708 189.708

1 2 3

1

4

2.9

A Study on the Effect of the Blade Properties on Coupled Bending-Torsion Vibrations of Rotating Uniform Blades

In Sec. 2.5, the geometric and material properties affecting the vibration characteristics of a uniform cantilever blade were investigated and represented by two 4 ω2 y2 L 2 dimensionless parameters λb = µL and χ = GJ ( ) . In the problem of the EIx2 EIx2 r rotating blade, the same blade properties affect the vibration characteristics, besides the rotational motion. The combined effects of the blade properties and the rotational motion on the vibrations characteristics are studied in this section. To understand the influence of the rotational motion, one has to go back to the blade model. It is found that the contributing potential energy terms to the governing equations are the same as the cantilever blade case because the gravitational energy does not include any of the degrees of freedom in question, as mentioned earlier. The kinetic energy term is what distinguishes the rotating blade, however. Taking a closer look at the kinetic energy explression in Eq. 2.102, which is also written below for convenience, it is found that the rotational motion contributes to


54

the kinetic energy by two terms, which are coefficients of Ω and Ω2 .

1 = Ω2 2

T = T0 + T1 + T2 1 2 02 0 0 ˙ Jz1m + Ix2m β + Ix2m w + Ω 2Ix2m βw − 2Ix2m β w˙ + 2 1 2 2 02 ˙ ˙ µw˙ − 2rx µw˙ β + Jy2m β + Ix2m w˙ (2.102) 2

The product Ix2m Ω2 is a common factor in the effective terms of the coefficient of Ω2 in the kinetic energy 3 , while the product Ix2m Ω is a common factor in the coefficients of Ω. Both of these products represent the rotational motion, as well as the blade properties related to the rotational motion, Ix2m in that case. This means that the significance of the rotational motion effect on the blade vibration characteristics is dependent on the mass moment of inertia of the blade cross-section about the airfoil chord and the rotational speed. Whether Ix2m Ω2 or Ix2m Ω has the greater influence on the blade vibrations is a question left to the magnitude of their coefficients to decide. Beginning with the product Ix2m Ω2 and denoting it by f for convenience, the assumed modes discretized characteristic equation is solved for the vibrations frequencies of different blades rotating at different angular speeds but having equal values of f . It was found that rotating blades having equal values of f had the same vibrational characteristics. These results imply that the product Ix2m Ω has a negligible influence on the blade characteristics. This is consistent with the literature results as it can be seen that the kinetic energy term including that product is the term contributing to the gyroscopic forces which are negligible [17, 24]. The parameter λb was plotted on the vertical axis against χ for blades having different values of f in Fig. 2.5. For the blade in our case study with χ = 856, the rotation effect on the vibration frequencies of that blade is almost negligible at this value of χ. However, the plot can be used to know at which values of the structural properties the rotation begins to have a notable effect. The significance of the rotational effects for a certain blade is not only dependent on its χ value, but also on the rotating speed Ω and the mass moment of inertia of the blade about the x2 axis Ix2m . As f increases for a certain blade, the rotational effects are demonstrated at even smaller values of χ. The product 12 Jz1m Ω2 does not contribute to the governing equations because it does not include any generalized coordinates. 3

55


First Torsional Mode 3

10

Second Uncoupled Bending Mode

f = 25

f=0

Second Coupled Mode

2

λb

f = 200 f = 89

f = 200

10

f = 89 First Uncoupled Bending Mode

f = 25

1

10

f=0

First Coupled Mode 0

10

0

10

1

10

2

3

10

10

4

2

4

χ

10

5

10

6

10

7

10

ω Figure 2.5: The parameter λb = µL for rotating uniform blades with different EIx2 GJy2 L 2 values of χ = EIx2 ( r ) and f = Ix2m Ω2

Chapter 3 Non-Uniform Blade Structural Dynamic Model In the previous chapter, the governing equations of motion for uniform cantilever and rotating blades with asymmetric cross-sections were derived using Hamilton’s principal. Exact expressions for the natural mode shapes of coupled bending-torsional vibrations and the vibration frequencies were obtained for the case of the cantilever blade. However, the exact solution technique could not be employed with the rotating blade. Thus, the assumed modes method was used to get an approximate solution for the mode shapes and vibration natural frequencies of a rotating blade in coupled torsional-bending vibrations. In this chapter, the governing partial differential equations for a rotating non-uniform blade in coupled bending-torsional vibrations are derived.

3.1

Non-Uniform Rotating Blade Model

In this section, the energy expressions for the rotating non-uniform blade are derived so that the governing equations can be obtained by Hamilton’s principal. The procedures for deriving the governing equations are the same except for the fact that the blade properties are a function of the blade length.

3.1.1

Potential energy

Proceeding from the kinematic analysis conducted in the previous chapter in Sec. 2.1, it is found that in the case of a rotating blade, the potential energy originates from 56

CHAPTER 3. NON-UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 57 the elastic deformations and earth gravity. Elastic potential energy remains the same as in the case of the non-rotating blade. 1 1 Ve = EIx2 (y)w002 + GJy2 (y)β 02 2 2

(3.1)

Assuming the origin point lies at the nacelle of the wind turbine, the gravitational potential can be calculated by Vg = µh.g (3.2) where g is the gravity vector [0 g 0]T and h is the distance between a certain point P on the blade and the turbine nacelle. Since this work deals with linear deflections, the changes in the gravitational potential energy due to deflections are ignored and only those changes due to the rotational motion are included. Vg = g y µ(y) cos(Ω t) − g rx (y) µ(y) sin(Ω t)

(3.3)

where the angle Ω t is assumed to be zero when the blade is in the upward vertical position. The gravitational potential energy will be ignored when deriving the governing PDEs as it does not include any of the generalized coordinates so the variation of the gravitational potential energy is taken to be zero.

3.1.2

Kinetic Energy

Using the inertial velocity vector derived in Sec. 2.1 and following the same procedures in the non-rotating blade case, the kinetic energy per unit length for the rotating blade is

1 = Ω2 Jz1m 2

T = T0 + T1 + T2 1 ˙ 0 − 2Ix2m (y) β w˙ 0 + + Ix2m (y) β 2 + Ix2m (y) w02 + Ω 2Ix2m (y)βw 2 1 (3.4) µ(y) w˙ 2 − 2rx µ(y) w˙ β˙ + Jy2m (y) β˙ 2 + Ix2m (y) w˙ 02 2

where

1 T0 = Ω2 Jz1m + Ix2m (y)β 2 + Ix2m (y)w02 2 and is called the centrifugal kinetic energy, 1 0 0 ˙ T1 = Ω 2Ix2m (y)βw − 2Ix2m (y)β w˙ 2

CHAPTER 3. NON-UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 58 which is the term that gives rise to the gyroscopic forces, and 1 2 2 02 ˙ ˙ T2 = µ(y)w˙ − 2rx (y)µ(y)w˙ β + Jy2m (y)β + Ix2m (y)w˙ 2 which is the kinetic energy term that arises due to the blade vibrations alone without any influence of the rotational motion.

3.1.3

Governing Partial Differential Equations by Hamilton’s principal

Hamilton’s equation yields the governing equations. −µ(y)w¨ + rx (y) µ(y) β¨ − 2Ix2m (y)Ωβ˙ 0 − Ix2m (y)Ω2 w00 + Ix2m (y)w¨ 00 − EIx2 (y) w0000 = 0 (3.5) Ix2m (y)Ω2 β + rx (y) µ(y)w¨ − Jy2m (y)β¨ − 2Ix2m (y)Ωw˙ 0 + GJy2 (y)β 00 = 0

(3.6)

Ignoring the frotary inertia terms, the governing equations are simplified to −µ(y)w¨ + rx (y) µ(y) β¨ − 2Ix2m (y)Ωβ˙ 0 − Ix2m (y)Ω2 w00 −EIx2 (y) w0000 = 0

Ix2m (y)Ω2 β + rx (y) µ(y)w¨ − Jy2m (y)β¨ − 2Ix2m (y)Ωw˙ 0 + GJy2 (y)β 00 = 0

(3.7) (3.8)

In the previous chapter, the characteristic equation of the uniform blade undergoing coupled uni-axial bending and torsional vibrations problem was derived from the determinant of the boundary conditions matrix and it was proved that it cannot be solved analytically. The same applies to the case of non-uniform rotating blades. Therefore, approximate techniques are employed to solve for the vibration characteristics of the non-uniform rotating blade. In the next section, the rotating blade problem is discretized using the assumed modes method.

CHAPTER 3. NON-UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 59

3.2

Assumed Modes Method

The vibrational displacements are expressed as the sum of N functions in space and time w(y, t) = β(y, t) =

N X i=1 N X

φwi (y) qi (t) φβi (y) qi (t)

(3.9)

i=1

where φwi (y) and φβi (y) are the trial functions used to approximate the spatial mode shapes, N is the number of employed trial functions and i is a counter that takes values from i to N , and qi (t) is the generalized coordinate used to describe the time response of the blade vibrations. It is expressed as qi (t) = ai eλi t

(3.10)

where λi is the time frequency of vibration and ai determines the contribution of the trial functions i to the total solution. It should be noted that though φwi (y) and φβi (y) are the trial functions used to represent the bending and torsional vibrations, yet they are multiplied by the same generalized coordinate qi (t) due to the coupling effect which makes them both effectively a single coupled bending-torsional mode. In the following section, the assumed modes discretization is applied to the case of a rotating blade.

3.2.1

Rotating Non-Uniform blade Case Assumed Modes Discretization

Substituting the discretized dependent variables in the energy expressions, the potential energy can be rewritten as Ve =

N X N X 1 i=1 j=1

2

EIx2 (y) φ00wi qi

φ00wj

1 qj + GJy2 (y) φ0βi qi φ0βj qj 2

(3.11)

CHAPTER 3. NON-UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 60 and the kinetic energy can be rewritten as T =

N X N X 1 i=1 j=1

2

Ω2 µ(y) ro2 + Ix2m (y)φβi qi φβj qj + Ix2m (y)φ0wi qi φ0wj qj

1 + Ω 2 Ix2m (y) φβi q˙i φ0wj qj − 2 Ix2m (y) φβi qi φ0wj q˙j 2 1 +µ(y) φwi q˙i φwj q˙j − 2 rx (y) µ(y) φwi q˙i φβj q˙j + µ(y)(r(y)2 + rx (y)2 ) φβi q˙i φβj q˙j 2 1 + Ix2m (y)φ0w i qi φ0w j qj (3.12) 2 Substituting the discretized energy expressions in Lagrange’s equations d dt

δ Tl δ q˙k

−


(3.13)

where Vl and Tl are the potential and kinetic energies integrated along the beam length. Equation 3.13 can be used to write N equations of motion. The governing equations can be written as Mq ¨ + Gq˙ + Kq = 0

(3.14)

where M is an N by N inertia matrix constructed from the coefficients of q¨i in which each element can be expressed as ˆ mij = 0

L

µ(y) φwi φwj − µ(y) rx (y) φwi φβj − µ(y) rx (y) φwj φβi + µ(y) (r(y)2 + rx (y)2 ) φβi φβj dy (3.15)

G is an N by N gyroscopic matrix constructed from the coefficients of q˙i in which element can be written as ˆ gij = 0

L

Ix2m (y) Ω φ0wi φβj − Ix2m (y) Ω φ0wj φβi dy

(3.16)

CHAPTER 3. NON-UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 61 and K is an N by N stiffness matrix constructed from the coefficients of qi in which each element can be expressed as ˆ kij = 0

L

Ix2m (y) Ω2 (φβi φβj + φ0wi φ0wj ) + GJy2 (y) φ0βi φ0βj + EIx2 (y) φ00wi φ00wj dy (3.17)

Substituting the general solution for the generalized coordinates qi (t) = ai eλi t

(2.85)

Equation 3.14 changes the differential equation to the algebraic equation (λ2 M + λG + K)a = 0

(3.18)

The values of λ can be solved for by finding the roots of the determinant of (λ2 M + λG + K), which constitutes the characteristic equation of the rotating non-uniform blade problem. The mode shapes can be found by solving the under-determined system of equations for each value of λ.

3.3

Controls Advanced Research Turbine (CART) Blade Case Study

In this section, the assumed modes discretization method is applied to the rotating non-uniform blade problem. The blade in this study is the CART wind turbine blade [31]. The distribution of the blade physical properties is shown in Appendix F. The properties distributions along the blade length are also plotted in Appendix F.

3.3.1


The exact mode shapes for a uniform cantilever blade whose properties are the integrated average of the fitted functions were used as trial functions. The results showed acceptable accuracy level when used with the case of the uniform rotating blade, which makes the employed trial functions eligible for use in the case of the

CHAPTER 3. NON-UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 62 non-uniform rotating blade. For example, ´L EIx2 =

0

EIx2 (y)dy = L

´L 0

(1.719E(+8)e−0.2214 y )dy = 3.84395E7N.m2 19.955

(3.19)

CHAPTER 3. NON-UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 63 Thus, the employed trial functions for the bending displacements are W1 (y) = −0.0003855 cos(0.006039y) − 0.9996 cos(0.09396y) + cosh(0.0939561y) −0.00002836 sin(0.006039y) + 0.7339 sin(0.09396y) − 0.734 sinh(0.09395y) (3.20) W2 (y) = −0.00241481 cos(0.0378124y) − 0.997585 cos(0.235187y) + cosh(0.235045y) −0.00281346 sin(0.0378124y) + 1.01838 sin(0.235187y) − 1.01854 sinh(0.235045y) (3.21)

W3 (y) = −0.00504467 cos(0.0787436y) − 0.994955 cos(0.339522y) + cosh(0.339095y) +11.8222 sin(0.0787436y) − 1.74068 sin(0.339522y) − 1.00246 sinh(0.339095y) (3.22)

W4 (y) = −0.00680078 cos(0.105753y) − 0.993199 cos(0.393573y) + cosh(0.392906y) +0.00780018 sin(0.105753y) + 0.995424 sin(0.393573y) − 0.999214 sinh(0.392906y) (3.23)

B1 (y) = 0.00491 cos(0.006039y) − 0.002467 cos(0.09396y) − 0.002447 cosh(0.09395y) +0.0003615 sin(0.006039y) + 0.001811 sin(0.09396y) + 0.001796 sinh(0.09395y) (3.24)

B2 (y) = 0.03075 cos(0.03781y) − 0.01576 cos(0.2351y) − 0.01499 cosh(0.235y) +0.03583 sin(0.03781y) + 0.01609 sin(0.2351y) + 0.01526 sinh(0.235y)

(3.25)

B3 (y) = 0.06411 cos(0.07874y) − 0.03374 cos(0.3395y) − 0.03037 cosh(0.339y)− 150.249 sin(0.07874y) − 0.05903 sin(0.3395y) + 0.03044 sinh(0.339y)

(3.26)

B4 (y) = 0.0862 cos(0.1057y) − 0.04615 cos(0.3935y) − 0.04006 cosh(0.3929y) −0.0989 sin(0.1057y) + 0.04626 sin(0.3935y) + 0.04003 sinh(0.3929y)

(3.27)


3.3.2

Results

The characteristic equation obtained from the determinant of the algebraic eigenvalue problem in Eq. 3.18 yields the coupled vibration frequencies. Taking the case of a rotating blade at a speed of 1 rad/sec for example, the inertia, gyroscopic and stiffness matrices were found to be   603.883 563.084 195.178 62.2911    563.084 1513.13 1258.98 897.879    M = (3.28)  195.178 1258.98 1712.43 1628.33   62.2911 897.879 1628.33 1829.38

   G=  

   K=  

 0. −0.13397 −1.004 −0.0002  0.13397 0. −0.88702 −0.5684   1.00395 0.88702 0. −0.31526   0.0002 0.5684 0.31526 0.

128092. 356645. 464060. 475022. 356645. 2.90968E6 4.68465E6 5.13103E6 464060. 4.68465E6 1.18873E7 1.42011E7 475022. 5.13103E6 1.42011E7 1.92374E7

(3.29)

     

(3.30)

Solving the characteristic equation for the eigen frequencies, the results are listed in Table 3.1. It is noted that the approximate natural frequencies computed by the assumed modes method for the rotating blade are always higher than those for the cantilever blade due to the stiffening effect discussed earlier. To verify the results in this work, they are compared to the results measured in the modal survey of the CART blade, as well as the ADAMS model results [31]. The first mode frequency computed in this work is 2.07102 Hz, while the first mode frequencies in the modal survey and the ADAMS model are both 2.06 Hz, which shows reasonable accuracy. The code lines used to discretize the non-uniform rotating blade problem by the assumed modes method is shown in Appendix E.

CHAPTER 3. NON-UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 65 Table 3.1: Computed approximate natural frequencies for a rotating blade using assumed modes approach Mode

1

3.4

Rotating Blade Frequency Hz

Rotating Speed rad/s

ADAMS model

Modal survey

2.07102 2.07102 2.07104 2.07105

0 1 2 3

2.06 -

2.06 -

A Study on the Effect of the Blade Properties on Coupled Bending-Torsion Vibrations of Rotating Non-Uniform Blades

In this section, the equations derived earlier are used to perform a parametric study for the rotating CART wind turbine blade. The parametric study includes the effect of the rotational motion and the blade structural parameters on the vibration characteristics. By performing dimensional analysis on all factors, two dimensionless 4 ω2 , where λb is a dimensionless numbers are obtained. One of them is λb = µL EIx2 parameter including the blade vibration frequency, the bending stiffness, mass per unit length and the blade length. Other important dimensionless quantities are the y2 slenderness ratio Lr and the blade stiffness ratio GJ . The dimensionless parameter EIx2 GJy2 L 2 is χ = EIx2 ( r ) , is the product of two other dimensionless quantities: the inverse of the blade slenderness ratio and the torsional-to-bending stiffness ratio. As the blade vibration frequency ω is taken to be the dependent variable ω and is included in λb , the dimensionless parameter λb is plotted on the vertical axis versus χ as shown in Fig. 3.1 and Fig. 3.2. The plots in Fig. 3.1 are for uniform and non-uniform blades together. For the case of the non-uniform blade, the fitted equations of the blade properties were averaged and then substituted in the dimensionless parameters. The plots are for the non-uniform blade are shown in Fig. 3.2, Fig. 3.3 and Fig. 3.4. In the plots, the uncoupled bending modes are zero-slope λb lines while the uncoupled torsion modes are represented as nonzero-slope λb lines. The intersection between the uncoupled torsional and bending λb lines always marks the change in the behavior of the coupled λb lines as they change from bending dominant to torsion dominant vibrations and vice versa, e.g., the first coupled mode line is asymptotic to the lowest uncoupled λb line for a certain value of χ. Generally, the lines representing

CHAPTER 3. NON-UNIFORM BLADE STRUCTURAL DYNAMIC MODEL 66 the nth coupled mode are asymptotic to the uncoupled mode of the nth lowest λb , for a given value of χ. It should be noted that the blade vibration characteristics is affected by the nature of the blade properties distribution as can be seen in the difference between the uniform and non-uniform blade plots in Fig. 3.1. It is found that the non-uniform and uniform plots are different in two aspects: First, the magnitude of λb lines is higher for the non-uniform blade; Second, the values of χ at which the transition between the zero-slope lines to the constant-slope lines takes place is different. The last parameter to be examined in this study is the rotational motion. Having a closer look at equations 3.15, 3.16 and 3.17, it is found that there are two added terms due to the rotational motion, one is a coefficient of Ω2 and the other one is a coefficient of Ω, which is the gyroscopic forces term. As discussed earlier in the previous chapter in section 2.9, the contribution of the Ω term has a negligible value. Thus, moving on to the Ω2 term, the parameter f = Ix2m Ω2 is found, where Ix2m is the mass moment of inertia of the blade about the x2 axis that extends along the airfoil chord. It accounts for both the rotational motion of the blade and the geometry of the blade cross-section. In the plots in Fig. 3.2, Fig. 3.3 and Fig. 3.4, the vibration frequencies of rotating blades increase with increasing the rotational speed and Ix2m for both uniform and non-uniform blades. The plot can be used to know at which values of the structural properties the rotation begins to have a notable effect. The significance of the rotational effects for a certain blade is not only dependent on its χ value, but also on the rotating speed Ω and the mass moment of inertia of the blade about the x2 axis Ix2m . As f increases for a certain blade, the rotational effects are demonstrated at even smaller values of χ.


First uniform blade uncoupled torsional mode Second uniform blade uncoupled bending mode 3

10

Second nonuniform blade coupled mode f = 200 f = 89 f = 25

λb

2

f=0

10

First nonuniform blade coupled mode Second uniform blade coupled mode 1

10

First uniform blade coupled mode

First uniform blade uncoupled bending mode 0

10

0

10

2

4

10

χ

6

10

10

Figure 3.1: λb -χ plot for both uniform and non-uniform rotating blades, where 4 ω2 y2 L 2 λb = µL and χ = GJ ( ) and f = Ix2m Ω2 EIx2 EIx2 r

Second nonuniform blade coupled mode 3

10

f = 200 f = 89 f = 25 f=0

λb

2

10

First nonuniform blade coupled mode 1

10

0

10

0

2

4

6

10 10 10 10 Figure 3.2: λ-χ plot for a non-uniform rotatingβ blade showing the first two coupled 4 ω2 y2 L 2 modes, where λb = µL and χ = GJ ( ) EIx2 EIx2 r


180

f = 200

160

140

f = 89

λ

120

100

f = 25 90 nonuniform coupled mode First 80

70

f=0 60 4 10

5

10

6

χ

7

10

10

Figure 3.3: λ-χ plot for a non-uniform rotating blade showing the first coupled 4 ω2 y2 L 2 mode, where λb = µL and χ = GJ ( ) EIx2 EIx2 r

λ

f = 200

f = 89

Second nonuniform coupled mode f = 25 f=0

4

10

5

10

6

χ

10

7

10

Figure 3.4: λ-χ plot for a non-uniform rotating blade showing the second coupled 4 ω2 y2 L 2 mode, where λb = µL and χ = GJ ( ) EIx2 EIx2 r

Chapter 4 Conclusion and Recommendations This work is an attempt to describe wind turbine blades and model them in a way that can be easily implemented in early design stages. Motivation and background information were presented in chapter 1. The importance of the renewable energy and the important role wind energy plays in the renewable energy industry was underlined. Challenges facing wind turbine blades design to optimize the generated power and the manufacturing cost were mentioned. The special coupled nature of the dynamic response of blades was explained in light of the blade airfoil cross-section geometric properties. Several literature models for blades were introduced, of which the Euler-Bernoulli model was chosen. Then, literature works studying the effects of blade properties and rotation on the dynamic response were reviewed. A conclusion was obtained that the fact that many blade properties were involved in predicting its dynamic behavior called for the need for a study investigating the collective effect of the blade parameters by combining them in fewer dimensionless parameters upon which different blades can be judged similar in terms of their dynamic response, a process also called as Buckingham Pi theory or dimensional analysis. In chapter 2, the problem of a uniform cantilever blade was introduced as a midway step to fully model the rotating non-uniform blade problem. After deriving the governing equations using Hamilton’s principal, the vibration characteristics of the uniform cantilever blade were solved for using an exact approach, besides the approximate assumed modes discretization approach. Both techniques showed similar results. Then, a study was developed to investigate the effect of the blade torsional and bending stiffnesses, distributed mass, radius of gyration and length on the coupled frequencies. The previously mentioned properties were combined 69

CHAPTER 4. CONCLUSION AND RECOMMENDATIONS

70

into two dimensionless parameters to study the dynamic response of the blade. The nature of the blade coupled vibrations and the dominating vibration type were successfully explained in light of the parametric study and the blade model. The second part of the chapter took the blade model one step further and extended it to include the rotational motion. The rotating uniform blade problem, however, could not be solved using the exact analytical approach because the gyroscopic terms in the governing equations added an odd-powered term to the characteristic equation, which makes it unsolvable by Cardan’s formula that was used in the exact solution technique. The rotational motion was found to increase the blade vibrations freuquencies, an effect that was found to increase with the rotational speed and the mass moment of inertia of the airfoil cross-section about the airfoil chord. Having proceeded further in the modeling process and known that the gyroscopic terms can be neglected when calculating the coupled vibration frequency, the oddpowered term in the characteristic equation can be neglected. Thus, the exact solution technique can also be applicable in the case of the rotating uniform blade in Eq. 2.109. Such measure, however, is not recommended in case of solving for the blade time response as the gyroscopic terms must contribute to the response phase lag. The phase angle of the blade response is known to be a determinant factor in some failure reasons like flutter [7]. In chapter 3, the model has fully matured to the non-uniform rotating wind turbine blade model. The vibrations characteristics of the blade were calculated from the characteristic equation obtained by the assumed modes discretization technique. A study was also conducted to find the effect of the blade parameters on its dynamic response. The previously obtained results were also applicable in the case of the rotating non-uniform blade. Moreover, another parameter had to be taken into consideration, i.e., the distribution of the blade properties along the blade length. It was found that the nature of the distribution affects the vibration frequencies. In this study, the integrated average was taken as an indicator of the blade characteristics that was effective to a reasonable degree in predicting the transitional stages in the coupled modes as the blade stiffness and slenderness ratios changed, as well as the blade properties at which the rotation effects start to be pronounced. A more accurate measure of the nature of the properties distribution can be a desired enhancement to the model at hand. The significance of this work lies in that it provides a systematic approach and guidelines to draw a similarity rule upon which blades can be judged similar in terms

CHAPTER 4. CONCLUSION AND RECOMMENDATIONS

71

of their dynamic behavior. The implications that can be drawn from such a study are important as they can be a very strong design tool. An important application in which parametric studies can be useful in the design process is flutter prevention. Patil et el. in [32] mentioned that wing stability against flutter is very sensitive to the ratio of bending and torsional stiffnesses. They also mentioned that the impact of thrust, whether it is stabilizing or destabilizing, is dependent on this ratio. The same idea is mentioned in Karpouzian’s work in [33], in which it is mentioned that the slenderness ratio and the stiffness ratio can implicate flutter critical speed and frequency. He also mentioned that the bending to torsional stiffness ratio determines whether coupled vibrations are torsional or bending dominated. In his introduction to the theory of aeroelasticity [7], Fung explained that the flutter critical frequency for a certain blade lies between the uncoupled bending and torsion frequencies for that blade. The book also elaborates on the effect of individual structural and geometric parameters on the critical speed and frequency of flutter of a blade. The amount of studies detailing the effect of each property of the blade on its dynamic response points to the need for efforts aimed at studying the collective influence of all of these parameters on the vibration characteristics. Hence, this work provides a systematic approach on how to attain a certain blade dynamic behavior using the insightful understanding of the interplay of the blade parameters and the dynamic response.

Bibliography [1] T. Burton, D. Sharpe, N. Jenkins, and E. Bossanyi. Wind Energy Handbook. John Wiley & Sons, Inc., 2001. [2] World wind energy report 2010. Technical report, World Wind Energy Association WWEA, April 2011. [3] New and Renweable Energy Authority. http://www.nrea.gov.eg/english1.html.

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[12] E. Dokumaci. An exact solution for coupled bending and torsion vibrations of uniform beams having single cross-sectional symmetry. Journal of Sound and Vibration, 119(3):443–449, 1987. [13] S. S. Rao. Vibrations of Continuous Systems. John Wiley and Sons, 2007. [14] M. R. M. Crespo Da Silva and C. C. Glynn. “nonlinear flexural-flexuraltorsional dynamics of inextensible beams. i. equations of motion”. Mechanics based design of structures and machines, 6(4):437–448, 1978. [15] D. H. Hodges and E. H. Dowell. Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. Technical report, Ames research center, NASA, 1974. [16] S. Timoshenko and D. H. Young. Vibration Problems in Engineering. Wiley, 1974. [17] K. B. Subrahmanyam, S. V. Kulkarni, and J. S. Rao. “coupled bending-torsion vibrations of rotating blades of asymmetric aerofoil cross-section with allowance for shear deflection and rotatry inertia by use of the reissner method”. Journal of Sound and Vibration, 75(1):17–36, 1981. [18] A. N. Bercin and M. Tanaka. “coupled flexural-torsional vibrations of timoshenko beams”. Journal of Sound and Vibration, 207(1):47–59, 1997. [19] A. R. Sahu. “theoretical frequency equation of bending vibrations of an exponentially tapered beam under rotation”. Journal of Vibration and Control, 7:775–780, 2001. [20] O. O. Ozgumus and M. O. Kaya. “energy expressions and free vibration analysis of a rotating double tapered timoshenko beam featuring bending-torsion coupling”. Internaional Journal of Engineering Science, 45:562–586, 2007. [21] A. Bazoune. “effect of tapering on natural frequencies of rotating beams””. Shock and Vibrations, 14:169–179, 2007. [22] M.-H. Hsu. “dynamic behaviour of wind turbine blades”. Mechanical Engineering Science, 222, 2008. [23] G. Altintas. “effect of material properties on vibrations of nonsymmetrical axially loaded thin-walled euler-bernoulli beam””. Mathematical and Computational Applications, 15:96–107, 2010.

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[24] M. O. Kaya and O. O. Ozgumus. “energy expressions and free vibration analysis of a rotating uniform timoshenko beam featuring bending-torsion coupling”. Journal of Vibration and Control, 16(6):915–934, 2010. [25] W. J. Book. Recursive Lagrangian Dynamics of Flexible Manipulator Arms via Transformation Matrices, volume Vol. 3. 1984. [26] H. Baruh. Analytical Dynamics. McGraw-Hill, 1999. [27] G. S. Bir. “users guide to bmodes (software for computing rotating beam coupled modes)”. National Renewable Energy Laboratory, 2007. [28] L. Meirovitch. Methods of Analytical Dynamics. McGraw-Hill, 1970. [29] P. K. Kundu, I. M. Cohen, and D. R. Dowling. Fluid Mechanics. Academic Press, 2011. [30] J. V. Uspensky. Theory of Equations. McGraw-Hill, 1948. [31] K. A. Stol. “geometry and structural properties for the controls advanced research turbine (cart) from model tuning”. National Renewable Energy Lab, 2004. [32] D. H. Hodges, M. J. Patil, and S. Chae. “effect of thrust on bending-torsion flutter of wings”. Journal of Aircraft, 39:371–376, 2002. [33] G. Karpouzian. Study of Bending-Torsion Flutter of High Aspect Ratio Wings. PhD thesis, University of Southern California, 1986.

Appendices

75

Appendix A Dimensional Analysis

76

1 In this chapter, dimensional analysis techniques are used in order to determine the dimensionless parameters upon which blades can be judged similar in terms of their vibrational behavior. The vibration frequency is taken to represent the blade vibration characteristics. The blade vibration frequency depends on physcial properties like the blade length L, distributed mass µ, polar radius of gyration of the blade cross-section r, bending stiffness per unit length EILx2 and torsional stiffness per unit length GJLy2 .

ω = f (µ, L, r,

EIx2 GJy2 , ) L L

µm Ll

rl EIx2 (ml2 t−2 ) L GJy2 (ml2 t−2 ) L Number of variables n = 6 Number of units j = 3 (Number of repeating variables) k = n − j = 3, 3Πs EIx2 Π1 (µ, , L, ω) L EIx2 , L, r) Π2 (µ, L EIx2 GJy2 Π3 (µ, , L, ) L L (1)

2 EIx2 , L, ω) L (ml−1 )a1 (ml2 t−2 )b1 (l)c1 (t−1 )

Π1 (µ,

a1 + b 1 = 0 −a1 + 2b1 + c1 = 0 −2b1 − 1 = 0 a1 = 1/2 b1 = −1/2 c1 = 3/2 Π21 =

µω 2 L4 = λb EIx2

(2)

EIx2 , L, r) L (ml−1 )a2 (ml2 t−2 )b2 (l)c2 (t−1 ) Π2 (µ,

a2 + b 2 = 0 −2a2 + 2b2 + c1 + 1 = 0 −2b2 = 0 a2 = 0 b2 = 0 c1 = −1 r Π2 = L

(3)

3 GJy2 EIx2 , L, ) L L (ml−1 )a3 (ml2 t−2 )b3 (l)c3 (ml2 t−2 )

Π3 (µ,

a3 + b 3 + 1 = 0 −a3 + 2b3 + c3 + 2 = 0 −2b3 − 2 = 0 a3 = 0 b3 = −1 c3 = 0 GJy2 Π3 = EIx2

(4)

Appendix B Partial Differential Equations with Variations

80

Kinematics (*Note that the conventions and names necountered in this text can be different from those used in the rest of this work, e.g., the elastic axis here is x.*)

u@x_, t_D; w@x_, t_D; v@x_, t_D; Α@x_, t_D; Εxx = D@u@x, tD, xD + z D@w@x, tD, 8x, 2; z u@x, tD - z D@w@x, tD, xD + y D@v@x, tD, xD r2@x_, t_D = : v@x, tD > - : >; - z Α@x, tD w@x, tD y Α@x, tD

r@x_, t_D = r1 + r2@x, tD Flatten; Vel = D@r@x, tD, tD Flatten; 1 Ti = Ρ Vel.VelH*.8x® xt, y® yt + ry, z® zt+rz 2

2

Dividing the kinetic energy term into 3 parts; T1a which consists of terms having which consists of terms having ΡW as a coefficient and terms having

1 2

1 2

Ρ W2 as a coefficient, T1b

Ρ W0 as a coefficient.

T1a =

FullSimplify BJx2 Cos@t WD2 + y2 Cos@t WD2 + x2 Sin@t WD2 + y2 Sin@t WD2 + 2 x z Cos@t WD2 Β@y, tD + 2 x z Sin@t WD2 Β@y, tD + z2 Cos@t WD2 Β@y, tD2 + z2 Sin@t WD2 Β@y, tD2 - 2 y z Cos@t WD2 wH1,0L @y, tD -

2 y z Sin@t WD2 wH1,0L @y, tD + z2 Cos@t WD2 wH1,0L @y, tD + z2 Sin@t WD2 wH1,0L @y, tD NF Expand 2

2

x2 + y2 + 2 x z Β@y, tD + z2 Β@y, tD2 - 2 y z wH1,0L @y, tD + z2 wH1,0L @y, tD

2

T1b = FullSimplify A

I- 2 y z Cos@t WD2 ΒH0,1L @y, tD - 2 y z Sin@t WD2 ΒH0,1L @y, tD + 2 z2 Cos@t WD2 ΒH0,1L @y, tD wH1,0L @y, tD + 2 z2 Sin@t WD2 ΒH0,1L @y, tD wH1,0L @y, tD - 2 x z Cos@t WD2 wH1,1L @y, tD - 2 x z Sin@t WD2 wH1,1L @y, tD 2 z2 Cos@t WD2 Β@y, tD wH1,1L @y, tD - 2 z2 Sin@t WD2 Β@y, tD wH1,1L @y, tDME Expand

- 2 y z ΒH0,1L @y, tD + 2 z2 ΒH0,1L @y, tD wH1,0L @y, tD - 2 x z wH1,1L @y, tD - 2 z2 Β@y, tD wH1,1L @y, tD T1c = FullSimplify BJz2 Cos@t WD2 ΒH0,1L @y, tD + z2 Sin@t WD2 ΒH0,1L @y, tD + 2

2

IwH0,1L @y, tD - x ΒH0,1L @y, tDM + z2 Cos@t WD2 wH1,1L @y, tD + z2 Sin@t WD2 wH1,1L @y, tD NF 2

2

2

wH0,1L @y, tD - 2 x wH0,1L @y, tD ΒH0,1L @y, tD + Ix2 + z2 M ΒH0,1L @y, tD + z2 wH1,1L @y, tD 2

2

2

PDEs derivation.nb

5

Ρ W T1b H*Total kinetic per unit volume, same as T10 but 2 2 2 reformulated so that it is more easily integrated with respect ot area.*L 1

T1to =

1 2

Ρ W2 T1a +

1

1

Ρ T1c +

Ρ W2 Jx2 + y2 + 2 x z Β@y, tD + z2 Β@y, tD2 - 2 y z wH1,0L @y, tD + z2 wH1,0L @y, tD N + 2

1 2 1 2

Ρ W I- 2 y z ΒH0,1L @y, tD + 2 z2 ΒH0,1L @y, tD wH1,0L @y, tD - 2 x z wH1,1L @y, tD - 2 z2 Β@y, tD wH1,1L @y, tDM + Ρ JwH0,1L @y, tD - 2 x wH0,1L @y, tD ΒH0,1L @y, tD + Ix2 + z2 M ΒH0,1L @y, tD + z2 wH1,1L @y, tD N 2

2

2

The kinetic energy per unit length T1. Since the blade is considered to be symmetric about the xaxis, then all area integral terms having z 1 were reduced to zero due to symmetry. In terms having x 1 , the x was changed to x + rx . So, rx remained and was multiplied by Μ and x terms were reduced to zero due to symmetry. Terms with z 2 were integrated to Ix2m after multiplying with the density Ρ. Terms with Ix 2 + z 2 ) were integrated to Jy2m after multiplying with the density Ρ. Ix2m - Mass moment of inertia about the x2 axis. Jy2m - Mass moment of inertia about the y2 axis. rx - Elastic center offset from the centroid in the x-direction. Ρ - mass per unit volume, density. Μ - mass per unit length. 1 T1 = 2

W2 JJzt1m + Ix2m Β@y, tD2 + Ix2m wH1,0L @y, tD + Μ rx2 N + 2

W I Ix2m ΒH0,1L @y, tD wH1,0L @y, tD - Ix2m Β@y, tD wH1,1L @y, tDM +

1 2

JΜ wH0,1L @y, tD 2

2 rx Μ wH0,1L @y, tD ΒH0,1L @y, tD + Jyt2m ΒH0,1L @y, tD + Ix2m wH1,1L @y, tD + Μ rx2 ΒH0,1L @y, tD N; 2

2

2

6

PDEs derivation.nb

Potential Energy

Elastic potential energy Εyy = DA- z wH1,0L @y, tD, yE 1 Εxy = 2 1 Εyz = 2 1 P2o = 2

ID@z Β@y, tD, yD + DA- z wH1,0L @y, tD, xEM ID@- x Β@y, tD + w@y, tD, yD + DA- z wH1,0L @y, tD, zEM El Εyy 2 + 2 G IΕxy

2

+ Εyz 2 M Expand

- z wH2,0L @y, tD 1 2

z ΒH1,0L @y, tD 1

2 1 2

x ΒH1,0L @y, tD

G x2 ΒH1,0L @y, tD + 2

1 P2 = 2

H*P2 =

1 2

G z2 ΒH1,0L @y, tD + 2

El Ix2 wH2,0L @y, tD + 2

1 2

1 2

El Ix2 wH2,0L @y,tD + 2

1 2

El z2 wH2,0L @y, tD

2

G Jy2 ΒH1,0L @y, tD ; 2

1 2

G

Jy2 ΒH1,0L @y,tD

2

+

1 2

G rx2 A ΒH1,0L @y,tD

2

;*L

Gravitational Potential Energy Pgo =

x Cos@t WD - y Sin@t WD + z ICos@t WD Β@y, tD + Sin@t WD wH1,0L @y, tDM 0 g Ρ TransposeB F. y Cos@t WD + x Sin@t WD + z ISin@t WD Β@y, tD - Cos@t WD wH1,0L @y, tDM 0 z + w@y, tD - x Β@y, tD

99g y Ρ Cos@t WD + g x Ρ Sin@t WD + g z Ρ Sin@t WD Β@y, tD - g z Ρ Cos@t WD wH1,0L @y, tD== Pg = g y Μ Cos@t WD + g rx Μ Sin@t WD g y Μ Cos@t WD + g rx Μ Sin@t WD

Expand

PDEs derivation.nb

Partial Differential Equations