structural dynamic modeling of wind turbine blades

Seediscussions,stats,andauthorprofilesforthispublicationat:http://www.researchgate.net/publication/276935868

StructuralDynamicModelingofWindTurbineBlades

RESEARCH·MAY2015

DOI:10.13140/RG.2.1.1965.9683

READS

22

1AUTHOR:

MaryBastawrous

TechnischeUniversiteitEindhoven

3PUBLICATIONS0CITATIONS

SEEPROFILE

Availablefrom:MaryBastawrous

Retrievedon:03November2015

http://www.researchgate.net/publication/276935868_Structural_Dynamic_Modeling_of_Wind_Turbine_Blades?enrichId=rgreq-572d3481-8de3-48d8-99d2-c068b49a6db4&enrichSource=Y292ZXJQYWdlOzI3NjkzNTg2ODtBUzoyMzExMTYxOTE4MjU5MjNAMTQzMjExMzgwMjU1Nw%3D%3D&el=1_x_2

http://www.researchgate.net/publication/276935868_Structural_Dynamic_Modeling_of_Wind_Turbine_Blades?enrichId=rgreq-572d3481-8de3-48d8-99d2-c068b49a6db4&enrichSource=Y292ZXJQYWdlOzI3NjkzNTg2ODtBUzoyMzExMTYxOTE4MjU5MjNAMTQzMjExMzgwMjU1Nw%3D%3D&el=1_x_3

http://www.researchgate.net/?enrichId=rgreq-572d3481-8de3-48d8-99d2-c068b49a6db4&enrichSource=Y292ZXJQYWdlOzI3NjkzNTg2ODtBUzoyMzExMTYxOTE4MjU5MjNAMTQzMjExMzgwMjU1Nw%3D%3D&el=1_x_1

http://www.researchgate.net/profile/Mary_Bastawrous2?enrichId=rgreq-572d3481-8de3-48d8-99d2-c068b49a6db4&enrichSource=Y292ZXJQYWdlOzI3NjkzNTg2ODtBUzoyMzExMTYxOTE4MjU5MjNAMTQzMjExMzgwMjU1Nw%3D%3D&el=1_x_4


http://www.researchgate.net/institution/Technische_Universiteit_Eindhoven?enrichId=rgreq-572d3481-8de3-48d8-99d2-c068b49a6db4&enrichSource=Y292ZXJQYWdlOzI3NjkzNTg2ODtBUzoyMzExMTYxOTE4MjU5MjNAMTQzMjExMzgwMjU1Nw%3D%3D&el=1_x_6


Faculty of Postgraduate Studies and Scientific Research

German University in Cairo

Structural Dynamic Modeling ofWind Turbine Blades

A thesis submitted in partial fulfillment of the requirements for thedegree 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


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?


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

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

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

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

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

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

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

2.4 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 Bending-

Torsion Vibrations of Cantilever Uniform Blades . . . . . . . . . . . . 39

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


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

2.6.3 Governing Partial Differential Equations by Hamilton’s principal 46

2.7 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 . . . . . . . . . . . . . . . . . . . . . . . 51

2.8.1 Assumed Modes Approximate Solution . . . . . . . . . . . . . 51

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

2.8.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . 52


Torsion Vibrations of Rotating Uniform Blades . . . . . . . . . . . . 53

3 Non-Uniform Blade Structural Dynamic Model 56

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


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

3.1.3 Governing Partial Differential Equations by Hamilton’s principal 58

3.2 Assumed Modes Method . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.1 Rotating Non-Uniform blade Case Assumed Modes Discretiza-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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

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

3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


Torsion Vibrations of Rotating Non-Uniform Blades . . . . . . . . . . 65

4 Conclusion and Recommendations 69

IX

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 Aerody-

namics by John D. Anderson, McGraw-Hill, 1984. . . . . . . . . . . . 3

1.3 Lift and drag forces acting on an airfoil cross-section. From: Aero-

dynamics 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 = µω2 L4

E Ix2for blades with different χ = λb

λt= GJy2

E Ix2

(Lr

)2. 41

2.5 The parameter λb = µL4ω2

EIx2for rotating uniform blades with different

values of χ = GJy2EIx2

(Lr)2 and f = Ix2mΩ2 . . . . . . . . . . . . . . . . . 55

3.1 λb-χ plot for both uniform and non-uniform rotating blades, where

λb = µL4ω2

EIx2and χ = GJy2

EIx2(Lr)2 and f = Ix2m Ω2 . . . . . . . . . . . . . 67

XI

3.2 λ-χ plot for a non-uniform rotating blade showing the first two cou-

pled modes, where λb = µL4ω2

EIx2and χ = GJy2

EIx2(Lr)2 . . . . . . . . . . . . 67

3.3 λ-χ plot for a non-uniform rotating blade showing the first coupled

mode, where λb = µL4ω2

EIx2and χ = GJy2

EIx2(Lr)2 . . . . . . . . . . . . . . . 68

3.4 λ-χ plot for a non-uniform rotating blade showing the second coupled

mode, where λb = µL4ω2

EIx2and χ = GJy2

EIx2(Lr)2 . . . . . . . . . . . . . . . 68

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 − 0.6628y2 − 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 ap-

proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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.


P A point on the blade.1h Position of point P relative [xyz]1.

xpypzp 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.


w(y, t) Bending deflection of point P .

[xyz]3 Frame rotated by angle β about y2.

XIV


β(y, t) torsional angle about the y2 axis.

L Blade length.1T2 Transformation matrix between frames 1 and 2.2T3 Transformation matrix between frames 2 and 3.1A2 Link Transformation matrix between frames 1 and 1.1E2 Elastic Transformation matrix between frames 1 and 1.

(′) 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.

ε 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.

[xyz]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.

Jy2 area polar moment of inertia of the blade cross section about the y2 axis.

Ix2 area moment of inertia of the blade cross section about the x2 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.

Jy2m mass polar moment of inertia of the blade cross section about the y2 axis.

Ix2m mass moment of inertia of the blade cross section about the x2 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.

w amplitude of the bending mode shape.

XV

b amplitude of the torsional mode shape.

r mass radius of gyration of the blade cross section about the y2 axis.

λb A dimensionless parameter equal to λb = µω2 L4

E Ix2.

χ A dimensionless parameter equal to χ = λbλt

= GJy2E Ix2

(Lr)2.

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 Impor-

tance

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 inter-

ested 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”


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 Aerodynamicsby 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


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: Aerody-namics 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].

∂p

∂rc= ρair

V 2r

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


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

l =1

2CLρV

2r c (1.2)

d =1

2CDρV

2r c (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

2Cp ρair AU

3 (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


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 dif-

ference 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 vibra-

tion 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


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 at-

tention 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 slender-

ness 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


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.

Trailing Edge

Shear Center

Centroid

Chord

Leading Edge

Figure 1.4: Geometrical properties of airfoils.

Dokumaci and Rao have explored linear models of beams with small deforma-

tions [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 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 vi-

brations 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 Timo-

shenko 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,


which makes modeling a blade as an Euler-Bernoulli beam a reasonable approxi-

mation 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 ma-

trices 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 tur-

bine 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, be-

sides 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 cou-

pled 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 Pois-

son’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 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. Di-

mensional analysis techniques, as well as the problem physics, are employed to ex-

tract 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 en-

ergy 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 so-

lution. 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

0T1 =

cos(Ωt) − sin(Ωt) 0 0

sin(Ωt) cos(Ωt) 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 1h = [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 2h = [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


β

β

z3 z2

x2

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 0w] in the

x2, y2 and z2 directions, respectively. The homogeneous transformation matrix from

frame 1 to frame 2 can be expressed as [25]

1T2 = 1A21E2 =

1 0 0 0

0 1 0 y

0 0 1 0

0 0 0 1

1 0 0 0

0 1 −w′(y, t) 0

0 w′(y, t) 1 w(y, t)

0 0 0 1

(2.2)

where w′(y, t) is used to express ∂w(y,t)∂y

. The homogeneous transformation matrix1A2 represents the displacement due to the link between the two frames and 1E2

is due to the elastic deformations. Similarly, the transformation matrix 2T3 can be

expressed as

2T3 =

1 0 β(y, t) 0

0 1 0 0

−β(y, t) 0 1 0

0 0 0 1

(2.3)

Assuming no warping occurs in torsion so that 3h, after torsional deformation,

stays the same as 2h before torsion, the position of P relative to the rotating frame


after deformation can be expressed as

1h = 1T33h = 1T2

2T33h (2.4)

1h =

x+ zβ

y − zw′ + xβ w′

z + w − x β1

(2.5)

Assuming that the blade is undergoing small deformations, the product β w′ is

neglected since its value is of a much smaller order of magnitude compared to the

rest of the expression. So, 1h is simplified to

1h =

x+ zβ

y − z w′z + w − x β

1

(2.6)

and the velocity of P can be written as

1h =

zβ

−z w′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

0h =0 T11 h =

x cos(Ωt)− y sin(Ωt) + z cos(Ωt)β + z sin(Ωt)w′

y cos(Ωt) + x sin(Ωt) + z sin(Ωt)β − z cos(Ωt)w′

z + w − xβ1

(2.8)


and the velocity as

0h =

−yΩ cos(Ωt)− xΩ sin(Ωt)− zΩ sin(Ωt)β + z cos(Ωt)β + zΩ cos(Ωt)w′ + z sin(Ωt)w′

xΩ cos(Ωt)− yΩ sin(Ωt) + zΩ cos(Ωt)β + z sin(Ωt)β + zΩ sin(Ωt)w′ − z cos(Ωt)w′

w − xβ0

(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 grav-

itational energy are ignored. The strain energy of the deflections can be calculated

by [26]

V0 =

ˆ ε

0

σ dε (2.10)

where V0 is the strain energy, σ is the stress and ε 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

V0 =

ˆ εyy

0

E εyy dεyy + 2G

(ˆ εxy

0

εxy dεxy +

ˆ εyx

0

εyx dεyx +

ˆ εyz

0

εyz dεyz +

ˆ εzy

0

εzy dεzy +

ˆ εzx

0

εzx dεzx +

ˆ εxz

0

εxz dεxz

)(2.11)

where E and G are the blade’s tensile elastic modulus and shear elastic modulus,

respectively, εyy is the axial strain in the direction of the elastic axis, and the notation

εij 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

u =

zβ

−z w′w − x β

(2.12)


The elastic strain is defined as [8]

εij =1

2

(∂uj∂i

+∂ui∂j

)i,j = x, y and z. (2.13)

For example,

εyx =1

2

(∂ux∂y

+∂uy∂x

)=

1

2(zβ′(y, t))

Therefore, the elastic strain is expressed as εyx

εyy

εyz

=

12zβ′(y, t)

−zw′′(y, t)12xβ′(y, t)

(2.14)

The strain εyz was calculated to be zero. The relations

εyx = εxy

εyz = εzy

εzx = εxz

were used. Substituting Eq. 2.14 into Eq. 2.11, the elastic potential energy per unit

volume becomes

V0 =1

2E z2w′′2 +

1

2Gβ′2 (z2 + x2) (2.15)

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 [xyz]2 in which x2,

y2 and z2 axes are parallel to the x2, y2 and z2 axes respectively. However, the only

difference is that the y2 axis extends along the centroidal axis instead of the elastic

axis, which simplifies the area integrations as they will vanish about the x2 and

z2 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


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.

x2 + rx = x2 y2 = y2 z2 = z2 (2.16)

Integrating equation 2.15 with respect to the blade’s cross-section area dA = dx2 dz2

to get the elastic potential energy per unit length yields

ˆV0 d V =

ˆ L

0

(1

2Ew′′2

ˆz2dA

)dy +

ˆ L

0

(1

2Gβ′2

ˆ ((x+ rx)

2 + z2)dA

)dy

=

ˆ L

0

(1

2E Ix2w

′′2 +1

2GJy2β

′2)dy (2.17)

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 y2 axis Jy2 by

Jy2 = Jy2 + Ar2x (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]

T =

ˆ1

2ρ~hT . ~h dV (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


becomes

T =

ˆ1

2ρ(w2 + z2 w2 − 2x w β + (x2 + z2) β2

)dV (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

T =

ˆ1

2ρ(w2 + z2w2 − 2(x+ rx)wβ + ((x+ rx)

2 + z2) β2)

dV (2.21)

The kinetic energy per unit length is

T =

ˆ L

0

(1

2µ(w(y, t)2 − 2rxw(y, t)β(y, t)

)+ µ(r2 + r2

x)β(y, t)2 + Ix2mw′(y, t)2

)dy

(2.22)

where µ is the material’s density per unit length calculated by µ =´Aρ dA = ρA,

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 = Jy2m + µr2x Ix2m = Ix2m (2.23)

where Jy2m is the polar mass moment of inertia about the y2 axis and Ix2m is the

mass moment of inertia about the x2 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 me-

chanics [11]. In analytical mechanics, the concept of virtual displacements is used.

Virtual displacements can be defined as slight variations in the generalized coordi-

nates that are consistent with the system constraints and happen instantaneously,

which means that the variation of time occurring in parallel with the virtual dis-

placements is zero, δt = 0. Virtual displacements are assumed to be completely


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 displace-

ments, 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

definite integral I =´ t2t1

(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 ar-

bitrariness, 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 t2

t1

δV dt =

ˆ t2

t1

ˆ L

0

(GJy2β′δβ′ + EIx2w

′′δw′′) d y d t (2.25)


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

t1

ˆ L

0

GJy2β′δβ′dydt =

ˆ t2

t1

ˆ L

0

(GJy2β′δβ) |L0 dt−

ˆ t2

t1

ˆ L

0

(GJy2β′)′δβdydt (2.26)

ˆ t2

t1

ˆ L

0

EIx2w′′δw′′dydt =

ˆ t2

t1

(EIx2w

′′δw′|L0)dt−

ˆ t2

t1

ˆ L

0

(EIx2w′′)′δw′dy dt

=

ˆ t2

t1

(EIx2w

′′δw′|L0)dt−

ˆ t2

t1

((EIx2w

′′)′δw|L0)dt+

ˆ t2

t1

ˆ L

0

(EIx2w′′)′′δwdydt

(2.27)

The variation of the potential energy in Hamilton’s principal is

ˆ t2

t1

δV dt =

ˆ t2

t1

ˆ L

0

((EIx2w′′)′′δw − (GJy2β

′)′δβ) dy dt+

GJy2β′δβ|L0 + EIx2w

′′δw′|L0 − (EIx2w′′)′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 t2

t1

δTd t =

ˆ t2

t1

ˆ L

0

(µwδw − µrxβδw − rxwδβ + J0mβδβ + Ix2mw′δw′

)d y d t

(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

t1

ˆ L

0

F (y)δG(y, t)K(y, t)dydt

and ˆ t2

t1

ˆ L

0

F (y)δG′(y, t)K ′(y, t)dydt


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

Eq. 2.31.

ˆ L

0

ˆ t2

t1

µ(y)δwwdt dy =

ˆ L

0

µwδw|t2t1dy −ˆ L

0

ˆ t2

t1

µwδwdtdy (2.30)

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

ˆ L

0

Ix2mδw′w′d y d t =

ˆ t2

t1

Ix2mw′δw|L0 dt−ˆ t2

t1

ˆ L

0

(Ix2mw′)′δwdy dt

=

ˆ t2

t1

Ix2mw′δw|L0 dt−ˆ L

0

(Ix2mw′)′δw|t2t1d y +

ˆ t2

t1

ˆ L

0

(Ix2mw′)′δwdy dt (2.31)

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 t2

t1

δT =

ˆ t2

t1

ˆ L

0

(−µw + µrxβ + (Ix2mw′)

′)δw +

(µrxw − Jy2mβ

)δβdy dt

−ˆ t2

t1

Ix2mw′δw|L0 dt (2.32)

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

ˆ t2

t1

ˆ L

0

(−µw + µrxβ + (Ix2mw′)

′ − (EIx2w′′)′′)δw+(

µrxw − Jy2mβ + (GJy2β′)′)δβdy dt

+

ˆ t2

t1

(−Ix2mw′δw|L0 −GJy2β

′δβ|L0 − EIx2w′′δw′|L0 + (EIx2w

′′)′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


modes of vibration that may cause wrinkling of the blade, the rotary inertia terms

Ix2mw′ are ignored. The governing PDEs of the fixed cantilever blade become

−µw + µrxβ − (EIx2w′′)′′ = 0

µrxw − Jy2mβ + (GJy2β′)′ = 0 (2.34)

The boundary conditions are

GJy2β′δβ|L0 = 0 (2.35)

EIx2w′′δw′|L0 = 0 (2.36)

(EIx2w′′)′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 EIx2w′′ at the blade tip and root. Finally, the boundary

condition in Eq. 2.37 imposes conditions on the bending deflection and the shear

force (EIx2w′′)′ 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)

w′(0, t) = 0 (2.40)

GJy2β′(L, t) = 0 (2.41)

EIx2w′′(L, t) = 0 (2.42)

(EIx2w′′)′(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,


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 can-

not be solved in a similar manner, which is why numerical approximate techniques

are used in such cases. The assumed modes method is the approximate numeri-

cal 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 dif-

ferential 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


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

µω2W (y)− µrxω2B(y)− (EIx2W′′(y))′′ = 0

−µrxω2W (y) + Jy2mω2B(y) + (GJy2B

′(y))′ = 0 (2.46)

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

W (y) = w expsyL (2.47)

B(y) = b expsyL (2.48)

where w and b are the amplitudes of the mode shapes. Thus, Eq. 2.46 can be

rewritten as,(ω2 µ− E Ix2

s4

L4 −µ rx ω2

−ω2 µ rx ω2 (Jy2m + µr2x) + s2

L2GJy2

)w

b

= 0 (2.49)

Now, let

r =

√Jmy2

µ(2.50)

λb =µω2 L4

E Ix2

(2.51)

λt =µ r2 ω2 L2

GJy2

(2.52)

ε = 1 +r2x

r2(2.53)

The coefficient matrix in Eq. 2.49 becomes(λb − s4 −rxλb−λt rxr2 ε λt + s2

)(2.54)

where r is the polar mass radius of gyration of the blade about the centroidal axis y2,


λb and λt are dimensionless parameters that can be obtained by dimensionless anal-

ysis techniques [29], and ε is a dimensionless quantity measuring the offset between

the centroidal and elastic axes, where ε = 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 w and b are solved for. This

determinant constitutes the uncoupled characteristic equation of the ordinary dif-

ferential 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 ε 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 ε 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 − ε λt) + a(3k2 − λb − 2kελt) + k3 − k λb − k2 ε λt − λbλt = 0 (2.58)

Setting k to ε λt3

such that a2 vanishes, equation 2.58 becomes

a3 + P a + Q = 0 (2.59)


where

a = s2 +ε λt3

(2.60)

P = −λb(

1 +ε2λ2

t

3λb

)Q = −λbλt

(1− ε

3− 2 ε3 λ2

t

27λb

)(2.61)

Equation 2.59 can be solved by Cardan’s formula if the discriminant

∆ = 4P 3 + 27Q2 (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

(4 + (ε2 + 18 ε− 27)

λ2t

λb+ 4ε3(

λ2t

λb)2

)(2.63)

In this paragraph, it is proved that the discriminant in Eq. 2.63 is negative. Notice

the term between brackets in the discriminant is a quadratic polynomial inλ2tλb

. The

discriminant in Eq. 2.63 is negative if the term

ε2 + 18 ε− 27

is positive. Solving for the roots of that term, it is found that it has two roots at

ε = ±1.4. Since the physics of the problem implies that ε must be at least one, only

the positive root needs to considered. The sign of the discriminant is investigated

in the intervals 1 ≤ ε ≤ 1.4 and ε ≥ 1.4. The positive root is at approximately

1.4, which marks a change of sign in the term (ε2 + 18 ε − 27) at that value of

ε. 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 ε = -9 before

the root, concluding that the value of that term must be increasing after the global

minimum at -9. Thus, the term (ε2 + 18 ε − 27) must be negative in the interval

1 ≤ ε ≤ 1.4 and positive when ε ≥ 1.4, which makes the discriminant δ negative

in the interval ε ≥ 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


interval 1 ≤ ε ≤ 1.4 as well.

λ2t/λb =

27− (−9 + ε)3/2√−1 + ε− 18ε− ε2

8ε2

λ2t/λb =

27 + (−9 + ε)3/2√−1 + ε− 18ε− ε2

8ε2(2.64)

To find out the nature of the roots at the desired interval, the discriminant

∆′ = (−9 + ε)3/2√−1 + ε

that can be found in the roots of theλ2tλb

polynomial in equations 2.64 is investigated

in the desired interval of ε. Equation 2.64 has complex roots in the interval 1 ≤ε ≤ 1.4 because the discriminant ∆′ 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 ≤ ε ≤ 1.4. This proves that the value of ∆ in Eq. 2.63 is negative in

the interval ε ≥ 1. Hence the roots are all real and can be solved for by Cardan’s

formula yielding

a1 = 2

√−P3

cos(φ

3) (2.65)

a2 = 2

√−P3

cos(φ

3+

2π

3) (2.66)

a3 = 2

√−P3

cos(φ

3+

4π

3) (2.67)

where

cos(φ) =

√27Q

2P√−P (2.68)

si = ±√ai −

ε λt3

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 (expsi 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

1The number of positive real roots of an equation with real coefficients f(x) = a0xn + a1 x

n−1 +


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

W (y) = A1 cosh(s1 y

L) + A2 sinh(

s1 y

L) + A3 cos(

s2 y

L) + A4 sin(

s2 y

L)+

A5 cos(s3 y

L) + A6 sin(

s3 y

L)

B(y) = B1 cosh(s1 y

L) +B2 sinh(

s1 y

L) +B3 cos(

s2 y

L) +B4 sin(

s2 y

L)+

B5 cos(s3 y

L) +B6 sin(

s3 y

L) (2.71)

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( s1 y)L

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)

or

B1 =A1 rx λt

r2 (s2 + ε λt)(2.73)

As the value of rx approaches zero, the value of ε decreases at an even faster rate

because of the r2x 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.

2expı θ = cos(θ) + ı sin(θ)


terms of the constants A1, A2, A3, A4, B5, and B6 as

W (y) = A1 cosh(s1 y

L

)+ A2 sinh

(s1 y

L

)+ A3 cos

(s2 y

L

)+ A4 sin

(s2 y

L

)+

r2 (s23 + ε λt)

rx λt

(B5 cos

(s3 y

L

)+B6 sin

(s3 y

L

))B(y) =

−(s41 − λb)rx λb

(A1 cosh

(s1 y

L

)+ A2 sinh

(s1 y

L

))+−(s4

1 − λb)rx λb

(A3 cos(s2 y

L

)+ A4 sin

(s2 y

L

)+B5 cos

(s3 y

L

)+B6 sin

(s3 y

L

)(2.74)

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 1

0 s1L

0s21 cosh(s1)

L2

s21 sinh(s1)

L2 − s22 cos(s2)

L2

s31 sinh(s1)

L3

s31 cosh(s1)

L3

s32 sin(s2)

L3

− 1rx

+s41rxλb

0 − 1rx

+s42rxλb

− s1 sinh(s1)Lrx

+s51 sinh(s1)

Lrxλb− s1 cosh(s1)

Lrx+

s51 cosh(s1)

Lrxλb

s2 sin(s2)Lrx

− s52 sin(s2)

Lrxλb

0 − r2

rx− rx +

r2s23rxλt

0s2L

0 − r2s3Lrx− rxs3

L+

r2s33Lrxλt

− s22 sin(s2)

L2

r2s23 cos(s3)

L2rx+

rxs23 cos(s3)

L2 − r2s43 cos(s3)

L2rxλt

r2s23 sin(s3)

L2rx+

rxs23 sin(s3)

L2 − r2s43 sin(s3)

L2rxλt

− s32 cos(s2)

L3 − r2s33 sin(s3)

L3rx− rxs33 sin(s3)

L3 +r2s53 sin(s3)

L3rxλt

r2s33 cos(s3)

L3rx+

rxs33 cos(s3)

L3 − r2s53 cos(s3)

L3rxλt

0 1 0

− s2 cos(s2)Lrx

+s52 cos(s2)

Lrxλb− s3 sin(s3)

Ls3 cos(s3)

L


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 combi-

nation 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 approxi-

mates the conservative distributed parameter system at hand. The assumed modes

method aims at developing the discretized modal equations of motion by first dis-

cretizing 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 homo-

geneous uniform wind turbine blade with an asymmetric cross-section. The Euler-

Bernoulli 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


time as

w(y, t) =N∑i=1

φwi(y) qi(t)

β(y, t) =N∑i=1

φβi(y) qi(t) (2.76)

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 displace-

ment 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 equa-

tion 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


modes method. In the previous sections, the elastic potential energy per unit length

of a fixed cantilever uniform blade was expressed as

Ve =

ˆ L

0

(1

2E Ix2w

′′2 +1

2GJy2β

′2)dy (2.17)

while the kinetic energy per unit length was expressed as

T =

ˆ L

0

(1

2µ(w(y, t)2 − 2rxw(y, t)β(y, t)

)+ µ(r2 + r2

x)β(y, t)2 + Ix2mw′(y, t)2

)dy

(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∑i=1

N∑j=1

1

2

(µφwi qi φwj qj − 2 rx µφwi qi φβj qj +

1

2µ(r2 + r2

x)φβi qi φβj qj

+1

2Ix2mφ

′w i qi φ

′w jqj

)(2.78)

Ve =N∑i=1

N∑j=1

1

2EIx2 φ

′′wi qi φ

′′wj qj +

1

2GJy2 φ

′βi qi φ

′βjqj

(2.79)

The discretized energy expressions are integrated with respect to the blade length

and are substituted in Lagrange’s equations

d

d t

(δ Tlδqk

)− δTlδqk

+δVlδqk

= Qk

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

M q +Kq = Qk (2.81)

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


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.

mij =

ˆ L

0

(µφwi φwj − µ rx φwi φβj − µ rx φwj φβi + µ (r2 + r2

x)φβi φβj)dy

(2.82)

Similarly, K is the stiffness matrix with dimensions NxN 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.

kij =

ˆ L

0

(GJy2 φ

′βi φ

′βj + EIx2 φ

′′wi φ

′′wj

)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

(λ2M +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 λ2M + K = 0 must be equal to zero which yields the characteristic

equation of the system, a polynomial of λ of degree N2. 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


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 = 102.097 kg/m

rx = 0.09 m

GJy2 = 1.09E7 N.m2

EIx2 = 3.8441E7 N.m2

r = 0.3632 m

L = 19.955 m

(2.87)

2.4.1 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 prop-

erties 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 characteris-

tic 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)


Table 2.1: Computed exact natural frequencies using exact solution approach

Mode Exact Frequency(rad/s)

1 5.416962 33.90793 94.81264 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)


−0.00281346 sin(0.0378124y) + 1.01838 sin(0.235187y)− 1.01854 sinh(0.235045y)

(2.89)


+11.8222 sin(0.0787436y)− 1.74068 sin(0.339522y)− 1.00246 sinh(0.339095y)

(2.90)


+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)


+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)


−0.0989 sin(0.1057y) + 0.04626 sin(0.3935y) + 0.04003 sinh(0.3929y) (2.95)


0 5 10 15 200

0.5

1

Blade Length (m)

First Bending Mode

0 5 10 15 20−1

0

1

Blade Length (m)

Second Bending Mode

0 5 10 15 200

0.5

1

Blade Length (m)

Third Bending Mode

0 5 10 15 20−1

0

1

Blade Length (m)

Fourth Bending Mode

0 5 10 15 200

2

4x 10

−3

Blade Length (m)

First Torsional Mode

0 5 10 15 200

0.02

0.04

Blade Length (m)

Second Torsional Mode

0 5 10 15 20−20

−10

0

Blade Length (m)

Third Torsional Mode

0 5 10 15 20−0.1

0

0.1

Blade Length (m)

Fourth Torsional Mode

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.

(λ2M +K)a = 0 (2.86)


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.

mij =

ˆ L

0


x)φβi φβj)dy

(2.82)

kij =

ˆ L

0

(GJy2 φ

′βi φ

′βj + EIx2 φ

′′wi φ

′′wj

)dy (2.83)

2.4.2.1 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 =

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

(2.96)

K =

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

(2.97)

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


Table 2.2: Computed approximate natural frequencies using assumed modes ap-proach

Mode Exact Frequency(rad/s) Approximate Frequency(rad/s) %

1 5.41696 5.424212 33.9079 34.16223 94.8126 95.84114 185.181 189.708

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 Proper-

ties 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

like the slenderness ratio rL

, the stiffness ratio GJy2EIx2

, λb which is the uncoupled


bending mode shape spatial frequency λb = ω2µL4

EIx2, andλt which is the uncoupled

torsional mode shape spatial frequency λt = ω2µL2r2

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

axis in Fig. 2.4, where χ = λbλt

= GJy2E Ix2

(Lr

)2is a dimensionless parameter constituted

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 uncou-

pled 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


100

101

102

103

104

105

10−

2

100

102

104

106

χ =

(GJ y2

/EI x2

)(L

/r)2

λb

Firs

t Ben

ding

Mod

e

Sec

ond

Ben

ding

Mod

e

Fou

rth

Ben

ding

Mod

e

Thi

rd B

endi

ng M

ode

Firs

tT

orsi

onal

Mod

e

Sec

ond

Tor

sion

alM

ode

Thi

rdT

orsi

onal

Mod

e

Fou

rth

Tor

sion

alM

ode

Sec

ond

Cou

pled

Mod

e

Thi

rd C

oupl

edM

ode

Fou

rth

Cou

pled

Mod

e

Firs

t Cou

pled

Mod

e

Fig

ure

2.4:

The

par

amet

erλb

=µω2L4

EI x

2fo

rbla

des

wit

hdiff

eren

tχ

=λb

λt

=GJy2

EI x

2

( L r

) 2 .


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 vibra-

tions 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 vi-

bration 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 vibra-

tions frequencies which are represented by the constant non-zero-slope lines. Fixing

the product µL4

EIx2and varying the torsional stiffness GJy2 results in a constant-slope

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


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 Extending the Governing Equations to Rotat-

ing Blades

Earlier in this chapter, the structural model for a uniform cantilever blade in cou-

pled bending-torsional vibrations was developed based on the Euler-Bernoulli beam

model. The governing partial differential equations for a blade in coupled bending-

torsion 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 vi-

brations. 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.


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.

Ve =1

2EIx2w

′′2 +1

2GJy2β

′2 (2.98)

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


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)w′(y, t)

(2.100)

Performing the area integrations with respect to the centroidal axes, the gravita-

tional 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 verti-

cal 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 pro-

cedures in the non-rotating blade case, the kinetic energy per unit length for the

rotating blade is

T = T0 + T1 + T2

=1

2Ω2(Jz1m + Ix2mβ

2 + Ix2mw′2)+

1

2Ω(

2Ix2mβw′ − 2Ix2mβw

′)

+

1

2

(µw2 − 2rxµwβ + Jy2mβ

2 + Ix2mw′2)

(2.102)

where

T0 =1

2Ω2(Jz1m + Ix2mβ

2 + Ix2mw′2)

and is called the centrifugal kinetic energy,

T1 =1

2Ω(


′)

which is the term that gives rise to the gyroscopic forces, and

T2 =1

2


2 + Ix2mw′2)

which is the kinetic energy term that arises due to the beam vibrations without any

influence of the rotational motion.


2.6.3 Governing Partial Differential Equations by Hamil-

ton’s principal

Taking the variations of equations 2.102, 2.98 and 2.101, it is found that the gravita-

tional 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 − 2Ix2mΩβ′ − Ix2mΩ2w′′ + Ix2mw′′ − EIx2w

′′′′ = 0 (2.103)

Ix2mΩ2β + rx µw − Jy2mβ − 2Ix2mΩw′ + GJy2β′′ = 0 (2.104)

Ignoring the rotary inertia terms, the governing equations are simplified to

−µw + rx µ β − 2Ix2mΩβ′ − Ix2mΩ2w′′ − EIx2w′′′′ = 0

Ix2mΩ2β + rx µw − Jy2mβ − 2Ix2mΩw′ + GJy2β′′ = 0 (2.105)

2.7 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)


Dividing by eλt and substituting W (y) and B(y) by WesLy and Be

sLy, respectively,

B

(rx µλ

2 − 2Ix2m s λΩ

L

)+ W

(−El Ix2 s

4

L4+Ix2m s

2 Ω2

L2− µλ2 − Ix2ms2 Ω2

L2

)= 0

W

(rx µλ2 − 2Ix2m s λΩ

L

)+ B

(GJy2 s

2

L2− Jy2m λ

2 − rx2µλ2 + Ix2m Ω2

)= 0

(2.107)

The system of algebraic equations in Eq. 2.107 can be rewritten in matrix form as(−EIx2 s4

L4 + Ix2m s2 λ2

L2 − µλ2 − Ix2m s2 Ω2

L2 rx µλ2 − 2Ix2m s λΩ

L

rx µλ2 − 2Ix2m s λΩL

GJy2 s2

L2 − µr2 λ2 − µ r2x λ

2 + Ix2m Ω2

)(W

B

)= 0 (2.108)

Thus, the characteristic equation will be the determinant of the coefficient matrix

in Eq. 2.108

s6

(−E Ix2GJy2

L6

)+ s4

(Ix2mGJy2 λ

2

L4+E Ix2, µ(r2 + r2

x)λ2

L4−

(E Ix2 +GJy2) Ix2m Ω2

L4

)+ s2

(−Ix2mµ(r2 + r2x)λ

4

L2− GJy2λ

2µ

L2− 3I2

x2m λ2 Ω2

L2

+Ix2m µ(r2 + r2

x)λ2Ω2

L2− I2

x2mΩ4

L2

)− s

(4 Ix2m rx λ

3 µΩ

L

)− Ix2m λ

2 µΩ2 + µ2r2 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


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 Rayleigh-

Ritz 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∑i=1

φwi(y) qi(t) β(y, t) =N∑i=1

φβi(y) qi(t) (2.110)



values from i to N , qi(t) is the generalized coordinate used to describe the time


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 Uniform Rotating Blade Case Assumed Modes Dis-

cretization

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

Ve =1

2EIx2w

′′2 +1

2GJy2β

′2 (3.1)



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

T =1

2Ω2(Jz1m + Ix2mβ

2 + Ix2mw′2)+

1

2Ω(


′)

+

1

2


2 + Ix2mw′2)

(3.4)

Substituting the discretized dependent variables in the energy expressions, the po-

tential energy can be rewritten as

Ve =N∑i=1

N∑j=1

1

2EIx2 φ

′′wi qi φ

′′wj qj +

1

2GJy2 φ

′βi qi φ

′βjqj

(2.112)


and the kinetic energy can be rewritten as

T =N∑i=1

N∑j=1

1

2Ω2(µ r2

o + Ix2mφβi qi φβj qj + Ix2mφ′wi qi φ

′wj qj

)+

1

2Ω(2 Ix2m φβi qi φ

′wj qj − 2 Ix2m φβi qi φ

′wj qj

)+µφwi qi φwj qj − 2 rx µφwi qi φβj qj +

1

2µ(r2 + r2

x)φβi qi φβj qj

+1

2Ix2mφ

′w i qi φ

′w jqj

(2.113)

Substituting the discretized energy expressions in Lagrange’s equations

d

d t

(δ Tlδqk

)− δTlδqk

+δVlδqk

= Qk (2.114)

: k = 1 , 2 , ... , N (2.80)


length. Equation 2.80 can be used to write N equations of motion. The governing

modal equations can be written as

M q + Gq + Kq = 0 (2.115)

where M is an N by N inertia matrix constructed from the coefficients of qi in which

each element can be expressed as

mij =

ˆ L

0


x)φβi φβj)dy

(2.116)

G is an N by N gyroscopic matrix constructed from the coefficients of qi in which

element can be written as

gij =

ˆ L

0

(Ix2m Ωφ′wi φβj − Ix2m Ωφ′wj φβi

)dy (2.117)

and K is an N by N stiffness matrix constructed from the coefficients of qi in which


kij =

ˆ L

0

(Ix2m Ω2 (φβi φβj + φ′wi φ

′wj) + GJy2 φ

′βi φ

′βj + EIx2 φ

′′wi φ

′′wj

)dy (2.118)


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

(λ2M + λG+K)a = 0 (2.119)

The matrix (λ2M + 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 (λ2M + λG +

K), which constitutes the characteristic equation of the assumed modes discretized

rotating blade problem. The mode shapes can be found by solving the under-

determined 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 = 102.097 kg/m

rx = 0.09 m

GJy2 = 1.09E7 N.m2

EIx2 = 3.844E7 N.m2

r = 0.363152 m

Ix2m = 2.7847 kg.m

L = 19.955 m

(2.120)

2.8.1 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 equa-

tions 2.116, 2.117 and 2.118.


2.8.1.1 Trial functions selection

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 stiff-

ness matrices are shown as follows

M =

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

(2.121)

G =

0. 1.08938 −1.57943 −0.304428

−1.08938 0. 5.14569 7.88767

1.57943 −5.14569 0. 0.41833

0.304428 −7.88767 −0.41833 0.

(2.122)

K =

58466. −20300.6 32599.7 −45491.2

−20300.6 2.29857E6 −412730. 526419.

32599.7 −412730. 1.82743E7 28.1941

−45491.2 526419. 28.1941 7.05323E7

(2.123)

The characteristic equation obtained from the determinant of the algebraic eigen-

value 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 frequen-

cies 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.


Table 2.3: Computed approximate natural frequencies for a rotating blade usingassumed modes approach

Mode Fixed Blade Frequency Rotating Blade Frequency Rotating Speedrad/s rad/s rad/s

5.42424 11 5.42421 5.42433 2

5.42448 3

34.1622 12 34.1622 34.1623 2

34.1625 3

95.8411 13 95.8411 95.8412 2

95.8414 3

189.708 14 189.708 189.708 2

189.708 3



of Rotating Uniform Blades

In Sec. 2.5, the geometric and material properties affecting the vibration charac-

teristics of a uniform cantilever blade were investigated and represented by two

dimensionless parameters λb = µL4ω2

EIx2and χ = GJy2

EIx2(Lr)2. In the problem of the

rotating blade, the same blade properties affect the vibration characteristics, be-

sides 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 gov-

erning 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


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

T = T0 + T1 + T2

=1

2Ω2(Jz1m + Ix2mβ

2 + Ix2mw′2)+

1

2Ω(


′)

+

1

2


2 + Ix2mw′2)

(2.102)

The product Ix2mΩ2 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 Ix2mΩ2 or Ix2mΩ has the

greater influence on the blade vibrations is a question left to the magnitude of their

coefficients to decide.

Beginning with the product Ix2mΩ2 and denoting it by f for convenience, the

assumed modes discretized characteristic equation is solved for the vibrations fre-

quencies 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 Ix2mΩ has a

negligible influence on the blade characteristics. This is consistent with the litera-

ture 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.

3The product 12Jz1mΩ2 does not contribute to the governing equations because it does not

include any generalized coordinates.


100

101

102

103

104

105

106

107

100

101

102

103

χ

λ b

f = 0

f = 0

f = 25

f = 25

f = 89

f = 89

f = 200

f = 200

Second UncoupledBending Mode

First UncoupledBending Mode

First CoupledMode

SecondCoupled Mode

First TorsionalMode

Figure 2.5: The parameter λb = µL4ω2

EIx2for rotating uniform blades with different

values of χ = GJy2EIx2

(Lr)2 and f = Ix2mΩ2

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 prin-

cipal. 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 ro-

tating 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 dif-

ferential 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.


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.

Ve =1

2EIx2(y)w′′2 +

1

2GJy2(y)β′2 (3.1)



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 verti-

cal 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 pro-

cedures in the non-rotating blade case, the kinetic energy per unit length for the

rotating blade is

T = T0 + T1 + T2

=1

2Ω2(Jz1m + Ix2m(y) β2 + Ix2m(y)w′2

)+

1

2Ω(

2Ix2m(y)βw′ − 2Ix2m(y) β w′)

+

1

2

(µ(y) w2 − 2rx µ(y) wβ + Jy2m(y) β2 + Ix2m(y) w′2

)(3.4)

where

T0 =1

2Ω2(Jz1m + Ix2m(y)β2 + Ix2m(y)w′2

)and is called the centrifugal kinetic energy,

T1 =1

2Ω(

2Ix2m(y)βw′ − 2Ix2m(y)βw′)


which is the term that gives rise to the gyroscopic forces, and

T2 =1

2

(µ(y)w2 − 2rx(y)µ(y)wβ + Jy2m(y)β2 + 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 Hamil-

ton’s principal

Hamilton’s equation yields the governing equations.

−µ(y)w + rx(y)µ(y) β − 2Ix2m(y)Ωβ′ − Ix2m(y)Ω2w′′ + Ix2m(y)w′′ − EIx2(y)w′′′′ = 0

(3.5)

Ix2m(y)Ω2β + rx(y)µ(y)w − Jy2m(y)β − 2Ix2m(y)Ωw′ + GJy2(y)β′′ = 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′ − Ix2m(y)Ω2w′′

−EIx2(y)w′′′′ = 0 (3.7)

Ix2m(y)Ω2β + rx(y)µ(y)w − Jy2m(y)β − 2Ix2m(y)Ωw′ + GJy2(y)β′′ = 0 (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 charac-

teristics of the non-uniform rotating blade. In the next section, the rotating blade

problem is discretized using the assumed modes method.


3.2 Assumed Modes Method

The vibrational displacements are expressed as the sum of N functions in space and

time

w(y, t) =N∑i=1

φwi(y) qi(t)

β(y, t) =N∑i=1

φβi(y) qi(t) (3.9)



values from i to N , and qi(t) is the generalized coordinate used to describe the time


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 po-

tential energy can be rewritten as

Ve =N∑i=1

N∑j=1

1

2EIx2(y)φ′′wi qi φ

′′wj qj +

1

2GJy2(y)φ′βi qi φ

′βjqj

(3.11)


and the kinetic energy can be rewritten as

T =N∑i=1

N∑j=1

1

2Ω2(µ(y) r2

o + Ix2m(y)φβi qi φβj qj + Ix2m(y)φ′wi qi φ′wj qj

)+

1

2Ω(2 Ix2m(y)φβi qi φ

′wj qj − 2 Ix2m(y)φβi qi φ

′wj qj

)+µ(y)φwi qi φwj qj − 2 rx(y)µ(y)φwi qi φβj qj +

1

2µ(y)(r(y)2 + rx(y)2)φβi qi φβj qj

+1

2Ix2m(y)φ′w i qi φ

′w jqj

(3.12)

Substituting the discretized energy expressions in Lagrange’s equations

d

d t

(δ Tlδqk

)− δTlδqk

+δVlδqk

= Qk

k = 1 , 2 , ... , N (3.13)


length. Equation 3.13 can be used to write N equations of motion. The governing

equations can be written as

M q + Gq + Kq = 0 (3.14)

where M is an N by N inertia matrix constructed from the coefficients of qi in which


mij =

ˆ L

0

(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 qi in which

element can be written as

gij =

ˆ L

0

(Ix2m(y) Ωφ′wi φβj − Ix2m(y) Ωφ′wj φβi

)dy (3.16)


and K is an N by N stiffness matrix constructed from the coefficients of qi in which


kij =

ˆ L

0

(Ix2m(y) Ω2 (φβi φβj + φ′wi φ

′wj) + GJy2(y)φ′βi φ

′βj + EIx2(y)φ′′wi φ

′′wj

)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

(λ2M + λG+K)a = 0 (3.18)

The values of λ can be solved for by finding the roots of the determinant of (λ2M +

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 Trial functions selection

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


non-uniform rotating blade. For example,

EIx2 =

´ L0EIx2(y)dy

L=

´ L0

(1.719E(+8)e−0.2214 y)dy

19.955= 3.84395E7N.m2 (3.19)


Thus, the employed trial functions for the bending displacements are


−0.00002836 sin(0.006039y) + 0.7339 sin(0.09396y)− 0.734 sinh(0.09395y) (3.20)


−0.00281346 sin(0.0378124y) + 1.01838 sin(0.235187y)− 1.01854 sinh(0.235045y)

(3.21)


+11.8222 sin(0.0787436y)− 1.74068 sin(0.339522y)− 1.00246 sinh(0.339095y)

(3.22)


+0.00780018 sin(0.105753y) + 0.995424 sin(0.393573y)− 0.999214 sinh(0.392906y)

(3.23)


+0.0003615 sin(0.006039y) + 0.001811 sin(0.09396y) + 0.001796 sinh(0.09395y)

(3.24)


+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)


−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 eigen-

value 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

M =

603.883 563.084 195.178 62.2911

563.084 1513.13 1258.98 897.879

195.178 1258.98 1712.43 1628.33

62.2911 897.879 1628.33 1829.38

(3.28)

G =

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.

(3.29)

K =

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

(3.30)

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.


Table 3.1: Computed approximate natural frequencies for a rotating blade usingassumed modes approach

Mode Rotating Blade Frequency Rotating Speed ADAMS model Modal surveyHz rad/s

2.07102 0 2.06 2.062.07102 1 - -

1 2.07104 2 - -2.07105 3 - -



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

numbers are obtained. One of them is λb = µL4ω2

EIx2, where λb is a dimensionless

parameter including the blade vibration frequency, the bending stiffness, mass per

unit length and the blade length. Other important dimensionless quantities are the

slenderness ratio rL

and the blade stiffness ratio GJy2EIx2

. The dimensionless parameter

is χ = GJy2EIx2

(Lr)2, 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


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 = Ix2mΩ2 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.


100

102

104

106

100

101

102

103

χ

λ b

f = 89

f = 0

f = 200

f = 25

First uniform blade uncoupled torsional mode

Second uniform blade uncoupled bending mode

Second nonuniform blade coupled mode

Second uniform blade coupled mode

First uniform blade uncoupled bending mode

First uniform blade coupled mode

First nonuniform blade coupled mode

Figure 3.1: λb-χ plot for both uniform and non-uniform rotating blades, whereλb = µL4ω2

EIx2and χ = GJy2

EIx2(Lr)2 and f = Ix2m Ω2

100

102

104

106

100

101

102

103

β

λ b

f = 89

f = 0

f = 200

f = 25

First nonuniform blade coupled mode

Second nonuniform blade coupled mode

Figure 3.2: λ-χ plot for a non-uniform rotating blade showing the first two coupledmodes, where λb = µL4ω2

EIx2and χ = GJy2

EIx2(Lr)2


104

105

106

107

100

60

70

80

90

120

140

160

180

χ

λ

First nonuniform coupled mode

f = 200

f = 89

f = 0

f = 25

Figure 3.3: λ-χ plot for a non-uniform rotating blade showing the first coupledmode, where λb = µL4ω2

EIx2and χ = GJy2

EIx2(Lr)2

104

105

106

107

χ

λ

f = 200

f = 89

f = 0

f = 25

Second nonuniform coupled mode

Figure 3.4: λ-χ plot for a non-uniform rotating blade showing the second coupledmode, where λb = µL4ω2

EIx2and χ = GJy2

EIx2(Lr)2

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 mid-

way 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 odd-

powered 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 Associ-

ation WWEA, April 2011.

[3] New and Renweable Energy Authority. National Strategy.

http://www.nrea.gov.eg/english1.html.

[4] “manufacturing wind turbine components in egypt”. How We Made It In

Africa, July 26 2010. http://www.howwemadeitinafrica.com/manufacture-

wind-turbine-components-in-egypt/155/.

[5] J. D. Anderson. Fundamentals of Aerodynamics. McGraw-Hill, 1984.

[6] M. O. L. Hansen. Aerodynamics of Wind Turbines. Earthscan, second edition,

2008.

[7] Y. C. Fung. An Introduction to the Theory of Aeroelasticity. Courier Dover,

2002.

[8] A. H. Nayfeh and P. F. Pai. Linear and Nonlinear Structural Mechanics. John

Wiley and Sons, 2004.

[9] M. R. M. Crespo Da Silva. “a comprehensive analysis of the dynamics of a

helicopter rotor blade”. International Journal of Solid Structures, 35(7-8):619–

635, 1998.

[10] D. H. Hodges and G. A. Pierce. Introduction to Structural Dynamics and Aeroe-

lasticity. Cambridge University Press, 2002.

[11] L. Meirovitch. Fundamentals of Vibrations. McGraw-Hill, 2001.

72

BIBLIOGRAPHY 73

[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-flexural-

torsional 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 timo-

shenko beams”. Journal of Sound and Vibration, 207(1):47–59, 1997.

[19] A. R. Sahu. “theoretical frequency equation of bending vibrations of an ex-

ponentially 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 anal-

ysis 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 Engineer-

ing Science, 222, 2008.

[23] G. Altintas. “effect of material properties on vibrations of nonsymmetrical

axially loaded thin-walled euler-bernoulli beam””. Mathematical and Compu-

tational Applications, 15:96–107, 2010.

BIBLIOGRAPHY 74

[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 EIx2L

and torsional stiffness

per unit length GJy2L

.

ω = f (µ, L, r,EIx2L

,GJy2L

)

µ m

L l

r l

EIx2L

(ml2t−2)

GJy2L

(ml2t−2)

Number of variables n = 6

Number of units j = 3 (Number of repeating variables)

k = n− j = 3, 3Πs

Π1(µ,EIx2L

,L, ω)

Π2(µ,EIx2L

,L, r)

Π3(µ,EIx2L

,L,GJy2L

)

(1)

2

Π1(µ,EIx2L

,L, ω)

(ml−1)a1(ml2t−2)b1(l)c1(t−1)

a1 + b1 = 0

−a1 + 2b1 + c1 = 0

−2b1 − 1 = 0

a1 = 1/2

b1 = −1/2

c1 = 3/2

Π21 =

µω2L4

EIx2= λb (2)

Π2(µ,EIx2L

,L, r)

(ml−1)a2(ml2t−2)b2(l)c2(t−1)

a2 + b2 = 0

−2a2 + 2b2 + c1 + 1 = 0

−2b2 = 0

a2 = 0

b2 = 0

c1 = −1

Π2 =r

L(3)

3

Π3(µ,EIx2L

,L,GJy2L

)

(ml−1)a3(ml2t−2)b3(l)c3(ml2t−2)

a3 + b3 + 1 = 0

−a3 + 2b3 + c3 + 2 = 0

−2b3 − 2 = 0

a3 = 0

b3 = −1

c3 = 0

Π3 =GJy2EIx2

(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<D - y D@v@x, tD, 8x, 2<D;Εxy =

1

2z D@Α@x, tD, xD;

Εxz =-1

2y D@Α@x, tD, xD;

Elastic potential energy

Ue0 =1

2El Εxx

2 + 2 G IΕxy2 + Εxz

2M;Ue0 =

1

2El Εxx

2 + 2 G IΕxy2 + Εxz

2M . 8x ® xt, y ® yt + ry, z ® zt + rz< Expand;

Integration wrt area

Ue =

1

2El A uH1,0L@xt, tD2

+1

2G J ΑH1,0L@xt, tD2

- El A ry uH1,0L@xt, tD vH2,0L@xt, tD +

1

2El IIzz v

H2,0L@xt, tD2+ El A rz uH1,0L@xt, tD wH2,0L@xt, tD -

El ry rz vH2,0L@xt, tD wH2,0L@xt, tD +1

2El IIyy w

H2,0L@xt, tD2;

Taking the variation

D@Ue, tD .9uH1,1L@xt, tD ® ∆u ', ΑH1,1L@xt, tD ® ∆Α', vH2,1L@xt, tD ® ∆v '', wH2,1L@xt, tD ® ∆w '',

uH1,0L@xt, tD ® u', ΑH1,0L@xt, tD ® Α', vH2,0L@xt, tD ® v'', wH2,0L@xt, tD ® w''=;

∆Ue = A El u¢ ∆u ¢+ G J Α¢ ∆Α

¢- A El ry ∆u ¢ v¢¢ + A El rz ∆u ¢ w¢¢ - A El ry u¢ ∆v ¢¢

+

El IIzz v¢¢ ∆v ¢¢

- El ry rz w¢¢ ∆v ¢¢+ A El rz u¢ ∆w ¢¢

- El ry rz v¢¢ ∆w ¢¢+ El IIyy w

¢¢ ∆w ¢¢;

Integrating with respect to time and space(along the elastic axis x)

ii∆Ue = àt1

t2à0

LIH H-A El u'L' + HA El ry v''L' - HA El rz w''L'L ∆u +

H- HGg J Α'L' L ∆Α + HHEl IIzz v''L'' - HEl A ry u'L'' - HEl A ry rz w''L''L ∆v +

IIEl IIyy w''M'' + HEl A rz u'L'' - HEl A ry rz v''L''M ∆w M âxt ât +

àt1

t2

A El u'L

0

-A El ry v''L

0

+A El rz w''L

0

∆u + Gg J Α' ∆ΑL

0

+ El IIzz v'' ∆v 'L

0

- HEl IIzz v''L' ∆vL

0

+ El IIyy w'' ∆w 'L

0

- IEl IIyy w''M' ∆wL

0

-El A ry u' ∆v 'L

0

+ HEl A ry u'L' ∆vL

0

- El A ry rz w'' ∆v 'L

0

+ HEl A ry rz w''L' ∆vL

0

+ El A rz u' ∆w 'L

0

- HEl A rz u'L' ∆wL

0

- El A ry rz v'' ∆w 'L

0

+ HEl A ry rz v''L' ∆wL

0

ât

2 PDEs with variations.nb

Kinetic Energy

r1 = : xy

z>;

r2@x_, t_D = :u@x, tDv@x, tDw@x, tD

> - :-z D@w@x, tD, xD + y D@v@x, tD, xD

-z Α@x, tDy Α@x, tD

>;r@x_, t_D = r1 + r2@x, tD Flatten;

Vel = D@r@x, tD, tD Flatten;

Ti =1

2Ρ Vel.VelH*.8x® xt, y® yt + ry, z® zt+rz<*L Expand

1

2Ρ uH0,1L@x, tD2

+

1

2Ρ vH0,1L@x, tD2

+

1

2Ρ wH0,1L@x, tD2

+ z Ρ vH0,1L@x, tD ΑH0,1L@x, tD -

y Ρ wH0,1L@x, tD ΑH0,1L@x, tD +

1

2y2 Ρ Α

H0,1L@x, tD2+

1

2z2 Ρ Α

H0,1L@x, tD2- y Ρ uH0,1L@x, tD vH1,1L@x, tD +

1

2y2 Ρ vH1,1L@x, tD2

+ z Ρ uH0,1L@x, tD wH1,1L@x, tD - y z Ρ vH1,1L@x, tD wH1,1L@x, tD +

1

2z2 Ρ wH1,1L@x, tD2

Differentiating the kinetic energy wrt time to get the variations and simplifying notations

VT = D@Ti , tD Expand

Ρ uH0,1L@x, tD uH0,2L@x, tD + Ρ vH0,1L@x, tD vH0,2L@x, tD +

z Ρ ΑH0,1L@x, tD vH0,2L@x, tD + Ρ wH0,1L@x, tD wH0,2L@x, tD - y Ρ Α

H0,1L@x, tD wH0,2L@x, tD +

z Ρ vH0,1L@x, tD ΑH0,2L@x, tD - y Ρ wH0,1L@x, tD Α

H0,2L@x, tD + y2 Ρ ΑH0,1L@x, tD Α

H0,2L@x, tD +

z2 Ρ ΑH0,1L@x, tD Α

H0,2L@x, tD - y Ρ uH0,2L@x, tD vH1,1L@x, tD + z Ρ uH0,2L@x, tD wH1,1L@x, tD -

y Ρ uH0,1L@x, tD vH1,2L@x, tD + y2 Ρ vH1,1L@x, tD vH1,2L@x, tD - y z Ρ wH1,1L@x, tD vH1,2L@x, tD +

z Ρ uH0,1L@x, tD wH1,2L@x, tD - y z Ρ vH1,1L@x, tD wH1,2L@x, tD + z2 Ρ wH1,1L@x, tD wH1,2L@x, tD

The kinetic energy time derivation in simplified notation,

VarT = Ρ uH0,1L@x, tD uH0,2L@x, tD + Ρ vH0,1L@x, tD vH0,2L@x, tD +

z Ρ ΑH0,1L@x, tD vH0,2L@x, tD + Ρ wH0,1L@x, tD wH0,2L@x, tD - y Ρ ΑH0,1L@x, tD wH0,2L@x, tD +

z Ρ vH0,1L@x, tD ΑH0,2L@x, tD - y Ρ wH0,1L@x, tD ΑH0,2L@x, tD + y2 Ρ ΑH0,1L@x, tD ΑH0,2L@x, tD +

z2 Ρ ΑH0,1L@x, tD ΑH0,2L@x, tD - y Ρ uH0,2L@x, tD vH1,1L@x, tD + z Ρ uH0,2L@x, tD wH1,1L@x, tD -

y Ρ uH0,1L@x, tD vH1,2L@x, tD + y2 Ρ vH1,1L@x, tD vH1,2L@x, tD - y z Ρ wH1,1L@x, tD vH1,2L@x, tD +

z Ρ uH0,1L@x, tD wH1,2L@x, tD - y z Ρ vH1,1L@x, tD wH1,2L@x, tD + z2 Ρ wH1,1L@x, tD wH1,2L@x, tD .9uH0,1L@x, tD ® u

, u@x, tD ® u, uH1,1L@x, tD ® u

', uH2,0L@x, tD ® u'', uH2,1L@x, tD ® u

'',

vH1,0L@x, tD ® v', v@x, tD ® v, vH1,1L@x, tD ® v ', wH1,0L@x, tD ® w',

ΑH0,1L@x, tD ® Α , vH0,1L@x, tD ® v

, wH0,1L@x, tD ® w

, w@x, tD ® w, wH1,1L@x, tD ® w

',

wH2,0L@x, tD ® w'', wH2,1L@x, tD ® w '', ΑH1,0L@x, tD ® Α', Α@x, tD ® Α,

ΑH1,1L@x, tD ® Α ', ΑH2,0L@x, tD ® Α'', vH1,2L@x, tD ® ∆v

' , wH1,2L@x, tD ® ∆w

' ,

ΑH0,2L@x, tD ® ∆Α , uH0,2L@x, tD ® ∆u

, vH0,2L@x, tD ® ∆v

, wH0,2L@x, tD ® ∆w

=Ρ u

∆u

+ Ρ v

∆v

+ z Ρ Α

∆v

+ Ρ w

∆w

- y Ρ Α

∆w

+ z Ρ v

∆Α

- y Ρ w

∆Α

+ y2 Ρ Α

∆Α

+ z2 Ρ Α

∆Α

-

y Ρ ∆u v ¢

+ z Ρ ∆u w ¢

- y Ρ u

∆v ¢

+ y2 Ρ v ¢

∆v ¢

- y z Ρ w ¢

∆v ¢

+ z Ρ u

∆w ¢

- y z Ρ v ¢

∆w ¢

+ z2 Ρ w ¢

∆w ¢

Integrate wrt area

PDEs with variations.nb 3

VaT = Μ u

∆u

+ Μ v

∆v

+ Μ z Α

∆v

+ Μ w

∆w

- Μ y Α

∆w

+ Μ z v

∆Α

- Μ y w

∆Α

+ Μ J Α

∆Α

- Μ y ∆u v ¢

+

Μ z ∆u w ¢

- Μ y u

∆v ¢

+ Μ IIzz v ¢

∆v ¢

- Μ y z w ¢

∆v ¢

+ Μ z u

∆w ¢

- Μ y z v ¢

∆w ¢

+ Μ IIyy w ¢

∆w ¢ .

8x ® xt, y ® yt + ry, z ® zt + rz< Expand

Μ u

∆u

+ Μ v

∆v

+ rz Μ Α

∆v

+ zt Μ Α

∆v

+ Μ w

∆w

- ry Μ Α

∆w

- yt Μ Α

∆w

+ rz Μ v

∆Α

+ zt Μ v

∆Α

-

ry Μ w

∆Α

- yt Μ w

∆Α

+ J Μ Α

∆Α

- ry Μ ∆u v ¢

- yt Μ ∆u v ¢

+ rz Μ ∆u w ¢

+ zt Μ ∆u w ¢

- ry Μ u

∆v ¢

-

yt Μ u

∆v ¢

+ Μ IIzz v ¢

∆v ¢

- ry rz Μ w ¢

∆v ¢

- rz yt Μ w ¢

∆v ¢

- ry zt Μ w ¢

∆v ¢

- yt zt Μ w ¢

∆v ¢

+

rz Μ u

∆w ¢

+ zt Μ u

∆w ¢

- ry rz Μ v ¢

∆w ¢

- rz yt Μ v ¢

∆w ¢

- ry zt Μ v ¢

∆w ¢

- yt zt Μ v ¢

∆w ¢

+ Μ IIyy w ¢

∆w ¢

∆T = Μ u

∆u

+ Μ v

∆v

+ rz Μ Α

∆v

+ Μ w

∆w

- ry Μ Α

∆w

+ rz Μ v

∆Α

- ry Μ w

∆Α

+ J Μ Α

∆Α

- ry Μ ∆u v ¢

+

rz Μ ∆u w ¢

- ry Μ u

∆v ¢

+ Μ IIzz v ¢

∆v ¢

- ry rz Μ w ¢

∆v

+ rz Μ u

∆w ¢

- ry rz Μ v ¢

∆w ¢

+ Μ IIyy w ¢

∆w ¢

ii∆T =

à0

Làt1

t2I-Μ I∆u u..

+ ∆v v..

+ ∆w w..M - Μ J I∆Α Α

..M - Μ rz Iv.. ∆ Α + Α..

∆ vM + ry Μ Iw.. ∆ Α + Α..

∆ wM + IΜ IIzz v..'M'

∆v + IΜ IIyy w..'M' ∆ w + Μ ry I∆u v

..'M - Μ rz I∆u w

..'M - IΜ ry u

..M' ∆ v +

IΜ rz u..M' ∆ w - IΜ ry rz v

..'M' ∆ w - IΜ ry rz w

..'M' ∆ v M âx ât +

àt1

t2

-Μ IIzz v..' ∆ v

L

0

- Μ IIyy w..' ∆ w

L

0

+ Μ ry u..

∆ vL

0

- Μ rz u..

∆ wL

0

+ Μ ry rz v..' ∆ w

L

0

+ Μ ry rz w..' ∆ v

L

0

ât

Integration wrt area, where Μ[x] is the density per unit length of the blade

Ordering terms according to orders of magnitude, O(u) = O(Α) =O(rz w) = O(ry v), neglecting terms with less orders of magnitude.

H*∆T = Μ II u

-ry v ¢

-rz w ¢ M∆u

+ v

∆v

+w

∆w

+Irz v

+ry w

+ J Α M∆Α

+IIzz v

¢∆v '+IIyy w

¢∆w 'M*L

Μ Jv ∆v

+ w

∆w

+ Irz v + ry w

+ J Α M ∆Α

+ ∆u

Iu - ry v ¢

- rz w ¢M + IIzz v

¢∆v ¢

+ IIyy w ¢

∆w ¢N

4 PDEs with variations.nb

Hamilton's

H*∆L = ∆T - ∆UeExpandFullSimplify*L∆ Μ u

2+ ∆ Μ v

2+ ∆ Μ w

2+ rz ∆ Μ v

Α

+ ry ∆ Μ w

Α

+ J ∆ Μ Α 2

- A El u¢∆u ¢

- G J Α¢

∆Α¢

+ ∆ Μ IIzz Hv ¢L2+

∆ Μ IIyy Hw ¢L2- ∆ Μ u

Iry v ¢+ rz w

¢M + A El ry ∆u ¢ v¢¢- A El rz ∆u ¢ w¢¢

- El IIzz v¢¢

∆v ¢¢- El IIzz w

¢¢∆w ¢¢

Integrating wrt time and x,

ii∆T =

à0

Làt1

t2II-Μ u..

+ Μ ry v..' - Μ rz w

..'M ∆u + I-Μ v

..- Μ rz Α

..+ IΜ IIzz v

..'M' - IΜ ry u

..M' - IΜ ry rz w..'M'M

∆v + I-Μ w..

+ ry Μ Α..

+ IΜ IIyy w..'M' + IΜ rz u

..M' - IΜ ry rz v..'M' M

∆w + I-Μ J Α..

- Μ rz v..

+ ry Μ w..M ∆ΑM âx ât +

àt1

t2

-Μ IIzz v..'

L

0

+ Μ ry u.. L

0

+ Μ ry rz w..'

L

0

∆v + - Μ IIyy w..'

L

0

- Μ rz u.. L

0

+ Μ ry rz v..'

L

0

∆w ât

ii∆Ue = àt1

t2à0

LIH H-A El u'L' + HA El ry v''L' - HA El rz w''L'L ∆u +

H- HGg J Α'L' L ∆Α + HHEl IIzz v''L'' - HEl A ry u'L'' - HEl A ry rz w''L''L ∆v +

IIEl IIyy w''M'' + HEl A rz u'L'' - HEl A ry rz v''L''M ∆w M âxt ât +

àt1

t2

A El u'L

0

-A El ry v''L

0

+A El rz w''L

0

∆u + Gg J Α' ∆ΑL

0

+ El IIzz v'' ∆v 'L

0

- HEl IIzz v''L' ∆vL

0

+ El IIyy w'' ∆w 'L

0

- IEl IIyy w''M' ∆wL

0

-El A ry u' ∆v 'L

0

+ HEl A ry u'L' ∆vL

0

- El A ry rz w'' ∆v 'L

0

+ HEl A ry rz w''L' ∆vL

0

+ El A rz u' ∆w 'L

0

- HEl A rz u'L' ∆wL

0

- El A ry rz v'' ∆w 'L

0

+ HEl A ry rz v''L' ∆wL

0

ât

PDEs

-Μ u..

+ Μ ry v..' - Μ rz w

..'+(A El u')' - (A El ry v'')' + (A El rz w'')' = fu

-Μ v..

- Μ rz Α..

+ IΜ IIzz v..'M' - IΜ ry u

..M' - IΜ ry rz w..'M' -

HEl IIzz v''L'' + HEl A ry u'L'' + HEl A ry rz w''L'' =

fv

-Μ w..

+ ry Μ Α..

+ IΜ IIyy w..'M' + IΜ rz u

..M' - IΜ ry rz v..'M' -

IEl IIyy w''M'' - HEl A rz u'L'' + HEl A ry rz v''L''= fw

-Μ J Α..

- Μ rz v..

+ ry Μ w..

+ Μ J Α..

+ Μ rz v..

- ry Μ w..

= fΑ

PDEs with variations.nb 5

Appendix C

Partial Differential Equations

Derivation Code

86

Overview:The PDEs for a 2-dof vibration of a rotating cantilevered blade are derived in this notebook. The blade canbend in the z-axis direction and rotate about the y0axis (elastic axis). The kinetic and potential energies arederived from the blade kinematics. Eventually, the blade PDEs are derived by the EulerLagrangeEquationscommand in Mathematica offered by the variational methods package. The PDEs are derived for the freeresponse case. The position of a point P' lying on the blade at (x,y,z) is denoted by r1 relative to the inertialframe and by r2 relative to the rotating frame.

Kinematics

Defining the blade's degrees of freedom

Β@y_, t_D; H*Torsional angle about the

elastic axis Hy-axisL as a function of y and time. *Lw@y_, t_D; H*Bending deflection in the z-

direction of the blade as a function of y and time*L

Transformation matrices between the inertial frame 0, the rotating frame attached to the root of the rotatingblade 1, the frame attached to a point P at a distance y away from the root of the beam after deformation inbending and torsion 3.

2 PDEs derivation.nb

h =

x0z1

; H*The position of a point P' relative to frame 3*L

D1 =

Cos@W tD -Sin@W tD 0 0

Sin@W tD Cos@W tD 0 0

0 0 1 0

0 0 0 1

; H*The transformation matrix between the inertial

frame 0 and the rotating frame 1. W is the angular speed of the blade rotation.*L

D2 =

1 0 0 0

0 1 0 y

0 0 1 0

0 0 0 1

.

1 0 0 0

0 1 -D@w@y, tD, yD 0

0 D@w@y, tD, yD 1 w@y, tD0 0 0 1

.

1 0 Β@y, tD 0

0 1 0 0

0 - Β@y, tD 0 1 0

0 0 0 1

;

H*The transformation matrix between frames 1 and 3. The first matrix represents the link

transformation from frame 1 to frame 2 before deformation. The second matrix represents

the transformation due to bending deflections between frames 1 and 2. The third matrix

represent the transformation between frames 2 and 3 due to torsional deformation.*Lr1 = D1 .D2. h;

r1 MatrixForm;

r1 = r1 . Β@y, tD wH1,0L@y, tD ® 0 ;H*The product Β@y,tD wH1,0L@y,tDis approximated as zero since the deformations are to be very small.*L

r1 MatrixForm

r2 = D2.h;

r2 MatrixForm;

x Cos@t WD - y Sin@t WD + z ICos@t WD Β@y, tD + Sin@t WD wH1,0L@y, tDMy Cos@t WD + x Sin@t WD + z ISin@t WD Β@y, tD - Cos@t WD wH1,0L@y, tDM

z + w@y, tD - x Β@y, tD1

Calculating the velocity of the point P' relative to the inertial frame 0

v1 = D@r1, tD;D@r1, tD MatrixForm

-y W Cos@t WD - x W Sin@t WD + z I-W Sin@t WD Β@y, tD + Cos@t WD ΒH0,1L@y, tD + W Cos@t WD wH1,0L@y, tD + Sin@t WD wH1,1L@y, tDM

x W Cos@t WD - y W Sin@t WD + z IW Cos@t WD Β@y, tD + Sin@t WD ΒH0,1L@y, tD + W Sin@t WD wH1,0L@y, tD - Cos@t WD wH1,1L@y, tDM

wH0,1L@y, tD - x ΒH0,1L@y, tD

0

Kinetic Energy

Calculating the kinetic energy per unit volume of the rotating blade.

PDEs derivation.nb 3

T1o = CollectB1

2Ρ [email protected] Flatten , 9W

2, W=F

:1

2Ρ W

2 Jx2 Cos@t WD2+ y2 Cos@t WD2

+ x2 Sin@t WD2+ y2 Sin@t WD2

+ 2 x z Cos@t WD2Β@y, tD + 2 x z

Sin@t WD2Β@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, tD2+ z2 Sin@t WD2 wH1,0L@y, tD2N +

1

2Ρ W

I-2 y z Cos@t WD2Β

H0,1L@y, tD - 2 y z Sin@t WD2Β

H0,1L@y, tD + 2 z2 Cos@t WD2Β

H0,1L@y, tD wH1,0L@y, tD +

2 z2 Sin@t WD2Β

H0,1L@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 WD2Β@y, tD wH1,1L@y, tD - 2 z2 Sin@t WD2

Β@y, tD wH1,1L@y, tDM +

1

2Ρ Jz2 Cos@t WD2

ΒH0,1L@y, tD2

+ z2 Sin@t WD2Β

H0,1L@y, tD2+ IwH0,1L@y, tD - x Β

H0,1L@y, tDM2+

z2 Cos@t WD2 wH1,1L@y, tD2+ z2 Sin@t WD2 wH1,1L@y, tD2N>

Dividing the kinetic energy term into 3 parts; T1a which consists of terms having 12

Ρ W2 as a coefficient, T1b

which consists of terms having ΡW as a coefficient and terms having 12

Ρ W0 as a coefficient.

T1a =

FullSimplifyBJx2 Cos@t WD2+ y2 Cos@t WD2

+ x2 Sin@t WD2+ y2 Sin@t WD2

+ 2 x z Cos@t WD2Β@y, tD + 2 x z

Sin@t WD2Β@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, tD2+ z2 Sin@t WD2 wH1,0L@y, tD2NF Expand

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

T1b = FullSimplifyAI-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 WD2Β

H0,1L@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 WD2Β@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 Β

H0,1L@y, tD wH1,0L@y, tD - 2 x z wH1,1L@y, tD - 2 z2 Β@y, tD wH1,1L@y, tD

T1c = FullSimplifyBJz2 Cos@t WD2Β

H0,1L@y, tD2+ z2 Sin@t WD2

ΒH0,1L@y, tD2

+

IwH0,1L@y, tD - x ΒH0,1L@y, tDM2

+ z2 Cos@t WD2 wH1,1L@y, tD2+ z2 Sin@t WD2 wH1,1L@y, tD2NF

wH0,1L@y, tD2- 2 x wH0,1L@y, tD Β

H0,1L@y, tD + Ix2 + z2M ΒH0,1L@y, tD2

+ z2 wH1,1L@y, tD2


T1to =1

2Ρ W

2 T1a +1

2Ρ T1c +

1

2Ρ W T1b H*Total kinetic per unit volume, same as T10 but

reformulated so that it is more easily integrated with respect ot area.*L1

2Ρ W

2 Jx2 + y2 + 2 x z Β@y, tD + z2 Β@y, tD2- 2 y z wH1,0L@y, tD + z2 wH1,0L@y, tD2N +

1

2Ρ W I-2 y z Β

H0,1L@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 +

1

2Ρ JwH0,1L@y, tD2

- 2 x wH0,1L@y, tD ΒH0,1L@y, tD + Ix2 + z2M Β

H0,1L@y, tD2+ z2 wH1,1L@y, tD2N

The kinetic energy per unit length T1. Since the blade is considered to be symmetric about the x-axis, then all area integral terms having z 1 were reduced to zero due to symmetry. In terms hav-ing x 1 , the x was changed to x + rx . So, rx remained and was multiplied by Μ and x terms werereduced to zero due to symmetry. Terms with z 2 were integrated to Ix2m after multiplying withthe 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.

T1 =1

2W2 JJzt1m + Ix2m Β@y, tD2

+ Ix2m wH1,0L@y, tD2+ Μ rx2N +

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

1

2JΜ wH0,1L@y, tD2

-

2 rx Μ wH0,1L@y, tD ΒH0,1L@y, tD + Jyt2m Β

H0,1L@y, tD2+ Ix2m wH1,1L@y, tD2

+ Μ rx2 ΒH0,1L@y, tD2N;


Potential Energy

Elastic potential energy

Εyy = DA-z wH1,0L@y, tD, yE

Εxy =1

2ID@z Β@y, tD, yD + DA-z wH1,0L@y, tD, xEM

Εyz =1

2ID@-x Β@y, tD + w@y, tD, yD + DA-z wH1,0L@y, tD, zEM

P2o =1

2El Εyy2

+ 2 G IΕxy 2+ Εyz 2M Expand

-z wH2,0L@y, tD

1

2z Β

H1,0L@y, tD

-

1

2x Β

H1,0L@y, tD

1

2G x2 Β

H1,0L@y, tD2+

1

2G z2 Β

H1,0L@y, tD2+

1

2El z2 wH2,0L@y, tD2

P2 =1

2El Ix2 wH2,0L@y, tD2

+1

2G Jy2 Β

H1,0L@y, tD2;

H*P2 =1

2El Ix2 wH2,0L@y,tD2

+1

2G Jy2 ΒH1,0L@y,tD2

+1

2G rx2 A ΒH1,0L@y,tD2

;*L

Gravitational Potential Energy

Pgo =

Ρ TransposeB0g

0

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

z + w@y, tD - x Β@y, tD Expand

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


Partial Differential Equations

<< VariationalMethods`

EulerEquationsB1

2W2 JJzt1m + Ix2m Β@y, tD2

+ Ix2m wH1,0L@y, tD2+ Μ rx2N + W I Ix2m Β

H0,1L@y, tD wH1,0L@y, tD -

Ix2m Β@y, tD wH1,1L@y, tDM +1

2JΜ wH0,1L@y, tD2

- 2 rx Μ wH0,1L@y, tD ΒH0,1L@y, tD +

Μ r2 ΒH0,1L@y, tD2

+ Ix2m wH1,1L@y, tD2+ Μ rx2 Β

H0,1L@y, tD2N -

1

2El Ix2 wH2,0L@y, tD2

+1

2G Jy2 Β

H1,0L@y, tD2- H g y Μ Cos@t WD - g rx Μ Sin@t WDL ,

8w@y, tD, Β@y, tD<, 8y, t<F Expand

9-Μ wH0,2L@y, tD + rx Μ ΒH0,2L@y, tD - 2 Ix2m W Β

H1,1L@y, tD -

Ix2m W2 wH2,0L@y, tD + Ix2m wH2,2L@y, tD - El Ix2 wH4,0L@y, tD 0,

Ix2m W2

Β@y, tD + rx Μ wH0,2L@y, tD - r2 Μ ΒH0,2L@y, tD - rx2 Μ Β

H0,2L@y, tD -

2 Ix2m W wH1,1L@y, tD + G Jy2 ΒH2,0L@y, tD 0=


Appendix D

Coupled Vibrations Exact

Analytical Solution Code

94

Overview

In this notebook, the plot Λb is plotted against Β = ΛbΛt

as an indicator of the influence of coupled and stiffness

ratio on the natural frequency. The particular solution used here is the same as the one used in the paper byDokumaci.

w@y_D; H*Bending deflection mode shape *LΒ@y_D; H* Torsional angle mode shape *Lw@y_, t_D = w@yD ãä Ω t;

H*The bending deflection is decoupled to two functions in space and time respectively.*LΒ@y_, t_D = Β@yD ãä Ω t;

H*The torsional angle is decoupled to two functions in space and time respectively.*L

The general solution for the coupled bending - torsional vibrations as derived in Dokumaci1 - AnalyticalSolution.nb.

w@y_D =

A1 CoshBs1y

LF + A2 SinhBs1

y

LF + A3 CosBs2

y

LF + A4 SinBs2

y

LF + A5 CosBs3

y

LF + A6 SinBs3

y

LF;

Β@y_D = B1 CoshBs1y

LF + B2 SinhBs1

y

LF + B3 CosBs2

y

LF +

B4 SinBs2y

LF + B5 CosBs3

y

LF + B6 SinBs3

y

LF;

The governing ODEs of the coupled bending-torsional vibrations as written in the paper by DokumaciΜ - mass per unit lengthEl - Elastic modulusG - Torsional rigidityJ - Polar area moment of inertia about the z-axisIx - Area moment of inertia about the x-axisL - Blade lengthrx - Shear center offset from the cross-section centroidr - Polar radius of gyrationΩ - Coupled vibration frequency

Λb = Ω2 ΜL4

El Ix

Λt = Ω2 Μ r2 L2

G J

The governing ODEs of the coupled bending-torsional vibrations as written in the paper by DokumaciΜ - mass per unit lengthEl - Elastic modulusG - Torsional rigidityJ - Polar area moment of inertia about the z-axisIx - Area moment of inertia about the x-axisL - Blade lengthrx - Shear center offset from the cross-section centroidr - Polar radius of gyrationΩ - Coupled vibration frequency

Λb = Ω2 ΜL4

El Ix

Λt = Ω2 Μ r2 L2

G J

eom1 = -Λb w@yD - rx Λb Β@yD + L4 D@w@yD, 8y, 4<D Expand

eom2 = 1 +rx2

r2Λt Β@yD + L2 D@Β@yD, 8y, 2<D +

rx

r2Λt w@yD Expand

After deriving the general solution for the mode shapes, we find that we have 12 unknown constants and only 6boundary conditions. After substituting the general solution in the ODEs above, we find that we can get 12equations by grouping the coefficients of the trigonmetric functions in the ODEs and equating them to zero.However, only 6 equations have to be selected out of the 12. The selection criteria are formulated such that theuncoupled solution satisfy the literature results.

e1 = CoefficientBeom1, CoshBs1y

LFF 0

e2 = CoefficientBeom1, SinhBs1y

LFF 0

e3 = CoefficientBeom1, CosBs2y

LFF 0

e4 = CoefficientBeom1, SinBs2y

LFF 0

e5 = CoefficientBeom2, CosBs3y

LFF 0

e6 = CoefficientBeom2, SinBs3y

LFF 0

Sol = Solve@8e1, e2, e3, e4, e5, e6<, 8B1, B2, B3, B4, A5, A6<DH*Solving the equations, we eliminate 6 out of 12 constant.*L

w@yD = w@yD . Sol

Β@yD = Β@yD . Sol

Rewriting the boundary conditions in terms of the remaining constants.

2 CoupledVibrationExactSolution.nb

BC1 = w@0D . Sol

BC2 = w'@0D . Sol

BC3 = w''@LD . Sol

BC4 = w'''@LD . Sol

BC5 = Β@0D . Sol

BC6 = Β'@LD . Sol

Constructing the matrix of coefficients for the constants A1, A2, A3, A4, B5 and B6 in the coupled bending-torsional mode shape boundary conditions.

coupledmat =

Coefficient@BC1, A1D Coefficient@BC1, A2D Coefficient@BC1, A3D Coefficient@BC1, A4D Coefficient






;

mc = Flatten@coupledmatD;cmat = Partition@mc, 6D;cmat MatrixForm

CoupledVibrationExactSolution.nb 3

Substituting the numerical values in mbdyn' s case study.

Clear@L, Μ, rx, G, El, r, cmatn, Γ, Ε, ΩD

Λb = Ω2 ΜL4

El Ix;

Λt = Ω2 Μ r2L2

G J;

Α = 1 +rx2

r2;

P = -Λb 1 +Α2 Λt2

3 Λb;

Q = -Λb Λt 1 -Α

3-

2 Α3 Λt2

27 Λb;

Φ = ArcCosB27 Q

2 P -PF;

a1 = 2-P

3CosB

Φ

3F;

a2 = 2-P

3CosB

Φ

3+2 Π

3F;

a3 = 2-P

3CosB

Φ

3+ 2

2 Π

3F;

Clear@s1, s2, s3, Γ, ΕD

s1 = a1 -Α Λt

3

2 14

;

s3 = a3 -Α Λt

3

2 14

; s2 = a2 -Α Λt

3

2 14

;

mu@y_D = 232.6 ã0.06363 y - 0.6628 y^2 - 27.95 y;


rxx@y_D = 0.08718 + 0.005621 [email protected] yD + 0.02088 [email protected] yD;rr@y_D = H0.3616 + 0.07763 [email protected] yDL;Gg@y_D = 4.423 107 ã-0.1995 y;

Ell@y_D = 1.719 108 ã-0.2214 y ;

Ix2mm@y_D = 12.88 ã-0.2294 y;

L = 19.955;

ro = SqrtBNIntegrateAmu@yD * y2, 8y, 0, L<ENIntegrate@mu@yD, 8y, 0, L<D

F

rx = NIntegrate@rxx@yD, 8y, 0, L<D L

Μ = NIntegrate@mu@yD, 8y, 0, L<D L

G = NIntegrate@Gg@yD, 8y, 0, L<D H J L LIx2m = NIntegrate@Ix2mm@yD, 8y, 0, L<D L

El = Ε NIntegrate@Ell@yD, 8y, 0, L<D H Ix LLr = NIntegrate@rr@yD, 8y, 0, L<D L

cmatn = cmat

cmatn MatrixForm;

H*Recalling the coupled coefficient matrix after setting numerical

values for the constants and saving the resultant matrix in cmatn*L


Solving for the vibration frequencies at Ε = 1 for example.

Clear@Γ, ΕDΕ = 100;

Γ = 1;

Plot@Det@cmatnD, 8Ω, 140, 500<DΩ . 8FindRoot@Det@cmatnD, 8Ω, 10<D,

FindRoot@Det@cmatnD, 8Ω, 30<D, FindRoot@Det@cmatnD, 8Ω, 100<D,FindRoot@Det@cmatnD, 8Ω, 200<D, FindRoot@Det@cmatnD, 8Ω, 170<D,FindRoot@Det@cmatnD, 8Ω, 230<D, FindRoot@Det@cmatnD, 8Ω, 270<D,FindRoot@Det@cmatnD, 8Ω, 450<D , FindRoot@Det@cmatnD, 8Ω, 500<D<

D@Det@cmatnD, ΩD . Ω ® 5.5446201612398704`

D@Det@cmatnD, ΩD . Ω ® 34.70343670432565`

D@Det@cmatnD, ΩD . Ω ® 100.44248735785521`

D@Det@cmatnD, ΩD . Ω ® 150.01834720330547`

200 250 300 350 400 450 500

-500 000

500 000

1 ´ 106

85.41696, 33.9079, 94.8126, 185.181, 212.838, 212.838, 212.838, 452.574, 498.507<

-7.19443 ´ 10-9

0.0000437723

0.0610392

1.58229

Clear@Γ, Ε, Βnc, ΛbncD

Βnc @Ε_, Γ_D =Λb

Λt

Λbnc @Ω_, Ε_D = Λb

Βlc = MapThreadAΒnc,

990.0001, 0.001, 10-2.8, 10-2.6, 10-2.4, 10-2.2, 0.01, 10-1.8, 10-1.6, 10-1.4, 10-1.2, 0.1,

10-0.8, 10-.6, 10-0.4, 10-0.2, 1, 100.2, 100.4, 100.6, 100.8, 10, 101.2, 101.4, 101.6,

101.8, 100 , 102.2, 102.4, 1000, 10 000=, 81, 1, 1, 1, 1, 1, 1, 1, 1, 1,

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1<=E


firstΛbCoupled =

MapThreadAΛbnc, 980.05545574059041422`, 0.17536642264551425`, 0.22077322379308742`,

0.2779369776792456`, 0.34990183628938715`, 0.44050013817259465`, 0.554556468327618`,

0.6981445328260145`, 0.8789105017763282`, 1.106479988075812`, 1.3929702316332127`,

1.7536345578422814`, 2.207673015512618`, 2.7792515310075525`, 3.498782173291105`,

4.404529302671492` , 5.5446201612398704`, 6.97955247955464`, 8.785301821745769`,

11.057126359163574`, 13.914114840066924`, 17.504329468538124`, 22.009739656003`,

27.647773762298492`, 34.65618318035894`, 43.19103680767989`, 52.63302008726953`,

58.91951670313392`, 60.7601103649714`, 61.73955154406432`, 61.907366951701945`<,90.0001, 0.001, 10-2.8, 10-2.6, 10-2.4, 10-2.2, 0.01, 10-1.8, 10-1.6, 10-1.4,

10-1.2, 0.1, 10-0.8, 10-.6, 10-0.4, 10-0.2, 1, 100.2, 100.4, 100.6,

100.8, 10, 101.2, 101.4, 101.6, 101.8, 100 , 102.2, 102.4, 1000, 10 000==E

secondΛbCoupled =

MapThreadAΛbnc, 980.34753515848623767`, 1.0990014895482287`, 1.3835599385811617`,

1.741796841123232`, 2.192788464591716`, 2.7605494527531724`, 3.4753105428082094`,

4.375126156532354`, 5.507896301850886`, 6.933908064315541`, 8.729027084551273`,

10.988698540677055`, 13.832947934775701`, 17.412595616020898`, 21.916884667267468`,

27.58254977495702`, 34.70343670432565`, 43.63378759308861`, 54.694423185771996`,

62.78743877812869`, 63.21557941378641`, 63.330377019127695`, 63.4338887884413`,

63.58547906896941`, 63.86663956187842`, 64.51633156649038`, 66.64826277983792`,

74.94463236169192`, 91.47016399314931`, 176.6406102173198`, 185.7210746623806`<, 90.0001, 0.001, 10-2.8, 10-2.6, 10-2.4, 10-2.2, 0.01, 10-1.8, 10-1.6,

10-1.4, 10-1.2, 0.1, 10-0.8, 10-.6, 10-0.4, 10-0.2, 1, 100.2, 100.4, 100.6,

100.8, 10, 101.2, 101.4, 101.6, 101.8, 100 , 102.2, 102.4, 1000, 10 000==E

thirdΛbCoupled =

MapThreadAΛbnc, 980.9731083465472835`, 3.0772309171863297`, 3.8739977595328075`,

4.877061369306674`, 6.13983082447647`, 7.729537775899698`, 9.73080903781806`,

12.250157682275562`, 15.421623645217299`, 19.413847094405774`, 24.438913817929162`,

30.76335494261762`, 38.721631111504294`, 48.73156829077164`, 61.26122388545168`,

63.40918872835164`, 63.44248735785521`, 63.50509978384452`, 63.74801496958866`,

69.8457993254728`, 87.185083048919`, 109.1650036744073`, 135.9166104196782`,

164.5101235272911`, 179.86720321531104`, 183.62777279026665`, 184.88578392971118`,

185.46917771316794`, 185.80659760953316`, 189.2562875450527`, 309.59969285249554`<,90.0001, 0.001, 10-2.8, 10-2.6, 10-2.4, 10-2.2, 0.01, 10-1.8, 10-1.6, 10-1.4,

10-1.2, 0.1, 10-0.8, 10-.6, 10-0.4, 10-0.2, 1, 100.2, 100.4, 100.6,

100.8, 10, 101.2, 101.4, 101.6, 101.8, 100 , 102.2, 102.4, 1000, 10 000==E

fourthΛbCoupled =

MapThreadAΛbnc, 981.9069035187487506`, 6.030129875234055`, 7.591460369575675`,

9.557035684438748`, 19.888684683686183`, 15.146576294931156`, 19.06803447161629`,

24.004476342437467`, 30.218305734376127`, 38.039395334931996`, 47.88163850870627`,

63.425369426612534`, 63.39586901205343`, 63.40901616971091`, 63.48182644544462`,

77.17258355159576`, 97.01834720330547`, 121.82577466026827`, 152.38396839377938`,

183.21681527360505`, 189.13144098747705`, 190.58016727189283`, 192.76485817999796`,

200.38078479254432`, 230.2097191012146`, 280.2940676197781`, 305.8880863261297`,

308.2530898578358`, 308.926198361812`, 309.4934434049744`, 433.4364988400108`<,90.0001, 0.001, 10-2.8, 10-2.6, 10-2.4, 10-2.2, 0.01, 10-1.8, 10-1.6, 10-1.4,

10-1.2, 0.1, 10-0.8, 10-.6, 10-0.4, 10-0.2, 1, 100.2, 100.4, 100.6,

100.8, 10, 101.2, 101.4, 101.6, 101.8, 100 , 102.2, 102.4, 1000, 10 000==E


interFirstCoupledMode = Riffle@Βlc, firstΛbCoupledD Flatten;

firstCoupledMode = Partition@interFirstCoupledMode, 2D

interSecondCoupledMode = Riffle@Βlc, secondΛbCoupledD Flatten;

secondCoupledMode = Partition@interSecondCoupledMode , 2D;

interThirdCoupledMode = Riffle@Βlc, thirdΛbCoupledD Flatten;

thirdCoupledMode = Partition@interThirdCoupledMode, 2D;

interFourthCoupledMode = Riffle@Βlc, fourthΛbCoupledD Flatten;

fourthCoupledMode = Partition@interFourthCoupledMode , 2D;


ListLogLogPlotA8firstCoupledMode, secondCoupledMode, thirdCoupledMode, fourthCoupledMode<,PlotRange ® 980.6, 30 000<, 95, 105==, Joined ® True, PlotStyle ® BlackE


Clear@Γ, Ε, Βn, ΛbnD

Βn @Ε_, Γ_D =Λb

Λt

Λbn @Ω_, Ε_D = Λb

ΒlBending = MapThread@Βn, 880.0001, 1000 <, 81, 1<<D

firstΛbBending =

MapThread@Λbn, 88 0.05545574153739154`, 175.36645259176498`<, 80.0001, 1000 <<DsecondΛbBending =

MapThread@Λbn, 880.3475351998796204`, 1099.0027987014741`<, 80.0001, 1000 <<DthirdΛbBending =

MapThread@Λbn, 88 0.9731086229196968`, 3077.2396591762135`<, 80.0001, 1000 <<DfourthΛbBending =

MapThread@Λbn, 881.9069045200233559` , 6030.161563743895`<, 80.0001, 1000 <<DΒlTorsion = MapThread@Βn, 881, 1<, 80.001, 1000 <<D

firstΛbTorsion = MapThread@Λbn, 882.004760008676603`, 2004.7600086766026`<, 81, 1 <<D

secondΛbTorsion = MapThread@Λbn, 886.014280026029809`, 6014.280026029809`<, 81, 1 <<D

thirdΛbTorsion = MapThread@Λbn, 8810.023800043383016`, 10023.800043383016`<, 81, 1 <<D


fourthΛbTorsion = MapThread@Λbn, 8814.033320060736221`, 14033.320060736221`<, 81, 1 <<D

interFirstBendingMode = Riffle@ΒlBending, firstΛbBendingD Flatten;

firstBendingMode = Partition@interFirstBendingMode, 2D;interFirstTorsionMode = Riffle@ΒlTorsion, firstΛbTorsionD Flatten;

firstTorsionMode = Partition@interFirstTorsionMode, 2D;

interSecondBendingMode = Riffle@ΒlBending, secondΛbBendingD Flatten;

secondBendingMode = Partition@interSecondBendingMode , 2D;interSecondTorsionMode = Riffle@ΒlTorsion, secondΛbTorsionD Flatten;

secondTorsionMode = Partition@interSecondTorsionMode , 2D;

interThirdBendingMode = Riffle@ΒlBending, thirdΛbBendingD Flatten;

thirdBendingMode = Partition@interThirdBendingMode, 2D;interThirdTorsionMode = Riffle@ΒlTorsion, thirdΛbTorsionD Flatten;

thirdTorsionMode = Partition@interThirdTorsionMode, 2D;

interFourthBendingMode = Riffle@ΒlBending, fourthΛbBendingD Flatten;

fourthBendingMode = Partition@interFourthBendingMode , 2D;interFourthTorsionMode = Riffle@ΒlTorsion, fourthΛbTorsionD Flatten;

fourthTorsionMode = Partition@interFourthTorsionMode , 2D;


Plotting the mode shapes Ω ® 5.54462 Ω ® 34.7034Ω ® 63.4425 Ω ® 97.0183

Γ = 1; Ε = 1;

H*First Bending and Torsional Mode*L

Clear@Ω, A1DΩ = 5.54462;

A1 = 1;

constantsFirstMode = Solve@Flatten@8BC1 0, BC2 0, BC3 0, BC4 0, BC5 0, BC5 0<D, 8A2, A3, A4, B5, B6<D;

firstBendingModeShape = w@yD . Ω ® 5.54462 . constantsFirstMode ;

Plot@firstBendingModeShape , 8y, 0, L<DfirstTorsionalModeShape = Β@yD . Ω ® 5.54462 . constantsFirstMode ;

Plot@firstTorsionalModeShape , 8y, 0, L<D


H*Second Bending and Torsional Modes*L

Clear@Ω, A1DΩ = 34.7034;

A1 = 1

constantsSecondMode = Solve@Flatten@8BC1 0, BC2 0, BC3 0, BC4 0, BC5 0, BC5 0<D, 8A2, A3, A4, B5, B6<D;

secondBendingModeShape = w@yD . constantsSecondMode ;

Plot@secondBendingModeShape , 8y, 0, L<DsecondTorsionalModeShape = Β@yD . constantsSecondMode ;

Plot@secondTorsionalModeShape , 8y, 0, L<DH*Third Bending and Torsional Mode*L

Clear@Ω, A1DΩ = 63.4425;

A1 = 1

constantsThirdMode = Solve@Flatten@8BC1 0, BC2 0, BC3 0, BC4 0, BC5 0, BC5 0<D, 8A2, A3, A4, B5, B6<D;

thirdBendingModeShape = w@yD . constantsThirdMode ;

Plot@thirdBendingModeShape , 8y, 0, L<DthirdTorsionalModeShape = Β@yD . constantsThirdMode ;

Plot@thirdTorsionalModeShape , 8y, 0, L<D

H*Fourth Bending Mode*L

Clear@Ω, A1DΩ = 97.0183;

A1 = 1

constantsFourthMode = Solve@Flatten@8BC1 0, BC2 0, BC3 0, BC4 0, BC5 0, BC5 0<D, 8A2, A3, A4, B5, B6<D;

fourthBendingModeShape = w@yD . constantsFourthMode ;

Plot@fourthBendingModeShape , 8y, 0, L<DfourthTorsionalModeShape = Β@yD . constantsFourthMode ;

Plot@fourthTorsionalModeShape , 8y, 0, L<D


Appendix E

Assumed Modes Discretization

Code

108

Assumed Modes Discretization for a Non-Uniform RotatingBlade

Defining Trial Functions as the exact solution for a uniform blade with the average properties

W@y_D = A3 CosBs2 y

LF +

IB5 r2 s32 - B5 r2 Λt - B5 rx2 ΛtM CosB s3 y

LF

rx Λt+ A1 CoshB

s1 y

LF +

A4 SinBs2 y

LF +

IB6 r2 s32 - B6 r2 Λt - B6 rx2 ΛtM SinB s3 y

LF

rx Λt+ A2 SinhB

s1 y

LF;

Β@y_D =

IA3 s24 - A3 ΛbM CosB s2 y

LF

rx Λb+ B5 CosB

s3 y

LF +

IA1 s14 - A1 ΛbM CoshB s1 y

LF

rx Λb+

IA4 s24 - A4 ΛbM SinB s2 y

LF

rx Λb+ B6 SinB

s3 y

LF +

IA2 s14 - A2 ΛbM SinhB s1 y

LF

rx Λb;

Λb = Ω2 ΜL4

El;

Λt = Ω2 Μ r2L2

G;

Α = 1 +rx2

r2;

P = -Λb 1 +Α2 Λt2

3 Λb;

Q = -Λb Λt 1 -Α

3-

2 Α3 Λt2

27 Λb;

Φ = ArcCosB27 Q

2 P -PF;

a1 = 2-P

3CosB

Φ

3F;

a2 = 2-P

3CosB

Φ

3+2 Π

3F;

a3 = 2-P

3CosB

Φ

3+ 2

2 Π

3F;

s1 = a1 -Α Λt

3

2 14

;

s3 = a3 -Α Λt

3

2 14

;

s2 = a2 -Α Λt

3

2 14

;

BC1 = W@0D ;

BC2 = W'@0D;BC3 = W''@LD;BC4 = W'''@LD;BC5 = Β@0D;BC6 = Β'@LD;

d@t_D; d1@t_D; d2@t_D; d3@t_D; d4@t_D; d@tD = 8d1@tD, d2@tD, d3@tD, d4@tD<;

2 AssumedModes.nb

mu@y_D = 232.6 ã0.06363 y - 0.6628 y^2 - 27.95 y;

rxx@y_D = 0.08718 + 0.005621 [email protected] yD + 0.02088 [email protected] yD;rr@y_D = H0.3616 + 0.07763 [email protected] yDL;Gg@y_D = 4.423 107 ã-0.1995 y;

Ell@y_D = 1.719 108 ã-0.2214 y ;

Ix2mm@y_D = 12.88 ã-0.2294 y;

L = 19.955;

H*Jy2m@y_D = -0.001525 y^4 + 0.07912 y^3 -1.283 y^2 +4.641 y +32.01;*LJy2m@y_D = 34.85 ã0.1006 y - 0.4936 y2 - 2.979 y;

L = 19.955;

ro = SqrtBNIntegrateAmu@yD * y2, 8y, 0, L<ENIntegrate@mu@yD, 8y, 0, L<D

F

rx = NIntegrate@rxx@yD, 8y, 0, L<D L

Μ = NIntegrate@mu@yD, 8y, 0, L<D L

G = NIntegrate@Gg@yD, 8y, 0, L<D HL LIx2m = NIntegrate@Ix2mm@yD, 8y, 0, L<D L

El = Ε NIntegrate@Ell@yD, 8y, 0, L<D L

r = NIntegrate@rr@yD, 8y, 0, L<D L

Exact frequencies from the exact solution

H*Ε = 10-4*L

Ω1 =

80.05147689484063991`, 0.32260016869804164`, 0.9032897024119771`, 1.7700869830281618`<;H*Ε = 10-3

*LΩ2 = 80.16278421589629868`, 1.0201504903312566`, 2.8564473957390693`, 5.597486766203126`<;H*Ε = 10-2.8

*LΩ3 = 80.20493317075327874`, 1.2842927081509274`, 3.5960497528089803`, 7.046802151503228`<;H*Ε = 10-2.6

*LΩ4 = 80.2579955458978555`, 1.6168273934386141`, 4.527149511669745`, 8.871365981587278`<;H*Ε = 10-2.4

*LΩ5 = 80.3247970880381118`, 2.0354624315829537`, 5.699325787235554`, 18.46186621678898`<;H*Ε = 10-2.2

*LΩ6 = 80.4088951864486579`, 2.5624900691045585`, 7.174990564894016`, 14.059956993668727`<;H*Ε = 10-2

*LΩ7 = 80.5147682988286992`, 3.225973261589984`, 9.032707023489781`, 17.700178107159562`<;H*Ε = 10-1.8

*LΩ8 = 80.6480544093989621`, 4.061238535490123`, 11.371362572991385`, 22.282683173195434`<;H*Ε = 10-1.6

*LΩ9 = 80.8158512000591143`, 5.112754057956717`, 14.31541328926235`, 28.051182697881067`<;

AssumedModes.nb 3

Ω9 = 80.8158512000591143`, 5.112754057956717`, 14.31541328926235`, 28.051182697881067`<;H*Ε = 10-1.4

*LΩ10 = 81.0270938839868564`, 6.436491292734335`, 18.02146753899715`, 35.31218964032004`<;H*Ε = 10-1.2

*LΩ11 = 81.2930307511178825`, 8.10289265531048`, 22.68653492883803`, 44.45084745845464`<;H*Ε = 10-1

*LΩ12 = 81.627821606928113`, 10.200596221345258`, 28.558324914065462`, 67.09725756976944`<;H*Ε = 10-0.8

*LΩ13 = 82.049290687096452`, 12.841100922094371`, 35.94800477271191`, 67.0648824458346`<;H*Ε = 10-0.6

*LΩ14 = 82.579873568975992`, 16.164587467027562`, 45.2456258979318`, 67.09403344782945`<;H*Ε= 10-0.4

*LΩ15 = 83.247807350695184`, 20.347126112046965`, 56.93527913835524`, 67.11123131817821`<;H*Ε = 10-0.2

*LΩ16 = 84.088625143675724`, 25.60944681575679`, 67.06674194868862`, 71.69897876643674`<;H*Ε = 100*LΩ17 = 85.1470297106591865`, 32.227084968008796`, 67.11580002102171`, 90.14171688841705`<;H*Ε = 100.2*LΩ18 = 86.479236234521975`, 40.539706464388054`, 67.15707171774199`, 113.2882909051066`<;H*Ε = 100.4*LΩ19 = 88.155888450175745`, 50.94082531113244`, 67.25924709108726`, 142.121191587708`<;H*Ε = 100.6*LΩ20 = 810.265659218066308`, 63.31424817588562`, 68.08334574465447`, 176.83087148305262`<;H*Ε = 100.8*LΩ21 = 812.919628098067692`, 66.76563458368898`, 81.18739253076973`, 198.49361666919165`<;H*Ε = 10*LΩ22 = 816.25643289455957`, 66.96119841690717`, 101.68052636080965`, 200.86131011510454`<;H*Ε = 101.2*LΩ23 = 820.447857211026246`, 67.06428466889649`, 127.15799502018912`, 202.41480168216367`<;H*Ε = 101.4*LΩ24 = 825.703362458452393`, 67.18489289478981`, 157.31978136366007`, 205.95558037181786`<;H*Ε = 101.6*LΩ25 = 832.26763146115732`, 67.3832117330641`, 183.94159513975413`, 221.52556519821425`<;H*Ε = 101.8*LΩ26 = 840.384038239666886`, 67.78472328808967`, 192.90619270006414`, 264.86520463624805`<;H*Ε = 102*LΩ27 = 850.05838129123676`, 68.84349520870325`, 195.21342458039803`, 316.0217565228044`<;H*Ε = 102.2*LΩ28 = 859.43357587088149`, 72.99291197577938`, 196.1191655525916`, 325.5548194819424`<;H*Ε = 102.4*LΩ29 = 863.587468691413356`, 85.87699723588034`, 196.58167614674102`, 326.84548153247846`<;H*Ε = 103*L

4 AssumedModes.nb

H*Ε = 103*LΩ30 = 865.31878788117646`, 166.13495208000055`, 197.82635548207577`, 327.69736726365227`<;H*Ε = 104*LΩ31 = 865.55342238833991`, 196.65706123899398`, 327.8505082553225`, 458.9658302100301`<;

Ωi = 8Ω1, Ω2, Ω3, Ω4, Ω5, Ω6, Ω7, Ω8, Ω9, Ω10, Ω11, Ω12, Ω13, Ω14, Ω15, Ω16,

Ω17, Ω18, Ω19, Ω20, Ω21, Ω22, Ω23, Ω24, Ω25, Ω26, Ω27, Ω28, Ω29, Ω30, Ω31<;

Εi = 8-4, -3, -2.8, -2.6, -2.4, -2.2, -2, -1.8, -1.6, -1.4, -1.2, -1, -0.8, -0.6,

-0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.4, 3, 4<;

ForAWi = 8<; Βi = 8<; i = 1, i < 32, i++,

ForAWj = 8<; Βj = 8<; j = 1, j < 5, j++,

Clear@b1, b2, b3, b4, b5, b6D;b1 = BC1 . 9Ω ® Ωi@@i, jDD, Ε ® 10Εi @@iDD=; b2 = BC2 . 9Ω ® Ωi@@i, jDD, Ε ® 10Εi @@iDD=;b3 = BC3 . 9Ω ® Ωi@@i, jDD, Ε ® 10Εi @@iDD=; b4 = BC4 . 9Ω ® Ωi@@i, jDD, Ε ® 10Εi @@iDD=;b5 = BC5 . 9Ω ® Ωi@@i, jDD, Ε ® 10Εi @@iDD=; b6 = BC6 . 9Ω ® Ωi@@i, jDD, Ε ® 10Εi @@iDD=;constant = Chop@Solve@8b1 0, b2 0, b4 0, b5 0, b6 0<, 8A2, A3, A4, B5, B6<DD;A1 = 1; Wj = AppendAWj, d@tD@@jDD

FlattenAW@yD . 9Ω ® Ωi@@i, jDD, Ε ® 10Εi @@iDD= . constantE ChopE; Βj = AppendAΒj,

d@tD@@jDD FlattenA Β@yD . 9Ω ® Ωi@@i, jDD, Ε ® 10Εi @@iDD= . constantE ChopEE;Wi = Append@Wi, Flatten@WjDD;Βi = Append@Βi, Flatten@ΒjDDE

Introducing the rotational motion

Clear@omega, OmDomega = 0; Om = 8omega<;For@m = 1, m < 31, m++, omega = omega + 1; Om = Append@Flatten@OmD, omegaDD

Εi = 8-4, -3, -2.8, -2.6, -2.4, -2.2, -2, -1.8, -1.6, -1.4, -1.2, -1, -0.8, -0.6,

-0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.4, 3, 4<;

Assumed modes

AssumedModes.nb 5

lamO = 8<; Clear@TLy, VLyDForBn = 1, n < 4, n++, Clear@lambdaD; lambda = 8<;

ForBk = 1, k < 32, k++, TLy = 0; VLy = 0; Clear@ΕD; Ε = 10Εi@@kDD;

ForAi = 1, i < 5, i++,

ForAj = 1, j < 5, j++,

TLy = TLy + 0.5 * mu@yD * Irxx@yD2 + rr@yD2M * D@Βi@@k, iDD, tD * D@Βi@@k, jDD, tD +

0.5 * mu@yD * D@Wi@@k, iDD, tD * D@Wi@@k, jDD, tD -

rxx@yD * mu@yD * D@Wi@@k, iDD, tD * D@Βi@@k, jDD, tD + 0.5 * Om@@nDD2* Imu@yD ro2 +

Ix2mm@yD * HΒi@@k, iDD * Βi@@k, jDD + D@Wi@@k, iDD, yD * D@Wi@@k, jDD, yDLM + Ix2mm@yDOm@@nDD H-Βi@@k, iDD * D@D@Wi@@k, jDD, tD, yD + D@Βi@@k, iDD, tD * D@Wi@@k, jDD, yDL;

VLy = VLy + 0.5 * 10Εi@@kDD Ell@yD * D@Wi@@k, iDD, 8y, 2<D * D@Wi@@k, jDD, 8y, 2<D +

0.5 * Gg@yD * D@Βi @@k, iDD, yD * D@Βi@@k, jDD, yDEE;mass =

Coefficient@D@TLy, d1'@tDD, d1'@tDD Coefficient@D@TLy, d1'@tDD, d2'@tDD Coefficient@D@TLyCoefficient@D@TLy, d2'@tDD, d1'@tDD Coefficient@D@TLy, d2'@tDD, d2'@tDD Coefficient@D@TLyCoefficient@D@TLy, d3'@tDD, d1'@tDD Coefficient@D@TLy, d3'@tDD, d2'@tDD Coefficient@D@TLyCoefficient@D@TLy, d4'@tDD, d1'@tDD Coefficient@D@TLy, d4'@tDD, d2'@tDD Coefficient@D@TLy

; stiffness =

Coefficient@D@TLy, d1@tDD, d1@tDD + Coefficient@D@VLy, d1@tDD, d1@tDD Coefficient@D@TLy, d1




; Gyro =

Coefficient@D@TLy, d1@tDD, d1'@tDD + Coefficient@D@VLy, d1@tDD, d1'@tDD Coefficient@D@TLy,Coefficient@D@TLy, d2@tDD, d1'@tDD + Coefficient@D@VLy, d2@tDD, d1'@tDD Coefficient@D@TLy,Coefficient@D@TLy, d3@tDD, d1'@tDD + Coefficient@D@VLy, d3@tDD, d1'@tDD Coefficient@D@TLy,Coefficient@D@TLy, d4@tDD, d1'@tDD + Coefficient@D@VLy, d4@tDD, d1'@tDD Coefficient@D@TLy,

;

M = NIntegrate@mass, 8y, 0, L<D; S = NIntegrate@stiffness, 8y, 0, L<D;Gy = NIntegrate@Gyro, 8y, 0, L<D; mat = PartitionAFlattenA-Λ2 M + Λ I Gy + SE, 4E;solution = Solve@Det@matD 0, ΛD;inter = solution@@5 ;; 8DD; lambda = Append@lambda, Flatten@interDDF;

lamO = Append@lamO, lambdaDF

Clear@ΛD

6 AssumedModes.nb

Clear@lam, lam1, lam2, inter1, ΛDlam = 8<;For @a = 1, a < 5, a++, For@lam2 = 8<; n = 1, n < 32, n++,

For@lam1 = 8<; c = 1, c < 5, c++, Λ = Λ . lamO@@a, n, cDD; inter1 = Λ;

lam1 = Append @lam1, inter1DD; lam2 = Append@lam2, lam1DD; lam = Append@lam, lam2DD

ForBlb1 = 8<; a = 1, a < 5, a++,

For Bfirst = 8<; Be = 8<; i = 1, i < 32, i++, Clear@l, beta, Ε, Λb, ΛtD;

Λb = l2 ΜL4

El; Λt = l2 Μ r2

L2

G; Ε = 10Εi@@iDD; l = lam@@a, i, 1DD; beta =

Λb

Λt;

Be = Append@Be, beta D; first = Append@first, ΛbDF; lb1 = Append@lb1, firstDF

ForBlb2 = 8<; a = 1, a < 5, a++,

For Bsecond = 8<; Be = 8<; i = 1, i < 32, i++, Clear@l, beta, Ε, Λb, ΛtD;

Λb = l2 ΜL4

El; Λt = l2 Μ r2

L2


Λb

Λt;

Be = Append@Be, beta D; second = Append@second, ΛbDF; lb2 = Append@lb2, secondDF

ForBlb3 = 8<; a = 1, a < 5, a++,

For Bthird = 8<; Be = 8<; i = 1, i < 32, i++, Clear@l, beta, Ε, Λb, ΛtD;

Λb = l2 ΜL4

El; Λt = l2 Μ r2

L2


Λb

Λt;

Be = Append@Be, beta D; third = Append@third, ΛbDF; lb3 = Append@lb3, thirdDF

ForBlb4 = 8<; a = 1, a < 5, a++,

For Bfourth = 8<; Be = 8<; i = 1, i < 32, i++, Clear@l, beta, Ε, Λb, ΛtD;

Λb = l2 ΜL4

El; Λt = l2 Μ r2

L2


Λb

Λt;

Be = Append@Be, beta D; fourth = Append@fourth, ΛbDF; lb4 = Append@lb4, fourthDF

Assumed Modes Discretization for a Uniform Roating Blade with the AverageProperties

Clear@ΛD; TLy = 0; VLy = 0;

AssumedModes.nb 7

lamO = 8<; Clear@TLy, VLyDForBn = 1, n < 5, n++, Clear@lambdaD; lambda = 8<;

ForBk = 1, k < 32, k++, TLy = 0; VLy = 0; Clear@ΕD; Ε = 10Εi@@kDD;

ForAi = 1, i < 5, i++,

ForAj = 1, j < 5, j++,

TLy = TLy + 0.5 * Μ * Ir2 + rx2M * D@Βi@@k, iDD, tD * D@Βi@@k, jDD, tD + 0.5 * Μ * D@Wi@@k, iDD,tD * D@Wi@@k, jDD, tD - rx * Μ * D@Wi@@k, iDD, tD * D@Βi@@k, jDD, tD + 0.5 * Om@@nDD2

*

IΜ ro2 + Ix2m * HΒi@@k, iDD * Βi@@k, jDD + D@Wi@@k, iDD, yD * D@Wi@@k, jDD, yDLM +

Ix2m Om@@nDD H-Βi@@k, iDD * D@D@Wi@@k, jDD, tD, yD + D@Βi@@k, iDD, tD * D@Wi@@k, jDD, yDL;VLy = VLy + 0.5 * 10Εi@@kDD El * D@Wi@@k, iDD, 8y, 2<D * D@Wi@@k, jDD, 8y, 2<D +

0.5 * G * D@Βi @@k, iDD, yD * D@Βi@@k, jDD, yDEE;mass =

Coefficient@D@TLy, d1'@tDD, d1'@tDD Coefficient@D@TLy, d1'@tDD, d2'@tDD Coefficient@D@TLyCoefficient@D@TLy, d2'@tDD, d1'@tDD Coefficient@D@TLy, d2'@tDD, d2'@tDD Coefficient@D@TLyCoefficient@D@TLy, d3'@tDD, d1'@tDD Coefficient@D@TLy, d3'@tDD, d2'@tDD Coefficient@D@TLyCoefficient@D@TLy, d4'@tDD, d1'@tDD Coefficient@D@TLy, d4'@tDD, d2'@tDD Coefficient@D@TLy

; stiffness =





; Gyro =

Coefficient@D@TLy, d1@tDD, d1'@tDD + Coefficient@D@VLy, d1@tDD, d1'@tDD Coefficient@D@TLy,Coefficient@D@TLy, d2@tDD, d1'@tDD + Coefficient@D@VLy, d2@tDD, d1'@tDD Coefficient@D@TLy,Coefficient@D@TLy, d3@tDD, d1'@tDD + Coefficient@D@VLy, d3@tDD, d1'@tDD Coefficient@D@TLy,Coefficient@D@TLy, d4@tDD, d1'@tDD + Coefficient@D@VLy, d4@tDD, d1'@tDD Coefficient@D@TLy,

;

M = NIntegrate@mass, 8y, 0, L<D; S = NIntegrate@stiffness, 8y, 0, L<D;Gy = NIntegrate@Gyro, 8y, 0, L<D; mat = PartitionAFlattenA-Λ2 M + Λ I Gy + SE, 4E;solution = Solve@Det@matD 0, ΛD;inter = solution@@5 ;; 8DD; lambda = Append@lambda, Flatten@interDDF;

lamO = Append@lamO, lambdaDF

8 AssumedModes.nb

Appendix F

Controls Advanced Research

Turbine (CART) Blade Data

Table F.1: Distributed CART blade characteristics.

Dist., m/L Iz/L Ix/L xcg xelast GJy2 EIx2(m) (kg/m) (kg.m) (kg.m) (m) (m) (N.m2) (N.m2)

0 282.92 29.47 12.33 -0.09 0.008 4.13E+07 1.65E+080.448 290.24 33.11 11.97 -0.116 -0.019 4.11E+07 1.61E+081.057 261.88 34.19 10.57 -0.156 -0.056 3.84E+07 1.42E+082.276 201.28 31.97 7.35 -0.238 -0.131 2.86E+07 9.87E+073.496 186.52 35.48 5.82 -0.314 -0.205 2.27E+07 7.84E+074.715 169.1 35.67 4.41 -0.379 -0.279 1.67E+07 5.92E+075.985 149.28 29.02 3.38 -0.37 -0.272 1.29E+07 4.54E+077.255 133.19 24.71 2.54 -0.378 -0.265 9.74E+06 3.41E+078.525 111.74 17.58 1.86 -0.341 -0.249 7.24E+06 2.50E+079.795 96.86 14.34 1.33 -0.338 -0.234 5.29E+06 1.79E+0711.065 78.57 9.81 0.92 -0.303 -0.219 3.71E+06 1.23E+0712.335 65.03 7.54 0.61 -0.296 -0.204 2.53E+06 8.19E+0613.605 49.68 4.87 0.38 -0.265 -0.189 1.63E+06 5.14E+0614.875 37.59 3.4 0.23 -0.257 -0.174 1.01E+06 3.02E+0616.145 25.01 1.98 0.12 -0.229 -0.159 5.70E+05 1.62E+0617.415 16.01 1.35 0.06 -0.216 -0.144 3.17E+05 8.68E+0518.685 10.73 0.92 0.03 -0.185 -0.129 1.69E+05 4.68E+0519.955 6.02 0.71 0.02 -0.172 -0.114 7.58E+04 2.09E+05

The properties distributions along the blade length are plotted in Fig. F.1,

Fig. F.2, Fig. F.4, Fig. F.3, Fig. F.5 and Fig. F.6. In order to be able to include the

117

APPENDIX F. CONTROLS ADVANCED RESEARCH TURBINE (CART) BLADE DATA118

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18x 10

7

Blade length (m)

Stif

fnes

s E

Ix2

(N.m

)

datafitted curve

Figure F.1: Distribution of bending stiffness EIx2 along the blade length and theexponential fit EIx2(y) = 1.719E + 8e−0.2214 y .

influence of the properties distribution in the vibrational behavior of the blade, the

properties distribution is expressed as a function of the blade length. Curve fitting

was done using the curve fitting toolbox in Matlab. Considering the physical prop-

erties distribution shown in Table F.1 and plotted, it was judged that the properties

distribution along the blade length would be best fitted by the exponential function

except in the offset and polar radius of gyration cases. The functions are compared

to the distributed data in order to judge the fit accuracy in Fig. F.1, Fig. F.2,

Fig. F.4, Fig. F.3 and Fig. F.6. It is noted that all of the fits are exponential except

for the offset and the radius of gyration.

µ(y) = 232.6e0.06363 y − 0.6628y2 − 27.95y (F.1)

rx = 0.08718 + 0.005621 sin(0.5485y) + 0.02088 sin(0.2574y) (F.2)

r(y) = 0.3616 + 0.07763 sin(0.2905 y) (F.3)

GJy2(y) = GJy2(y) = 4.423E + 7e−0.1995 y (F.4)

EIx2(y) = 1.719E + 8e−0.2214 y (F.5)

Ix2m(y) = 12.88e−0.2294 y (F.6)


0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

7

Blade length (m)

Pol

ar S

tiffn

ess

(N.m

)

datafitted curve

Figure F.2: The torsional stiffness distribution along the blade length and the ex-ponential fit utilized GJy2(y) = 4.423E + 7e−0.1995 y.

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

250

300

Blade Length (m)

mas

s/le

ngth

(K

g/m

)

datafitted curve

Figure F.3: Distribution of mass per unit length µ along the blade length and theexponential fit µ(y) = 232.6e0.06363 y − 0.6628y2 − 27.95y.


0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

Blade length (m)

Ix2m

(K

g.m

)

datafitted curve

Figure F.4: Distribution of mass moment of inertia Ix2m along the blade length andthe exponential fit Ix2m(y) = 12.88e−0.2294 y.

0 5 10 15 200.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Blade Length (m)

She

ar c

ente

r of

fset

rx (

m)

datafitted curve

Figure F.5: Distribution of shear center offset rx along the blade length and the fitutilized is rx = 0.08718 + 0.005621 sin(0.5485y) + 0.02088 sin(0.2574y) .


0 2 4 6 8 10 12 14 16 18 20

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

Blade length (m)

Rad

ius

of g

yrat

ion

(m)

datafitted curve

Figure F.6: Distribution of the polar radius of gyration of the blade cross-sectionabout the centroidal axis along the blade length and the fit utilized r(y) = 0.3616 +0.07763 sin(0.2905 y) .

structural dynamic modeling of wind turbine blades

Documents