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CHAPTER 16

Tower design and analysis

Biswajit Basu Trinity College Dublin, Ireland.

This chapter addresses some of the design and analysis issues of interest to struc-tural and wind engineers involved in ensuring the serviceability and survivability of wind turbine towers. Wind turbine towers are fl exible multi-body entities con-sisting of rotor blades which collect the energy contained within the wind, and the tower which supports the weight of the rotor system and nacelle and transfers all gravity and environment loading to the foundation. Two themes on the design and analysis aspects of the tower have been presented. The fi rst is the mathematical representation of the behaviour of wind turbine towers when subjected to wind loading and the second is the suppression of the vibrations caused by this wind action. The fi rst theme focuses on a series of mathematical models representing the rotor blades, the tower with the added mass of the nacelle, and the coupled rotor blade and tower system which are used to determine the free and forced vibration characteristics of the structure. Response estimation for the rotating blades includes the effects of centrifugal stiffening, dynamic gravity effects due to rotation and rotationally sampled turbulence. A gust factor approach is also presented for design of the wind turbine towers. The second theme considers the mitigation of vibrations under dynamic wind action by adding energy dampers to the system, and fi nding the optimal properties of these dampers in order to maxi-mise the reduction of vibration. Modelling and analysis of offshore towers have also been discussed.

1 I ntroduction

With the exponential growth in the wind energy market, turbines with larger rotor diameter and hence taller towers are becoming more common. This has a crucial impact on the design and analysis of wind turbine towers. The primary function of the wind turbine tower is to elevate the turbine rotor for a horizontal axis wind

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528 Wind Power Generation and Wind Turbine Design

turbine (HAWT) and support the mechanical and electrical system housed in the nacelle. Wind speed increases with altitude and also tends to become less turbu-lent. As a result more energy can be extracted with taller towers. However, this comes at a price of higher cost of construction and installation. Choice of tower height is based on a tradeoff between increased energy production at a particular site and the increase in the cost of construction.

The principal types of towers currently in use are the free standing type using steel tubes ( Fig. 1a ), lattice (or truss) towers ( Fig. 1b ) and concrete towers. For smaller turbines, guyed towers are also used. Tower height is typically 1–1.5 times the rotor diameter. Tower selection is greatly infl uenced by the character-istics of the site. The stiffness of the tower is a major factor in wind turbine system dynamics because of the possibility of coupled vibrations between the rotor and tower. In addition, there are several other factors which affect the selec-tion of the type of tower and its design, such as the mode of erection and fabrica-tion, sizes of crane required for construction, noise, impact on avian population and aesthetics. Among the different type of towers, tubular towers are more com-mon and they are also preferable due to aesthetics and in minimizing impact on avian population.

One of the primary considerations in the tower design is the overall tower stiff-ness, which in turn affects its natural frequency. From a structural dynamics point of view, a stiff tower whose fundamental natural frequency is higher than that of the blade passing frequency (rotor’s rotational speed times the number of blades) is preferable. This type of tower has the advantage of being relatively unaffected by the motions of the rotor-turbine itself. However, the cost may be prohibitive due to a larger mass and hence more material requirement.

(a) (b)

Figure 1: (a) Free standing tubular wind turbine tower; (b) lattice wind turbine tower.


Tower Design and Analysis 529

Towers are usually classifi ed based on the relative natural frequencies of the tower and the rotor blades. Opposite to the stiff towers, soft towers are those whose fundamental natural frequency is lower than the blade passing frequency. A further subdivision differentiates a soft and a soft–soft tower. A soft tower’s natural fre-quency is above the rotor frequency but below the blade passing frequency while a soft–soft tower has its natural frequency below both the rotor frequency and the blade passing frequency. These kinds of towers (soft and soft–soft) are generally less expensive than the stiffer ones, since they are lighter. However, they require particular attention and need careful dynamic analysis of the entire system to ensure that no resonances are excited by any motions in the rest of the turbine.

2 Analysis of towers

2.1 Tower blade coupling

Design engineers are interested in understanding and analyzing the coupled dynam-ics of wind turbine towers with associated components, especially with proliferation of such systems worldwide for renewable energy production. As wind turbines are becoming larger in size and are being placed in varying global wind environments, knowledge of the dynamic behaviour is important. The behaviour of the subcompo-nents of the system (the tower and rotor blades) as well as the dynamic interaction of those components with each other is vital to ensure the serviceability and survivabi-lity of such expensive power generating infrastructure. Following a conventional and simplifi ed design analysis, the mass of the components (nacelle and rotor blades) can be simply lumped at the top of the tower, and as long as the fundamental fre-quencies of the tower and blades are far apart, a stochastic forced vibration analysis could be carried out. While the simplicity of this is attractive, the fl exibility of large rotor systems may result in either economically ineffi cient design due to the conser-vatism required to accommodate the uncertainties of component interaction or an unsafe design due to ignoring the coupling effects.

Published literature available regarding the dynamic interaction of wind turbine components, especially from the point of view of the structural design of the tower with the interaction of the mechanical rotor blade system is growing. Harrison et al. [ 1 ] state that the motion of the tower is strongly connected to the motion of the blades, as the blades transfer an axial force onto the low speed drive shaft which is ultimately transferred into the nacelle base plate at the top of the tower.

The dynamic characteristics of a multi-body system have traditionally been determined by the substructure synthesis or component mode synthesis method [ 2 , 3 ]. In coupled analyses, it is fi rst necessary to obtain the free vibration charac-teristics of all sub-entities, prior to dynamic coupling. The free vibration proper-ties of a tower carrying a rigid nacelle mass at the top may be evaluated by techniques such as the discrete parameter method, the fi nite element method or by using closed form solutions. The discrete parameter method was used by Wu and Yang [ 4 ] in a study on the control of transmission towers under the action of sto-chastic wind loading. Lavassas et al. [ 5 ] also used this technique to assess the



accuracy and reliability of more computationally expensive fi nite element analy-ses of wind turbine tower. Recent studies using the fi nite element technique for free vibration analyses of structures in wind engineering include Bazeos et al. [ 6 ] and Dutta et al. [ 7 ]. Murtagh et al. [ 8 ] derived an expression in closed form to yield the eigenvalues and eigenvectors of a tower-nacelle system comprising of a prismatic cantilever beam with a rigid mass at its free end.

2.2 Rotating blades

The free vibration properties of realistic wind turbine blades are computationally more diffi cult to obtain, and models are usually mathematically complicated due to the complex geometry of the blade and the effects of blade rotation. Baumgart [ 9 ] used a combination of fi nite elements and virtual work, accounting for the complex geometry of the blade to obtain the modal parameters. Naguleswaran [ 10 ] proposed an approach to determine the free vibration characteristics of a spanwise rotating beam subjected to centrifugal stiffening. This model [ 10 ] can be used in many industrial fi elds, such as wind turbine blades, aircraft rotor blades and tur-bine rotor blades. Naguleswaran [ 10 ] and Banerjee [ 11 ] both used the Frobenius method to obtain the natural frequencies of spanwise rotating uniform beams for several cases of boundary conditions. Chung and Yoo [ 12 ] used the fi nite element method to obtain the dynamic properties of a rotating cantilever, whereas Lee et al. [ 13 ] carried out experimental studies on the same. All studies indicate that the natural frequencies rise as the rotational frequency of the blade increases. Vari-ous software codes have been developed by engineers to dynamically analyse the various components of a wind turbine tower. Buhl [ 14 ] presented guidelines for the use of the software code ADAMS in free and forced vibrations of wind turbine towers.

Under the action of rotation, the free vibration parameters of the blades are affected by two axial phenomena. The fi rst is centrifugal stiffening and the second is blade gravity (self weight) effects. In order to fi nd the free vibration properties of the blades, each blade can be discretized into a lumped parameter system com-prising of ‘ n ’ degrees of freedom. The eigenvalues of a blade undergoing fl apping motion may be obtained from the eigenvalue analysis:

2B B B[ ] [ ] = 0K Mw′ −

(1 )

where B B BG[ ] [ ]K = K K′ + represents the modifi ed stiffness matrix due to the geo-metric stiffness matrix [ K BG ], accounting for the effect of axial load, w B is the natural frequency, [ K B ] is the fl exural stiffness matrix and [ M B ] is the mass matrix. The mass matrix may be formulated as a diagonal matrix with the mass m i at each discrete node i .

The geometric stiffness matrix contains force contributions due to blade rota-tion which are always tensile, and contributions from the self weight of the blade, which may be either tensile or compressive, depending on blade position. The geometric stiffness matrix is



1 1

1 1

1 1 2

1 1 2BG

1

1

1 1

1 1

… 0

… 0

[ ]

0 0

n

n

n n n

n n n

N N

l l

N N N+

l l lK =

N

l

N N N+

l l l

−

−

− −

− −

−⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

� � �

(2 )

where N i is the axial force at node ‘ i ’ and l i is the length of beam segment between the nodes ‘ i ’ and ‘ i + 1’. The magnitude of the tensile centrifugal axial force, CT( x ), along the axis of a continuous blade, may be found from the expression given by Naguleswaran [ 10 ] as

2 2

B B B H HCT( ) 0.5 ( 2 2 )x m L L R R x x= Ω + − − (3 )

where Bm represents the mass per unit length of the blade, Ω is the rotational frequency of the blade, and x is the distance along the blade from the hub. This continuous force distribution is discretized into nodal values (CT i ) and used to form the geometric stiffness matrix. The component of nodal blade gravity force (self weight), G i , acting axially may be obtained from geometry and depends on the angle q that the longitudinal axis of the blade makes with the horizontal global axis, in the plane of rotation. Values of N i are obtained from the expression:

i i iCTN G= ± (4 )

with the sign convention that tensile forces are positive and compressive forces are negative.

2.3 Forced vibration analysis

Forced vibration analyses of structures may either be carried out in the time or frequency domain, with each having its own distinct merits. Analysis through the time domain allows for the inclusion of behavioural non-linearity and response coupling. Due to limited availability of actual input time-histories as measured in the fi eld, the designer has to generate relevant artifi cial time-histories using widely published spectral density functions. The method for generating the artifi cial time-histories can be divided into three categories, the fi rst based on a fast Fourier trans-form (FFT) algorithm, the second based on wavelets and other time–frequency algorithms and the third based on time-series techniques such as Auto-Regressive Moving Average (ARMA) method. Suresh Kumar and Stathopoulos [ 15 ] simu-lated both Gaussian and non-Gaussian wind pressure time-histories based on the FFT algorithm. Kitagawa and Nomura [ 16 ] recently used wavelet theory to gen-erate wind velocity time-histories by assuming that eddies of varying scale and strength may be represented on the time axis by wavelets of corresponding scales.



In an investigation on the buffeting of long-span bridges, Minh et al. [ 17 ] used the digital fi ltering ARMA method to numerically generate time-histories of wind turbulence.

In simulating drag force time-histories on the tower, information on spatial cor-relation, or coherence is necessary to be included. Coherence relates the similarity of signals measured over a spatial distance within a random fi eld. Coherence is of great importance, especially if gust eddies are smaller than the height of a struc-ture. Some of the earliest investigations into the spatial correlation of wind forces were carried out by Panofsky and Singer [ 18 ] and Davenport [ 19 ] and later aug-mented by Vickery [ 20 ] and Brook [ 21 ]. Recent publications involving lateral coherence in wind engineering include Højstrup [ 22 ], Sørensen et al. [ 23 ] and Minh et al. [ 17 ].

2.4 Rotationally sampled spectra

In order to simulate the drag force time-histories on the rotating blades, a special type of wind velocity spectrum is needed. Connell [ 24 ] reported that a rotating blade is subjected to an atypical fl uctuating wind velocity spectrum, known as a rotationally sampled spectrum. Due to the rotation of the blades, the spectral energy distribution is altered, with variance shifting from the lower frequencies to peaks located at integer multiples of the rotational frequency. Kristensen and Frandsen [ 25 ], following on from work by Rosenbrock [ 26 ], developed a simple model to predict the power spectrum associated with a rotating blade, and this was signifi cantly different to a spectrum without the rotation considered. Though liter-ature on this topic is limited, Madsen and Frandsen [ 27 ], Verholek [ 28 ], Hardesty et al. [ 29 ] and Sørensen et al. [ 23 ] are some relevant references on this topic.

Rotationally sampled spectra are used to quantify the energy as a function of frequency for rotor blades within a turbulent wind fl ow for representing the redis-tribution of spectral energy due to rotation. The required redistribution of spectral energy can be achieved by identifying the specifi c frequencies 1 Ω , 2Ω , 3 Ω , and 4 Ω ( Ω being the rotational frequency of the blades), and then deriving the Fourier coeffi cients for those frequencies according to specifi c standard deviation values. These values can be obtained based on some measurements or assumption related to the rotational turbulence spectra. Madsen and Frandsen [ 27 ] observed that the peaks of redistributed spectral energy in a rotationally sampled spec-trum tend to become more pronounced as distance increases along the blade, away from the hub.

The typical rotationally sampled turbulence spectra are shown in Fig. 2 [ 30 ]. It has been assumed for the spectra that the variance values increase by an arbitrary value of 10%, for each successive blade node radiating out from the hub. It is also assumed that 30% of the total variance at each node is localized into peaks at 1 Ω , 2 Ω , 3Ω , and 4Ω (15%, 7.5%, 4.5% and 3% of the total energy is allocated to the different peaks). Nodal fl uctuating velocity time-histories with specifi c energy–frequency relationships can be simulated from the spectra in Fig. 2 using a discrete Fourier transform (DFT) technique.



Using the loading from the rotationally sampled spectra of turbulence and using a mode-acceleration method, Murtagh et al. [ 31 ] estimated the wind-induced dynamic time-history response of tapered rotating wind turbine blades. The mode-acceleration method was initially implemented by Williams [ 32 ] and Craig [ 33 ] reported that it has superior convergence characteristics compared to the mode-displacement method. Singh [ 34 ] presented a method for obtaining the spectral response of a non-classically damped system, based on the mode-acceleration technique. Akgun [ 35 ] presented an augmented algorithm based on the mode-acceleration method which has improved convergence for computation of stresses in large models.

2.5 Loading on tower-nacelle

The tower can be modelled as a lumped mass multi-degree-of-freedom (MDOF) fl exible entity, which includes a lumped mass at the top of the tower, to represent the mass of the nacelle and the effect of the blades. An eigenvalue analysis can be performed to obtain the natural frequencies and mode shapes. As the tower-nacelle is a MDOF system, it is convenient to obtain modal force time-histories associated with each mode for analysis. This allows the spatial correlation or coherence of drag forces along the height of the tower to be included. Nigam and Narayanan [ 36 ] presented an expression for the modal fl uctuating drag force power spectrum, for a continuous line-like structure, which can be used following modifi cation for a discretized MDOF system [ 30 ].

The wind velocity auto and cross power spectral density (PSD) terms may be evaluated as

( ) ( ) ( )coh( , ; )V V Vk kVk lVlV l

S f S f S f k l f= (5 )

Figure 2 : Rotationally sampled turbulence spectra.



with S vkvk ( f ) and S vkvk ( f ) being the velocity PSD functions at nodes k and l respec-tively and coh( k , l ; f ) is the spatial coherence function between nodes k and l . The terms S vkvk ( f ) and S vkvk ( f ) are functions of frequency f and may be calculated using the Kaimal spectra [ 37 ]. A coherence function suggested by Davenport [ 19 ], coh( k , l ; f ), which relates the frequency dependent spatial correlation between nodes k and l , is represented as

s

coh( , ; ) expk l

k l fL

⎛ ⎞−= −⎜ ⎟⎝ ⎠

(6 )

where | k – l | is the spatial separation and L S is a length scale given by

S

v̂L

fD=

(7 )

with

ˆ 0.5( )k lv v v= + (8 )

and D is a decay constant. The fl uctuating component of the modal force acting on the tower may be obtained by employing the DFT technique. The mean nodal drag force component is obtained by transforming the nodal mean drag force time-histories into modal force time-histories using the modal matrix. The mean modal drag force is added to the modal fl uctuating component to obtain the total modal drag force time-history.

2.6 Response of tower including blade–tower interaction

In order to couple the tower and rotating blades, equations of motion for the tower that includes the blade shear forces is necessary to be considered. This is represented by

{ } { }T T T B[ ] ( ) +[ ] ( ) +[ ]{ ( )} = { ( )}+{ ( )}TM x t C x t K x t F t V t�� V (9 )

where [ M T ], [ K T ] and [ C T ] are the mass, stiffness and damping matrices of the tower-nacelle respectively, { ( )},{ ( )},{ ( )}x t x t x t� �� are the displacement, velocity and acceleration vectors respectively, { F T ( t )} is the total wind drag loading vector act-ing on the tower and B{ ( )}V t′ is the effective blade base shear vector transmitted from the root of the rotating blades and acting at the top of the tower. The set of equations cannot be solved directly in time domain as the base shear is dependent on the motion of the tower (due to coupling) and hence is not known explicitly. An alternative way to solve the equations is to convert the set into a set of algebraic equations by FFT and subsequently solve by inverse FFT [ 30 ].

A numerical example [ 30 ] is presented for a steel wind turbine tower of height 60 m with three blades of rotor radius 30 m. The total mass of the nacelle and rotor system is 19,876 kg. The average wind speed at the top of the tower is 20 m/s. Figure 3 shows the displacement response time-history at the top of tower when



the blades masses are lumped on the top of the tower thus, ignoring the tower–blade interaction. The maximum observed tower tip response is 0.108 m.

The forced vibration response of the coupled tower–blade model is also calcu-lated for a rotational frequency of 1.57 rad/s. Figure 4 presents a Fourier transform of the simulated fl uctuating wind velocity acting at the tip of the blade. An increase in energy is clearly observable at integer products of the rotational frequency.

Figure 5 illustrates the computed blade tip displacement time-history. The maxi-mum observed displacement is approximately 0.75 m. Figure 6 presents the total

0 10 20 30 40 50 60 70 80 90 1000

0.02

0.04

0.06

0.08

0.1

0.12

Time (s)

Tot

al R

espo

nse

(m)

Figure 3: Displacement time-history at the top of the tower ignoring blade rotation.

0 5 10 15 20100

101

102

103

104

Frequency (rads-1)

Fou

rier

Am

plitu

de

Figure 4 : Fourier transform amplitude of wind velocity.



base shear time-history due to the forced vibration of the three rotating blades. A maximum base shear force of nearly 150 kN is observed. The three rotating blades are now coupled to the tower-nacelle and the maximum tower tip displace-ment response is found to be 0.385 m, as presented in the displacement time-history in Fig. 7 . Thus, inclusion of blade–tower interaction results in a 256% increase in peak tip displacement of the tower compared to the case excluding blade–tower interaction.

0 20 40 60 80 100-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

Tot

al R

espo

nse

(m)

Figure 5: Blade tip displacement time-history.

0 20 40 60 80 100-1.5

-1

-0.5

0

0.5

1

1.5

2x 105

Time (s)

Tot

al B

ase

She

ar (

N)

Figure 6: Base shear time-history.



In the approach by Murtagh et al. [ 30 ], the coupled system equation of motion is primarily cast in the frequency domain via Fourier transform. This allows the coupling of the tower and the blades. The time domain along-wind response of the coupled assembly is ultimately obtained by inverse Fourier transform. There are a number of merits behind this type of approach. The technique is relatively simple, especially compared with a more computationally expensive fi nite element formu-lation. The approach may be used in a preliminary quantitative design, which may subsequently be validated by a more rigorous analysis. The dynamic properties of the coupled system are available using the dynamic properties of each of the two sub-systems, which is an extension of the substructure synthesis approach.

3 Design of tower

A complete dynamic analysis of the tower taking into account the effect of the rota-tion of the blades (rotors) and the nacelle mounted at the top is necessary for ensuring the safety and operational serviceability. However, such a detailed dynamic analysis may be time consuming and rigorous at a preliminary design stage when the initial confi guration has to be chosen based on the design forces and displacements. Hence, for an initial assessment it may be more attractive to use an approximate simplifi ed approach while taking account of the stochasticity in the wind loading (and hence in the response of the tower) and the rotor–tower interaction.

Gust response factor (GRF) approach is a simple technique used by struc-tural engineers in the along-wind design of fl exible structures and incorporates the stochastic and dynamic effects. This technique is now well developed due to the contributions of Davenport [ 38 ] and Velozzi and Cohen [ 39 ]. GRF is the ratio of the maximum or peak response quantity to the mean response quantity.

0 10 20 30 40 50 60 70 80 90 100-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time (s)

Tot

al R

espo

nse

(m)

Figure 7 : Displacement time-history at the top of the tower with blade interaction.



Hence, when this factor is applied to the responses from the mean wind loading, it yields the maximum design values.

The methodology developed by Davenport and, Velozzi and Cohen calculated the GRF using a ratio of displacements, and while this yielded accurate maximum expected response for displacement, it was found to fall short in providing esti-mates of other response parameters, such as bending moment and shear force. Following the work by [ 38 , 39 ] several new models of the GRF have been pro-posed by Holmes [ 40 ] and Zhou and Kareem [ 41 ], with the latter being based on base bending moment, rather than displacement. The GRF methodology has also become the basis of most modern design codes worldwide [ 42 ].

3.1 Gust factor approach

The traditional Davenport-type GRF assumes that the fl exible structure may be rep-resented by a single degree-of-freedom (SDOF) representing the fundamental mode of vibration, and this is usually suffi cient. However, if a structural system like a wind turbine tower (with coupled tower–rotor interaction) has more than one mode contributing to the response, the traditional GRF methodology may yield inaccurate representations of the energy contained in the response. Thus an extension of the traditional GRF methodology to include the effects of higher modes in the derivation of the GRF is required for application in the case of a wind turbine tower.

A GRF for evaluating the along-wind response of wind turbine towers has been proposed by Murtagh et al. [ 43 ]. The approach presented differs from the conven-tional GRF methods as the GRF contains contributions from two resonant modes, mainly due to rotor blade–tower interaction effects. The wind turbine tower model considered contains two inter-connected fl exible sub-systems, representing the tower and a three-bladed rotor system. It is assumed that all the blades vibrate identically in the fl apwise mode (out-of-plane) coupled with the tower. Each com-ponent is initially modelled as a separate degree-of-freedom (DOF) and these are coupled together to form an equivalent reduced order model of the coupled tower–rotor system considering the fi rst two dominant modes. Thus, the resonant compo-nent of the response contains energy output from the two modes of the coupled system. This is an approximate way to account for the effect of the blades fed back to the tower including the coupled tower–blade interaction. The GRF is obtained for both tower tip displacement and base bending moment through numerical integration, with a closed form expression included for the former.

3.2 Displacement GRF

The displacement GRF [ 43 ], G DISP , is obtained as a ratio of the expected maxi-mum displacement response, X MAX ( t ) divided by the mean displacement, x , with the latter being represented by the equation:

CS,1-TT D,1 CS,2-TT D,2

CS,1 CS,2

= +f f

xK K

Φ Φ

(10 )



with Φ cs, j –TT ( j = 1, 2) being the j th coupled system (CS) mode shape component at the top of the tower, CS, jK is the j th modal stiffness of the coupled system and

D, jf is the j th modal mean drag force. Because modal/generalized quantities are used in eqn ( 10 ), it is assumed that the free vibration parameters obtained from the tower–rotor system are from a classically damped one. The modal mean drag force on a structure (i.e. the tower or the blade) is obtained as

2D, D CS,

0

1( ) ( ) ( ) ( )d

2

H

j jf C z B z v z z zr⎛ ⎞= Φ⎜ ⎟⎝ ⎠∫ (11 )

where H is the length over which drag is to be calculated (i.e. the total height of the tower or the length of the blade), C D ( z ) is the drag coeffi cient, B ( z ) is the width of the tower (or blades), and ( )v z is the mean wind velocity and Φ CS,j ( z ) is the j th mode shape component of the coupled system, all as a function of the spatial vari-able z . The expected maximum displacement may be obtained as the product of a peak factor, Ψ (using fi rst passage analysis, as in [ 44 ]) and the root mean square (RMS) of the displacement response at the top of the tower, s X . This RMS dis-placement response, which includes a second mode of vibration, may be obtained by taking the square root of the area under the displacement response PSD func-tion, S XX ( f ) The PSD function S XX ( f ) is found as the sum of the products of the modal wind drag force PSD functions with their appropriate squared amplitude of the modal mechanical admittance functions [ 43 ].

The modal drag force PSD function may be obtained from the expression:

2MF MF VV D 1 D 2 1 2 1 2

0 0

CS, 1 CS, 2 1 2 1 2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( , ; )d d

H H

j j

j j

S f S f C z C z B z B z v z v z

z z R z z f z z

r= ∫ ∫F F

(12)

where S VV ( f ) denotes the wind velocity PSD function at the top of the tower [ 37 ], r is the density of air, and R ( z 1 , z 2 ; f ) is the spatial coherence function between elevations z 1 and z 2 [ 19 ]. The mechanical admittance function at the top of the tower due to a unit force at that point for the j th mode may be obtained as

CS, TT CS,D, 2 2 2

CS, CS, CS, CS, CS,

( )4 1 ( / ) 2 ( / )

j jj

j j j j j

FH f

f M f f i f fx−=

⎡ ⎤− +⎣ ⎦

F

p (13 )

where F CS, j is the j th modal force due to a unit force placed at the top of the tower, f CS, j is the j th natural frequency, CS, jM is the j th modal mass

2CS, 0 CS,( ( ) ( )d )H

j jM m z z z= ∫ Φ with m ( z ) as the mass distribution of the structure and, x CS. j is the j th modal damping ratio.

Two procedures have been proposed by Murtagh [ 43 ] based on how the value of s X . may be calculated. It may be computed by numerically evaluating an integral or it may also be obtained in closed form based on some approximation. For the closed form calculation, a method of decomposition can be employed, in which it is assumed that the variance of the displacement response PSD function may be separated into two components: a background component and a resonant component. Contrary to



the conventional GRF approach, in the proposed methodology [ 43 ], there are two contributions for the resonant component. The square of the non-dimensionalized form of background component of the gust factor 2

BG can be expressed as

2 2 2 2 2 2CS,1 TT CS,1 CS,2 CS,2 CS,22

B MF1MF1 MF2MF24 4 2 2 4 4 2 2CS,1 CS,1 CS,2 CS,20 0

( )d ( )d16 16

TTF FG S f f S f f

f M x f M x

∞ ∞− −Φ Ψ Φ Ψ

= +π π∫ ∫

( 14 )

The integral in eqn ( 14 ) may be evaluated numerically, or by assuming the integrand to be a white noise, or from a known value of turbulence intensity.

The resonant component of the gust factor comprises of two non-dimensionalized terms representing contributions of the fi rst and second modes of vibration, 2

R,1G and 2

R,2G , respectively. These terms are given by the expressions:

2 2 2CS, -TT CS, MF MF CS,2

R, 3 3 2 2CS, j CS, j CS, j

( )= , = 1,2

64p f M x x

j j j j j jj

S fG j

Φ Φ Ψ

(15 )

where Ψj is the peak factor associated with mode ‘ j ’. Thus, the closed form solution for the displacement GRF, G DISP-CF , is obtained as

2 2 2

DISP CF B R,1 R,21G G G G− = + + + (16 )

where G B and G R, j represent the background and resonant components of the displacement GRF, respectively.

3.3 Bending moment GRF

A GRF also has been derived based on the bending moment GRF [ 41 ] at the tower base, G BM by [ 43 ] which is presented for comparison. Similar to the displacement GRF, G BM will contain contributions from two modes of vibration and is obtained as the ratio of the expected maximum base bending moment, Y MAX ( t ) (= Ψ s BM ), to the mean base bending moment, 2

0 D( 0.5 ( ) ( ) ( ) d ).Hy C z B z v z z zr∫ The RMS of the base bending moment, s BM , is obtained from the equation:

1/ 22 22

BM MF MF D,1 0

( ) ( ) dj j j jj

S f H f fs∞

=

⎛ ⎞= Γ⎜ ⎟

⎝ ⎠∑ ∫ (17 )

where jΓ is given by

2CS, CS,

0

(2 ) ( ) ( ) dH

j j jf m z z z zΓ = π Φ∫ (18 )

The base bending moment GRF, G BM may be obtained as

BMBM 1G

y

s= + Ψ (19 )



A series of numerical examples are presented from [ 43 ] to investigate the magnitude of GRFs obtained for the model which allows for blade–tower interac-tion, and these are compared with GRF values obtained from an equivalent SDOF model which ignores blade–tower interaction by lumping the mass of the blades in with that of the nacelle. A tower (steel) of height 50 m with rotor (GFR epoxy) diameter of 60 m is considered with the details available in [ 43 ]. Four different rotational frequencies of the rotor blades were considered. As rotational frequency of the blades increases, the fundamental frequency of the blades also increases, and this leads to increase in the natural frequencies of the coupled systems.

Tables 1 and 2 show the GRFs obtained for the lumped mass equivalent SDOF and two DOF tower–blade interaction models for a mean wind velocity of 20 m/s at the top of the tower. A time of 600 s was used to obtain the GRFs, as used in Eurocode 1 (CEN 2004) [ 45 ]. Included in these tables are the displacement GRFs obtained by numerical integration and in closed form, G DISP-NI and G DISP-CF , respectively, and the base bending moment GRF obtained using numerical integra-tion, G BM-NI . It may be noted that the second mode affects the background and the resonant components and changes the response obtained from the classical gust factor approach.

It is evident from Tables 1 and 2 that the choice of modelling strategy, i.e. lumped mass SDOF or two DOF blade/tower interaction, has a bearing on the magnitudes of both the displacement and base bending moment GRFs obtained. When the blades are stationary ( Ω = 0 rad/s) in the two DOF case, the values of G DISP-NI and G BM-NI obtained differ from the SDOF model values of G DISP-NI and G BM-NI by over 10 and 8%, respectively. These differences remain nearly constant until the case of Ω = 3.14 rad/s where they are equal to 5 and 8%, respectively.

The values of G DISP-NI and G DISP-CF showed a close match in most cases, though it was observed that when the two modes were closest together ( Ω = 0 rad/s),

Table 1: GRFs for SDOF lumped mass model.

G DISP-NI 2.275

G DISP-CF 2.291

G B 1.019

G R,1 0.792

G BM-NI 2.429

Table 2: GRFs for coupled model with blade–tower interaction.

Ω (rad/s) G DISP-NI G DISP-CF G B G R,1 G R,2 G BM-NI

0.000 2.507 2.356 1.032 0.850 0.268 2.6330.785 2.509 2.370 1.044 0.837 0.266 2.5991.570 2.503 2.392 1.070 0.833 0.257 2.5063.140 2.381 2.327 1.059 0.753 0.170 2.225



G DISP-CF yielded a difference of 6% from G DISP-NI . The difference in the value dropped to less than 1% when the modes move further apart at Ω = 3.14 rad/s. It was also observed from Tables 1 and 2 that the displacement and bending moment GRFs obtained showed some disagreement, with the values of G BM-NI being higher than those of G DISP-NI . The largest disagreements were observed at the single DOF model and the two DOF model case of Ω = 0 rad/s, where differences of 7 and 5% were observed.

4 Vibration control of tower

As the wind turbines grow bigger in size and become fl exible with the increase in rotor diameter, it is not only enough to estimate the design forces and ensure the safety of the wind turbine. Additionally, it is necessary to control the vibration response of the fl exible wind turbine tower. It has been observed that wind-induced accelerations may be the reason for the unavailability of wind turbine with increased downtime and may cause damage to the acceleration sensitive subcomponents and devices in a wind turbine [ 46 ]. Hence, it is important to consider structural vibration control strategies for wind turbine towers for operational reliability of wind turbines.

Vibration control strategies for fl exible and tall structures susceptible to large wind-induced oscillations in general are becoming increasingly important, partic-ularly with the current tendency to build higher and lighter. HAWTs are no excep-tion, having experienced a dramatic increase in scale in the past decade. This is particularly evident in offshore wind turbines, with rotor diameter measuring over 120 m. As the design approach is based on strength considerations, stiffness does not increase proportionally with increase in height and these fl exible turbines may experience large-scale blade and tower deformations having non-linear character-istics, which may prove detrimental to the functioning of the turbine. Thus, there is distinct merit in investigating the vibratory control of both wind turbine blades, e.g. using blade pitch [ 47 , 48 ] and towers [ 49 ], using an external energy damper.

Among the several structural vibration controllers available, tuned mass damper (TMD) as a passive vibration control device has become popular. It suppresses vibration by acting as an energy dissipator. Considerable amount of literature now exists on the use of TMDs for fl exible structures [ 50– 52 ]. Use of a TMD for sup-pression of vibration in a wind turbine tower including blade–tower interaction has been studied by Murtagh et al. [ 49 ]. They provided a simple analytical framework in order to qualitatively investigate the effect of a TMD on the fore-aft response of a wind turbine tower.

4.1 Response of tower with a TMD

The displacement response of a wind turbine tower including blade–tower interac-tion and rotationally sampled turbulence acting on the rotor blades, and with an attached TMD may be expressed as [ 49 ]:

DAMPT T T T B[ ]{ ( )}+[ ]{ ( )}+[ ]{ ( )} = { ( )}+{ ( )}+{ ( )}M x t C x t K x t F t V t F t�� V

(20 )



where [ M T ], [ K T ] and [ C T ] are the mass, stiffness and damping matrices of the tower/nacelle, respectively, { ( )},{ ( )},{ ( )}x t x t x t� �� are the time-dependent displace-ment, velocity and acceleration vectors respectively, { F T ( t )} is the total wind drag loading acting on the tower, B{ ( )}V t′ is the effective blade base shear acting at the top of the tower and { F DAMP ( t )} is the damping force brought about by the action of the TMD. Details on how to calculate the effective blade base shear time-histories and total wind drag loadings may be found in Murtagh et al. [ 30 ].

The response time-histories of the tower can be obtained following a modal decomposition of the tower response, transforming the set of equations in eqn ( 20 ) in a Fourier domain and subsequently applying an inverse FFT [ 49 ].

4.2 Design of TMD

For designing a TMD two important parameters need to be considered, the damp-ing ratio and the tuning ratio. For an effi cient performance of a TMD these two ratios need to be optimized.

A number of approximate and empirical expressions are available for the evalu-ation of the optimum damping ratio of the TMD. Given below is the simple expression by Luft [ 51 ] for the optimum damping ratio of the TMD:

,opt 2D

mx = (21 )

where m is the mass ratio of the damper (i.e. mass of the damper to the entire mass of the assembly). In order to tune the damper, its natural frequency is obtained as the product of a tuning ratio n , times the natural frequency of the coupled tower–blades system, i.e.:

D

CS,1

wn

w=

(22 )

where w CS,1 is the fundamental frequency of the coupled tower-rotating blades assembly. It is possible to derive a closed form expression for the optimum tun-ing ratio of the TMD attached to a damped structure based on the “fi xed- point” theory of Den Hartog [ 53 ] which had been proposed for the case of undamped structural systems subjected to sinusoidal excitation. In the optimal design of a TMD attached to an undamped structural system subjected to sinusoidal excitation [ 53 , 54 ], two “fi xed-point” frequencies were obtained at which the transmissibil-ity of vibration is independent of the damping in the TMD. It was also observed that the amplitude of the response transfer functions at the two fi xed points was unequal and had a contrasting effect with the change in the tuning ratio. For a structure subjected to an external force which has wide banded energy content or which has dominant energy at the natural period of the structure, the maximum response reduction is achieved when the area under the transfer function curve is at a minimum. This implies that the values of the transfer function at the fi xed points should be equal and the value of the tuning ratio for which this occurs is the



optimal tuning ratio of the TMD. Ghosh and Basu [ 55 ] extended the theory based on “fi xed-points” to obtain closed form expression for optimal tuning ratio in case of a damped structure. This was used by Murtagh et al. [ 49 ] designing an optimal TMD for a wind turbine tower. The expression for the optimal tuning parameter n opt for a wind turbine tower with damping ratio x n in the fundamental mode of vibration is [ 49 , 55 ]:

2 21 4 (2 1)opt 3(1 )

n nx m xn

m

− − −=

+ (23 )

The optimal tuning ratio together with an optimal damping ratio in the TMD will minimize the maxima of the displacement transfer function of a wind turbine tower.

Murtagh et al. [ 49 ] considered a tower of hub height 60 m and blades with radius 30 m for a three-bladed wind turbine and designed a TMD for suppression of the tip displacement. The mean wind speed at the top of the tower was assumed to be 20 m/s. The fi rst three modal damping ratios of the tower were assumed to be 1% of the critical. A mass ratio of 1% was assumed for the TMD, giving the damper a damping ratio of 5% of critical. Thus, when used in conjunction with eqn ( 23 ), an optimal tuning ratio of 0.99 is obtained. The forced vibration responses of the coupled tower–blades model including and excluding the TMD were calcu-lated and compared. Two rotational frequencies of the rotor system were consid-ered, and the blades are perturbed under the action of rotationally sampled wind turbulence [ 30 ]. The design parameters of the dampers designed for the two cases are presented in Table 3 .

Figure 8 presents the tip displacement transfer function amplitudes obtained for the coupled tower and rotating blades model ( Ω = 15 rev/min) with and without the damper. When contrasting the two transfer functions obtained, it is evident that the presence of the damper causes the peak to split and decrease substantially in magnitude. Figure 9 presents the simulated wind-induced response of the coupled blade–tower model, at the top of the tower, including and excluding the damper. From this fi gure, it is evident that the damper has been effective in suppressing the vibrations, particularly in the earliest portion of the time-history, where the

Table 3: Properties of the TMD.

Rotational frequency (rev/min)

15 30

Mass ratio (%) 1 1Tuning ratio 0.99 0.99Natural frequency (rad/s) 4.45 4.55Mass (kg) 997 997Stiffness constant (kN/m) 20.64 19.74Damping constant (kNs/m) 0.45 0.44Damping ratio (%) 5 5



maximum tower tip displacement observed without the damper of about 0.4 m, reduced to approximately 0.32 m when the damper was included.

5 Wind tunnel testing

Wind tunnel testing of scaled model in order to experimentally investigate aero-elastic and aerodynamic phenomena associated with structures has proved to be a

Figure 8 : Transfer function for the coupled tower-nacelle and rotating blades model.

Figure 9: Simulated displacement response at the top of the tower.



valuable approach for wind engineers. Ever since the fi rst major building study in a boundary layer wind tunnel was conducted by Cermak and Davenport in the 1960s, engineers have been able to inexpensively investigate turbulence-induced pheno-mena. The results provide vital information necessary to ensure the serviceability and survivability of fl exible structures like a wind turbine.

Considerable experimental literature now exists regarding wind tunnel testing of structures in general. Aerodynamic studies are primarily focused on evaluation of drag and lift coeffi cients, such as those by Carril et al. [ 56 ] and Gioffrè et al. [ 57 ]. Aeroelastic scale model studies, similar to those by Ruscheweyh [ 58 ] and Kim and You [ 59 ], examine the link between structural geometrical form and aeroelastic phe-nomena, such as vortex shedding. Passive and active dampers are also proving to be valuable devices in the mitigation of wind-induced structural vibration, and the wind tunnel provides an excellent means to develop and test control strategies [ 60 , 61 ]. While there is very limited literature available on wind tunnel testing of wind tur-bines, this kind of testing can be very useful for system identifi cation [ 62 ], design, and analysis of wind turbines and associated vibration control systems.

Figure 10 shows a model assembly of wind turbine constructed at the Depart-ment of Civil Engineering, Trinity College Dublin, Ireland being tested in the wind tunnel facility at National University of Ireland, Galway [ 63 ]. The model assembly was composed of three main components: the tower, the nacelle and motor, and the rotor system. The model was designed so that the fundamental frequencies of the rotor blades and the tower were close to each other, ensuring signifi cant dynamic coupling between the two subcomponents. The model was immersed in a turbulent wind fl ow and the responses were recorded. The recorded bending strain at the base of the tower and the corresponding Fourier amplitude spectrum are shown in Figs 11 and 12 for the case of a stationary wind turbine.

Figure 10 : Wind turbine tower model installed in test section of wind tunnel.



6 Offshore towers

Recent expansion in the wind energy sector has seen an associated growth in energy production from offshore wind farms. Hence, turbines are becoming larger with taller towers and are being moved further out to sea. As a result the wind

0 1 2 3 4 5 6 7 8 9 10-150

-100

-50

0

50

100

150

Time (s)

Flu

ctua

ting

Mic

ro-S

trai

n

Figure 11 : Strain time-history recorded at the tower base point for rotational speed of 0 rad/s .

0 2 4 6 8 10 12 14 16 18 20102

103

104

105

106

Frequency (Hz)

Fou

rier

Am

plitu

de

Figure 12: Fourier amplitude of strain response at tower base point for rotational speed of 0 rad/s.



turbine towers are subjected to ever greater wind and wave forces. Thus, it is necessary to analyse the dynamics and minimize the response of wind turbine towers to simultaneous actions of joint wind and wave loadings, instead of just the wind loading as in the onshore case.

6.1 Simple model for offshore towers

A model for analysis of an offshore wind turbine tower can in general be repre-sented by a discrete MDOF system [ 64 ]. A simple schematic model of an off-shore tower is shown in Fig. 13 [ 65 ]. The response of such an MDOF system under joint wind and wave loading subjected at the nodes can be calculated by a time-history integration using a standard technique like Runge-Kutta of suit-able order. A fatigue analysis can be performed using the rainfl ow counting

Figure 13 : Structural model.



method and Miner’s rule in accordance with [ 66 ] following Colwell and Basu [ 65 ].

6.2 Wave loading

Following the collection of data and analysis carried out under the Joint North Sea Wave Observation Project (JONSWAP) [ 67 ], it was found that the wave spec-trum continues to develop through non-linear, wave–wave interactions even for very long times and distances compared to the Pierson–Moskowitz spectrum. The wave excitation for an offshore wind turbine tower can be modelled using the JONSWAP spectrum which takes into account the higher peak of the energy spec-trum in a storm. Also, for the same total energy as compared with the Pierson–Moskowitz wave energy spectra, it takes into account the occurrence of frequency shift of the spectra maximum. The spectrum takes the form

2 2 2m m

42exp[( ) / 2 ]m

5

5S ( ) exp

4

g w w s whh

waw g

ww−

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦ (24 )

where h is the function of water surface elevation. Equation ( 24 ) defi nes a station-ary Gaussian process of standard deviation equal to 1. In eqn ( 24 ), g is the peak enhancement factor (3.3 for the North sea), g is the acceleration of gravity and w is the circular wave frequency. The wave data from the JONSWAP project was used to calculate the values of the constants in eqn ( 24 ) as follows:

0.222100.076

U

Fga

⎛ ⎞= ⎜ ⎟⎝ ⎠

(25 )

1/ 32

m10

22g

U Fw

⎡ ⎤= ⎢ ⎥

⎣ ⎦ (26 )

and

m

m

0.07,

0.09,

w ws

w w≤⎧

= ⎨ >⎩ (27 )

where U 10 is the mean wind speed 10 m from the sea surface, F (fetch) is the uninter-rupted distance over which the wind blows (measured in the direction of the wind) without a signifi cant change of direction. The fetch varies in its non-dimensional form as follows [ 68 ]:

1 4210

10 10gF

U− < <

(28 )

The wave force acting on the offshore wind turbine structure can be calculated by using the linearized Morison equation [ 69 ] and from the wave surface elevation time-history calculated based on the wave spectrum (for details see [ 65 ]).



6.3 Joint distribution of wind and waves

The JONSWAP spectrum defi ned in the previous section is a stationary Gaussian process and can be mapped into the process of the sea state defi ned by the sig-nifi cant wave height and mean zero-crossing wave period ( H s , T z ) by letting the dimensionless time be t / T z and the dimensionless process be X/(l0)

1/2 = 4X/Hs, [ 68 ]. The wind speed at 10 m, U 10 , and the signifi cant wave height, H s , from the JONSWAP spectrum can be related through the integral of eqn ( 24 ):

( )0

0

dShhl w w∞

= ∫ (29 )

where (l0)1/2 is the standard deviation of surface displacement. If a sea contains a

narrow range of wave frequencies, H s is related to the standard deviation of the sea surface displacement [ 70 ]:

s 04H l= ( 30 )

The time-histories used for analysis in the joint distribution of wave period and height are approximated by the linear combination of trigonometric polynomials [ 71 ].

Simulated wave surface elevation time-history for ‘moderate’ wave excitation with target and simulated PSD have been presented for the purpose of illustration in Figs 14 and 15 which have been taken from the investigation carried out by [ 65 ]. The wave surface elevation time-history has been simulated with a joint dependence on

Figure 14 : Time-series for the ‘moderate’ wave excitation.



a wind loading with mean wind speed of 18 m/s at a level of 10 m. Also, shown in Fig. 16 is the wave force time-history at node 3 of the structural model in Fig. 13 for ‘moderate’ wave excitation [ 71 ].

6.4 Vibration control of offshore towers

As in the case with onshore wind turbine towers several structural vibration con-trol strategies could be adopted. The use of passive control devices such as TMD,

Figure 15: PSD function of ‘moderate’ wave elevation.

0 10 20 30 40 50 60 70 80 90 100

-4

-3

-2

-1

0

1

2

3

Time (s)

Nod

al D

rag

For

ce (

106 N

)

Figure 16: Wave force at node 3 for ‘moderate’ wave excitation.



TLD or tuned liquid column damper (TLCD) can be useful to suppress undesirable structural vibrations. The use of a TLCD for the vibration control of offshore towers has been investigated by Colwell and Basu [ 65 ]. The simple model in the previous section was used to analyse the dynamic response of the system under joint wind and wave loading. The assumed wind velocity at a height of 10 m was 18 m/s. The blades were 60 m in length and individually weighed 9.5 tonnes. The blades rotated at a frequency equal to the fundamental natural frequency of the MDOF system. The system was subjected to joint wind and wave loading with the wind turbulence simulated from Kaimal spectra and the wave excitation simulated from JONSWAP spectra. The details on the damper parameters are avail-able in [ 65 ]. Figure 17 compares the total displacement response at the top of the tower with and without the TLCD. It has been concluded that the passive damper has a signifi cant benefi cial impact in suppressing the structural vibrations by about 60%. The maximum design bending moment for the theoretical simulation at the base of the structure is reduced from 6.2607 × 10 4 to 4.0101 × 10 4 kNm.

7 Conclusions

Several aspects of dynamic analysis and design of wind turbine towers have been discussed in this chapter. With the increase in rotor diameter and the height of the towers, the analysis of wind turbine towers becomes crucial and needs special attention particularly in the view of tower–rotor coupling. The important physi-cal behaviour and phenomena to be accounted for are centrifugal stiffening of blades, gravity effects, rotationally sampled turbulence and the tower–blade cou-pled dynamic interaction. Numerical results have indicated that responses of the

0 5 10 15 20 25 30 35 40 45 50-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (s)

Tot

al R

espo

nse

(m)

MDOF with TLCDMDOF without TLCD

Figure 17 : Time-history response of a rotating offshore turbine under wind and ‘moderate’ wave excitation with and without TLCD.



tower may be severely underestimated if the tower–blade coupling is not taken into consideration properly. A simplifi ed gust factor approach for calculating the response of wind turbine towers has been discussed. Two gust factors based on dis-placement and bending moment response of the tower have been presented. They have again highlighted the importance of tower–rotor coupling with the need for response specifi c gust factors. The importance of wind tunnel testing has been dis-cussed and emphasized. Since vibration poses a critical challenge in wind turbine towers, vibration control strategies with TMD and other passive dampers have been discussed. Mathematical models have been presented for both onshore and offshore wind turbine towers. Joint wind and wave loading have been modelled for analyzing the offshore towers. It has been observed that vibration control damp-ers signifi cantly reduce the motions in a wind turbine tower with possible design, maintenance and operational benefi ts.

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