FEM Modelling of a Static Wind Turbine

8/12/2019 FEM Modelling of a Static Wind Turbine

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FEM Modelling - SG2860

Project - Wind Turbine

Course responsible

Anders ERIKSSON

Author

Pierre-Alexandre BEAUFORT

February 2014


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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Overview of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1 Structure modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Euler-beam approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Wind loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Comsolstructure modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Natural eigenmodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1 Analytical eigenmodes of the blades. . . . . . . . . . . . . . . . . . . . . 53.2 Analytical eigenmodes of the tower . . . . . . . . . . . . . . . . . . . . . 63.3 Comsoleigenmodes of the wind turbine . . . . . . . . . . . . . . . . . . . 6

4 Wind-response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.1 Wind modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Wind importation inComsol . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

A.1 handCalculationBlades.m . . . . . . . . . . . . . . . . . . . . . . . . . 17

A.2 handCalculationTower.m . . . . . . . . . . . . . . . . . . . . . . . . . . 17A.3 velocity.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18A.4 testVelocity.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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Introduction

Our society still needs more energy. Yet, the nuclear energy is little by little disapproved, spe-cially since Fukushima event. For example, the German government has decided to close somenuclear plants. Therefore, the energy research is focused on renewable energies, like the windturbines. However, even if the wind turbines convert the wind energy in electricity, it may not

be done for too large wind speeds.

The purpose of this report is to present the analysis of the structure of a wind turbine, withwind loading. It is based on [2].

First, we begin with a scientific description of the problem. We describe the structure modellingwith its assumptions and equations. We also introduce the wind modelling.

Then, we explain how we model a wind turbine within a FEM solver, Comsol Multiphysics4.3-b. We underline the modelling of the assumptions, that are important.

Afterwards, we focus on the eigenmodes of the structure. We begin by calculating analyticallythe eigenmodes of the blades and then those of the tower. We end with the Comsol computa-tion of the structural eigenmodes and we perform a comparison with the analytical calculations.

Finally, we study the wind-response of the structure, while a quite fast wind blows. We firstpresent the wind modelling and explain then how we import a wind within Comsol. Obviously,some simulations are performed and are analyzed.

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1 Overview of the problem

The problem consists in the structure-response of a wind turbine which is stopped.

First of all, it is interesting to study the eigenmodes of the wind turbine without wind loading.These eigenmodes are the natural frequencies of vibration of the structure. This information

allows us to prevent to phenomena of structural resonance.

The next step is to apply a wind loading on the structure. Since the wind turbine is stopped,we assume that the wind loading is the result of a wind speed which implies that an usual windturbine has to be stopped. Therefore, if we know the characteristics of wind in a windy region,we can predict the safety to install this type of wind turbines there.

Before going through these studies, we have to define the structure modeling, with its as-sumptions. From this modeling, we will present some equations about the structural behavior.Afterwards, we will model the wind knowing some of its parameters, in order to produce the

corresponding wind loading on the structure.

1.1 Structure modelling

Usually, a wind turbine consists of a vertical tower that supports a nacelle, which contains theturbine. This nacelle is thus attached to a rigid hub, which is connected to 3 blades. Figure1(a) displays the representation of an usual wind turbine. We consider that the wind turbineis parked with one vertical blade.

We can model this kind of wind turbine as a vertical cantilever structure(i) (i.e. the tower)supporting 3 other cantilevers (i.e. the blades). Obviously, the blades and tower are ratherlong in comparison with their cross-section. Therefore, we may assume they are Euler-Bernoullibeams; their main structure-response is then defined by flexural modes. For the shake of thesimplicity, we model the nacelle and hub as lumped masses at the top of the tower and thecenter of rotation of the blades. Figure 1(b)displays the modeling of the wind turbine.

Blade

Tower

Nacelle

Hub

(a) Usual construction.

Beam Elements

Lumped Masses

Fixed Support

RigidBeam

Element

(b) Structural model-ing

Figure 1: Wind turbine

(i)Cantilever: A long beam fixed at only one end.

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Since we assume the wind turbine is stopped, we can ignore the centrifugal effect on the blades.

Finally - by simplicity - we will assume the tower and blades sections have constant character-istics(ii).

1.2 Euler-beam approximation

We have assumed that the whole structure is composed of Euler beams, with two added mass-points. The displacement u(x; t) : [m]of an Euler-beam is described by the equation (1):

m(x)2u

t2 +

2

x2

EI(x)

2u

x2

=p(x; t) (1)

wherem(x)is the mass density[ kgm

],EI(x)the flexural stiffness[m3 kg

s2 ]andp(x; t)the external

load [kgs2

].

If we assume that the blades have vibration frequencies far from those at which the tower would

vibrate if the blades were rigid, on one hand we may consider separately the blades from thetower. Then, the eigenmodes of a blade are given by solving (1). On the other hand, we mayconsider the tower as a cantilever of length L, mass m and inertia Iwith a lumped mass Mof inertial Jat its head. According to [4], the eigenmodes of such a cantilever(iii) are given bysolving the implicit equation (2):

(1 (L)4RMRJ) cosh(L) cos(L) ((L)RM) + ((L)3Rj)cosh(L)sin(L)+((L)RM) (L)Rj)cos(L)sinh(L) + (1 + (L)4RMRJ) = 0 (2)

where RM= MmL

, RJ= JmL3

and 4 = 42f2m

EI .

1.3 Wind loadingObviously, the wind loading is related to the wind speed:

Fw =CD U

2A

2 (3)

where Fw is the wind load [N] on a given area A: [m2], with the density [ kg

m3] of the air(iv),

CD a drag coefficient [/] and Uthe wind speed [ms] on the area. Actually, we are going to as-

sume that the wind loading is significant only on the blades, and we can neglige it on the tower.

The wind speed can be modelled by a stochastic process, owing to a Power Spectral Density

function(v)

. From a PSD function, it is possible to derive a wind speed by doing an inverseFourier transform. However, the exposed area of the wind turbine will be discretized in severalones. Each stochastic process is related to the other ones. Indeed, the wind speed between twoadjacent areas cannot have a large difference between their respective wind speed. Therefore,a coherence function(vi) will ensure that all the stochastic processes generating the wind speedare not independent.

(ii)i.e. constant mass and stiffness distributions, through the cross-section of the beam.(iii)If the cantilever has a non-uniform mass and stiffness, the problem is analytically harder.(iv)We will use the value for a temperature of 20 Celsius degree, i.e. = 1.2041[kgm3], according toWikipediahttp://en.wikipedia.org/wiki/Density_of_air

(v)PSD function is a measure of the frequency content of a signal (here, the wind speed). It is thus defined

as a function of the frequency f, with usual parameters and , a mean and standard deviation of the signal,respectively.(vi)A coherence function has 2 variables: a distance d and a frequency f. It equals 1 when d = 0and decreases

as dincreases.

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2 Comsol structure modelling

We consider the same wind turbine as [2]. The tower is a uniform cylindrical shell of60[m]height, with a 3[m] outer diameter and 15 103[m] wall thickness and built with structuralsteel. Since the blades are stopped, we can approximate them as cuboids. The blades are then30[m] long, uniform hollow rectangular sections, with outer width of 2.8[m], outer depth of

0.8[m], wall thickness of10 10

3[m] and built with aluminum. The nacelle has a length of4[m]and the hub has a diameter of6[m]. They are both built with a rigid structural steel(vii).We remind that the nacelle and the hub are represented by two lumped masses: the nacelle is23 20000[kg] at the top of the tower and the hub is 1

3 20000[kg] at the center of rotation of

the blades.

Components Density: [ kgm3

] Youngs modulus: [GPa] Poissons ratio: [/]Tower 7850 210 0.33Blades 2100 650 0.33

Nacelle-Hub (7850) 200 0.49

Table 1: Properties of the materials.

In Comsol, we use the model Beam in 3D. The wind turbine is drawn by using ParametricCurvein the Geometrysection. The tower is then a vertical line and the nacelle is an horizon-tal line starting at the head of the tower. The hub is represented by three lines starting at theend of the nacelle and each separated by an angle of 2

3, with one vertical. The blades are the

extensions of the hub lines. The yz-plane is parallel to the blades plane. Obviously, the width

of the blades is defined in the blades plane.

The material properties are set in Materials section, by choosing the different ones throughthebuilt-inlibrary and modifying some values inMaterial Contents, according to the table1.

Afterwards, we define Edge Load for the tower and blades, by defining each time a force perunit volume -g_const*beam.rho along the z-direction. We define two Point Load represent-ing the lumped masses. At the bottom of the tower, we define a Fixed Constraint, accordingto the definition of a cantilever.

Since we are using an Euler-beam approximation of our problem, it is important to well definethe cross section of each beam. In order to do that, we define Cross section data for eachbeam. The tower is a pipe cross section. About the nacelle, we suppose that its cross section iscircular, with a diameter equals to this of the tower. The hub has a rectangular cross section,with same dimension as the blades; we have to define the y-direction of this cross sectionperpendicularly to the longest side of the rectangle. In order to get such a y-direction, wedefine a reference point in the Section Orientationsection. For each blade, the cross sectionis Box. The y-direction is still perpendicular to the longest side.

(vii)It means that the Poissons ratio is near 0.5.

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3 Natural eigenmodes

In this section, we derive analytically the eigenmodes of the blades. It is relevant, since we haveassumed that the wind loading is significant only on the blades, which seems legit as they aresupposed to have a larger windage than the tower.

However, we are interested in the eigenmodes of the whole structure, in order to avoid anystructural resonance, due to the wind for example. We will then compute the eigenmodes ofthe tower by solving the implicit equation (2).

Afterwards, we will compute the eigenmodes of the structure with Comsol. We will comparethe results with our previous hand calculations and attempt to valid the Comsolsimulation. Inthis way, we will be able to analyze further results from Comsolabout this wind turbine.

3.1 Analytical eigenmodes of the blades

From our assumption that the blades are each an individual cantilever, we know that theeigenmodes of the blades are related to (1). In order to get them, we have to solve thisdifferential equation by variables separation, i.e. by letting that u(x; t) = (x)(t); (1)becomes then:

=2 =(EI(x))

m(x)

The blades are uniform beams = u(x) =cst and EI(x) =cst. As we are only interested inthe eigenmodes of the blades, we have only to solve:

(4) 4= 0with 4 =

2m

EI

.

The general solution of this differential equation is:

(x) =a sin(x) + b cos(x) + c sinh(x) + d cosh(x)

wherea, b, c, dare determined by the boundary conditions. In the case of a cantilever, they are:

u(x= 0) = 0 = (0) = 0 = d= b u(x= 0) = 0 = (0) = 0 = c= a M(x= L) = 0 = (x= L) = 0 (bending moment)

= a(sin(L) + sinh(L)) + b(cos(L) + cosh(L)) = 0 V(x= L) = 0 = (x= L) = 0 (shear)

= a(cos(L) + cosh(L)) b(sin(L) sinh(L))From these boundary conditions, we obtain a linear system of 2 unknowns (a, b) with 2 equa-tions. A trivial solution is given by a= b = 0, but it means that the blades do not move. Another solution is given by the implicit equation:

1 + cos(L)cosh(L) = 0 (4)

The 4 first solutions are:

L = {1.875;4.694;7.855;10.996}The length of each blade is L = 30[m]. Then, we have to compute the inertiaI. For a beamwith a Boxcross section:

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that are respectively near 1.3, 1.8[Hz]and 3.7, 4.6[Hz]that are themselves quite near(xi).

However, we get the eigenfrequency3.8[Hz]describing a bending of the blades (see figure 2(c)),which was predicted by the analytical solution. Besides, the eigenfrequency 5.4[Hz]describinga bending of the tower (see figure 2(d)) is just between 4.6 and 6.4[Hz]; this can be producedagain by a combined effect with the blades.

(a) Torsion of the tower. (b) Eigenmode produce by the combination tower-blades.

(c) Blades eigenmode that was predicted. (d) Tower eigenmode.

Figure 2: Eigenmodes of the wind turbine of section2, with von Mises stresses. From left to

right, from top to bottom: 1.56, 2.19, 3.81 and 5.46 [Hz].

In conclusion, the assumption that the blades and tower have eigenfrequencies that are farenough, is not really sharp, about the analytical derivations. Nevertheless, it is enough to beconvinced that the Comsol model is correct since we are able to explain the Comsol eigenfre-quencies from the analytical derivations.

(xi)which means that the eigenmodes of the tower and blades are not so far

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4 Wind-response

Through this section, we perform Comsol simulations of wind loading that are applied to aparked wind turbine. We remind that we have assumed that the wind loading is significantonly on the blades. Moreover, we suppose that the wind speed is such that the wind turbinehas to be stopped, for safety. Here, we will consider that the wind turbine is parked with a

vertical up blade.

Before doing any simulation, we have to derive analytically our wind modeling. Then, we willshortly explain how we introduce the wind effect within Comsol.

4.1 Wind modelling

Let us consider the wind speed at one point. Even if we know that the wind is due to differencesof atmospheric pressures, we are not able to predict exactly the wind speed in a single point.This kind of phenomenon is generally modelled as a stochastic process, since it is not 100%

deterministic(xii)

.

Actually, the wind speed in a point can be view as a signal, with a certain frequency content.In a stochastic process, a nondeterministic signal is at least characterized by a mean and astandard deviation. Hence, if we know the mean and standard deviation of the wind speedin a point, we are then able to build a PSD function.

Once we assume that the wind speedu(t)in a point has a period T, owing to an inverse discreteFourier transformation:

u(t) = +

N2

n=1

ancos

2nT t

+ bnsin

2nT t

(5)

We know the following relationships between the variance(xiii) and (5):

2 =1

2

N21

n=1

(a2n+ b2n) + a

2N2

and the PSD function:

2 N2

n=1PSD( n

T)

T

From these relationships, we get:

PSD(fn) T2

a2n+ b

2n

(6)

We can rewrite (5) as:

u(t) = +

N2

n=1

a2n+ b

2ncos

2n

T t n

(7)

with n a random phase angle.

(xii)theory of chaos(xiii)square of the standard deviation

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Then, by using (6) in (7), we express the wind speed in a point:

u(t) = +

N2

n=1

2PSD nT

T cos

2n

T t n

(8)

However, we want to get the wind speed in several points. We cannot use (8) for differentpoints, since the wind speed is continuous through the space dimension. This implies thusthat the wind speed in different points is not independent. Therefore, we are going to use acoherence function in order to generate a coherent field of wind speed points.

Let U(t) be a NP 1 vector, which describes the wind speed in Np points at time t. Theequation (8) becomes:

Ui(t) = + 2

N2

n=1

AMPi(fn) cos(2fnti(fn)) (9)

where:

AMPi(fn) = ||Vi(fn)||2

i(fn) =phase(Vi(fn))

Vi(fn) =i

j=1

Hijexp(ijn)

H11 =S11

Hjj =

Sjj

j1k=1

H2jk

Hij =

Sij j1k=1

HikHjk

Hjj

Sii= PSDi

Sij = cohijSiiSjj

with PSDi the value of the PSD function in the i-th point for a certain frequency f and cohijthe value of the coherence function between the i-th and j-th points for a certain frequency f.

Now, we are able to model a wind speed field in different points from a given PSD and a givencoherence functions. We will use these defined by [1]:

PSD(f) = 42L

1 +6fL

5

3 coh(d, f) = exp

12

fd

2+

0.12d

L

2

with L a length scale and d: [m] the distance between two points.

TheMATLABfunctionvelocity.m(xiv) performs the computation of wind speed points, for giventimes t, mean , standard deviation , scale length L, Ndiscrete frequencies, a period T andcoordinates(yi; zi). Here, we consider the same PSD function for every point.

(xiv)see appendixA.3

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4.2 Wind importation inComsol

The MATLAB script testVelociy.m(xv) sets up the parameters and data of the wind speed andthe wind turbine in order to use the function velocity.m and then it writes the results indifferent files.txtfor different points. We have then time-histories wind speed of a blade pointin one .txtfile.

Once the files are written, we have to import them in Comsol. We define thus anInterpolationfunction per .txtfile, in the Global Definitionssection. We use a Cubic splineinterpola-tion and aConstantextrapolation of the time histories wind speed. It is thus better to computetime-histories through the whole time of the simulation. We associate each Interpolationfunction to a part of one blade, which the wind speed is the value at the midpoint of thispart except for the point values at the end of the blades; we need then to define points in theGeometrysection, in order to divide our beam-blades in several parts.

Now, we are able to compute the wind loading on each part of blades. For doing this, wedefineEdge Loadfor every part, with-rhoAIR*drag*bip

2

j

(t)/2*wB[Nm

](xvi) along the x-direction,according to (3). bipj is the Interpolation function corresponding to the j-th point of thei-th blade and wB the width of the blades. Actually, we have assumed that the wind velocityis perpendicular to the blades plane. This is the configuration with the most important windloading since the blades have a larger windage in the direction that is perpendicular to theblades plane.

4.3 Simulations

We perform here some simulations of the structural response when the wind has a speed thatis large enough to force the wind turbine being on stand-by, for safety. We keep the Comsol

wind turbine model that we developed in the section 2; we just add the wind loading as weexplained within the previous subsection. Our wind has an average speed of30[m

s](xvii) and a

standard deviation of1[ms](xviii). We work with a discrete spectrum of 500 frequencies and a

period of1000[s]. We use 10 point-histories along each blade, from one end to the other one.The length scale is 340: the same as [3]. The corresponding wind speed is displayed by figure3in several points of the 3 blades.

(xv)see appendixA.4(xvi)the drag coefficient is 2, as in[2](xvii)According to SETIS about the Wind power generation, an usual wind turbine has to stop forwind speed around 25[m s1]. http://setis.ec.europa.eu/setis-deliverables/technology-mapping/technology-map-chapters-2011/wind-power-generation

(xviii)in order to stay near the critical speed

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http://setis.ec.europa.eu/setis-deliverables/technology-mapping/technology-map-chapters-2011/wind-power-generationhttp://setis.ec.europa.eu/setis-deliverables/technology-mapping/technology-map-chapters-2011/wind-power-generationhttp://setis.ec.europa.eu/setis-deliverables/technology-mapping/technology-map-chapters-2011/wind-power-generationhttp://setis.ec.europa.eu/setis-deliverables/technology-mapping/technology-map-chapters-2011/wind-power-generation


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(a) Point at the hub on the up blade. (b) Point next to the one at the hub on the up blade.

(c) Point at the hub on the left blade. (d) Midpoint on the right blade.

Figure 3: Interpolation of time histories. We observe that figures 3(a) and 3(b) has roughlythe same time histories, while the other ones are different: this is the effect of the coherencefunction.

Standard settings

First, we run Comsol with the Time Dependentsolver, with range(0,0.1,1). We begin bycomputing the solution only during the first second in order to observe the effect of an instan-taneous wind loading. Indeed, initially velocity.m did not compute ramp values for the firsttime histories of the wind speed. It corresponds thus to a very sudden strong wind. Figure4

displays the results of the Comsol computations.

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(a) Total displacement (m). (b) von Mises stresses (N/(m*m)).

Figure 4: Results of the wind loading simulation within the first second of a very sudden strongwind.

We see on the figure4(a)that the up blade has a higher total displacement than the averageof the 3 blades, that is larger than the tower displacement. It is logical since the blades areconnected to the top of the tower which is fixed to the ground. This implies that if the towermoves a little, then the blades move more. Besides, we remind that the tower is not directlyaffected by the wind loading; the wind speed is applied only the blades. Notice by the way thatthe 3 curves are increasing.

On the figure4(b), the von Mises stresses of the blades are oscillating at a rough frequency of4[Hz]; this is near two Comsol eigenmodes(xix) we computed in the section3. However we donot observe an oscillation of the tower at this frequency. Analytically, we did not get towereigenmode about 4[Hz], but we got 3.7[Hz] for the blades. Notice that these oscillations arerelated to the total displacement: the blades curves are increasing with periodic drops of4[Hz],while we do not observe such a behavior for the tower curve. The von Mises stresses of thetower seem to increase. Finally, we see that the wind speed is sharp at t = 0[s]: the von Misesstresses of the blades are about 4 107[ N

m2]at onlyt = 0.15[s]. Moreover, after only one second

the total displacement of the blades is already about 30[cm].

Let us runComsollonger, till one minute in order to have an overview of the structural responsethrough time. Figure5displays the results of this computation.

(xix)3.8and 4.2[Hz], which both includes blades bending

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(a) Total displacement (m). (b) von Mises stresses (N/(m*m)).

Figure 5: Results of the wind loading simulation within the first minute of a very sudden strongwind.

We observe that the 3 total displacements on figure 5(a) are oscillating at an approximativeperiod of6[s]. This corresponds to a frequency of0.16[Hz], which is a Comsol eigenmode wegot in the section3. This eigenmode describes a bending of the tower. It seems that the totaldisplacement of the blades are here produced by the total displacement of the tower, which iscaused by the wind loading that is applied on the blades.

This is confirmed by the figure5(b), since the von Mises stresses of the tower are also oscillatingat roughly 0.16[Hz] and not these of the blades. Indeed, we notice that the blades curves areoscillating at a larger frequency, but are damped.

Realistic settings

Now, we are going to consider a more realistic situation. First, since the wind speed wassuddenly around 30[m

s] at t= 0[s], we add a ramp values from 0 to 30 [m

s] for the wind speed

during the 30 first seconds. Besides, the wind speed has a period of100[s] only, with still 500discrete frequencies. The time histories of the wind speed in one point of one blade is displayedby figure6.

Figure 6: Wind speed around the middle of the left blade.

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On the other hand, we noticed that the whole structure was oscillating because of the towerdue to the wind loading applied on the blades. Therefore, we could minimize this oscillationby adding a kind of damper on the tower. We are going to consider a massless damper alongthe whole tower, that we will model by the Rayleighcoefficients through Comsol.

We assume that the tower is a spring-mass-dashpot system. Its position is described by the

following differential equation:

d2x

dt2 + 2n

dx

dt + 2nx= p(x; t) (10)

where x : [m] is the position, : [/] is the damp coefficient, n : [s1] is the relative stiffness

coefficient and p: [N] the external load. Here, we arbitrarily decide to choose the value fora small damping effect at the frequency f = 0.16[Hz](xx). Such a value seems to be = 0.1according to the figure7.

Figure 7: Damping effect for different values of.

In Comsol, we can set this value by defining the Rayleigh coefficient:

= 2

+

2

We arbitrarily set = 0[s1] and thus we assume that the damping effect of the tower is dueto its stiffness. We get = 0.19894[s]. In the section Linear Elastic Material, we addDampingthat we applied on the tower, with the value = 0.2[s].

Let us run Comsol for this situation. Figure8displays some results.

(xx)The Comsol eigenmode corresponding to the bending of the tower in the quasi-static case (without wind),but also to oscillations of the tower when the wind has a speed of30[m

s].

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(a) Total displacement (m) within the first second. (b) von Mises stresses (N/(m*m)) within the firstsecond.

(c) Total displacement (m) within the 90 first sec-

onds.

(d) von Mises stresses (N/(m*m)) within the 90 first

seconds.

Figure 8: Results of a wind loading simulation, with a ramp values at the beginning and amassless damper along the tower.

Within the first second, we observe the same behaviors about the total displacement, but nowit is only about 4[mm] for the tower and 3.5[cm] for the up blade. About the von Misesstresses, we notice that now the tower has the most important stresses, while the up blade hassmaller stresses than the average of the blades. Besides, the von Mises stresses of the bladesare oscillating, but not at the same frequency. The average of the blades has a frequency of

9.5[Hz](xxi)

. Again, this oscillation is related to the total displacement. Finally, see that thevon Mises stresses are about 106 while in the previous simulation they were about 107.

Within the 90 first seconds, we can observe the ramp values of the wind speed and then akind of steady state of the wind speed. The oscillations of the tower are damped or at leastthey have a smaller amplitude than the previous simulation, thanks to the damper. The bladesstill oscillates more than the tower, but with a smaller amplitude than before. The von Misesstresses oscillate barely and they are still more important for the tower than for the blades. Weobserve a slightly shift between the up blade and the average: the up one has smaller stresses.

(xxi)This frequency is not within the 10 first frequencies.

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A Appendix

A.1 handCalculationBlades.m

1 % M A TL A B s c r i pt t ha t c a lc u la t es t he 4 f ir s t e i ge m od e s o f t h e b l ad e s

2

3 f ormat long4

5 % s o lu t io n s t o t he i m pl i ci t e q ua t io n

6 n F = [ 1 . 8 7 5 10 4 0 6 8 7 12 , 4 . 6 9 4 0 9 1 1 3 29 7 4 2 , 7 . 8 5 4 7 5 7 48 2 3 7 6 , 1 0 . 9 9 5 54 0 7 3 4 8 75 ] ;

7 s an it yC HE CK = 1 + c o s (nF).* c os h ( n F )8

9 % c a l c u at i n g t h e c o r r e sp o n d i ng e i g e n fr e q u e n cy

10 f o r i = 1 : l e n g t h (nF)

11

12 b e ta L = n F (i ) ;

13 L = 3 0;

14 be ta = b e ta L / L ;

15

16 t hB = 1 0 e -3 ;

17 w B = 2 . 8;

18 dB = . 8;

19 r ho = 2 70 0;

20 A = 2 * t h B * ( wB + d B ) - 4 * t hB ^ 2 ;

21 M = r ho * A* L ;

22

23 E = 70 e9 ;

24 I z = ( t hB * d B ^3 + t hB ^ 3 * ( wB - 2 * t hB ) ) / 6 + ( t hB * ( wB - 2 * t hB ) * ( dB - t h B ) ^2 ) / 2;

25 I y = ( t hB * w B ^3 + t hB ^ 3 * ( dB - 2 * t hB ) ) / 6 + ( t hB * ( dB - 2 * t hB ) * ( wB - t h B ) ^2 ) / 2;

26 I = min ( I z , I y ) ;

27 o m e g a Sq u a r e = be ta ^ 4 * E * I/ M ;28 o m eg a = s q r t (omegaSquare);29 f = o m eg a / (2 *p i )30

31 en d

A.2 handCalculationTower.m

1 % h a n d C a c u l at i o n T o we r c o m p ut e s t h e e i g e n f re q e n c i es o f t h e t o w er

2 f u n c t i o n [ ] = h a n d C a cu l a t i o nT o w e r ( )3

4 % i n it i al t r ia l s f o r t he i m pl i ci t e q ua t io n

5 x = 0 :1 00 ;

6 % f o r e v er y t r ia l

7 f o r i = 1 : l e n g t h ( x ) - 1

8

9 b e ta H ol d = x ( i );

10 b e ta H ol d 2 = x ( i +1 ) ;

11 t ol = 1 e -9 ;

12 C OU NT = 0 ;

13 M AX = 1 00 0;

14 d el ta = t ol + 1;

15

16 % n e w to n m e th o d , s e c a nt v e r s io n

17 w h i l e d e lt a > t ol & & C O UN T < M AX18

19 b e t aH = b e t a H o l d - f u n ( b e t aH o l d ) * ( b e ta H o ld - b e t a H o l d2 ) / ( f u n ( b e t a H ol d ) - f u n ( b e t a H ol d 2 ) ) ;

20 d e lt a = abs ( b e t a H - b e t a H o l d ) ;21 b e t a H ol d 2 = b e t a Ho l d ;

22 b e ta H ol d = b e ta H ;

23

24 C OU NT = C OU NT + 1 ;

25 en d26 s a n i t yC H E C K = f u n ( b e ta H )

27

17


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28

29 % c a l c u l a ti n g t h e c o r r e s p o n d in g e i g e n fr e q u e n cy

30 H = 6 0;

31 be ta = b e ta H / H ;

32

33 r ho = 7 85 0;

34 r o = 1 . 5;

35 r i = r o - 15 e -3 ;

36 A = p i *(ro^2-ri^2);37 H = 6 0;

38 M = A * H* r ho ;

39

40 E = 2 10 e 9;

41 I = p i *(ro^4-ri^4)/16;42 o m e g a Sq u a r e = be ta ^ 4 * E * I/ M ;43

44 o m eg a = s q r t (omegaSquare);

45

46 f = o m eg a / (2 *p i )47

48 en d

49 en d50

51

52 % i m p l i ci t e q u a ti o n

53 f u n c t i o n f = f un ( x)54

55 r ho T = 7 8 50 ;

56 At = p i *(1.5^2-(1.5-15e-3)^2);57 H = 6 0;

58 m = A t *H * r ho T ;

59

60 L = 3 0;

61 t hB = 1 0 e -3 ;

62 w B = 2 . 8;

63 dB = . 8;

64 r ho = 2 10 0;

65 A = 2 * t h B * ( wB + d B ) - 4 * t hB ^ 2

66 M b = 3 * r ho * A * L;

67 J z = ( t hB * d B ^3 + t hB ^ 3 * ( wB - 2 * t hB ) ) / 6 + ( t hB * ( wB - 2 * t hB ) * ( dB - t h B ) ^2 ) / 2;

68 J y = ( t hB * w B ^3 + t hB ^ 3 * ( dB - 2 * t hB ) ) / 6 + ( t hB * ( dB - 2 * t hB ) * ( wB - t h B ) ^2 ) / 2;

69 J= min (Jy,Jz)*Mb/20000;70 M = Mb + 2 00 00 ;

71

72 R = M /( m *H ) ;

73 r = J / ( m* H ^ 3) ;

74

75 f = ( 1 - x ^4 * R *r ) * (c os h ( x ) * c o s (x))-(x*R+x^3*r)*( c os h ( x ) * s i n (x))+(x*R-x^3*r)*( c o s ( x ) * s i n h (x))+(1+

x ^ 4 * R * r ) ;

76 en d

A.3 velocity.m

1 %%

2 % U = v e lo c it y ( mu , s i gm a , L , T ) i s a f u nc t io n t h at c o mp u te s t he v e lo c it y o f t he

3 % w i nd i n a c e rt a in p o in t s .

4 %@PRE:

5 % * t i s a 1 x (N + 1) v ec to r t ha t r ep re se nt s t he t im e ~ [ s ]

6 % * m u i s a s ca la r t ha t i s t he w in d v el oc it y a ro un d t he h ub ~ [ m /s ]

7 % * s i g m a i s s c al a r t h a t i s t he d e vi a ti o n o f t he v e lo c it y i n t h i s p o i n t

8 % * L i s a s ca la r t ha t d es ig ns a l en gt h s ca le ~ [ m ]

9 % * N i s a s ca la r t ha t i s t h e n um be r o f f re qu en ci es u se d f or t he i nv er se D FT

10

% * T i s a s ca la r t ha t d es cr ib es a n a rb it ra ry p er io d a bo ut t he w in d v el oc it y ~ [ s ]11 % * Y i s a v ec to r t ha t c on ta in s t he y c oo rd in at es o f t h e p oi nt s

12 % * Z i s a v ec to r t ha t c on ta in s t he z c oo rd in at es o f t h e p oi nt s

13 % @ P O S T :

14 % U i s a l e ng t h (Y ) x (N + 1) m a tr i x t h a t c o nt a in s t he w in d s pe e d i n c e rt a in p o in t s

18


21/23

15 % t h r ou g h t i m e

16 %%

17 f u n c t i o n U = v e l o ci t y ( t , mu , s i g ma , L , N , T , Y , Z)18

19 N = N / 2; % n u m be r o f u s e fu l f r e q u en c i e s

20 M = l e n g t h (Y); % n u m be r o f p o in t s21 F = ( 1: N )/ T ;% d i s c re t e d f r e q ue n c i e s

22 F re q = o n es ( M , 1) * F ;

23

24

25 P HI = rand (M,N)*2* p i ;26 P SY = z e r o s ( M , N ) ;

27

28 V = z e r o s ( M , N ) ; % c o l u mn s = f r e q u e n cy A N D r o w s = s p ac e29 f o r f = 1 : N %frequency30

31 S = d i a g (PSD(F(f))*ones(M,1));

32 H = z e r o s ( M , M ) ;33

34 f o r k = 1 : M %space

35

36 f o r l=1:(k-1) %sum37 H (k , k) = H (k , k) - H (k , l) ^ 2;

38 en d39 H (k , k) = H (k , k) + S (k , k) ;

40 H ( k ,k ) = s q r t ( H ( k , k ) ) ;41

42 f o r j = 1 : M %space

43

44 i f j ~ = k45 r = s q r t( ( Y ( j ) - Y (k ) ) ^ 2 + ( Z ( j ) - Z ( k ) ) ^2 ) ;46 S ( j , k ) = C O H ( r , F (f ) ) * P S D ( F ( f) ) ;

47 en d48

49 i f j >k50 f o r l=1:(k-1) %sum

51 H ( j ,k ) = H ( j ,k ) - H ( j ,l ) * H( k , l) ;

52 en d53 H (j , k) = H (j , k) + S (j , k) ;

54 H ( j ,k ) = H ( j ,k ) / H( k , k) ;

55 en d56

57 i f j


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A.4 testVelocity.m

1 % M A TL A B s c r i pt t ha t p r od u ce s a w i nd s p ee d a nd w r it e t he d at a i n s e ve r al . t xt f i le s

2

3 % w i n d p a r a m e t e rs

4 mu = 3 0;

5 s ig ma = 1 ;

6

7 % b l ad e p a ra m et e rs

8 L = 3 40 ;

9 lB = 3 0;

10 o ff se t = 3 ;

11

12 % d i sc r et e p o in t s ~ t im e h i st o ri e s

13 n PT S = 1 0; % n u mb e r o f p o in t s a l on g a b l ad e

14 d x = l B / ( n PT S - 1 ) ;

15 X = z e r o s(1,3*nPTS);

16 Y = X;

17 Z = Y;

18 a l ph a 1 = p i /2 ;19 a l ph a 2 = a l ph a 1 + 2* p i /3 ;

20 a l ph a 3 = a l ph a 2 + 2* p i /3 ;

21 A L P HA = [ a l ph a 1 , a l p h a2 , a l p h a3 ] ;

22 f o r i=1:nPTS % c o o r di n a t e s o f d i s c re t e p o i nt s23 f o r j=1:3

24 k = j + ( i- 1) * 3;

25 Y ( k ) = ( ( i - 1 ) * dx + o f f s e t ) * c o s (ALPHA(j));26 Z ( k ) = ( ( i - 1 ) * dx + o f f s e t ) * s i n (ALPHA(j));27 en d28 en d

29 Y( abs (Y)


23/23

Bibliography

[1] Iec 61400-1 "wind turbines. part 1: Design requirements", 2005.

[2] Chad Van der Woude and Dr. Sriram Narasimhan. Dynamic structural modelling ofwind turbines using comsol multiphysics. In Proceedings of the COMSOL Conference 2010Boston, 2010.

[3] Martin O. L. Hansen. Aerodynamics of Wind Turbines.

[4] Murtagh and Broderick. Simple models for natural frequencies and mode shapes of towersupporting utilities.

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