Gasch R., Twele J. (Eds.) Wind Power Plants: Fundamentals, Design, Construction and Operation

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R. Gasch and J. Twele (eds.), Wind Power Plants: Fundamentals, Design, Construction 272

and Operation, DOI 10.1007/978-3-642-22938-1_8, © Springer-Verlag Berlin Heidelberg 2012

8 Structural dynamics

Wind turbines are structures which have the tendency to vibrate easily due to the

nacelle mass placed on top of the slender, elastic mast. They are heavily excited

by the varying loads caused by wind, waves, earthquake, rotation of the rotor,

switching and control procedures. The vibration behaviour has a big influence on

the deformations, the inner stresses and the resulting ultimate limit state, fatigue

and operating life of the wind turbine.

This chapter offers a qualitative description of the fundamental aspects of the

dynamic behaviour and the design calculations of the wind turbine as a complete

system. In chapter 9 design and analysis procedures are exemplarily presented for

the three typical wind turbine components tower, hub and rotor blade.

Many issues from other chapters of this book are drawn on here. This is visual-

ized in the presentation of the design procedure, Fig. 8-1, by referring to the

respective chapter numbers. After choosing a wind turbine specific design guide-

line (section 9.1) the relevant load cases are determined according to the site

conditions (section 4.5 and 16.1) and the type of wind turbine (chapter 3).

General standards

and guidelines

Specific wind energy

standards and guidelines

Standards, guidelines

Guideline

Environmental

conditions

Load cases

Operating

conditions

Wind

field

Dynamic simulation of the system “wind turbine”

Hydro-

dynamics

Aero-

dynamics

Structural

dynamics

Electr.

system

Control

system

Structural loads,

(time series or spectra, extreme values,

load collective)

Site

Wind tur-

bine type

(Statist.) mechan. component model

(FEM, analytical or empirical)

Ultimate limit state verification

fracture, buckling, fatigue

Natural frequencies

and damping

Deformations

Serviceability analysis

(geometry, free of resonance, dynamic stability)

16.14.2 5, 6

4.5 3

8.3

9.2.1 9.2.2

9.2.3

9.1

8.1

8.2

Validation

by measure-

ments

8.4

General standards

and guidelines

Specific wind energy

standards and guidelines

Standards, guidelines

Guideline

Environmental

conditions

Load cases

Operating

conditions

Wind

field

Dynamic simulation of the system “wind turbine”

Hydro-

dynamics

Aero-

dynamics

Structural

dynamics

Electr.

system

Control

system

Structural loads,

(time series or spectra, extreme values,

load collective)

Site

Wind tur-

bine type

(Statist.) mechan. component model

(FEM, analytical or empirical)

Ultimate limit state verification

fracture, buckling, fatigue

Natural frequencies

and damping

Deformations

Serviceability analysis

(geometry, free of resonance, dynamic stability)

16.14.2 5, 6

4.5 3

8.3

9.2.1 9.2.2

9.2.3

9.1

8.1

8.2

Validation

by measure-

ments

8.4

Fig. 8-1 Design procedure for wind turbines; determination of design load cases from site, type

of wind turbine and applied guidelines, dynamic simulation and final scope of verification

(numbers refer to corresponding chapters of this book)

The internal forces and moments at different component sections as well as the

global deformations and natural frequencies are then determined numerically

through simulation of the structural dynamics for the complete wind turbine sys-

tem (section 8.3). Typically, this done by simultaneously calculating the aspects of

the wind field, the aerodynamics, hydrodynamics and structural dynamics as well

as the electrical system and the supervisory and control system in the time domain

in one single model. In a second step, detailed models are used to determine the

8 Structural dynamics

273

stresses on the component level according to general technical standards (section

9.1) in order to perform the various analyses (section 9.2). The operating and per-

formance behaviour as well as the modelling of the structural dynamics are

validated by measurements on erected wind turbines (section 8.4)

Sections 8.3 and 8.4 of this chapter will discuss the simulation and its valida-

tion by measurements. Prior to that, the following two sections will present the

basic aspects of dynamic excitations and the phenomenology of the vibrational

behaviour of the wind turbine system.

8.1 Dynamic excitations

Types of vibration excitation

For the design and the fatigue analysis of most wind turbine components, the

dynamic behaviour in the frequency range of up to approx. 10 Hz is relevant.

Phenomena of higher frequencies occur, for example, in the electrical system, the

drive train and the gearbox (teeth contact frequencies, shaft and casing elasticity,

bearing clearance). The description of the behaviour often relates to two time

ranges of different length. Short-term events, so-called transient phenomena, such

as switching procedures or impulse effects, are investigated using short time series

of up to approximately 1 minute. By contrast, the operating characteristics for

stochastic excitation are analysed in time series between 10 minutes and 1 hour in

which the statistic environmental conditions, e.g. the mean wind speed and turbu-

lence or the swell, may be considered as constant, cf. Fig. 4-23.

The excitations, i.e. the external loads acting on the wind turbine, may be dis-

tinguished according to their time history, cf. table 8.1:

- constant (quasi-steady)

- cyclic (periodic)

- stochastic (random)

- short-time (transient)

Moreover, it is helpful to distinguish the exciting forces according to their physi-

cal origin. There are mass forces, inertia forces and gravitational forces due to the

operating conditions as well as aerodynamic and hydrodynamic loads from the

environmental conditions. In particular, for transient manoeuvres and malfunction

there are additional relevant electrical and electromechanical forces, frictional and

braking forces, hydraulic or electro-mechanic actuating forces. An overview of the

most important exciting forces is given in table 8.1.

8.1 Dynamic excitations

274

Table 8.1 Classification of some exemplary exciting forces according to time history and origin

Origin

Time history

Type of force Source Operating

condition

Constant

(quasi-steady)

Gravity force,

Centrifugal force,

Mean thrust

Weight,

Rotor revolution,

Mean wind

Normal operation

Cyclic

(periodic)

Mass unbalance,

Aerodynamic

forces

Unbalance,

Tower dam,

Oblique flow,

Blade passage

tion,

Malfunction

Stochastic

(random)

hydrodynamic

forces

Turbulence of the

wind,

Swell,

Earthquake

Normal operation

Short-time

(transient) braking forces,

Aerodynamic

forces

Manoeuvre,

Malfunction,

tions

8.1.1 Mass, inertia and gravitational forces

Gravitational forces and excitations due to weight

For the non-rotating components of the wind turbine, e.g. nacelle, tower, founda-

tion, there are constant steady loads due to the dead weight. However, the internal

forces and moments in the tower sections should be determined for the deformed

system using the 2

order theory (buckling) because the horizontal displacement

of the nacelle induces additional bending moments.

The dead weight of rotating components, e.g. the blades, has different effects.

Here, the direction of the gravitational force relative to the blade is changing per-

manently, resulting in strong alternating loads with the frequency of the rotational

speed (1ȍ).

Normal opera-

Aerodynamic and

Frictional and Shut down of the

wind turbine,

Yawing of the

nacelle

Extreme condi-

8 Structural dynamics

275

m g

ȍt

Achs

m gm g

ȍt

Achs

m g

axis

m g

ȍt

Achs

m gm g

ȍt

Achs

m g

axis

Fig. 8-2 Alternating loads due to blade weight in edgewise and flapwise direction (r

= center of

gravity radius)

The highest alternating bending moment at the blade root due to weight occurs in

the edgewise direction, Fig. 8-2

trgmM

sin

Sedgewise

. (8.1)

The inclination of the rotor shaft by the angle

axis

increases the distance between

blade tip and tower and is often chosen between 4° and 5°. Thus, the blade weight

also causes small 1ȍ-excitations of the flapwise bending moment at the blade root

trgmM

sinsin

axisSflapwise

. (8.2)

Centrifugal forces and mass unbalance

For constant rotational speed the centrifugal forces F cause only steady tensional

loading at the rotating blade . This is also the case if one blade is slightly heavier

than the others having an excess mass ǻm, Fig. 8-3.

Relative to the rotational axis of the rotor (i.e. in the inertial system) the star of

the three centrifugal forces F is basically balanced. Only the excess mass ǻm

causes a rotating excitation force

:' ' rmF . (8.3)

Its horizontal component excites lateral vibrations of nacelle and tower,

trmF ::' sin

shor

. (8.4)

8.1 Dynamic excitations

276

There is also a vertical excitation, but since in this direction the tower is very stiff

the resulting movements are negligible.

horz

= ǻF sin ȍt

ǻF= ǻm r

ȍt

ǻF

F= m r

ǻm

horz

= ǻF sin ȍt

ǻF= ǻm r

ȍt

ǻF

F= m r

ǻm

hor

horz

= ǻF sin ȍt

ǻF= ǻm r

ȍt

ǻF

F= m r

ǻm

horz

= ǻF sin ȍt

ǻF= ǻm r

ȍt

ǻF

F= m r

ǻm

hor

Fig. 8-3 Centrifugal forces F in blades and additional centrifugal force ǻF from mass unbalance

ǻm r

, causing lateral vibrations of tower and nacelle

8.1.2 Aerodynamic and hydrodynamic loads

There is a wide variety of aerodynamic excitations of the wind turbine. In the

following, we will first describe some periodic loads, which origin from determi-

nistic disturbances of the uniform inflow (deterministic wind field). After that,

stochastic wind loads (turbulent wind field) and transient aerodynamic forces are

presented. When considering the aerodynamics it must be kept in mind that due to

the dependency of the air density on temperature and altitude, the loads may vary

between up to 20 and 30% depending on the site conditions and season.

Excitations by the vertical wind profile

In section 4.2 the properties of the surface boundary layer concerning the vertical

wind profile, the gustiness and, the turbulence of the wind were already presented.

Apart from these two aspects further phenomena, e.g. oblique flow cause strong

aerodynamic excitation of the wind turbine. To simplify matters we consider that

the vertical wind profile in the rotor area increases linear with increasing height,

Fig. 8-4. Viewed from the individual rotating blade at the flow (rotating observer),

the wind speed in a blade section varies periodically around the mean value v

between v

= v

+ ǻv at the top position and v

= v

- ǻv at the bottom position,

according to v

= v

+ ǻv cos ȍt.

8 Structural dynamics

277

+ ǻv

- ǻv

ǻv

dA + ǻdA

dA - ǻdA

ǻv

–

ȍt

+ ǻv

- ǻv

ǻv

dA + ǻdA

dA - ǻdA

ǻv

–

ȍt

+ ǻv

- ǻv

ǻv

dA + ǻdA

dA - ǻdA

ǻv

–

ȍt

+ ǻv

- ǻv

ǻv

dA + ǻdA

dA - ǻdA

ǻv

–

ȍt

dL + 'dL

dL - 'dL

+ ǻv

- ǻv

ǻv

dA + ǻdA

dA - ǻdA

ǻv

–

ȍt

+ ǻv

- ǻv

ǻv

dA + ǻdA

dA - ǻdA

ǻv

–

ȍt

+ ǻv

- ǻv

ǻv

dA + ǻdA

dA - ǻdA

ǻv

–

ȍt

+ ǻv

- ǻv

ǻv

dA + ǻdA

dA - ǻdA

ǻv

–

ȍt

dL + 'dL

dL - 'dL

Fig. 8-4 Variation of the flow conditions in the plane of rotation and lift force dL, due to the

vertical wind profile and the angular position of the blade

The angle of attack Į

and the relative velocity w oscillate correspondingly. Only

the circumferential speed u is independent of the angular blade position ȍt. There-

fore, the aerodynamic forces at the rotating blade fluctuate with the angular fre-

quency 1ȍ around the mean value.

Higher harmonics of the rotational frequency, which are caused by the non-

linear curves of the c

- and c

coefficients and the square dependency of lift and

drag on the relative velocity w, have weak effects.

For nacelle and tower (inertial system) of a wind turbine with a three-bladed

rotor, these fluctuations are completely balanced out by the superposition of the

values from the three blades arranged at 120° to one another. The rotor acts like a

disk where the pressure point lies above the rotation axis – it is shifted upwards a

little due to the vertical wind profile. This causes a constant tilting moment around

the lateral nacelle axis. By contrast, viewed from the rotating main shaft, this iner-

tially fixed pressure point causes for every revolution alternating bending stress

which oscillates with the rotor’s angular frequency (1ȍ).

The two-bladed rotor behaves quite differently. Here the fluctuating blade

loads are not equalized with respect to the inertial system of nacelle and tower,

Fig. 8-5. If the blades are vertical (0°) a strong tilting (and yawing) moment is

produced in the inertial system, which disappears at the horizontal blade position

(90°) but again reaches its maximum value at 180°. This cycle is repeated once

again until at 360° one revolution is finished.

8.1 Dynamic excitations

278

Nick

Rotation angle

ȍ t

2ʌ

(360

)

ʌ/2

(90

)

(180

)

3/2 ʌ

(270

)

Nick

Rotation angle

ȍ t

2ʌ

(360

)

ʌ/2

(90

)

(180

)

3/2 ʌ

(270

)

Nick

Rotation angle

ȍ t

2ʌ

(360

)

ʌ/2

(90

)

(180

)

3/2 ʌ

(270

)

tilt

Nick

Rotation angle

ȍ t

(360

)

ʌ/2

(90

)

(180

)

3/2

(270

)

Nick

Rotation angle

ȍ t

(360

)

ʌ/2

(90

)

(180

)

3/2

(270

)

Nick

Rotation angle

ȍ t

(360

)

ʌ/2

(90

)

(180

)

3/2

(270

)

tilt

Fig. 8-5 Bending moment (twice per revolution) around the lateral nacelle axis for a two-bladed

wind turbine due to the vertical wind profile

The resulting tilting moment, as well as the yawing moment, fluctuates with twice

the angular frequency which produces high fatigue loads for nacelle frame and

tower. In the rotating main shaft there is again alternating bending stress with the

rotor’s angular frequency (1ȍ).

Since in the two-bladed wind turbine the natural frequencies of the tower-

nacelle system depend on the angular blade position there are parameter-excited

vibrations which cause dynamics that are far more problematic than in the three-

bladed wind turbine.

Excitations by tower dam, tower wake and Kármán vortex street

The tower dam and tower wake are caused by the tower being an obstacle that

retards and deflects the flow, which then separates behind the tower and becomes

very turbulent. These separations occur periodically, alternately right and left, and

are washed downstream which causes the so-called Kármán vortex street, Fig. 8-6.

This may excite lateral vibrations of the tower if the vortices shed with a fre-

quency which is close to the tower’s natural bending frequency. In reality, these

conditions only occur during erection before the heavy nacelle is mounted on the

tower. The tower dam of wind turbines with an upwind rotor (and the even more

severe tower wake of the downwind rotor turbines) has strong effects on the vibra-

tions of blade and tower-nacelle system. The short-term collapse of the aero-

dynamic forces at the blade when the blade approaches the tower once per revolution

provokes a strong excitation with the 1ȍ angular frequency with many additional

8 Structural dynamics

279

harmonics (2ȍ, 3ȍ, 4ȍ, etc.); with growing order the intensity decreases only

slightly.

In the inertial tower-nacelle system there is a loading impulse from each blade

passage, Fig. 8-7. For a three-bladed rotor, the tower-nacelle vibrations are excited

with 3ȍ, three times the angular frequency, and its multiples, 6ȍ, 9ȍ, 12ȍ, etc.

Correspondingly, the excitation frequencies for the two-bladed rotor are 2ȍ, 4ȍ,

6ȍ, 8ȍ, etc.

Fig. 8-6 Tower dam, wake and Kármán vortex street

S(t) Thrust

Rotation angle

2ʌ

(360

)

ȍ t

ʌ/3

(60

)

(180

)

5/6 ʌ

(300

)

ȍ t

S(t) Thrust

Rotation angle

2ʌ

(360

)

ȍ t

ʌ/3

(60

)

(180

)

5/6 ʌ

(300

)

ȍ t

T (t)

(60°) (180°) (300°) (360°)

S(t) Thrust

Rotation angle

(360

)

ȍ t

ʌ/3

(60

)

(180

)

5/6

(300

)

ȍ t

S(t) Thrust

Rotation angle

(360

)

ȍ t

ʌ/3

(60

)

(180

)

5/6

(300

)

ȍ t

T (t)

(60°) (180°) (300°) (360°)

Fig. 8-7 Effects of tower dam: thrust versus rotation angle for a three-bladed wind turbine

Oblique flow and aerodynamic unbalance

There is almost always a slight oblique inflow (i.e. inclined inflow) at the rotor, for

instance due to effects of a hill, delayed yawing of the nacelle or the inclination of

the rotor axis to the horizontal plane. The blade loads vary with the rotor’s angular

frequency 1ȍ, and the tower-nacelle system experiences the blade passing

8.1 Dynamic excitations

280

frequency 3ȍ. Calculating the aerodynamics of a rotor operating under oblique

inflow is very complicated.

Errors in the blade angle adjustment, e.g. during assembly, as well as produc-

tion tolerances of the outer blade shape, may lead to different aerodynamic forces

for the individual blades of a rotor. These aerodynamic unbalances periodically act

on the tower-nacelle system in axial and lateral direction with the angular fre-

quency 1ȍ, which may excite heavy vibrations.

Gusts and turbulent wind field

In order to determine the energy yield the fluctuating wind was classified using

1- to 10-min averages (histogram, distribution function, cf. chapter 4). The wind

turbine displays quasi-steady behaviour for processes and changes in the time

scale of minutes.

Gusts typically have a passing time of 3 to 20 s and a spatial extension in the

lateral direction of 10 to 100 m. They excite the relevant natural frequencies of the

structures in the range between 0.2 and 10 Hz. During operation, the rotor has a

period of 2 to 5 s per revolution (i.e. 12 to 30 rpm). Typically, the rotor is hit by

the gust eccentrically; therefore, every rotor blade passes several times through the

structure of the gust, Fig. 8-8. This phenomena of the blade rotating through a par-

tial gust is called ”eddy slicing“ or ”rotational sampling“. As with the tower dam,

the gusts provoke a harmonic excitation of 1ȍ, 2ȍ, 3ȍ, etc. [1]. Since these load-

ings occur once or several times per revolution,, the excitations are also called 1P,

2P, 3P, etc. excitations. This is clearly visible after transforming the turbulence

spectra in Fig. 8-9 from the inertial system of the hub to the rotating system of the

blade. The broad band energy of the turbulent excitation is concentrated near the

rotational frequency of 0.5 Hz and its harmonics. As could be assumed from the

time curve shown in Fig. 8-8, this effect is more distinct the farther away the point

is from the hub centre.

Apart from the amplitude height of these cyclic loads induced by turbulence

their extreme high number of occurrences is crucial for the fatigue loading of an

operating wind turbine. During the expected lifetime of 20 years, the rotor of a

modern wind turbine revolves approx. 100,000,000 times typically causing 5·10

to 1·10

load cycles. This is an order of magnitude far beyond the load cycle

numbers occurring in other technical applications, e.g. automobiles or airplanes,

cf. section 9.2.3.

8 Structural dynamics

281

Rotational angle ȍ t

Wind speed v(t)

6ʌ2ʌ

8ʌ

4ʌ

ȍ t

Point 3

Point 2

Point 1

Point 2

Point 3

v·T

Rotational angle ȍ t

Wind speed v(t)

6ʌ2ʌ

8ʌ

4ʌ

ȍ t

Point 3

Point 2

Point 1

Point 2

Point 3

v·T

Fig. 8-8 Left: time history of wind speed for three points on the rotor blade during the passage

of a partial wind gust: point 1 inertial at hub centre, point 2 rotating in the middle of the blade,

point 3 rotating near blade tip; right: spatial structure of the eccentric partial wind gust passing

the rotor, front view and side view

r = 10m

0.001

Frequency (log. scale) in Hz

r = 20m

0.01

0.1

Power spectral density

(log. scale) [m

/s]

Frequency (log. scale) in 1 /revolution

1ȍ 2ȍ

3ȍ 4ȍ

r = 0m

0.5

1.5

r = 10m

0.001

Frequency (log. scale) in Hz

r = 20m

0.01

0.1

Power spectral density

(log. scale) [m

/s]

Frequency (log. scale) in 1 /revolution

1ȍ 2ȍ

3ȍ 4ȍ

r = 0m

0.5

1.5

Fig. 8-9 Transformation of the turbulence spectrum from the inertial coordinate system of the

hub (r = 0 m) into the rotating coordinate system of the rotor blade for 2 different radial positions

(r = 10 m and r = 20 m) (example: rotor diameter 40 m, rotor speed 30 rpm), [1]