Structural analysis of a wind turbine and its drive train using the exible multibody simulation technique

J. Peeters1 , D. Vandepitte2 , P. Sas2

1

Hansen Transmissions International, Leonardo da Vincilaan 1, B-2650 Edegem, Belgium e-mail: jpeeters@hansentransmissions.com
2

K.U.Leuven, Department of Mechanical Engineering, Celestijnenlaan 300 B, B-3001, Heverlee, Belgium

Abstract
This article demonstrates the application of a generic methodology, based on the exible multibody simulation technique, for the dynamic analysis of a wind turbine and its drive train, including a gearbox. The analysis of the complete wind turbine is limited up to 10 Hz, whereas the study of the drive train includes frequencies up to 1500 Hz. Both studies include a normal modes analysis. The analysis of the drive train includes additionally a response calculation for an excitation from the meshing gears, a Campbell analysis for the identication of possible resonance behaviour and a simulation of a transient load case, which occurs as a sudden torque variation caused by a disturbance in the electrical grid.

Introduction

During the last decades, the interest for using renewable energy sources for electricity generation increased [1]. One of its results is a boom in the wind turbine industry since ten years. Figure 1 shows how the global installed wind power capacity reached 59.3 GW at the end of 2005, of which about 20% had been installed in that year. This rapid growth is expected to continue in the coming years and to drive new technological improvements to further increase the capacity and reduce the cost of wind turbines. In their design calculations, the wind turbine manufacturers use dedicated software codes to simulate the load levels and variations on all components in their machines. Peeters [3] gives an overview of the existing traditional simulation codes. He concludes that the concept of the structural model of a wind turbine in all these codes is more or less similar and that the behaviour of the complete drive train (from rotor hub to generator) is typically represented by only one degree of freedom (DOF). This DOF represents the rotation of the generator and, consequently, the torsion in the drive train. Peeters describes additionally the consequences of using this limited structural model for the simulation of drive train loads. The output of the traditional simulations lacks insight in the dynamic behaviour of the internal drive train components. De Vries [4] also raises the lack of insight in local loads and stresses in a drive train and the insufcient understanding of the design loads. He relates furthermore a series of gearbox failures in wind turbines to these consequences of simulating with a limited structural model. More insight can be gained from a more detailed simulation approach. Peeters [3] presents a generic methodology for this, which is based on three multibody system (MBS) modelling approaches.

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Figure 1: The evolution of the global cumulative installed wind power capacity from 1995 to 2005 [2]. The rst approach is limited to the analysis of torsional vibrations only. The second technique offers a more realistic representation of the bearings and the gears in the drive train and its generic implementation can be used for both helical and spur gears in parallel and planetary gear stages. The third method is the extension to a exible MBS analysis, which yields information about the elastic deformation of the drive train components in addition to their large overall rigid-body motion. The implementation of the different models was done in LMS DADS Revision 9.6 [5] (DADS). This article presents an application of the second and third technique for the analysis of a drive train in a wind turbine. This example includes: a normal modes analysis for the determination of eigenmodes and eigenfrequencies (1) of the wind turbine and (2) in the drive train a response calculation for an excitation from the meshing gears a Campbell analysis for the identication of possible resonance behaviour a simulation of a transient load case

2 Drive train layout and model implementation

Figure 2 shows the wind turbine for the present application. It is a generic example of which the results are representative for a modern wind turbine with a gearbox. The drive train has one main bearing integrated in the gearbox carrying the wind turbine rotor. The generator is a doubly fed induction generator (DFIG) and the gearbox design is a combination of two planetary stages with one high speed parallel stage. The wind turbine rotor is connected to the planet carrier of the rst planetary stage. This stage has spur gears and its ring wheel is xed in the gearbox housing. This housing is assumed to be rigid as well as its connection to the bed plate. This latter frame supports also the generator and rests on the yaw bearing, which connects the complete nacelle with the tower. The second gear stage in the gearbox is a helical planetary stage. Its planet carrier is driven by the sun of the rst stage and its ring wheel is also xed in the gearbox housing. The sun of this stage drives the gear of the third stage, which is a parallel stage with helical gears. The

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pinion of this stage rotates at the speed of the generator. A brake disk is mounted on this output shaft and a exible coupling connects it with the input shaft of the generator. Since only the parallel gear stage causes a change in the direction of rotation, the high speed pinion and the generator rotate in the opposite direction of the rotor. The reaction torque of the gearbox on the bed plate is therefore slightly larger than the input torque. This torque acts in the same direction as the rotation of the rotor, which is clockwise when looking at the wind turbine in the direction of the wind. The torque on the generator support acts in the other direction.

rotor hub g g g g

high speed planetary stage g g g high speed parallel stage g g g g exible coupling      h low speed planetary stage h h h h h h generator h h bed plate brake disk yaw bearing

Figure 2: Simplied representation of the three bladed wind turbine with a zoom on its drive train. The model of the wind turbine consists of: 1. a model of the gearbox, which is a system of rigid bodies with six DOFs per body: the formulations for the gear contacts and the bearings are based on a synthesis of the work of Kahraman [6] and of Lin and Parker [7]; 2. a brake disk, which is modelled as an additional body clamped on the pinion; 3. a exible coupling between the high speed shaft of the gearbox and the generator, which is considered as very exible in all directions, except for its torsional deformation and, therefore, it is modelled as a torsional spring; 4. a rigid body with only one rotational DOF to represent the generator: the generator is furthermore free to rotate;

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5. a model of the rotor and the tower, which differs depending on the frequency range of the analysis (cfr. below). 1. [ 0 - 10 Hz] For the calculation of the wind turbines response in a frequency range up to 10 Hz, which is typically the limit in the traditional simulations, it is necessary to include a realistic representation of the structural properties of the rotor and the tower. This is done by using a exible multibody approach for these bodies. Both the rotor and the tower have six rigid-body DOFs and, furthermore, an extra set of DOFs to represent the internal deformations of these components. The latter DOFs of the so-called exible bodies are derived from an FE model of the rotor and the tower respectively, using the CMS technique [8, 9]. The set of component modes is composed such that it accurately represents the dynamic behaviour of the individual components up to 10 Hz. This requires a consideration of four pairs of normal bending modes for each blade and the rst ten normal modes of the tower. The remainder of this article describes the results of a normal modes analysis for this model. 2. [ 10 - 1500 Hz] The analysis of the drive train loads in a frequency range up to 1.5 kHz1 requires - in theory - the consideration of all rotor and tower modes up to approximately 3 kHz. However, since this is an impracticable task and since both the rotor and the tower have more than ten modes below this frequency range, it is assumed that they will act as a large inertia with respect to an excitation at higher frequencies. Therefore, instead of adding two exible bodies for the rotor and the tower, only one rigid body with six DOFs is included to represent the large inertia of the rotor. The tower is considered as a rigid ground, which supports the gearbox and the generator. This implies no need for an additional body to represent the tower. The remainder of this article describes three types of analysis for this model: a normal modes analysis, a frequency response analysis and a transient load simulation.

Discussion of the analyses

3.1 Low frequency range: [ 0 - 10 Hz]

Table 1 summarises the normal modes calculated for the exible multibody model of the wind turbine in the frequency range [ 0 - 10 Hz]. These results correspond to the typical results, which can be calculated in a traditional wind turbine simulation code. The 8th, 14th and 23rd mode are the respective rotor torsion modes or drive train modes. These modes have the biggest inuence on the torque in the drive train and are determined by a combined effect of the following parameters, as demonstrated by Peeters [3]: 1. the (distributed) rotor inertia and exibility (including the pitch angle and rotor position) 2. the drive train stiffness 3. the generator inertia 4. the coupling between the drive train and the tower top (e.g. the gearbox support and the generator characteristic) 5. the (distributed) tower inertia and exibility
This work focusses on a range up to 1.5 kHz, which is considered to be more than sufcient, since 1 kHz is generally the maximum for the gear mesh excitation frequencies.
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No. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Description rigid-body mode (drive train rotation) 1st tower longitudinal 1st tower transversal 1st asymmetric rotor ap/yaw (A) 1st asymmetric rotor ap/tilt (B) 1st symmetric rotor ap (C) 1st asymmetric rotor edge (B) 1st asymmetric rotor edge (C) 1st rotor torsion (A) 2nd asymmetric rotor ap/tilt + 2nd tower bending 2nd asymmetric rotor ap/yaw + 2nd tower bending 2nd asymmetric rotor ap/yaw + 1st tower torsion 2nd symmetric rotor ap 2nd asymmetric rotor ap/tilt 2nd rotor torsion 2nd asymmetric rotor edge (B) 2nd asymmetric rotor edge (C) 3rd asymmetric rotor ap/yaw + tower torsion 3rd asymmetric rotor ap/tilt 3rd symmetric rotor ap 3rd asymmetric rotor ap/yaw + 3rd tower bending 3rd asymmetric rotor ap/tilt + 3rd tower bending 3rd asymmetric rotor ap/yaw + 2nd tower torsion 3rd rotor torsion 3rd asymmetric rotor edge (B) 3rd asymmetric rotor edge (C) 4th asymmetric rotor ap/yaw 4th symmetric rotor ap 4th asymmetric rotor ap/tilt

Eigenfrequency (Hz) 0 0.33 0.34 1.03 1.11 1.17 1.49 1.50 2.03 2.45 2.53 2.70 3.10 3.18 4.42 4.44 4.46 4.88 5.63 6.10 6.87 6.90 7.67 8.71 9.28 9.55 9.74 10.1 10.3

Table 1: Results of a normal modes analysis of the wind turbine model with one blade in horizontal position, all blades pitched for normal operation and a pinned generator (= free to rotate).

3.2
3.2.1

Medium frequency range: [ 10 - 1500 Hz]

Normal modes analysis

Table 2 shows the eigenfrequencies calculated for the rigid multibody model of the wind turbine in the frequency range [ 10 - 1500 Hz]. The eigenfrequencies are classied according to the location of the nodes in the corresponding mode shapes and, additionally, for the planetary stages according to their type. The types of eigenmodes in the planetary stages are [7]: rotational modes: the corresponding mode shapes have pure rotation of the carrier, ring and sun and all planets have the same motion in phase. translational modes: the corresponding mode shapes have pure translation of the carrier, ring and sun. out-of-plane modes: the corresponding mode shapes have an out-of-plane component in translation and/or rotation of the carrier, ring, sun and/or planets.

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No. 1 2 3,4 5 38,39 44 45 51 52 53 No.

Global mode 0 5.6 32 68 1033 1506 1527 2390 2396 2560

No. 17 18 19 21 25 32 33 43 50

Parallel stage 346 406 409 430 562 750 832 1238 1731 Low speed planetary stage Rotational Trans- Out-of-plane mode lational mode (m=1) mode (m=2) 205 227 305 423 513 660 731 1003 1011 1640

6 7 14 23,24 25 27 34 40 42 46 48 54 55,56 57 58,59

High speed planetary stage Rotational Translational Out-of-plane mode mode mode (m=1) (m=2) 81 140 302 527 568 630 864 1093 1151 1558 1564 3048 3054 3066 3123

No.

9 11,12,13 15 20 22 29 31 35 37 49

Table 2: Eigenfrequencies (Hz) of the rigid multibody model of the wind turbine in the frequency range [ 0 - 1500 Hz] (m denotes the multiplicity of the modes in the planetary stages). The category of global modes are those modes, for which the deformation cannot be attributed to a single gear stage: The global mode at 5.6 Hz corresponds to a torsional deformation of the drive train or a so-called drive train mode. However, as discussed in section 3.1, it is inaccurate to calculate such modes without the consideration of the rotor exibility. Therefore, this mode is not considered as a relevant result. The double mode at 32 Hz corresponds to a tilting motion of the rotor in the main bearing. Again, an omission of the rotor exibility for the calculation of this mode is inaccurate, since the apwise rotor modes have a determining inuence. Consequently, also this double mode is not considered as a relevant result. The global mode no. 5 at 68 Hz corresponds to the torsional deformation of the exible coupling. This is the rst relevant internal eigenmode for this particular drive train. Therefore, in the remainder of this text, it is considered to be sufcient to focus in the frequency response analyses on a frequency range starting from 50 Hz.

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The out-of-plane modes in the low speed planetary stage (at 227 Hz and 443 Hz) are further not considered to be relevant, since they cannot be excited by lack of axial forces on the spur gears in this stage. 3.2.2 Frequency response analysis

This section describes how the results of the normal modes analysis should be interpreted in order to avoid resonance and, furthermore, it demonstrates how the drive train loads can be simulated for a sinusoidal load excitation. Both analyses imply a calculation of the frequency response. In order to avoid resonance, it is important to know which eigenmodes can considerably inuence the loads in the drive train. This means that in one way or another there should be a coupling between these modes and a source of excitation. One known excitation source is the once-per-tooth pattern in a gear mesh, which excites at the so-called gear mesh frequency [10]. The coupling with such an excitation can be estimated from a detailed interpretation of the mode shapes, although, this is not straightforward. An easier method is to calculate a frequency response function (FRF) between the gear mesh excitation and a specic load in the drive train and check which eigenmodes lead to amplied load levels. This is demonstrated for the gear mesh excitation in the high speed planetary stage. The excitation is considered here as a variation of an input torque at the high speed sun. The input torque is a multisine with a minimal frequency of 50 Hz, according to the statement above. The upper frequency limit is 2.0 kHz, which guarantees a proper excitation at all frequencies in the range from 50 to 1.5 kHz. Figure 3 shows the spectrum of the multisine signal. The DADS solver calculates the output loads for this signal using a time integration procedure. A maximum time step of 0.0001 second is chosen. The numerical integration is only possible with a certain amount of damping in the drive train, which is included by adding a viscous damper element to the DADS model. The damping denes the amplitudes of the response, especially at the eigenfrequencies. Since the determination of the damping values is not within the scope of this analysis, the responses are only considered qualitatively.

PSD [dB/Hz]

Frequency [Hz] Figure 3: Power spectrum of the torque excitation signal used in the FRF calculations. Figure 4 shows the FRFs calculated for the torque excitation of the sun in the high speed planetary stage. The torque on the high speed pinion and on the two suns are the respective outputs in these calculations. Note that the FRF to the torque at the high speed sun is a direct FRF. The maximum frequency in these plots is only 1.0 kHz, since no relevant amplied torque levels are identied at higher frequencies. The analysis of these results, leads to the following conclusions:

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PSD [dB/Hz]

Frequency [Hz]
(a) FRF to the torque at the high speed pinion.

PSD [dB/Hz]

Frequency [Hz]
(b) FRF to the torque at the high speed sun.

PSD [dB/Hz]

Frequency [Hz]
(c) FRF to the torque at the low speed sun.

Figure 4: Response calculation for a gear mesh excitation in the high speed planetary stage, which is applied as a sinusoidal torque excitation of the high speed sun.

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1. For the high speed pinion, mainly the local modes at 430 Hz and at 562 Hz are dominant in the torque response (cfr. gure 4(a)). 2. The direct FRF to the torque on the sun of the high speed planetary stage, indicates that mainly the local modes in the parallel stage, at 346 Hz and 430 Hz respectively, can lead to a considerable torque amplication (cfr. gure 4(b)). This indicates the importance of analysing the drive train as a whole. The analysis of an individual gear stage yields insight in its local modes, but it does not permit to determine the mutual interaction between dynamic loads in different stages. 3. The torque spectrum at the low speed sun is dominated by one local mode in the parallel stage (346 Hz), one translational mode in the low speed planetary stage (305 Hz) and by six local modes in the range [ 500 - 700 Hz]. 4. The global modes at 68 Hz and 140 Hz contribute also to the torque response in the different gear stages, albeit considerably less than the so-called local modes. The excitation signal used in this analysis has a broadband spectrum in order to get an overall idea about which eigenmodes can lead to amplied torque levels. However, for a particular speed of the drive train, the gear mesh excitation frequency in the high speed planetary stage is exactly known. This permits to determine whether it coincides with an important eigenfrequency. Figure 5 demonstrates this procedure. It is a Campbell diagram, which indicates how the gear mesh frequency varies with the rotational speed of the rotor in the wind turbine. In addition to the actual gear mesh frequency, its rst and second harmonic are also plotted as excitation frequencies. The eigenfrequencies included in this gure are the horizontal lines, which correspond to the dominating peaks from the direct FRF in gure 4(b). These frequencies are considered as the only important frequencies for this excitation with respect to possible torque amplications in this gear stage. The intersection of lines indicate possible resonances. For a xed-speed wind turbine the focus can be limited to a single speed. However, modern variable-speed wind turbines require the consideration of a certain speed range. For the present example, two cursors indicate such a speed range for the wind turbine rotor from 10 to 20 RPM. In this range, the following intersections are found: 1. the gear mesh frequency intersects the eigenfrequency at 140 Hz; 2. the rst harmonic intersects the eigenfrequency at 302 Hz; 3. the second harmonic intersects four eigenfrequencies above 300 Hz. The results from this analysis are valuable input for assessing whether or not a drive train resonance can occur as a result of this excitation. The same analysis can be performed for other excitations and with the focus on other loads. It is obvious that avoiding all intersections between excitations and eigenfrequencies in a quite broad speed range is impossible. However, keeping in mind that the actual gear mesh frequencies are usually more important than their harmonics, the insights from the calculated FRFs yield already valuable information for the evaluation of a drive train design in order to avoid severe resonances. To gain more experience in this evaluation and, more generally, to gain further condence in the present analysis techniques and their results, it is recommended to perform sufcient experimental validation measurements. A more detailed numerical investigation of the intersections in the Campbell diagram at a particular frequency, requires a prediction of the amplied load levels. This simulation requires an accurate consideration of the drive train damping values and an appropriate description of the excitation signal, which represents the transmission error in the gear mesh. Since this is not within the scope of the present article, the load simulation at resonance is not further elaborated.

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Eigenfrequencies [Hz]

Rotor speed [RPM] Figure 5: Campbell diagram which indicates the position of the gear mesh frequency and its harmonics (inclined lines) of the second gear stage for a varying rotor speed, in comparison to the dominant eigenfrequencies (horizontal lines) from the direct torque response function in gure 4(b). The cursors (vertical lines) indicate the speed range during operation. 3.2.3 Simulation of a transient load case

This section demonstrates how the load simulation with a detailed drive train model yields the desired insight in its behaviour and gives much more and more accurate information than obtained with the simulations in the traditional wind turbine design codes. For the present demonstration, the simulation of a transient load case is used. An accurate description of the external forces during a transient load case requires a complete model of the wind turbine system, including a model for the generator, for the aerodynamics, . . . Since these models are not available, an assumption is made here for the load excitation. This is sufcient for the present illustration. An integration of the presented drive train model in a traditional wind turbine design code can help in further rening the load simulations. The simulated transient load case includes a sudden torque variation at the generator with a high amplitude. This phenomenon can have various causes, such as disturbances in the electrical grid as described by Soens et al [11] and Seman et al [12, 13]: e.g. frequency disturbances, a voltage dip or swell and a network short circuit. In the present example, the torque variation occurs during a start-up of the wind turbine. This mean that the generator torque would normally be increasing as shown in gure 6(a). Firstly, the simulation is done for this reference signal, i.e. without the sudden torque peak. The slope of this signal equals 1 kNm/second and the time series has a length of 1 second. Subsequently, the torque variation is added at t = 0.5 s, which is visible in gure 6(b). The shape and the duration of the torque variation may largely differ for various grid disturbances and is furthermore highly dependent on the type of generator. Here, a damped sinusoidal variation with a frequency of 20 Hz is considered; it has a duration of two periods (100 ms) and a maximum amplitude of 9.1 kNm. This example is based on the description in [13] of the generator torque variation during a network short circuit in a DFIG with an over-current protection system.

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(a) 1st load case: a normal start-up (reference signal)

(b) 2nd load case: a grid disturbance during start-up

Figure 6: Generator torque used in the simulations of two transient load cases. Figure 7 shows a comparison of the simulations calculated for the two load cases. In this example, the focus is put on: 1. the level of the torque, which acts on the pinion of the high speed stage 2. the rotational acceleration of the pinion in its bearings 3. the axial displacement of the pinion in its bearings The comparison of the two load cases in gure 7 yields the following conclusions: The sudden torque peak in the generator torque, as a result of the grid disturbance, causes a torque peak at the pinion. The level of this latter peak equals 2.5 kNm, which is about 3.5 times lower than the level of the torque peak in the generator. Further experimental validation of the numerical models and a proper consideration of the damping in the drive train is required to assess the accuracy of this absolute level. This level depends furthermore on the type of coupling used in the drive train, as explained below. The sudden impact in the drive train excites moreover the 1st drive train mode of the wind turbine. As a result, various torque reversals occur during this start-up. This may lead to backlashing in the bearings (cfr. below), which should be investigated for the bearing design. Note that a proper simulation of the 1st drive train mode requires the consideration of the rotor and the tower exibility as described in section 3.1. The grid disturbance and resulting torque variation cause the pinion to accelerate rapidly in its bearings. The acceleration peak level in this example is about 30 times higher than during the normal startup. This should also be considered with care in the design of the bearings as well as the negative acceleration, which follows rapidly after the positive acceleration peak. In the resulting variation of the acceleration, the eigenmode at 68 Hz is clearly visible, which corresponds to the deformation of the exible coupling. Since the investigated model includes linear bearing models, the axial displacement of the pinion follows the torque variation. This implies that the displacement peaks when the torque peak occurs. Subsequently, it becomes negative. Under the assumption that the drive train is at no load conditions for a displacement value of 0 m the change in sign for the displacement corresponds to passing the clearance at the pinion. The simulation of this so-called backlashing may be further improved by using a non-linear bearing model, including clearance.

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(a) Torque at the pinion of the high speed stage

(b) Rotational acceleration of the pinion (high speed stage) in its bearings

(c) Axial displacement of the pinion (high speed stage) in its bearings (0 m corresponds to no load)

Figure 7: Comparison of the results calculated for two transient load cases (dashed: normal start-up; solid: grid disturbance).

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The presented analyses are further elaborated by investigating the inuence of the stiffness of the exible coupling on the level of the torque and the acceleration at the pinion. This demonstrates how the multibody simulation approach can be used as an effective tool to assess the inuence of design changes. Figure 8 shows the simulated torque and acceleration signals for the original coupling (A), for a coupling with a stiffness value of 10% (B) and of 1000% (C) of the original. The absolute stiffness values of the three couplings are shown in table 3, including the dimensions of a hypothetical steel shaft which has this stiffness. Coupling A B C Stiffness [MNm/rad] 1.4 0.14 14 Length [m] 0.5 1 0.3 Diameter [mm] 97 65 155

Table 3: Stiffness values for the three exible couplings used in the transient simulation. The dimensions (length and diameter) of a hypothetical steel shaft with a corresponding stiffness are included.

(a) Torque at the pinion of the high speed stage

(b) Rotational acceleration of the pinion (high speed stage) in its bearings

Figure 8: Comparison of the simulations for three different couplings: solid: original coupling (A); dashed: lower stiffness (B); dotted: higher stiffness (C).

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A comparison of the simulations for the three couplings yields the following conclusions. The level of the torque peak at the pinion decreases when a coupling with a lower stiffness (B) is used. In the present example, the peak is 1.6 times lower than for the original coupling (A). On the other hand, using a coupling with a higher stiffness (C), yields a higher torque peak. In the present example, the maximum torque level is 5% higher. The maximum level of the rotational acceleration of the pinion in its bearings is for both additional couplings lower than for the original coupling.

Conclusions

This article starts from the statement that the output of the simulations with traditional wind turbine design codes lacks insight in the dynamic behaviour of the internal drive train components. It demonstrates subsequently how the multibody simulation technique can be used to gain more insight. The particular example consists of a drive train with a single main bearing integrated in the gearbox and a DFIG. It is a three stage gearbox with one spur planetary gear stage, one helical planetary gear stage and one helical parallel gear stage. The dynamic analysis of this drive train is split up into the frequency ranges [ 0 - 10 Hz] and [ 10 1500 Hz]. For the study in the former range, a exible multibody model with an accurate description of the rotor, the tower and the complete drive train is analysed. All eigenmodes of this model are described and those modes which have the biggest inuence on the torque - the so-called drive train modes - are indicated. In the latter frequency range, the focus is limited on the local eigenmodes in the drive train. The analysed model includes a rigid body with a large inertia to represent the rotor and no additional body for the tower, since this acts as a rigid supporting structure in this frequency range. The coupling with the generator is included as a torsional spring element between the gearbox output shaft and the discrete mass of the generator. This nal model of the drive train in the wind turbine has about 70 DOFs and has its rst relevant eigenfrequency at 68 Hz. A frequency response analysis in the range [ 50 - 1500 Hz] for a torque excitation at the gear mesh in the high speed planetary gear stage, indicates the importance of the different eigenmodes for the torque in the drive train during this excitation. Based on the calculated FRFs, a set of eigenmodes, which can lead to amplied torque levels, is identied and, subsequently, compared with possible excitation frequencies in a Campbell diagram to identify possible drive train resonances. In addition, the frequency response analyses demonstrate how the loads in the drive train can be simulated for a sinusoidal load excitation. Finally, the investigation of two transient load cases demonstrates how a disturbance, which causes a torque variation with a high amplitude in the generator torque, yields a torque peak on the pinion of the high speed stage, high rotational acceleration levels for the bearings of the pinion and an oscillating axial displacement of the pinion in its bearings. A sensitivity analysis indicates how a exible coupling with a lower stiffness value can reduce the amplitude of the torque peak on the pinion. This article demonstrates the simulation of a transient load case for a particular disturbance in the electricity grid. Since the accuracy of the predicted load levels in a transient analysis depend highly on the accuracy of the load excitation, it is recommended to further investigate the occurrence and the appropriate description of such excitations during realistic load cases. The most efcient solution for this problem seems to be an integration of the extended drive train model in one of the existing wind turbine design codes, since these codes contain usually most of the desired features to describe the applied loads on the wind turbine.

References
[1] W. Dhaeseleer. Energie vandaag en morgen : Beschouwingen over energievoorziening en -gebruik. Leuven : Acco, 2005.

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[2] GWEC. Global Wind Report. 2006. [3] J. Peeters. Simulation of dynamica drive train loads in a wind turbine. PhD dissertation, K.U.Leuven, Department of Mechanical Engineering, Division PMA, Leuven (Heverlee), Belgium, 2006. available online: http://hdl.handle.net/1979/344. [4] E. De Vries. Trouble spots: Gearbox failures and design solutions. Renewable Energy World, 9(2):37 47, 2006. [5] LMS. DADS Revision 9.6 Documentation. LMS International, Belgium, 2000. [6] A. Kahraman. Effect of axial vibrations on the dynamics of a helical gear pair. Journal of Vibration and Acoustics, 115:3339, 1993. [7] J. Lin and R. G. Parker. Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration. Journal of Vibration and Acoustics, 121:316321, 1999. [8] R. R. J. Craig. Structural dynamics. John Wiley & Sons, Inc., 1981. [9] R. R. J. Craig. A review of time-domain and frequency-domain component-mode synthesis methods. International Journal of Analytical and Experimental Modal Analysis, 2(2):5972, 1987. [10] J. D. Smith. Gears and their vibration. Marcel Dekker, New York and MacMillan, London, 1983. [11] J. Soens, J. Driesen, and R. Belmans. Interaction between Electrical Grid Phenomena and the Wind Turbines Behaviour. Proceedings of The International Conference on Noise and Vibration Engineering (ISMA2004), pp. 3969-3988, Leuven, Belgium, September 20-22, 2004. [12] S. Seman, J. Niiranen, S. Kanerva, and A. Arkkio. Analysis of a 1.7 MVA Doubly Fed Wind-Power Induction Generator during Power Systems Disturbances. Proceedings of the Nordic Workshop on Power and Industrial Electronics (NORPIE/2004), Trondheim, Norway, June 14-16, 2004. [13] S. Seman, S. Kanerva, J. Niiranen, and A. Arkkio. Transient Analysis of Doubly Fed Power Induction Generator Using Coupled Field-Circuit Model. Proceedings of the 16th International Conference on Electrical Machines, Krakow, Poland, September 5-8, 2004.

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