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WIND ENERGY Wind Energ. (in press) Published online in Wiley Interscience (www.interscience.wiley.com). DOI: 10.1002/we.173

Research Article

Analysis of Internal Drive Train Dynamics in a Wind Turbine

Joris L. M. Peeters*, Dirk Vandepitte and Paul Sas, Department of Mechanical Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 41, B-3001 Heverlee (Leuven), Belgium

Key words: wind turbine; drive train; gearbox; dynamic loads; multibody system

Three types of multibody models are presented for the investigation of the internal dynamics of a drive train in a wind turbine.The ?rst approach is limited to the analysis of torsional vibrations only. Then a rigid multibody model is presented with special focus on the representation of the bearings and gears in the drive train. The generic model implementation can be used for parallel as well as planetary gear stages with both helical and spur gears. Examples for different gear stages describe the use of the presented formulations. Furthermore, the in?uence of the helix angle and the ?exibility of the bearings on the results of eigenmode calculations are discussed. The eigenmodes of a planetary stage are classi?ed as rotational, translational or out-of-plane modes.Thirdly, the extension to a ?exible multibody model is presented as a method to include directly the drive train components’ ?exibilities. Finally, a comparison of two different modelling techniques is discussed for a wind turbine’s drive train with a helical parallel gear stage and two planetary gear stages. In addition, the response calculation for a torque input at the generator demonstrates which eigenmodes can be excited through this path. Copyright ? 2005 John Wiley & Sons, Ltd.

Introduction

Traditional design calculations for wind turbines are based on the output of speci?c aeroelastic simulation codes as described by Molenaar and Dijkstra.1 The output of these codes gives the mechanical loads on the wind turbine components caused by external forces such as the wind, the electricity grid and (for offshore applications) sea waves. Since the focus in the traditional codes lies mainly on the rotor loads and the dynamic behaviour of the overall wind turbine, the model of the drive train in the wind turbine is reduced to only a few degrees of freedom. This means that for the design of the drive train the simulated load time series need to be further processed to loads on the individual components, such as gears and bearings. Furthermore, the limitation of the model implies that vibrations of these internal drive train components are not taken into account and, as a consequence, dynamic loads on these components cannot be simulated. Instead, application factors according to DIN 39902 and DIN ISO 2813 are typically used for the processing of the simulated load time series to loads on the gears and bearings respectively. For existing wind turbines nowadays, this approach seems acceptable from the point of view that the internal drive train dynamics are in a frequency range well above the overall wind turbine dynamics. However, this argument does not cover the complete range of phenomena that can occur in the drive train. After all, not only external low-frequency excitation of the drive train is possible, but also internal excitation at higher frequencies exists. For instance, excitation at gear meshing frequencies or from generator fault transients might introduce energy in the range of the internal eigenfre-

* Correspondence to: J. L. M. Peeters, Department of Mechanical Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 41, B-3001 Heverlee (Leuven), Belgium E-mail: joris.peeters@mech.kuleuven.be

Copyright ? 2005 John Wiley & Sons, Ltd.

Received 22 November 2004 Revised 25 August 2005 Accepted 19 September 2005

J. L. M. Peeters, D. Vandepitte and P. Sas

quencies. This indicates the importance of further insight into the internal dynamics of the drive train, which implies the need for additional numerical simulation methods. Moreover, extra insight from additional analyses might be useful for vibration monitoring and noise radiation calculations. The multibody simulation technique is a well-established method to analyse in detail the loads on internal components of drive trains. This article investigates the use of this technique for the dynamic analysis of a wind turbine drive train with a gearbox. Models with different levels of complexity are analysed in this investigation and all models are implemented in the multibody software package DADS.4 The ?rst subsection of section two describes the simplest level of modelling, where exactly one degree of freedom (DOF) per drive train component is used to simulate only torsional vibrations in the drive train. These models are further called ‘torsional (multibody) models’. The second subsection of section two presents more elaborate models, where all individual drive train components have six DOFs, further called ‘rigid multibody models’. The interactions between the bodies, which represent the gear and bearing ?exibilities, are modelled by linear springs. Their implementation is based on a synthesis of the work presented by Kahraman5,6 on helical gears and by Parker and co-workers7,8 on planetary gears. This synthesis makes it possible to analyse a single-stage helical planetary gear set, as already introduced by Kahraman.9 In addition, the formulations presented in this subsection yield three-dimensional, generic models to simulate the dynamics of complete gearboxes integrated in a wind turbine drive train. Finally, the third subsection of section two discusses a further extension of the multibody model to a ‘?exible multibody model’ in which the drive train components are modelled as ?nite element models instead of rigid bodies, adding the possibility of calculating stresses and deformations in the drive train components continuously in time. Every addition to the model leads to speci?c additional information about the internal dynamics of the drive train but makes the modelling and simulation more complex. Therefore, depending on the aim of the analysis, a designer has to decide how much detail is required in the models. The presentation of the step-by-step increase in complexity enables the drive train designer, and in particular the gearbox designer, to get an overview of the advantages and limitations of the different levels of modelling. Each level can be used as a separate tool in speci?c design phases to estimate the signi?cance of dynamic loads. Section three presents an application of the different modelling techniques on a drive train of a wind turbine with two planetary gear stages and one parallel helical gear stage. First the parallel stage is analysed separately as an example of the ?exible multibody simulation technique. Then an individual discussion of the high-speed helical planetary gear stage introduces the ‘out-of-plane’ modes. These analyses focus mainly on the calculation of mode shapes and corresponding frequencies. Finally, the complete drive train is implemented as a purely torsional and as a rigid multibody model. The results of eigenmode calculations for both models are compared and the use of frequency response function (FRF) calculations in the latter model is demonstrated to estimate how the individual modes contribute to the response on speci?c excitations. Section four summarizes the main conclusions of the presented research and gives an overview of ongoing work.

The Multibody Simulation Technique

In a multibody model of a drive train, each body represents an individual drive train component which can translate in three directions and rotate around three axes (six DOFs). The different bodies can be connected using the appropriate joints or stiffnesses. The speci?c implementation of these links for particular models is discussed in the following subsections, but a general overview of the ?exibilities in the different models is ?rst given here. 1. Tooth ?exibility. All teeth in contact of a gear pair under load exhibit bending deformation, which can be represented as a tooth stiffness between the gears (gear mesh stiffness). 2. Component ?exibility. All individual components which transfer torque in a drive train will deform under different load components such as axial, torsional, shear and bending loads. This can be represented as a stiffness between the bodies or as an integrated stiffness in a ?exible multibody model.

Copyright ? 2005 John Wiley & Sons, Ltd. Wind Energ. (in press)

Internal Drive Train Dynamics

3. Bearing ?exibility. All bearings will deform under load, which is represented as a stiffness between the bodies and their housing. In this article the housing is considered to be rigid. However, ?exibility of the gear unit housing may be added in a similar fashion as the component ?exibilities.

A Purely Torsional Multibody Model

A ?rst approach in modelling the internal dynamics of a drive train is only focusing on torsional vibrations. In a torsional multibody model, all bodies have exactly one DOF, namely the rotation around their axis of symmetry. The other ?ve DOFs are ?xed, so they can be left out of the equations of motion, and the coupling of two bodies involves only two DOFs (q1, q2). Only the torsional inertia is needed as input for the rigid bodies; furthermore, their torsional stiffness and the gear mesh stiffnesses are the only ?exibilities taken into account. Torsional models can be used for the dynamic analysis of the torque in the drive train; the other force components can only be derived by further processing the torque. The torsional ?exibility of a shaft (Kshaft) between two bodies is included in the equations of torque as shown in equation (1). Material damping is neglected in this model. T1 = - T2 = Kshaft (q 2 - q1 ) (1)

Gear contact forces between two wheels are modelled by a linear spring acting in the plane of action along the contact line (normal to the tooth surface at the point of contact). This spring couples the two DOFs of the wheels and includes the transmission ratio between them. Since the equations of motion for a torsional model are based on torque and rotations, the gear contact forces are written in this form as presented in Figure 1. The

{ { { { {

2

T1 = - Td = - Ftooth contact ? r1 = -k gear ? (r1q1 - r2q 2 ) ? r1

r1 r2 u Td (a) (b) (c) (d) (e)

(a) = -k gear ? (r1 ) ? (q1 - u ? q 2 ) (b) (c)

base circle radius of pinion base circle radius of gear wheel r transmission ratio ? 2 ? ? r1 ? positive driving torque applied to the pinion deformation along the line of contact (>0) torsional stiffness referred to the pinion torsional deformation torsional stiffness torsional deformation referred to the gear wheel

1 ? T2 = - T1 ? u = -k gear ? (r2 ) 2 ? ? ?q 2 - u ? q1 ?

(d)

(e)

Figure 1. A torsional model for the gear contact forces between a driving pinion and a driven gear wheel. Td is a positive driving torque applied to the pinion causing a negative reaction torque T1 on the pinion and a positive reaction torque T2 on the gear wheel

Copyright ? 2005 John Wiley & Sons, Ltd. Wind Energ. (in press)

J. L. M. Peeters, D. Vandepitte and P. Sas

force on the teeth of both gears (Ftooth contact) is equal in magnitude, resulting in a higher torque on the larger gear. The direction of this force is such that the resulting torque on the driving wheel is always opposite to the input torque. The stiffness value kgear is de?ned according to DIN 3990 as the normal distributed tooth force in the normal plane causing the deformation of one or more engaging tooth pairs, over a distance of 1 mm, normal to the evolvent pro?le in the normal plane; this deformation results from the bending of the teeth in contact between the two gear wheels, of which one is ?xed and the other is loaded. In the gear contact model the time-varying components due to a static transmission error excitation or a ?uctuation in the number of tooth pairs in contact are not considered. Furthermore, no damping or friction forces are included. From a physical understanding it is clear that the presented spring will only work under compression. To ensure that this limitation will not be exceeded during simulation, the following extra assumption is made here. No contact loss between the gears will occur, something that could happen for a system with backlash when the dynamic mesh force becomes larger than the static force transmitted. This assumption is valid for heavily to moderately loaded gears.5 q1 and q2 in Figure 1 are de?ned as the rotations of the pinion and the gear wheel in their respective reference frame. For a parallel gear stage these reference frames are ?xed to the gearbox housing. However, the same formulation is valid when the reference frame of a wheel follows the rotation of a component, which implies a kinematic coupling between the wheel and this component. This makes it also applicable for a planetary gear stage where the reference frame of a planet follows the rotation of the planet carrier. Thus, by keeping the gear contact formulation independent from the de?nition of the reference frame, it can be used as a generic module for all possible gear set-ups. This independence can be implemented straightforwardly in the multibody software package DADS, since co-ordinate systems can be created and referenced freely. When using the formulation for a wheel with internal teeth, the base circle radius should be taken negative. A detailed discussion of the implementation of a torsional model in DADS and the numerical validation of this implementation using the software DRESP is described by Peeters et al.10,11 DRESP is a simulation program of the FVA (Germany) for torsional vibrations only.12 The modelling approach described in this subsection is considered to be the state of the art for most industrial applications. Flexibility is assumed to be concentrated in shafts and gear teeth. Bearings are considered to be rigid in radial and axial directions. As a conclusion, two simple examples of torsional models are discussed. Figure 2 shows a parallel gear stage with the necessary input for this model and the results of an eigenmode calculation. The same example will be discussed in the next subsection for a rigid multibody approach and was ?rst presented by Kahraman.5 It is clear that there are only two DOFs in the torsional model: on the one hand the coupled rotation of the gears in their bearings and on the other hand the deformation of the teeth. The results of the calculation show that only the second DOF yields a non-zero eigenfrequency. The second example was ?rst presented by Lin and Parker7 and is a model of a planetary gear stage with three planets as shown in Figure 3. The three planets are identical as well as all sun–planet and planet–ring mesh stiffnesses. Furthermore, the ring wheel is constrained as non-rotating and therefore connected through a torsional spring with the rigid housing. Again the same example will be discussed in the next subsection for a rigid multibody approach. Here the eigenmode calculation yields ?ve non-zero eigenfrequencies; two of them form a double pole, resulting from the symmetry in the planetary stage. The in?uence of the corresponding mode shapes on any torque ?uctuation is zero and therefore these two modes do not matter in a torsional analysis.

model input kgear (N/m) r1 (mm) r2 (mm) J1 (kg · m2) J2 (kg · m2) 2 · 108 50 50 2·9 · 10-3 2·9 · 10-3 eigenfrequencies (1) (2) 0 Hz 2955 Hz

Figure 2. A torsional model for a parallel gear stage. Both the input and the output shaft have a free boundary

Copyright ? 2005 John Wiley & Sons, Ltd.

Wind Energ. (in press)

Internal Drive Train Dynamics model input kgear (N/m) rsun (mm) rplanet (mm) rring wheel (mm) rplanet carrier (mm) Jsun (kg · m2) Jplanet (kg · m2) Jring wheel (kg · m2) Jplanet carrier (kg · m2) mplanet (kg) kring-housing (Nm/rad) 5 · 108 38·7 50·2 -137·5 96·9 2·34 · 10-3 6·14 · 10-3 227 · 10-3 197 · 10-3 0·66 76 · 106 eigenfrequencies (1) (2) (3) (4) (5) (6) 0 Hz 2253 Hz 6283 Hz 6449 Hz 6449 Hz 11241 Hz

Figure 3. A torsional model for a planetary gear stage with three planets and a ?xed ring wheel. Both the planet carrier and the sun have a free boundary

A Rigid Multibody Model with Discrete Flexible Elements

The extension of a torsional model to a rigid multibody model adds the possibility to investigate the in?uence of bearing stiffnesses on the internal dynamics of the drive train. Furthermore, the analysis can also yield insight into dynamic bearing loads, which are coupled with the displacements of the bodies in their bearings. All drive train components are still treated as rigid bodies but now have a full set of six DOFs instead of only one. This implies that the linkages in the multibody model, representing the bearing and tooth ?exibilities, now need to couple 12 DOFs. The presented modelling techniques are based on a synthesis of the work presented by Kahraman5,6 on helical gears and by Parker and co-workers7,8 on planetary gears. Linear springs are used here to model the bearing and gear mesh stiffnesses. An individual formulation of these models yields two three-dimensional plug-in components, ready to use in a generic modelling approach for the drive train. This means that both models are suitable for a rather simple parallel gear stage as well as for a more complex helical planetary stage with any number and any positioning of the planets. Furthermore, the analyses with these models are not limited to single gear stages; an extension to complete gearboxes is straightforward. First the formulations of the bearing and the gear mesh model are discussed separately. Then a real parallel helical gear stage is analysed as an example of their application. In addition, the in?uences of the helix angle and the bearing ?exibilities in this model are investigated in two sensitivity analyses. A second example discusses the application of the presented formulations for a planetary gear stage. Again the impact of the bearing ?exibilities on the results is investigated. Modelling of the Bearing Flexibility The six DOFs of the rigid bodies need appropriate constraints in the bearing model. This model is represented by a linear spring and implemented as a 6 ? 6 stiffness matrix de?ned in the XYZ co-ordinate system as shown in Figure 4. Damping is neglected and all bearings are assumed to have an axisymmetric behaviour without coupling between the individual DOFs. Therefore all off-diagonal terms are zero and both the radial and tilt stiffnesses are equal. Practically, the bearing component in a multibody model connects the XYZ co-ordinate system ?xed to a certain body with the X?Y?Z? system ?xed to this body’s reference frame. This reference frame can be for instance the ?xed housing; however, it can also be the planet carrier, e.g. for the planet bearings. Modelling of the Gear Mesh Stiffness The contact forces working on the teeth of two gears in contact cause bending of the teeth. This deformation is represented in the model by a linear spring acting in the plane of action along the contact line (normal to the tooth surface at the point of contact). This formulation was already introduced for the purely torsional equivalent of the gear mesh model, but now this spring involves a coupling between 12 DOFs instead of only two. The assumptions postulated for the gear mesh model subsection one are still valid. For the sake of completeness they are repeated in the list of assumptions made here.

Copyright ? 2005 John Wiley & Sons, Ltd. Wind Energ. (in press)

J. L. M. Peeters, D. Vandepitte and P. Sas ?kradial ? ? ? Kb = ? ? ? ? ? ? Fb = K b ? q q = [x y z r X rY q]

T

0 kradial

... O k axial

... O ktilt

...

O ktilt

0? M˙ ˙ M˙ ˙ M˙ 0˙ ˙ 0˙ ?

where x, y, z, r X , r Y and q are the projections in the XYZ system of the translations and rotations from the position and orientation of the gear in its reference frame X ?Y ?Z ?. Figure 4. Schematic representation of a bearing model: a linear spring connects the XYZ co-ordinate system ?xed to the body with the X?Y?Z? system ?xed to this body’s reference frame. The spring is modelled by a symmetric 6 ? 6 stiffness matrix and Fb is the force working on the gear in the XYZ system

1. The gear mesh model is a linear time-invariant model. Static transmission error excitation is not considered and therefore no phasing relationships between gear meshes are included. Furthermore, a variable stiffness caused by a ?uctuation in the number of tooth pairs in contact is assumed negligible. The validity of these assumptions for the presented linear analyses can be justi?ed as in subsection one. 2. Sliding of teeth in contact and corresponding friction forces are neglected as well as any other possible damping in the system. 3. Occurrence of tooth separation is considered non-existent and consequently the modelling of gear backlash is not included. This implies that the spring is always under compression. 4. Coriolis accelerations of gears that are rotating and simultaneously translating (e.g. planets on their carrier) are neglected and all gyroscopic effects as described by Lin and Parker7 are excluded. These assumptions are valid for wind turbine applications, since planetary gear stages in wind turbines are only rarely used as high-speed stages. Formulation of the gear contact forces is based on the model approach shown in Figure 5. ? Co-ordinate systems X1 ? Y1 ? Z1 ? and X2 ? Y2 ? Z2 ? are oriented with X? along the centreline pointing from gear 1 to gear 2; Z? is lying along the axis of rotation. These co-ordinate systems are ?xed to the reference frames of the respective wheels (as introduced in subsection one and Figure 4). ? X1Y1Z1 and X2Y2Z2 are ?xed to the respective gears and in their starting position they coincide with the corresponding X?Y?Z?. ? ft is the pressure angle of the gear mesh, which is an input parameter. It is de?ned as the angle measured from the centreline towards the normal on the contact line in the corresponding X?Y?Z?. The sign of this angle changes when the driving direction of the system changes. ? y1 ? and y 2 ? are the angles measured respectively from X1 ? to X1 and from X2 ? to X2 along the corresponding Z?: y1 = ft - y 1 ? and y2 = ft - y 2 ?. ? b is the helix angle, which is positive when the teeth of gear 1 are turned ‘left’ from a reference position where b = 0; b > 0 in Figure 5. The compression of the linear spring (d) can be written as a function of the vectors q1 and q2 as introduced in Figure 4. Since the spring works always under compression, d should be positive. d = (x1 sin y 1 - x 2 sin y 2 - y1 cos y 1 + y2 cos y 2 - u1 - u2 ) sign(f t ) cos b -(z1 - z2 + w X1 sin y 1 + w X 2 sin y 2 - y Y 1 cos y 1 - y Y 2 cos y 2 ) sign(f t ) sin b with

Copyright ? 2005 John Wiley & Sons, Ltd. Wind Energ. (in press)

(2)

Internal Drive Train Dynamics Gear1 is the driving wheel (Tinput < 0) m1, I1, J1 r1: base circle radius Gear2 is the driven wheel m2, I2, J2 r2: base circle radius kgear is the gear mesh stiffness as de?ned in subsection one

Figure 5. The gear mesh model of two helical gears in contact

[ x1 y1 z1 w X1 w Y1 u1 ] = diag(1, 1, 1, r1 , r1 , r1 ,)q1 [ x 2 y2 z2 w X2 w Y2 u2 ] = diag(1, 1, 1, r2 , r2 , r2 ,)q2

The stiffness value of the linear spring, kgear, is the same as de?ned in subsection one, namely the ratio of the contact force on a tooth over the resulting displacement of the contact point. The spring force causes forces and moments on the gears, which can be projected in the XYZ co-ordinate systems and thus written as FX1 = -dk gear sin y 1 cosb sign(f t ) FY 1 = dk gear cos y 1 cosb sign(f t ) FZ1 = dk gear sinb sign(f t ) TX1 = dk gear r1 siny 1 sinb sign(f t ) TY 1 = -dk gear r1 cos y 1 sinb sign(f t ) TZ1 = dk gear r1 cosb sign(f t )

Copyright ? 2005 John Wiley & Sons, Ltd.

FX 2 = dk gear sin y 2 cosb sign(f t ) FY 2 = -dk gear cos y 2 cosb sign(f t ) FZ 2 = -dk gear sinb sign(f t ) TX 2 = dk gear r2 sin y 2 sinb sign(f t ) TY 2 = -dk gear r2 cos y 2 sinb sign(f t ) TZ 2 = dk gear r2 cosb sign(f t )

Wind Energ. (in press)

J. L. M. Peeters, D. Vandepitte and P. Sas

By writing the force components and the compression d in the XYZ systems, the gear contact formulation is a generic module. Its implementation in DADS is a user-de?ned subroutine which can be used to couple any two gears in all possible gear set-ups, keeping in mind that a wheel with internal teeth is given a negative radius. The formulation of the gear forces in matrix form yields ? F1 ? = k ?k 11 k 12 ? ?q1 ? gear ? ? ? F2 ˙ ? ?k 21 k 22 ˙ ?? ? q2 ˙ ? where F1 = [ FX1 FY 1 FZ1 TX1 TY 1 TZ1 ]T F2 = [ FX 2 FY 2 FZ 2 TX 2 TY 2 TZ 2 ]T

?-c2 bs2y 1 ? ?c2 bcy 1sy 1 ? ?cbsbsy 1 k 11 = ? ?r1cbsbs2y 1 ? ?-r1cbsbcy 1sy 1 ? 2 ? ?r1c bsy 1 ?r 2 bsy 1sy 2 ? ?-c2 bcy 1sy 2 ? ?-cbsbsy 2 = ? ?-r1cbsbsy 1sy 2 ? ?r1cbsbcy 1sy 2 ? 2 ? ?-r1c bsy 2 c2 bcy 1sy 1 -c2 bc2y 1 -cbsbcy 1 -r1cbsbcy 1sy 1 r1cbsbc2y 1 -r1c2bcy 1 -c2 bcy 2sy 1 c2 bcy 1cy 2 cbsbcy 2 r1cbsbcy 2sy 1 -r1cbsbcy 1cy 2 r1c2bcy 2 -c2 bcy 1sy 2 c2 bcy 1cy 2 cbsbcy 1 -r 2 cbsbcy 1sy 2 r 2 cbsbcy 1cy 2 -r 2 c2bcy 1 c2 bcy 2sy 2 -c2 bc2y 2 -cbsbcy 2 r 2 cbsbcy 2sy 2 -r 2 cbsbc2y 2 r 2 c2 bcy 2 cbsbsy 1 -cbsbcy 1 -s2 b -r1s2bsy 1 r1s2bcy 1 -r1cbsb -cbsbsy 1 cbsbcy 1 s2 b r1s2bsy 1 -r1s2bcy 1 r1cbsb -cbsbsy 2 cbsbcy 2 s2 b -r 2s2bsy 2 r 2s2bcy 2 -r 2 cbsb cbsbsy 2 -cbsbcy 2 -s2 b r 2s2 bsy 2 -r 2s2 bcy 2 r 2 cbsb cbsbs2y 1 -cbsbcy 1sy 1 -s2 bsy 1 -r1s2bs2y 1 r1s2bcy 1sy 1 -r1cbsbsy 1 cbsbsy 1sy 2 -cbsbcy 1sy 2 -s2 bsy 2 -r1s2bsy 1y 2 r1s2bcy 1sy 2 -r1cbsbsy 2 -cbsbsy 1sy 2 cbsbcy 2sy 1 -s2 bsy 1 -r 2s2bsy 1sy 2 r 2s2bcy 2sy 1 -r 2 cbsbsy 1 -cbsbs2y 2 cbsbcy 2sy 2 s2 bsy 2 -r 2s2 bs2y 2 r 2 s 2 bcy 2 sy 2 -r 2 cbsbsy 2 -cbsbcy 1sy 1 cbsbc2y 1 s2 bcy 1 r1s2bcy 1sy 1 -r1s2bc2y 1 r1cbsbcy 1 -cbsbcy 2sy 1 cbsbcy 1cy 2 s2 bcy 2 r1s2bcy 2sy 1 -r1s2bcy 1cy 2 r1cbsbcy 2 cbsbcy 1sy 2 -cbsbcy 1cy 2 s2 bcy 1 r 2s2bcy 1y 2 -r 2s2bcy 1cy 2 r 2 cbsbcy 1 cbsbcy 2sy 2 -cbsbc2y 2 -s2 bcy 2 r 2s2 bcy 2sy 2 -r 2s2 bc2y 2 r 2 cbsbcy 2 c2 bsy 1 ? ˙ -c2 bcy 1 ˙ ˙ -cbsb ˙ ˙ -r1cbsbsy 1˙ ˙ r1cbsbcy 1 ˙ ˙ -r1c2b ˙ ? c2 bsy 1 ? ˙ -c2 bcy 1 ˙ ˙ -cbsb ˙ ˙ -r1cbsbsy 1˙ ˙ r1cbsbcy 1 ˙ ˙ 2 -r1c b ˙ ? -c2 bsy 2 ? ˙ c2 bcy 2 ˙ ˙ ˙ cbsb ˙ -r 2 cbsbsy 2˙ ˙ r 2 cbsbcy 2 ˙ ˙ -r 2 c 2 b ˙ ? -c2 bsy 2 ? ˙ c2 bcy 2 ˙ ˙ cbsb ˙ ˙ r 2 cbsbsy 2˙ ˙ r 2 cbsbcy 2˙ ˙ -r 2 c 2 b ˙ ?

(3)

k 12

?r 2 bsy 1sy 2 ? ?-c2 bcy 2sy 1 ? ?-cbsbsy 1 k 21 = ? ?r 2 cbsbsy 1sy 2 ? ?r 2 cbsbcy 2sy 1 ? 2 ? ?r 2 c bsy 1 ?-c2 bs2y 2 ? ?c2 bcy 2sy 2 ? ?cbsbsy 2 = ? ?r 2 cbsbs2y 2 ? ?r 2 cbsbcy 2sy 2 ? 2 ? ?-r 2 c bsy 2

k 22

with cb = cos b, sb = sin b, cy = cos y and sy = sin y. When b π 0, it is possible to have no zero components in the k ij matrices. At that moment, all DOFs of both gears are coupled with each other. The application of this method is given below in an example of a parallel and a planetary gear stage. Example of a Rigid Multibody Model for a Parallel Gear Stage The analysis of a multibody model of the parallel helical gear system introduced by Kahraman5 is described. The presented model consists of a helical gear pair mounted on two rigid shafts supported by rolling element bearings assembled in a rigid housing. Figure 6 shows this set-up with the necessary input parameters for the model. Here the helix angle b is variable and its in?uence on the calculated results is examined in a sensitivity analysis. Furthermore, the same gear system with b = 0° and kb = ? was used as an example of a torsional model in subsection one. As a result, conclusions concerning the advantages of the more detailed approach presented here can be drawn. This is discussed in the sensitivity analysis of the bearing stiffnesses.

Copyright ? 2005 John Wiley & Sons, Ltd. Wind Energ. (in press)

Internal Drive Train Dynamics model input kgear (N/m) pressure angle (°) r1 (mm) r2 (mm) m1 (kg) m2 (kg) J1 (kg · m2) J2 (kg · m2) I1 (kg · m2) I2 (kg · m2) kbax (N/m) kbrad (N/m) kbtilt (Nm/rad) 2 · 108 20 50 50 2·0 2·0 2·9 · 10-3 2·9 · 10-3 1·45 · 10-3 1·45 · 10-3 3·5 · 108 1·0 · 108 277·5 · 103

Figure 6. Helical gear system. The two bearings supporting each of the shafts are represented by one stiffness matrix per shaft with equivalent radial (kbrad), axial (kbax) and tilt (kbtilt) stiffness values. The helix angle b is varied through the analysis and both the input and the output shaft are free at their boundaries

Table I. Eigenfrequencies (Hz) for the model of the helical gear pair in Figure 6 1 b = 0° b = 20° D (%) 0 0 – 2 1125 1058 -6·0 3 1125 1125 0 4 1566 1519 -3·0 5 2105 2105 0 6 2105 2105 0 7 2105 2105 0 8 2202 2173 -1·3 9 2202 2202 0 10 2202 2202 0 11 2202 2202 0 12 3972 4150 +4·5

In?uence of the Helix Angle b. Table I shows the eigenfrequencies calculated for the helical gear pair in Figure 6 for b = 0° and 20°. These values match almost perfectly with the results calculated by Kahraman,5 which proves the validity of the model implementation in the frictionless case. Figure 7(a) shows how the eigenfrequencies change with changing b. Only w2, w4, w8 and w12 are in?uenced and the largest relative change is observed for w2, which is only 6% for b = 20° (Table I). Thus the in?uence of the helix angle is rather small; as a result, a simpli?cation of a parallel helical gear system to a spur gear pair can be justi?ed when calculating only the eigenfrequencies. The effect of the helix angle on the corresponding mode shapes is shown in Figure 8 for w2, w4, w8 and w12. In?uence of the Bearing Stiffness Values (kbax, kbrad, kbtilt). The fourth mode shape in Figure 8 (for b = 0°) corresponds to the eigenmode that was calculated with a torsional multibody model in Figure 2 and which has the biggest impact on the torque. Remarkable is the drop in frequency (2955 ? 1566 Hz) for this mode as a result of adding realistic bearing ?exibilities in the rigid multibody model, which were lacking in the torsional model. Users of torsional models are aware of this limitation in their models and therefore often use gear mesh stiffness reduction factors based on their experience. However, the new formulation gives directly more accurate predictions for the torque dynamics. In addition, the results are no longer limited to the torque DOF only. Several new modes appear which lie in the same frequency range (Table I). This underlines the importance of the rigid multibody approach. Figure 7(b) demonstrates the statement that increasing all bearing stiffnesses for the helical gear system to in?nity yields a purely torsional equivalent of the model. For this purpose the real bearing stiffnesses are multiplied by a stiffness factor, which is taken equal for kbax, kbrad and kbtilt, since the focus is not on the individual sensitivity of these values. All eigenfrequencies increase towards in?nity, except for w4 (1566 Hz); this freCopyright ? 2005 John Wiley & Sons, Ltd. Wind Energ. (in press)

J. L. M. Peeters, D. Vandepitte and P. Sas

(a) The effect of the helix angle b on the eigenfrequencies.

(b) The effect of multiplying kbax, kbrad and kbtilt with an equal stiffness factor on the eigenfrequencies (b = 0°)

Figure 7. Sensitivity analyses on the helical gear pair in Figure 6

quency approaches asymptotically the torsional eigenfrequency (2955 Hz), which corresponds to the conclusions above. Example of a Multibody Model for a Planetary Gear Stage Given the generic formulation of the presented methodology, a similar approach can be used for planetary gear systems. There is no limitation on the number of planets or on their positioning around the sun. The sun can be constrained by a bearing model or can be modelled as ?oating, depending on the application. Furthermore, the choice of which component is constrained as non-rotating is an input parameter for the model. As an example, three planetary gear systems introduced by Lin and Parker7 are discussed here, respectively having three, four and ?ve planets. A torsional equivalent of the presented three-planet system was used in subsection one. From a comparison between the results the advantages of the rigid multibody approach for a planetary gear system become clear. Figure 9 shows the input parameters for the planetary gear systems and the multibody model of the fourplanet system. In all three systems the planets are identical as well as all bearing stiffnesses and all sun–planet and planet–ring gear mesh stiffnesses. The ring wheel is in all cases non-rotating and therefore constrained with a torsional stiffness to the rigid housing. Lin and Parker limited their analysis to planar vibrations in the gear system, since the helix angle is zero and consequently no forces are acting out-of-plane. A limitation in our model implementation to planar vibrations is made by constraining all components with bearings that are in?nitely stiff for the displacement and rotations out-of-plane (axial and tilt stiffness). Table II divides the eigenfrequencies of the three systems according to the classi?cation by Lin and Parker. They distinguished three categories of planar modes for planetary systems with N planets, satisfying equation (4) for the positioning around the sun.13

? sin y

N

=

? cos y

N

=0

(4)

1. Six rotational modes always have multiplicity m = 1 for various numbers of planets N. The mode shapes ( R1-6) have pure rotation of the carrier, ring and sun and all planets have the same motion in phase. 2. Six translational modes always have multiplicity m = 2 for different N. Here the six mode shape pairs ( T 1a ,b -6a ,b) have pure translation of the carrier, ring and sun.

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Internal Drive Train Dynamics

b = 0° w2 = 1125 Hz

b = 0° w4 = 1566 Hz

b = 20° w2 = 1058 Hz (a) Mode 2

b = 20° w4 = 1519 Hz (b) Mode 4

b = 0° w8 = 2202 Hz

b = 0° w12 = 3972 Hz

b = 20° w8 = 2173 Hz (c) Mode 8

b = 20° w12 = 4150 Hz (d) Mode 12

Figure 8. Natural mode shapes for the helical gear pair (wireframe, undeformed; solid, deformed)

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J. L. M. Peeters, D. Vandepitte and P. Sas Sun Mass (kg) J (· 10-3 kgm2) Base circle radius (mm) Mesh stiffness Bearing Stiffness Torsional Stiffness Nominal Pressure Angle Helix Angle (a) Model input parameters. (b) The four-planet system. Figure 9. Planetary gear systems introduced by Lin and Parker.7 In all systems the planet carrier and the sun are free at their boundaries Table II. Eigenfrequencies (Hz) for the planetary gear systems with respectively three, four and ?ve planets as introduced by Lin and Parker7 N Mode shape m=1 R1 R2 R3 R4 R5 R6 T 1,a,b T 2,a,b T 3,a,b T 4,a,b T 5,a,b T 6,a,b P1 P2 P3 3 0 1425 2032 2644 7500 11744 770 1101 1989 2238 7060 9582 4 0 1496 2060 2611 7800 13050 759 1092 1947 2328 7245 10390 1959 6444 6497 5 0 1538 2082 2602 8086 14237 745 1073 1921 2421 7427 11136 1959 6444 6497 0·4 2·34 38·7 Planet Carrier Ring 2·35 227 -137·5

0·66 5·43 6·14 197 50·2 96·9 kgear = 5 · 108 N/m krad = 108 N/m kring-housing = 76 · 106 Nm/rad 24·6° b = 0°

m=2

m=N-3

3. Three planet modes exist for N > 3 and have multiplicity m = N - 3. The carrier, ring and sun have no rotation or translation in the corresponding mode shapes ( P1-3). Figure 10 shows an example of a mode shape for each category. The classi?cation of the DADS results is based on animations of the mode shapes, and the corresponding eigenfrequencies were validated with those from Lin and Parker. They correlated well, which further enhances the con?dence in the use of the formulation for the planetary gear systems used in wind turbines. In?uence of the Bearing Stiffness Values (krad). The difference between the torsional model of the three-planet system in subsection one and the model presented here is the addition of a realistic radial bearing

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Internal Drive Train Dynamics

(a) rotational mode ( R 2: 1496 Hz)

(b) translational mode ( T 1a: 759 Hz)

(c) planet mode ( P1: 1959 Hz)

Figure 10. Different types of mode shapes for a planetary gear system with four planets rigid multibody model (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 0 ( R1) 770 ( T 1) 1101 ( T 2) 1425 ( R 2) 1989 ( T 3) 2032 ( R 3) 2238 ( T 4) 2644 ( R 4) 7060 ( T 5) 7500 ( R 5) 9582 ( T 6) 11744 ( R 6) (b) (a) (a) The effect of multiplying the radial bearing stiffnesses with a factor. (b) Comparison of the eigenfrequencies calculated with a rigid multibody model and with a torsional model. ( R ) are the rotational modes (m = 1); ( T ) are the translational modes (m = 2). The arrows indicate how the eigenfrequencies shift from the rigid model results to the torsional model results when the radial bearing stiffness values are increased towards in?nity. Figure 11. In?uence of the bearing stiffness values on the eigenfrequencies of the planetary gear model with three planets torsional model (Hz) 0 2227 ? 6192 6442 ? 6442 11210 ?

?

R4

R3 T1a,b R2

R1

?exibility for the sun, the planets, the carrier and the ring wheel. This addition has a major impact on the results, which is demonstrated in Figure 11(a). For a stiffness factor of unity the curves give the eigenfrequencies of the rigid multibody model. The higher stiffness factors correspond to a gradual increase in the radial bearing stiffnesses. The eigenfrequencies corresponding to the ?rst four rotational modes and the ?rst translational double mode approach asymptotically the results from the torsional model when the stiffness values approach in?nity. This phenomenon is numerically shown in Figure 11(b), where the eigenfrequencies calculated with a rigid multibody model are compared with the results for a torsional model. Again the remarkable difference and, furthermore, the additional modes found with the former model underline the importance of this approach.

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J. L. M. Peeters, D. Vandepitte and P. Sas

A Flexible Multibody Model

Typically, multibody models consist of rigid bodies linked by joints and stiffnesses. The stiffness values can include an equivalent discretized stiffness for the ?exibility of the individual components. However, the reduction to an equivalent stiffness and the discretization method complicate the modelling, especially for more complex systems. As a result, complex bodies are in practice often assumed to be rigid and no ?exibility is further taken into account. Considering the component’s ?exibility as a property of the body would lead to a more realistic understanding of the models. Moreover, it may give further insight into the role of this ?exibility in the overall dynamic behaviour. Estimating this in?uence for non-conventional, rather ?exible parts used today in wind turbines is barely possible with the traditional multibody formulation. Therefore a ?exible multibody formulation is presented which makes the modelling more complex but also enables one to calculate (dynamic) deformations of a body on top of its motion as a rigid component. In such models the linkages between the bodies represent only the stiffness of the coupling, such as the gear mesh or the bearing stiffness. The extension to the ?exible multibody formulation is no straightforward adaptation of the traditional method. The additional DOFs to represent the deformations of an individual body are introduced by a ?nite element approach. The direct ?nite element analysis is typically used on the level of individual components, whereas the multibody simulation technique is on the level of the coupling between individual rigid components. A combination of both methods can be made by including reduced ?nite element models in the multibody models. These are further called ?exible multibody models. For all bodies in a traditional multibody model with an increased level of interest for its ?exibility, a ?nite element model is built with as much detail as needed. Typically, these models can have a large number of DOFs, of the order of magnitude of 10,000 up to 100,000. The reduction to a smaller set of DOFs, of the order of magnitude of one up to 10, which can be imported into the multibody model is done with the component mode synthesis (CMS) technique.14–18 The Craig–Bampton19 method is a well-established CMS technique which is supported by DADS and used in the present research. This reduction involves the creation of a set of static constraint modes and ?xed interface normal modes with their corresponding eigenfrequencies. These modes represent all additional DOFs of the body, and the body’s deformation is a linear combination of them. The static modes represent the deformation related to loads and displacements in the body’s interface nodes, whereas the normal modes are related to dynamic deformations. MSC/NASTRAN is used for the calculation of the modes, and all ?nite element models are kept linear. Accurate modelling for this purpose, especially the reduction to an appropriate full set of modes, requires some modelling experience. The coupling of the reduced ?nite element models in a multibody model is possible in their interface nodes by using the presented formulations for the gear mesh and bearing stiffnesses. Not only is the ?exibility of a body included in this approach, but also its mass distribution is closer to reality, because the analyst has the option of deriving ?nite element models directly from CAD models with a very realistic representation of the geometry. As a result, mass values and other inertial properties are no longer input parameters but are automatically calculated from the reduced models. The ?exible multibody simulation has an extra advantage for postprocessing of the results. When the deformation of a body at a certain moment during simulation is known for the reduced component, this result can be transferred back to the ?nite element program. Here it can be further processed into internal stresses and strains for this component, which can for instance be used for strength and fatigue analysis. Baumjohann et al.20 demonstrated this method for the calculation of stresses in wind turbine blades, but it is not discussed further in this article. As an example of the ?exible multibody formulation, the next section presents the effect of including the individual ?exibilities of the pinion and the wheel in the parallel high-speed stage in a wind turbine. Conclusions concerning the in?uence of these additional ?exibilities are discussed there.

Analysis of a Drive Train in a Wind Turbine

Drive trains in modern wind turbines typically have a gearbox with up to three gear stages. Here an application of the presented methods is given for a drive train consisting of two planetary gear stages and one parallel gear stage. The models are based on an existing wind turbine drive train, but the input parameters are

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Internal Drive Train Dynamics

slightly modi?ed owing to con?dentiality. The results are still representative for a drive train in a wind turbine. The section starts with an application of two methods which have been presented but not yet applied in this article. Subsection one discusses the implementation of the ?exible multibody method for the high-speed parallel stage. Subsection two investigates the application of the rigid multibody methods for the high-speed planetary stage, which has helical teeth. This leads to the introduction of a fourth category of mode shapes. Finally, subsection three describes two models of the complete drive train: a purely torsional and a rigid multibody model.

High-speed Parallel Stage

The high-speed stage of the gearbox is a helical gear pair. For this stage the three presented modelling techniques are implemented. First a torsional model is built, only taking into account the gear mesh stiffness. Then a rigid multibody model is implemented by adding the bearing stiffnesses. Finally, this model is extended to a ?exible multibody model which integrates the components’ ?exibilities. In all three models, both the gear and the pinion are free to rotate in their bearings. Figure 12 shows these three models and Table III shows a comparison of the calculated eigenfrequencies. The eigenfrequency 1479 Hz calculated with the torsional model drops to 702 Hz in the rigid multibody model, indicating again the impact of the bearing ?exibilities on the torque dynamics. Furthermore, other additional modes are found and are given with the main component of their corresponding mode shape. By adding the components’ ?exibilities in the ?exible multibody model, the eigenfrequencies decrease further. However, the impact of these ?exibilities on the frequencies is smaller and for some modes negligible. For instance, the eigenfrequencies of the axial translation modes (w2, w3) hardly change, since both shafts are very stiff in the longitudinal direction. On the other hand, the

Z Y (a) Torsional model X (b) Rigid multibody model Figure 12. High-speed parallel gear stage Table III. Comparison of the eigenfrequencies (Hz) of the helical gear pair calculated with a torsional model, a rigid multibody model (MBM) and a ?exible multibody model No. Torsional mode 0 – – 1479 Rigid MBM 0 400 510 702 755 761 775 801 821 1003 1197 1947 Flexible MBM 0 395 497 518 519 635 696 702 718 848 1012 1634 Corresponding shape model (c) Flexible multibody model

1 2 3 4 5 6 7 8 9 10 11 12

Rigid body mode x-Translation pinion x-Translation gear y–z-Rotation pinion y–z-Rotation pinion y–z-Rotation gear y–z-Rotation gear y–z-Translation gear y–z-Translation gear y–z-Translation pinion y–z-Translation pinion x-Rotation pinion and gear

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J. L. M. Peeters, D. Vandepitte and P. Sas Table IV. Rigid multibody model of the high-speed helical planetary gear stage and the results of an eigenmode calculation: the eigenfrequencies (Hz) are divided into three categories based on their corresponding mode shapes Rotational mode (m = 1) R2 R3 R4 728 1067 1524 Translational mode (m = 2) T2 T3 T4 T5 187 535 1143 1318

R1 0

R5 2265

T1 147

T6 1545

Out-of-plane mode O1 140 O2 437 O3 (m = 2) 570 O4 602 O5 (m = 2) 2617 O6 2947 O7 (m = 2) 2962 O8 2977 O9 (m = 2) 3100

bending ?exibility of the rather long and slender high-speed pinion causes a considerable decrease in the eigenfrequencies corresponding to its y–z-rotation modes (w4, w5). This example shows how the ?exible multibody technique makes it possible to evaluate the effect of the components’ ?exibilities without reducing them to discrete stiffnesses. Furthermore, the eigenmodes of the components are also taken into account, which is impossible in a rigid multibody model.

High-speed Planetary Stage

The high-speed planetary gear stage in the wind turbine consists of three planets with helical teeth and is modelled as a rigid multibody model. In this model the sun is ?oating in the radial direction and both the planet carrier and the sun are free to rotate. The ring wheel is ?xed; this ?xation is not modelled as a connection through a torsional spring with its housing (as in the example of subsection two of section two), but instead its DOFs are removed from the equations of motion. This yields ?ve rotational modes instead of six in the results of an eigenmode calculation, which are shown in Table IV. The categorization presented in subsection two of section two for a planetary system is used, but an extra category of out-of-plane modes is introduced, since the out-of-plane motion is not ?xed here. The relevance of the out-of-plane modes is indicated by the fact that they lie in the same frequency range as the other modes, which could interfere with the range of e.g. the gear mesh excitations. Furthermore, these excitations imply out-of-plane forces because of the helix angle, which enables energy input in the out-of-plane modes.

Model of the Complete Drive Train

This subsection discusses the analysis of the complete drive train with a purely torsional and a rigid multibody model. Figure 13 shows the latter model, which demonstrates the layout of the drive train. 1. The low-speed planetary gear stage consists of three planets and its ring wheel is ?xed to the gearbox housing. The wind turbine’s rotor is considered rigid and its large inertia is added to the inertia of the planet carrier, which can rotate freely. The helix angle of this stage is zero and its sun is connected to the planet carrier of the second planetary stage through an appropriate stiffness matrix. 2. The high-speed planetary stage was presented and investigated separately in subsection two. Its sun is connected to the gear wheel of the parallel stage through an appropriate stiffness matrix. 3. The parallel gear stage was discussed in subsection one. The high-speed pinion is the output of the gearbox; the inertias and ?exibilities of the brake disc, the high-speed coupling and the generator are added to complete the drive train. The generator can rotate freely in its stator. No compensation factors for the bearing ?exibilities are used in the purely torsional model, which means that the bearings are considered rigid. Table V shows the results of an eigenmode calculation for this model.

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Internal Drive Train Dynamics

Figure 13. Rigid multibody model of the complete drive train

Table V. The eigenfrequencies (Hz) calculated for a torsional multibody model of the complete drive train No. Global mode Parallel stage High-speed planetary stage Low-speed planetary stage

1 2 3 4 5 6 7 8 9 10 11 12 13

0 8·6 83 630 733 861 861 1368 1397 1397 1514 1780 7982

The eigenfrequencies are categorized according to the location of the nodes in the corresponding mode shapes; several modes cannot be classi?ed in this way and are called ‘global’. The same categorization is used in Table VI for the eigenfrequencies calculated for the rigid multibody model. In addition, these frequencies are ordered according to the type of the corresponding mode shapes. A comparison of both table yields the following conclusions. 1. The impact of the bearing ?exibilities on the torque dynamics cannot be neglected, since the corresponding eigenfrequencies shift considerably when they are taken into account. The rigid multibody approach gives a straightforward method to include these ?exibilities. 2. A consideration of more than only the torsional DOFs gives more insight into the dynamic behaviour of the drive train. Extra modes are found in the same frequency range and could e.g. be excited by radial or axial loads in the gear contact.

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J. L. M. Peeters, D. Vandepitte and P. Sas Table VI. The eigenfrequencies (Hz) calculated for a rigid multibody model of the complete drive train, which are categorized according to the location of the nodes in the corresponding mode shapes and their type No. 1 2 5 14 29 44 61 63 Global mode 0 7 77 261 634 811 1835 2349 No. 18 23 30 31 32 33 37 45 46 53 62 Parallel stage 401 511 706 755 761 775 801 821 1006 1197 1950

No.

High-speed planetary stage Rotational mode (m = 1) Translational mode (m = 2) Out-of-plane mode

No.

Low-speed planetary stage Rotational mode (m = 1) Translational mode (m = 2) 33 252 253 364 399 407 (?2) 488 782 801 808 (?6) 1007 1060 1217 >4000 Out-of-plane mode

6 7, 8 9, 10 21 24, 25 26, 27 28 50 51, 52 56, 57 58 59, 60 64, 65 66 67, 68 69 70, 71

140 147 187 451 535 570 (?2) 601 1071 1143 1318 1527 1545 2617 2947 2962 2978 3100

3, 4 11 12, 13 15 16, 17 19, 20 22 34 35, 36 38–43 47, 48 49 54, 55 72, 73

Only those modes which in?uence the loads in the drive train are of importance. This means that in one way or another there should be a coupling with an excitation. This can be an internal local excitation (e.g. gear mesh frequency) or an external excitation (e.g. the wind spectrum or as a result of a generator fault transient). The coupling with such an excitation source 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 input of an excitation and a speci?c load in the drive train. This is demonstrated for the torque in three positions of the drive train. The torque at the generator is used as input and the torque in the two suns and in the high-speed pinion as outputs. The input torque is a multisine with a frequency range from 1 to 2000 Hz with a spectrum as shown in Figure 14(a). This range should cover all possible internal excitations, since 1000 Hz can generally be considered as a maximum for the gear mesh excitation frequencies. Furthermore, the output time series can only be calculated with a certain amount of damping, which de?nes the ampliCopyright ? 2005 John Wiley & Sons, Ltd. Wind Energ. (in press)

Internal Drive Train Dynamics

(a) Power spectrum of the multisine input signal

(b) FRF from the generator torque to the torque at the high-speed pinion.

(c) FRF from the generator torque to the torque at the high speed sun.

(d) FRF from the generator torque to the torque at the low-speed sun.

Figure 14. Response calculation for a multisine applied as generator torque

tudes of the response, especially at the eigenfrequencies. The determination of the damping values is not within the scope of this article and therefore the responses are only considered qualitatively in this analysis. Figure 14 shows the FRFs from the generator input to the respective torque signals. Focusing on the results above 10 Hz, for reasons which are explained below, leads to the following conclusions. 1. For the high-speed pinion it is mainly the global mode at 77 Hz which is excited by a torque input at the generator. Locally, the modes at 706 and 1006 Hz are dominant. 2. For the high-speed planetary stage the global mode at 77 Hz is less dominating. It is the local modes at 147, 187, 535 and 570 Hz that are clearly excited. 3. For the low-speed planetary stage, mainly the global modes at 77 and 261 Hz determine the dynamic response. The presented results describe the behaviour of the drive train in a wind turbine without detailed consideration of its boundaries. For example, the ?exibilities of the rotor and the tower, which are known to be determining for the dynamic behaviour in a frequency range below 10 Hz, are not included. Furthermore, the generator controller is simpli?ed as a free boundary, which is not always a valid assumption. The analysis of these limitations needs further model extensions and is still part of ongoing research. However, from a comparison of the presented results with those calculated for the individual stages separately (which was part of a

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J. L. M. Peeters, D. Vandepitte and P. Sas

more in-depth analysis not further elaborated in this article), an interesting insight is concluded. All modes calculated for the rigid multibody models of the individual stages are present in the results of an analysis for the complete drive train with a quasi-equal eigenfrequency, except for the second and ?fth rotational modes of the planetary stages, which become ‘global modes’. This means that the internal eigenfrequencies for the gearbox, ‘global modes’ excluded, are hardly in?uenced by its boundaries and can be predicted locally.

Conclusions

Certain phenomena in wind turbine drive trains can occur in a higher frequency range than traditionally simulated. This article describes how a drive train can be modelled when further insight into its internal dynamics is needed. The multibody simulation technique is presented, which implies a split of the drive train into different parts: the bearings, the gear contacts and the drive train components all need individual consideration. Furthermore, different modelling approaches for a multibody model are investigated. A purely torsional model has only one DOF per drive train component and only the torsional ?exibilities, such as the gear mesh stiffness and the components’ torsion, are directly included. The torsional analysis gives insight into the dynamic torque variations in the drive train. Here the in?uence of the bearing ?exibilities is dif?cult to assess and bearing loads can only be derived by further processing the torque. More accurate modelling of the bearings is included in the presented rigid multibody model, where all bodies have six DOFs and where the bearings and gear meshes are modelled by linear springs. A generic implementation of this approach in DADS makes it applicable for parallel and planetary gear stages as well as for complete gearboxes with several gear stages. Application of this implementation for a parallel and a planetary gear stage shows the use of the presented techniques. In addition, a sensitivity analysis for the helical parallel gear stage indicates the minor in?uence of the helix angle on the eigenfrequencies. In contrast, two other sensitivity analyses clearly demonstrate the big impact of the bearing ?exibilities on the results for the parallel and the planetary gear stage. Taking into account the realistic ?exibility of all bearings causes a remarkable drop in eigenfrequencies and, furthermore, several other modes in the same frequency range are found by considering the extra DOFs. This underlines the importance of the rigid multibody approach, which still has the limitation that no internal stresses and strains of the drive train components can be calculated. Therefore the ?exible multibody model is presented as a further extension in which the components’ ?exibilities are taken into account through a ?nite element approach. Finite element models of individual drive train components are reduced using the component mode synthesis technique according to Craig and Bampton19 and can replace the rigid bodies in a multibody model. The eigenmode calculation for a ?exible multibody model of the helical parallel gear stage in the wind turbine shows how some eigenfrequencies decrease only slightly in comparison with the results for a rigid model, where other modes are affected much more. The gearbox in this wind turbine has an additional high-speed planetary stage with helical gears. The axial forces in such a stage imply out-of-plane translations and rotations, which leads to the introduction of out-of-plane modes, next to the rotational and translational modes in a planetary stage. The presented formulations and examples give an overview of the advantages, limitations and modelling consequences of the different multibody approaches. The application on the complete drive train emphasizes the importance of the bearing ?exibilities for an accurate prediction of the eigenfrequencies. Furthermore, the use of FRFs demonstrates how the response for a certain input can be calculated. For the torque in the parallel and the low-speed planetary stage the modes identi?ed as ‘global modes’ dominate the response for a torque input at the generator. In contrast, the same input excites mainly the internal modes of the high-speed planetary stage. These analyses lead to qualitative conclusions only, since damping is not considered in this article but only applied for numerical reasons. Other limitations of the presented formulations are the exclusion of the static transmission error (i.e. a variable gear mesh stiffness), non-linear stiffnesses, friction and other possible non-linear effects in the drive train. The investigation of the effect of each individual issue is part of ongoing research. In addition, further analyses examine the in?uence of the generator controller and the ?exible rotor and tower on the (low-frequency) dynamics of the drive train.

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Internal Drive Train Dynamics

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