Analysis of Internal Drive Train Article Dynamics in a...

Analysis of Internal Drive TrainDynamics in a Wind TurbineJoris L. M. Peeters*, Dirk Vandepitte and Paul Sas, Department of Mechanical Engineering,Katholieke Universiteit Leuven, Kasteelpark Arenberg 41, B-3001 Heverlee (Leuven), Belgium

Three types of multibody models are presented for the investigation of the internal dynam-ics of a drive train in a wind turbine.The first approach is limited to the analysis of torsionalvibrations only. Then a rigid multibody model is presented with special focus on the repre-sentation of the bearings and gears in the drive train. The generic model implementationcan 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. Further-more, the influence of the helix angle and the flexibility of the bearings on the results ofeigenmode calculations are discussed. The eigenmodes of a planetary stage are classifiedas rotational, translational or out-of-plane modes.Thirdly, the extension to a flexible multi-body model is presented as a method to include directly the drive train components’ flexi-bilities. Finally, a comparison of two different modelling techniques is discussed for a windturbine’s drive train with a helical parallel gear stage and two planetary gear stages. Inaddition, the response calculation for a torque input at the generator demonstrates whicheigenmodes can be excited through this path. Copyright © 2005 John Wiley & Sons, Ltd.

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

Received 22 November 2004Copyright © 2005 John Wiley & Sons, Ltd. Revised 25 August 2005

Accepted 19 September 2005

Research Article

* Correspondence to: J. L. M. Peeters, Department of Mechanical Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 41, B-3001 Heverlee (Leuven), BelgiumE-mail: [email protected]

IntroductionTraditional design calculations for wind turbines are based on the output of specific aeroelastic simulationcodes as described by Molenaar and Dijkstra.1 The output of these codes gives the mechanical loads on thewind turbine components caused by external forces such as the wind, the electricity grid and (for offshoreapplications) sea waves. Since the focus in the traditional codes lies mainly on the rotor loads and the dynamicbehaviour of the overall wind turbine, the model of the drive train in the wind turbine is reduced to only a fewdegrees of freedom. This means that for the design of the drive train the simulated load time series need to befurther processed to loads on the individual components, such as gears and bearings. Furthermore, the limita-tion of the model implies that vibrations of these internal drive train components are not taken into accountand, as a consequence, dynamic loads on these components cannot be simulated. Instead, application factorsaccording to DIN 39902 and DIN ISO 2813 are typically used for the processing of the simulated load timeseries to loads on the gears and bearings respectively. For existing wind turbines nowadays, this approachseems acceptable from the point of view that the internal drive train dynamics are in a frequency range wellabove the overall wind turbine dynamics. However, this argument does not cover the complete range of phe-nomena that can occur in the drive train. After all, not only external low-frequency excitation of the drive trainis possible, but also internal excitation at higher frequencies exists. For instance, excitation at gear meshingfrequencies or from generator fault transients might introduce energy in the range of the internal eigenfre-

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

quencies. This indicates the importance of further insight into the internal dynamics of the drive train, whichimplies the need for additional numerical simulation methods. Moreover, extra insight from additional analy-ses 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 internalcomponents of drive trains. This article investigates the use of this technique for the dynamic analysis of awind turbine drive train with a gearbox. Models with different levels of complexity are analysed in this inves-tigation and all models are implemented in the multibody software package DADS.4 The first subsection ofsection two describes the simplest level of modelling, where exactly one degree of freedom (DOF) per drivetrain 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, whereall individual drive train components have six DOFs, further called ‘rigid multibody models’. The interactionsbetween the bodies, which represent the gear and bearing flexibilities, are modelled by linear springs. Theirimplementation is based on a synthesis of the work presented by Kahraman5,6 on helical gears and by Parkerand co-workers7,8 on planetary gears. This synthesis makes it possible to analyse a single-stage helical plane-tary gear set, as already introduced by Kahraman.9 In addition, the formulations presented in this subsectionyield three-dimensional, generic models to simulate the dynamics of complete gearboxes integrated in a windturbine drive train. Finally, the third subsection of section two discusses a further extension of the multibodymodel to a ‘flexible multibody model’ in which the drive train components are modelled as finite elementmodels instead of rigid bodies, adding the possibility of calculating stresses and deformations in the drive traincomponents continuously in time. Every addition to the model leads to specific additional information aboutthe 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. Thepresentation of the step-by-step increase in complexity enables the drive train designer, and in particular thegearbox 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 specific design phases to estimate the significance of dynamicloads.

Section three presents an application of the different modelling techniques on a drive train of a wind turbinewith two planetary gear stages and one parallel helical gear stage. First the parallel stage is analysed separatelyas an example of the flexible multibody simulation technique. Then an individual discussion of the high-speedhelical planetary gear stage introduces the ‘out-of-plane’ modes. These analyses focus mainly on the calcula-tion of mode shapes and corresponding frequencies. Finally, the complete drive train is implemented as a purelytorsional and as a rigid multibody model. The results of eigenmode calculations for both models are comparedand the use of frequency response function (FRF) calculations in the latter model is demonstrated to estimatehow the individual modes contribute to the response on specific excitations. Section four summarizes the mainconclusions of the presented research and gives an overview of ongoing work.

The Multibody Simulation TechniqueIn a multibody model of a drive train, each body represents an individual drive train component which cantranslate in three directions and rotate around three axes (six DOFs). The different bodies can be connectedusing the appropriate joints or stiffnesses. The specific implementation of these links for particular models isdiscussed in the following subsections, but a general overview of the flexibilities in the different models is firstgiven here.

1. Tooth flexibility. All teeth in contact of a gear pair under load exhibit bending deformation, which can berepresented as a tooth stiffness between the gears (gear mesh stiffness).

2. Component flexibility. All individual components which transfer torque in a drive train will deform underdifferent load components such as axial, torsional, shear and bending loads. This can be represented as astiffness between the bodies or as an integrated stiffness in a flexible multibody model.

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

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

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

A Purely Torsional Multibody ModelA first 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 sym-metry. The other five DOFs are fixed, so they can be left out of the equations of motion, and the coupling oftwo 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 flexibilities taken into account.Torsional models can be used for the dynamic analysis of the torque in the drive train; the other force com-ponents can only be derived by further processing the torque. The torsional flexibility of a shaft (Kshaft) betweentwo bodies is included in the equations of torque as shown in equation (1). Material damping is neglected inthis model.

(1)

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

T T K1 2 2 1= - = -( )shaft q q

Internal Drive Train Dynamics


Figure 1. A torsional model for the gear contact forces between a driving pinion and a driven gear wheel. Td is apositive 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

T T F r

r r r

r u

T T u ru

d1 1

1 1 2 2 1

12

1 2

2 1 22

2 11

= - = - ◊= - ◊ -( ) ◊

= - ◊ ( ) ◊ - ◊( )

= - ◊ = - ◊ ( ) ◊ - ◊ÊË

ˆ¯

tooth contact

gear

gear

gear

k q q

k q q

k q q

{

(a){

(b)

{

(c)

{

(d) {

(e)

r1 base circle radius of pinionr2 base circle radius of gear wheel

u transmission ratio

Td positive driving torque applied to the pinion

(a) deformation along the line of contact (>0)(b) torsional stiffness referred to the pinion(c) torsional deformation

(d) torsional stiffness referred to the gear wheel(e) torsional deformation

r

r2

1

ÊË

ˆ¯

force on the teeth of both gears (Ftooth contact) is equal in magnitude, resulting in a higher torque on the largergear. The direction of this force is such that the resulting torque on the driving wheel is always opposite to theinput torque. The stiffness value kgear is defined according to DIN 3990 as the normal distributed tooth forcein the normal plane causing the deformation of one or more engaging tooth pairs, over a distance of 1mm,normal to the evolvent profile in the normal plane; this deformation results from the bending of the teeth incontact between the two gear wheels, of which one is fixed and the other is loaded. In the gear contact modelthe time-varying components due to a static transmission error excitation or a fluctuation in the number oftooth pairs in contact are not considered. Furthermore, no damping or friction forces are included. From aphysical understanding it is clear that the presented spring will only work under compression. To ensure thatthis limitation will not be exceeded during simulation, the following extra assumption is made here. No contactloss between the gears will occur, something that could happen for a system with backlash when the dynamicmesh force becomes larger than the static force transmitted. This assumption is valid for heavily to moderatelyloaded gears.5

q1 and q2 in Figure 1 are defined as the rotations of the pinion and the gear wheel in their respective refer-ence frame. For a parallel gear stage these reference frames are fixed to the gearbox housing. However, thesame formulation is valid when the reference frame of a wheel follows the rotation of a component, whichimplies a kinematic coupling between the wheel and this component. This makes it also applicable for a plane-tary gear stage where the reference frame of a planet follows the rotation of the planet carrier. Thus, by keepingthe gear contact formulation independent from the definition of the reference frame, it can be used as a genericmodule for all possible gear set-ups. This independence can be implemented straightforwardly in the multi-body software package DADS, since co-ordinate systems can be created and referenced freely. When usingthe formulation for a wheel with internal teeth, the base circle radius should be taken negative. A detailed dis-cussion of the implementation of a torsional model in DADS and the numerical validation of this implemen-tation 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 indus-trial applications. Flexibility is assumed to be concentrated in shafts and gear teeth. Bearings are consideredto be rigid in radial and axial directions. As a conclusion, two simple examples of torsional models are dis-cussed. Figure 2 shows a parallel gear stage with the necessary input for this model and the results of an eigen-mode calculation. The same example will be discussed in the next subsection for a rigid multibody approachand was first presented by Kahraman.5 It is clear that there are only two DOFs in the torsional model: on theone 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 secondexample was first presented by Lin and Parker7 and is a model of a planetary gear stage with three planets asshown 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 springwith the rigid housing. Again the same example will be discussed in the next subsection for a rigid multibodyapproach. Here the eigenmode calculation yields five non-zero eigenfrequencies; two of them form a doublepole, resulting from the symmetry in the planetary stage. The influence of the corresponding mode shapes onany torque fluctuation is zero and therefore these two modes do not matter in a torsional analysis.



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

model input

kgear (N/m) 2 ·108

r1 (mm) 50r2 (mm) 50J1 (kg ·m2) 2·9 ·10-3

J2 (kg ·m2) 2·9 ·10-3

eigenfrequencies

(1) 0Hz(2) 2955Hz

A Rigid Multibody Model with Discrete Flexible ElementsThe extension of a torsional model to a rigid multibody model adds the possibility to investigate the influenceof bearing stiffnesses on the internal dynamics of the drive train. Furthermore, the analysis can also yieldinsight 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 onlyone. This implies that the linkages in the multibody model, representing the bearing and tooth flexibilities, nowneed to couple 12 DOFs. The presented modelling techniques are based on a synthesis of the work presentedby Kahraman5,6 on helical gears and by Parker and co-workers7,8 on planetary gears. Linear springs are usedhere to model the bearing and gear mesh stiffnesses. An individual formulation of these models yields twothree-dimensional plug-in components, ready to use in a generic modelling approach for the drive train. Thismeans that both models are suitable for a rather simple parallel gear stage as well as for a more complex helicalplanetary stage with any number and any positioning of the planets. Furthermore, the analyses with thesemodels 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 para-llel helical gear stage is analysed as an example of their application. In addition, the influences of the helixangle and the bearing flexibilities in this model are investigated in two sensitivity analyses. A second examplediscusses the application of the presented formulations for a planetary gear stage. Again the impact of thebearing flexibilities 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 representedby a linear spring and implemented as a 6 ¥ 6 stiffness matrix defined in the XYZ co-ordinate system as shownin Figure 4. Damping is neglected and all bearings are assumed to have an axisymmetric behaviour withoutcoupling between the individual DOFs. Therefore all off-diagonal terms are zero and both the radial and tiltstiffnesses are equal. Practically, the bearing component in a multibody model connects the XYZ co-ordinatesystem fixed to a certain body with the X¢Y¢Z¢ system fixed to this body’s reference frame. This reference framecan be for instance the fixed 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 deformationis represented in the model by a linear spring acting in the plane of action along the contact line (normal tothe tooth surface at the point of contact). This formulation was already introduced for the purely torsionalequivalent of the gear mesh model, but now this spring involves a coupling between 12 DOFs instead of onlytwo. The assumptions postulated for the gear mesh model subsection one are still valid. For the sake of com-pleteness they are repeated in the list of assumptions made here.



Figure 3. A torsional model for a planetary gear stage with three planets and a fixed ring wheel. Both the planetcarrier and the sun have a free boundary

model input

kgear (N/m) 5 ·108

rsun (mm) 38·7rplanet (mm) 50·2rring wheel (mm) -137·5rplanet carrier (mm) 96·9Jsun (kg ·m2) 2·34 ·10-3

Jplanet (kg ·m2) 6·14 ·10-3

Jring wheel (kg ·m2) 227 ·10-3

Jplanet carrier (kg ·m2) 197 ·10-3

mplanet (kg) 0·66kring-housing (Nm/rad) 76 ·106

eigenfrequencies

(1) 0Hz(2) 2253Hz(3) 6283Hz(4) 6449Hz(5) 6449Hz(6) 11241Hz

1. The gear mesh model is a linear time-invariant model. Static transmission error excitation is not consideredand therefore no phasing relationships between gear meshes are included. Furthermore, a variable stiffnesscaused by a fluctuation in the number of tooth pairs in contact is assumed negligible. The validity of theseassumptions for the presented linear analyses can be justified as in subsection one.

2. Sliding of teeth in contact and corresponding friction forces are neglected as well as any other possibledamping in the system.

3. Occurrence of tooth separation is considered non-existent and consequently the modelling of gear backlashis 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 assumptionsare valid for wind turbine applications, since planetary gear stages in wind turbines are only rarely used ashigh-speed stages.

Formulation of the gear contact forces is based on the model approach shown in Figure 5.

• Co-ordinate systems X¢1Y¢1Z¢1 and X¢2Y¢2Z¢2 are oriented with X¢ along the centreline pointing from gear 1 to gear2; Z¢ is lying along the axis of rotation. These co-ordinate systems are fixed to the reference frames of therespective wheels (as introduced in subsection one and Figure 4).

• X1Y1Z1 and X2Y2Z2 are fixed to the respective gears and in their starting position they coincide with the cor-responding X¢Y¢Z¢.

• ft is the pressure angle of the gear mesh, which is an input parameter. It is defined as the angle measuredfrom the centreline towards the normal on the contact line in the corresponding X¢Y¢Z¢. The sign of this anglechanges when the driving direction of the system changes.

• y ¢1 and y ¢2 are the angles measured respectively from X¢1 to X1 and from X¢2 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 positionwhere b = 0; b > 0 in Figure 5.

The compression of the linear spring (d) can be written as a function of the vectors and as introduced inFigure 4. Since the spring works always under compression, d should be positive.

(2)

with

d y y y y f bw y w y y y y y f b

= - - + - -( ) ( )

- - + + - -( ) ( )

x x y y u u

z z X X Y Y

1 1 2 2 1 1 2 2 1 2

1 2 1 1 2 2 1 1 2 2

sin sin cos cos cos

sin sin cos cos sin

sign

sign

t

t

q2q1



Figure 4. Schematic representation of a bearing model: a linear spring connects the XYZ co-ordinate system fixed tothe body with the X¢Y¢Z¢ system fixed to this body’s reference frame. The spring is modelled by a symmetric 6 ¥ 6

stiffness matrix and is the force working on the gear in the XYZ systemFb

K

k

k

k

k

k

F K q

q x y z

b

radial

radial

axial

tilt

tilt

b b

X YT

X

=

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙

= ◊

= [ ]

¢ ¢ ¢

0 0

0

0

. . . . . . . . .

O M

O M

O M

r r qr r qwhere x, y, z, , and 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 .

Y

The stiffness value of the linear spring, kgear, is the same as defined in subsection one, namely the ratio of thecontact force on a tooth over the resulting displacement of the contact point. The spring force causes forcesand moments on the gears, which can be projected in the XYZ co-ordinate systems and thus written as

F

F

F

T r

T r

T r

FX

Y

Z

X

Y

Z

X1 1

1 1

1

1 1 1

1 1 1

1 1

2 2= - ( )

= ( )

= ( )

= ( )

= - ( )

= ( )

= (dk y b f

dk y b f

dk b f

dk y b f

dk y b f

dk b f

dk y b fgear t

gear t

gear t

gear t

gear t

gear t

gear tsign

sign

sign

sign

sign

sign

signsin cos

cos cos

sin

sin sin

cos sin

cos

sin cos ))

= - ( )

= - ( )

= ( )

= - ( )

= ( )

F

F

T r

T r

T r

Y

Z

X

Y

Z

2 2

2

2 2 2

2 2 2

2 2

dk y b f

dk b f

dk y b f

dk y b f

dk b f

gear t

gear t

gear t

gear t

gear t

sign

sign

sign

sign

sign

cos cos

sin

sin sin

cos sin

cos

x y z u r r r q

x y z u r r r q

X Y

X Y

1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2

w w

w w

[ ] = ( )

[ ] = ( )

diag 1, 1, 1, , , ,

diag 1, 1, 1, , , ,



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

Gear1 is the driving wheel (Tinput < 0)m1, I1, J1

r1: base circle radius

Gear2 is the driven wheelm2, I2, J2

r2: base circle radius

kgear is the gear mesh stiffness as defined insubsection one

By writing the force components and the compression d in the XYZ systems, the gear contact formulation is ageneric module. Its implementation in DADS is a user-defined subroutine which can be used to couple anytwo gears in all possible gear set-ups, keeping in mind that a wheel with internal teeth is given a negativeradius. The formulation of the gear forces in matrix form yields

(3)

where

with cb = cosb, sb = sinb, cy = cosy and sy = siny. When b π 0, it is possible to have no zero componentsin the matrices. At that moment, all DOFs of both gears are coupled with each other. The application ofthis 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 influence 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 themore detailed approach presented here can be drawn. This is discussed in the sensitivity analysis of the bearingstiffnesses.

k ij

k

b y b y y b b y b b y b b y y b y

b y y b y b b y b b y y b b y b y

b b y b b y11

2 21

21 1 1

21 1 1

21

21 1

2 21 1 1 1

21

21

1=

- -

- - - -

-

c s c c s c s s c s s c s c s c s

c c s c c c s c c s c s c s c c c

c s s c s c 112 2

12

1

12

1 1 1 1 12

1 12 2

1 12

1 1 1 1

1 1 1 12

1 12

- - -

- - - -

-

s s s c c s

c s s c s c s s s s s s c s c s s

c s c s c s c s c

b b y b y b b

b b y b b y y b y b y b y y b b y

b b y y b b y b y

s

r r r r r r

r r r 11 12

1 1 12 2

1 1 1

12

1 12

1 1 1 1 1 1 12

12

21 2

r r r

r r r r r r

r

s c s s c c s c

c s c c c s c s s c s c c

s s

b y y b y b b y

b y b y b b b b y b b y b

k

b y y

-

- - - -

È

Î

ÍÍÍÍÍÍÍÍÍÍ

˘

˚

˙˙˙˙˙˙˙˙˙˙

=

-- - -

- - -

-

c c s c s s c s s s c s c s c s

c c s c c c c s c c s c s c s c c c c

c s s c s c s

2

22 1 1 1 2 2 1

21

21 2

21 2 1 1 2 1 2 1

2 22

b y y b b y b b y y b b y y b y

b y y b y y b b y b b y y b b y y b y

b b y b b y bb b y b y b b

b b y y b b y y b y b y y b y y b b y

b b y y b b y y b

- -

- - -

- -

s s s c c s

c s s s c s c s s s s s s c s c s s

c s c s c s c c s c

22

22

1 1 2 1 2 1 12

1 12

1 2 12

2 1 1 1

1 1 2 1 1 2 12

r r r r r r

r r r yy b y y b y y b b y


k

b y

1 12

1 2 12

1 2 1 1

12

2 12

2 1 1 2 1 2 12

21

21

r r r

r r r r r r

r

s c s s c c c s c


s s

-

- - -

È

Î


˘

˚

˙˙˙˙˙˙˙˙˙˙

=

yy b y y b b y b b y y b b y y b y

b y y b y y b b y b b y y b b y y b y

b b y b b y

22

1 2 2 1 2 1 22

2

22 1

21 2 2 2 1 1 2

22

1 1

- - - -

- -

-

c c s c s s c s s s c s c s c s

c c s c c c c s c c s c s c s c c c c

c s s c s c ss s s s c c s

c s s s c s c s s s s s s c c s s

c s c s c s c c s c

2 21

21

2 1 2 2 1 2 22

2 22

1 2 22

1 2 2 2

2 2 1 2 1 2 22

b b y b y b b

b b y y b b y y b y b y y b y y b b y

b b y y b b y y b

-

- - - -r r s r r r r

r r r yy b y y b y y b b y


k

b

2 22

2 1 22

1 2 2 2

22

1 22

1 2 2 1 2 1 22

22

2 2

r r r

r r r r r r

s c s s c c c s c


c s

-

- - - -

È

Î


˘

˚

˙˙˙˙˙˙˙˙˙˙

=

- yy b y y b b y b b y b b y y b y

b y y b y b b y b b y y b b y b y

b b y b b y b b

22

2 2 22

2 2 22

2

22 2

2 22 2 2 2

22

22

2 22 2

c c s c s s c s s c s c s c s

c c s c c c s c c s c s c s c c c

c s s c s c s s

- -

- - -

- - ss s c c s

c s s c s c s s s s s s c s c s s

c s c s c s c s c c

y b y b b

b b y b b y y b y b y b y y b b y

b b y y b b y b y b y y

22

2

22

2 2 2 2 22

2 22 2

2 22

2 2 2 2

2 2 2 22

2 22

2 22

2

-

-

- -

r r r r r r

r r r r s s 22 22 2

2 2 2

22

2 22

2 2 2 2 2 2 22

-

- - -

È

Î


˘

˚

˙˙˙˙˙˙˙˙˙˙

r r

r r r r r r

s c c s c


b y b b y


F F F F T T T

F F F F T T T

X Y Z X Y ZT

X Y Z X Y ZT

1 1 1 1 1 1 1

2 2 2 2 2 2 2

= [ ]

= [ ]

F

F

q

q1

2

11 12

21 22

1

2

ÈÎÍ

˘˚̇

= ÈÎÍ

˘˚̇ÈÎÍ

˘˚̇

kk kk kgear



Influence of the Helix Angle b. Table I shows the eigenfrequencies calculated for the helical gear pair inFigure 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 theeigenfrequencies change with changing b. Only w2, w4, w8 and w12 are influenced and the largest relative changeis observed for w2, which is only 6% for b = 20° (Table I). Thus the influence of the helix angle is rather small;as a result, a simplification of a parallel helical gear system to a spur gear pair can be justified when calculat-ing only the eigenfrequencies. The effect of the helix angle on the corresponding mode shapes is shown inFigure 8 for w2, w4, w8 and w12.

Influence 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 hasthe biggest impact on the torque. Remarkable is the drop in frequency (2955 Æ 1566 Hz) for this mode as aresult of adding realistic bearing flexibilities in the rigid multibody model, which were lacking in the torsionalmodel. Users of torsional models are aware of this limitation in their models and therefore often use gear meshstiffness reduction factors based on their experience. However, the new formulation gives directly more accu-rate 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 ofthe rigid multibody approach.

Figure 7(b) demonstrates the statement that increasing all bearing stiffnesses for the helical gear system toinfinity yields a purely torsional equivalent of the model. For this purpose the real bearing stiffnesses are mul-tiplied by a stiffness factor, which is taken equal for kbax, kbrad and kbtilt, since the focus is not on the individualsensitivity of these values. All eigenfrequencies increase towards infinity, except for w4 (1566 Hz); this fre-



Figure 6. Helical gear system. The two bearings supporting each of the shafts are represented by one stiffness matrixper 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

model input

kgear (N/m) 2 ·108

pressure angle (°) 20r1 (mm) 50r2 (mm) 50m1 (kg) 2·0m2 (kg) 2·0J1 (kg ·m2) 2·9 ·10-3

J2 (kg ·m2) 2·9 ·10-3

I1 (kg ·m2) 1·45 ·10-3

I2 (kg ·m2) 1·45 ·10-3

kbax (N/m) 3·5 ·108

kbrad (N/m) 1·0 ·108

kbtilt (Nm/rad) 277·5 ·103

Table I. Eigenfrequencies (Hz) for the model of the helical gear pair in Figure 6

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

b = 0° 0 1125 1125 1566 2105 2105 2105 2202 2202 2202 2202 3972b = 20° 0 1058 1125 1519 2105 2105 2105 2173 2202 2202 2202 4150D (%) – -6·0 0 -3·0 0 0 0 -1·3 0 0 0 +4·5

quency approaches asymptotically the torsional eigenfrequency (2955 Hz), which corresponds to the conclu-sions 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 gearsystems. There is no limitation on the number of planets or on their positioning around the sun. The sun canbe constrained by a bearing model or can be modelled as floating, depending on the application. Furthermore,the choice of which component is constrained as non-rotating is an input parameter for the model. As anexample, three planetary gear systems introduced by Lin and Parker7 are discussed here, respectively havingthree, four and five planets. A torsional equivalent of the presented three-planet system was used in subsec-tion one. From a comparison between the results the advantages of the rigid multibody approach for a plane-tary gear system become clear.

Figure 9 shows the input parameters for the planetary gear systems and the multibody model of the four-planet system. In all three systems the planets are identical as well as all bearing stiffnesses and all sun–planetand planet–ring gear mesh stiffnesses. The ring wheel is in all cases non-rotating and therefore constrainedwith a torsional stiffness to the rigid housing. Lin and Parker limited their analysis to planar vibrations in thegear system, since the helix angle is zero and consequently no forces are acting out-of-plane. A limitation inour model implementation to planar vibrations is made by constraining all components with bearings that areinfinitely stiff for the displacement and rotations out-of-plane (axial and tilt stiffness). Table II divides theeigenfrequencies of the three systems according to the classification by Lin and Parker. They distinguishedthree categories of planar modes for planetary systems with N planets, satisfying equation (4) for the posi-tioning around the sun.13

(4)

1. Six rotational modes always have multiplicity m = 1 for various numbers of planets N. The mode shapes( ) 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 ( ) have pure translation of the carrier, ring and sun.T1a b 6a b, ,-

R1 6-

sin cosy yN N= =ÂÂ 0



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

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

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



b = 0° w2 = 1125Hz b = 0° w4 = 1566Hz

b = 20° w4 = 1519Hz(b) Mode 4

b = 20° w2 = 1058Hz(a) Mode 2

b = 0° w8 = 2202Hz b = 0° w12 = 3972Hz

b = 20° w8 = 2173Hz(c) Mode 8

b = 20° w12 = 4150Hz(d) Mode 12

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

3. Three planet modes exist for N > 3 and have multiplicity m = N - 3. The carrier, ring and sun have no rota-tion or translation in the corresponding mode shapes ( ).

Figure 10 shows an example of a mode shape for each category. The classification of the DADS results isbased on animations of the mode shapes, and the corresponding eigenfrequencies were validated with thosefrom Lin and Parker. They correlated well, which further enhances the confidence in the use of the formula-tion for the planetary gear systems used in wind turbines.

Influence 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

P1 3-



Figure 9. Planetary gear systems introduced by Lin and Parker.7 In all systems the planet carrier and the sun are freeat their boundaries

(b) The four-planet system.

Table II. Eigenfrequencies (Hz) for the planetary gear systems with respectivelythree, four and five planets as introduced by Lin and Parker7

ModeN

shape 3 4 5

m = 1 0 0 0

1425 1496 1538

2032 2060 2082

2644 2611 2602

7500 7800 8086

11744 13050 14237

m = 2 770 759 745

1101 1092 1073

1989 1947 1921

2238 2328 2421

7060 7245 7427

9582 10390 11136

m = N - 3 1959 1959

6444 6444

6497 6497P3

P2

P1

T 6,a,b

T 5,a,b

T 4,a,b

T 3,a,b

T 2,a,b

T 1,a,b

R6

R5

R4

R3

R2

R1

Sun Planet Carrier Ring

Mass (kg) 0·4 0·66 5·43 2·35J (· 10-3 kgm2) 2·34 6·14 197 227Base circle radius (mm) 38·7 50·2 96·9 -137·5Mesh stiffness kgear = 5 ·108 N/mBearing Stiffness krad = 108 N/mTorsional Stiffness kring-housing = 76 ·106 Nm/radNominal Pressure Angle 24·6°Helix Angle b = 0°

(a) Model input parameters.

flexibility for the sun, the planets, the carrier and the ring wheel. This addition has a major impact on theresults, which is demonstrated in Figure 11(a). For a stiffness factor of unity the curves give the eigenfre-quencies of the rigid multibody model. The higher stiffness factors correspond to a gradual increase in theradial bearing stiffnesses. The eigenfrequencies corresponding to the first four rotational modes and the firsttranslational double mode approach asymptotically the results from the torsional model when the stiffnessvalues approach infinity. This phenomenon is numerically shown in Figure 11(b), where the eigenfrequenciescalculated with a rigid multibody model are compared with the results for a torsional model. Again the remark-able difference and, furthermore, the additional modes found with the former model underline the importanceof this approach.



Figure 10. Different types of mode shapes for a planetary gear system with four planets

(a) rotational mode ( : 1496Hz)

R2 (b) translational mode ( :759Hz)

T a1

(c) planet mode ( : 1959Hz)P1

T1a,b

R4

R3

R2

R1

(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. ( ) are therotational modes (m = 1); ( ) are the translational modes (m = 2). The arrows indicate how the eigenfrequenciesshift from the rigid model results to the torsional model results when the radial bearing stiffness values are increasedtowards infinity.

Figure 11. Influence of the bearing stiffness values on the eigenfrequencies of the planetary gear model with three planets

TR

rigid multibody torsionalmodel model(Hz) (Hz)

1 0 ( ) 02 770 ( ) 22273 1101 ( ) 61924 1425 ( ) 64425 1989 ( ) 64426 2032 ( ) 112107 2238 ( )8 2644 ( )9 7060 ( )10 7500 ( )11 9582 ( )12 11744 ( )

(b)

R6

T 6

R5

T 5

R4

T 4

R3

T 3

R2

T 2

T 1

R1

�

�

�

�

A Flexible Multibody ModelTypically, multibody models consist of rigid bodies linked by joints and stiffnesses. The stiffness values caninclude an equivalent discretized stiffness for the flexibility of the individual components. However, the reduc-tion to an equivalent stiffness and the discretization method complicate the modelling, especially for morecomplex systems. As a result, complex bodies are in practice often assumed to be rigid and no flexibility isfurther taken into account. Considering the component’s flexibility as a property of the body would lead to amore realistic understanding of the models. Moreover, it may give further insight into the role of this flexibil-ity in the overall dynamic behaviour. Estimating this influence for non-conventional, rather flexible parts usedtoday in wind turbines is barely possible with the traditional multibody formulation. Therefore a flexible multi-body 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 linkagesbetween the bodies represent only the stiffness of the coupling, such as the gear mesh or the bearing stiffness.

The extension to the flexible multibody formulation is no straightforward adaptation of the traditionalmethod. The additional DOFs to represent the deformations of an individual body are introduced by a finiteelement approach. The direct finite 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 compo-nents. A combination of both methods can be made by including reduced finite element models in the multi-body models. These are further called flexible multibody models. For all bodies in a traditional multibodymodel with an increased level of interest for its flexibility, a finite element model is built with as much detailas needed. Typically, these models can have a large number of DOFs, of the order of magnitude of 10,000 upto 100,000. The reduction to a smaller set of DOFs, of the order of magnitude of one up to 10, which can beimported into the multibody model is done with the component mode synthesis (CMS) technique.14–18 TheCraig–Bampton19 method is a well-established CMS technique which is supported by DADS and used in thepresent research. This reduction involves the creation of a set of static constraint modes and fixed interfacenormal modes with their corresponding eigenfrequencies. These modes represent all additional DOFs of thebody, and the body’s deformation is a linear combination of them. The static modes represent the deformationrelated to loads and displacements in the body’s interface nodes, whereas the normal modes are related todynamic deformations. MSC/NASTRAN is used for the calculation of the modes, and all finite element modelsare 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 finite element models in a multibody modelis possible in their interface nodes by using the presented formulations for the gear mesh and bearing stiff-nesses. Not only is the flexibility of a body included in this approach, but also its mass distribution is closerto reality, because the analyst has the option of deriving finite element models directly from CAD models witha very realistic representation of the geometry. As a result, mass values and other inertial properties are nolonger input parameters but are automatically calculated from the reduced models.

The flexible multibody simulation has an extra advantage for postprocessing of the results. When the defor-mation of a body at a certain moment during simulation is known for the reduced component, this result canbe transferred back to the finite element program. Here it can be further processed into internal stresses andstrains for this component, which can for instance be used for strength and fatigue analysis. Baumjohann etal.20 demonstrated this method for the calculation of stresses in wind turbine blades, but it is not discussedfurther in this article. As an example of the flexible multibody formulation, the next section presents the effectof including the individual flexibilities of the pinion and the wheel in the parallel high-speed stage in a windturbine. Conclusions concerning the influence of these additional flexibilities are discussed there.

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



slightly modified owing to confidentiality. 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 thisarticle. Subsection one discusses the implementation of the flexible multibody method for the high-speed par-allel stage. Subsection two investigates the application of the rigid multibody methods for the high-speed plan-etary 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 multibodymodel.

High-speed Parallel StageThe high-speed stage of the gearbox is a helical gear pair. For this stage the three presented modelling tech-niques are implemented. First a torsional model is built, only taking into account the gear mesh stiffness. Thena rigid multibody model is implemented by adding the bearing stiffnesses. Finally, this model is extended toa flexible multibody model which integrates the components’ flexibilities. In all three models, both the gearand the pinion are free to rotate in their bearings. Figure 12 shows these three models and Table III shows acomparison of the calculated eigenfrequencies. The eigenfrequency 1479 Hz calculated with the torsionalmodel drops to 702 Hz in the rigid multibody model, indicating again the impact of the bearing flexibilitieson the torque dynamics. Furthermore, other additional modes are found and are given with the main compo-nent of their corresponding mode shape. By adding the components’ flexibilities in the flexible multibodymodel, the eigenfrequencies decrease further. However, the impact of these flexibilities on the frequencies issmaller 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



Table III. Comparison of the eigenfrequencies (Hz) of the helical gear pair cal-culated with a torsional model, a rigid multibody model (MBM) and a flexiblemultibody model

No. Torsional Rigid Flexible Corresponding shape modelmode MBM MBM

1 0 0 0 Rigid body mode2 – 400 395 x-Translation pinion3 – 510 497 x-Translation gear4 1479 702 518 y–z-Rotation pinion5 755 519 y–z-Rotation pinion6 761 635 y–z-Rotation gear7 775 696 y–z-Rotation gear8 801 702 y–z-Translation gear9 821 718 y–z-Translation gear10 1003 848 y–z-Translation pinion11 1197 1012 y–z-Translation pinion12 1947 1634 x-Rotation pinion and gear

X

Y

Z

Figure 12. High-speed parallel gear stage

(a) Torsional model (b) Rigid multibody model (c) Flexible multibody model

bending flexibility of the rather long and slender high-speed pinion causes a considerable decrease in the eigen-frequencies corresponding to its y–z-rotation modes (w4, w5). This example shows how the flexible multibodytechnique makes it possible to evaluate the effect of the components’ flexibilities without reducing them to dis-crete stiffnesses. Furthermore, the eigenmodes of the components are also taken into account, which is impos-sible in a rigid multibody model.

High-speed Planetary StageThe high-speed planetary gear stage in the wind turbine consists of three planets with helical teeth and is mod-elled as a rigid multibody model. In this model the sun is floating in the radial direction and both the planetcarrier and the sun are free to rotate. The ring wheel is fixed; this fixation is not modelled as a connectionthrough a torsional spring with its housing (as in the example of subsection two of section two), but insteadits DOFs are removed from the equations of motion. This yields five rotational modes instead of six in theresults of an eigenmode calculation, which are shown in Table IV. The categorization presented in subsectiontwo 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 fixed here. The relevance of the out-of-plane modes is indicated by thefact 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 TrainThis subsection discusses the analysis of the complete drive train with a purely torsional and a rigid multibodymodel. 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 fixed to the gearboxhousing. The wind turbine’s rotor is considered rigid and its large inertia is added to the inertia of the planetcarrier, which can rotate freely. The helix angle of this stage is zero and its sun is connected to the planetcarrier 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 con-nected 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 flexibilities of the brake disc, the high-speed coupling and the generator are added to com-plete the drive train. The generator can rotate freely in its stator.

No compensation factors for the bearing flexibilities are used in the purely torsional model, which meansthat the bearings are considered rigid. Table V shows the results of an eigenmode calculation for this model.



Rotational mode (m = 1) Translational mode (m = 2)

0 728 1067 1524 2265 147 187 535 1143 1318 1545

Out-of-plane mode

(m = 2) (m = 2) (m = 2) (m = 2)

140 437 570 602 2617 2947 2962 2977 3100

O9O8O7O6O5O4O3O2O1

T 6T 5T 4T 3T 2T 1R5R4R3R2R1

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

The eigenfrequencies are categorized according to the location of the nodes in the corresponding mode shapes;several modes cannot be classified in this way and are called ‘global’. The same categorization is used in TableVI for the eigenfrequencies calculated for the rigid multibody model. In addition, these frequencies are orderedaccording to the type of the corresponding mode shapes. A comparison of both table yields the following conclusions.

1. The impact of the bearing flexibilities on the torque dynamics cannot be neglected, since the correspondingeigenfrequencies shift considerably when they are taken into account. The rigid multibody approach givesa straightforward method to include these flexibilities.

2. A consideration of more than only the torsional DOFs gives more insight into the dynamic behaviour of thedrive train. Extra modes are found in the same frequency range and could e.g. be excited by radial or axialloads in the gear contact.



Figure 13. Rigid multibody model of the complete drive train

Table V. The eigenfrequencies (Hz) calculated for a torsional multibody modelof the complete drive train

No. Global mode Parallel stage High-speed Low-speedplanetary stage planetary stage

1 02 8·63 834 6305 7336 8617 8618 13689 1397

10 139711 151412 178013 7982

Only those modes which influence the loads in the drive train are of importance. This means that in oneway or another there should be a coupling with an excitation. This can be an internal local excitation (e.g. gearmesh 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 specific load in the drive train. This is demonstrated for the torque inthree positions of the drive train. The torque at the generator is used as input and the torque in the two sunsand 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. Further-more, the output time series can only be calculated with a certain amount of damping, which defines the ampli-



Table VI. The eigenfrequencies (Hz) calculated for a rigid multibody model of the complete drive train, which are cate-gorized according to the location of the nodes in the corresponding mode shapes and their type

No. Parallel stage

18 40123 51130 70631 75532 76133 77537 80145 82146 100653 119762 1950

No. High-speed planetary stage

Rotational Translational Out-of-planemode mode mode

(m = 1) (m = 2)

6 1407, 8 1479, 10 18721 45124, 25 53526, 27 570 (¥2)28 60150 107151, 52 114356, 57 131858 152759, 60 154564, 65 261766 294767, 68 296269 297870, 71 3100

No. Low-speed planetary stage

Rotational Translational Out-of-planemode mode mode

(m = 1) (m = 2)

3, 4 3311 25212, 13 25315 36416, 17 39919, 20 407 (¥2)22 48834 78235, 36 80138–43 808 (¥6)47, 48 100749 106054, 55 121772, 73 >4000

No. Global mode

1 02 75 77

14 26129 63444 81161 183563 2349

tudes of the response, especially at the eigenfrequencies. The determination of the damping values is not withinthe scope of this article and therefore the responses are only considered qualitatively in this analysis. Figure14 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 thegenerator. 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 dynamicresponse.

The presented results describe the behaviour of the drive train in a wind turbine without detailed consider-ation of its boundaries. For example, the flexibilities of the rotor and the tower, which are known to be deter-mining for the dynamic behaviour in a frequency range below 10 Hz, are not included. Furthermore, thegenerator controller is simplified as a free boundary, which is not always a valid assumption. The analysis ofthese limitations needs further model extensions and is still part of ongoing research. However, from a com-parison of the presented results with those calculated for the individual stages separately (which was part of a



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

(a) Power spectrum of the multisine inputsignal

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

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

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

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

ConclusionsCertain phenomena in wind turbine drive trains can occur in a higher frequency range than traditionally sim-ulated. This article describes how a drive train can be modelled when further insight into its internal dynam-ics is needed. The multibody simulation technique is presented, which implies a split of the drive train intodifferent parts: the bearings, the gear contacts and the drive train components all need individual considera-tion. Furthermore, different modelling approaches for a multibody model are investigated. A purely torsionalmodel has only one DOF per drive train component and only the torsional flexibilities, such as the gear meshstiffness and the components’ torsion, are directly included. The torsional analysis gives insight into thedynamic torque variations in the drive train. Here the influence of the bearing flexibilities is difficult to assessand bearing loads can only be derived by further processing the torque. More accurate modelling of the bear-ings is included in the presented rigid multibody model, where all bodies have six DOFs and where the bear-ings and gear meshes are modelled by linear springs. A generic implementation of this approach in DADSmakes it applicable for parallel and planetary gear stages as well as for complete gearboxes with several gearstages. Application of this implementation for a parallel and a planetary gear stage shows the use of the pre-sented techniques. In addition, a sensitivity analysis for the helical parallel gear stage indicates the minor influ-ence of the helix angle on the eigenfrequencies. In contrast, two other sensitivity analyses clearly demonstratethe big impact of the bearing flexibilities on the results for the parallel and the planetary gear stage. Takinginto account the realistic flexibility of all bearings causes a remarkable drop in eigenfrequencies and, further-more, several other modes in the same frequency range are found by considering the extra DOFs. This under-lines the importance of the rigid multibody approach, which still has the limitation that no internal stresses andstrains of the drive train components can be calculated. Therefore the flexible multibody model is presented asa further extension in which the components’ flexibilities are taken into account through a finite elementapproach. Finite element models of individual drive train components are reduced using the component modesynthesis technique according to Craig and Bampton19 and can replace the rigid bodies in a multibody model.The eigenmode calculation for a flexible multibody model of the helical parallel gear stage in the wind turbineshows 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-speedplanetary stage with helical gears. The axial forces in such a stage imply out-of-plane translations and rota-tions, which leads to the introduction of out-of-plane modes, next to the rotational and translational modes ina planetary stage.

The presented formulations and examples give an overview of the advantages, limitations and modellingconsequences of the different multibody approaches. The application on the complete drive train emphasizesthe importance of the bearing flexibilities for an accurate prediction of the eigenfrequencies. Furthermore, theuse of FRFs demonstrates how the response for a certain input can be calculated. For the torque in the para-llel and the low-speed planetary stage the modes identified as ‘global modes’ dominate the response for a torqueinput at the generator. In contrast, the same input excites mainly the internal modes of the high-speed plane-tary stage. These analyses lead to qualitative conclusions only, since damping is not considered in this articlebut only applied for numerical reasons. Other limitations of the presented formulations are the exclusion ofthe static transmission error (i.e. a variable gear mesh stiffness), non-linear stiffnesses, friction and other pos-sible non-linear effects in the drive train. The investigation of the effect of each individual issue is part ofongoing research. In addition, further analyses examine the influence of the generator controller and the flexi-ble rotor and tower on the (low-frequency) dynamics of the drive train.



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