A Dynamic Study to Single-Stage Helical Planetary Gear Systems of Wind Turbines Including

Gravity and Gear Backlash Effect

Kuo Jao Huang1, Jeng Shiun Chen

2

1, 2 Department of Mechanical Engineering, Chung Hua University, Hsinchu, Taiwan

e-mail: kjhuang@chu.edu.tw1,

, m10108018@chu.edu.tw2

Abstract: Giant MW scale wind turbines generally must

situate in the harsh operation condition due to extremely

fluctuating wind and grid safety complexity. System

operation reliability is greatly affected by their gear

transmission lines. Recently, nonlinear dynamic effect

including gravity and gear backlashes has been highly

concerned. In this study, dynamic analyses are executed

on a single-stage helical planetary system using a 3D

discrete stiffness model. Since the meshing backlashes of

gears are incorporated, the nonlinear phenomenon of

contact separation, back-contact, or even double side

contact may occur. The models of bearing stiffness,

weight, and inertia of wind rotor and generator are also

included. Accordingly, the dynamic responses of the gear

system can be analyzed under different operation

conditions. The dynamic contact forces between gear pairs

and planet bearings are calculated. Accordingly, the

nonlinear dynamic characteristics of planetary gearing are

investigated by incorporating the gear backlashes and

gravity models. The result shows the essentiality of the

incorporation of gear backlashes and component gravity

effect in the dynamic analysis of the gear system.

Keywords: Planetary gear system, Backlash, Gravity, Dynamic

analysis, Contact force, Nonlinear

1 Introduction

Large scale wind turbines necessarily operate in remote

location at the harsh condition due to fluctuating wind and

grid safety consideration. Therefore high system reliability

is demanded which is greatly dependent on the

performance of planetary gear transmission trains from the

operation experience. For a long time, planetary gear

systems (PGSs) possess the fine features of power density,

precision, and noise and vibration and thus are widely

applied in vehicles, aircrafts, energies, and robots [1-3].

Recently, with a arising essential role in the wind energy

industry, rigorous studies on PGS behavior under dynamic

conditions are endeavored for seeking the enhancement of

their design and produce techniques. Especially, dynamic

performance of the PGSs under varying excitation PGS is

urgently demanded. Both the discrete and the finite

element methods are applied. Generally, the equivalent

discrete model is adopted for its simplicity for computing

cost in which an equivalent spring for the modeling of

mesh pairs between mating gears or bearings. Some of

numerous works to obtain the equivalent stiffness of spur

gear pairs are referred [4, 5]. In addition, Hedlund [6]

calculated the tooth deflection of helical gears by both

including Hertzian contact and tooth foundation

flexibilities [7].

In the dynamic aspect, the work [8] is a pioneer in the

PGS dynamic analysis. Latter, August and Kasuba [9]

found that dynamic responses of PGS are critically

affected by the variation of the meshing stiffness and

fixity design of their sun gears. Kahraman [10] presented

a 3D discrete model to investigate the helical PGS

dynamics in which modal shapes and dynamic forces

caused from the static transmission error was investigated.

Velex and Flamand [11] indicated the stiffness of meshing

gear pairs has a more significant effect on planetary

dynamic behavior than the other components. In addition,

Sadda and Velex [12] considered torsional, flexural, and

axial displacements of components basing on the finite

element procedure in which a complex planetary gearing

was simplified to a twelve degrees of freedom (DOFs)

discrete system. In addition, plentiful studies on planetary

gear dynamics of a research team in recent years were

presented. The authors analyzed the modal behaviors of

PGSs including three- and four-planets of equally spacing

and diametrical symmetry with planet meshing phase

difference or not using linear or nonlinear models. For

example, Lin and Parker [13] calculated natural

frequencies of PGSs in which the nonlinearity due to

meshing stiffness discontinuity of gear pairs was

discussed. In the study [14], using the modal analyses and

mesh phasing properties, design rules are analytically

derived to suppress specific harmonics of plant modal

response of a PGS. Recently, using a nonlinear discrete

vibration model, Al-shyyab and Kahraman [15]

investigated influences of time-varying meshing stiffness,

contact ratio, and backlash on the dynamic responses of

single-stage. Additionally, Farshidianfar et al. [16]

investigated the nonlinear vibration of single-stage PGS

using several analyzing methods in calculating its

dynamic spectrum and modal property. Two-dimensional

model discrete models are commonly adopted in gear

dynamics. Difficulty and deviation from practical

application may occur. Therefore, an approach using the

3D nonlinear and time varying discrete model is intended

for the benefit of gear dynamic analyses.

Gear transmission has gained the wide confidence for

its fine performance and high reliability. However,

unexpected challenge occurs with its increasing role in the

wind energy applications. Lack of its operation reliability

is found especially in the speed increasing planetary

gearbox of the transmission line. Thus, performance

enhancement of PGSs is urgently demanded. Most of time,

a wind turbine needs operating in the dynamic and harsh

The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.PS6.006

condition due to the wind source and electric grid safety

complexity. A different and deeper insight of its dynamics

of PGSs is urgently required. Recently, nonlinear dynamic

effect due to gravity and gear backlashes attract a great

concern. In this study, an approach using a 3D time

varying nonlinear discrete dynamic model is proposed.

The dynamic contact forces between gear pairs and planet

bearings are calculated. by incorporating the gear

backlashes and gravity models. Besides, experimental

result of a PGS by NREL is used for verification [17].

Finally, the dynamic contact forces of planetary gearing

are also discussed under various bearing stiffness

assignments. The effect of gear backlashes and gravity on

the dynamic behavior of the planetary gear system is

emphasized.

2 Configuration and Meshing Stiffness

Generally, PGSs are constituted by plural external sun-

planet gear pairs and internal planet-ring gear pairs. Figure

1(a) shows the configuration and components of a PGS

used in the transmission line for speed-up in wind turbines.

Figure 1(b) is its discrete analysis model. The ring gear is

fixed. The input shaft is coupled to the carrier and the

output shaft is connected to the sun gear for power output.

Fig. 1. (a) Configuration of PGS; (b) its discrete analysis model

2.1 Spring for Stiffness of Helical Gear Pairs

The equivalent discrete springs of gear pairs are tangent to

the base circles accounting for the varying stiffness of

external and internal mating pairs in the PGS as shown in

Fig. 2. Using the derived involute profile equation, the

stiffness models in this study are obtained by more widely

incorporating gear design parameters including tooth

number, pressure angle, helical angle, instant contact

points, and nonstandard tool setting. ksi represents the

resulted equivalent stiffness of the ith contact tooth pair

per face width in the mating spur gears that includes

bending deformation of mating teeth, flexible foundation

of body [7], and Hertz contact deformation [8]. Basing on

the mesh stiffness ksi of spur gears, the mesh stiffness khi

of the ith helical tooth pair is expressed as

, , ,0

iL

h i si j i jk k dL (1)

where ,si jk is the stiffness of the ith tooth pair per face

width for the corresponding equivalent spur gear at instant

j, Li,j is the length of contact line of the ith tooth pair at the

jth instant. For a helical gear pair with a number tn of

tooth pairs in contact, its equivalent mesh stiffness is

,

1

tn

h h i

i

k k

(2)

Fig. 2. Stiffness springs of gear pairs: (a) external; (b) internal

2.2 Hand Sign of Gear Pairs and Spring Orientation

Essentially, the geometric and contact force relation of

helical gear pairs are much complicated comparing to the

spur pairs. The equivalent spring is defined tangent to the

base circles of meshing gear, but its orientation is

dependent on the driving/driven, helix hand sign, and

rotation direction. Figure 2 shows the orientation of the

equivalent spring and the helical gear pairs under a

condition. Figure 2(a) illustrates the spring for a driving

sun gear of right hand (RH) at counterclockwise (CCW)

rotation. Once that change rotation direction, orientation

angles and tangent points of the spring are changed. The

angle of ,s j between contact line and y axis is expressed

as

, ,

,

, ,

, torque on sun gear is CCW

( ), torque on sun gear is CW

s j s j

s j

s j s j

(3)

The orientation of the spring for internal pair is shown

in Fig. 2(b). Their angle of ,s j between contact line and

y axis is expressed as

, ,

,

, ,

torque on planet gear is CCW

( ) torque on planet gear is CW

r j r j

r j

r j r j

(4)

3 Gear Dynamic Equation

3.1 Lagrangian for PGS

The discrete dynamic model of a PGS is derived using the

Lagrange equation. In the gearing, three key action pairs

are sun-planet gear pair (SPj), planet-ring gear pair (RPj),

and planet-pin bearing pair (PCj). The additional

components have input and output shafts, bearings,

housing, and boundary conditions, the dynamic equation

of the PGS can be resulted. The Lagrange equation for

resulting dynamic equation is written as

ii

d L LQ

dt q q

(5)

where Lagrangian L T V in which T is total kinetic

energy, V total strain energy, iq generalized coordinates,

and iQ generalized forces. Thus, kinetic energy T is the

sum of kinetic energy of the components. For the PGS, it

is

1

pN

cs c r j s ss

j

T T T T T T T

(6)

in which , , , , ,cs c r j s ssT T T T T T are the componential

kinetic energies of the input shaft, carrier, ring gear,

planets , sun gear, and output shaft, respectively.

(a) (b)

rJgJ

r-p internal

s-p external

Then, V represents total strain energy from the

deformation parts in the PGS which is composed as

1

[ ( )]pN

cs cj rj sj ss

j

V V V V V V

(7)

where , , , ,cs cj rj sj ssV V V V V are the strain energies of the

components in the input shaft to carrier, planet-pin pairs,

ring-planet pairs, sun-planet pair, and output shaft,

respectively.

3.2 Rigid Body Motion and Elastic Deformation

The PGS dynamic model in this study can incorporate the

effect of rigid body motion. Here the rotational degree of

freedom corresponding to about the z axis is considered

since the other DOFs are stationary. Using the Lagrange

equation in Eq. (5), the equation of motion of the PGS

including rigid body motion effect is written as

1 1 1 1 1 1 1Μ X +C X + K X = F (8)

The subscript 1 denotes the equation used for a single

stage planetary gear system. M1, C1, K1, and F1 are

respectively the system mass, damping, stiffness, and

force terms. Also the displacement vector X1 of the gear

system is written as

1 1 2 ...

ps r n c R

X x x x x x x x (9)

Each elemental vector *x has six element representing its

three translation and there rotation DOFs as

* * *

T

* * * * x y zx y z x (10)

in which the subscript * denotes s, r, c, 1, 2,..,np, and c, R

for the corresponding parts. Then, the displacements in Eq.

(8) of the DOFs to the elastic deformation Xf and rigid

body motion XR are separately given as

0 0

0 0 0 0

ff fR f fff fff f

Rf RR R RR R

Μ Μ X FC KX X+ + =

Μ Μ X FX X (11)

where submatrices Mff and Kff are DOFs of deformation in

the PGS, MRR is accounting for the rigid DOFs. Also, MfR

and MRf is the coupling of elastic deformation and rigid

body motion which are given as

1 diag m m m m ... m m mpff ss s r p pN c cs

Μ (12)

T

Rf fRΜ Μ (13)

3.3 Excitations by Force and Gross Acceleration

Figure 3 is a time chart of y(t) representing the external

excitations of the gross motion acceleration or applied

loading. The speed-time chart is shown in Fig. 3.The PGS

is starting up at the time instant of t0 from the initial speed

of ω0. Then, it undergoes a constant acceleration or

deceleration and achieving the maximum speed of ω1 at

time t1. Next, a constant speed is maintained until to the

time of t2. After that, it enters to a constant deceleration to

the speed of ω2 at time t3. At the final period, the PGS

runs at ω2 until to the end of t4. In this analysis, the

starting time t0 =0. Time ratios αi are α1=t1/ t4, α2=(t3-t2)/t4,

α3=t3/t4, and α4=(t4-t3)/t4, respectively. Thus, as time ratio

1 0 , it is an extreme case of impact stepwise up.

Fig. 3. Dynamic excitation

How external excitations are implementing on PGS is

described below. Excitation vector F1 in Eq. (8) can

include the transmitted torque, inertial force from gross

motion, transmission error, and weight. It is specially

described how gross motion effect is included. Firstly, Eq.

(10) is rewritten as

ff f fR R ff f ff f f Μ X Μ X C X K X F (14)

Rf f RR R R Μ X Μ X F (15)

Case 1: For gross acceleration excitation,Ry X ,

corresponding to the gross motion given, dynamic elastic

displacement Xff can be obtained by directly using

numerical integration on Eq. (15) as

ff f ff f ff f f fR R Μ X C X K X F Μ X (16)

Case 2: For applied loading excitation, y=FR given,

corresponding to gross motion given, Eq. (15) can be

rewritten as

1( )R RR R Rf f

X Μ F Μ X (17)

Substituting Eq. (17) to Eq. (14) yields

1 1( )ff fR RR Rf f ff f ff f f fR RR R

Μ Μ Μ Μ X C X K X F Μ Μ F (18)

3.3 Backlashes and Gravity to Contact Forces

Figure 2 shows the contact relation, deformation, and

contact forces of an external contact pair between the sun

gear and the jth planet gear denoted as SPj. As the

expression in Eq. (10), displacement vectors of sun gear

and the jth planet gear are respectively written as

T

T

,s s s s xs ys zs

j j j j xj yj zj

x y z

x y z

x

x

(19)

The PGS in Fig. 1 is used in transmission line for speed-

up in wind turbines, the power is input from the main

shaft connecting to the carrier. Thus, the jth planet gear is

driver and the sun gear is driven gear. The helix sign and

rotation direction of a SPj contact pair is shown in Fig. 2.

The dynamic deformation of dsj is an equivalent spring of

the jth SP is given as

( ) ( sin sin cos cos )cos

( sin sin cos cos )sin

sj s sj j sj s sj j sj s zs j zj sj

s j s xs sj j xj sj s ys sj j yj sj sj

d t x x y y r r

z z r r r r

(20)

Here, the spring is compressed for positive sign of dsj and

stretched for minus sign.

Next, the time varying meshing stiffness is denoted as

ksj and the meshing backlash is bsj. There are 4 contact

conditions discussed as illustrated in Figs. 4(a)-4(d). Thus,

calculation of the dynamic contact forces is explained as:

(i) Regular contact (dsj >0): As shown in Fig. 4(a), the sun

gear drives the jth planet. In this kind of contact condition,

the meshing stiffness of sun-planet pair is modeled as

equivalent spring 1, ksj,1. The gear meshing follows the

action line a (blue). Thus, the dynamic meshing force is,

,1 ( )sj sj sjf k t d , ,1( )sj sjk k t (21)

(ii) No contact (dsj<0,sj sjd b ): As the gear pair has

backlash bsj, contact of the meshing pair may separate and

lose its contact as shown in Fig. 4(b). The equivalent

stiffness ksj=0. At the moment, no dynamic contact force

exists. Thus,

0sjf , 0sjk (22)

(iii) Back contact ( 0 , sj sj sjd d b ): As shown in Fig.

4(c), the tooth pair not only loses its regular contact but

also significantly rebounds such that the back

deformation is larger than the backlash bsj. Oppositely,

the sun gear drives the planet gear. Once that, the

backside of the planet tooth surface is driven by the

mating sun tooth. The equivalent meshing stiffness is the

spring 2, i.e., ksj=ksj,2. The meshing is along the action

line 2 (red). The meshing stiffness and direction of the

dynamic contact are different from that of the regular

contact. Then dynamic force of back contact condition is

( )( )sj sj sj sjf k t d b , ,2sj sjk k (23)

(iv) Double sides contact: Figure 4(d) shows the

condition of the double side contact which may occur

due to clearance, eccentricity, manufacturing errors, and

deformation. Once it appears, the wedge occurs. Both

the mesh condition incorporates both conditions of case

(i) and (ii), simultaneously. Thus, gear contact follows

action line 1 and 2 of regular and back contact

conditions. Two action lines have phase difference.

Dynamic force of two contacts are respectively given as

,1 ,1( )sj sj sjf k t d , ,2 ,2 ( )( )sj sj sj sjf k t d b (24)

(a) (b) (c)

(d)

(d) (e) (f)

Fig. 4. (a) Regular mesh; (b) mesh separation; (c ) back

contact; (d) action line and equivalent springs; (e) kinds of

mesh stiffness and mesh phase relation.

4 Results and Discussion

An example of PGS with three planets as the gearing type

shown in Fig. 1 is analyzed that is the first stage of a

speed-up gearbox used in 2.2MW wind turbines. The gear

data of the PGS are given as: teeth numbers of gears

zs=19, z

p=34, and z

r=89. The normal module mn = 16mm,

normal pressure angle αn=20o, helical angle β=7

o, face

width b=330mm. Its discrete analysis model was built and

shown in Fig. 1(b).

4.1 Transient Dynamic Response of PGS

The transient response without including gravity and

backlash effect is solved. A transmitted torque of 3×105

Nm is exerted on the input shaft coupling with the carrier.

The moment inertia Jr on the input shaft is 87500kg·m2

and which on the sun gear is Jg=1600kg·m2. The operation

speeds of this analysis are assigned as ω1=60rpm and ω0

=ω2=0rpm. The total operation time is t4 =1.5s. Besides,

α1=α2, α3=1. The six DOFs of translational and angular

displacements of the second planet gear are shown in Fig.

5. Then, Figure 6 depicts the dynamic meshing forces

among the meshing pairs (SPj, RPj, PCj; i=1, 2, 3).

Fig. 5. Dynamic displacements of the 2nd planet gear.

(a) SP pair (b) RP pair

(c) PC pair (d) max contact force Fig. 6. Dynamic contact forces in meshing pairs: (a) SPj; (b)

RPj; (c) PCj.

4.2 Steady State Response of the PGS

A dynamic steady state result can be faster arrived by a

suitable initial condition assignment. In this analysis, the

initial values are assigned using the static deformation

caused by the applied torque and component weight. The

attained results are obviously same as the steady state

responses through a transient analysis to the steady state,

but much more computing cost is needed. When the static

bsj bsj

response is used to be the initial values, the steady state

responses can be efficiently achieved in 2 second. The

dynamic responses of three different conditions to gravity

and backlashes are solved. Figures 7 to 9 depict the

dynamic meshing forces among the meshing pairs under

the 3 condition considerations by different incorporation

of gravity and backlash effect. Figure 7 shows the result

without the gravity and clearance consideration. Under

that, the meshing of gear pair and bearing are regular.

Always, the meshing pairs remain continuous contact.

Thus, the dynamic response in Fig. 7 performs well. Then,

the result in Fig. 8 is the condition that gravity effect is

included but not the gear backlashes. In this analysis

condition, the deformation caused by static force of

gravity is combined with deformation from the applying

loading. Caused by gross rotation of the planet carrier,

static gravity has dynamic effect on the gear and bearing

pairs. Thus with its rotation proceeds, the dynamic

response is sine wave characteristic. It is shown below

that the low frequency sine wave deformation may make

the backlash effect occur. Figure 9 is the case of both

considering the gravity and clearance. Under that, the

dynamic effect of the gravity and backlash combines.

Separation or back contact in the meshing mating pairs

may occur. Once the contact of action pairs cannot

maintain, their contact forces disappear. These meshing

pairs are free constraint. The relative displacement of

meshing pair is large. Thus, a very big dynamic meshing

force may appear once the meshing pairs return a regular

contact since a sudden large contact force can be induced.

(a) SP pair (b) RP pair

(c) PC pair (d) max contact force

Fig. 7. Dynamic forces without gravity and clearance effect

(a) SP pair (b) RP pair

(c) PC pair (d) max dynamic force

Fig. 8. Dynamic forces with gravity but without backlash effect

(a) SP pair (b) RP pair

(c) PC pair (d) max dynamic force

Fig. 9. Dynamic forces with gravity and backlash effect

4.3 Result Verification by a NREL’s PGS

For verification, dynamic contact forces using the

proposed method are compared with the experimental

results of another gearbox in a 0.75MW wind turbine

which is heavily studied by NREL [18]. The first stage of

the gearing is a PGS (21-39-99). Gear data are below:

module mn=10mm and helical angle of 7.5o. The dynamic

contact forces of SPj and RPj pairs are given in Fig. 10(a)

and 10(b) which shows both the amplitudes and change

tendency are close. The analysis result of this study is

verified. Noticeably, the amplitudes of PCj pair are about

double of the NREL’s. Actually, they are rational since

designs of planet bearing supports of two studies are

different. The support of bearing design for plant gears in

this study is cantilevered, but it is a simply beam in

NREL’s.

(a) SP pair (b) RP pair

(c) PC pair (d) max dynamic force

Fig. 10. Analysis contact forces of meshing pairs of another PGS

compare with the experimental results of NREL

4.5 Various Bearing Stiffness Including Gravity and

Backlash

By incorporating the gravity and gear backlashes in PGSs,

the effect of bearing stiffness on the dynamic behavior is

discussed. Figure 11 depicts the dynamic contact forces of

the gear and bearing meshing pairs (SPj, RPj, PCj; j=1, 2,

3) by assigning various base values of bearing stiffness

which are 1e10, 1e11, and 1e12, respectively. The result

shows the nonlinear dynamic characteristic arisen from

the gear backlashes combining with gravity can be

diminished by increasing the bearing stiffness. The effect

of gravity inducing non-uniformity of the dynamic contact

forces due to sine wave change among the meshing pairs

(SPj, RPj, PCj) to the individual planet gears can be

reduced. Thus, the phenomenon of meshing separation or

back contact due to rebound of the gear and bearing pairs

has a higher opportunity to be avoided. For explanation,

the contact characteristic of SPj pair is discussed below.

As the showing in Fig. 11(a), the maximum dynamic

contact force of SPj pair is 460kN for the base value of

bearing stiffness of 1x1010

. Then, it is decreased to 370kN

for the base value of 1x1011

in Fig. 11(b). It is even

decreased to 202kN for the value of 1x1012

. Under the

bigger value of 1x1012

, mesh separation or back contact in

mesh gear and bearing pairs caused by the combining

effect due to backlashes with the gravity become small.

The nonlinear phenomenon may not occur so as the form

of dynamic forces is similar to the result of case in Fig. 8

that only the gravity effect included. However, such large

a value of bearing stiffness of 1x1012

is almost impossible

to be realized in the gear design practice.

(a) Stiff=1e10 (b) Stiff=1e11

(c) Stiff=1e12 (d) max dynamic force

Fig. 11. Dynamic contact forces of different bearing stiffness

5 Conclusions

This study used a 3D discrete dynamic model of a helical

PGS in wind turbines under dynamic excitations

concerning the effect of change process of transmitted

loading and rotation speed. The dynamic contact forces

between gear pairs and planet bearings were resulted. The

discrete analysis model is formulated basing on the

equivalent springs for representation of varying meshing

stiffness of gear or bearing pairs. The physical models of

bearings, shafts, and housing frames were also

incorporated. After torque and rotation speed assigned,

dynamic transient and steady-state responses of the PGS

were analyzed. The dynamic contact forces between

meshing pairs of gears and bearings were resulted. The

analyzed contact forces of another PGS compared with the

experimental result by NREL. The proposed approach is

able to include models of bearing stiffness, weight, and

inertia of turbine rotor and generator. Accordingly,

dynamic responses of the planetary system are analyzed

under kinds of design considerations and operation

conditions. On the emphasis of gear backlashes and

gravity, the dynamic contact forces of planetary gearing

are discussed under different bearing stiffness assignments.

The result indicates that incorporation of gear mesh

backlashes with gravity effect is essential in the dynamic

study of the planetary gear system. The obtained and

further results are expected to be applied in wind turbines

concerning the dynamic excitation condition during the

design or operation phrases.

Acknowledgment

The authors would like to thank the financial support from

the National Science Council of Taiwan, R.O.C. under

grant MOST-102-2221-E-216-004.

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