2010 - Guo, Parker - Nonlinear Dynamics and Stability of Wind Turbine Planetary Gear Sets

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Dynamic modeling and analysis of a spurplanetary gear involving tooth wedging

ARTICLE in EUROPEAN JOURNAL OF MECHANICS - A/SOLIDS APRIL 2010

Impact Factor: 1.68 DOI: 10.1016/j.euromechsol.2010.05.001

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Yi Guo

National Renewable Energy Laboratory

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Robert G. Parker

Virginia Polytechnic Institute and State Uni

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Retrieved on: 01 February 2016
https://www.researchgate.net/profile/Yi_Guo14?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_4https://www.researchgate.net/profile/Yi_Guo14?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_5https://www.researchgate.net/?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_1https://www.researchgate.net/profile/Robert_Parker8?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_7https://www.researchgate.net/institution/Virginia_Polytechnic_Institute_and_State_University?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_6https://www.researchgate.net/profile/Robert_Parker8?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_5https://www.researchgate.net/profile/Robert_Parker8?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_4https://www.researchgate.net/profile/Yi_Guo14?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_7https://www.researchgate.net/institution/National_Renewable_Energy_Laboratory?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_6https://www.researchgate.net/profile/Yi_Guo14?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_5https://www.researchgate.net/profile/Yi_Guo14?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_4https://www.researchgate.net/?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_1https://www.researchgate.net/publication/228422578_Dynamic_modeling_and_analysis_of_a_spur_planetary_gear_involving_tooth_wedging?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_3https://www.researchgate.net/publication/228422578_Dynamic_modeling_and_analysis_of_a_spur_planetary_gear_involving_tooth_wedging?enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4&el=1_x_2


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Nonlinear dynamics and stability of wind turbine planetary gear sets

under gravity effects

Yi Guo a,*, Jonathan Keller a, Robert G. Parker b

a National Wind Technology Center, National Renewable Energy Laboratory, Mail Stop: 3811, 15013 Denver West Parkway, Golden, CO 80401-3305, USAb Department of Mechanical Engineering, Virgina Tech, USA

a r t i c l e i n f o

Article history:

Received 23 August 2012

Accepted 20 February 2014

Available online 12 March 2014

Keywords:

Dynamics

Wind turbine

Planetary gear

a b s t r a c t

This paper investigates the dynamics of wind turbine planetary gear sets under the effect of gravity using

a modied harmonic balance method that includes simultaneous excitations. This modied method

along with arc-length continuation and Floquet theory is applied to a lumped-parameter planetary gear

model including gravity, uctuating mesh stiffness, bearing clearance, and nonlinear tooth contact to

obtain the dynamic response of the system. The calculated dynamic responses compare well with time

domain-integrated mathematical models and experimental results. Gravity is a fundamental vibration

source in wind turbine planetary gear sets and plays an important role in the system dynamic response

compared to excitations from tooth meshing alone. Gravity causes nonlinear effects induced by tooth

wedging and bearing-raceway contacts. Tooth wedging, also known as a tight mesh, occurs when a gear

tooth comes into contact on the drive-side and back-side simultaneously and it is a source of planet-

bearing failures. Clearance in carrier bearings decreases bearing stiffness and signicantly reduces the

lowest resonant frequencies of the translational modes. Gear tooth wedging can be prevented if the

carrier-bearing clearance is less than the tooth backlash.

2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

The National Renewable Energy Laboratory (NREL) Gearbox

Reliability Collaborative (GRC) was established by the U.S. Depart-

mentof Energy in 2006. Its key goal is tounderstand the rootcauses

of premature gearbox failures (Musial et al., 2007) through a

combined approach of dynamometer testing, eld testing, and

modeling (Link et al., 2011), resulting in improved wind turbine

gearbox reliability and a reduction in the cost of energy. As a part of

the GRC program, this paper investigates gravity-induced dynamic

behaviors of planetary gear sets in wind turbine drivetrains that

could reduce gearbox life. Planetary gear sets have been used inwind turbines for decades because of their compact design and

high efciency. Despite these advantages, planetary gear sets

generate considerable noise and vibration. Vibration causing high

dynamic loads may result in gear tooth and bearing failures (Musial

et al., 2007). Fatigue failures are a concern in long life-cycle appli-

cations. Analyzing the dynamics of wind turbine planetary gear

drivetrains is important in improving gearbox life and reducing

noise and vibration.

The majority of wind turbines use a horizontal-axis congura-

tion; thus, gravity becomes a periodic excitation source in the

rotating carrier frame. Prior study of gravity by the authors was

performed with a static analysis and focused on the effect of gravity

upon bearing force and tooth wedging in a planetary spur gear set

(Guo and Parker, 2010). It was found that tooth wedging, an

abnormal contact situation where the tooth is in contact with both

the drive-side and back-side anks simultaneously, was caused by

gravity. Tooth wedging increases planet-bearing forces and disturbs

load sharing among the planets, which could lead to prematurebearing failure.

Signicant in-plane translational gear component motions in

planetary systems lead to tooth wedging. It is the combined effect

of gravity and bearing clearance nonlinearity. Bearing clearance

results in greater translational vibration, while gravity is the

dominant excitation source causing the large motions that lead to

tooth wedging. For heavy planetary gear sets, tooth wedging is

likely to occur. Tooth wedging in planetary gear sets leads to un-

equal load sharing and excessive planet-bearing loads by disturbing

the symmetry of the planet gears. This may cause bearing failure

and tooth damage (Guo and Parker, 2010).* Corresponding author. Tel.: 1 303 384 7187; fax: 1 303 384 6901.

E-mail addresses: [email protected],[email protected](Y. Guo).

Contents lists available atScienceDirect

European Journal of Mechanics A/Solids

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . co m / l o c a t e / e j m s o l

http://dx.doi.org/10.1016/j.euromechsol.2014.02.013

0997-7538/

2014 Elsevier Masson SAS. All rights reserved.

European Journal of Mechanics A/Solids 47 (2014) 45 e57
http://-/?-http://-/?-http://-/?-http://-/?-https://www.researchgate.net/publication/237293035_Improving_Wind_Turbine_Gearbox_Reliability?el=1_x_8&enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4https://www.researchgate.net/publication/237293035_Improving_Wind_Turbine_Gearbox_Reliability?el=1_x_8&enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4https://www.researchgate.net/publication/237293035_Improving_Wind_Turbine_Gearbox_Reliability?el=1_x_8&enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-mailto:[email protected]:[email protected]://www.sciencedirect.com/science/journal/09977538http://www.elsevier.com/locate/ejmsolhttp://dx.doi.org/10.1016/j.euromechsol.2014.02.013http://dx.doi.org/10.1016/j.euromechsol.2014.02.013http://dx.doi.org/10.1016/j.euromechsol.2014.02.013https://www.researchgate.net/publication/237293035_Improving_Wind_Turbine_Gearbox_Reliability?el=1_x_8&enrichId=rgreq-aeb06dd6-64e4-4cea-a719-a91c4c826871&enrichSource=Y292ZXJQYWdlOzIyODQyMjU3ODtBUzo5ODkxNDg3OTQxMDE4M0AxNDAwNTk0NTU4MTA4http://dx.doi.org/10.1016/j.euromechsol.2014.02.013http://dx.doi.org/10.1016/j.euromechsol.2014.02.013http://dx.doi.org/10.1016/j.euromechsol.2014.02.013http://www.elsevier.com/locate/ejmsolhttp://www.sciencedirect.com/science/journal/09977538http://crossmark.crossref.org/dialog/?doi=10.1016/j.euromechsol.2014.02.013&domain=pdfmailto:[email protected]:[email protected]://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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Research on tooth separation is well-established for automotive

and helicopter applications. Tooth separation was observed in spur

gear pair experiments (Blankenship and Kahraman, 1996).Botman

(1976)experimentally observed tooth separation in planetary gear

sets. Using nite element and lumped-parameter models,

Ambarisha and Parker (2007)predicted tooth separation and other

nonlinear phenomena in a planetary gear set in a helicopter

gearbox. Velex and Flamand (1996) investigated tooth separation at

critical speeds. Bahk and Parker (2011) derived closed-form solu-

tions for the dynamic response of planetary gear sets with tooth

separation based on a purely torsional model.

Nonlinear dynamics induced by bearing clearance has been

studied for relatively small geared systems.Kahraman and Singh

(1991)observed chaos in the dynamic response of a geared rotor-

bearing system with bearing clearance and backlash. Gurkan and

Ozguven (2007)studied the effects of backlash and bearing clear-

ance in a geared exible rotor and the interactions between these

two nonlinearities. Guo and Parker (2012a) investigated the

nonlinear effects and instability caused by bearing clearance in

helicopter planetary gear sets. Dynamic effects of bearing clearance

in wind turbine planetary gear sets have not been studied in the

past because the wind turbine operating speed was believed to be

well below the frequency range of drivetrain dynamics. However,bearing clearance reduces some gearbox resonances signicantly.

The nite element program developed byVijayakar (1991)uses

a combined surface integral and nite element approach to capture

tooth deformation and contact loads in geared systems. This nite

element model includes bearing clearance, tooth separation, tooth

wedging, uctuating mesh stiffness, and gravity. Numerical inte-

gration is widely adopted to compute dynamic responses of me-

chanical systems in the time domain.Ambarisha and Parker (2007)

used numerical integration to study nonlinear dynamics and the

impacts of mesh phasing on vibration reduction of planetary gear

sets. Velex and Flamand used numerical integration results of a

planetarygear setwith time-varying mesh stiffness as a benchmark

to evaluate results from a Ritz method.

The harmonic balance method (Thomsen, 2003) calculatesnonlinear, frequency domain, steady-state response of mechanical

systems.Zhu and Parker (2005)used this method to study clutch

engagement loss in a belt-pulley system.Al-shyyab and Kahraman

(2005a,b) investigated primary resonances, subharmonic reso-

nances, and chaos in a multimesh gear train caused byuctuating

gear mesh stiffness.Bahk and Parker (2011) employed harmonic

balance to analyze planetary gear dynamics based on a purely

rotational model. Use of the harmonic balance method reduces

computational time for lightly damped or physically unstable

systems by avoiding the long transient decay time before a steady

state is reached. Compared to numerical integration and nite

element analysis, which are widely adopted approaches to

compute dynamic responses, the computation time of the har-

monic balance method is onee

two orders of magnitude lower.Harmonic balance often employs arc-length continuation (Nayfeh

and Balachandran, 1995) and Floquet theory (Raghothama and

Narayanan, 1999; Seydel, 1994) to calculate nonlinear reso-

nances in the dynamic response, including unstable solutions that

numerical integration and nite element analysis are unable to

obtain. The established harmonic balance formulation is only

suitable for systems with one fundamental excitation frequency

and its higher harmonics. However, wind turbine drivetrains have

simultaneous internal and external excitations, including uctu-

ating mesh stiffness, gravity, bending-moment-induced excita-

tions in the rotating carrier frame, wind shear, tower shadow, and

other aero-induced excitations.

The major objectives of this study are to: 1) develop a modied

harmonic balance method to obtain the dynamic response of wind

turbine planetary gear sets considering gravity, uctuating mesh

stiffness, bearing clearance, and nonlinear tooth contact; 2) validate

the proposed method by comparing the calculated results against

experimental data and a numerical integration approach; and 3)

investigate the gravity-induced dynamic behaviors using the

developed approach, which includes tooth wedging, tooth contact

loss, and bearing-raceway contacts.

2. Gearbox description

This study investigates both a 750-kW wind turbine planetary

gear (PG-A) used by the GRC (Link et al., 2011) and a 550-kW wind

turbine planetary gear (PG-B) (Guo and Parker, 2010;Larsen et al.,

2003; Rasmussen et al., 2004). These drivetrains have a main

bearing that supports the main shaft and rotor weight, and two

trunnion mounts that support the gearbox. These two gearboxes

have similar congurations and are representative of the majority

of three-point-mounted wind turbine drivetrains.

PG-A is congured in a helical planetary gear arrangement with

two parallel stages as shown inFig. 1.Within the gearbox, there are

two cylindrical roller bearings supporting the carrier, each with

275 mm of clearance. The planetary gear set is arranged in an in-

phase bridged carrier design with three equally spaced planets.

The sunpinion shaft is connected to the intermediate stage through

a spline joint that partially oats the sun pinion. The ring gear is

bolted to the front and rear of the gearbox housing. The rated tor-

que is 322,610 Nm and the rated speed of the input shaft is 22.2 rpm

(Guo et al., 2012).

PG-B is congured in a spur planetary gear arrangement with

two parallel stages. Like PG-A, PG-B has three equally spaced

planets. The rated torque is 180,000 Nm and the rated speed is

30 rpm. Additional key parameters of these two gearboxes are

listed inTables 1and 2. A large ring gear mass in these designs

is a result of a typical wind turbine gearbox arrangement

whereby much of the gearbox housing is rigidly connected to

the ring.

3. Mathematical models

3.1. Lumped-parameter model for planetary gear sets

A previously developed and validated lumped-parameter model

was adopted for this paper (Guo and Parker, 2010, 2012a). As

depicted in Fig. 2(a), the carrier, ring, sun, and planets are rigid

bodies, each having two translational and one rotational degree of

freedom. The carrier rotating frame is used as the general co-

ordinates for all of the components of planetary gear sets. This two-

dimensional model has 3(N 3) degrees of freedom, where Nis the

number of planets. The model includes gravity, uctuating mesh

stiffness, bearing clearance, tooth contact loss, and tooth wedging.

Fig. 1. The GRC gearbox (PG-A) con

guration.

Y. Guo et al. / European Journal of Mechanics A/Solids 47 (2014) 45e5746
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Bearings are modeled using circumferentially distributed radial

springs with a uniform clearance. This study focuses on the carrier

bearing with clearance as shown in Fig. 2(b), which has dynamic

effects on low-speed resonances (Guo and Parker, 2012a).

The coordinates are shown in Fig. 2(a). Throughout this paper,

the subscriptsc,r,s,p denote the carrier, ring, sun, and planet; the

subscripts B, g, m denote the bearing, gravity, and gear mesh; andsuperscripts b, d denote drive-side and back-side tooth contact.

Translational displacements in the x and y directions,xw,yw,w c,

r, s, are assigned to the carrier, ring, and sun, respectively, with

regard to the rotating carrier frame. The originO is at the center of

the planetary gear set. The radial and tangential displacements of

thejth planet are denoted by xj,hj,j 1, .,Nwith respect to the

carrier and oriented for each planet as shown in Fig. 2(a). The

rotational displacements areuvrvqv,vc,r,s, 1,.,N, where qvis

the rotation in radians and rv is the base circle radius for the sun,

ring, and planets and the radius to the planet center for the carrier.

The masses and moments of inertias of the carrier, ring, sun, and

planets are denoted by mk,Ik, k c,r,s,p. Quantitieskwx,kwy,kwudenote the bearing stiffnesses of the carrier, ring, and sun supports

inx,y, andudirections. The torsional stiffnesses of the carrier, ring,and sun supports equal kwur

2w, wherekwu is the torsional stiffness

with units of force/length and rw,w c,r,s is the base radius. The

jth planet-bearing stiffness is kpj. The mesh stiffnesses at the jth

sun-planet and ring-planet mesh areksj,krj. The radial stiffnesses of

the bearing that connects the carrier to ring gear in the x and y

directions arekcrx, kcry. The nondimensionalized equations of mo-

tion of planetary gear sets are

~Mz00 ~Cz0 ~fd

ms; z ~f

b

ms; z ~fBs; z

~Fs

z x

L; ~M

M

M; ~C

C

MLu; ~f

d

m fdmMLu2

; ~fb

m fbmMLu2

;

~fB fBMLu2

; ~F FMLu2

(1)

where s utand z x/L, whereu is characteristic frequency,L is

the characteristic length, and Mis the characteristic mass. Deriva-

tions of the mass matrix M and the displacement vector x are

detailed in Guo and Parker (2010). C U1Tdiag2znUnU1

where znare the damping ratios and Unare the natural frequencies

of the linear system where all bearings are in contact and the mesh

stiffnesses are averaged over a mesh cycle; U is the ortho-

normalized modal matrix (UTMUI). The nonlinear forces fall into

three categories: the drive- and back-side tooth mesh force vectors

fdmand fbm, and bearing force vector fB(Guo and Parker, 2010). The

external force F(t) Fs Fg(t). Fs includes the static torques applied

to each component. Fg(t) is the gravity force vector. Gravitational

force acting on the carrier, ring, sun, and planets is periodic in the

rotating carrier frame, resulting in a fundamental external excita-

tion source.

Fgt hfxcg;f

ycg;0;f

xrg;f

yrg;0;f

xsg;f

ysg;0;f

x1g

;fh1g

;0;.;fxNg;fh

Ng;0iT

(2)

Quantities fx

wg

;fywgw c; r; sandf

x

jg

;fh

jg

j 1; .; N denote the

gravity force acting on the carrier, ring, sun, and planets 1eN. These

are

fxwg mwgsinUct

fywg mwgcosUct; w c; r; s (3)

fxjg

mjgsinUct jj

fh

jg mjgcos

Uct jj

; j 1; .; N

(4)

where the variable Uc um/Nr (if the ring is xed) denotes the

carrier rotation frequency. um denotes the mesh frequency. Nrde-

notes the number of teeth on the ring.The mesh stiffness uctuates as the number of teeth in

contact changes and is also an important internal excitation

source of geared systems. These excitations are included through

time-varying mesh stiffnesses. The mesh stiffnesses are calcu-

lated using the Calyx program (Nayfeh and Balachandran, 1995).

Its mean amplitudes are listed inTables 1and 2 (Appendix). This

program analyzes gear tooth contact and rolling element contact

by using a combined analytical/nite element analysis detailed

in Vijayakar (1991). Its results have been compared against

studies of gear dynamics (Singh, 2010, 2011; Guo and Parker,

2012b).

3.2. Finite element model

The two-dimensional nite element model includes 36 quad-

rilateral elements per tooth, four nodes per element, and each node

has three degrees of freedom. It uses a combined surface integral

and nite element method to capture tooth deformation and con-

tact loads in geared systems (Kahraman and Blankenship, 1994).

The software developed byVijayakar (1991)intrinsically evaluates

time-varying tooth contact forces that are specied externally with

conventional simulation tools. Fluctuating mesh stiffness over a

mesh cycle, tooth wedging, and tooth separation are included

intrinsically in the nite element model. The nite element model

also includes clearance nonlinearity at the carrier-ring bearing.

Other bearings are modeled as linear stiffnesses without clearance

in the nite element and analytical models. The nite element

approach has been validated by experiments of geared systems

(Parker et al., 2000a,b;Kahraman and Vijayakar, 2001).It is used to

benchmark the established lumped-parameter model when

experimental data is unavailable.

4. Extended harmonic balance method

The extended harmonic balance method is used to obtain the

dynamic responses of the model in Eq. (1). The formulation in-

cludes two excitation sources with excitation frequencies U1 and

U2. The coupling effects between these two excitations are

considered by including their side bands. Other excitation sources

can be considered in a similar way. The response z is expanded

into a Fourier series and assumed to include theR1,R2,R3R4R5, and

R6R7R8 harmonics of excitation frequencies U1 and U2 and their

Table 1

System parameters for the 750-kW Gearbox Reliability Collaborative (GRC) plane-

tarygearbox (PG-A). The ringmass includes the gearboxhousing and parallel stages.

Sun Ring Carrier Planet

Mass (kg) 181.6 2633 759.9 104

Moment of Inertia (kg-m2) 3.2 144.2 59.1 3.2

Number of Teeth 21 99 e 39

Pitch Diameter (mm) 215.6 1016.4 e 400.4

Root Diameter (mm) 186.0 1047.7 e

372.9Average Mesh Stiffness (N/m) ksp 16.9 10

9,krp 19.2 109

Bearing Stiffness (N/m) 100 102 106 5 109 6.8 109

Carrier-Bearing Stiffness (N/m) 3.2 109

Carrier-Bearing Clearance (mm) 0.275

Torsional Support Stiffness (N/m) 45.8 106 57.4 106 0 0

Y. Guo et al. / European Journal of Mechanics A/Solids 47 (2014) 45e57 47
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side bandsmU1lU2and pU1qU2. Each componentzhin z then

has a total of 2(R1 R2 R3R4R5 R6R7R8) 1 terms and is

expressed as

zh zh;1 XR1

i 1

zh;2icos iU1s zh;2i1sin iU1sXR2i 1

hzh;2iL1cosjU2s zh;2iL11 sin iU2s

i

XR5i 1

XR4m 1

XR3l 1

nzh;2ai;m;lcosimU1lU2s

zh;2ai;m;l1 sin imU1lU2so

XR8i 1

XR7p 1

XR6q 1

nzh;2bi;p;qcosipU1qU2s

zh;2bi;p;q1sin ipU1qU2so

(5)

where a(i, m, l) l L2 (m 1)R3 (i 1)R3R4 and b(i, p,

q) q L3 (p 1)R6 (i 1)R6R7. L1 R1, L2 R1 R2,

L3 R1 R2 R3R4R5,L4 2(R1 R2 R3R4R5 R6R7R8) 1.

The time domain is discretized into n 1 evenly distributed

time intervals [s1, ., sn]. Each component zh in the response z is

extended into a vector in the time domain aszhs1; .; zhsnT.

The response vector transforms intoz Gzby dening a function

G that maps the response from the frequency domain to the time

domain. Additional terms G1eG4are dened inAppendix A.

G

2664

n

G1 G2 PR4

m1

PR3l1

G3m;l;R5PR7

p1

PR6q1

G4p;q;R8

n

3775

(7)

The response derivatives are then transformed into

z00 GU21A1z GU22A2z G

PR4m 1

PR3l 1

mU1lU22A3z

GXR7

p 1

XR6q 1

pU1qU22A4z

z0

GU

1B

1z

GU

2B

2z

G X

R4

m 1X

R3

l 1mU

1lU

2B

3z

GXR7

p 1

XR6q 1

pU1qU2B4z (8)

where the operatorsAand B are also dened inAppendix A.

The nonlinear force vectors, detailed inGuo and Parker (2012a),

are transformed as

~fd

m Gfd

m; ~f

b

m Gfb

m; ~fB GfB; ~F GF (9)

By substituting Eqs.(8) and (9)into the equations of motion, Eq.

(1)yields

Table 2

System parameters for the 550-kW planetary gearbox (PG-B). The ring mass in-

cludes the gearbox housing and parallel stages.

Sun Ring Carrier Planet

Mass (kg) 51 4000 1330 114

Moment of Inertia (kg-m2) 61.1 2484 314.7 51.9

Number of Teeth 16 68 e 26

Pitch Diameter (mm) 224 952 e 364

Root Diameter (mm) 202 980 e

329Average Mesh Stiffness (N/m) ksp 3.95 10

9,krp 5.29 109

Bearing Stiffne ss (N/m) 100 e 4 109 5.3 109

Gearbox Trunnion Stiffness (N/m) 126 106

Carrier-Bearing Stiffness (N/m) 3.6 109

Carrier-Bearing Clearance (mm) 1

Torsional Support Stiffness (N/m) 3 106 24.4 106 0 0

z

266664

z1;1; .; z1;2L11|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}xcU1

;.; z1;2L21|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}xcU2

;.; z1;2L31; .; z1;L4|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}xcU1;U2

;.;

z3N3;1; .; z3N3;2L11|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}uNU1

;.; z3N3;2L21|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}uNU2

;.; z3N3;2L31; .; z3N3;L4|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}uNU1;U2

377775

T

(6)

Y. Guo et al. / European Journal of Mechanics A/Solids 47 (2014) 45e5748
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G

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Documents

2010 - Guo, Parker - Nonlinear Dynamics and Stability of Wind Turbine Planetary Gear Sets