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7/24/2019 2010 - Guo, Parker - Nonlinear Dynamics and Stability of Wind Turbine Planetary Gear Sets
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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/228422578
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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2 AUTHORS:
Yi Guo
National Renewable Energy Laboratory
21PUBLICATIONS 138CITATIONS
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Robert G. Parker
Virginia Polytechnic Institute and State Uni
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Available from: Robert G. Parker
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_27/24/2019 2010 - Guo, Parker - Nonlinear Dynamics and Stability of Wind Turbine Planetary Gear Sets
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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://-/?-7/24/2019 2010 - Guo, Parker - Nonlinear Dynamics and Stability of Wind Turbine Planetary Gear Sets
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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
8