research article dynamic reliability analysis of...
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Research ArticleDynamic Reliability Analysis of Gear TransmissionSystem of Wind Turbine in Consideration of Randomness ofLoadings and Parameters
Lei Wang1 Tao Shen1 Chen Chen1 and Huitao Chen12
1 School of Automation Chongqing University Chongqing 400044 China2 School of Mechanical and Power Engineering Henan Polytechnic University Jiaozuo Henan 454150 China
Correspondence should be addressed to Lei Wang leiwang08cqueducn
Received 30 December 2013 Accepted 19 January 2014 Published 6 March 2014
Academic Editor Weichao Sun
Copyright copy 2014 Lei Wang et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A dynamic model of gear transmission system of wind turbine is built with consideration of randomness of loads and parametersThe dynamic response of the system is obtained using the theory of random sampling and the Runge-Kutta method According torain flow counting principle the dynamic meshing forces are converted into a series of luffing fatigue load spectra The amplitudeand frequency of the equivalent stress are obtained using equivalent method of Geber quadratic curve Moreover the dynamicreliability model of components and system is built according to the theory of probability of cumulative fatigue damageThe systemreliability with the random variation of parameters is calculated and the influence of random parameters on dynamic reliability ofcomponents is analyzed In the end the results of the proposed method are compared with that of Monte Carlo methodThis papercan be instrumental in the design of wind turbine gear transmission system with more advantageous dynamic reliability
1 Introduction
Wind turbine generators usually work in a severe environ-ment and suffer from the impact of random wind withvarying directions and varying loads as well as the stronggust year after year As a vital part of the transmissionsystem of a wind turbine generator the gear transmissionsystem needs to withstand random dynamic loads andmuch higher fatigue cycles than any other transmissionsystems thus making it possess the highest failure rate[1] However results of the general design and evaluationmethod of the gear transmission system inwhich the randomwind load is processed roughly as static load using statisticmethod are not satisfied in solving the high failure rateproblem of gear transmission system which is a fundamentalfact to restrict the life span of the whole wind turbinegenerator
Many scholars worldwide have done many deepresearches on the random vibration and dynamic reliabilityof random construction caused by random excitations [2ndash7]However their researches are relatively simple in choosing
research objects which can hardly conduct the design ofdynamic reliability of gear transmission system Recently thedynamic issue of gear transmission system of wind turbinegenerator attracts more and more attention
Peeters [8] and his fellows built the flexible multibodydynamics model of a wind turbine transmission systemby applying multibody dynamics software and studied thenatural frequency and vibrationmode of the system Caichaoet al [9] built the nonlinear dynamics model of wind turbinegearbox and analyzed the dynamic characteristic Qin etal [10 11] studied the dynamic characteristic of the windturbine transmission system with the dynamic torque inputcaused by simulated natural wind data However thesestudies did not consider the randomness of external loadsthe uncertainty of gear transmission system material andthe geometric parameters Nor did they analyze the dynamicreliability of the system In actual wind farm due to the fiercework environment and the uncertainty during the processingand assembly of the gears the external excitations and theparameters of the gear transmission system are all random
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 261767 10 pageshttpdxdoiorg1011552014261767
2 Mathematical Problems in Engineering
So it is of great practical significance to develop a method toanalyze the dynamic characteristics and the reliability underrandom wind conditions
In this paper we studied the gear transmission system of a15MWwind turbine Elasticmodulusmass density workingtooth width pitch circle diameter and comprehensive trans-mission error were taken as random variables The dynamicmeshing force of gears was obtained by using the theoryof random sampling and the Runge-Kutta method takinginto consideration the influence of external random load Onthis basis statistical processing of dynamic meshing forcewas done using rain flow counting principle and equivalentmethod of Geber quadratic curve Dynamic reliability ofcomponents and the whole system were calculated accordingto the theory of probability of cumulative fatigue damage Inthe end the variation of the system dynamic reliability overtime under variable parameters was studiedThe effect of thisvariation to the system dynamic reliability was analyzed andthe results were comparedwith those ofMonteCarlomethod
2 Dynamic Model of GearTransmission System
21 Dynamic Model of Gear Transmission System This paperstudies the gear transmission system of a 15MW wind tur-bine generator which contains one level of NGW planetarygear and two levels of parallel shaft gear The structurediagram is shown in Figure 1
Torsional vibration model of gear transmission systemis built using centralized parameter method as is shownin Figure 2 Variation of meshing stiffness comprehensivetransmission error and other factors are taken into consid-eration in this model The planetary gears are assumed tobe uniformly distributed and have the same physical andgeometrical parameters
In Figure 2 119906119888 119906119904 119906119901119894 119906119895(119894 = 1 2 3 119895 = 1 2 3 4)
represent the torsion displacement of planet carrier sun gearplanetary gears and medium and high speed level gearsrespectively 119896
119904119901119894 119896119903119901119894
represent the meshing stiffness of sungear and planetary gear 119894 and themeshing stiffness of annulargear and planetary gear 119894 respectively 119896
1199041represents the
torsional stiffness of the connecting shaft between the sungear and gear 1 119896
23represents the torsional stiffness of the
connecting shaft between gear 2 and gear 3 11989612 11989634represent
themeshing stiffness of themedium speed gears and the highspeed gears 119888
119904119901119894 119888119903119901119894
represent the meshing damping of sungear annular gear and planetary gear 119894 119888
1199041represents the
torsional damper of connecting shaft between the sun gearand gear 1 119888
23represents the torsional damper of connecting
shaft between gear 2 and gear 3 11988812 11988834represent the meshing
damping of the medium speed level gears and the high speedlevel gears 119890
119904119901119894 119890119903119901119894
represent the transmission error of sungear annular gear and planetary gear 119894 119890
12 11989034represent the
comprehensive transmission error of the medium speed andhigh speed level gears
In the gear transmission system the meshing of the gearpair and the gear meshing force and meshing displacementare all happening in the direction of the meshing line In
1
2
3
4
s
r
c
Tin
Tout
ks1
k23
pi
Figure 1 Schematic of gear transmission system of wind turbine 119901planetary gear 119903 internal gear 119888 planet carrier 119904 sun gear 1 smallgear at medium-level speed 2 large gear at medium-level speed 3small gear at high-level speed 4 large gear at high-level speed 119879ininput torque and 119879out output torque
order to simplify the following analysis and calculation onthe torsional vibration system we replace the generalizedcoordinates in form of gear angular displacement with theones in form of line displacement along the meshing line Setthe rotation angular displacement of each gear is 120579
119894 respec-
tively (119894 = 119888 119901119894 119904 1 2 3 4) According to the newly definedgeneralized coordinates the rotation freedom of sun gear isconverted to microdisplacement 119906
119904 119906119904
= 119903119904120579119904(119903119904is the radius
of the base circle of the sun gear) which is in the directionof the planetary gear meshing line Similarly the rotationfreedom of planetary gear is converted to microdisplacement119906119888
= 119903119888120579119888 also in the direction of meshing line and so on
The analysis of elastic deformation of each meshing forceis as follows
The elastic deformation in the direction of meshing forcebetween the 119894th planetary gear and the sun gear is
120575119904119901119894
= 119903119887119888cos120572119904119901
120579119888
minus 119903119887119901119894
120579119901119894
minus 119903119887119904
120579119904
minus 119890119904119901119894
= 119906119888cos120572119904119901
minus 119906119901119894
minus 119906119904
minus 119890119904119901119894
(1)
Its first derivative is
120575119904119901119894
= 119903119887119888cos120572119904119901
120579119888
minus 119903119887119901119894
120579119901119894
minus 119903119887119904
120579119904
minus 119890119904119901119894
= 119888cos120572119904119901
minus 119901119894
minus 119904
minus 119890119904119901119894
(2)
The elastic deformation in the direction of meshing forcebetween the 119894th planetary gear and the internal ring gear is
120575119903119901119894
= 119903119887119901119894
120579119901119894
minus 119903119887119903
120579119903
+ 119903119887119888cos120572119903119901
120579119888
minus 119890119903119901119894
= 119906119901119894
minus 119906119903
+ 119906119888cos120572119903119901
minus 119890119903119901119894
(3)
Its first derivative is
120575119903119901119894
= 119903119887119901119894
120579119901119894
minus 119903119887119903
120579119903
+ 119903119887119888cos120572119903119901
120579119888
minus 119890119903119901119894
= 119901119894
minus 119903
+ 119888cos120572119903119901
minus 119890119903119901119894
(4)
The elastic deformation in the direction of meshing forcebetween the spur gear 1 and the spur gear 2 is
12057512
= 1199031198871
1205791
+ 1199031198872
1205792
minus 11989012
= 1199061
+ 1199062
minus 11989012
(5)
Mathematical Problems in Engineering 3
X
Y
Z
c
r
k34
u4
e34c34
k12
k23
u3
u1
u2
us
ucup1
e12c12
c23
1
2
3
4
Xc
Yccrp2
erp2
crp3
krp2
krp3
up2
up3Tin
esp2
erp1crp1
csp2
csp1
esp1
csp3
ksp2esp3
erp3ksp3
ksp1
krp1
cs1
ks1
Figure 2 Torsional vibration model of gear transmission system
Its first derivative is
12057512
= 1199031198871
1205791
+ 1199031198872
1205792
minus 11989012
= 1
+ 2
minus 11989012
(6)
The elastic deformation in the direction of meshing forcebetween the spur gear 3 and the spur gear 4 is
12057534
= 1199031198873
1205793
+ 1199031198874
1205794
minus 11989034
= 1199063
+ 1199064
minus 11989034
(7)
Its first derivative is
12057534
= 1199031198873
1205793
+ 1199031198874
1205794
minus 11989034
= 3
+ 4
minus 11989034
(8)
So in the gear transmission system the relative displace-ments in the direction of meshing line of all gear pairs are
120575119904119901119894
= 119906119888cos120572119904119901
minus 119906119901119894
minus 119906119904
minus 119890119904119901119894
120575119903119901119894
= 119906119901119894
+ 119906119888cos120572119903119901
minus 119890119903119901119894
12057511989912
= 1199061
+ 1199062
minus 11989012
12057511989934
= 1199063
+ 1199064
minus 11989034
(9)
Equation (10) is the vibration differential equations of thesystem based on Lagrange equation Consider
(
119868119888
1199032
119887119888
) 119888
+
3
sum
119894=1
119888119903119901119894
cos120572119903119901119894
120575119903119901119894
+
3
sum
119894=1
119888119904119901119894
cos120572119904119901119894
120575119904119901119894
+ 119888119906119888
119888
+
3
sum
119894=1
119896119903119901119894
(119905) cos120572119903119901119894
120575119903119901119894
+
3
sum
119894=1
119896119904119901119894
(119905) cos120572119904119901119894
120575119904119901119894
+ 119896119906119888
119906119888
=
119879in119903119887119888
(
119868119901119894
1199032
119887119901119894
) 119901119894
minus 119888119904119901119894
120575119904119901119894
+ 119888119903119901119894
120575119903119901119894
minus 119896119904119901119894
(119905) 120575119904119901119894
+ 119896119903119901119894
(119905) 120575119903119901119894
= 0
(
119868119904
1199032
119887119904
) 119904
minus
3
sum
119894=1
119888119904119901119894
120575119904119901119894
+ 1198881199041
(
119904
1199032
119887119904
minus
1
119903119887119904
1199031198871
)
minus
3
sum
119894=1
119896119904119901119894
(119905) 120575119904119901119894
+ 1198961199041
(
119906119904
1199032
119887119904
minus
1199061
119903119887119904
1199031198871
) =
119879119904
119903119887119904
(
1198681
1199032
1198871
) 1
+ 11988812
12057511989912
+ 1198881199041
(
1
1199032
1198871
minus
119904
119903119887119904
1199031198871
)
+ 11989612
(119905) 12057511989912
+ 1198961199041
(
1199061
1199032
1198871
minus
119906119904
119903119887119904
1199031198871
) =
1198791
1199031198871
(
1198682
1199032
1198872
) 2
+ 11988812
12057511989912
+ 11988823
(
2
1199032
1198872
minus
3
1199031198872
1199031198873
)
+ 11989612
(119905) 12057511989912
+ 11989623
(
1199062
1199032
1198872
minus
1199063
1199031198872
1199031198873
) =
1198792
1199031198872
(
1198683
1199032
1198873
) 3
+ 11988834
12057511989934
+ 11988823
(
3
1199032
1198873
minus
2
1199031198872
1199031198873
)
+ 11989634
(119905) 12057511989934
+ 11989623
(
1199063
1199032
1198873
minus
1199062
1199031198872
1199031198873
) =
1198793
1199031198873
(
1198684
1199032
1198874
) 4
+ 11988834
12057511989934
+ 11989634
(119905) 12057511989934
= minus
119879out1199031198874
(10)
where 119903119887119888 119903119887119904 119903119887119901119894 119903119887119895represent the base circle radii of planet
carrier sun gear planetary gear the medium speed gearand the high speed gear respectively 119879in 119879out are the inputtorque and output torque of the system respectively
4 Mathematical Problems in Engineering
Equation (10) can be simplified as matrix form
119872 + 119862 + 119870 (119905) 119909 = 119879 (119905) (11)
where 119909 represents generalized displacement vector of thesystem 119909 = [119906
119888 1199061199011
1199061199012
1199061199013
119906119904 1199061 1199062 1199063]119879 119872 119862 119870(119905) are
9 order matrixes of mass damp and time varying stiffnesses119879(119905) is the vector of external load caused by external inputtorque
22 Solving of Equations The common approaches to solvethe equations of the dynamic model of gear transmission sys-tem are analyticalmethod and numerical simulationmethodThe former includes piecewise linearizationmethod and har-monic balance method while the latter includes Newmark-120573 method and Runge-Kutta method Unfortunately theirobjects are all determined systems thus making it impossibleto solve the dynamic response of random system by directlyapplying these existing methods In this paper the randomproblem is converted into a determined one by sampling therandom parameters in every moment
The specific steps are as follows
(1) Determine the elasticmodulusmass density workingtooth width pitch circle diameter and the distribu-tion of comprehensive transmission error of the gearmaterial
(2) Divide the external excitation into119873portions equallyand determine each integration time step Δ119905 based on119873
(3) Assume that the rest of the parameters are determinedwhen studying the influence of the response broughtby variation of one single parameter Sample thevarying parameter at each sampling time
(4) Obtain the dynamic response at one moment bycalculating the dynamic equations with the sampleresults using fixed step Runge-Kutta method
(5) Sample the parameters of the next moment andcalculate the dynamic response at this moment
(6) Change to another parameter and repeat (2)ndash(5)
After getting the statistical characteristics of vibrationdisplacement and vibration velocity of the system at eachmoment the dynamic meshing force of each gear pair can bederived from the following equation
119882119894119895
= 119896119894119895
sdot (119883119894119895
minus 119909119894119895
minus 119890119894119895
) + 119888119894119895
(119894119895
minus 119894119895
minus 119890119894119895
) (12)
in which 119896119894119895 119888119894119895 and 119883
119894119895 respectively are the meshing stiff-
ness damping coefficient and relative displacement betweengears 119894 and 119895 119909
119894119895is the equivalent displacement of center
displacement between the meshing lines of gears 119894 and 119895 119890119894119895is
the comprehensive meshing error of gears 119894 and 119895
3 Analysis of System Excitations
31 External Excitation The randomness of system loadis mainly caused by external wind load The variation
of external excitation of the gear transmission system isdetermined by the random wind velocity In this paperstochastic volatility (SV) model is built to obtain the randomwind velocity sequence in the wind farm Then the externalexcitation of the transmission system is calculated accordingto the theory of aerodynamic
SV model is a method of time series analysis whichis used in research on analyzing wind velocity The mainfeature of SV model is to regard volatility as an implicitvariable that cannot be observedThe basic form of SVmodelis [12]
V119905
= 120576119905
+ 119864 (119910119905120595119905minus1
) = 120590119905119911119905
ln (1205902
119905) = 119886 + 120593 ln (120590
2
119905minus1) + 120590120578120578119905
(13)
where V119905is the amplitude of volatility 120576
119905is kurtosis 119864(119910
119905|
120595119905minus1
) is the conditional mean of V119905calculated from the
information sampled at 119905 minus 1 120590119905is the conditional mean
square deviation 119911119905follows a normal distribution with 0
mean and 1 variance 119886 is a constant which reflects the averagevolatility 120593 is a parameter which reflects sustainability 120590
120578
is the mean square deviation of volatility disturbance 120578119905 119911119905
follow independent normal distributions with 0 mean and 1variance
The randomwind velocity simulated by SVmodel is takenas the input of the gear transmission system of wind turbineBased on the aerodynamic theory the input power of thetransmission system is [10]
119901in =
1
2
1205881198782V3119905119862119901
(14)
where 119901in is the input power of transmission system 120588 is airdensity 119878 is the sweeping area of wind turbine 119862
119901is wind
energy utilization factor V119905is the wind velocity simulated
from SV model far from wind turbinesThe external excitation of the system is the torque ripple
caused by random wind velocity The torques from the inputand output sides respectively are
119879in =
119901in120596
119879out =
119879in119894
(15)
where 120596 is the angular velocity of wind turbines 119894 is thetransmission ratio of the gear transmission system
32 Stiffness Excitation Stiffness excitation is a parametricexcitation caused by the variation of meshing stiffness duringthe meshing process Due to many influencing factors duringmachining and assembling the size and material of thegear transmission components vary randomly such as elasticmodulus and working tooth width In this paper gearrsquosstiffness is assumed to be a superposition of a sine wave anda randomwaveThe former is expressed by limited harmonicwaves of Fourier series and the latter is expressed by standard
Mathematical Problems in Engineering 5
normal distribution function Therefore the comprehensivemeshing stiffness of gears is as follows
119896 (119905) = 119896119898
+
119898
sum
119895=1
[1198961198951cos 119895120596119905 + 119896
1198952sin 119895120596119905] + 120576
1 (16)
where 119896119898is the average meshing stiffness of the gear pairs
1198961198951
and 1198961198952
are the meshing stiffness of harmonic waves 120596
is meshing frequency 1205761is stiffness fluctuation caused by
the variation of elastic modulus which follows a normaldistribution
33 Error Excitation Meshing error is a displacement excita-tion which is related to the machining accuracy of the gearsThe gear error and base pitch error can be expressed as asuperposition of a sine wave and a random wave as follows[13]
119890 (119905) = 119890119898
+ 119890119903sin(
2120587120596119905
119879
+ 120593) + 1205762 (17)
where 119890119898
and 119890119903are the offset and amplitude of the gear
meshing error 119879 120596 120593 are the meshing period of the gearpair meshing frequency and initial phase angle 120576
2is the
fluctuation of comprehensive transmission error caused bymachining and assembling which is assumed to follow anormal distribution In this paper the gear accuracy ispresumed to be grade 6 and parameters involved are basedon GBT 10095-1988 standard
4 Analysis of System Dynamic Reliability
41 Random Fatigue Load Spectrum of Gear TransmissionSystem Load-time history of each gear pair can be obtainedby the dynamic gear transmission model built before Thento analyze the fatigue reliability of the system the load-time history is converted into a series of complete cyclesThe main converting methods are peak counting methodcycle counting method rain flow counting method and soforth
In this paper we count the dynamic meshing force ofeach gear pair circularly according to the rain flow countingprinciple [14ndash16] in order to obtain the frequency of luffingfatigue load As is shown in Figure 5 the mean stress ofthe gear pairs follows a normal distribution approximatelyand the amplitude of the stress follows Weibull distributionapproximately
In order to analyze the fatigue life of the transmissionsystem the equivalent amplitude and frequency of thesystem stress are obtained by using equivalent method ofGeber quadratic curve The Geber quadratic curve formulais [17]
119878eqv = 119878119886
1205902
119887
1205902
119887minus 11987810158402
119898
(18)
where 119878119886is the amplitude of stress after the conversion 119878
1015840
119898is
the mean stress of 119878-119873 curve of the given material 119878eqv is theequivalent stress corresponding to 119878eqv with equal lifetime
42 Dynamic Reliability Model of Gear Transmission Com-ponents and System Fatigue failure of the components iscaused by the accumulation of material internal damage Asthe number of stress cycles increases the material internaldamage exacerbates and the structural life decreases Theoryof probability fatigue damage is based on the fatigue damageevolution which demonstrates the irreversibility and therandomness of fatigue damage The main reason of therandomness of fatigue damage lies in the characteristics ofthe material the geometric dimensions of the test pieces andthe uncertainty of external load
The decay rate of the material ultimate stress generallyfollows distribution as [18ndash20]
119889120590119906
119889119899
=
minus119891 (119878max 119891119888 119903)
119888120590119888
119906
(19)
where 119878max is the maximin cyclic stress 119891119888is the cycle
frequency 119903 is cyclic stress ratio 119888 is a constantThe remaining ultimate stress of the component material
after 119899 cycles is
120590119906
(119899) = 1205901199060
1 minus [1 minus (
119878max1205901199060
)
119888
]
119899
119899119894
1119888
(20)
where 120590119906(119899) is the remaining ultimate stress after 119899 cycles 120590
1199060
is the ultimate stress whenmaterials are in good condition 119899119894
is the number of ultimate cyclesThe damage index of component under the level 119894 luffing
cyclic stress after 119899 cycles is
Δ119863 =
119899
sum
119894=1
(1 minus (120590119906
(119899119894) 120590119906
(119899119894minus1
))119888
)
(1 minus (119878max 119894120590119906
(119899119894minus1
))119888
)
(21)
where 120590119906(119899119894) and 120590
119906(119899119894minus1
) are the remaining ultimate stressunder the level 119894 and level 119894 minus 1 stress 119878max 119894 is the maximumstress of level 119894 stress cycles
Suppose 120590119906(119899119894) (119894 = 1 2 119899) are independent random
variables from each other 120590119906
= (1205901199061
1205901199062
120590119906119899
) whicharemeans of 120590
119906(1198991) 120590119906(1198992) 120590
119906(119899119899) in (6) respectively are
expanded into the Taylor series Then the approximate mean120583Δ119863
and standard deviation 120590Δ119863
of the damage index Δ119863
are obtained by choosing the linear terms from the Taylorexpansion Consider
120583Δ119863
= 119892 (120590119863
) +
119899
sum
119894=1
(120590119906(119899119894) minus 120590119906119894
)
120597119892
120597120590119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+
1
2
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+ sdot sdot sdot
120590Δ119863
= radic119864(Δ119863119894
minus 120583Δ119863
)2
= radic
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
100381610038161003816100381610038161003816100381610038161003816120590119906119894
(22)
6 Mathematical Problems in Engineering
Table 1 Geometric parameters of gear transmission system of wind turbine
Low-level speedNumber of sun gear
teeth 119885119904
Number of planetarygear teeth 119885
119901
Number of internalgear teeth 119885
119903
Number ofmolds 119898
Meshing angle120572119904119901(∘)
Meshing angle120572119903119901(∘)
27 44 117 13 230000 210000Medium-levelspeed
Number of driving gears 1198851
Number of driven gears 1198852
Meshing angle 120572 (∘)104 23 210000
High-level speed Number of driving gears 1198853
Number of driven gears 1198854
Meshing angle 120572 (∘)98 25 210000
In engineering application random index of fatiguecumulative damage is supposed to follow a lognormal distri-bution That makes the distribution of 119863 be
119891119863
(119863) =
1
radic2120587120590119863
119863
exp[minus
(ln119863 minus 120583119863
)2
21205902
119863
] 119863 gt 0
0 119863 le 0
(23)
where 120583119863is the logarithmic mean of cumulative damage 119863
120590119863is the logarithmic mean square deviation of 119863Based on the theory of probability fatigue cumulative
damage the structural dynamic reliability of one moment is
119877 = 119877 (119863 lt 1198630) = int
1198630
0
119891119863
(119910) 119889119910 (24)
where 1198630is the limit of damage index
According to Figure 1 in the gear transmission systemthe gears are connected in series thus the whole systemwill fail if one of the gears fails In other words the systemreliability is based on the reliability of each gear Thereforethe reliability model gear transmission system is as follows
119877 (119905) = 119877119888
(119905) sdot
3
prod
119894=1
119877119901119894
(119905) sdot 119877119903
(119905) sdot 119877119904
(119905) sdot
4
prod
119895=1
119877119895
(119905) (25)
where 119894 is the number of planetary gears 119895 is the number ofmedium speed level and high speed level gears
43 Calculation of System Dynamic Reliability The reliabilityof the gear transmission system is calculated using Matlabsoftware Based on the analysis before the steps of theprogram are as follows
(1) Take the random input torque of the gear transmis-sion system as the extern excitation Get the dynamicmeshing force and its statistic characteristics by usingthe numerical integration method
(2) Process the data of meshing force by rain flowcountingmethodThen calculate the equivalent stressamplitude and frequency by using equivalent methodof Geber quadratic curve
(3) Calculate structural fatigue damage under the luffingstress
(4) Calculate cumulative fatigue damage of arbitrary time119905 under several stress cycles
0 2 4 6 8 100
02
04
06
Time t (s)
med
ium
-leve
l spe
ed g
ears
F12
(N)
Mea
n dy
nam
ic m
esh
forc
e of times106
Figure 3 Mean dynamic meshing force of medium-level speedgears
(5) Calculate the structural limit value of fatigue damage
(6) By giving a random cumulative damage index applythe equation of dynamic reliability to calculate thereliability of each gear when the tooth surface reachesthe contact fatigue limit and the tooth root reaches thebending fatigue limit
(7) Calculate the dynamic reliability of the gear transmis-sion system using (25)
5 Analysis of Examples
The study object of the example research is the gear trans-mission system of a 15MW wind turbine Here are someparameters used in this research the rated power of the windturbine is 15MW the impeller diameter is 70m the designedimpeller speed is 148 rmin average wind speed of the windfarm is 143ms wind density is 121 kgm3 wind energyutilization factor is 032 system transmission ratio is 9453Suppose the strength of the material and the coefficient ofperformance both follow a normal distribution while otherparameters are constant Suppose the material of planetarygear is 40Cr and the material of medium-level speed andhigh-level speed gear is 20CrMnTi Other parameters of thesystem are shown in Table 1
By solving the dynamic equation (10) of the systemvibration displacement and vibration velocity of the gearsat each moment are obtained as well as their statisticalcharacteristics By solving (11) the meshing forces of eachgear are obtained Figure 3 shows the curve of mean dynamicmeshing force of medium-level speed gears Figure 4 shows
Mathematical Problems in Engineering 7
8000
6000
4000
2000
00
12
34
56
Freq
uenc
y
times106
times106
959
756
453
150
(a) mean (N)
(b) amplitude (N)
Figure 4 Luffing load spectrum of medium-level speed gears
092
096
1
0 5 10 15 20
094
098
E
b
e
120588d0
Time t (y)
Mea
n re
liabi
lity120583R(t)
Figure 5 Dynamic reliability of system when variation coefficientof random parameter is 0
the luffing load spectrum of medium-level speed gears basedon the theory of rain flow counting method
Wedefine the ratio of themean square error and themeanof the systemparameters as their variation coefficient Figures5ndash7 show the variation of systemdynamic reliability over timewith the variation coefficient being 0 01 and 03 respectivelyAs is demonstrated in Figure 5 the comprehensive transmis-sion error 119890 has the greatest influence on the system reliabilityfollowed by the elastic modulus of gear material 119864 contacttooth width 119861 and pitch circle diameter 119889
0 Mass density 120588
has the least influence By comparing Figures 6 and 7 wecan also learn that with the variation of random parametersincreases the system gets more reliable
Table 2 shows the dynamic reliability of each componentin the transmission system as the comprehensive transmis-sion error 119890 and mass density 120588 vary randomly when 119905 =
63times108 sThe table also shows that when the comprehensive
transmission error 119890 and mass density 120588 are 0 01 and 03respectively in the whole transmission system planetarygear system has the highest dynamic reliability followed by
1
5 10 15 20084
092
0
088
096
E
b
e
120588
d0
Mea
n re
liabi
lity120583R(t)
Time t (y)
Figure 6 Dynamic reliability of system when variation coefficientof random parameter is 01
5 10 15 200080
090
1
085
095M
ean
relia
bilit
y120583R(t)
E
b
e
120588
d0
Time t (y)
Figure 7 Dynamic reliability of system when variation coefficientof random parameter is 03
the medium speed level gears while high speed level gearis the least reliable In the planetary gear system internalgears have the highest reliability followed by the planetarygear while the sun gear is the least reliable In the mediumand high speed level gears large gears are more dynamic-reliable than the small ones The dynamic reliability of thegear transmission system reduces and the dispersion degreeof the system increases with the increase of the parametersrsquovariation
We obtained the statistical properties of the dynamicreliability of the high speed level gears through 20000simulations when 119905 = 63 times 10
8 s using Monte Carlo methodand compared the results with this paper as is shown inTable 3 The method proposed in this paper is more accuratethan Monte Carlo method
6 Conclusions
In this paper the dynamic reliability of the gear transmissionsystem of a 15MW wind turbine with consideration of
8 Mathematical Problems in Engineering
Table2Dyn
amicreliabilityof
each
compo
nent
ofplanetarygear
syste
mwith
rand
omparameters
Influ
encing
factors
Varia
tion
coeffi
cient
Reliabilityof
sun
gear
Reliabilityof
planetarygear
Reliabilityof
internalgear
Reliabilityof
large
gearsin
medium-le
velspeed
Reliabilityof
small
gearsin
medium-le
velspeed
Reliabilityof
large
gearsinhigh
-level
speed
Reliabilityof
small
gearsinhigh
-level
speed
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
120588
00987387
0007365
0991021
0007327
0995148
0007324
0979066
0007367
0971572
0007631
0975882
0007483
0963271
0007790
005
0980301
0007366
0984813
0007327
0991275
0007324
0978731
0007369
0965088
0007638
0971755
0007489
0950338
0007795
01
0953633
0007371
0959300
0007332
0975381
0007335
0950280
0007371
0931572
0007652
0942372
0007506
0927510
0007803
119890
00962833
0007802
0970332
0007789
0983727
0007756
0958207
0007803
0936281
0007817
0952283
0007804
0932588
0007832
005
0948471
0007815
0955783
0007795
0970115
0007758
0944342
0007815
0920175
0007822
0947502
0007815
0901392
0007863
01
0911502
0007843
0927009
0007804
0943928
0007781
0909252
0007845
0883011
0007827
0897641
0007844
0877252
0007904
Mathematical Problems in Engineering 9
Table 3 The comparison of dynamic reliability of big gear of high speed gear system
Random parameters Variation coefficient Proposed method Monte Carlo methodMean of 119877(119905) Root mean square of 119877(119905) Mean of 119877(119905) Root mean square of 119877(119905)
119861 01 0943257 0008450 0940832 0008541119864 01 0940144 0008671 0939124 0008454120588 01 0944946 0008377 0945271 00082721198890
01 0946826 0008498 0939567 0008157119890 01 0909033 0008870 0897354 0008625
randomness of load and system parameters is analyzedby applying the theory of probability of cumulative fatiguedamage The main contributions and conclusions of thispaper are the following
(1) The dynamic model of the gear transmission of windturbine is built In consideration of the randomness of theload and gear parameters the dynamic response of thesystem is obtained by utilizing the random sampling methodand Runge-Kutta method The statistical properties of themeshing force of components in the gear transmission systemare obtained by statistic method
(2) By applying the method of rain flow counting thetime history of the components meshing force is convertedinto a series of luffing load spectra and the equivalent stressamplitude and frequency are calculated according to theequivalent method of Geber quadratic curve
(3) The dynamic reliability model of the transmissionsystem and gear components are built according to theprinciple of probability fatigue damage cumulative Variationof the system reliability over time is calculated when theparameters vary and the effect of the parameter variation tothe system reliability is analyzed Results show that (i) thecomprehensive transmission error has the largest influenceon system dynamic reliability while the mass density hasthe least influence (ii) the dynamic reliability of the geartransmission system reduces and the dispersion degreeincreases with the increase of the variation of the parameters(iii) for the gear transmission system of the 15MW windturbine planetary gear system has the highest dynamicreliability followed by the medium speed level gears whilehigh speed level gear is the least reliable At the same timein the planetary gear system internal gears have the highestreliability followed by the planetary gear while the sun gearis the least reliable In themedium and high speed level gearslarge gears are more dynamic-reliable than the small ones
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Major State BasicResearch Development Program 973 (no 2012CB215202)the National Natural Science Foundation of China (no51205046) and the Fundamental Research Funds for the
Central Universities The constructive comments providedby the anonymous reviewers and the editors are also greatlyappreciated
References
[1] W Musial S Butterfield and B McNiff ldquoImproving windturbine gearbox reliabilityrdquo in Proceedings of the EuropeanWindEnergy Conference Milan Italy May 2007
[2] L Katafygiotis and S H Cheung ldquoWedge simulation methodfor calculating the reliability of linear dynamical systemsrdquoProbabilistic Engineering Mechanics vol 19 no 3 pp 229ndash2382004
[3] L Katafygiotis and S H Cheung ldquoDomain decompositionmethod for calculating the failure probability of linear dynamicsystems subjected to gaussian stochastic loadsrdquo Journal ofEngineering Mechanics vol 132 no 5 pp 475ndash486 2006
[4] P Liu and Q-F Yao ldquoEfficient estimation of dynamic reliabilitybased on simple additive rules of probabilityrdquo EngineeringMechanics vol 27 no 4 pp 1ndash4 2010
[5] H-W Qiao Z-Z Lu A-R Guan and X-H Liu ldquoDynamicreliability analysis of stochastic structures under stationaryrandom excitation using hermite polynomials approximationrdquoEngineering Mechanics vol 26 no 2 pp 60ndash64 2009
[6] A Lupoi P Franchin and M Schotanus ldquoSeismic risk eval-uation of RC bridge structuresrdquo Earthquake Engineering ampStructural Dynamics vol 32 no 8 pp 1275ndash1290 2003
[7] P Franchin ldquoReliability of uncertain inelastic structures underearthquake excitationrdquo Journal of Engineering Mechanics vol130 no 2 pp 180ndash191 2004
[8] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006
[9] Z Caichao H Zehao T Qian and T Yonghu ldquoAnalysis ofnonlinear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005
[10] D T Qin Z K Xing and J H Wang ldquoOptimization designof system parameters of the gear transmission of wind turbinebased on dynamics and reliabilityrdquo Chinese Journal of Mechani-cal Engineering vol 44 no 7 pp 24ndash31 2008
[11] D-T Qin X-G Gu J-H Wang and J-G Liu ldquoDynamicanalysis and optimization of gear trains in amegawatt level windturbinerdquo Journal of Chongqing University vol 32 no 4 pp 408ndash414 2009
[12] X-L Jiang and C-F Wang ldquoStochastic volatility models basedBayesian method and their applicationrdquo Systems Engineeringvol 23 no 10 pp 22ndash28 2005
[13] H T Chen X L Wu D T Qin J Yang and Z Zhou ldquoEffectsof gear manufacturing error on the dynamic characteristics of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
So it is of great practical significance to develop a method toanalyze the dynamic characteristics and the reliability underrandom wind conditions
In this paper we studied the gear transmission system of a15MWwind turbine Elasticmodulusmass density workingtooth width pitch circle diameter and comprehensive trans-mission error were taken as random variables The dynamicmeshing force of gears was obtained by using the theoryof random sampling and the Runge-Kutta method takinginto consideration the influence of external random load Onthis basis statistical processing of dynamic meshing forcewas done using rain flow counting principle and equivalentmethod of Geber quadratic curve Dynamic reliability ofcomponents and the whole system were calculated accordingto the theory of probability of cumulative fatigue damage Inthe end the variation of the system dynamic reliability overtime under variable parameters was studiedThe effect of thisvariation to the system dynamic reliability was analyzed andthe results were comparedwith those ofMonteCarlomethod
2 Dynamic Model of GearTransmission System
21 Dynamic Model of Gear Transmission System This paperstudies the gear transmission system of a 15MW wind tur-bine generator which contains one level of NGW planetarygear and two levels of parallel shaft gear The structurediagram is shown in Figure 1
Torsional vibration model of gear transmission systemis built using centralized parameter method as is shownin Figure 2 Variation of meshing stiffness comprehensivetransmission error and other factors are taken into consid-eration in this model The planetary gears are assumed tobe uniformly distributed and have the same physical andgeometrical parameters
In Figure 2 119906119888 119906119904 119906119901119894 119906119895(119894 = 1 2 3 119895 = 1 2 3 4)
represent the torsion displacement of planet carrier sun gearplanetary gears and medium and high speed level gearsrespectively 119896
119904119901119894 119896119903119901119894
represent the meshing stiffness of sungear and planetary gear 119894 and themeshing stiffness of annulargear and planetary gear 119894 respectively 119896
1199041represents the
torsional stiffness of the connecting shaft between the sungear and gear 1 119896
23represents the torsional stiffness of the
connecting shaft between gear 2 and gear 3 11989612 11989634represent
themeshing stiffness of themedium speed gears and the highspeed gears 119888
119904119901119894 119888119903119901119894
represent the meshing damping of sungear annular gear and planetary gear 119894 119888
1199041represents the
torsional damper of connecting shaft between the sun gearand gear 1 119888
23represents the torsional damper of connecting
shaft between gear 2 and gear 3 11988812 11988834represent the meshing
damping of the medium speed level gears and the high speedlevel gears 119890
119904119901119894 119890119903119901119894
represent the transmission error of sungear annular gear and planetary gear 119894 119890
12 11989034represent the
comprehensive transmission error of the medium speed andhigh speed level gears
In the gear transmission system the meshing of the gearpair and the gear meshing force and meshing displacementare all happening in the direction of the meshing line In
1
2
3
4
s
r
c
Tin
Tout
ks1
k23
pi
Figure 1 Schematic of gear transmission system of wind turbine 119901planetary gear 119903 internal gear 119888 planet carrier 119904 sun gear 1 smallgear at medium-level speed 2 large gear at medium-level speed 3small gear at high-level speed 4 large gear at high-level speed 119879ininput torque and 119879out output torque
order to simplify the following analysis and calculation onthe torsional vibration system we replace the generalizedcoordinates in form of gear angular displacement with theones in form of line displacement along the meshing line Setthe rotation angular displacement of each gear is 120579
119894 respec-
tively (119894 = 119888 119901119894 119904 1 2 3 4) According to the newly definedgeneralized coordinates the rotation freedom of sun gear isconverted to microdisplacement 119906
119904 119906119904
= 119903119904120579119904(119903119904is the radius
of the base circle of the sun gear) which is in the directionof the planetary gear meshing line Similarly the rotationfreedom of planetary gear is converted to microdisplacement119906119888
= 119903119888120579119888 also in the direction of meshing line and so on
The analysis of elastic deformation of each meshing forceis as follows
The elastic deformation in the direction of meshing forcebetween the 119894th planetary gear and the sun gear is
120575119904119901119894
= 119903119887119888cos120572119904119901
120579119888
minus 119903119887119901119894
120579119901119894
minus 119903119887119904
120579119904
minus 119890119904119901119894
= 119906119888cos120572119904119901
minus 119906119901119894
minus 119906119904
minus 119890119904119901119894
(1)
Its first derivative is
120575119904119901119894
= 119903119887119888cos120572119904119901
120579119888
minus 119903119887119901119894
120579119901119894
minus 119903119887119904
120579119904
minus 119890119904119901119894
= 119888cos120572119904119901
minus 119901119894
minus 119904
minus 119890119904119901119894
(2)
The elastic deformation in the direction of meshing forcebetween the 119894th planetary gear and the internal ring gear is
120575119903119901119894
= 119903119887119901119894
120579119901119894
minus 119903119887119903
120579119903
+ 119903119887119888cos120572119903119901
120579119888
minus 119890119903119901119894
= 119906119901119894
minus 119906119903
+ 119906119888cos120572119903119901
minus 119890119903119901119894
(3)
Its first derivative is
120575119903119901119894
= 119903119887119901119894
120579119901119894
minus 119903119887119903
120579119903
+ 119903119887119888cos120572119903119901
120579119888
minus 119890119903119901119894
= 119901119894
minus 119903
+ 119888cos120572119903119901
minus 119890119903119901119894
(4)
The elastic deformation in the direction of meshing forcebetween the spur gear 1 and the spur gear 2 is
12057512
= 1199031198871
1205791
+ 1199031198872
1205792
minus 11989012
= 1199061
+ 1199062
minus 11989012
(5)
Mathematical Problems in Engineering 3
X
Y
Z
c
r
k34
u4
e34c34
k12
k23
u3
u1
u2
us
ucup1
e12c12
c23
1
2
3
4
Xc
Yccrp2
erp2
crp3
krp2
krp3
up2
up3Tin
esp2
erp1crp1
csp2
csp1
esp1
csp3
ksp2esp3
erp3ksp3
ksp1
krp1
cs1
ks1
Figure 2 Torsional vibration model of gear transmission system
Its first derivative is
12057512
= 1199031198871
1205791
+ 1199031198872
1205792
minus 11989012
= 1
+ 2
minus 11989012
(6)
The elastic deformation in the direction of meshing forcebetween the spur gear 3 and the spur gear 4 is
12057534
= 1199031198873
1205793
+ 1199031198874
1205794
minus 11989034
= 1199063
+ 1199064
minus 11989034
(7)
Its first derivative is
12057534
= 1199031198873
1205793
+ 1199031198874
1205794
minus 11989034
= 3
+ 4
minus 11989034
(8)
So in the gear transmission system the relative displace-ments in the direction of meshing line of all gear pairs are
120575119904119901119894
= 119906119888cos120572119904119901
minus 119906119901119894
minus 119906119904
minus 119890119904119901119894
120575119903119901119894
= 119906119901119894
+ 119906119888cos120572119903119901
minus 119890119903119901119894
12057511989912
= 1199061
+ 1199062
minus 11989012
12057511989934
= 1199063
+ 1199064
minus 11989034
(9)
Equation (10) is the vibration differential equations of thesystem based on Lagrange equation Consider
(
119868119888
1199032
119887119888
) 119888
+
3
sum
119894=1
119888119903119901119894
cos120572119903119901119894
120575119903119901119894
+
3
sum
119894=1
119888119904119901119894
cos120572119904119901119894
120575119904119901119894
+ 119888119906119888
119888
+
3
sum
119894=1
119896119903119901119894
(119905) cos120572119903119901119894
120575119903119901119894
+
3
sum
119894=1
119896119904119901119894
(119905) cos120572119904119901119894
120575119904119901119894
+ 119896119906119888
119906119888
=
119879in119903119887119888
(
119868119901119894
1199032
119887119901119894
) 119901119894
minus 119888119904119901119894
120575119904119901119894
+ 119888119903119901119894
120575119903119901119894
minus 119896119904119901119894
(119905) 120575119904119901119894
+ 119896119903119901119894
(119905) 120575119903119901119894
= 0
(
119868119904
1199032
119887119904
) 119904
minus
3
sum
119894=1
119888119904119901119894
120575119904119901119894
+ 1198881199041
(
119904
1199032
119887119904
minus
1
119903119887119904
1199031198871
)
minus
3
sum
119894=1
119896119904119901119894
(119905) 120575119904119901119894
+ 1198961199041
(
119906119904
1199032
119887119904
minus
1199061
119903119887119904
1199031198871
) =
119879119904
119903119887119904
(
1198681
1199032
1198871
) 1
+ 11988812
12057511989912
+ 1198881199041
(
1
1199032
1198871
minus
119904
119903119887119904
1199031198871
)
+ 11989612
(119905) 12057511989912
+ 1198961199041
(
1199061
1199032
1198871
minus
119906119904
119903119887119904
1199031198871
) =
1198791
1199031198871
(
1198682
1199032
1198872
) 2
+ 11988812
12057511989912
+ 11988823
(
2
1199032
1198872
minus
3
1199031198872
1199031198873
)
+ 11989612
(119905) 12057511989912
+ 11989623
(
1199062
1199032
1198872
minus
1199063
1199031198872
1199031198873
) =
1198792
1199031198872
(
1198683
1199032
1198873
) 3
+ 11988834
12057511989934
+ 11988823
(
3
1199032
1198873
minus
2
1199031198872
1199031198873
)
+ 11989634
(119905) 12057511989934
+ 11989623
(
1199063
1199032
1198873
minus
1199062
1199031198872
1199031198873
) =
1198793
1199031198873
(
1198684
1199032
1198874
) 4
+ 11988834
12057511989934
+ 11989634
(119905) 12057511989934
= minus
119879out1199031198874
(10)
where 119903119887119888 119903119887119904 119903119887119901119894 119903119887119895represent the base circle radii of planet
carrier sun gear planetary gear the medium speed gearand the high speed gear respectively 119879in 119879out are the inputtorque and output torque of the system respectively
4 Mathematical Problems in Engineering
Equation (10) can be simplified as matrix form
119872 + 119862 + 119870 (119905) 119909 = 119879 (119905) (11)
where 119909 represents generalized displacement vector of thesystem 119909 = [119906
119888 1199061199011
1199061199012
1199061199013
119906119904 1199061 1199062 1199063]119879 119872 119862 119870(119905) are
9 order matrixes of mass damp and time varying stiffnesses119879(119905) is the vector of external load caused by external inputtorque
22 Solving of Equations The common approaches to solvethe equations of the dynamic model of gear transmission sys-tem are analyticalmethod and numerical simulationmethodThe former includes piecewise linearizationmethod and har-monic balance method while the latter includes Newmark-120573 method and Runge-Kutta method Unfortunately theirobjects are all determined systems thus making it impossibleto solve the dynamic response of random system by directlyapplying these existing methods In this paper the randomproblem is converted into a determined one by sampling therandom parameters in every moment
The specific steps are as follows
(1) Determine the elasticmodulusmass density workingtooth width pitch circle diameter and the distribu-tion of comprehensive transmission error of the gearmaterial
(2) Divide the external excitation into119873portions equallyand determine each integration time step Δ119905 based on119873
(3) Assume that the rest of the parameters are determinedwhen studying the influence of the response broughtby variation of one single parameter Sample thevarying parameter at each sampling time
(4) Obtain the dynamic response at one moment bycalculating the dynamic equations with the sampleresults using fixed step Runge-Kutta method
(5) Sample the parameters of the next moment andcalculate the dynamic response at this moment
(6) Change to another parameter and repeat (2)ndash(5)
After getting the statistical characteristics of vibrationdisplacement and vibration velocity of the system at eachmoment the dynamic meshing force of each gear pair can bederived from the following equation
119882119894119895
= 119896119894119895
sdot (119883119894119895
minus 119909119894119895
minus 119890119894119895
) + 119888119894119895
(119894119895
minus 119894119895
minus 119890119894119895
) (12)
in which 119896119894119895 119888119894119895 and 119883
119894119895 respectively are the meshing stiff-
ness damping coefficient and relative displacement betweengears 119894 and 119895 119909
119894119895is the equivalent displacement of center
displacement between the meshing lines of gears 119894 and 119895 119890119894119895is
the comprehensive meshing error of gears 119894 and 119895
3 Analysis of System Excitations
31 External Excitation The randomness of system loadis mainly caused by external wind load The variation
of external excitation of the gear transmission system isdetermined by the random wind velocity In this paperstochastic volatility (SV) model is built to obtain the randomwind velocity sequence in the wind farm Then the externalexcitation of the transmission system is calculated accordingto the theory of aerodynamic
SV model is a method of time series analysis whichis used in research on analyzing wind velocity The mainfeature of SV model is to regard volatility as an implicitvariable that cannot be observedThe basic form of SVmodelis [12]
V119905
= 120576119905
+ 119864 (119910119905120595119905minus1
) = 120590119905119911119905
ln (1205902
119905) = 119886 + 120593 ln (120590
2
119905minus1) + 120590120578120578119905
(13)
where V119905is the amplitude of volatility 120576
119905is kurtosis 119864(119910
119905|
120595119905minus1
) is the conditional mean of V119905calculated from the
information sampled at 119905 minus 1 120590119905is the conditional mean
square deviation 119911119905follows a normal distribution with 0
mean and 1 variance 119886 is a constant which reflects the averagevolatility 120593 is a parameter which reflects sustainability 120590
120578
is the mean square deviation of volatility disturbance 120578119905 119911119905
follow independent normal distributions with 0 mean and 1variance
The randomwind velocity simulated by SVmodel is takenas the input of the gear transmission system of wind turbineBased on the aerodynamic theory the input power of thetransmission system is [10]
119901in =
1
2
1205881198782V3119905119862119901
(14)
where 119901in is the input power of transmission system 120588 is airdensity 119878 is the sweeping area of wind turbine 119862
119901is wind
energy utilization factor V119905is the wind velocity simulated
from SV model far from wind turbinesThe external excitation of the system is the torque ripple
caused by random wind velocity The torques from the inputand output sides respectively are
119879in =
119901in120596
119879out =
119879in119894
(15)
where 120596 is the angular velocity of wind turbines 119894 is thetransmission ratio of the gear transmission system
32 Stiffness Excitation Stiffness excitation is a parametricexcitation caused by the variation of meshing stiffness duringthe meshing process Due to many influencing factors duringmachining and assembling the size and material of thegear transmission components vary randomly such as elasticmodulus and working tooth width In this paper gearrsquosstiffness is assumed to be a superposition of a sine wave anda randomwaveThe former is expressed by limited harmonicwaves of Fourier series and the latter is expressed by standard
Mathematical Problems in Engineering 5
normal distribution function Therefore the comprehensivemeshing stiffness of gears is as follows
119896 (119905) = 119896119898
+
119898
sum
119895=1
[1198961198951cos 119895120596119905 + 119896
1198952sin 119895120596119905] + 120576
1 (16)
where 119896119898is the average meshing stiffness of the gear pairs
1198961198951
and 1198961198952
are the meshing stiffness of harmonic waves 120596
is meshing frequency 1205761is stiffness fluctuation caused by
the variation of elastic modulus which follows a normaldistribution
33 Error Excitation Meshing error is a displacement excita-tion which is related to the machining accuracy of the gearsThe gear error and base pitch error can be expressed as asuperposition of a sine wave and a random wave as follows[13]
119890 (119905) = 119890119898
+ 119890119903sin(
2120587120596119905
119879
+ 120593) + 1205762 (17)
where 119890119898
and 119890119903are the offset and amplitude of the gear
meshing error 119879 120596 120593 are the meshing period of the gearpair meshing frequency and initial phase angle 120576
2is the
fluctuation of comprehensive transmission error caused bymachining and assembling which is assumed to follow anormal distribution In this paper the gear accuracy ispresumed to be grade 6 and parameters involved are basedon GBT 10095-1988 standard
4 Analysis of System Dynamic Reliability
41 Random Fatigue Load Spectrum of Gear TransmissionSystem Load-time history of each gear pair can be obtainedby the dynamic gear transmission model built before Thento analyze the fatigue reliability of the system the load-time history is converted into a series of complete cyclesThe main converting methods are peak counting methodcycle counting method rain flow counting method and soforth
In this paper we count the dynamic meshing force ofeach gear pair circularly according to the rain flow countingprinciple [14ndash16] in order to obtain the frequency of luffingfatigue load As is shown in Figure 5 the mean stress ofthe gear pairs follows a normal distribution approximatelyand the amplitude of the stress follows Weibull distributionapproximately
In order to analyze the fatigue life of the transmissionsystem the equivalent amplitude and frequency of thesystem stress are obtained by using equivalent method ofGeber quadratic curve The Geber quadratic curve formulais [17]
119878eqv = 119878119886
1205902
119887
1205902
119887minus 11987810158402
119898
(18)
where 119878119886is the amplitude of stress after the conversion 119878
1015840
119898is
the mean stress of 119878-119873 curve of the given material 119878eqv is theequivalent stress corresponding to 119878eqv with equal lifetime
42 Dynamic Reliability Model of Gear Transmission Com-ponents and System Fatigue failure of the components iscaused by the accumulation of material internal damage Asthe number of stress cycles increases the material internaldamage exacerbates and the structural life decreases Theoryof probability fatigue damage is based on the fatigue damageevolution which demonstrates the irreversibility and therandomness of fatigue damage The main reason of therandomness of fatigue damage lies in the characteristics ofthe material the geometric dimensions of the test pieces andthe uncertainty of external load
The decay rate of the material ultimate stress generallyfollows distribution as [18ndash20]
119889120590119906
119889119899
=
minus119891 (119878max 119891119888 119903)
119888120590119888
119906
(19)
where 119878max is the maximin cyclic stress 119891119888is the cycle
frequency 119903 is cyclic stress ratio 119888 is a constantThe remaining ultimate stress of the component material
after 119899 cycles is
120590119906
(119899) = 1205901199060
1 minus [1 minus (
119878max1205901199060
)
119888
]
119899
119899119894
1119888
(20)
where 120590119906(119899) is the remaining ultimate stress after 119899 cycles 120590
1199060
is the ultimate stress whenmaterials are in good condition 119899119894
is the number of ultimate cyclesThe damage index of component under the level 119894 luffing
cyclic stress after 119899 cycles is
Δ119863 =
119899
sum
119894=1
(1 minus (120590119906
(119899119894) 120590119906
(119899119894minus1
))119888
)
(1 minus (119878max 119894120590119906
(119899119894minus1
))119888
)
(21)
where 120590119906(119899119894) and 120590
119906(119899119894minus1
) are the remaining ultimate stressunder the level 119894 and level 119894 minus 1 stress 119878max 119894 is the maximumstress of level 119894 stress cycles
Suppose 120590119906(119899119894) (119894 = 1 2 119899) are independent random
variables from each other 120590119906
= (1205901199061
1205901199062
120590119906119899
) whicharemeans of 120590
119906(1198991) 120590119906(1198992) 120590
119906(119899119899) in (6) respectively are
expanded into the Taylor series Then the approximate mean120583Δ119863
and standard deviation 120590Δ119863
of the damage index Δ119863
are obtained by choosing the linear terms from the Taylorexpansion Consider
120583Δ119863
= 119892 (120590119863
) +
119899
sum
119894=1
(120590119906(119899119894) minus 120590119906119894
)
120597119892
120597120590119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+
1
2
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+ sdot sdot sdot
120590Δ119863
= radic119864(Δ119863119894
minus 120583Δ119863
)2
= radic
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
100381610038161003816100381610038161003816100381610038161003816120590119906119894
(22)
6 Mathematical Problems in Engineering
Table 1 Geometric parameters of gear transmission system of wind turbine
Low-level speedNumber of sun gear
teeth 119885119904
Number of planetarygear teeth 119885
119901
Number of internalgear teeth 119885
119903
Number ofmolds 119898
Meshing angle120572119904119901(∘)
Meshing angle120572119903119901(∘)
27 44 117 13 230000 210000Medium-levelspeed
Number of driving gears 1198851
Number of driven gears 1198852
Meshing angle 120572 (∘)104 23 210000
High-level speed Number of driving gears 1198853
Number of driven gears 1198854
Meshing angle 120572 (∘)98 25 210000
In engineering application random index of fatiguecumulative damage is supposed to follow a lognormal distri-bution That makes the distribution of 119863 be
119891119863
(119863) =
1
radic2120587120590119863
119863
exp[minus
(ln119863 minus 120583119863
)2
21205902
119863
] 119863 gt 0
0 119863 le 0
(23)
where 120583119863is the logarithmic mean of cumulative damage 119863
120590119863is the logarithmic mean square deviation of 119863Based on the theory of probability fatigue cumulative
damage the structural dynamic reliability of one moment is
119877 = 119877 (119863 lt 1198630) = int
1198630
0
119891119863
(119910) 119889119910 (24)
where 1198630is the limit of damage index
According to Figure 1 in the gear transmission systemthe gears are connected in series thus the whole systemwill fail if one of the gears fails In other words the systemreliability is based on the reliability of each gear Thereforethe reliability model gear transmission system is as follows
119877 (119905) = 119877119888
(119905) sdot
3
prod
119894=1
119877119901119894
(119905) sdot 119877119903
(119905) sdot 119877119904
(119905) sdot
4
prod
119895=1
119877119895
(119905) (25)
where 119894 is the number of planetary gears 119895 is the number ofmedium speed level and high speed level gears
43 Calculation of System Dynamic Reliability The reliabilityof the gear transmission system is calculated using Matlabsoftware Based on the analysis before the steps of theprogram are as follows
(1) Take the random input torque of the gear transmis-sion system as the extern excitation Get the dynamicmeshing force and its statistic characteristics by usingthe numerical integration method
(2) Process the data of meshing force by rain flowcountingmethodThen calculate the equivalent stressamplitude and frequency by using equivalent methodof Geber quadratic curve
(3) Calculate structural fatigue damage under the luffingstress
(4) Calculate cumulative fatigue damage of arbitrary time119905 under several stress cycles
0 2 4 6 8 100
02
04
06
Time t (s)
med
ium
-leve
l spe
ed g
ears
F12
(N)
Mea
n dy
nam
ic m
esh
forc
e of times106
Figure 3 Mean dynamic meshing force of medium-level speedgears
(5) Calculate the structural limit value of fatigue damage
(6) By giving a random cumulative damage index applythe equation of dynamic reliability to calculate thereliability of each gear when the tooth surface reachesthe contact fatigue limit and the tooth root reaches thebending fatigue limit
(7) Calculate the dynamic reliability of the gear transmis-sion system using (25)
5 Analysis of Examples
The study object of the example research is the gear trans-mission system of a 15MW wind turbine Here are someparameters used in this research the rated power of the windturbine is 15MW the impeller diameter is 70m the designedimpeller speed is 148 rmin average wind speed of the windfarm is 143ms wind density is 121 kgm3 wind energyutilization factor is 032 system transmission ratio is 9453Suppose the strength of the material and the coefficient ofperformance both follow a normal distribution while otherparameters are constant Suppose the material of planetarygear is 40Cr and the material of medium-level speed andhigh-level speed gear is 20CrMnTi Other parameters of thesystem are shown in Table 1
By solving the dynamic equation (10) of the systemvibration displacement and vibration velocity of the gearsat each moment are obtained as well as their statisticalcharacteristics By solving (11) the meshing forces of eachgear are obtained Figure 3 shows the curve of mean dynamicmeshing force of medium-level speed gears Figure 4 shows
Mathematical Problems in Engineering 7
8000
6000
4000
2000
00
12
34
56
Freq
uenc
y
times106
times106
959
756
453
150
(a) mean (N)
(b) amplitude (N)
Figure 4 Luffing load spectrum of medium-level speed gears
092
096
1
0 5 10 15 20
094
098
E
b
e
120588d0
Time t (y)
Mea
n re
liabi
lity120583R(t)
Figure 5 Dynamic reliability of system when variation coefficientof random parameter is 0
the luffing load spectrum of medium-level speed gears basedon the theory of rain flow counting method
Wedefine the ratio of themean square error and themeanof the systemparameters as their variation coefficient Figures5ndash7 show the variation of systemdynamic reliability over timewith the variation coefficient being 0 01 and 03 respectivelyAs is demonstrated in Figure 5 the comprehensive transmis-sion error 119890 has the greatest influence on the system reliabilityfollowed by the elastic modulus of gear material 119864 contacttooth width 119861 and pitch circle diameter 119889
0 Mass density 120588
has the least influence By comparing Figures 6 and 7 wecan also learn that with the variation of random parametersincreases the system gets more reliable
Table 2 shows the dynamic reliability of each componentin the transmission system as the comprehensive transmis-sion error 119890 and mass density 120588 vary randomly when 119905 =
63times108 sThe table also shows that when the comprehensive
transmission error 119890 and mass density 120588 are 0 01 and 03respectively in the whole transmission system planetarygear system has the highest dynamic reliability followed by
1
5 10 15 20084
092
0
088
096
E
b
e
120588
d0
Mea
n re
liabi
lity120583R(t)
Time t (y)
Figure 6 Dynamic reliability of system when variation coefficientof random parameter is 01
5 10 15 200080
090
1
085
095M
ean
relia
bilit
y120583R(t)
E
b
e
120588
d0
Time t (y)
Figure 7 Dynamic reliability of system when variation coefficientof random parameter is 03
the medium speed level gears while high speed level gearis the least reliable In the planetary gear system internalgears have the highest reliability followed by the planetarygear while the sun gear is the least reliable In the mediumand high speed level gears large gears are more dynamic-reliable than the small ones The dynamic reliability of thegear transmission system reduces and the dispersion degreeof the system increases with the increase of the parametersrsquovariation
We obtained the statistical properties of the dynamicreliability of the high speed level gears through 20000simulations when 119905 = 63 times 10
8 s using Monte Carlo methodand compared the results with this paper as is shown inTable 3 The method proposed in this paper is more accuratethan Monte Carlo method
6 Conclusions
In this paper the dynamic reliability of the gear transmissionsystem of a 15MW wind turbine with consideration of
8 Mathematical Problems in Engineering
Table2Dyn
amicreliabilityof
each
compo
nent
ofplanetarygear
syste
mwith
rand
omparameters
Influ
encing
factors
Varia
tion
coeffi
cient
Reliabilityof
sun
gear
Reliabilityof
planetarygear
Reliabilityof
internalgear
Reliabilityof
large
gearsin
medium-le
velspeed
Reliabilityof
small
gearsin
medium-le
velspeed
Reliabilityof
large
gearsinhigh
-level
speed
Reliabilityof
small
gearsinhigh
-level
speed
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
120588
00987387
0007365
0991021
0007327
0995148
0007324
0979066
0007367
0971572
0007631
0975882
0007483
0963271
0007790
005
0980301
0007366
0984813
0007327
0991275
0007324
0978731
0007369
0965088
0007638
0971755
0007489
0950338
0007795
01
0953633
0007371
0959300
0007332
0975381
0007335
0950280
0007371
0931572
0007652
0942372
0007506
0927510
0007803
119890
00962833
0007802
0970332
0007789
0983727
0007756
0958207
0007803
0936281
0007817
0952283
0007804
0932588
0007832
005
0948471
0007815
0955783
0007795
0970115
0007758
0944342
0007815
0920175
0007822
0947502
0007815
0901392
0007863
01
0911502
0007843
0927009
0007804
0943928
0007781
0909252
0007845
0883011
0007827
0897641
0007844
0877252
0007904
Mathematical Problems in Engineering 9
Table 3 The comparison of dynamic reliability of big gear of high speed gear system
Random parameters Variation coefficient Proposed method Monte Carlo methodMean of 119877(119905) Root mean square of 119877(119905) Mean of 119877(119905) Root mean square of 119877(119905)
119861 01 0943257 0008450 0940832 0008541119864 01 0940144 0008671 0939124 0008454120588 01 0944946 0008377 0945271 00082721198890
01 0946826 0008498 0939567 0008157119890 01 0909033 0008870 0897354 0008625
randomness of load and system parameters is analyzedby applying the theory of probability of cumulative fatiguedamage The main contributions and conclusions of thispaper are the following
(1) The dynamic model of the gear transmission of windturbine is built In consideration of the randomness of theload and gear parameters the dynamic response of thesystem is obtained by utilizing the random sampling methodand Runge-Kutta method The statistical properties of themeshing force of components in the gear transmission systemare obtained by statistic method
(2) By applying the method of rain flow counting thetime history of the components meshing force is convertedinto a series of luffing load spectra and the equivalent stressamplitude and frequency are calculated according to theequivalent method of Geber quadratic curve
(3) The dynamic reliability model of the transmissionsystem and gear components are built according to theprinciple of probability fatigue damage cumulative Variationof the system reliability over time is calculated when theparameters vary and the effect of the parameter variation tothe system reliability is analyzed Results show that (i) thecomprehensive transmission error has the largest influenceon system dynamic reliability while the mass density hasthe least influence (ii) the dynamic reliability of the geartransmission system reduces and the dispersion degreeincreases with the increase of the variation of the parameters(iii) for the gear transmission system of the 15MW windturbine planetary gear system has the highest dynamicreliability followed by the medium speed level gears whilehigh speed level gear is the least reliable At the same timein the planetary gear system internal gears have the highestreliability followed by the planetary gear while the sun gearis the least reliable In themedium and high speed level gearslarge gears are more dynamic-reliable than the small ones
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Major State BasicResearch Development Program 973 (no 2012CB215202)the National Natural Science Foundation of China (no51205046) and the Fundamental Research Funds for the
Central Universities The constructive comments providedby the anonymous reviewers and the editors are also greatlyappreciated
References
[1] W Musial S Butterfield and B McNiff ldquoImproving windturbine gearbox reliabilityrdquo in Proceedings of the EuropeanWindEnergy Conference Milan Italy May 2007
[2] L Katafygiotis and S H Cheung ldquoWedge simulation methodfor calculating the reliability of linear dynamical systemsrdquoProbabilistic Engineering Mechanics vol 19 no 3 pp 229ndash2382004
[3] L Katafygiotis and S H Cheung ldquoDomain decompositionmethod for calculating the failure probability of linear dynamicsystems subjected to gaussian stochastic loadsrdquo Journal ofEngineering Mechanics vol 132 no 5 pp 475ndash486 2006
[4] P Liu and Q-F Yao ldquoEfficient estimation of dynamic reliabilitybased on simple additive rules of probabilityrdquo EngineeringMechanics vol 27 no 4 pp 1ndash4 2010
[5] H-W Qiao Z-Z Lu A-R Guan and X-H Liu ldquoDynamicreliability analysis of stochastic structures under stationaryrandom excitation using hermite polynomials approximationrdquoEngineering Mechanics vol 26 no 2 pp 60ndash64 2009
[6] A Lupoi P Franchin and M Schotanus ldquoSeismic risk eval-uation of RC bridge structuresrdquo Earthquake Engineering ampStructural Dynamics vol 32 no 8 pp 1275ndash1290 2003
[7] P Franchin ldquoReliability of uncertain inelastic structures underearthquake excitationrdquo Journal of Engineering Mechanics vol130 no 2 pp 180ndash191 2004
[8] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006
[9] Z Caichao H Zehao T Qian and T Yonghu ldquoAnalysis ofnonlinear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005
[10] D T Qin Z K Xing and J H Wang ldquoOptimization designof system parameters of the gear transmission of wind turbinebased on dynamics and reliabilityrdquo Chinese Journal of Mechani-cal Engineering vol 44 no 7 pp 24ndash31 2008
[11] D-T Qin X-G Gu J-H Wang and J-G Liu ldquoDynamicanalysis and optimization of gear trains in amegawatt level windturbinerdquo Journal of Chongqing University vol 32 no 4 pp 408ndash414 2009
[12] X-L Jiang and C-F Wang ldquoStochastic volatility models basedBayesian method and their applicationrdquo Systems Engineeringvol 23 no 10 pp 22ndash28 2005
[13] H T Chen X L Wu D T Qin J Yang and Z Zhou ldquoEffectsof gear manufacturing error on the dynamic characteristics of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
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Mathematical Problems in Engineering
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
X
Y
Z
c
r
k34
u4
e34c34
k12
k23
u3
u1
u2
us
ucup1
e12c12
c23
1
2
3
4
Xc
Yccrp2
erp2
crp3
krp2
krp3
up2
up3Tin
esp2
erp1crp1
csp2
csp1
esp1
csp3
ksp2esp3
erp3ksp3
ksp1
krp1
cs1
ks1
Figure 2 Torsional vibration model of gear transmission system
Its first derivative is
12057512
= 1199031198871
1205791
+ 1199031198872
1205792
minus 11989012
= 1
+ 2
minus 11989012
(6)
The elastic deformation in the direction of meshing forcebetween the spur gear 3 and the spur gear 4 is
12057534
= 1199031198873
1205793
+ 1199031198874
1205794
minus 11989034
= 1199063
+ 1199064
minus 11989034
(7)
Its first derivative is
12057534
= 1199031198873
1205793
+ 1199031198874
1205794
minus 11989034
= 3
+ 4
minus 11989034
(8)
So in the gear transmission system the relative displace-ments in the direction of meshing line of all gear pairs are
120575119904119901119894
= 119906119888cos120572119904119901
minus 119906119901119894
minus 119906119904
minus 119890119904119901119894
120575119903119901119894
= 119906119901119894
+ 119906119888cos120572119903119901
minus 119890119903119901119894
12057511989912
= 1199061
+ 1199062
minus 11989012
12057511989934
= 1199063
+ 1199064
minus 11989034
(9)
Equation (10) is the vibration differential equations of thesystem based on Lagrange equation Consider
(
119868119888
1199032
119887119888
) 119888
+
3
sum
119894=1
119888119903119901119894
cos120572119903119901119894
120575119903119901119894
+
3
sum
119894=1
119888119904119901119894
cos120572119904119901119894
120575119904119901119894
+ 119888119906119888
119888
+
3
sum
119894=1
119896119903119901119894
(119905) cos120572119903119901119894
120575119903119901119894
+
3
sum
119894=1
119896119904119901119894
(119905) cos120572119904119901119894
120575119904119901119894
+ 119896119906119888
119906119888
=
119879in119903119887119888
(
119868119901119894
1199032
119887119901119894
) 119901119894
minus 119888119904119901119894
120575119904119901119894
+ 119888119903119901119894
120575119903119901119894
minus 119896119904119901119894
(119905) 120575119904119901119894
+ 119896119903119901119894
(119905) 120575119903119901119894
= 0
(
119868119904
1199032
119887119904
) 119904
minus
3
sum
119894=1
119888119904119901119894
120575119904119901119894
+ 1198881199041
(
119904
1199032
119887119904
minus
1
119903119887119904
1199031198871
)
minus
3
sum
119894=1
119896119904119901119894
(119905) 120575119904119901119894
+ 1198961199041
(
119906119904
1199032
119887119904
minus
1199061
119903119887119904
1199031198871
) =
119879119904
119903119887119904
(
1198681
1199032
1198871
) 1
+ 11988812
12057511989912
+ 1198881199041
(
1
1199032
1198871
minus
119904
119903119887119904
1199031198871
)
+ 11989612
(119905) 12057511989912
+ 1198961199041
(
1199061
1199032
1198871
minus
119906119904
119903119887119904
1199031198871
) =
1198791
1199031198871
(
1198682
1199032
1198872
) 2
+ 11988812
12057511989912
+ 11988823
(
2
1199032
1198872
minus
3
1199031198872
1199031198873
)
+ 11989612
(119905) 12057511989912
+ 11989623
(
1199062
1199032
1198872
minus
1199063
1199031198872
1199031198873
) =
1198792
1199031198872
(
1198683
1199032
1198873
) 3
+ 11988834
12057511989934
+ 11988823
(
3
1199032
1198873
minus
2
1199031198872
1199031198873
)
+ 11989634
(119905) 12057511989934
+ 11989623
(
1199063
1199032
1198873
minus
1199062
1199031198872
1199031198873
) =
1198793
1199031198873
(
1198684
1199032
1198874
) 4
+ 11988834
12057511989934
+ 11989634
(119905) 12057511989934
= minus
119879out1199031198874
(10)
where 119903119887119888 119903119887119904 119903119887119901119894 119903119887119895represent the base circle radii of planet
carrier sun gear planetary gear the medium speed gearand the high speed gear respectively 119879in 119879out are the inputtorque and output torque of the system respectively
4 Mathematical Problems in Engineering
Equation (10) can be simplified as matrix form
119872 + 119862 + 119870 (119905) 119909 = 119879 (119905) (11)
where 119909 represents generalized displacement vector of thesystem 119909 = [119906
119888 1199061199011
1199061199012
1199061199013
119906119904 1199061 1199062 1199063]119879 119872 119862 119870(119905) are
9 order matrixes of mass damp and time varying stiffnesses119879(119905) is the vector of external load caused by external inputtorque
22 Solving of Equations The common approaches to solvethe equations of the dynamic model of gear transmission sys-tem are analyticalmethod and numerical simulationmethodThe former includes piecewise linearizationmethod and har-monic balance method while the latter includes Newmark-120573 method and Runge-Kutta method Unfortunately theirobjects are all determined systems thus making it impossibleto solve the dynamic response of random system by directlyapplying these existing methods In this paper the randomproblem is converted into a determined one by sampling therandom parameters in every moment
The specific steps are as follows
(1) Determine the elasticmodulusmass density workingtooth width pitch circle diameter and the distribu-tion of comprehensive transmission error of the gearmaterial
(2) Divide the external excitation into119873portions equallyand determine each integration time step Δ119905 based on119873
(3) Assume that the rest of the parameters are determinedwhen studying the influence of the response broughtby variation of one single parameter Sample thevarying parameter at each sampling time
(4) Obtain the dynamic response at one moment bycalculating the dynamic equations with the sampleresults using fixed step Runge-Kutta method
(5) Sample the parameters of the next moment andcalculate the dynamic response at this moment
(6) Change to another parameter and repeat (2)ndash(5)
After getting the statistical characteristics of vibrationdisplacement and vibration velocity of the system at eachmoment the dynamic meshing force of each gear pair can bederived from the following equation
119882119894119895
= 119896119894119895
sdot (119883119894119895
minus 119909119894119895
minus 119890119894119895
) + 119888119894119895
(119894119895
minus 119894119895
minus 119890119894119895
) (12)
in which 119896119894119895 119888119894119895 and 119883
119894119895 respectively are the meshing stiff-
ness damping coefficient and relative displacement betweengears 119894 and 119895 119909
119894119895is the equivalent displacement of center
displacement between the meshing lines of gears 119894 and 119895 119890119894119895is
the comprehensive meshing error of gears 119894 and 119895
3 Analysis of System Excitations
31 External Excitation The randomness of system loadis mainly caused by external wind load The variation
of external excitation of the gear transmission system isdetermined by the random wind velocity In this paperstochastic volatility (SV) model is built to obtain the randomwind velocity sequence in the wind farm Then the externalexcitation of the transmission system is calculated accordingto the theory of aerodynamic
SV model is a method of time series analysis whichis used in research on analyzing wind velocity The mainfeature of SV model is to regard volatility as an implicitvariable that cannot be observedThe basic form of SVmodelis [12]
V119905
= 120576119905
+ 119864 (119910119905120595119905minus1
) = 120590119905119911119905
ln (1205902
119905) = 119886 + 120593 ln (120590
2
119905minus1) + 120590120578120578119905
(13)
where V119905is the amplitude of volatility 120576
119905is kurtosis 119864(119910
119905|
120595119905minus1
) is the conditional mean of V119905calculated from the
information sampled at 119905 minus 1 120590119905is the conditional mean
square deviation 119911119905follows a normal distribution with 0
mean and 1 variance 119886 is a constant which reflects the averagevolatility 120593 is a parameter which reflects sustainability 120590
120578
is the mean square deviation of volatility disturbance 120578119905 119911119905
follow independent normal distributions with 0 mean and 1variance
The randomwind velocity simulated by SVmodel is takenas the input of the gear transmission system of wind turbineBased on the aerodynamic theory the input power of thetransmission system is [10]
119901in =
1
2
1205881198782V3119905119862119901
(14)
where 119901in is the input power of transmission system 120588 is airdensity 119878 is the sweeping area of wind turbine 119862
119901is wind
energy utilization factor V119905is the wind velocity simulated
from SV model far from wind turbinesThe external excitation of the system is the torque ripple
caused by random wind velocity The torques from the inputand output sides respectively are
119879in =
119901in120596
119879out =
119879in119894
(15)
where 120596 is the angular velocity of wind turbines 119894 is thetransmission ratio of the gear transmission system
32 Stiffness Excitation Stiffness excitation is a parametricexcitation caused by the variation of meshing stiffness duringthe meshing process Due to many influencing factors duringmachining and assembling the size and material of thegear transmission components vary randomly such as elasticmodulus and working tooth width In this paper gearrsquosstiffness is assumed to be a superposition of a sine wave anda randomwaveThe former is expressed by limited harmonicwaves of Fourier series and the latter is expressed by standard
Mathematical Problems in Engineering 5
normal distribution function Therefore the comprehensivemeshing stiffness of gears is as follows
119896 (119905) = 119896119898
+
119898
sum
119895=1
[1198961198951cos 119895120596119905 + 119896
1198952sin 119895120596119905] + 120576
1 (16)
where 119896119898is the average meshing stiffness of the gear pairs
1198961198951
and 1198961198952
are the meshing stiffness of harmonic waves 120596
is meshing frequency 1205761is stiffness fluctuation caused by
the variation of elastic modulus which follows a normaldistribution
33 Error Excitation Meshing error is a displacement excita-tion which is related to the machining accuracy of the gearsThe gear error and base pitch error can be expressed as asuperposition of a sine wave and a random wave as follows[13]
119890 (119905) = 119890119898
+ 119890119903sin(
2120587120596119905
119879
+ 120593) + 1205762 (17)
where 119890119898
and 119890119903are the offset and amplitude of the gear
meshing error 119879 120596 120593 are the meshing period of the gearpair meshing frequency and initial phase angle 120576
2is the
fluctuation of comprehensive transmission error caused bymachining and assembling which is assumed to follow anormal distribution In this paper the gear accuracy ispresumed to be grade 6 and parameters involved are basedon GBT 10095-1988 standard
4 Analysis of System Dynamic Reliability
41 Random Fatigue Load Spectrum of Gear TransmissionSystem Load-time history of each gear pair can be obtainedby the dynamic gear transmission model built before Thento analyze the fatigue reliability of the system the load-time history is converted into a series of complete cyclesThe main converting methods are peak counting methodcycle counting method rain flow counting method and soforth
In this paper we count the dynamic meshing force ofeach gear pair circularly according to the rain flow countingprinciple [14ndash16] in order to obtain the frequency of luffingfatigue load As is shown in Figure 5 the mean stress ofthe gear pairs follows a normal distribution approximatelyand the amplitude of the stress follows Weibull distributionapproximately
In order to analyze the fatigue life of the transmissionsystem the equivalent amplitude and frequency of thesystem stress are obtained by using equivalent method ofGeber quadratic curve The Geber quadratic curve formulais [17]
119878eqv = 119878119886
1205902
119887
1205902
119887minus 11987810158402
119898
(18)
where 119878119886is the amplitude of stress after the conversion 119878
1015840
119898is
the mean stress of 119878-119873 curve of the given material 119878eqv is theequivalent stress corresponding to 119878eqv with equal lifetime
42 Dynamic Reliability Model of Gear Transmission Com-ponents and System Fatigue failure of the components iscaused by the accumulation of material internal damage Asthe number of stress cycles increases the material internaldamage exacerbates and the structural life decreases Theoryof probability fatigue damage is based on the fatigue damageevolution which demonstrates the irreversibility and therandomness of fatigue damage The main reason of therandomness of fatigue damage lies in the characteristics ofthe material the geometric dimensions of the test pieces andthe uncertainty of external load
The decay rate of the material ultimate stress generallyfollows distribution as [18ndash20]
119889120590119906
119889119899
=
minus119891 (119878max 119891119888 119903)
119888120590119888
119906
(19)
where 119878max is the maximin cyclic stress 119891119888is the cycle
frequency 119903 is cyclic stress ratio 119888 is a constantThe remaining ultimate stress of the component material
after 119899 cycles is
120590119906
(119899) = 1205901199060
1 minus [1 minus (
119878max1205901199060
)
119888
]
119899
119899119894
1119888
(20)
where 120590119906(119899) is the remaining ultimate stress after 119899 cycles 120590
1199060
is the ultimate stress whenmaterials are in good condition 119899119894
is the number of ultimate cyclesThe damage index of component under the level 119894 luffing
cyclic stress after 119899 cycles is
Δ119863 =
119899
sum
119894=1
(1 minus (120590119906
(119899119894) 120590119906
(119899119894minus1
))119888
)
(1 minus (119878max 119894120590119906
(119899119894minus1
))119888
)
(21)
where 120590119906(119899119894) and 120590
119906(119899119894minus1
) are the remaining ultimate stressunder the level 119894 and level 119894 minus 1 stress 119878max 119894 is the maximumstress of level 119894 stress cycles
Suppose 120590119906(119899119894) (119894 = 1 2 119899) are independent random
variables from each other 120590119906
= (1205901199061
1205901199062
120590119906119899
) whicharemeans of 120590
119906(1198991) 120590119906(1198992) 120590
119906(119899119899) in (6) respectively are
expanded into the Taylor series Then the approximate mean120583Δ119863
and standard deviation 120590Δ119863
of the damage index Δ119863
are obtained by choosing the linear terms from the Taylorexpansion Consider
120583Δ119863
= 119892 (120590119863
) +
119899
sum
119894=1
(120590119906(119899119894) minus 120590119906119894
)
120597119892
120597120590119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+
1
2
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+ sdot sdot sdot
120590Δ119863
= radic119864(Δ119863119894
minus 120583Δ119863
)2
= radic
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
100381610038161003816100381610038161003816100381610038161003816120590119906119894
(22)
6 Mathematical Problems in Engineering
Table 1 Geometric parameters of gear transmission system of wind turbine
Low-level speedNumber of sun gear
teeth 119885119904
Number of planetarygear teeth 119885
119901
Number of internalgear teeth 119885
119903
Number ofmolds 119898
Meshing angle120572119904119901(∘)
Meshing angle120572119903119901(∘)
27 44 117 13 230000 210000Medium-levelspeed
Number of driving gears 1198851
Number of driven gears 1198852
Meshing angle 120572 (∘)104 23 210000
High-level speed Number of driving gears 1198853
Number of driven gears 1198854
Meshing angle 120572 (∘)98 25 210000
In engineering application random index of fatiguecumulative damage is supposed to follow a lognormal distri-bution That makes the distribution of 119863 be
119891119863
(119863) =
1
radic2120587120590119863
119863
exp[minus
(ln119863 minus 120583119863
)2
21205902
119863
] 119863 gt 0
0 119863 le 0
(23)
where 120583119863is the logarithmic mean of cumulative damage 119863
120590119863is the logarithmic mean square deviation of 119863Based on the theory of probability fatigue cumulative
damage the structural dynamic reliability of one moment is
119877 = 119877 (119863 lt 1198630) = int
1198630
0
119891119863
(119910) 119889119910 (24)
where 1198630is the limit of damage index
According to Figure 1 in the gear transmission systemthe gears are connected in series thus the whole systemwill fail if one of the gears fails In other words the systemreliability is based on the reliability of each gear Thereforethe reliability model gear transmission system is as follows
119877 (119905) = 119877119888
(119905) sdot
3
prod
119894=1
119877119901119894
(119905) sdot 119877119903
(119905) sdot 119877119904
(119905) sdot
4
prod
119895=1
119877119895
(119905) (25)
where 119894 is the number of planetary gears 119895 is the number ofmedium speed level and high speed level gears
43 Calculation of System Dynamic Reliability The reliabilityof the gear transmission system is calculated using Matlabsoftware Based on the analysis before the steps of theprogram are as follows
(1) Take the random input torque of the gear transmis-sion system as the extern excitation Get the dynamicmeshing force and its statistic characteristics by usingthe numerical integration method
(2) Process the data of meshing force by rain flowcountingmethodThen calculate the equivalent stressamplitude and frequency by using equivalent methodof Geber quadratic curve
(3) Calculate structural fatigue damage under the luffingstress
(4) Calculate cumulative fatigue damage of arbitrary time119905 under several stress cycles
0 2 4 6 8 100
02
04
06
Time t (s)
med
ium
-leve
l spe
ed g
ears
F12
(N)
Mea
n dy
nam
ic m
esh
forc
e of times106
Figure 3 Mean dynamic meshing force of medium-level speedgears
(5) Calculate the structural limit value of fatigue damage
(6) By giving a random cumulative damage index applythe equation of dynamic reliability to calculate thereliability of each gear when the tooth surface reachesthe contact fatigue limit and the tooth root reaches thebending fatigue limit
(7) Calculate the dynamic reliability of the gear transmis-sion system using (25)
5 Analysis of Examples
The study object of the example research is the gear trans-mission system of a 15MW wind turbine Here are someparameters used in this research the rated power of the windturbine is 15MW the impeller diameter is 70m the designedimpeller speed is 148 rmin average wind speed of the windfarm is 143ms wind density is 121 kgm3 wind energyutilization factor is 032 system transmission ratio is 9453Suppose the strength of the material and the coefficient ofperformance both follow a normal distribution while otherparameters are constant Suppose the material of planetarygear is 40Cr and the material of medium-level speed andhigh-level speed gear is 20CrMnTi Other parameters of thesystem are shown in Table 1
By solving the dynamic equation (10) of the systemvibration displacement and vibration velocity of the gearsat each moment are obtained as well as their statisticalcharacteristics By solving (11) the meshing forces of eachgear are obtained Figure 3 shows the curve of mean dynamicmeshing force of medium-level speed gears Figure 4 shows
Mathematical Problems in Engineering 7
8000
6000
4000
2000
00
12
34
56
Freq
uenc
y
times106
times106
959
756
453
150
(a) mean (N)
(b) amplitude (N)
Figure 4 Luffing load spectrum of medium-level speed gears
092
096
1
0 5 10 15 20
094
098
E
b
e
120588d0
Time t (y)
Mea
n re
liabi
lity120583R(t)
Figure 5 Dynamic reliability of system when variation coefficientof random parameter is 0
the luffing load spectrum of medium-level speed gears basedon the theory of rain flow counting method
Wedefine the ratio of themean square error and themeanof the systemparameters as their variation coefficient Figures5ndash7 show the variation of systemdynamic reliability over timewith the variation coefficient being 0 01 and 03 respectivelyAs is demonstrated in Figure 5 the comprehensive transmis-sion error 119890 has the greatest influence on the system reliabilityfollowed by the elastic modulus of gear material 119864 contacttooth width 119861 and pitch circle diameter 119889
0 Mass density 120588
has the least influence By comparing Figures 6 and 7 wecan also learn that with the variation of random parametersincreases the system gets more reliable
Table 2 shows the dynamic reliability of each componentin the transmission system as the comprehensive transmis-sion error 119890 and mass density 120588 vary randomly when 119905 =
63times108 sThe table also shows that when the comprehensive
transmission error 119890 and mass density 120588 are 0 01 and 03respectively in the whole transmission system planetarygear system has the highest dynamic reliability followed by
1
5 10 15 20084
092
0
088
096
E
b
e
120588
d0
Mea
n re
liabi
lity120583R(t)
Time t (y)
Figure 6 Dynamic reliability of system when variation coefficientof random parameter is 01
5 10 15 200080
090
1
085
095M
ean
relia
bilit
y120583R(t)
E
b
e
120588
d0
Time t (y)
Figure 7 Dynamic reliability of system when variation coefficientof random parameter is 03
the medium speed level gears while high speed level gearis the least reliable In the planetary gear system internalgears have the highest reliability followed by the planetarygear while the sun gear is the least reliable In the mediumand high speed level gears large gears are more dynamic-reliable than the small ones The dynamic reliability of thegear transmission system reduces and the dispersion degreeof the system increases with the increase of the parametersrsquovariation
We obtained the statistical properties of the dynamicreliability of the high speed level gears through 20000simulations when 119905 = 63 times 10
8 s using Monte Carlo methodand compared the results with this paper as is shown inTable 3 The method proposed in this paper is more accuratethan Monte Carlo method
6 Conclusions
In this paper the dynamic reliability of the gear transmissionsystem of a 15MW wind turbine with consideration of
8 Mathematical Problems in Engineering
Table2Dyn
amicreliabilityof
each
compo
nent
ofplanetarygear
syste
mwith
rand
omparameters
Influ
encing
factors
Varia
tion
coeffi
cient
Reliabilityof
sun
gear
Reliabilityof
planetarygear
Reliabilityof
internalgear
Reliabilityof
large
gearsin
medium-le
velspeed
Reliabilityof
small
gearsin
medium-le
velspeed
Reliabilityof
large
gearsinhigh
-level
speed
Reliabilityof
small
gearsinhigh
-level
speed
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
120588
00987387
0007365
0991021
0007327
0995148
0007324
0979066
0007367
0971572
0007631
0975882
0007483
0963271
0007790
005
0980301
0007366
0984813
0007327
0991275
0007324
0978731
0007369
0965088
0007638
0971755
0007489
0950338
0007795
01
0953633
0007371
0959300
0007332
0975381
0007335
0950280
0007371
0931572
0007652
0942372
0007506
0927510
0007803
119890
00962833
0007802
0970332
0007789
0983727
0007756
0958207
0007803
0936281
0007817
0952283
0007804
0932588
0007832
005
0948471
0007815
0955783
0007795
0970115
0007758
0944342
0007815
0920175
0007822
0947502
0007815
0901392
0007863
01
0911502
0007843
0927009
0007804
0943928
0007781
0909252
0007845
0883011
0007827
0897641
0007844
0877252
0007904
Mathematical Problems in Engineering 9
Table 3 The comparison of dynamic reliability of big gear of high speed gear system
Random parameters Variation coefficient Proposed method Monte Carlo methodMean of 119877(119905) Root mean square of 119877(119905) Mean of 119877(119905) Root mean square of 119877(119905)
119861 01 0943257 0008450 0940832 0008541119864 01 0940144 0008671 0939124 0008454120588 01 0944946 0008377 0945271 00082721198890
01 0946826 0008498 0939567 0008157119890 01 0909033 0008870 0897354 0008625
randomness of load and system parameters is analyzedby applying the theory of probability of cumulative fatiguedamage The main contributions and conclusions of thispaper are the following
(1) The dynamic model of the gear transmission of windturbine is built In consideration of the randomness of theload and gear parameters the dynamic response of thesystem is obtained by utilizing the random sampling methodand Runge-Kutta method The statistical properties of themeshing force of components in the gear transmission systemare obtained by statistic method
(2) By applying the method of rain flow counting thetime history of the components meshing force is convertedinto a series of luffing load spectra and the equivalent stressamplitude and frequency are calculated according to theequivalent method of Geber quadratic curve
(3) The dynamic reliability model of the transmissionsystem and gear components are built according to theprinciple of probability fatigue damage cumulative Variationof the system reliability over time is calculated when theparameters vary and the effect of the parameter variation tothe system reliability is analyzed Results show that (i) thecomprehensive transmission error has the largest influenceon system dynamic reliability while the mass density hasthe least influence (ii) the dynamic reliability of the geartransmission system reduces and the dispersion degreeincreases with the increase of the variation of the parameters(iii) for the gear transmission system of the 15MW windturbine planetary gear system has the highest dynamicreliability followed by the medium speed level gears whilehigh speed level gear is the least reliable At the same timein the planetary gear system internal gears have the highestreliability followed by the planetary gear while the sun gearis the least reliable In themedium and high speed level gearslarge gears are more dynamic-reliable than the small ones
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Major State BasicResearch Development Program 973 (no 2012CB215202)the National Natural Science Foundation of China (no51205046) and the Fundamental Research Funds for the
Central Universities The constructive comments providedby the anonymous reviewers and the editors are also greatlyappreciated
References
[1] W Musial S Butterfield and B McNiff ldquoImproving windturbine gearbox reliabilityrdquo in Proceedings of the EuropeanWindEnergy Conference Milan Italy May 2007
[2] L Katafygiotis and S H Cheung ldquoWedge simulation methodfor calculating the reliability of linear dynamical systemsrdquoProbabilistic Engineering Mechanics vol 19 no 3 pp 229ndash2382004
[3] L Katafygiotis and S H Cheung ldquoDomain decompositionmethod for calculating the failure probability of linear dynamicsystems subjected to gaussian stochastic loadsrdquo Journal ofEngineering Mechanics vol 132 no 5 pp 475ndash486 2006
[4] P Liu and Q-F Yao ldquoEfficient estimation of dynamic reliabilitybased on simple additive rules of probabilityrdquo EngineeringMechanics vol 27 no 4 pp 1ndash4 2010
[5] H-W Qiao Z-Z Lu A-R Guan and X-H Liu ldquoDynamicreliability analysis of stochastic structures under stationaryrandom excitation using hermite polynomials approximationrdquoEngineering Mechanics vol 26 no 2 pp 60ndash64 2009
[6] A Lupoi P Franchin and M Schotanus ldquoSeismic risk eval-uation of RC bridge structuresrdquo Earthquake Engineering ampStructural Dynamics vol 32 no 8 pp 1275ndash1290 2003
[7] P Franchin ldquoReliability of uncertain inelastic structures underearthquake excitationrdquo Journal of Engineering Mechanics vol130 no 2 pp 180ndash191 2004
[8] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006
[9] Z Caichao H Zehao T Qian and T Yonghu ldquoAnalysis ofnonlinear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005
[10] D T Qin Z K Xing and J H Wang ldquoOptimization designof system parameters of the gear transmission of wind turbinebased on dynamics and reliabilityrdquo Chinese Journal of Mechani-cal Engineering vol 44 no 7 pp 24ndash31 2008
[11] D-T Qin X-G Gu J-H Wang and J-G Liu ldquoDynamicanalysis and optimization of gear trains in amegawatt level windturbinerdquo Journal of Chongqing University vol 32 no 4 pp 408ndash414 2009
[12] X-L Jiang and C-F Wang ldquoStochastic volatility models basedBayesian method and their applicationrdquo Systems Engineeringvol 23 no 10 pp 22ndash28 2005
[13] H T Chen X L Wu D T Qin J Yang and Z Zhou ldquoEffectsof gear manufacturing error on the dynamic characteristics of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Complex AnalysisJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Equation (10) can be simplified as matrix form
119872 + 119862 + 119870 (119905) 119909 = 119879 (119905) (11)
where 119909 represents generalized displacement vector of thesystem 119909 = [119906
119888 1199061199011
1199061199012
1199061199013
119906119904 1199061 1199062 1199063]119879 119872 119862 119870(119905) are
9 order matrixes of mass damp and time varying stiffnesses119879(119905) is the vector of external load caused by external inputtorque
22 Solving of Equations The common approaches to solvethe equations of the dynamic model of gear transmission sys-tem are analyticalmethod and numerical simulationmethodThe former includes piecewise linearizationmethod and har-monic balance method while the latter includes Newmark-120573 method and Runge-Kutta method Unfortunately theirobjects are all determined systems thus making it impossibleto solve the dynamic response of random system by directlyapplying these existing methods In this paper the randomproblem is converted into a determined one by sampling therandom parameters in every moment
The specific steps are as follows
(1) Determine the elasticmodulusmass density workingtooth width pitch circle diameter and the distribu-tion of comprehensive transmission error of the gearmaterial
(2) Divide the external excitation into119873portions equallyand determine each integration time step Δ119905 based on119873
(3) Assume that the rest of the parameters are determinedwhen studying the influence of the response broughtby variation of one single parameter Sample thevarying parameter at each sampling time
(4) Obtain the dynamic response at one moment bycalculating the dynamic equations with the sampleresults using fixed step Runge-Kutta method
(5) Sample the parameters of the next moment andcalculate the dynamic response at this moment
(6) Change to another parameter and repeat (2)ndash(5)
After getting the statistical characteristics of vibrationdisplacement and vibration velocity of the system at eachmoment the dynamic meshing force of each gear pair can bederived from the following equation
119882119894119895
= 119896119894119895
sdot (119883119894119895
minus 119909119894119895
minus 119890119894119895
) + 119888119894119895
(119894119895
minus 119894119895
minus 119890119894119895
) (12)
in which 119896119894119895 119888119894119895 and 119883
119894119895 respectively are the meshing stiff-
ness damping coefficient and relative displacement betweengears 119894 and 119895 119909
119894119895is the equivalent displacement of center
displacement between the meshing lines of gears 119894 and 119895 119890119894119895is
the comprehensive meshing error of gears 119894 and 119895
3 Analysis of System Excitations
31 External Excitation The randomness of system loadis mainly caused by external wind load The variation
of external excitation of the gear transmission system isdetermined by the random wind velocity In this paperstochastic volatility (SV) model is built to obtain the randomwind velocity sequence in the wind farm Then the externalexcitation of the transmission system is calculated accordingto the theory of aerodynamic
SV model is a method of time series analysis whichis used in research on analyzing wind velocity The mainfeature of SV model is to regard volatility as an implicitvariable that cannot be observedThe basic form of SVmodelis [12]
V119905
= 120576119905
+ 119864 (119910119905120595119905minus1
) = 120590119905119911119905
ln (1205902
119905) = 119886 + 120593 ln (120590
2
119905minus1) + 120590120578120578119905
(13)
where V119905is the amplitude of volatility 120576
119905is kurtosis 119864(119910
119905|
120595119905minus1
) is the conditional mean of V119905calculated from the
information sampled at 119905 minus 1 120590119905is the conditional mean
square deviation 119911119905follows a normal distribution with 0
mean and 1 variance 119886 is a constant which reflects the averagevolatility 120593 is a parameter which reflects sustainability 120590
120578
is the mean square deviation of volatility disturbance 120578119905 119911119905
follow independent normal distributions with 0 mean and 1variance
The randomwind velocity simulated by SVmodel is takenas the input of the gear transmission system of wind turbineBased on the aerodynamic theory the input power of thetransmission system is [10]
119901in =
1
2
1205881198782V3119905119862119901
(14)
where 119901in is the input power of transmission system 120588 is airdensity 119878 is the sweeping area of wind turbine 119862
119901is wind
energy utilization factor V119905is the wind velocity simulated
from SV model far from wind turbinesThe external excitation of the system is the torque ripple
caused by random wind velocity The torques from the inputand output sides respectively are
119879in =
119901in120596
119879out =
119879in119894
(15)
where 120596 is the angular velocity of wind turbines 119894 is thetransmission ratio of the gear transmission system
32 Stiffness Excitation Stiffness excitation is a parametricexcitation caused by the variation of meshing stiffness duringthe meshing process Due to many influencing factors duringmachining and assembling the size and material of thegear transmission components vary randomly such as elasticmodulus and working tooth width In this paper gearrsquosstiffness is assumed to be a superposition of a sine wave anda randomwaveThe former is expressed by limited harmonicwaves of Fourier series and the latter is expressed by standard
Mathematical Problems in Engineering 5
normal distribution function Therefore the comprehensivemeshing stiffness of gears is as follows
119896 (119905) = 119896119898
+
119898
sum
119895=1
[1198961198951cos 119895120596119905 + 119896
1198952sin 119895120596119905] + 120576
1 (16)
where 119896119898is the average meshing stiffness of the gear pairs
1198961198951
and 1198961198952
are the meshing stiffness of harmonic waves 120596
is meshing frequency 1205761is stiffness fluctuation caused by
the variation of elastic modulus which follows a normaldistribution
33 Error Excitation Meshing error is a displacement excita-tion which is related to the machining accuracy of the gearsThe gear error and base pitch error can be expressed as asuperposition of a sine wave and a random wave as follows[13]
119890 (119905) = 119890119898
+ 119890119903sin(
2120587120596119905
119879
+ 120593) + 1205762 (17)
where 119890119898
and 119890119903are the offset and amplitude of the gear
meshing error 119879 120596 120593 are the meshing period of the gearpair meshing frequency and initial phase angle 120576
2is the
fluctuation of comprehensive transmission error caused bymachining and assembling which is assumed to follow anormal distribution In this paper the gear accuracy ispresumed to be grade 6 and parameters involved are basedon GBT 10095-1988 standard
4 Analysis of System Dynamic Reliability
41 Random Fatigue Load Spectrum of Gear TransmissionSystem Load-time history of each gear pair can be obtainedby the dynamic gear transmission model built before Thento analyze the fatigue reliability of the system the load-time history is converted into a series of complete cyclesThe main converting methods are peak counting methodcycle counting method rain flow counting method and soforth
In this paper we count the dynamic meshing force ofeach gear pair circularly according to the rain flow countingprinciple [14ndash16] in order to obtain the frequency of luffingfatigue load As is shown in Figure 5 the mean stress ofthe gear pairs follows a normal distribution approximatelyand the amplitude of the stress follows Weibull distributionapproximately
In order to analyze the fatigue life of the transmissionsystem the equivalent amplitude and frequency of thesystem stress are obtained by using equivalent method ofGeber quadratic curve The Geber quadratic curve formulais [17]
119878eqv = 119878119886
1205902
119887
1205902
119887minus 11987810158402
119898
(18)
where 119878119886is the amplitude of stress after the conversion 119878
1015840
119898is
the mean stress of 119878-119873 curve of the given material 119878eqv is theequivalent stress corresponding to 119878eqv with equal lifetime
42 Dynamic Reliability Model of Gear Transmission Com-ponents and System Fatigue failure of the components iscaused by the accumulation of material internal damage Asthe number of stress cycles increases the material internaldamage exacerbates and the structural life decreases Theoryof probability fatigue damage is based on the fatigue damageevolution which demonstrates the irreversibility and therandomness of fatigue damage The main reason of therandomness of fatigue damage lies in the characteristics ofthe material the geometric dimensions of the test pieces andthe uncertainty of external load
The decay rate of the material ultimate stress generallyfollows distribution as [18ndash20]
119889120590119906
119889119899
=
minus119891 (119878max 119891119888 119903)
119888120590119888
119906
(19)
where 119878max is the maximin cyclic stress 119891119888is the cycle
frequency 119903 is cyclic stress ratio 119888 is a constantThe remaining ultimate stress of the component material
after 119899 cycles is
120590119906
(119899) = 1205901199060
1 minus [1 minus (
119878max1205901199060
)
119888
]
119899
119899119894
1119888
(20)
where 120590119906(119899) is the remaining ultimate stress after 119899 cycles 120590
1199060
is the ultimate stress whenmaterials are in good condition 119899119894
is the number of ultimate cyclesThe damage index of component under the level 119894 luffing
cyclic stress after 119899 cycles is
Δ119863 =
119899
sum
119894=1
(1 minus (120590119906
(119899119894) 120590119906
(119899119894minus1
))119888
)
(1 minus (119878max 119894120590119906
(119899119894minus1
))119888
)
(21)
where 120590119906(119899119894) and 120590
119906(119899119894minus1
) are the remaining ultimate stressunder the level 119894 and level 119894 minus 1 stress 119878max 119894 is the maximumstress of level 119894 stress cycles
Suppose 120590119906(119899119894) (119894 = 1 2 119899) are independent random
variables from each other 120590119906
= (1205901199061
1205901199062
120590119906119899
) whicharemeans of 120590
119906(1198991) 120590119906(1198992) 120590
119906(119899119899) in (6) respectively are
expanded into the Taylor series Then the approximate mean120583Δ119863
and standard deviation 120590Δ119863
of the damage index Δ119863
are obtained by choosing the linear terms from the Taylorexpansion Consider
120583Δ119863
= 119892 (120590119863
) +
119899
sum
119894=1
(120590119906(119899119894) minus 120590119906119894
)
120597119892
120597120590119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+
1
2
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+ sdot sdot sdot
120590Δ119863
= radic119864(Δ119863119894
minus 120583Δ119863
)2
= radic
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
100381610038161003816100381610038161003816100381610038161003816120590119906119894
(22)
6 Mathematical Problems in Engineering
Table 1 Geometric parameters of gear transmission system of wind turbine
Low-level speedNumber of sun gear
teeth 119885119904
Number of planetarygear teeth 119885
119901
Number of internalgear teeth 119885
119903
Number ofmolds 119898
Meshing angle120572119904119901(∘)
Meshing angle120572119903119901(∘)
27 44 117 13 230000 210000Medium-levelspeed
Number of driving gears 1198851
Number of driven gears 1198852
Meshing angle 120572 (∘)104 23 210000
High-level speed Number of driving gears 1198853
Number of driven gears 1198854
Meshing angle 120572 (∘)98 25 210000
In engineering application random index of fatiguecumulative damage is supposed to follow a lognormal distri-bution That makes the distribution of 119863 be
119891119863
(119863) =
1
radic2120587120590119863
119863
exp[minus
(ln119863 minus 120583119863
)2
21205902
119863
] 119863 gt 0
0 119863 le 0
(23)
where 120583119863is the logarithmic mean of cumulative damage 119863
120590119863is the logarithmic mean square deviation of 119863Based on the theory of probability fatigue cumulative
damage the structural dynamic reliability of one moment is
119877 = 119877 (119863 lt 1198630) = int
1198630
0
119891119863
(119910) 119889119910 (24)
where 1198630is the limit of damage index
According to Figure 1 in the gear transmission systemthe gears are connected in series thus the whole systemwill fail if one of the gears fails In other words the systemreliability is based on the reliability of each gear Thereforethe reliability model gear transmission system is as follows
119877 (119905) = 119877119888
(119905) sdot
3
prod
119894=1
119877119901119894
(119905) sdot 119877119903
(119905) sdot 119877119904
(119905) sdot
4
prod
119895=1
119877119895
(119905) (25)
where 119894 is the number of planetary gears 119895 is the number ofmedium speed level and high speed level gears
43 Calculation of System Dynamic Reliability The reliabilityof the gear transmission system is calculated using Matlabsoftware Based on the analysis before the steps of theprogram are as follows
(1) Take the random input torque of the gear transmis-sion system as the extern excitation Get the dynamicmeshing force and its statistic characteristics by usingthe numerical integration method
(2) Process the data of meshing force by rain flowcountingmethodThen calculate the equivalent stressamplitude and frequency by using equivalent methodof Geber quadratic curve
(3) Calculate structural fatigue damage under the luffingstress
(4) Calculate cumulative fatigue damage of arbitrary time119905 under several stress cycles
0 2 4 6 8 100
02
04
06
Time t (s)
med
ium
-leve
l spe
ed g
ears
F12
(N)
Mea
n dy
nam
ic m
esh
forc
e of times106
Figure 3 Mean dynamic meshing force of medium-level speedgears
(5) Calculate the structural limit value of fatigue damage
(6) By giving a random cumulative damage index applythe equation of dynamic reliability to calculate thereliability of each gear when the tooth surface reachesthe contact fatigue limit and the tooth root reaches thebending fatigue limit
(7) Calculate the dynamic reliability of the gear transmis-sion system using (25)
5 Analysis of Examples
The study object of the example research is the gear trans-mission system of a 15MW wind turbine Here are someparameters used in this research the rated power of the windturbine is 15MW the impeller diameter is 70m the designedimpeller speed is 148 rmin average wind speed of the windfarm is 143ms wind density is 121 kgm3 wind energyutilization factor is 032 system transmission ratio is 9453Suppose the strength of the material and the coefficient ofperformance both follow a normal distribution while otherparameters are constant Suppose the material of planetarygear is 40Cr and the material of medium-level speed andhigh-level speed gear is 20CrMnTi Other parameters of thesystem are shown in Table 1
By solving the dynamic equation (10) of the systemvibration displacement and vibration velocity of the gearsat each moment are obtained as well as their statisticalcharacteristics By solving (11) the meshing forces of eachgear are obtained Figure 3 shows the curve of mean dynamicmeshing force of medium-level speed gears Figure 4 shows
Mathematical Problems in Engineering 7
8000
6000
4000
2000
00
12
34
56
Freq
uenc
y
times106
times106
959
756
453
150
(a) mean (N)
(b) amplitude (N)
Figure 4 Luffing load spectrum of medium-level speed gears
092
096
1
0 5 10 15 20
094
098
E
b
e
120588d0
Time t (y)
Mea
n re
liabi
lity120583R(t)
Figure 5 Dynamic reliability of system when variation coefficientof random parameter is 0
the luffing load spectrum of medium-level speed gears basedon the theory of rain flow counting method
Wedefine the ratio of themean square error and themeanof the systemparameters as their variation coefficient Figures5ndash7 show the variation of systemdynamic reliability over timewith the variation coefficient being 0 01 and 03 respectivelyAs is demonstrated in Figure 5 the comprehensive transmis-sion error 119890 has the greatest influence on the system reliabilityfollowed by the elastic modulus of gear material 119864 contacttooth width 119861 and pitch circle diameter 119889
0 Mass density 120588
has the least influence By comparing Figures 6 and 7 wecan also learn that with the variation of random parametersincreases the system gets more reliable
Table 2 shows the dynamic reliability of each componentin the transmission system as the comprehensive transmis-sion error 119890 and mass density 120588 vary randomly when 119905 =
63times108 sThe table also shows that when the comprehensive
transmission error 119890 and mass density 120588 are 0 01 and 03respectively in the whole transmission system planetarygear system has the highest dynamic reliability followed by
1
5 10 15 20084
092
0
088
096
E
b
e
120588
d0
Mea
n re
liabi
lity120583R(t)
Time t (y)
Figure 6 Dynamic reliability of system when variation coefficientof random parameter is 01
5 10 15 200080
090
1
085
095M
ean
relia
bilit
y120583R(t)
E
b
e
120588
d0
Time t (y)
Figure 7 Dynamic reliability of system when variation coefficientof random parameter is 03
the medium speed level gears while high speed level gearis the least reliable In the planetary gear system internalgears have the highest reliability followed by the planetarygear while the sun gear is the least reliable In the mediumand high speed level gears large gears are more dynamic-reliable than the small ones The dynamic reliability of thegear transmission system reduces and the dispersion degreeof the system increases with the increase of the parametersrsquovariation
We obtained the statistical properties of the dynamicreliability of the high speed level gears through 20000simulations when 119905 = 63 times 10
8 s using Monte Carlo methodand compared the results with this paper as is shown inTable 3 The method proposed in this paper is more accuratethan Monte Carlo method
6 Conclusions
In this paper the dynamic reliability of the gear transmissionsystem of a 15MW wind turbine with consideration of
8 Mathematical Problems in Engineering
Table2Dyn
amicreliabilityof
each
compo
nent
ofplanetarygear
syste
mwith
rand
omparameters
Influ
encing
factors
Varia
tion
coeffi
cient
Reliabilityof
sun
gear
Reliabilityof
planetarygear
Reliabilityof
internalgear
Reliabilityof
large
gearsin
medium-le
velspeed
Reliabilityof
small
gearsin
medium-le
velspeed
Reliabilityof
large
gearsinhigh
-level
speed
Reliabilityof
small
gearsinhigh
-level
speed
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
120588
00987387
0007365
0991021
0007327
0995148
0007324
0979066
0007367
0971572
0007631
0975882
0007483
0963271
0007790
005
0980301
0007366
0984813
0007327
0991275
0007324
0978731
0007369
0965088
0007638
0971755
0007489
0950338
0007795
01
0953633
0007371
0959300
0007332
0975381
0007335
0950280
0007371
0931572
0007652
0942372
0007506
0927510
0007803
119890
00962833
0007802
0970332
0007789
0983727
0007756
0958207
0007803
0936281
0007817
0952283
0007804
0932588
0007832
005
0948471
0007815
0955783
0007795
0970115
0007758
0944342
0007815
0920175
0007822
0947502
0007815
0901392
0007863
01
0911502
0007843
0927009
0007804
0943928
0007781
0909252
0007845
0883011
0007827
0897641
0007844
0877252
0007904
Mathematical Problems in Engineering 9
Table 3 The comparison of dynamic reliability of big gear of high speed gear system
Random parameters Variation coefficient Proposed method Monte Carlo methodMean of 119877(119905) Root mean square of 119877(119905) Mean of 119877(119905) Root mean square of 119877(119905)
119861 01 0943257 0008450 0940832 0008541119864 01 0940144 0008671 0939124 0008454120588 01 0944946 0008377 0945271 00082721198890
01 0946826 0008498 0939567 0008157119890 01 0909033 0008870 0897354 0008625
randomness of load and system parameters is analyzedby applying the theory of probability of cumulative fatiguedamage The main contributions and conclusions of thispaper are the following
(1) The dynamic model of the gear transmission of windturbine is built In consideration of the randomness of theload and gear parameters the dynamic response of thesystem is obtained by utilizing the random sampling methodand Runge-Kutta method The statistical properties of themeshing force of components in the gear transmission systemare obtained by statistic method
(2) By applying the method of rain flow counting thetime history of the components meshing force is convertedinto a series of luffing load spectra and the equivalent stressamplitude and frequency are calculated according to theequivalent method of Geber quadratic curve
(3) The dynamic reliability model of the transmissionsystem and gear components are built according to theprinciple of probability fatigue damage cumulative Variationof the system reliability over time is calculated when theparameters vary and the effect of the parameter variation tothe system reliability is analyzed Results show that (i) thecomprehensive transmission error has the largest influenceon system dynamic reliability while the mass density hasthe least influence (ii) the dynamic reliability of the geartransmission system reduces and the dispersion degreeincreases with the increase of the variation of the parameters(iii) for the gear transmission system of the 15MW windturbine planetary gear system has the highest dynamicreliability followed by the medium speed level gears whilehigh speed level gear is the least reliable At the same timein the planetary gear system internal gears have the highestreliability followed by the planetary gear while the sun gearis the least reliable In themedium and high speed level gearslarge gears are more dynamic-reliable than the small ones
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Major State BasicResearch Development Program 973 (no 2012CB215202)the National Natural Science Foundation of China (no51205046) and the Fundamental Research Funds for the
Central Universities The constructive comments providedby the anonymous reviewers and the editors are also greatlyappreciated
References
[1] W Musial S Butterfield and B McNiff ldquoImproving windturbine gearbox reliabilityrdquo in Proceedings of the EuropeanWindEnergy Conference Milan Italy May 2007
[2] L Katafygiotis and S H Cheung ldquoWedge simulation methodfor calculating the reliability of linear dynamical systemsrdquoProbabilistic Engineering Mechanics vol 19 no 3 pp 229ndash2382004
[3] L Katafygiotis and S H Cheung ldquoDomain decompositionmethod for calculating the failure probability of linear dynamicsystems subjected to gaussian stochastic loadsrdquo Journal ofEngineering Mechanics vol 132 no 5 pp 475ndash486 2006
[4] P Liu and Q-F Yao ldquoEfficient estimation of dynamic reliabilitybased on simple additive rules of probabilityrdquo EngineeringMechanics vol 27 no 4 pp 1ndash4 2010
[5] H-W Qiao Z-Z Lu A-R Guan and X-H Liu ldquoDynamicreliability analysis of stochastic structures under stationaryrandom excitation using hermite polynomials approximationrdquoEngineering Mechanics vol 26 no 2 pp 60ndash64 2009
[6] A Lupoi P Franchin and M Schotanus ldquoSeismic risk eval-uation of RC bridge structuresrdquo Earthquake Engineering ampStructural Dynamics vol 32 no 8 pp 1275ndash1290 2003
[7] P Franchin ldquoReliability of uncertain inelastic structures underearthquake excitationrdquo Journal of Engineering Mechanics vol130 no 2 pp 180ndash191 2004
[8] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006
[9] Z Caichao H Zehao T Qian and T Yonghu ldquoAnalysis ofnonlinear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005
[10] D T Qin Z K Xing and J H Wang ldquoOptimization designof system parameters of the gear transmission of wind turbinebased on dynamics and reliabilityrdquo Chinese Journal of Mechani-cal Engineering vol 44 no 7 pp 24ndash31 2008
[11] D-T Qin X-G Gu J-H Wang and J-G Liu ldquoDynamicanalysis and optimization of gear trains in amegawatt level windturbinerdquo Journal of Chongqing University vol 32 no 4 pp 408ndash414 2009
[12] X-L Jiang and C-F Wang ldquoStochastic volatility models basedBayesian method and their applicationrdquo Systems Engineeringvol 23 no 10 pp 22ndash28 2005
[13] H T Chen X L Wu D T Qin J Yang and Z Zhou ldquoEffectsof gear manufacturing error on the dynamic characteristics of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
normal distribution function Therefore the comprehensivemeshing stiffness of gears is as follows
119896 (119905) = 119896119898
+
119898
sum
119895=1
[1198961198951cos 119895120596119905 + 119896
1198952sin 119895120596119905] + 120576
1 (16)
where 119896119898is the average meshing stiffness of the gear pairs
1198961198951
and 1198961198952
are the meshing stiffness of harmonic waves 120596
is meshing frequency 1205761is stiffness fluctuation caused by
the variation of elastic modulus which follows a normaldistribution
33 Error Excitation Meshing error is a displacement excita-tion which is related to the machining accuracy of the gearsThe gear error and base pitch error can be expressed as asuperposition of a sine wave and a random wave as follows[13]
119890 (119905) = 119890119898
+ 119890119903sin(
2120587120596119905
119879
+ 120593) + 1205762 (17)
where 119890119898
and 119890119903are the offset and amplitude of the gear
meshing error 119879 120596 120593 are the meshing period of the gearpair meshing frequency and initial phase angle 120576
2is the
fluctuation of comprehensive transmission error caused bymachining and assembling which is assumed to follow anormal distribution In this paper the gear accuracy ispresumed to be grade 6 and parameters involved are basedon GBT 10095-1988 standard
4 Analysis of System Dynamic Reliability
41 Random Fatigue Load Spectrum of Gear TransmissionSystem Load-time history of each gear pair can be obtainedby the dynamic gear transmission model built before Thento analyze the fatigue reliability of the system the load-time history is converted into a series of complete cyclesThe main converting methods are peak counting methodcycle counting method rain flow counting method and soforth
In this paper we count the dynamic meshing force ofeach gear pair circularly according to the rain flow countingprinciple [14ndash16] in order to obtain the frequency of luffingfatigue load As is shown in Figure 5 the mean stress ofthe gear pairs follows a normal distribution approximatelyand the amplitude of the stress follows Weibull distributionapproximately
In order to analyze the fatigue life of the transmissionsystem the equivalent amplitude and frequency of thesystem stress are obtained by using equivalent method ofGeber quadratic curve The Geber quadratic curve formulais [17]
119878eqv = 119878119886
1205902
119887
1205902
119887minus 11987810158402
119898
(18)
where 119878119886is the amplitude of stress after the conversion 119878
1015840
119898is
the mean stress of 119878-119873 curve of the given material 119878eqv is theequivalent stress corresponding to 119878eqv with equal lifetime
42 Dynamic Reliability Model of Gear Transmission Com-ponents and System Fatigue failure of the components iscaused by the accumulation of material internal damage Asthe number of stress cycles increases the material internaldamage exacerbates and the structural life decreases Theoryof probability fatigue damage is based on the fatigue damageevolution which demonstrates the irreversibility and therandomness of fatigue damage The main reason of therandomness of fatigue damage lies in the characteristics ofthe material the geometric dimensions of the test pieces andthe uncertainty of external load
The decay rate of the material ultimate stress generallyfollows distribution as [18ndash20]
119889120590119906
119889119899
=
minus119891 (119878max 119891119888 119903)
119888120590119888
119906
(19)
where 119878max is the maximin cyclic stress 119891119888is the cycle
frequency 119903 is cyclic stress ratio 119888 is a constantThe remaining ultimate stress of the component material
after 119899 cycles is
120590119906
(119899) = 1205901199060
1 minus [1 minus (
119878max1205901199060
)
119888
]
119899
119899119894
1119888
(20)
where 120590119906(119899) is the remaining ultimate stress after 119899 cycles 120590
1199060
is the ultimate stress whenmaterials are in good condition 119899119894
is the number of ultimate cyclesThe damage index of component under the level 119894 luffing
cyclic stress after 119899 cycles is
Δ119863 =
119899
sum
119894=1
(1 minus (120590119906
(119899119894) 120590119906
(119899119894minus1
))119888
)
(1 minus (119878max 119894120590119906
(119899119894minus1
))119888
)
(21)
where 120590119906(119899119894) and 120590
119906(119899119894minus1
) are the remaining ultimate stressunder the level 119894 and level 119894 minus 1 stress 119878max 119894 is the maximumstress of level 119894 stress cycles
Suppose 120590119906(119899119894) (119894 = 1 2 119899) are independent random
variables from each other 120590119906
= (1205901199061
1205901199062
120590119906119899
) whicharemeans of 120590
119906(1198991) 120590119906(1198992) 120590
119906(119899119899) in (6) respectively are
expanded into the Taylor series Then the approximate mean120583Δ119863
and standard deviation 120590Δ119863
of the damage index Δ119863
are obtained by choosing the linear terms from the Taylorexpansion Consider
120583Δ119863
= 119892 (120590119863
) +
119899
sum
119894=1
(120590119906(119899119894) minus 120590119906119894
)
120597119892
120597120590119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+
1
2
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
1003816100381610038161003816100381610038161003816100381610038161003816120590119906119894
+ sdot sdot sdot
120590Δ119863
= radic119864(Δ119863119894
minus 120583Δ119863
)2
= radic
119899
sum
119894=1
(120590119906
(119899119894) minus 120590119906119894
)2 1205972
119892
1205971205902
119906(119899119894)
100381610038161003816100381610038161003816100381610038161003816120590119906119894
(22)
6 Mathematical Problems in Engineering
Table 1 Geometric parameters of gear transmission system of wind turbine
Low-level speedNumber of sun gear
teeth 119885119904
Number of planetarygear teeth 119885
119901
Number of internalgear teeth 119885
119903
Number ofmolds 119898
Meshing angle120572119904119901(∘)
Meshing angle120572119903119901(∘)
27 44 117 13 230000 210000Medium-levelspeed
Number of driving gears 1198851
Number of driven gears 1198852
Meshing angle 120572 (∘)104 23 210000
High-level speed Number of driving gears 1198853
Number of driven gears 1198854
Meshing angle 120572 (∘)98 25 210000
In engineering application random index of fatiguecumulative damage is supposed to follow a lognormal distri-bution That makes the distribution of 119863 be
119891119863
(119863) =
1
radic2120587120590119863
119863
exp[minus
(ln119863 minus 120583119863
)2
21205902
119863
] 119863 gt 0
0 119863 le 0
(23)
where 120583119863is the logarithmic mean of cumulative damage 119863
120590119863is the logarithmic mean square deviation of 119863Based on the theory of probability fatigue cumulative
damage the structural dynamic reliability of one moment is
119877 = 119877 (119863 lt 1198630) = int
1198630
0
119891119863
(119910) 119889119910 (24)
where 1198630is the limit of damage index
According to Figure 1 in the gear transmission systemthe gears are connected in series thus the whole systemwill fail if one of the gears fails In other words the systemreliability is based on the reliability of each gear Thereforethe reliability model gear transmission system is as follows
119877 (119905) = 119877119888
(119905) sdot
3
prod
119894=1
119877119901119894
(119905) sdot 119877119903
(119905) sdot 119877119904
(119905) sdot
4
prod
119895=1
119877119895
(119905) (25)
where 119894 is the number of planetary gears 119895 is the number ofmedium speed level and high speed level gears
43 Calculation of System Dynamic Reliability The reliabilityof the gear transmission system is calculated using Matlabsoftware Based on the analysis before the steps of theprogram are as follows
(1) Take the random input torque of the gear transmis-sion system as the extern excitation Get the dynamicmeshing force and its statistic characteristics by usingthe numerical integration method
(2) Process the data of meshing force by rain flowcountingmethodThen calculate the equivalent stressamplitude and frequency by using equivalent methodof Geber quadratic curve
(3) Calculate structural fatigue damage under the luffingstress
(4) Calculate cumulative fatigue damage of arbitrary time119905 under several stress cycles
0 2 4 6 8 100
02
04
06
Time t (s)
med
ium
-leve
l spe
ed g
ears
F12
(N)
Mea
n dy
nam
ic m
esh
forc
e of times106
Figure 3 Mean dynamic meshing force of medium-level speedgears
(5) Calculate the structural limit value of fatigue damage
(6) By giving a random cumulative damage index applythe equation of dynamic reliability to calculate thereliability of each gear when the tooth surface reachesthe contact fatigue limit and the tooth root reaches thebending fatigue limit
(7) Calculate the dynamic reliability of the gear transmis-sion system using (25)
5 Analysis of Examples
The study object of the example research is the gear trans-mission system of a 15MW wind turbine Here are someparameters used in this research the rated power of the windturbine is 15MW the impeller diameter is 70m the designedimpeller speed is 148 rmin average wind speed of the windfarm is 143ms wind density is 121 kgm3 wind energyutilization factor is 032 system transmission ratio is 9453Suppose the strength of the material and the coefficient ofperformance both follow a normal distribution while otherparameters are constant Suppose the material of planetarygear is 40Cr and the material of medium-level speed andhigh-level speed gear is 20CrMnTi Other parameters of thesystem are shown in Table 1
By solving the dynamic equation (10) of the systemvibration displacement and vibration velocity of the gearsat each moment are obtained as well as their statisticalcharacteristics By solving (11) the meshing forces of eachgear are obtained Figure 3 shows the curve of mean dynamicmeshing force of medium-level speed gears Figure 4 shows
Mathematical Problems in Engineering 7
8000
6000
4000
2000
00
12
34
56
Freq
uenc
y
times106
times106
959
756
453
150
(a) mean (N)
(b) amplitude (N)
Figure 4 Luffing load spectrum of medium-level speed gears
092
096
1
0 5 10 15 20
094
098
E
b
e
120588d0
Time t (y)
Mea
n re
liabi
lity120583R(t)
Figure 5 Dynamic reliability of system when variation coefficientof random parameter is 0
the luffing load spectrum of medium-level speed gears basedon the theory of rain flow counting method
Wedefine the ratio of themean square error and themeanof the systemparameters as their variation coefficient Figures5ndash7 show the variation of systemdynamic reliability over timewith the variation coefficient being 0 01 and 03 respectivelyAs is demonstrated in Figure 5 the comprehensive transmis-sion error 119890 has the greatest influence on the system reliabilityfollowed by the elastic modulus of gear material 119864 contacttooth width 119861 and pitch circle diameter 119889
0 Mass density 120588
has the least influence By comparing Figures 6 and 7 wecan also learn that with the variation of random parametersincreases the system gets more reliable
Table 2 shows the dynamic reliability of each componentin the transmission system as the comprehensive transmis-sion error 119890 and mass density 120588 vary randomly when 119905 =
63times108 sThe table also shows that when the comprehensive
transmission error 119890 and mass density 120588 are 0 01 and 03respectively in the whole transmission system planetarygear system has the highest dynamic reliability followed by
1
5 10 15 20084
092
0
088
096
E
b
e
120588
d0
Mea
n re
liabi
lity120583R(t)
Time t (y)
Figure 6 Dynamic reliability of system when variation coefficientof random parameter is 01
5 10 15 200080
090
1
085
095M
ean
relia
bilit
y120583R(t)
E
b
e
120588
d0
Time t (y)
Figure 7 Dynamic reliability of system when variation coefficientof random parameter is 03
the medium speed level gears while high speed level gearis the least reliable In the planetary gear system internalgears have the highest reliability followed by the planetarygear while the sun gear is the least reliable In the mediumand high speed level gears large gears are more dynamic-reliable than the small ones The dynamic reliability of thegear transmission system reduces and the dispersion degreeof the system increases with the increase of the parametersrsquovariation
We obtained the statistical properties of the dynamicreliability of the high speed level gears through 20000simulations when 119905 = 63 times 10
8 s using Monte Carlo methodand compared the results with this paper as is shown inTable 3 The method proposed in this paper is more accuratethan Monte Carlo method
6 Conclusions
In this paper the dynamic reliability of the gear transmissionsystem of a 15MW wind turbine with consideration of
8 Mathematical Problems in Engineering
Table2Dyn
amicreliabilityof
each
compo
nent
ofplanetarygear
syste
mwith
rand
omparameters
Influ
encing
factors
Varia
tion
coeffi
cient
Reliabilityof
sun
gear
Reliabilityof
planetarygear
Reliabilityof
internalgear
Reliabilityof
large
gearsin
medium-le
velspeed
Reliabilityof
small
gearsin
medium-le
velspeed
Reliabilityof
large
gearsinhigh
-level
speed
Reliabilityof
small
gearsinhigh
-level
speed
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
120588
00987387
0007365
0991021
0007327
0995148
0007324
0979066
0007367
0971572
0007631
0975882
0007483
0963271
0007790
005
0980301
0007366
0984813
0007327
0991275
0007324
0978731
0007369
0965088
0007638
0971755
0007489
0950338
0007795
01
0953633
0007371
0959300
0007332
0975381
0007335
0950280
0007371
0931572
0007652
0942372
0007506
0927510
0007803
119890
00962833
0007802
0970332
0007789
0983727
0007756
0958207
0007803
0936281
0007817
0952283
0007804
0932588
0007832
005
0948471
0007815
0955783
0007795
0970115
0007758
0944342
0007815
0920175
0007822
0947502
0007815
0901392
0007863
01
0911502
0007843
0927009
0007804
0943928
0007781
0909252
0007845
0883011
0007827
0897641
0007844
0877252
0007904
Mathematical Problems in Engineering 9
Table 3 The comparison of dynamic reliability of big gear of high speed gear system
Random parameters Variation coefficient Proposed method Monte Carlo methodMean of 119877(119905) Root mean square of 119877(119905) Mean of 119877(119905) Root mean square of 119877(119905)
119861 01 0943257 0008450 0940832 0008541119864 01 0940144 0008671 0939124 0008454120588 01 0944946 0008377 0945271 00082721198890
01 0946826 0008498 0939567 0008157119890 01 0909033 0008870 0897354 0008625
randomness of load and system parameters is analyzedby applying the theory of probability of cumulative fatiguedamage The main contributions and conclusions of thispaper are the following
(1) The dynamic model of the gear transmission of windturbine is built In consideration of the randomness of theload and gear parameters the dynamic response of thesystem is obtained by utilizing the random sampling methodand Runge-Kutta method The statistical properties of themeshing force of components in the gear transmission systemare obtained by statistic method
(2) By applying the method of rain flow counting thetime history of the components meshing force is convertedinto a series of luffing load spectra and the equivalent stressamplitude and frequency are calculated according to theequivalent method of Geber quadratic curve
(3) The dynamic reliability model of the transmissionsystem and gear components are built according to theprinciple of probability fatigue damage cumulative Variationof the system reliability over time is calculated when theparameters vary and the effect of the parameter variation tothe system reliability is analyzed Results show that (i) thecomprehensive transmission error has the largest influenceon system dynamic reliability while the mass density hasthe least influence (ii) the dynamic reliability of the geartransmission system reduces and the dispersion degreeincreases with the increase of the variation of the parameters(iii) for the gear transmission system of the 15MW windturbine planetary gear system has the highest dynamicreliability followed by the medium speed level gears whilehigh speed level gear is the least reliable At the same timein the planetary gear system internal gears have the highestreliability followed by the planetary gear while the sun gearis the least reliable In themedium and high speed level gearslarge gears are more dynamic-reliable than the small ones
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Major State BasicResearch Development Program 973 (no 2012CB215202)the National Natural Science Foundation of China (no51205046) and the Fundamental Research Funds for the
Central Universities The constructive comments providedby the anonymous reviewers and the editors are also greatlyappreciated
References
[1] W Musial S Butterfield and B McNiff ldquoImproving windturbine gearbox reliabilityrdquo in Proceedings of the EuropeanWindEnergy Conference Milan Italy May 2007
[2] L Katafygiotis and S H Cheung ldquoWedge simulation methodfor calculating the reliability of linear dynamical systemsrdquoProbabilistic Engineering Mechanics vol 19 no 3 pp 229ndash2382004
[3] L Katafygiotis and S H Cheung ldquoDomain decompositionmethod for calculating the failure probability of linear dynamicsystems subjected to gaussian stochastic loadsrdquo Journal ofEngineering Mechanics vol 132 no 5 pp 475ndash486 2006
[4] P Liu and Q-F Yao ldquoEfficient estimation of dynamic reliabilitybased on simple additive rules of probabilityrdquo EngineeringMechanics vol 27 no 4 pp 1ndash4 2010
[5] H-W Qiao Z-Z Lu A-R Guan and X-H Liu ldquoDynamicreliability analysis of stochastic structures under stationaryrandom excitation using hermite polynomials approximationrdquoEngineering Mechanics vol 26 no 2 pp 60ndash64 2009
[6] A Lupoi P Franchin and M Schotanus ldquoSeismic risk eval-uation of RC bridge structuresrdquo Earthquake Engineering ampStructural Dynamics vol 32 no 8 pp 1275ndash1290 2003
[7] P Franchin ldquoReliability of uncertain inelastic structures underearthquake excitationrdquo Journal of Engineering Mechanics vol130 no 2 pp 180ndash191 2004
[8] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006
[9] Z Caichao H Zehao T Qian and T Yonghu ldquoAnalysis ofnonlinear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005
[10] D T Qin Z K Xing and J H Wang ldquoOptimization designof system parameters of the gear transmission of wind turbinebased on dynamics and reliabilityrdquo Chinese Journal of Mechani-cal Engineering vol 44 no 7 pp 24ndash31 2008
[11] D-T Qin X-G Gu J-H Wang and J-G Liu ldquoDynamicanalysis and optimization of gear trains in amegawatt level windturbinerdquo Journal of Chongqing University vol 32 no 4 pp 408ndash414 2009
[12] X-L Jiang and C-F Wang ldquoStochastic volatility models basedBayesian method and their applicationrdquo Systems Engineeringvol 23 no 10 pp 22ndash28 2005
[13] H T Chen X L Wu D T Qin J Yang and Z Zhou ldquoEffectsof gear manufacturing error on the dynamic characteristics of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Table 1 Geometric parameters of gear transmission system of wind turbine
Low-level speedNumber of sun gear
teeth 119885119904
Number of planetarygear teeth 119885
119901
Number of internalgear teeth 119885
119903
Number ofmolds 119898
Meshing angle120572119904119901(∘)
Meshing angle120572119903119901(∘)
27 44 117 13 230000 210000Medium-levelspeed
Number of driving gears 1198851
Number of driven gears 1198852
Meshing angle 120572 (∘)104 23 210000
High-level speed Number of driving gears 1198853
Number of driven gears 1198854
Meshing angle 120572 (∘)98 25 210000
In engineering application random index of fatiguecumulative damage is supposed to follow a lognormal distri-bution That makes the distribution of 119863 be
119891119863
(119863) =
1
radic2120587120590119863
119863
exp[minus
(ln119863 minus 120583119863
)2
21205902
119863
] 119863 gt 0
0 119863 le 0
(23)
where 120583119863is the logarithmic mean of cumulative damage 119863
120590119863is the logarithmic mean square deviation of 119863Based on the theory of probability fatigue cumulative
damage the structural dynamic reliability of one moment is
119877 = 119877 (119863 lt 1198630) = int
1198630
0
119891119863
(119910) 119889119910 (24)
where 1198630is the limit of damage index
According to Figure 1 in the gear transmission systemthe gears are connected in series thus the whole systemwill fail if one of the gears fails In other words the systemreliability is based on the reliability of each gear Thereforethe reliability model gear transmission system is as follows
119877 (119905) = 119877119888
(119905) sdot
3
prod
119894=1
119877119901119894
(119905) sdot 119877119903
(119905) sdot 119877119904
(119905) sdot
4
prod
119895=1
119877119895
(119905) (25)
where 119894 is the number of planetary gears 119895 is the number ofmedium speed level and high speed level gears
43 Calculation of System Dynamic Reliability The reliabilityof the gear transmission system is calculated using Matlabsoftware Based on the analysis before the steps of theprogram are as follows
(1) Take the random input torque of the gear transmis-sion system as the extern excitation Get the dynamicmeshing force and its statistic characteristics by usingthe numerical integration method
(2) Process the data of meshing force by rain flowcountingmethodThen calculate the equivalent stressamplitude and frequency by using equivalent methodof Geber quadratic curve
(3) Calculate structural fatigue damage under the luffingstress
(4) Calculate cumulative fatigue damage of arbitrary time119905 under several stress cycles
0 2 4 6 8 100
02
04
06
Time t (s)
med
ium
-leve
l spe
ed g
ears
F12
(N)
Mea
n dy
nam
ic m
esh
forc
e of times106
Figure 3 Mean dynamic meshing force of medium-level speedgears
(5) Calculate the structural limit value of fatigue damage
(6) By giving a random cumulative damage index applythe equation of dynamic reliability to calculate thereliability of each gear when the tooth surface reachesthe contact fatigue limit and the tooth root reaches thebending fatigue limit
(7) Calculate the dynamic reliability of the gear transmis-sion system using (25)
5 Analysis of Examples
The study object of the example research is the gear trans-mission system of a 15MW wind turbine Here are someparameters used in this research the rated power of the windturbine is 15MW the impeller diameter is 70m the designedimpeller speed is 148 rmin average wind speed of the windfarm is 143ms wind density is 121 kgm3 wind energyutilization factor is 032 system transmission ratio is 9453Suppose the strength of the material and the coefficient ofperformance both follow a normal distribution while otherparameters are constant Suppose the material of planetarygear is 40Cr and the material of medium-level speed andhigh-level speed gear is 20CrMnTi Other parameters of thesystem are shown in Table 1
By solving the dynamic equation (10) of the systemvibration displacement and vibration velocity of the gearsat each moment are obtained as well as their statisticalcharacteristics By solving (11) the meshing forces of eachgear are obtained Figure 3 shows the curve of mean dynamicmeshing force of medium-level speed gears Figure 4 shows
Mathematical Problems in Engineering 7
8000
6000
4000
2000
00
12
34
56
Freq
uenc
y
times106
times106
959
756
453
150
(a) mean (N)
(b) amplitude (N)
Figure 4 Luffing load spectrum of medium-level speed gears
092
096
1
0 5 10 15 20
094
098
E
b
e
120588d0
Time t (y)
Mea
n re
liabi
lity120583R(t)
Figure 5 Dynamic reliability of system when variation coefficientof random parameter is 0
the luffing load spectrum of medium-level speed gears basedon the theory of rain flow counting method
Wedefine the ratio of themean square error and themeanof the systemparameters as their variation coefficient Figures5ndash7 show the variation of systemdynamic reliability over timewith the variation coefficient being 0 01 and 03 respectivelyAs is demonstrated in Figure 5 the comprehensive transmis-sion error 119890 has the greatest influence on the system reliabilityfollowed by the elastic modulus of gear material 119864 contacttooth width 119861 and pitch circle diameter 119889
0 Mass density 120588
has the least influence By comparing Figures 6 and 7 wecan also learn that with the variation of random parametersincreases the system gets more reliable
Table 2 shows the dynamic reliability of each componentin the transmission system as the comprehensive transmis-sion error 119890 and mass density 120588 vary randomly when 119905 =
63times108 sThe table also shows that when the comprehensive
transmission error 119890 and mass density 120588 are 0 01 and 03respectively in the whole transmission system planetarygear system has the highest dynamic reliability followed by
1
5 10 15 20084
092
0
088
096
E
b
e
120588
d0
Mea
n re
liabi
lity120583R(t)
Time t (y)
Figure 6 Dynamic reliability of system when variation coefficientof random parameter is 01
5 10 15 200080
090
1
085
095M
ean
relia
bilit
y120583R(t)
E
b
e
120588
d0
Time t (y)
Figure 7 Dynamic reliability of system when variation coefficientof random parameter is 03
the medium speed level gears while high speed level gearis the least reliable In the planetary gear system internalgears have the highest reliability followed by the planetarygear while the sun gear is the least reliable In the mediumand high speed level gears large gears are more dynamic-reliable than the small ones The dynamic reliability of thegear transmission system reduces and the dispersion degreeof the system increases with the increase of the parametersrsquovariation
We obtained the statistical properties of the dynamicreliability of the high speed level gears through 20000simulations when 119905 = 63 times 10
8 s using Monte Carlo methodand compared the results with this paper as is shown inTable 3 The method proposed in this paper is more accuratethan Monte Carlo method
6 Conclusions
In this paper the dynamic reliability of the gear transmissionsystem of a 15MW wind turbine with consideration of
8 Mathematical Problems in Engineering
Table2Dyn
amicreliabilityof
each
compo
nent
ofplanetarygear
syste
mwith
rand
omparameters
Influ
encing
factors
Varia
tion
coeffi
cient
Reliabilityof
sun
gear
Reliabilityof
planetarygear
Reliabilityof
internalgear
Reliabilityof
large
gearsin
medium-le
velspeed
Reliabilityof
small
gearsin
medium-le
velspeed
Reliabilityof
large
gearsinhigh
-level
speed
Reliabilityof
small
gearsinhigh
-level
speed
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
120588
00987387
0007365
0991021
0007327
0995148
0007324
0979066
0007367
0971572
0007631
0975882
0007483
0963271
0007790
005
0980301
0007366
0984813
0007327
0991275
0007324
0978731
0007369
0965088
0007638
0971755
0007489
0950338
0007795
01
0953633
0007371
0959300
0007332
0975381
0007335
0950280
0007371
0931572
0007652
0942372
0007506
0927510
0007803
119890
00962833
0007802
0970332
0007789
0983727
0007756
0958207
0007803
0936281
0007817
0952283
0007804
0932588
0007832
005
0948471
0007815
0955783
0007795
0970115
0007758
0944342
0007815
0920175
0007822
0947502
0007815
0901392
0007863
01
0911502
0007843
0927009
0007804
0943928
0007781
0909252
0007845
0883011
0007827
0897641
0007844
0877252
0007904
Mathematical Problems in Engineering 9
Table 3 The comparison of dynamic reliability of big gear of high speed gear system
Random parameters Variation coefficient Proposed method Monte Carlo methodMean of 119877(119905) Root mean square of 119877(119905) Mean of 119877(119905) Root mean square of 119877(119905)
119861 01 0943257 0008450 0940832 0008541119864 01 0940144 0008671 0939124 0008454120588 01 0944946 0008377 0945271 00082721198890
01 0946826 0008498 0939567 0008157119890 01 0909033 0008870 0897354 0008625
randomness of load and system parameters is analyzedby applying the theory of probability of cumulative fatiguedamage The main contributions and conclusions of thispaper are the following
(1) The dynamic model of the gear transmission of windturbine is built In consideration of the randomness of theload and gear parameters the dynamic response of thesystem is obtained by utilizing the random sampling methodand Runge-Kutta method The statistical properties of themeshing force of components in the gear transmission systemare obtained by statistic method
(2) By applying the method of rain flow counting thetime history of the components meshing force is convertedinto a series of luffing load spectra and the equivalent stressamplitude and frequency are calculated according to theequivalent method of Geber quadratic curve
(3) The dynamic reliability model of the transmissionsystem and gear components are built according to theprinciple of probability fatigue damage cumulative Variationof the system reliability over time is calculated when theparameters vary and the effect of the parameter variation tothe system reliability is analyzed Results show that (i) thecomprehensive transmission error has the largest influenceon system dynamic reliability while the mass density hasthe least influence (ii) the dynamic reliability of the geartransmission system reduces and the dispersion degreeincreases with the increase of the variation of the parameters(iii) for the gear transmission system of the 15MW windturbine planetary gear system has the highest dynamicreliability followed by the medium speed level gears whilehigh speed level gear is the least reliable At the same timein the planetary gear system internal gears have the highestreliability followed by the planetary gear while the sun gearis the least reliable In themedium and high speed level gearslarge gears are more dynamic-reliable than the small ones
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Major State BasicResearch Development Program 973 (no 2012CB215202)the National Natural Science Foundation of China (no51205046) and the Fundamental Research Funds for the
Central Universities The constructive comments providedby the anonymous reviewers and the editors are also greatlyappreciated
References
[1] W Musial S Butterfield and B McNiff ldquoImproving windturbine gearbox reliabilityrdquo in Proceedings of the EuropeanWindEnergy Conference Milan Italy May 2007
[2] L Katafygiotis and S H Cheung ldquoWedge simulation methodfor calculating the reliability of linear dynamical systemsrdquoProbabilistic Engineering Mechanics vol 19 no 3 pp 229ndash2382004
[3] L Katafygiotis and S H Cheung ldquoDomain decompositionmethod for calculating the failure probability of linear dynamicsystems subjected to gaussian stochastic loadsrdquo Journal ofEngineering Mechanics vol 132 no 5 pp 475ndash486 2006
[4] P Liu and Q-F Yao ldquoEfficient estimation of dynamic reliabilitybased on simple additive rules of probabilityrdquo EngineeringMechanics vol 27 no 4 pp 1ndash4 2010
[5] H-W Qiao Z-Z Lu A-R Guan and X-H Liu ldquoDynamicreliability analysis of stochastic structures under stationaryrandom excitation using hermite polynomials approximationrdquoEngineering Mechanics vol 26 no 2 pp 60ndash64 2009
[6] A Lupoi P Franchin and M Schotanus ldquoSeismic risk eval-uation of RC bridge structuresrdquo Earthquake Engineering ampStructural Dynamics vol 32 no 8 pp 1275ndash1290 2003
[7] P Franchin ldquoReliability of uncertain inelastic structures underearthquake excitationrdquo Journal of Engineering Mechanics vol130 no 2 pp 180ndash191 2004
[8] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006
[9] Z Caichao H Zehao T Qian and T Yonghu ldquoAnalysis ofnonlinear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005
[10] D T Qin Z K Xing and J H Wang ldquoOptimization designof system parameters of the gear transmission of wind turbinebased on dynamics and reliabilityrdquo Chinese Journal of Mechani-cal Engineering vol 44 no 7 pp 24ndash31 2008
[11] D-T Qin X-G Gu J-H Wang and J-G Liu ldquoDynamicanalysis and optimization of gear trains in amegawatt level windturbinerdquo Journal of Chongqing University vol 32 no 4 pp 408ndash414 2009
[12] X-L Jiang and C-F Wang ldquoStochastic volatility models basedBayesian method and their applicationrdquo Systems Engineeringvol 23 no 10 pp 22ndash28 2005
[13] H T Chen X L Wu D T Qin J Yang and Z Zhou ldquoEffectsof gear manufacturing error on the dynamic characteristics of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
8000
6000
4000
2000
00
12
34
56
Freq
uenc
y
times106
times106
959
756
453
150
(a) mean (N)
(b) amplitude (N)
Figure 4 Luffing load spectrum of medium-level speed gears
092
096
1
0 5 10 15 20
094
098
E
b
e
120588d0
Time t (y)
Mea
n re
liabi
lity120583R(t)
Figure 5 Dynamic reliability of system when variation coefficientof random parameter is 0
the luffing load spectrum of medium-level speed gears basedon the theory of rain flow counting method
Wedefine the ratio of themean square error and themeanof the systemparameters as their variation coefficient Figures5ndash7 show the variation of systemdynamic reliability over timewith the variation coefficient being 0 01 and 03 respectivelyAs is demonstrated in Figure 5 the comprehensive transmis-sion error 119890 has the greatest influence on the system reliabilityfollowed by the elastic modulus of gear material 119864 contacttooth width 119861 and pitch circle diameter 119889
0 Mass density 120588
has the least influence By comparing Figures 6 and 7 wecan also learn that with the variation of random parametersincreases the system gets more reliable
Table 2 shows the dynamic reliability of each componentin the transmission system as the comprehensive transmis-sion error 119890 and mass density 120588 vary randomly when 119905 =
63times108 sThe table also shows that when the comprehensive
transmission error 119890 and mass density 120588 are 0 01 and 03respectively in the whole transmission system planetarygear system has the highest dynamic reliability followed by
1
5 10 15 20084
092
0
088
096
E
b
e
120588
d0
Mea
n re
liabi
lity120583R(t)
Time t (y)
Figure 6 Dynamic reliability of system when variation coefficientof random parameter is 01
5 10 15 200080
090
1
085
095M
ean
relia
bilit
y120583R(t)
E
b
e
120588
d0
Time t (y)
Figure 7 Dynamic reliability of system when variation coefficientof random parameter is 03
the medium speed level gears while high speed level gearis the least reliable In the planetary gear system internalgears have the highest reliability followed by the planetarygear while the sun gear is the least reliable In the mediumand high speed level gears large gears are more dynamic-reliable than the small ones The dynamic reliability of thegear transmission system reduces and the dispersion degreeof the system increases with the increase of the parametersrsquovariation
We obtained the statistical properties of the dynamicreliability of the high speed level gears through 20000simulations when 119905 = 63 times 10
8 s using Monte Carlo methodand compared the results with this paper as is shown inTable 3 The method proposed in this paper is more accuratethan Monte Carlo method
6 Conclusions
In this paper the dynamic reliability of the gear transmissionsystem of a 15MW wind turbine with consideration of
8 Mathematical Problems in Engineering
Table2Dyn
amicreliabilityof
each
compo
nent
ofplanetarygear
syste
mwith
rand
omparameters
Influ
encing
factors
Varia
tion
coeffi
cient
Reliabilityof
sun
gear
Reliabilityof
planetarygear
Reliabilityof
internalgear
Reliabilityof
large
gearsin
medium-le
velspeed
Reliabilityof
small
gearsin
medium-le
velspeed
Reliabilityof
large
gearsinhigh
-level
speed
Reliabilityof
small
gearsinhigh
-level
speed
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
120588
00987387
0007365
0991021
0007327
0995148
0007324
0979066
0007367
0971572
0007631
0975882
0007483
0963271
0007790
005
0980301
0007366
0984813
0007327
0991275
0007324
0978731
0007369
0965088
0007638
0971755
0007489
0950338
0007795
01
0953633
0007371
0959300
0007332
0975381
0007335
0950280
0007371
0931572
0007652
0942372
0007506
0927510
0007803
119890
00962833
0007802
0970332
0007789
0983727
0007756
0958207
0007803
0936281
0007817
0952283
0007804
0932588
0007832
005
0948471
0007815
0955783
0007795
0970115
0007758
0944342
0007815
0920175
0007822
0947502
0007815
0901392
0007863
01
0911502
0007843
0927009
0007804
0943928
0007781
0909252
0007845
0883011
0007827
0897641
0007844
0877252
0007904
Mathematical Problems in Engineering 9
Table 3 The comparison of dynamic reliability of big gear of high speed gear system
Random parameters Variation coefficient Proposed method Monte Carlo methodMean of 119877(119905) Root mean square of 119877(119905) Mean of 119877(119905) Root mean square of 119877(119905)
119861 01 0943257 0008450 0940832 0008541119864 01 0940144 0008671 0939124 0008454120588 01 0944946 0008377 0945271 00082721198890
01 0946826 0008498 0939567 0008157119890 01 0909033 0008870 0897354 0008625
randomness of load and system parameters is analyzedby applying the theory of probability of cumulative fatiguedamage The main contributions and conclusions of thispaper are the following
(1) The dynamic model of the gear transmission of windturbine is built In consideration of the randomness of theload and gear parameters the dynamic response of thesystem is obtained by utilizing the random sampling methodand Runge-Kutta method The statistical properties of themeshing force of components in the gear transmission systemare obtained by statistic method
(2) By applying the method of rain flow counting thetime history of the components meshing force is convertedinto a series of luffing load spectra and the equivalent stressamplitude and frequency are calculated according to theequivalent method of Geber quadratic curve
(3) The dynamic reliability model of the transmissionsystem and gear components are built according to theprinciple of probability fatigue damage cumulative Variationof the system reliability over time is calculated when theparameters vary and the effect of the parameter variation tothe system reliability is analyzed Results show that (i) thecomprehensive transmission error has the largest influenceon system dynamic reliability while the mass density hasthe least influence (ii) the dynamic reliability of the geartransmission system reduces and the dispersion degreeincreases with the increase of the variation of the parameters(iii) for the gear transmission system of the 15MW windturbine planetary gear system has the highest dynamicreliability followed by the medium speed level gears whilehigh speed level gear is the least reliable At the same timein the planetary gear system internal gears have the highestreliability followed by the planetary gear while the sun gearis the least reliable In themedium and high speed level gearslarge gears are more dynamic-reliable than the small ones
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Major State BasicResearch Development Program 973 (no 2012CB215202)the National Natural Science Foundation of China (no51205046) and the Fundamental Research Funds for the
Central Universities The constructive comments providedby the anonymous reviewers and the editors are also greatlyappreciated
References
[1] W Musial S Butterfield and B McNiff ldquoImproving windturbine gearbox reliabilityrdquo in Proceedings of the EuropeanWindEnergy Conference Milan Italy May 2007
[2] L Katafygiotis and S H Cheung ldquoWedge simulation methodfor calculating the reliability of linear dynamical systemsrdquoProbabilistic Engineering Mechanics vol 19 no 3 pp 229ndash2382004
[3] L Katafygiotis and S H Cheung ldquoDomain decompositionmethod for calculating the failure probability of linear dynamicsystems subjected to gaussian stochastic loadsrdquo Journal ofEngineering Mechanics vol 132 no 5 pp 475ndash486 2006
[4] P Liu and Q-F Yao ldquoEfficient estimation of dynamic reliabilitybased on simple additive rules of probabilityrdquo EngineeringMechanics vol 27 no 4 pp 1ndash4 2010
[5] H-W Qiao Z-Z Lu A-R Guan and X-H Liu ldquoDynamicreliability analysis of stochastic structures under stationaryrandom excitation using hermite polynomials approximationrdquoEngineering Mechanics vol 26 no 2 pp 60ndash64 2009
[6] A Lupoi P Franchin and M Schotanus ldquoSeismic risk eval-uation of RC bridge structuresrdquo Earthquake Engineering ampStructural Dynamics vol 32 no 8 pp 1275ndash1290 2003
[7] P Franchin ldquoReliability of uncertain inelastic structures underearthquake excitationrdquo Journal of Engineering Mechanics vol130 no 2 pp 180ndash191 2004
[8] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006
[9] Z Caichao H Zehao T Qian and T Yonghu ldquoAnalysis ofnonlinear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005
[10] D T Qin Z K Xing and J H Wang ldquoOptimization designof system parameters of the gear transmission of wind turbinebased on dynamics and reliabilityrdquo Chinese Journal of Mechani-cal Engineering vol 44 no 7 pp 24ndash31 2008
[11] D-T Qin X-G Gu J-H Wang and J-G Liu ldquoDynamicanalysis and optimization of gear trains in amegawatt level windturbinerdquo Journal of Chongqing University vol 32 no 4 pp 408ndash414 2009
[12] X-L Jiang and C-F Wang ldquoStochastic volatility models basedBayesian method and their applicationrdquo Systems Engineeringvol 23 no 10 pp 22ndash28 2005
[13] H T Chen X L Wu D T Qin J Yang and Z Zhou ldquoEffectsof gear manufacturing error on the dynamic characteristics of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table2Dyn
amicreliabilityof
each
compo
nent
ofplanetarygear
syste
mwith
rand
omparameters
Influ
encing
factors
Varia
tion
coeffi
cient
Reliabilityof
sun
gear
Reliabilityof
planetarygear
Reliabilityof
internalgear
Reliabilityof
large
gearsin
medium-le
velspeed
Reliabilityof
small
gearsin
medium-le
velspeed
Reliabilityof
large
gearsinhigh
-level
speed
Reliabilityof
small
gearsinhigh
-level
speed
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
Mean
RMS
120588
00987387
0007365
0991021
0007327
0995148
0007324
0979066
0007367
0971572
0007631
0975882
0007483
0963271
0007790
005
0980301
0007366
0984813
0007327
0991275
0007324
0978731
0007369
0965088
0007638
0971755
0007489
0950338
0007795
01
0953633
0007371
0959300
0007332
0975381
0007335
0950280
0007371
0931572
0007652
0942372
0007506
0927510
0007803
119890
00962833
0007802
0970332
0007789
0983727
0007756
0958207
0007803
0936281
0007817
0952283
0007804
0932588
0007832
005
0948471
0007815
0955783
0007795
0970115
0007758
0944342
0007815
0920175
0007822
0947502
0007815
0901392
0007863
01
0911502
0007843
0927009
0007804
0943928
0007781
0909252
0007845
0883011
0007827
0897641
0007844
0877252
0007904
Mathematical Problems in Engineering 9
Table 3 The comparison of dynamic reliability of big gear of high speed gear system
Random parameters Variation coefficient Proposed method Monte Carlo methodMean of 119877(119905) Root mean square of 119877(119905) Mean of 119877(119905) Root mean square of 119877(119905)
119861 01 0943257 0008450 0940832 0008541119864 01 0940144 0008671 0939124 0008454120588 01 0944946 0008377 0945271 00082721198890
01 0946826 0008498 0939567 0008157119890 01 0909033 0008870 0897354 0008625
randomness of load and system parameters is analyzedby applying the theory of probability of cumulative fatiguedamage The main contributions and conclusions of thispaper are the following
(1) The dynamic model of the gear transmission of windturbine is built In consideration of the randomness of theload and gear parameters the dynamic response of thesystem is obtained by utilizing the random sampling methodand Runge-Kutta method The statistical properties of themeshing force of components in the gear transmission systemare obtained by statistic method
(2) By applying the method of rain flow counting thetime history of the components meshing force is convertedinto a series of luffing load spectra and the equivalent stressamplitude and frequency are calculated according to theequivalent method of Geber quadratic curve
(3) The dynamic reliability model of the transmissionsystem and gear components are built according to theprinciple of probability fatigue damage cumulative Variationof the system reliability over time is calculated when theparameters vary and the effect of the parameter variation tothe system reliability is analyzed Results show that (i) thecomprehensive transmission error has the largest influenceon system dynamic reliability while the mass density hasthe least influence (ii) the dynamic reliability of the geartransmission system reduces and the dispersion degreeincreases with the increase of the variation of the parameters(iii) for the gear transmission system of the 15MW windturbine planetary gear system has the highest dynamicreliability followed by the medium speed level gears whilehigh speed level gear is the least reliable At the same timein the planetary gear system internal gears have the highestreliability followed by the planetary gear while the sun gearis the least reliable In themedium and high speed level gearslarge gears are more dynamic-reliable than the small ones
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Major State BasicResearch Development Program 973 (no 2012CB215202)the National Natural Science Foundation of China (no51205046) and the Fundamental Research Funds for the
Central Universities The constructive comments providedby the anonymous reviewers and the editors are also greatlyappreciated
References
[1] W Musial S Butterfield and B McNiff ldquoImproving windturbine gearbox reliabilityrdquo in Proceedings of the EuropeanWindEnergy Conference Milan Italy May 2007
[2] L Katafygiotis and S H Cheung ldquoWedge simulation methodfor calculating the reliability of linear dynamical systemsrdquoProbabilistic Engineering Mechanics vol 19 no 3 pp 229ndash2382004
[3] L Katafygiotis and S H Cheung ldquoDomain decompositionmethod for calculating the failure probability of linear dynamicsystems subjected to gaussian stochastic loadsrdquo Journal ofEngineering Mechanics vol 132 no 5 pp 475ndash486 2006
[4] P Liu and Q-F Yao ldquoEfficient estimation of dynamic reliabilitybased on simple additive rules of probabilityrdquo EngineeringMechanics vol 27 no 4 pp 1ndash4 2010
[5] H-W Qiao Z-Z Lu A-R Guan and X-H Liu ldquoDynamicreliability analysis of stochastic structures under stationaryrandom excitation using hermite polynomials approximationrdquoEngineering Mechanics vol 26 no 2 pp 60ndash64 2009
[6] A Lupoi P Franchin and M Schotanus ldquoSeismic risk eval-uation of RC bridge structuresrdquo Earthquake Engineering ampStructural Dynamics vol 32 no 8 pp 1275ndash1290 2003
[7] P Franchin ldquoReliability of uncertain inelastic structures underearthquake excitationrdquo Journal of Engineering Mechanics vol130 no 2 pp 180ndash191 2004
[8] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006
[9] Z Caichao H Zehao T Qian and T Yonghu ldquoAnalysis ofnonlinear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005
[10] D T Qin Z K Xing and J H Wang ldquoOptimization designof system parameters of the gear transmission of wind turbinebased on dynamics and reliabilityrdquo Chinese Journal of Mechani-cal Engineering vol 44 no 7 pp 24ndash31 2008
[11] D-T Qin X-G Gu J-H Wang and J-G Liu ldquoDynamicanalysis and optimization of gear trains in amegawatt level windturbinerdquo Journal of Chongqing University vol 32 no 4 pp 408ndash414 2009
[12] X-L Jiang and C-F Wang ldquoStochastic volatility models basedBayesian method and their applicationrdquo Systems Engineeringvol 23 no 10 pp 22ndash28 2005
[13] H T Chen X L Wu D T Qin J Yang and Z Zhou ldquoEffectsof gear manufacturing error on the dynamic characteristics of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table 3 The comparison of dynamic reliability of big gear of high speed gear system
Random parameters Variation coefficient Proposed method Monte Carlo methodMean of 119877(119905) Root mean square of 119877(119905) Mean of 119877(119905) Root mean square of 119877(119905)
119861 01 0943257 0008450 0940832 0008541119864 01 0940144 0008671 0939124 0008454120588 01 0944946 0008377 0945271 00082721198890
01 0946826 0008498 0939567 0008157119890 01 0909033 0008870 0897354 0008625
randomness of load and system parameters is analyzedby applying the theory of probability of cumulative fatiguedamage The main contributions and conclusions of thispaper are the following
(1) The dynamic model of the gear transmission of windturbine is built In consideration of the randomness of theload and gear parameters the dynamic response of thesystem is obtained by utilizing the random sampling methodand Runge-Kutta method The statistical properties of themeshing force of components in the gear transmission systemare obtained by statistic method
(2) By applying the method of rain flow counting thetime history of the components meshing force is convertedinto a series of luffing load spectra and the equivalent stressamplitude and frequency are calculated according to theequivalent method of Geber quadratic curve
(3) The dynamic reliability model of the transmissionsystem and gear components are built according to theprinciple of probability fatigue damage cumulative Variationof the system reliability over time is calculated when theparameters vary and the effect of the parameter variation tothe system reliability is analyzed Results show that (i) thecomprehensive transmission error has the largest influenceon system dynamic reliability while the mass density hasthe least influence (ii) the dynamic reliability of the geartransmission system reduces and the dispersion degreeincreases with the increase of the variation of the parameters(iii) for the gear transmission system of the 15MW windturbine planetary gear system has the highest dynamicreliability followed by the medium speed level gears whilehigh speed level gear is the least reliable At the same timein the planetary gear system internal gears have the highestreliability followed by the planetary gear while the sun gearis the least reliable In themedium and high speed level gearslarge gears are more dynamic-reliable than the small ones
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the Major State BasicResearch Development Program 973 (no 2012CB215202)the National Natural Science Foundation of China (no51205046) and the Fundamental Research Funds for the
Central Universities The constructive comments providedby the anonymous reviewers and the editors are also greatlyappreciated
References
[1] W Musial S Butterfield and B McNiff ldquoImproving windturbine gearbox reliabilityrdquo in Proceedings of the EuropeanWindEnergy Conference Milan Italy May 2007
[2] L Katafygiotis and S H Cheung ldquoWedge simulation methodfor calculating the reliability of linear dynamical systemsrdquoProbabilistic Engineering Mechanics vol 19 no 3 pp 229ndash2382004
[3] L Katafygiotis and S H Cheung ldquoDomain decompositionmethod for calculating the failure probability of linear dynamicsystems subjected to gaussian stochastic loadsrdquo Journal ofEngineering Mechanics vol 132 no 5 pp 475ndash486 2006
[4] P Liu and Q-F Yao ldquoEfficient estimation of dynamic reliabilitybased on simple additive rules of probabilityrdquo EngineeringMechanics vol 27 no 4 pp 1ndash4 2010
[5] H-W Qiao Z-Z Lu A-R Guan and X-H Liu ldquoDynamicreliability analysis of stochastic structures under stationaryrandom excitation using hermite polynomials approximationrdquoEngineering Mechanics vol 26 no 2 pp 60ndash64 2009
[6] A Lupoi P Franchin and M Schotanus ldquoSeismic risk eval-uation of RC bridge structuresrdquo Earthquake Engineering ampStructural Dynamics vol 32 no 8 pp 1275ndash1290 2003
[7] P Franchin ldquoReliability of uncertain inelastic structures underearthquake excitationrdquo Journal of Engineering Mechanics vol130 no 2 pp 180ndash191 2004
[8] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006
[9] Z Caichao H Zehao T Qian and T Yonghu ldquoAnalysis ofnonlinear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005
[10] D T Qin Z K Xing and J H Wang ldquoOptimization designof system parameters of the gear transmission of wind turbinebased on dynamics and reliabilityrdquo Chinese Journal of Mechani-cal Engineering vol 44 no 7 pp 24ndash31 2008
[11] D-T Qin X-G Gu J-H Wang and J-G Liu ldquoDynamicanalysis and optimization of gear trains in amegawatt level windturbinerdquo Journal of Chongqing University vol 32 no 4 pp 408ndash414 2009
[12] X-L Jiang and C-F Wang ldquoStochastic volatility models basedBayesian method and their applicationrdquo Systems Engineeringvol 23 no 10 pp 22ndash28 2005
[13] H T Chen X L Wu D T Qin J Yang and Z Zhou ldquoEffectsof gear manufacturing error on the dynamic characteristics of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
planetary gear transmission system of wind turbinerdquo AppliedMechanics and Materials vol 86 pp 518ndash522 2011
[14] H Wang B Xing and H Luo ldquoRanflow counting method andits application in fatigue life predictionrdquo Mining amp ProcessingEquipment vol 34 no 3 pp 95ndash97 2006
[15] Z-B Wang G-F Zhai X-Y Huang and D-X Yi ldquoCombi-nation forecasting method for storage reliability parametersof aerospace relays based on grey-artificial neural networksrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 9 pp 3807ndash3816 2013
[16] Y-K Lin L C-L Yeng S-I Chang and S-R Hsieh ldquoNetworkreliability evaluation for computer networks a case of the Tai-wan advanced research and education networkrdquo InternationalJournal of Innovative Computing Information and Control vol9 no 1 pp 257ndash268 2013
[17] R M Ignatishchev ldquoRefinement of outlook at contact strengthof gears and new potentiality of increase of their reliabilityrdquoProblemyMashinostroeniya i NadezhnostiMashin no 6 pp 68ndash70 1992
[18] J N Yang andM D Liu ldquoResidual strength degradation modeland theory of periodic proof tests for graphiteepoxy laminatesrdquoJournal of Composite Materials vol 11 no 2 pp 176ndash203 1977
[19] W Sun H Gao and O Kaynak ldquoFinite frequency 119867infin
controlfor vehicle active suspension systemsrdquo IEEE Transactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011
[20] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator inputdelayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of