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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

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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

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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Decision SciencesAdvances in

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

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

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