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Research ArticleAnalysis of Dynamic Behavior of Multiple-Stage Planetary GearTrain Used in Wind Driven Generator
Jungang Wang12 Yong Wang12 and Zhipu Huo12
1 School of Mechanical Engineering Shandong University Jingshi Road 17923 Jinan Shandong 250061 China2 Key Laboratory of High-Efficieny and Clean Mechanical Manufacture Ministry of Education School of Mechanical EngineeringShandong University Jinan Shandong 250061 China
Correspondence should be addressed to Yong Wang wjg mrnsinacn
Received 5 August 2013 Accepted 5 October 2013 Published 5 January 2014
Academic Editors A Durmus and M Q Fan
Copyright copy 2014 Jungang Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A dynamic model of multiple-stage planetary gear train composed of a two-stage planetary gear train and a one-stage parallel axisgear is proposed to be used in wind driven generator to analyze the influence of revolution speed and mesh error on dynamic loadsharing characteristic based on the lumped parameter theory Dynamic equation of the model is solved using numerical method toanalyze the uniform load distribution of the system It is shown that the load sharing property of the system is significantly affectedby mesh error and rotational speed load sharing coefficient and change rate of internal and external meshing of the system are ofobvious difference from each other The study provides useful theoretical guideline for the design of the multiple-stage planetarygear train of wind driven generator
1 Introduction
The planetary gear train having advantages of large trans-mission ratio simple construction compactness and smoothrunning has been widely applied in many machines In spiteof these advantages planetary gears may have undesirabledynamic behavior resulting in much noise vibration andother unacceptable performances A number of papers havebeen published on planetary gear dynamics which compriselumped-parameter models and deformable or hybrid modelsof varying complexity [1ndash7] Modal analyses were performedby Lin andParker [8] andBodas et al [9ndash13] who emphasizedthe structured modal properties of single-stage drives andshowed that only planet rotational and translational modescould exist It is important to understand the fundamentalcause of the unequal load sharing behavior in planetary trans-missions The input torque applied should theoretically beshared by each planet in an 119899-planet that is each sun-pinion-ring path should carry 1119899 of the total torque However inactual transmissions there is unequal load sharing betweenthe parallel paths Du et al [14] found the deformationcompatibility equations and the torque balance equations ofthe 2119870-119867-type planetary transmission system based on the
characteristic that the system is composed of a closed loopof power flow In consideration of the manufacturing errorassembly error and float of the parts the load sharing coeffi-cient of each planetary gearwas calculated by using the theoryof equivalent mesh error and equivalent mesh stiffness GuandVelex [15] presented an original lumped parametermodelof planetary gears to account for planet position errors andsimulate their contribution to the dynamic load sharingamongst the planets Singh [16] developed the concept ofan epicyclic load sharing map to describe the load sharingcharacteristics of every epicyclic gear set at any positionalerror and torque level A comprehensive experimental study[17] was conducted to study the load sharing behavior ofa family of epicyclical gear sets with varying number ofplanets Experiments were conducted at several error andtorque levels The results clearly showed the influence ofpositional errors and that the sensitivity of the epicyclical gearset increased as the number of planets increased A physicalexplanation [18] has been provided for the load sharingbehavior Load required to produce the needed deformationis the cause of the unequal load sharing This explains theeffectiveness of system float in reducing the load sharinginequality Lu et al [19] presented a calculative model for
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 627045 11 pageshttpdxdoiorg1011552014627045
2 The Scientific World Journal
single-stage planetary gear with the dynamic way to studythe load sharing behavior of each planetary gear and therelationship between error and load sharing was analyzed Yeet al [20] built an analytical model for NGW planetary geartrainwith unequalmodulus and pressure angles and analyzedthe load sharing behavior of each planet
Although the references available focused on differentfields most of them established mathematical model ofone-stage planetary gear train Dynamic model of multiple-stage planetary gear train is limitedly reported Few reportsabout dynamic model of multiple-stage planetary gear traincomposed of two-stage planetary gear train and one-stageparallel axis and its dynamic load sharing characteristics areconcerned
In this study a transmission scheme of load-split two-stage planetary gear used in wind driven generator is pro-posed Transmission ratio of the planetary gear train isobtained as well as the relationship between transmissionratio and characteristic parameter of planetary gear trainaccording to conversion mechanism method and generalrelationship among the speed of each unit in planetary geartrain Dynamic model of load-split multiple-stage gear traincomposed of a two-stage planetary gear train and a one-stage parallel axis gear is established on the basis of lumpedparameter theory and influence of revolution speed andmesherror on dynamic load sharing characteristic of the system isanalyzed
2 Load-Split Two-Stage Planetary Gear Train
21 Kinematic Scheme The kinematic scheme of load-splittwo-stage planetary gear is shown in Figure 1 which iscomposed of closed planetary gear train and differentialplanetary gear train Former basic units 1c (planetary carrier)and 1s (sun gear) are connected to units 2r (inner ring) and2c (planetary carrier) of differential gear train respectivelyTherefore load split is realized by first-stage and second-stage gear bearing input torque simultaneously In Figure 11r 1p1 1s and 1c are inner ring planetary gear sun gearand planetary carrier of first-stage planetary gear trainrespectively while 2r 2p1 2s and 2c represent correspondingunits of second-stage planetary gear train
22 Speed of Each Unit of First-Stage Planetary Gear TrainThe relationship between rotational speed of sun gear andthat of planetary carrier and inner ring of first-stage planetarygear train is shown in
1198991s = 1198941r1s1c1198991c + 119894
1c1s1r1198991r (1)
Equation (22) is obtained according to general relation-ship of relative gear ratio among each unit in planetarygear train principle and transmission type and characteristicparameter of first-stage planetary gear train
1198941r1s1c = 1 minus 119894
1c1s1r
1198941c1s1r = minus
1198851r
1198851s
1r
2r
1c
2c
1s 2s
OutIn
1p12p1
Figure 1 Kinematic scheme of load-split two-stage planetary gear
1198851r
1198851s= 1205821
1198991r = 0
(2)
We can come to (3) by (1) and (22)
1198991s = (1 + 1205821) 1198991c (3)
The relationship between rotational speed of planetarygear and that of planetary carrier and inner ring is expressedas (4) according to relationship of relative rotational speedamong each unit in the first-stage planetary gear train
1198991p = 1198941r1p1c1198991c + 119894
1c1p1r1198991r (4)
Similar to (22) (5) is obtained as follows
1198941r1p1c = 1 minus 119894
1c1p1r
1198941c1p1r =
1198851r
1198851p
1198851p =
1198851r minus 1198851s2
1198851r
1198851s= 1205821
1198991r = 0
(5)
Thus (6) can be obtained using (5) and (4)
1198991p = (1 minus
21205821
1205821minus 1
) 1198991c (6)
The Scientific World Journal 3
where 1205821is the characteristic parameter of planetary gear
train and 1205821= 1198851r1198851s1198851r1198851s and1198851p are tooth number of
inner ring sun gear and planetary gear respectively 1198991119895(119895 =
c s r p) and 119894111990911198861119887
(119886 = c s r p 119887 = c s r p 119909 = c s r p)represent the rotational speed and relative gear ratio of eachunit of first-stage planetary gear train respectively
23 Speed of Each Unit of Second-Stage Planetary Gear TrainThe relationship between the rotational speed of sun geartrain and that of planetary carrier and inner ring of second-stage planetary gear is expressed as follows
1198992s = 1198942r2s2c1198992c + 119894
2c2s2r1198992r (7)
Equation (8) can be obtained according to the transmis-sion characteristic of basic unit of second-stage planetary geartrain
1198942r2s2c1198992c =
1198992r2s1198992r2c1198992r2c = 119899
2r2s
1198942c2s2r1198992r =
1198992c2s1198992c2r1198992c2r = 119899
2c2s
(8)
Using (7) and (8) gives
1198992s = 119899
2r2s + 1198992c2s (9)
Equations (10) and (11) are obtained by the relativemovement relationship of planetary gear trainrsquos units wheninner ring and planetary carrier of second-stage planetarygear train are fixed respectively
1198992r = 0
1198992r2s =
1198992c
1198942r2c2s
1198942r2c2s =
1
1 minus 1198942c2s2r
1198942c2s2r = minus
1198852r
1198852s
1198852r
1198852s= 1205822
(10)
1198992c = 0
1198992c2s =
1198992r
1198942c2r2s
1198942c2r2s = minus
1198852s
1198852r
1198852r
1198852s= 1205822
(11)
By connecting (10) and (11) to (9) the relationship ofrotational speed between input and output units of second-stage planetary gear train is obtained as follows
1198992s = (1 + 1205822) 1198992c minus 12058221198992r (12)
And the relationship between rotational speeds of plane-tary gear is expressed as follows
1198992p = 1198942r1p2c1198992c + 119894
2c2p2r1198992r (13)
Similar to (22) (14) can be obtained as follows
1198942r2p2c = 1 minus 119894
2c2p2r
1198942c2p2r =
1198852r
1198852p
1198852p =
1198852r minus 1198852s2
1198852r
1198852s= 1205822
(14)
Substitution of (14) into (13) yields
1198992p =
minus1205822minus 1
1205822minus 1
1198992c +
21205822
1205822minus 1
1198992r (15)
Considering the scheme of Figure 1 (16) can be given asfollows
1198992r = 1198991c
1198992c = 1198991s
(16)
Substitution of (16) and (3) into (15) gives
1198992p =
21205822minus (1205822+ 1) (1 + 120582
1)
1205822minus 1
1198991c (17)
where 1205822represents the characteristic parameter of planetary
line of second-stage planetary gear train and 1205822= 1198852r1198852s
1198852r 1198852s and 119885
2p stand for tooth number of inner ringsun gear and planetary gear of second planetary gear trainrespectively 119899
2119895(119895 = c s r p) is the rotational speed of each
unit of second-stage planetary gear train and 1198942119909
21198862119887(119886 =
c s r p 119887 = c s r p 119909 = c s r p) is the relative gear ratioof corresponding unit
24 Transmission Ratio of the Planetary Gear Train Theexpressions of input and output rotational speed of load-splittwo-stage planetary gear train are given by substitution of (16)and (3) into (12) as follows
1198992s = [(1 + 1205822) (1 + 1205821) minus 1205822] 1198991c (18)
Thus transmission ratio formula of load-split two-stageplanetary gear train is obtained as
1198942s1c =
1198992s1198991c= (1 + 120582
1) (1 + 120582
2) minus 1205822 (19)
General transmission ratio in Figure 1 is related to charac-teristic parameters of the planetary gear trains 120582
1and 120582
2 Too
small values of1198851r and1198852r result in undersizing of the system
and decreasing of bearing capacity Values of characteristic
4 The Scientific World Journal
1 2 3 4 5 6 7 8
123456780
1020304050607080
3 4 5 6 734567
Tran
smiss
ion
ratio
i c1s
1
Characteristic parameter 1205821
Characteristic parameter 1205822
Figure 2 Relationship between transmission ratio and characteris-tic parameters of planetary gear train
parameters have to be reasonable Recommended interval of1205821and 120582
2is [3 8]
Relationship between transmission ratio and character-istic parameters in load-split two-stage planetary gear trainis shown in Figure 2 Transmission ratio rises with increasingvalues of characteristic parameters of the planetary gear trainandmaximum transmission ratio in the interval of [3 8] is 73
3 Dynamic Model
31 Model of the Multiple-Stage Transmission System Amul-tiple-stage gear train composed of a two-stage planetary geartrain and a one-stage parallel axis gear is shown in Figure 33g1 and 3g2 in Figure 3 stand for pinion gear and driven gearof parallel axis
Dynamicmodel of Figure 3 is shown in Figure 4 based onlumped parameter theory Since the first-stage planetary geartrain and the second-stage planetary gear train have the samebasic structure they can be represented by the single stagepurely torsional model shown in Figure 5
The linear displacements of all members of themultistagetransmission system are shown as follows
1199061c = 1199031c1205791c 119906
2c = 1199032c1205792c
1199061s = 1199031bs1205791s 119906
2s = 1199032bs1205792s
1199061r = 1199031br1205791r 119906
2r = 1199032br1205792r
1199061p119895 = 1199031bp1198951205791p119895 119906
2p119895 = 1199032bp1198951205792p119895
1199063g1 = 1199033bg11205793g1 119906
3g2 = 1199033bg21205793g2
(20)
Generalized masses of all members of the multistagetransmission system are shown as follows
1198981c =
1198681ce1199032
1c 119898
2c =1198682ce1199032
2c 119898
1s =1198681s1199032
1bs
1198982s =
1198682s1199032
2bs 119898
1r =1198681r1199032
1br 119898
2r =1198682r1199032
2br
1r
2r
1c
2c
1s 2s
Out
In3g1
3g2
1p12p1
Figure 3 Transmission system of load-split multiple-stage plane-tary gear train
1198981p119895 =
1198681p119895
1199032
1bp119895 119898
2p119895 =1198682p119895
1199032
2bp119895 119898
3g1 =1198683g1
1199032
3bg1
1198983g2 =
1198683g2
1199032
3bg2
(21)
32 Dynamic Equation of the Multistage Transmission SystemThe interaction force between sun gear and the 119895th planetarygear of the first-stage planetary gear train along the line ofaction is given as follows
1198651sp119895 = 1198701sp1198951198831sp119895 + 1198621sp1198951sp119895
1198831sp119895 = 1199061s + 1199061p119895 minus cos120572
1s1199061c minus 1198641sp119895
1sp119895 = 1s + 1p119895 minus cos120572
1s1c minus 1sp119895
(22)
The interaction force between the inner ring and the 119895thplanet gear of the first-stage planetary gear train along the lineof action can be expressed as follows
1198651rp119895 = 1198701rp1198951198831rp119895 + 1198621rp1198951rp119895
1198831rp119895 = 1199061p119895 minus 1199061r + cos120572
1r1199061c minus 1198641rp119895
1rp119895 = 1p119895 minus 1r + cos120572
1r1c minus 1rp119895
(23)
The interaction force between sun gear and the 119895thplanetary gear of the second-stage planetary gear train alongthe line of action is
1198652sp119895 = 1198702sp1198951198832sp119895 + 1198622sp1198952sp119895
1198832sp119895 = 1199062s + 1199062p119895 minus cos120572
2s1199062c minus 1198642sp119895
2sp119895 = 2s + 2p119895 minus cos120572
2s2c minus 2sp119895
(24)
The Scientific World Journal 5
E1rp2
E2rp1
E3g1g2
E2rp2
E1rp1K1sp2C1sp2
K2sp2 K2s3g1
K3g1g2
C2sp2
C2s3g1
C3g1g2
K2sp1C2sp1
C1s2c
C1c2r
K1rp2K1c2r
K1s2c
C1rp2 K2rp2C2rp2
K1rp1
C1rp1K2rp1C2rp1
K1sp1C1sp1
E1sp2 E2sp2E1sp1 E1sp1
1205791p1 1205792p1
1205792p21205791p2
1205791r
1205793g1
1205793g21205792r
1205791c1205792c
1205791s 1205792sTi
To
Figure 4 Dynamic model of load-split multiple-stage planetary gear train
Sun
Carrier
Ring
Planet n
ypn
ys
xs
xpn
os
op
rbs
rbr
rbpnKsp
Krp
Csp
Crp
Esp
Erp
rc
120572s
120572r
120579c120579r
120579s
120579pn
Figure 5 Torsional model of single-stage planetary gear
The interaction force between the inner ring and the 119895thplanet gear of the second-stage planetary gear train along theline of action can be expressed as follows
1198652rp119895 = 1198702rp1198951198832rp119895 + 1198622rp1198952rp119895
1198832rp119895 = 1199062p119895 minus 1199062r + cos120572
2r1199062c minus 1198642rp119895
2rp119895 = 2p119895 minus 2r + cos120572
2r2c minus 2rp119895
(25)
The interaction force between the pinion gear and drivengear of the third-stage parallel axis gear along the line ofaction can be expressed as follows
1198653g1g2 = 1198703g1g21198833g1g2 + 1198623g1g23g1g2
1198833g1g2 = 1199063g1 + 1199063g2 minus 1198643g1g2
3g1g2 = 3g1 + 3g2 minus 3g1g2
(26)
Fix the inner ring of the first-stage planetary gear trainand take the number of planetary gears of the planetary geartrain as 3 namely 1119873 = 2119873 = 3 According to the planetarymechanism modeling methods in [13] dynamic equation ofthe multistage transmission system shown in Figure 4 can bebuilt as shown in
1198981c1c minus cos120572
1s
1119873
sum
119895=1
1198651sp119895 + cos120572
1r
1119873
sum
119895=1
1198651rp119895
+
1198701c2r1199031c
(
1199061c1199031cminus
1199062r1199032br) +
1198621c2r1199031c
(
1c1199031cminus
2r1199032br) =
119879i1199031c
1198981s1s +
1119873
sum
119895=1
1198651sp119895 +
1198701s2c1199031bs
(
1199061s1199031bs
minus
1199062c1199032c)
+
1198621s2c1199031bs
(
1s1199031bs
minus
2c1199032c) = 0
1198981p11p1 + 1198651sp1 + 1198651rp1 = 0
1198981p21p2 + 1198651sp2 + 1198651rp2 = 0
1198981p11198731p1119873 + 1198651sp1119873 + 1198651rp1119873 = 0
1198982c2c minus cos120572
2s
2119873
sum
119895=1
1198652sp119895 + cos120572
2r
2119873
sum
119895=1
1198652rp119895
minus
1198701s2c1199032c
(
1199061s1199031bs
minus
1199062c1199032c) minus
1198621s2c1199032c
(
1s1199031bs
minus
2c1199032c) = 0
1198982s2s +
2119873
sum
119895=1
1198652sp119895 +
1198702s3g1
1199032bs
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
+
1198622s3g1
1199032bs
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198982p12p1 + 1198652sp1 + 1198652rp1 = 0
6 The Scientific World Journal
1198982p22p2 + 1198652sp2 + 1198652rp2 = 0
1198982p21198732p2119873 + 1198652sp2119873 + 1198652rp2119873 = 0
1198982r2r minus
2119873
sum
119895=1
1198652rp119895 minus
1198701c2r1199032br
(
1199061c1199031cminus
1199062r1199032br)
minus
1198621c2r1199032br
(
1c1199031cminus
2r1199032br) = 0
1198983g13g1 + 1198653g1g2 minus
1198702s3g1
1199033bg1
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
minus
1198622s3g1
1199033bg1
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198983g23g2 + 1198653g1g2 +
119879119900
1199033bg2
= 0
(27)
The equations of the dynamic model are given in thematrix form as
119872 + 119862 + 119870119906 = 119865 (28)
where the displacement vector the mass matrix the dampingmatrix the stiffness matrix and the load vector are givenrespectively as
119906 = [1199061c 1199061s 1199061p1 1199061p2 1199061p3 1199062c 1199062s 1199062p1 1199062p2 1199062p3 1199062r 1199063g1 1199063g2]
119879
119872 = diag (1198981c 1198981s 1198981p1 1198981p2 1198981p3 1198982c 1198982s 1198982p1 1198982p2 1198982p3 1198982r 1198983g1 1198983g2)
119862 =
[
[
[
[
[
[
[
[
[
[
[
[
1198621c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198621sp119895 + cos1205722
1r
3
sum
119895=1
1198621rp119895 minus cos1205721s3
sum
119895=1
1198621sp119895 1198621rp1 cos1205721r minus 1198621sp1 cos1205721s 1198621rp2 cos1205721r minus 1198621sp2 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198621sp1198951198621s2c1199032
1bs+
3
sum
119895=1
1198621sp119895 1198621sp1 sdot sdot sdot
1198621sp1 + 1198621rp1 0 sdot sdot sdot
1198621sp2 + 1198621rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119870 =
[
[
[
[
[
[
[
[
[
[
[
[
1198701c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198701sp119895 + cos1205722
1r
3
sum
119895=1
1198701rp119895 minus cos1205721s3
sum
119895=1
1198701sp119895 1198701rp1 cos1205721r minus 1198701sp1 cos1205721s 1198701rp2 cos1205721r minus 1198701s1199012 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198701sp1198951198701s2c1199032
1bs+
3
sum
119895=1
1198701sp119895 1198701sp1 sdot sdot sdot
1198701sp1 + 1198701rp1 0 sdot sdot sdot
1198701sp2 + 1198701rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119865 =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879119894
1199031cminus cos120572
1s(3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895) + cos120572
1r(3
sum
119895=1
1198621rp1198951119903p119895 +
3
sum
119895=1
1198701rp1198951198641rp119895)
3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895
1198621sp11119904p1 + 1198701sp11198641sp1 + 1198621rp11rp1 + 1198701rp11198641rp1
1198621sp21sp2 + 1198701sp21198641sp2 + 1198621rp21rp2 + 1198701rp21198641rp2
1198621sp31sp3 + 1198701sp31198641sp3 + 1198621rp31rp3 + 1198701rp31198641rp3
cos1205722r(3
sum
119895=1
1198702rp1198951198642rp119895 +
3
sum
119895=1
1198622rp1198952rp119895) minus cos120572
2s(3
sum
119895=1
1198702sp1198951198642sp119895 +
3
sum
119895=1
1198622sp1198952sp119895)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(29)
The Scientific World Journal 7
4 Load Sharing Characteristic of Load-SplitMultiple-Stage Planetary Gear Train
41 Calculation of Load Sharing Coefficient Use numericalintegration method for solving the dynamic equation (28) ofthe system obtain the responses to displacement and velocityof the system and substitute the responses into (22)ndash(25) andthen obtain the engaging forces 119865
1sp119895 1198651rp119895 1198652sp119895 and 1198652rp119895Make 119863
1sp119895119896s and 1198631sp119895119896r respectively represent the load
sharing coefficients of the internal and external meshing ofall gear pairs of the first-stage planetary gear train and119863
2sp119894119896sand119863
2sp119894119896r as those of the internal and external meshing of allgear pairs of the second-stage planetary gear train then loadsharing coefficients are expressed as
1198631sp119895119896s =
1119873(1198651sp119895119896s)max
sum1119873
119895=1(1198651sp119895119896s)max
1198631rp119895119896r =
1119873(1198651rp119895119896r)max
sum1119873
119895=1(1198651rp119895119896r)max
1198632sp119894119896s =
2119873(1198652sp119895119896s)max
sum2119873
119894=1(1198652sp119895119896s)max
1198632rp119894119896r =
2119873(1198652rp119895119896r)max
sum2119873
119894=1(1198652rp119895119896r)max
(30)
where 119896s 119896r are meshing cycle numbers for internal andexternal meshing of the planetary gear pair
When 1198891sp119895 and 1198891rp119895 are used to stand for load sharing
coefficient of internal and externalmeshing of each first-stagegear and 119889
2sp119895 and 1198892rp119895 for that of each second-stage gearin system period respectively the expression can be given asfollows
1198891sp119895 =
100381610038161003816100381610038161198631sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198891rp119895 =
100381610038161003816100381610038161198631rp119895119896r minus 1
10038161003816100381610038161003816max + 1
1198892sp119895 =
100381610038161003816100381610038161198632sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198892rp119895 =
100381610038161003816100381610038161198632rp119895119896r minus 1
10038161003816100381610038161003816max + 1
(31)
The paper analyzes the transmission system as shown inFigure 4 The basic parameters of the transmission systemare shown in Tables 1 and 2 and other parameters can bedetermined by [21] Substitute the relevant parameters of thesystem into (28) for solution Use (30) and (31) to obtain theload sharing coefficients of the transmission system
42 Influence of Mesh Error on Load Sharing Coefficient ofthe System Load sharing property of planetary gear trainis significantly affected by manufacturing error installationerror and eccentric error which cannot be neglected inplanetary gear train Considering systemrsquos complexity it is
Table 1 Primary parameters of planetary gear train
Parameter Carrier Ring Sun gear Planetarygear
Pitch radius 1199031(mm) 468 726 210 258
Base circle 1199031119887(mm) mdash 68222 19734 24244
Mass1198721(kg) 204277 41034 34491 38839
Moment of inertia 1198691
(kgsdotm2) 46255 22657 762 1625
Pressure angle 1205721(∘) mdash 20 20 20
Pitch radius 1199032(mm)
1199032mm 345 550 140 205
Base circle 1199032119887(mm) mdash 51683 13156 19264
Mass1198722(kg) 121249 8056 13140 17654
Moment of inertia 1198692
(kgsdotm2) 15175 2565 129 512
Pressure angle 1205722(∘) mdash 20 20 20
Table 2 Primary parameters of parallel-shaft gears
Parameter Pinion gear Driven gearPitch radius 119903
3(mm) 292 100
Base circle 1199033119887(mm) 27439 9397
Mass1198723(kg) 20822 2414
Moment of inertia 1198693(kgsdotm2) 887 012
Pressure angle 1205723(∘) 20 20
assumed that equivalent mesh error of each stage planetarygear at the direction of meshing line is equal and values of10 20 30 40 and 50 120583m are given respectively Load sharingproperties of multiple-stage gear train under these fiveconditions are studied Relationships between load sharingcoefficient curves of internal and external meshing of first-stage and second-stage which are calculated according to(31) are drawn in Figure 6 with different mesh errors
Results below can be concluded according to Figure 6
(1) Each load-sharing coefficient increases with increas-ing mesh error
(2) Load sharing coefficient of internal-meshing is dif-ferent from that of external-meshing under differ-ent mesh errors Maximum external-meshing andinternal-meshing load sharing coefficients of first-stage planetary gear are 1579 and 1645 respectivelywhile those of second-stage planetary gear are 1630and 1665 respectively
(3) Compared to the differences in change rate of eachload sharing coefficient of second-stage planetarygear that of first-stage planetary gear is more evidentThe maximum difference in change rate of first-stageplanetary gear is 010150 120583m while that of second-stage planetary gear is only 000350 120583m
43 Influence of Revolution Speed on Load Sharing CoefficientTo analyze the influence of revolution speed of the first-stageplanetary gear on load sharing coefficient the revolution
8 The Scientific World Journal
10 20 30 40 50155
16
165
17
175
d1sp1d1sp2d1sp3
Gear error E1sp (120583m)
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
10 20 30 40 50155
16
165
17
175
d1rp1d1rp2d1rp3
Gear error E1rp (120583m)
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2sp1d2sp2d2sp3
Gear error E2sp (120583m)
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2rp1d2rp2d2rp3
Gear error E2rp (120583m)
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 6 Load-sharing coefficient curves of each planetary gear with different mesh errors
speed is set as 5 rmin 10 rmin 15 rmin 20 rmin and25 rmin respectively Equation (31) is used to calculatethe load sharing coefficient under different conditions andcurves are obtained in Figure 7
Influence of revolution speed on load-sharing coefficientcan be concluded below according to Figure 7
(1) Each load sharing coefficient increases with raisingthe revolution speed which indicates that load shar-ing capacity of planetary gear train is weakened andvibration is aggravated with increasing revolutionspeed
(2) At the variation interval of revolution speed thechange rate difference of load-sharing coefficientbetween internal and external meshing of first-stageplanetary gear train is significantly different thoseof first-stage planetary gears 1 2 and 3 are 177084 and 149 respectively Similar result can beconcluded in second-stage planetary gear train and
change rate differences of 147 271 and 276 ofsecond-stage planetary gears 1 2 and 3 are figuredout respectively
5 Conclusion
(1) The dynamic model is built to account for thedynamic behavior of multiple-stage planetary geartrain used in wind driven generator The model canprovide useful guideline for the dynamic design ofthe multiple-stage planetary gear train of wind drivengenerator
(2) Each load-sharing coefficient of the first-stage plan-etary gear varies more than that of the second-stageplanetary gear At the same mesh error second-stage internal-meshing load sharing coefficient is thelargest the first-stage internal-meshing load sharingcoefficient is the second largest and the first-stage
The Scientific World Journal 9
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1sp1d1sp2d1sp3
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1rp1d1rp2d1rp3
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2sp1d2sp2d2sp3
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2rp1d2rp2d2rp3
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 7 Load sharing coefficient curves of each planetary at different revolution speeds
external-meshing load sharing coefficient is the min-imum
(3) Load sharing property is weakened and transmissionsystemrsquos vibration is aggravated with increasing rev-olution speed At each interval of revolution speedinternal and external meshing load sharing coeffi-cients of the second-stage planetary gear train varymore than those of the first-stage planetary gear train
Nomenclature120579119894 Angular displacement of 119894th member(119894 = s p119899 r 119899 = 1 2 3)
119903b119894 Gear base radii 119894 = s p119899 r 119899 = 1 2 3119903c Radius of the circle passing through planet
centers119903119894c 119894th-stage radii of the circle passing
through planet centers 119894 = 1 2
120572s Sun-planet engaging angle120572r Ring-planet engaging angle119894119873 Total number of planet sets for the 119894th-
stage drive train 119894 = 1 21198681119895 Polar mass moment of inertia of 119895th
member for 1st-stage drive train 119895 =c s p1 p2 p1119873
1198721p Mass of 1st-stage planetary gear
1198681ce = 119868
1c + 11198731198721p1199032
1c1198682119895 Polar mass moment of inertia of 119895th
member for 2nd-stage drive train 119895 =c s p1 p2 p2119873
1198722p Mass of 2nd-stage planetary gear
1198682ce = 119868
2c + 21198731198722p1199032
2c119903119894b119895 Gear base radii of 119895th member for 119894th-
stage drive train 119894 = 1 2 3 119895 =
s r p1 p2 p119899 g1 g2
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
2 The Scientific World Journal
single-stage planetary gear with the dynamic way to studythe load sharing behavior of each planetary gear and therelationship between error and load sharing was analyzed Yeet al [20] built an analytical model for NGW planetary geartrainwith unequalmodulus and pressure angles and analyzedthe load sharing behavior of each planet
Although the references available focused on differentfields most of them established mathematical model ofone-stage planetary gear train Dynamic model of multiple-stage planetary gear train is limitedly reported Few reportsabout dynamic model of multiple-stage planetary gear traincomposed of two-stage planetary gear train and one-stageparallel axis and its dynamic load sharing characteristics areconcerned
In this study a transmission scheme of load-split two-stage planetary gear used in wind driven generator is pro-posed Transmission ratio of the planetary gear train isobtained as well as the relationship between transmissionratio and characteristic parameter of planetary gear trainaccording to conversion mechanism method and generalrelationship among the speed of each unit in planetary geartrain Dynamic model of load-split multiple-stage gear traincomposed of a two-stage planetary gear train and a one-stage parallel axis gear is established on the basis of lumpedparameter theory and influence of revolution speed andmesherror on dynamic load sharing characteristic of the system isanalyzed
2 Load-Split Two-Stage Planetary Gear Train
21 Kinematic Scheme The kinematic scheme of load-splittwo-stage planetary gear is shown in Figure 1 which iscomposed of closed planetary gear train and differentialplanetary gear train Former basic units 1c (planetary carrier)and 1s (sun gear) are connected to units 2r (inner ring) and2c (planetary carrier) of differential gear train respectivelyTherefore load split is realized by first-stage and second-stage gear bearing input torque simultaneously In Figure 11r 1p1 1s and 1c are inner ring planetary gear sun gearand planetary carrier of first-stage planetary gear trainrespectively while 2r 2p1 2s and 2c represent correspondingunits of second-stage planetary gear train
22 Speed of Each Unit of First-Stage Planetary Gear TrainThe relationship between rotational speed of sun gear andthat of planetary carrier and inner ring of first-stage planetarygear train is shown in
1198991s = 1198941r1s1c1198991c + 119894
1c1s1r1198991r (1)
Equation (22) is obtained according to general relation-ship of relative gear ratio among each unit in planetarygear train principle and transmission type and characteristicparameter of first-stage planetary gear train
1198941r1s1c = 1 minus 119894
1c1s1r
1198941c1s1r = minus
1198851r
1198851s
1r
2r
1c
2c
1s 2s
OutIn
1p12p1
Figure 1 Kinematic scheme of load-split two-stage planetary gear
1198851r
1198851s= 1205821
1198991r = 0
(2)
We can come to (3) by (1) and (22)
1198991s = (1 + 1205821) 1198991c (3)
The relationship between rotational speed of planetarygear and that of planetary carrier and inner ring is expressedas (4) according to relationship of relative rotational speedamong each unit in the first-stage planetary gear train
1198991p = 1198941r1p1c1198991c + 119894
1c1p1r1198991r (4)
Similar to (22) (5) is obtained as follows
1198941r1p1c = 1 minus 119894
1c1p1r
1198941c1p1r =
1198851r
1198851p
1198851p =
1198851r minus 1198851s2
1198851r
1198851s= 1205821
1198991r = 0
(5)
Thus (6) can be obtained using (5) and (4)
1198991p = (1 minus
21205821
1205821minus 1
) 1198991c (6)
The Scientific World Journal 3
where 1205821is the characteristic parameter of planetary gear
train and 1205821= 1198851r1198851s1198851r1198851s and1198851p are tooth number of
inner ring sun gear and planetary gear respectively 1198991119895(119895 =
c s r p) and 119894111990911198861119887
(119886 = c s r p 119887 = c s r p 119909 = c s r p)represent the rotational speed and relative gear ratio of eachunit of first-stage planetary gear train respectively
23 Speed of Each Unit of Second-Stage Planetary Gear TrainThe relationship between the rotational speed of sun geartrain and that of planetary carrier and inner ring of second-stage planetary gear is expressed as follows
1198992s = 1198942r2s2c1198992c + 119894
2c2s2r1198992r (7)
Equation (8) can be obtained according to the transmis-sion characteristic of basic unit of second-stage planetary geartrain
1198942r2s2c1198992c =
1198992r2s1198992r2c1198992r2c = 119899
2r2s
1198942c2s2r1198992r =
1198992c2s1198992c2r1198992c2r = 119899
2c2s
(8)
Using (7) and (8) gives
1198992s = 119899
2r2s + 1198992c2s (9)
Equations (10) and (11) are obtained by the relativemovement relationship of planetary gear trainrsquos units wheninner ring and planetary carrier of second-stage planetarygear train are fixed respectively
1198992r = 0
1198992r2s =
1198992c
1198942r2c2s
1198942r2c2s =
1
1 minus 1198942c2s2r
1198942c2s2r = minus
1198852r
1198852s
1198852r
1198852s= 1205822
(10)
1198992c = 0
1198992c2s =
1198992r
1198942c2r2s
1198942c2r2s = minus
1198852s
1198852r
1198852r
1198852s= 1205822
(11)
By connecting (10) and (11) to (9) the relationship ofrotational speed between input and output units of second-stage planetary gear train is obtained as follows
1198992s = (1 + 1205822) 1198992c minus 12058221198992r (12)
And the relationship between rotational speeds of plane-tary gear is expressed as follows
1198992p = 1198942r1p2c1198992c + 119894
2c2p2r1198992r (13)
Similar to (22) (14) can be obtained as follows
1198942r2p2c = 1 minus 119894
2c2p2r
1198942c2p2r =
1198852r
1198852p
1198852p =
1198852r minus 1198852s2
1198852r
1198852s= 1205822
(14)
Substitution of (14) into (13) yields
1198992p =
minus1205822minus 1
1205822minus 1
1198992c +
21205822
1205822minus 1
1198992r (15)
Considering the scheme of Figure 1 (16) can be given asfollows
1198992r = 1198991c
1198992c = 1198991s
(16)
Substitution of (16) and (3) into (15) gives
1198992p =
21205822minus (1205822+ 1) (1 + 120582
1)
1205822minus 1
1198991c (17)
where 1205822represents the characteristic parameter of planetary
line of second-stage planetary gear train and 1205822= 1198852r1198852s
1198852r 1198852s and 119885
2p stand for tooth number of inner ringsun gear and planetary gear of second planetary gear trainrespectively 119899
2119895(119895 = c s r p) is the rotational speed of each
unit of second-stage planetary gear train and 1198942119909
21198862119887(119886 =
c s r p 119887 = c s r p 119909 = c s r p) is the relative gear ratioof corresponding unit
24 Transmission Ratio of the Planetary Gear Train Theexpressions of input and output rotational speed of load-splittwo-stage planetary gear train are given by substitution of (16)and (3) into (12) as follows
1198992s = [(1 + 1205822) (1 + 1205821) minus 1205822] 1198991c (18)
Thus transmission ratio formula of load-split two-stageplanetary gear train is obtained as
1198942s1c =
1198992s1198991c= (1 + 120582
1) (1 + 120582
2) minus 1205822 (19)
General transmission ratio in Figure 1 is related to charac-teristic parameters of the planetary gear trains 120582
1and 120582
2 Too
small values of1198851r and1198852r result in undersizing of the system
and decreasing of bearing capacity Values of characteristic
4 The Scientific World Journal
1 2 3 4 5 6 7 8
123456780
1020304050607080
3 4 5 6 734567
Tran
smiss
ion
ratio
i c1s
1
Characteristic parameter 1205821
Characteristic parameter 1205822
Figure 2 Relationship between transmission ratio and characteris-tic parameters of planetary gear train
parameters have to be reasonable Recommended interval of1205821and 120582
2is [3 8]
Relationship between transmission ratio and character-istic parameters in load-split two-stage planetary gear trainis shown in Figure 2 Transmission ratio rises with increasingvalues of characteristic parameters of the planetary gear trainandmaximum transmission ratio in the interval of [3 8] is 73
3 Dynamic Model
31 Model of the Multiple-Stage Transmission System Amul-tiple-stage gear train composed of a two-stage planetary geartrain and a one-stage parallel axis gear is shown in Figure 33g1 and 3g2 in Figure 3 stand for pinion gear and driven gearof parallel axis
Dynamicmodel of Figure 3 is shown in Figure 4 based onlumped parameter theory Since the first-stage planetary geartrain and the second-stage planetary gear train have the samebasic structure they can be represented by the single stagepurely torsional model shown in Figure 5
The linear displacements of all members of themultistagetransmission system are shown as follows
1199061c = 1199031c1205791c 119906
2c = 1199032c1205792c
1199061s = 1199031bs1205791s 119906
2s = 1199032bs1205792s
1199061r = 1199031br1205791r 119906
2r = 1199032br1205792r
1199061p119895 = 1199031bp1198951205791p119895 119906
2p119895 = 1199032bp1198951205792p119895
1199063g1 = 1199033bg11205793g1 119906
3g2 = 1199033bg21205793g2
(20)
Generalized masses of all members of the multistagetransmission system are shown as follows
1198981c =
1198681ce1199032
1c 119898
2c =1198682ce1199032
2c 119898
1s =1198681s1199032
1bs
1198982s =
1198682s1199032
2bs 119898
1r =1198681r1199032
1br 119898
2r =1198682r1199032
2br
1r
2r
1c
2c
1s 2s
Out
In3g1
3g2
1p12p1
Figure 3 Transmission system of load-split multiple-stage plane-tary gear train
1198981p119895 =
1198681p119895
1199032
1bp119895 119898
2p119895 =1198682p119895
1199032
2bp119895 119898
3g1 =1198683g1
1199032
3bg1
1198983g2 =
1198683g2
1199032
3bg2
(21)
32 Dynamic Equation of the Multistage Transmission SystemThe interaction force between sun gear and the 119895th planetarygear of the first-stage planetary gear train along the line ofaction is given as follows
1198651sp119895 = 1198701sp1198951198831sp119895 + 1198621sp1198951sp119895
1198831sp119895 = 1199061s + 1199061p119895 minus cos120572
1s1199061c minus 1198641sp119895
1sp119895 = 1s + 1p119895 minus cos120572
1s1c minus 1sp119895
(22)
The interaction force between the inner ring and the 119895thplanet gear of the first-stage planetary gear train along the lineof action can be expressed as follows
1198651rp119895 = 1198701rp1198951198831rp119895 + 1198621rp1198951rp119895
1198831rp119895 = 1199061p119895 minus 1199061r + cos120572
1r1199061c minus 1198641rp119895
1rp119895 = 1p119895 minus 1r + cos120572
1r1c minus 1rp119895
(23)
The interaction force between sun gear and the 119895thplanetary gear of the second-stage planetary gear train alongthe line of action is
1198652sp119895 = 1198702sp1198951198832sp119895 + 1198622sp1198952sp119895
1198832sp119895 = 1199062s + 1199062p119895 minus cos120572
2s1199062c minus 1198642sp119895
2sp119895 = 2s + 2p119895 minus cos120572
2s2c minus 2sp119895
(24)
The Scientific World Journal 5
E1rp2
E2rp1
E3g1g2
E2rp2
E1rp1K1sp2C1sp2
K2sp2 K2s3g1
K3g1g2
C2sp2
C2s3g1
C3g1g2
K2sp1C2sp1
C1s2c
C1c2r
K1rp2K1c2r
K1s2c
C1rp2 K2rp2C2rp2
K1rp1
C1rp1K2rp1C2rp1
K1sp1C1sp1
E1sp2 E2sp2E1sp1 E1sp1
1205791p1 1205792p1
1205792p21205791p2
1205791r
1205793g1
1205793g21205792r
1205791c1205792c
1205791s 1205792sTi
To
Figure 4 Dynamic model of load-split multiple-stage planetary gear train
Sun
Carrier
Ring
Planet n
ypn
ys
xs
xpn
os
op
rbs
rbr
rbpnKsp
Krp
Csp
Crp
Esp
Erp
rc
120572s
120572r
120579c120579r
120579s
120579pn
Figure 5 Torsional model of single-stage planetary gear
The interaction force between the inner ring and the 119895thplanet gear of the second-stage planetary gear train along theline of action can be expressed as follows
1198652rp119895 = 1198702rp1198951198832rp119895 + 1198622rp1198952rp119895
1198832rp119895 = 1199062p119895 minus 1199062r + cos120572
2r1199062c minus 1198642rp119895
2rp119895 = 2p119895 minus 2r + cos120572
2r2c minus 2rp119895
(25)
The interaction force between the pinion gear and drivengear of the third-stage parallel axis gear along the line ofaction can be expressed as follows
1198653g1g2 = 1198703g1g21198833g1g2 + 1198623g1g23g1g2
1198833g1g2 = 1199063g1 + 1199063g2 minus 1198643g1g2
3g1g2 = 3g1 + 3g2 minus 3g1g2
(26)
Fix the inner ring of the first-stage planetary gear trainand take the number of planetary gears of the planetary geartrain as 3 namely 1119873 = 2119873 = 3 According to the planetarymechanism modeling methods in [13] dynamic equation ofthe multistage transmission system shown in Figure 4 can bebuilt as shown in
1198981c1c minus cos120572
1s
1119873
sum
119895=1
1198651sp119895 + cos120572
1r
1119873
sum
119895=1
1198651rp119895
+
1198701c2r1199031c
(
1199061c1199031cminus
1199062r1199032br) +
1198621c2r1199031c
(
1c1199031cminus
2r1199032br) =
119879i1199031c
1198981s1s +
1119873
sum
119895=1
1198651sp119895 +
1198701s2c1199031bs
(
1199061s1199031bs
minus
1199062c1199032c)
+
1198621s2c1199031bs
(
1s1199031bs
minus
2c1199032c) = 0
1198981p11p1 + 1198651sp1 + 1198651rp1 = 0
1198981p21p2 + 1198651sp2 + 1198651rp2 = 0
1198981p11198731p1119873 + 1198651sp1119873 + 1198651rp1119873 = 0
1198982c2c minus cos120572
2s
2119873
sum
119895=1
1198652sp119895 + cos120572
2r
2119873
sum
119895=1
1198652rp119895
minus
1198701s2c1199032c
(
1199061s1199031bs
minus
1199062c1199032c) minus
1198621s2c1199032c
(
1s1199031bs
minus
2c1199032c) = 0
1198982s2s +
2119873
sum
119895=1
1198652sp119895 +
1198702s3g1
1199032bs
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
+
1198622s3g1
1199032bs
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198982p12p1 + 1198652sp1 + 1198652rp1 = 0
6 The Scientific World Journal
1198982p22p2 + 1198652sp2 + 1198652rp2 = 0
1198982p21198732p2119873 + 1198652sp2119873 + 1198652rp2119873 = 0
1198982r2r minus
2119873
sum
119895=1
1198652rp119895 minus
1198701c2r1199032br
(
1199061c1199031cminus
1199062r1199032br)
minus
1198621c2r1199032br
(
1c1199031cminus
2r1199032br) = 0
1198983g13g1 + 1198653g1g2 minus
1198702s3g1
1199033bg1
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
minus
1198622s3g1
1199033bg1
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198983g23g2 + 1198653g1g2 +
119879119900
1199033bg2
= 0
(27)
The equations of the dynamic model are given in thematrix form as
119872 + 119862 + 119870119906 = 119865 (28)
where the displacement vector the mass matrix the dampingmatrix the stiffness matrix and the load vector are givenrespectively as
119906 = [1199061c 1199061s 1199061p1 1199061p2 1199061p3 1199062c 1199062s 1199062p1 1199062p2 1199062p3 1199062r 1199063g1 1199063g2]
119879
119872 = diag (1198981c 1198981s 1198981p1 1198981p2 1198981p3 1198982c 1198982s 1198982p1 1198982p2 1198982p3 1198982r 1198983g1 1198983g2)
119862 =
[
[
[
[
[
[
[
[
[
[
[
[
1198621c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198621sp119895 + cos1205722
1r
3
sum
119895=1
1198621rp119895 minus cos1205721s3
sum
119895=1
1198621sp119895 1198621rp1 cos1205721r minus 1198621sp1 cos1205721s 1198621rp2 cos1205721r minus 1198621sp2 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198621sp1198951198621s2c1199032
1bs+
3
sum
119895=1
1198621sp119895 1198621sp1 sdot sdot sdot
1198621sp1 + 1198621rp1 0 sdot sdot sdot
1198621sp2 + 1198621rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119870 =
[
[
[
[
[
[
[
[
[
[
[
[
1198701c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198701sp119895 + cos1205722
1r
3
sum
119895=1
1198701rp119895 minus cos1205721s3
sum
119895=1
1198701sp119895 1198701rp1 cos1205721r minus 1198701sp1 cos1205721s 1198701rp2 cos1205721r minus 1198701s1199012 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198701sp1198951198701s2c1199032
1bs+
3
sum
119895=1
1198701sp119895 1198701sp1 sdot sdot sdot
1198701sp1 + 1198701rp1 0 sdot sdot sdot
1198701sp2 + 1198701rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119865 =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879119894
1199031cminus cos120572
1s(3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895) + cos120572
1r(3
sum
119895=1
1198621rp1198951119903p119895 +
3
sum
119895=1
1198701rp1198951198641rp119895)
3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895
1198621sp11119904p1 + 1198701sp11198641sp1 + 1198621rp11rp1 + 1198701rp11198641rp1
1198621sp21sp2 + 1198701sp21198641sp2 + 1198621rp21rp2 + 1198701rp21198641rp2
1198621sp31sp3 + 1198701sp31198641sp3 + 1198621rp31rp3 + 1198701rp31198641rp3
cos1205722r(3
sum
119895=1
1198702rp1198951198642rp119895 +
3
sum
119895=1
1198622rp1198952rp119895) minus cos120572
2s(3
sum
119895=1
1198702sp1198951198642sp119895 +
3
sum
119895=1
1198622sp1198952sp119895)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(29)
The Scientific World Journal 7
4 Load Sharing Characteristic of Load-SplitMultiple-Stage Planetary Gear Train
41 Calculation of Load Sharing Coefficient Use numericalintegration method for solving the dynamic equation (28) ofthe system obtain the responses to displacement and velocityof the system and substitute the responses into (22)ndash(25) andthen obtain the engaging forces 119865
1sp119895 1198651rp119895 1198652sp119895 and 1198652rp119895Make 119863
1sp119895119896s and 1198631sp119895119896r respectively represent the load
sharing coefficients of the internal and external meshing ofall gear pairs of the first-stage planetary gear train and119863
2sp119894119896sand119863
2sp119894119896r as those of the internal and external meshing of allgear pairs of the second-stage planetary gear train then loadsharing coefficients are expressed as
1198631sp119895119896s =
1119873(1198651sp119895119896s)max
sum1119873
119895=1(1198651sp119895119896s)max
1198631rp119895119896r =
1119873(1198651rp119895119896r)max
sum1119873
119895=1(1198651rp119895119896r)max
1198632sp119894119896s =
2119873(1198652sp119895119896s)max
sum2119873
119894=1(1198652sp119895119896s)max
1198632rp119894119896r =
2119873(1198652rp119895119896r)max
sum2119873
119894=1(1198652rp119895119896r)max
(30)
where 119896s 119896r are meshing cycle numbers for internal andexternal meshing of the planetary gear pair
When 1198891sp119895 and 1198891rp119895 are used to stand for load sharing
coefficient of internal and externalmeshing of each first-stagegear and 119889
2sp119895 and 1198892rp119895 for that of each second-stage gearin system period respectively the expression can be given asfollows
1198891sp119895 =
100381610038161003816100381610038161198631sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198891rp119895 =
100381610038161003816100381610038161198631rp119895119896r minus 1
10038161003816100381610038161003816max + 1
1198892sp119895 =
100381610038161003816100381610038161198632sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198892rp119895 =
100381610038161003816100381610038161198632rp119895119896r minus 1
10038161003816100381610038161003816max + 1
(31)
The paper analyzes the transmission system as shown inFigure 4 The basic parameters of the transmission systemare shown in Tables 1 and 2 and other parameters can bedetermined by [21] Substitute the relevant parameters of thesystem into (28) for solution Use (30) and (31) to obtain theload sharing coefficients of the transmission system
42 Influence of Mesh Error on Load Sharing Coefficient ofthe System Load sharing property of planetary gear trainis significantly affected by manufacturing error installationerror and eccentric error which cannot be neglected inplanetary gear train Considering systemrsquos complexity it is
Table 1 Primary parameters of planetary gear train
Parameter Carrier Ring Sun gear Planetarygear
Pitch radius 1199031(mm) 468 726 210 258
Base circle 1199031119887(mm) mdash 68222 19734 24244
Mass1198721(kg) 204277 41034 34491 38839
Moment of inertia 1198691
(kgsdotm2) 46255 22657 762 1625
Pressure angle 1205721(∘) mdash 20 20 20
Pitch radius 1199032(mm)
1199032mm 345 550 140 205
Base circle 1199032119887(mm) mdash 51683 13156 19264
Mass1198722(kg) 121249 8056 13140 17654
Moment of inertia 1198692
(kgsdotm2) 15175 2565 129 512
Pressure angle 1205722(∘) mdash 20 20 20
Table 2 Primary parameters of parallel-shaft gears
Parameter Pinion gear Driven gearPitch radius 119903
3(mm) 292 100
Base circle 1199033119887(mm) 27439 9397
Mass1198723(kg) 20822 2414
Moment of inertia 1198693(kgsdotm2) 887 012
Pressure angle 1205723(∘) 20 20
assumed that equivalent mesh error of each stage planetarygear at the direction of meshing line is equal and values of10 20 30 40 and 50 120583m are given respectively Load sharingproperties of multiple-stage gear train under these fiveconditions are studied Relationships between load sharingcoefficient curves of internal and external meshing of first-stage and second-stage which are calculated according to(31) are drawn in Figure 6 with different mesh errors
Results below can be concluded according to Figure 6
(1) Each load-sharing coefficient increases with increas-ing mesh error
(2) Load sharing coefficient of internal-meshing is dif-ferent from that of external-meshing under differ-ent mesh errors Maximum external-meshing andinternal-meshing load sharing coefficients of first-stage planetary gear are 1579 and 1645 respectivelywhile those of second-stage planetary gear are 1630and 1665 respectively
(3) Compared to the differences in change rate of eachload sharing coefficient of second-stage planetarygear that of first-stage planetary gear is more evidentThe maximum difference in change rate of first-stageplanetary gear is 010150 120583m while that of second-stage planetary gear is only 000350 120583m
43 Influence of Revolution Speed on Load Sharing CoefficientTo analyze the influence of revolution speed of the first-stageplanetary gear on load sharing coefficient the revolution
8 The Scientific World Journal
10 20 30 40 50155
16
165
17
175
d1sp1d1sp2d1sp3
Gear error E1sp (120583m)
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
10 20 30 40 50155
16
165
17
175
d1rp1d1rp2d1rp3
Gear error E1rp (120583m)
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2sp1d2sp2d2sp3
Gear error E2sp (120583m)
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2rp1d2rp2d2rp3
Gear error E2rp (120583m)
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 6 Load-sharing coefficient curves of each planetary gear with different mesh errors
speed is set as 5 rmin 10 rmin 15 rmin 20 rmin and25 rmin respectively Equation (31) is used to calculatethe load sharing coefficient under different conditions andcurves are obtained in Figure 7
Influence of revolution speed on load-sharing coefficientcan be concluded below according to Figure 7
(1) Each load sharing coefficient increases with raisingthe revolution speed which indicates that load shar-ing capacity of planetary gear train is weakened andvibration is aggravated with increasing revolutionspeed
(2) At the variation interval of revolution speed thechange rate difference of load-sharing coefficientbetween internal and external meshing of first-stageplanetary gear train is significantly different thoseof first-stage planetary gears 1 2 and 3 are 177084 and 149 respectively Similar result can beconcluded in second-stage planetary gear train and
change rate differences of 147 271 and 276 ofsecond-stage planetary gears 1 2 and 3 are figuredout respectively
5 Conclusion
(1) The dynamic model is built to account for thedynamic behavior of multiple-stage planetary geartrain used in wind driven generator The model canprovide useful guideline for the dynamic design ofthe multiple-stage planetary gear train of wind drivengenerator
(2) Each load-sharing coefficient of the first-stage plan-etary gear varies more than that of the second-stageplanetary gear At the same mesh error second-stage internal-meshing load sharing coefficient is thelargest the first-stage internal-meshing load sharingcoefficient is the second largest and the first-stage
The Scientific World Journal 9
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1sp1d1sp2d1sp3
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1rp1d1rp2d1rp3
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2sp1d2sp2d2sp3
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2rp1d2rp2d2rp3
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 7 Load sharing coefficient curves of each planetary at different revolution speeds
external-meshing load sharing coefficient is the min-imum
(3) Load sharing property is weakened and transmissionsystemrsquos vibration is aggravated with increasing rev-olution speed At each interval of revolution speedinternal and external meshing load sharing coeffi-cients of the second-stage planetary gear train varymore than those of the first-stage planetary gear train
Nomenclature120579119894 Angular displacement of 119894th member(119894 = s p119899 r 119899 = 1 2 3)
119903b119894 Gear base radii 119894 = s p119899 r 119899 = 1 2 3119903c Radius of the circle passing through planet
centers119903119894c 119894th-stage radii of the circle passing
through planet centers 119894 = 1 2
120572s Sun-planet engaging angle120572r Ring-planet engaging angle119894119873 Total number of planet sets for the 119894th-
stage drive train 119894 = 1 21198681119895 Polar mass moment of inertia of 119895th
member for 1st-stage drive train 119895 =c s p1 p2 p1119873
1198721p Mass of 1st-stage planetary gear
1198681ce = 119868
1c + 11198731198721p1199032
1c1198682119895 Polar mass moment of inertia of 119895th
member for 2nd-stage drive train 119895 =c s p1 p2 p2119873
1198722p Mass of 2nd-stage planetary gear
1198682ce = 119868
2c + 21198731198722p1199032
2c119903119894b119895 Gear base radii of 119895th member for 119894th-
stage drive train 119894 = 1 2 3 119895 =
s r p1 p2 p119899 g1 g2
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
The Scientific World Journal 3
where 1205821is the characteristic parameter of planetary gear
train and 1205821= 1198851r1198851s1198851r1198851s and1198851p are tooth number of
inner ring sun gear and planetary gear respectively 1198991119895(119895 =
c s r p) and 119894111990911198861119887
(119886 = c s r p 119887 = c s r p 119909 = c s r p)represent the rotational speed and relative gear ratio of eachunit of first-stage planetary gear train respectively
23 Speed of Each Unit of Second-Stage Planetary Gear TrainThe relationship between the rotational speed of sun geartrain and that of planetary carrier and inner ring of second-stage planetary gear is expressed as follows
1198992s = 1198942r2s2c1198992c + 119894
2c2s2r1198992r (7)
Equation (8) can be obtained according to the transmis-sion characteristic of basic unit of second-stage planetary geartrain
1198942r2s2c1198992c =
1198992r2s1198992r2c1198992r2c = 119899
2r2s
1198942c2s2r1198992r =
1198992c2s1198992c2r1198992c2r = 119899
2c2s
(8)
Using (7) and (8) gives
1198992s = 119899
2r2s + 1198992c2s (9)
Equations (10) and (11) are obtained by the relativemovement relationship of planetary gear trainrsquos units wheninner ring and planetary carrier of second-stage planetarygear train are fixed respectively
1198992r = 0
1198992r2s =
1198992c
1198942r2c2s
1198942r2c2s =
1
1 minus 1198942c2s2r
1198942c2s2r = minus
1198852r
1198852s
1198852r
1198852s= 1205822
(10)
1198992c = 0
1198992c2s =
1198992r
1198942c2r2s
1198942c2r2s = minus
1198852s
1198852r
1198852r
1198852s= 1205822
(11)
By connecting (10) and (11) to (9) the relationship ofrotational speed between input and output units of second-stage planetary gear train is obtained as follows
1198992s = (1 + 1205822) 1198992c minus 12058221198992r (12)
And the relationship between rotational speeds of plane-tary gear is expressed as follows
1198992p = 1198942r1p2c1198992c + 119894
2c2p2r1198992r (13)
Similar to (22) (14) can be obtained as follows
1198942r2p2c = 1 minus 119894
2c2p2r
1198942c2p2r =
1198852r
1198852p
1198852p =
1198852r minus 1198852s2
1198852r
1198852s= 1205822
(14)
Substitution of (14) into (13) yields
1198992p =
minus1205822minus 1
1205822minus 1
1198992c +
21205822
1205822minus 1
1198992r (15)
Considering the scheme of Figure 1 (16) can be given asfollows
1198992r = 1198991c
1198992c = 1198991s
(16)
Substitution of (16) and (3) into (15) gives
1198992p =
21205822minus (1205822+ 1) (1 + 120582
1)
1205822minus 1
1198991c (17)
where 1205822represents the characteristic parameter of planetary
line of second-stage planetary gear train and 1205822= 1198852r1198852s
1198852r 1198852s and 119885
2p stand for tooth number of inner ringsun gear and planetary gear of second planetary gear trainrespectively 119899
2119895(119895 = c s r p) is the rotational speed of each
unit of second-stage planetary gear train and 1198942119909
21198862119887(119886 =
c s r p 119887 = c s r p 119909 = c s r p) is the relative gear ratioof corresponding unit
24 Transmission Ratio of the Planetary Gear Train Theexpressions of input and output rotational speed of load-splittwo-stage planetary gear train are given by substitution of (16)and (3) into (12) as follows
1198992s = [(1 + 1205822) (1 + 1205821) minus 1205822] 1198991c (18)
Thus transmission ratio formula of load-split two-stageplanetary gear train is obtained as
1198942s1c =
1198992s1198991c= (1 + 120582
1) (1 + 120582
2) minus 1205822 (19)
General transmission ratio in Figure 1 is related to charac-teristic parameters of the planetary gear trains 120582
1and 120582
2 Too
small values of1198851r and1198852r result in undersizing of the system
and decreasing of bearing capacity Values of characteristic
4 The Scientific World Journal
1 2 3 4 5 6 7 8
123456780
1020304050607080
3 4 5 6 734567
Tran
smiss
ion
ratio
i c1s
1
Characteristic parameter 1205821
Characteristic parameter 1205822
Figure 2 Relationship between transmission ratio and characteris-tic parameters of planetary gear train
parameters have to be reasonable Recommended interval of1205821and 120582
2is [3 8]
Relationship between transmission ratio and character-istic parameters in load-split two-stage planetary gear trainis shown in Figure 2 Transmission ratio rises with increasingvalues of characteristic parameters of the planetary gear trainandmaximum transmission ratio in the interval of [3 8] is 73
3 Dynamic Model
31 Model of the Multiple-Stage Transmission System Amul-tiple-stage gear train composed of a two-stage planetary geartrain and a one-stage parallel axis gear is shown in Figure 33g1 and 3g2 in Figure 3 stand for pinion gear and driven gearof parallel axis
Dynamicmodel of Figure 3 is shown in Figure 4 based onlumped parameter theory Since the first-stage planetary geartrain and the second-stage planetary gear train have the samebasic structure they can be represented by the single stagepurely torsional model shown in Figure 5
The linear displacements of all members of themultistagetransmission system are shown as follows
1199061c = 1199031c1205791c 119906
2c = 1199032c1205792c
1199061s = 1199031bs1205791s 119906
2s = 1199032bs1205792s
1199061r = 1199031br1205791r 119906
2r = 1199032br1205792r
1199061p119895 = 1199031bp1198951205791p119895 119906
2p119895 = 1199032bp1198951205792p119895
1199063g1 = 1199033bg11205793g1 119906
3g2 = 1199033bg21205793g2
(20)
Generalized masses of all members of the multistagetransmission system are shown as follows
1198981c =
1198681ce1199032
1c 119898
2c =1198682ce1199032
2c 119898
1s =1198681s1199032
1bs
1198982s =
1198682s1199032
2bs 119898
1r =1198681r1199032
1br 119898
2r =1198682r1199032
2br
1r
2r
1c
2c
1s 2s
Out
In3g1
3g2
1p12p1
Figure 3 Transmission system of load-split multiple-stage plane-tary gear train
1198981p119895 =
1198681p119895
1199032
1bp119895 119898
2p119895 =1198682p119895
1199032
2bp119895 119898
3g1 =1198683g1
1199032
3bg1
1198983g2 =
1198683g2
1199032
3bg2
(21)
32 Dynamic Equation of the Multistage Transmission SystemThe interaction force between sun gear and the 119895th planetarygear of the first-stage planetary gear train along the line ofaction is given as follows
1198651sp119895 = 1198701sp1198951198831sp119895 + 1198621sp1198951sp119895
1198831sp119895 = 1199061s + 1199061p119895 minus cos120572
1s1199061c minus 1198641sp119895
1sp119895 = 1s + 1p119895 minus cos120572
1s1c minus 1sp119895
(22)
The interaction force between the inner ring and the 119895thplanet gear of the first-stage planetary gear train along the lineof action can be expressed as follows
1198651rp119895 = 1198701rp1198951198831rp119895 + 1198621rp1198951rp119895
1198831rp119895 = 1199061p119895 minus 1199061r + cos120572
1r1199061c minus 1198641rp119895
1rp119895 = 1p119895 minus 1r + cos120572
1r1c minus 1rp119895
(23)
The interaction force between sun gear and the 119895thplanetary gear of the second-stage planetary gear train alongthe line of action is
1198652sp119895 = 1198702sp1198951198832sp119895 + 1198622sp1198952sp119895
1198832sp119895 = 1199062s + 1199062p119895 minus cos120572
2s1199062c minus 1198642sp119895
2sp119895 = 2s + 2p119895 minus cos120572
2s2c minus 2sp119895
(24)
The Scientific World Journal 5
E1rp2
E2rp1
E3g1g2
E2rp2
E1rp1K1sp2C1sp2
K2sp2 K2s3g1
K3g1g2
C2sp2
C2s3g1
C3g1g2
K2sp1C2sp1
C1s2c
C1c2r
K1rp2K1c2r
K1s2c
C1rp2 K2rp2C2rp2
K1rp1
C1rp1K2rp1C2rp1
K1sp1C1sp1
E1sp2 E2sp2E1sp1 E1sp1
1205791p1 1205792p1
1205792p21205791p2
1205791r
1205793g1
1205793g21205792r
1205791c1205792c
1205791s 1205792sTi
To
Figure 4 Dynamic model of load-split multiple-stage planetary gear train
Sun
Carrier
Ring
Planet n
ypn
ys
xs
xpn
os
op
rbs
rbr
rbpnKsp
Krp
Csp
Crp
Esp
Erp
rc
120572s
120572r
120579c120579r
120579s
120579pn
Figure 5 Torsional model of single-stage planetary gear
The interaction force between the inner ring and the 119895thplanet gear of the second-stage planetary gear train along theline of action can be expressed as follows
1198652rp119895 = 1198702rp1198951198832rp119895 + 1198622rp1198952rp119895
1198832rp119895 = 1199062p119895 minus 1199062r + cos120572
2r1199062c minus 1198642rp119895
2rp119895 = 2p119895 minus 2r + cos120572
2r2c minus 2rp119895
(25)
The interaction force between the pinion gear and drivengear of the third-stage parallel axis gear along the line ofaction can be expressed as follows
1198653g1g2 = 1198703g1g21198833g1g2 + 1198623g1g23g1g2
1198833g1g2 = 1199063g1 + 1199063g2 minus 1198643g1g2
3g1g2 = 3g1 + 3g2 minus 3g1g2
(26)
Fix the inner ring of the first-stage planetary gear trainand take the number of planetary gears of the planetary geartrain as 3 namely 1119873 = 2119873 = 3 According to the planetarymechanism modeling methods in [13] dynamic equation ofthe multistage transmission system shown in Figure 4 can bebuilt as shown in
1198981c1c minus cos120572
1s
1119873
sum
119895=1
1198651sp119895 + cos120572
1r
1119873
sum
119895=1
1198651rp119895
+
1198701c2r1199031c
(
1199061c1199031cminus
1199062r1199032br) +
1198621c2r1199031c
(
1c1199031cminus
2r1199032br) =
119879i1199031c
1198981s1s +
1119873
sum
119895=1
1198651sp119895 +
1198701s2c1199031bs
(
1199061s1199031bs
minus
1199062c1199032c)
+
1198621s2c1199031bs
(
1s1199031bs
minus
2c1199032c) = 0
1198981p11p1 + 1198651sp1 + 1198651rp1 = 0
1198981p21p2 + 1198651sp2 + 1198651rp2 = 0
1198981p11198731p1119873 + 1198651sp1119873 + 1198651rp1119873 = 0
1198982c2c minus cos120572
2s
2119873
sum
119895=1
1198652sp119895 + cos120572
2r
2119873
sum
119895=1
1198652rp119895
minus
1198701s2c1199032c
(
1199061s1199031bs
minus
1199062c1199032c) minus
1198621s2c1199032c
(
1s1199031bs
minus
2c1199032c) = 0
1198982s2s +
2119873
sum
119895=1
1198652sp119895 +
1198702s3g1
1199032bs
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
+
1198622s3g1
1199032bs
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198982p12p1 + 1198652sp1 + 1198652rp1 = 0
6 The Scientific World Journal
1198982p22p2 + 1198652sp2 + 1198652rp2 = 0
1198982p21198732p2119873 + 1198652sp2119873 + 1198652rp2119873 = 0
1198982r2r minus
2119873
sum
119895=1
1198652rp119895 minus
1198701c2r1199032br
(
1199061c1199031cminus
1199062r1199032br)
minus
1198621c2r1199032br
(
1c1199031cminus
2r1199032br) = 0
1198983g13g1 + 1198653g1g2 minus
1198702s3g1
1199033bg1
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
minus
1198622s3g1
1199033bg1
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198983g23g2 + 1198653g1g2 +
119879119900
1199033bg2
= 0
(27)
The equations of the dynamic model are given in thematrix form as
119872 + 119862 + 119870119906 = 119865 (28)
where the displacement vector the mass matrix the dampingmatrix the stiffness matrix and the load vector are givenrespectively as
119906 = [1199061c 1199061s 1199061p1 1199061p2 1199061p3 1199062c 1199062s 1199062p1 1199062p2 1199062p3 1199062r 1199063g1 1199063g2]
119879
119872 = diag (1198981c 1198981s 1198981p1 1198981p2 1198981p3 1198982c 1198982s 1198982p1 1198982p2 1198982p3 1198982r 1198983g1 1198983g2)
119862 =
[
[
[
[
[
[
[
[
[
[
[
[
1198621c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198621sp119895 + cos1205722
1r
3
sum
119895=1
1198621rp119895 minus cos1205721s3
sum
119895=1
1198621sp119895 1198621rp1 cos1205721r minus 1198621sp1 cos1205721s 1198621rp2 cos1205721r minus 1198621sp2 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198621sp1198951198621s2c1199032
1bs+
3
sum
119895=1
1198621sp119895 1198621sp1 sdot sdot sdot
1198621sp1 + 1198621rp1 0 sdot sdot sdot
1198621sp2 + 1198621rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119870 =
[
[
[
[
[
[
[
[
[
[
[
[
1198701c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198701sp119895 + cos1205722
1r
3
sum
119895=1
1198701rp119895 minus cos1205721s3
sum
119895=1
1198701sp119895 1198701rp1 cos1205721r minus 1198701sp1 cos1205721s 1198701rp2 cos1205721r minus 1198701s1199012 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198701sp1198951198701s2c1199032
1bs+
3
sum
119895=1
1198701sp119895 1198701sp1 sdot sdot sdot
1198701sp1 + 1198701rp1 0 sdot sdot sdot
1198701sp2 + 1198701rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119865 =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879119894
1199031cminus cos120572
1s(3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895) + cos120572
1r(3
sum
119895=1
1198621rp1198951119903p119895 +
3
sum
119895=1
1198701rp1198951198641rp119895)
3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895
1198621sp11119904p1 + 1198701sp11198641sp1 + 1198621rp11rp1 + 1198701rp11198641rp1
1198621sp21sp2 + 1198701sp21198641sp2 + 1198621rp21rp2 + 1198701rp21198641rp2
1198621sp31sp3 + 1198701sp31198641sp3 + 1198621rp31rp3 + 1198701rp31198641rp3
cos1205722r(3
sum
119895=1
1198702rp1198951198642rp119895 +
3
sum
119895=1
1198622rp1198952rp119895) minus cos120572
2s(3
sum
119895=1
1198702sp1198951198642sp119895 +
3
sum
119895=1
1198622sp1198952sp119895)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(29)
The Scientific World Journal 7
4 Load Sharing Characteristic of Load-SplitMultiple-Stage Planetary Gear Train
41 Calculation of Load Sharing Coefficient Use numericalintegration method for solving the dynamic equation (28) ofthe system obtain the responses to displacement and velocityof the system and substitute the responses into (22)ndash(25) andthen obtain the engaging forces 119865
1sp119895 1198651rp119895 1198652sp119895 and 1198652rp119895Make 119863
1sp119895119896s and 1198631sp119895119896r respectively represent the load
sharing coefficients of the internal and external meshing ofall gear pairs of the first-stage planetary gear train and119863
2sp119894119896sand119863
2sp119894119896r as those of the internal and external meshing of allgear pairs of the second-stage planetary gear train then loadsharing coefficients are expressed as
1198631sp119895119896s =
1119873(1198651sp119895119896s)max
sum1119873
119895=1(1198651sp119895119896s)max
1198631rp119895119896r =
1119873(1198651rp119895119896r)max
sum1119873
119895=1(1198651rp119895119896r)max
1198632sp119894119896s =
2119873(1198652sp119895119896s)max
sum2119873
119894=1(1198652sp119895119896s)max
1198632rp119894119896r =
2119873(1198652rp119895119896r)max
sum2119873
119894=1(1198652rp119895119896r)max
(30)
where 119896s 119896r are meshing cycle numbers for internal andexternal meshing of the planetary gear pair
When 1198891sp119895 and 1198891rp119895 are used to stand for load sharing
coefficient of internal and externalmeshing of each first-stagegear and 119889
2sp119895 and 1198892rp119895 for that of each second-stage gearin system period respectively the expression can be given asfollows
1198891sp119895 =
100381610038161003816100381610038161198631sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198891rp119895 =
100381610038161003816100381610038161198631rp119895119896r minus 1
10038161003816100381610038161003816max + 1
1198892sp119895 =
100381610038161003816100381610038161198632sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198892rp119895 =
100381610038161003816100381610038161198632rp119895119896r minus 1
10038161003816100381610038161003816max + 1
(31)
The paper analyzes the transmission system as shown inFigure 4 The basic parameters of the transmission systemare shown in Tables 1 and 2 and other parameters can bedetermined by [21] Substitute the relevant parameters of thesystem into (28) for solution Use (30) and (31) to obtain theload sharing coefficients of the transmission system
42 Influence of Mesh Error on Load Sharing Coefficient ofthe System Load sharing property of planetary gear trainis significantly affected by manufacturing error installationerror and eccentric error which cannot be neglected inplanetary gear train Considering systemrsquos complexity it is
Table 1 Primary parameters of planetary gear train
Parameter Carrier Ring Sun gear Planetarygear
Pitch radius 1199031(mm) 468 726 210 258
Base circle 1199031119887(mm) mdash 68222 19734 24244
Mass1198721(kg) 204277 41034 34491 38839
Moment of inertia 1198691
(kgsdotm2) 46255 22657 762 1625
Pressure angle 1205721(∘) mdash 20 20 20
Pitch radius 1199032(mm)
1199032mm 345 550 140 205
Base circle 1199032119887(mm) mdash 51683 13156 19264
Mass1198722(kg) 121249 8056 13140 17654
Moment of inertia 1198692
(kgsdotm2) 15175 2565 129 512
Pressure angle 1205722(∘) mdash 20 20 20
Table 2 Primary parameters of parallel-shaft gears
Parameter Pinion gear Driven gearPitch radius 119903
3(mm) 292 100
Base circle 1199033119887(mm) 27439 9397
Mass1198723(kg) 20822 2414
Moment of inertia 1198693(kgsdotm2) 887 012
Pressure angle 1205723(∘) 20 20
assumed that equivalent mesh error of each stage planetarygear at the direction of meshing line is equal and values of10 20 30 40 and 50 120583m are given respectively Load sharingproperties of multiple-stage gear train under these fiveconditions are studied Relationships between load sharingcoefficient curves of internal and external meshing of first-stage and second-stage which are calculated according to(31) are drawn in Figure 6 with different mesh errors
Results below can be concluded according to Figure 6
(1) Each load-sharing coefficient increases with increas-ing mesh error
(2) Load sharing coefficient of internal-meshing is dif-ferent from that of external-meshing under differ-ent mesh errors Maximum external-meshing andinternal-meshing load sharing coefficients of first-stage planetary gear are 1579 and 1645 respectivelywhile those of second-stage planetary gear are 1630and 1665 respectively
(3) Compared to the differences in change rate of eachload sharing coefficient of second-stage planetarygear that of first-stage planetary gear is more evidentThe maximum difference in change rate of first-stageplanetary gear is 010150 120583m while that of second-stage planetary gear is only 000350 120583m
43 Influence of Revolution Speed on Load Sharing CoefficientTo analyze the influence of revolution speed of the first-stageplanetary gear on load sharing coefficient the revolution
8 The Scientific World Journal
10 20 30 40 50155
16
165
17
175
d1sp1d1sp2d1sp3
Gear error E1sp (120583m)
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
10 20 30 40 50155
16
165
17
175
d1rp1d1rp2d1rp3
Gear error E1rp (120583m)
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2sp1d2sp2d2sp3
Gear error E2sp (120583m)
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2rp1d2rp2d2rp3
Gear error E2rp (120583m)
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 6 Load-sharing coefficient curves of each planetary gear with different mesh errors
speed is set as 5 rmin 10 rmin 15 rmin 20 rmin and25 rmin respectively Equation (31) is used to calculatethe load sharing coefficient under different conditions andcurves are obtained in Figure 7
Influence of revolution speed on load-sharing coefficientcan be concluded below according to Figure 7
(1) Each load sharing coefficient increases with raisingthe revolution speed which indicates that load shar-ing capacity of planetary gear train is weakened andvibration is aggravated with increasing revolutionspeed
(2) At the variation interval of revolution speed thechange rate difference of load-sharing coefficientbetween internal and external meshing of first-stageplanetary gear train is significantly different thoseof first-stage planetary gears 1 2 and 3 are 177084 and 149 respectively Similar result can beconcluded in second-stage planetary gear train and
change rate differences of 147 271 and 276 ofsecond-stage planetary gears 1 2 and 3 are figuredout respectively
5 Conclusion
(1) The dynamic model is built to account for thedynamic behavior of multiple-stage planetary geartrain used in wind driven generator The model canprovide useful guideline for the dynamic design ofthe multiple-stage planetary gear train of wind drivengenerator
(2) Each load-sharing coefficient of the first-stage plan-etary gear varies more than that of the second-stageplanetary gear At the same mesh error second-stage internal-meshing load sharing coefficient is thelargest the first-stage internal-meshing load sharingcoefficient is the second largest and the first-stage
The Scientific World Journal 9
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1sp1d1sp2d1sp3
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1rp1d1rp2d1rp3
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2sp1d2sp2d2sp3
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2rp1d2rp2d2rp3
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 7 Load sharing coefficient curves of each planetary at different revolution speeds
external-meshing load sharing coefficient is the min-imum
(3) Load sharing property is weakened and transmissionsystemrsquos vibration is aggravated with increasing rev-olution speed At each interval of revolution speedinternal and external meshing load sharing coeffi-cients of the second-stage planetary gear train varymore than those of the first-stage planetary gear train
Nomenclature120579119894 Angular displacement of 119894th member(119894 = s p119899 r 119899 = 1 2 3)
119903b119894 Gear base radii 119894 = s p119899 r 119899 = 1 2 3119903c Radius of the circle passing through planet
centers119903119894c 119894th-stage radii of the circle passing
through planet centers 119894 = 1 2
120572s Sun-planet engaging angle120572r Ring-planet engaging angle119894119873 Total number of planet sets for the 119894th-
stage drive train 119894 = 1 21198681119895 Polar mass moment of inertia of 119895th
member for 1st-stage drive train 119895 =c s p1 p2 p1119873
1198721p Mass of 1st-stage planetary gear
1198681ce = 119868
1c + 11198731198721p1199032
1c1198682119895 Polar mass moment of inertia of 119895th
member for 2nd-stage drive train 119895 =c s p1 p2 p2119873
1198722p Mass of 2nd-stage planetary gear
1198682ce = 119868
2c + 21198731198722p1199032
2c119903119894b119895 Gear base radii of 119895th member for 119894th-
stage drive train 119894 = 1 2 3 119895 =
s r p1 p2 p119899 g1 g2
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
4 The Scientific World Journal
1 2 3 4 5 6 7 8
123456780
1020304050607080
3 4 5 6 734567
Tran
smiss
ion
ratio
i c1s
1
Characteristic parameter 1205821
Characteristic parameter 1205822
Figure 2 Relationship between transmission ratio and characteris-tic parameters of planetary gear train
parameters have to be reasonable Recommended interval of1205821and 120582
2is [3 8]
Relationship between transmission ratio and character-istic parameters in load-split two-stage planetary gear trainis shown in Figure 2 Transmission ratio rises with increasingvalues of characteristic parameters of the planetary gear trainandmaximum transmission ratio in the interval of [3 8] is 73
3 Dynamic Model
31 Model of the Multiple-Stage Transmission System Amul-tiple-stage gear train composed of a two-stage planetary geartrain and a one-stage parallel axis gear is shown in Figure 33g1 and 3g2 in Figure 3 stand for pinion gear and driven gearof parallel axis
Dynamicmodel of Figure 3 is shown in Figure 4 based onlumped parameter theory Since the first-stage planetary geartrain and the second-stage planetary gear train have the samebasic structure they can be represented by the single stagepurely torsional model shown in Figure 5
The linear displacements of all members of themultistagetransmission system are shown as follows
1199061c = 1199031c1205791c 119906
2c = 1199032c1205792c
1199061s = 1199031bs1205791s 119906
2s = 1199032bs1205792s
1199061r = 1199031br1205791r 119906
2r = 1199032br1205792r
1199061p119895 = 1199031bp1198951205791p119895 119906
2p119895 = 1199032bp1198951205792p119895
1199063g1 = 1199033bg11205793g1 119906
3g2 = 1199033bg21205793g2
(20)
Generalized masses of all members of the multistagetransmission system are shown as follows
1198981c =
1198681ce1199032
1c 119898
2c =1198682ce1199032
2c 119898
1s =1198681s1199032
1bs
1198982s =
1198682s1199032
2bs 119898
1r =1198681r1199032
1br 119898
2r =1198682r1199032
2br
1r
2r
1c
2c
1s 2s
Out
In3g1
3g2
1p12p1
Figure 3 Transmission system of load-split multiple-stage plane-tary gear train
1198981p119895 =
1198681p119895
1199032
1bp119895 119898
2p119895 =1198682p119895
1199032
2bp119895 119898
3g1 =1198683g1
1199032
3bg1
1198983g2 =
1198683g2
1199032
3bg2
(21)
32 Dynamic Equation of the Multistage Transmission SystemThe interaction force between sun gear and the 119895th planetarygear of the first-stage planetary gear train along the line ofaction is given as follows
1198651sp119895 = 1198701sp1198951198831sp119895 + 1198621sp1198951sp119895
1198831sp119895 = 1199061s + 1199061p119895 minus cos120572
1s1199061c minus 1198641sp119895
1sp119895 = 1s + 1p119895 minus cos120572
1s1c minus 1sp119895
(22)
The interaction force between the inner ring and the 119895thplanet gear of the first-stage planetary gear train along the lineof action can be expressed as follows
1198651rp119895 = 1198701rp1198951198831rp119895 + 1198621rp1198951rp119895
1198831rp119895 = 1199061p119895 minus 1199061r + cos120572
1r1199061c minus 1198641rp119895
1rp119895 = 1p119895 minus 1r + cos120572
1r1c minus 1rp119895
(23)
The interaction force between sun gear and the 119895thplanetary gear of the second-stage planetary gear train alongthe line of action is
1198652sp119895 = 1198702sp1198951198832sp119895 + 1198622sp1198952sp119895
1198832sp119895 = 1199062s + 1199062p119895 minus cos120572
2s1199062c minus 1198642sp119895
2sp119895 = 2s + 2p119895 minus cos120572
2s2c minus 2sp119895
(24)
The Scientific World Journal 5
E1rp2
E2rp1
E3g1g2
E2rp2
E1rp1K1sp2C1sp2
K2sp2 K2s3g1
K3g1g2
C2sp2
C2s3g1
C3g1g2
K2sp1C2sp1
C1s2c
C1c2r
K1rp2K1c2r
K1s2c
C1rp2 K2rp2C2rp2
K1rp1
C1rp1K2rp1C2rp1
K1sp1C1sp1
E1sp2 E2sp2E1sp1 E1sp1
1205791p1 1205792p1
1205792p21205791p2
1205791r
1205793g1
1205793g21205792r
1205791c1205792c
1205791s 1205792sTi
To
Figure 4 Dynamic model of load-split multiple-stage planetary gear train
Sun
Carrier
Ring
Planet n
ypn
ys
xs
xpn
os
op
rbs
rbr
rbpnKsp
Krp
Csp
Crp
Esp
Erp
rc
120572s
120572r
120579c120579r
120579s
120579pn
Figure 5 Torsional model of single-stage planetary gear
The interaction force between the inner ring and the 119895thplanet gear of the second-stage planetary gear train along theline of action can be expressed as follows
1198652rp119895 = 1198702rp1198951198832rp119895 + 1198622rp1198952rp119895
1198832rp119895 = 1199062p119895 minus 1199062r + cos120572
2r1199062c minus 1198642rp119895
2rp119895 = 2p119895 minus 2r + cos120572
2r2c minus 2rp119895
(25)
The interaction force between the pinion gear and drivengear of the third-stage parallel axis gear along the line ofaction can be expressed as follows
1198653g1g2 = 1198703g1g21198833g1g2 + 1198623g1g23g1g2
1198833g1g2 = 1199063g1 + 1199063g2 minus 1198643g1g2
3g1g2 = 3g1 + 3g2 minus 3g1g2
(26)
Fix the inner ring of the first-stage planetary gear trainand take the number of planetary gears of the planetary geartrain as 3 namely 1119873 = 2119873 = 3 According to the planetarymechanism modeling methods in [13] dynamic equation ofthe multistage transmission system shown in Figure 4 can bebuilt as shown in
1198981c1c minus cos120572
1s
1119873
sum
119895=1
1198651sp119895 + cos120572
1r
1119873
sum
119895=1
1198651rp119895
+
1198701c2r1199031c
(
1199061c1199031cminus
1199062r1199032br) +
1198621c2r1199031c
(
1c1199031cminus
2r1199032br) =
119879i1199031c
1198981s1s +
1119873
sum
119895=1
1198651sp119895 +
1198701s2c1199031bs
(
1199061s1199031bs
minus
1199062c1199032c)
+
1198621s2c1199031bs
(
1s1199031bs
minus
2c1199032c) = 0
1198981p11p1 + 1198651sp1 + 1198651rp1 = 0
1198981p21p2 + 1198651sp2 + 1198651rp2 = 0
1198981p11198731p1119873 + 1198651sp1119873 + 1198651rp1119873 = 0
1198982c2c minus cos120572
2s
2119873
sum
119895=1
1198652sp119895 + cos120572
2r
2119873
sum
119895=1
1198652rp119895
minus
1198701s2c1199032c
(
1199061s1199031bs
minus
1199062c1199032c) minus
1198621s2c1199032c
(
1s1199031bs
minus
2c1199032c) = 0
1198982s2s +
2119873
sum
119895=1
1198652sp119895 +
1198702s3g1
1199032bs
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
+
1198622s3g1
1199032bs
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198982p12p1 + 1198652sp1 + 1198652rp1 = 0
6 The Scientific World Journal
1198982p22p2 + 1198652sp2 + 1198652rp2 = 0
1198982p21198732p2119873 + 1198652sp2119873 + 1198652rp2119873 = 0
1198982r2r minus
2119873
sum
119895=1
1198652rp119895 minus
1198701c2r1199032br
(
1199061c1199031cminus
1199062r1199032br)
minus
1198621c2r1199032br
(
1c1199031cminus
2r1199032br) = 0
1198983g13g1 + 1198653g1g2 minus
1198702s3g1
1199033bg1
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
minus
1198622s3g1
1199033bg1
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198983g23g2 + 1198653g1g2 +
119879119900
1199033bg2
= 0
(27)
The equations of the dynamic model are given in thematrix form as
119872 + 119862 + 119870119906 = 119865 (28)
where the displacement vector the mass matrix the dampingmatrix the stiffness matrix and the load vector are givenrespectively as
119906 = [1199061c 1199061s 1199061p1 1199061p2 1199061p3 1199062c 1199062s 1199062p1 1199062p2 1199062p3 1199062r 1199063g1 1199063g2]
119879
119872 = diag (1198981c 1198981s 1198981p1 1198981p2 1198981p3 1198982c 1198982s 1198982p1 1198982p2 1198982p3 1198982r 1198983g1 1198983g2)
119862 =
[
[
[
[
[
[
[
[
[
[
[
[
1198621c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198621sp119895 + cos1205722
1r
3
sum
119895=1
1198621rp119895 minus cos1205721s3
sum
119895=1
1198621sp119895 1198621rp1 cos1205721r minus 1198621sp1 cos1205721s 1198621rp2 cos1205721r minus 1198621sp2 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198621sp1198951198621s2c1199032
1bs+
3
sum
119895=1
1198621sp119895 1198621sp1 sdot sdot sdot
1198621sp1 + 1198621rp1 0 sdot sdot sdot
1198621sp2 + 1198621rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119870 =
[
[
[
[
[
[
[
[
[
[
[
[
1198701c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198701sp119895 + cos1205722
1r
3
sum
119895=1
1198701rp119895 minus cos1205721s3
sum
119895=1
1198701sp119895 1198701rp1 cos1205721r minus 1198701sp1 cos1205721s 1198701rp2 cos1205721r minus 1198701s1199012 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198701sp1198951198701s2c1199032
1bs+
3
sum
119895=1
1198701sp119895 1198701sp1 sdot sdot sdot
1198701sp1 + 1198701rp1 0 sdot sdot sdot
1198701sp2 + 1198701rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119865 =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879119894
1199031cminus cos120572
1s(3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895) + cos120572
1r(3
sum
119895=1
1198621rp1198951119903p119895 +
3
sum
119895=1
1198701rp1198951198641rp119895)
3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895
1198621sp11119904p1 + 1198701sp11198641sp1 + 1198621rp11rp1 + 1198701rp11198641rp1
1198621sp21sp2 + 1198701sp21198641sp2 + 1198621rp21rp2 + 1198701rp21198641rp2
1198621sp31sp3 + 1198701sp31198641sp3 + 1198621rp31rp3 + 1198701rp31198641rp3
cos1205722r(3
sum
119895=1
1198702rp1198951198642rp119895 +
3
sum
119895=1
1198622rp1198952rp119895) minus cos120572
2s(3
sum
119895=1
1198702sp1198951198642sp119895 +
3
sum
119895=1
1198622sp1198952sp119895)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(29)
The Scientific World Journal 7
4 Load Sharing Characteristic of Load-SplitMultiple-Stage Planetary Gear Train
41 Calculation of Load Sharing Coefficient Use numericalintegration method for solving the dynamic equation (28) ofthe system obtain the responses to displacement and velocityof the system and substitute the responses into (22)ndash(25) andthen obtain the engaging forces 119865
1sp119895 1198651rp119895 1198652sp119895 and 1198652rp119895Make 119863
1sp119895119896s and 1198631sp119895119896r respectively represent the load
sharing coefficients of the internal and external meshing ofall gear pairs of the first-stage planetary gear train and119863
2sp119894119896sand119863
2sp119894119896r as those of the internal and external meshing of allgear pairs of the second-stage planetary gear train then loadsharing coefficients are expressed as
1198631sp119895119896s =
1119873(1198651sp119895119896s)max
sum1119873
119895=1(1198651sp119895119896s)max
1198631rp119895119896r =
1119873(1198651rp119895119896r)max
sum1119873
119895=1(1198651rp119895119896r)max
1198632sp119894119896s =
2119873(1198652sp119895119896s)max
sum2119873
119894=1(1198652sp119895119896s)max
1198632rp119894119896r =
2119873(1198652rp119895119896r)max
sum2119873
119894=1(1198652rp119895119896r)max
(30)
where 119896s 119896r are meshing cycle numbers for internal andexternal meshing of the planetary gear pair
When 1198891sp119895 and 1198891rp119895 are used to stand for load sharing
coefficient of internal and externalmeshing of each first-stagegear and 119889
2sp119895 and 1198892rp119895 for that of each second-stage gearin system period respectively the expression can be given asfollows
1198891sp119895 =
100381610038161003816100381610038161198631sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198891rp119895 =
100381610038161003816100381610038161198631rp119895119896r minus 1
10038161003816100381610038161003816max + 1
1198892sp119895 =
100381610038161003816100381610038161198632sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198892rp119895 =
100381610038161003816100381610038161198632rp119895119896r minus 1
10038161003816100381610038161003816max + 1
(31)
The paper analyzes the transmission system as shown inFigure 4 The basic parameters of the transmission systemare shown in Tables 1 and 2 and other parameters can bedetermined by [21] Substitute the relevant parameters of thesystem into (28) for solution Use (30) and (31) to obtain theload sharing coefficients of the transmission system
42 Influence of Mesh Error on Load Sharing Coefficient ofthe System Load sharing property of planetary gear trainis significantly affected by manufacturing error installationerror and eccentric error which cannot be neglected inplanetary gear train Considering systemrsquos complexity it is
Table 1 Primary parameters of planetary gear train
Parameter Carrier Ring Sun gear Planetarygear
Pitch radius 1199031(mm) 468 726 210 258
Base circle 1199031119887(mm) mdash 68222 19734 24244
Mass1198721(kg) 204277 41034 34491 38839
Moment of inertia 1198691
(kgsdotm2) 46255 22657 762 1625
Pressure angle 1205721(∘) mdash 20 20 20
Pitch radius 1199032(mm)
1199032mm 345 550 140 205
Base circle 1199032119887(mm) mdash 51683 13156 19264
Mass1198722(kg) 121249 8056 13140 17654
Moment of inertia 1198692
(kgsdotm2) 15175 2565 129 512
Pressure angle 1205722(∘) mdash 20 20 20
Table 2 Primary parameters of parallel-shaft gears
Parameter Pinion gear Driven gearPitch radius 119903
3(mm) 292 100
Base circle 1199033119887(mm) 27439 9397
Mass1198723(kg) 20822 2414
Moment of inertia 1198693(kgsdotm2) 887 012
Pressure angle 1205723(∘) 20 20
assumed that equivalent mesh error of each stage planetarygear at the direction of meshing line is equal and values of10 20 30 40 and 50 120583m are given respectively Load sharingproperties of multiple-stage gear train under these fiveconditions are studied Relationships between load sharingcoefficient curves of internal and external meshing of first-stage and second-stage which are calculated according to(31) are drawn in Figure 6 with different mesh errors
Results below can be concluded according to Figure 6
(1) Each load-sharing coefficient increases with increas-ing mesh error
(2) Load sharing coefficient of internal-meshing is dif-ferent from that of external-meshing under differ-ent mesh errors Maximum external-meshing andinternal-meshing load sharing coefficients of first-stage planetary gear are 1579 and 1645 respectivelywhile those of second-stage planetary gear are 1630and 1665 respectively
(3) Compared to the differences in change rate of eachload sharing coefficient of second-stage planetarygear that of first-stage planetary gear is more evidentThe maximum difference in change rate of first-stageplanetary gear is 010150 120583m while that of second-stage planetary gear is only 000350 120583m
43 Influence of Revolution Speed on Load Sharing CoefficientTo analyze the influence of revolution speed of the first-stageplanetary gear on load sharing coefficient the revolution
8 The Scientific World Journal
10 20 30 40 50155
16
165
17
175
d1sp1d1sp2d1sp3
Gear error E1sp (120583m)
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
10 20 30 40 50155
16
165
17
175
d1rp1d1rp2d1rp3
Gear error E1rp (120583m)
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2sp1d2sp2d2sp3
Gear error E2sp (120583m)
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2rp1d2rp2d2rp3
Gear error E2rp (120583m)
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 6 Load-sharing coefficient curves of each planetary gear with different mesh errors
speed is set as 5 rmin 10 rmin 15 rmin 20 rmin and25 rmin respectively Equation (31) is used to calculatethe load sharing coefficient under different conditions andcurves are obtained in Figure 7
Influence of revolution speed on load-sharing coefficientcan be concluded below according to Figure 7
(1) Each load sharing coefficient increases with raisingthe revolution speed which indicates that load shar-ing capacity of planetary gear train is weakened andvibration is aggravated with increasing revolutionspeed
(2) At the variation interval of revolution speed thechange rate difference of load-sharing coefficientbetween internal and external meshing of first-stageplanetary gear train is significantly different thoseof first-stage planetary gears 1 2 and 3 are 177084 and 149 respectively Similar result can beconcluded in second-stage planetary gear train and
change rate differences of 147 271 and 276 ofsecond-stage planetary gears 1 2 and 3 are figuredout respectively
5 Conclusion
(1) The dynamic model is built to account for thedynamic behavior of multiple-stage planetary geartrain used in wind driven generator The model canprovide useful guideline for the dynamic design ofthe multiple-stage planetary gear train of wind drivengenerator
(2) Each load-sharing coefficient of the first-stage plan-etary gear varies more than that of the second-stageplanetary gear At the same mesh error second-stage internal-meshing load sharing coefficient is thelargest the first-stage internal-meshing load sharingcoefficient is the second largest and the first-stage
The Scientific World Journal 9
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1sp1d1sp2d1sp3
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1rp1d1rp2d1rp3
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2sp1d2sp2d2sp3
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2rp1d2rp2d2rp3
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 7 Load sharing coefficient curves of each planetary at different revolution speeds
external-meshing load sharing coefficient is the min-imum
(3) Load sharing property is weakened and transmissionsystemrsquos vibration is aggravated with increasing rev-olution speed At each interval of revolution speedinternal and external meshing load sharing coeffi-cients of the second-stage planetary gear train varymore than those of the first-stage planetary gear train
Nomenclature120579119894 Angular displacement of 119894th member(119894 = s p119899 r 119899 = 1 2 3)
119903b119894 Gear base radii 119894 = s p119899 r 119899 = 1 2 3119903c Radius of the circle passing through planet
centers119903119894c 119894th-stage radii of the circle passing
through planet centers 119894 = 1 2
120572s Sun-planet engaging angle120572r Ring-planet engaging angle119894119873 Total number of planet sets for the 119894th-
stage drive train 119894 = 1 21198681119895 Polar mass moment of inertia of 119895th
member for 1st-stage drive train 119895 =c s p1 p2 p1119873
1198721p Mass of 1st-stage planetary gear
1198681ce = 119868
1c + 11198731198721p1199032
1c1198682119895 Polar mass moment of inertia of 119895th
member for 2nd-stage drive train 119895 =c s p1 p2 p2119873
1198722p Mass of 2nd-stage planetary gear
1198682ce = 119868
2c + 21198731198722p1199032
2c119903119894b119895 Gear base radii of 119895th member for 119894th-
stage drive train 119894 = 1 2 3 119895 =
s r p1 p2 p119899 g1 g2
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
The Scientific World Journal 5
E1rp2
E2rp1
E3g1g2
E2rp2
E1rp1K1sp2C1sp2
K2sp2 K2s3g1
K3g1g2
C2sp2
C2s3g1
C3g1g2
K2sp1C2sp1
C1s2c
C1c2r
K1rp2K1c2r
K1s2c
C1rp2 K2rp2C2rp2
K1rp1
C1rp1K2rp1C2rp1
K1sp1C1sp1
E1sp2 E2sp2E1sp1 E1sp1
1205791p1 1205792p1
1205792p21205791p2
1205791r
1205793g1
1205793g21205792r
1205791c1205792c
1205791s 1205792sTi
To
Figure 4 Dynamic model of load-split multiple-stage planetary gear train
Sun
Carrier
Ring
Planet n
ypn
ys
xs
xpn
os
op
rbs
rbr
rbpnKsp
Krp
Csp
Crp
Esp
Erp
rc
120572s
120572r
120579c120579r
120579s
120579pn
Figure 5 Torsional model of single-stage planetary gear
The interaction force between the inner ring and the 119895thplanet gear of the second-stage planetary gear train along theline of action can be expressed as follows
1198652rp119895 = 1198702rp1198951198832rp119895 + 1198622rp1198952rp119895
1198832rp119895 = 1199062p119895 minus 1199062r + cos120572
2r1199062c minus 1198642rp119895
2rp119895 = 2p119895 minus 2r + cos120572
2r2c minus 2rp119895
(25)
The interaction force between the pinion gear and drivengear of the third-stage parallel axis gear along the line ofaction can be expressed as follows
1198653g1g2 = 1198703g1g21198833g1g2 + 1198623g1g23g1g2
1198833g1g2 = 1199063g1 + 1199063g2 minus 1198643g1g2
3g1g2 = 3g1 + 3g2 minus 3g1g2
(26)
Fix the inner ring of the first-stage planetary gear trainand take the number of planetary gears of the planetary geartrain as 3 namely 1119873 = 2119873 = 3 According to the planetarymechanism modeling methods in [13] dynamic equation ofthe multistage transmission system shown in Figure 4 can bebuilt as shown in
1198981c1c minus cos120572
1s
1119873
sum
119895=1
1198651sp119895 + cos120572
1r
1119873
sum
119895=1
1198651rp119895
+
1198701c2r1199031c
(
1199061c1199031cminus
1199062r1199032br) +
1198621c2r1199031c
(
1c1199031cminus
2r1199032br) =
119879i1199031c
1198981s1s +
1119873
sum
119895=1
1198651sp119895 +
1198701s2c1199031bs
(
1199061s1199031bs
minus
1199062c1199032c)
+
1198621s2c1199031bs
(
1s1199031bs
minus
2c1199032c) = 0
1198981p11p1 + 1198651sp1 + 1198651rp1 = 0
1198981p21p2 + 1198651sp2 + 1198651rp2 = 0
1198981p11198731p1119873 + 1198651sp1119873 + 1198651rp1119873 = 0
1198982c2c minus cos120572
2s
2119873
sum
119895=1
1198652sp119895 + cos120572
2r
2119873
sum
119895=1
1198652rp119895
minus
1198701s2c1199032c
(
1199061s1199031bs
minus
1199062c1199032c) minus
1198621s2c1199032c
(
1s1199031bs
minus
2c1199032c) = 0
1198982s2s +
2119873
sum
119895=1
1198652sp119895 +
1198702s3g1
1199032bs
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
+
1198622s3g1
1199032bs
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198982p12p1 + 1198652sp1 + 1198652rp1 = 0
6 The Scientific World Journal
1198982p22p2 + 1198652sp2 + 1198652rp2 = 0
1198982p21198732p2119873 + 1198652sp2119873 + 1198652rp2119873 = 0
1198982r2r minus
2119873
sum
119895=1
1198652rp119895 minus
1198701c2r1199032br
(
1199061c1199031cminus
1199062r1199032br)
minus
1198621c2r1199032br
(
1c1199031cminus
2r1199032br) = 0
1198983g13g1 + 1198653g1g2 minus
1198702s3g1
1199033bg1
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
minus
1198622s3g1
1199033bg1
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198983g23g2 + 1198653g1g2 +
119879119900
1199033bg2
= 0
(27)
The equations of the dynamic model are given in thematrix form as
119872 + 119862 + 119870119906 = 119865 (28)
where the displacement vector the mass matrix the dampingmatrix the stiffness matrix and the load vector are givenrespectively as
119906 = [1199061c 1199061s 1199061p1 1199061p2 1199061p3 1199062c 1199062s 1199062p1 1199062p2 1199062p3 1199062r 1199063g1 1199063g2]
119879
119872 = diag (1198981c 1198981s 1198981p1 1198981p2 1198981p3 1198982c 1198982s 1198982p1 1198982p2 1198982p3 1198982r 1198983g1 1198983g2)
119862 =
[
[
[
[
[
[
[
[
[
[
[
[
1198621c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198621sp119895 + cos1205722
1r
3
sum
119895=1
1198621rp119895 minus cos1205721s3
sum
119895=1
1198621sp119895 1198621rp1 cos1205721r minus 1198621sp1 cos1205721s 1198621rp2 cos1205721r minus 1198621sp2 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198621sp1198951198621s2c1199032
1bs+
3
sum
119895=1
1198621sp119895 1198621sp1 sdot sdot sdot
1198621sp1 + 1198621rp1 0 sdot sdot sdot
1198621sp2 + 1198621rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119870 =
[
[
[
[
[
[
[
[
[
[
[
[
1198701c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198701sp119895 + cos1205722
1r
3
sum
119895=1
1198701rp119895 minus cos1205721s3
sum
119895=1
1198701sp119895 1198701rp1 cos1205721r minus 1198701sp1 cos1205721s 1198701rp2 cos1205721r minus 1198701s1199012 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198701sp1198951198701s2c1199032
1bs+
3
sum
119895=1
1198701sp119895 1198701sp1 sdot sdot sdot
1198701sp1 + 1198701rp1 0 sdot sdot sdot
1198701sp2 + 1198701rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119865 =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879119894
1199031cminus cos120572
1s(3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895) + cos120572
1r(3
sum
119895=1
1198621rp1198951119903p119895 +
3
sum
119895=1
1198701rp1198951198641rp119895)
3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895
1198621sp11119904p1 + 1198701sp11198641sp1 + 1198621rp11rp1 + 1198701rp11198641rp1
1198621sp21sp2 + 1198701sp21198641sp2 + 1198621rp21rp2 + 1198701rp21198641rp2
1198621sp31sp3 + 1198701sp31198641sp3 + 1198621rp31rp3 + 1198701rp31198641rp3
cos1205722r(3
sum
119895=1
1198702rp1198951198642rp119895 +
3
sum
119895=1
1198622rp1198952rp119895) minus cos120572
2s(3
sum
119895=1
1198702sp1198951198642sp119895 +
3
sum
119895=1
1198622sp1198952sp119895)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(29)
The Scientific World Journal 7
4 Load Sharing Characteristic of Load-SplitMultiple-Stage Planetary Gear Train
41 Calculation of Load Sharing Coefficient Use numericalintegration method for solving the dynamic equation (28) ofthe system obtain the responses to displacement and velocityof the system and substitute the responses into (22)ndash(25) andthen obtain the engaging forces 119865
1sp119895 1198651rp119895 1198652sp119895 and 1198652rp119895Make 119863
1sp119895119896s and 1198631sp119895119896r respectively represent the load
sharing coefficients of the internal and external meshing ofall gear pairs of the first-stage planetary gear train and119863
2sp119894119896sand119863
2sp119894119896r as those of the internal and external meshing of allgear pairs of the second-stage planetary gear train then loadsharing coefficients are expressed as
1198631sp119895119896s =
1119873(1198651sp119895119896s)max
sum1119873
119895=1(1198651sp119895119896s)max
1198631rp119895119896r =
1119873(1198651rp119895119896r)max
sum1119873
119895=1(1198651rp119895119896r)max
1198632sp119894119896s =
2119873(1198652sp119895119896s)max
sum2119873
119894=1(1198652sp119895119896s)max
1198632rp119894119896r =
2119873(1198652rp119895119896r)max
sum2119873
119894=1(1198652rp119895119896r)max
(30)
where 119896s 119896r are meshing cycle numbers for internal andexternal meshing of the planetary gear pair
When 1198891sp119895 and 1198891rp119895 are used to stand for load sharing
coefficient of internal and externalmeshing of each first-stagegear and 119889
2sp119895 and 1198892rp119895 for that of each second-stage gearin system period respectively the expression can be given asfollows
1198891sp119895 =
100381610038161003816100381610038161198631sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198891rp119895 =
100381610038161003816100381610038161198631rp119895119896r minus 1
10038161003816100381610038161003816max + 1
1198892sp119895 =
100381610038161003816100381610038161198632sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198892rp119895 =
100381610038161003816100381610038161198632rp119895119896r minus 1
10038161003816100381610038161003816max + 1
(31)
The paper analyzes the transmission system as shown inFigure 4 The basic parameters of the transmission systemare shown in Tables 1 and 2 and other parameters can bedetermined by [21] Substitute the relevant parameters of thesystem into (28) for solution Use (30) and (31) to obtain theload sharing coefficients of the transmission system
42 Influence of Mesh Error on Load Sharing Coefficient ofthe System Load sharing property of planetary gear trainis significantly affected by manufacturing error installationerror and eccentric error which cannot be neglected inplanetary gear train Considering systemrsquos complexity it is
Table 1 Primary parameters of planetary gear train
Parameter Carrier Ring Sun gear Planetarygear
Pitch radius 1199031(mm) 468 726 210 258
Base circle 1199031119887(mm) mdash 68222 19734 24244
Mass1198721(kg) 204277 41034 34491 38839
Moment of inertia 1198691
(kgsdotm2) 46255 22657 762 1625
Pressure angle 1205721(∘) mdash 20 20 20
Pitch radius 1199032(mm)
1199032mm 345 550 140 205
Base circle 1199032119887(mm) mdash 51683 13156 19264
Mass1198722(kg) 121249 8056 13140 17654
Moment of inertia 1198692
(kgsdotm2) 15175 2565 129 512
Pressure angle 1205722(∘) mdash 20 20 20
Table 2 Primary parameters of parallel-shaft gears
Parameter Pinion gear Driven gearPitch radius 119903
3(mm) 292 100
Base circle 1199033119887(mm) 27439 9397
Mass1198723(kg) 20822 2414
Moment of inertia 1198693(kgsdotm2) 887 012
Pressure angle 1205723(∘) 20 20
assumed that equivalent mesh error of each stage planetarygear at the direction of meshing line is equal and values of10 20 30 40 and 50 120583m are given respectively Load sharingproperties of multiple-stage gear train under these fiveconditions are studied Relationships between load sharingcoefficient curves of internal and external meshing of first-stage and second-stage which are calculated according to(31) are drawn in Figure 6 with different mesh errors
Results below can be concluded according to Figure 6
(1) Each load-sharing coefficient increases with increas-ing mesh error
(2) Load sharing coefficient of internal-meshing is dif-ferent from that of external-meshing under differ-ent mesh errors Maximum external-meshing andinternal-meshing load sharing coefficients of first-stage planetary gear are 1579 and 1645 respectivelywhile those of second-stage planetary gear are 1630and 1665 respectively
(3) Compared to the differences in change rate of eachload sharing coefficient of second-stage planetarygear that of first-stage planetary gear is more evidentThe maximum difference in change rate of first-stageplanetary gear is 010150 120583m while that of second-stage planetary gear is only 000350 120583m
43 Influence of Revolution Speed on Load Sharing CoefficientTo analyze the influence of revolution speed of the first-stageplanetary gear on load sharing coefficient the revolution
8 The Scientific World Journal
10 20 30 40 50155
16
165
17
175
d1sp1d1sp2d1sp3
Gear error E1sp (120583m)
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
10 20 30 40 50155
16
165
17
175
d1rp1d1rp2d1rp3
Gear error E1rp (120583m)
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2sp1d2sp2d2sp3
Gear error E2sp (120583m)
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2rp1d2rp2d2rp3
Gear error E2rp (120583m)
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 6 Load-sharing coefficient curves of each planetary gear with different mesh errors
speed is set as 5 rmin 10 rmin 15 rmin 20 rmin and25 rmin respectively Equation (31) is used to calculatethe load sharing coefficient under different conditions andcurves are obtained in Figure 7
Influence of revolution speed on load-sharing coefficientcan be concluded below according to Figure 7
(1) Each load sharing coefficient increases with raisingthe revolution speed which indicates that load shar-ing capacity of planetary gear train is weakened andvibration is aggravated with increasing revolutionspeed
(2) At the variation interval of revolution speed thechange rate difference of load-sharing coefficientbetween internal and external meshing of first-stageplanetary gear train is significantly different thoseof first-stage planetary gears 1 2 and 3 are 177084 and 149 respectively Similar result can beconcluded in second-stage planetary gear train and
change rate differences of 147 271 and 276 ofsecond-stage planetary gears 1 2 and 3 are figuredout respectively
5 Conclusion
(1) The dynamic model is built to account for thedynamic behavior of multiple-stage planetary geartrain used in wind driven generator The model canprovide useful guideline for the dynamic design ofthe multiple-stage planetary gear train of wind drivengenerator
(2) Each load-sharing coefficient of the first-stage plan-etary gear varies more than that of the second-stageplanetary gear At the same mesh error second-stage internal-meshing load sharing coefficient is thelargest the first-stage internal-meshing load sharingcoefficient is the second largest and the first-stage
The Scientific World Journal 9
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1sp1d1sp2d1sp3
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1rp1d1rp2d1rp3
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2sp1d2sp2d2sp3
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2rp1d2rp2d2rp3
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 7 Load sharing coefficient curves of each planetary at different revolution speeds
external-meshing load sharing coefficient is the min-imum
(3) Load sharing property is weakened and transmissionsystemrsquos vibration is aggravated with increasing rev-olution speed At each interval of revolution speedinternal and external meshing load sharing coeffi-cients of the second-stage planetary gear train varymore than those of the first-stage planetary gear train
Nomenclature120579119894 Angular displacement of 119894th member(119894 = s p119899 r 119899 = 1 2 3)
119903b119894 Gear base radii 119894 = s p119899 r 119899 = 1 2 3119903c Radius of the circle passing through planet
centers119903119894c 119894th-stage radii of the circle passing
through planet centers 119894 = 1 2
120572s Sun-planet engaging angle120572r Ring-planet engaging angle119894119873 Total number of planet sets for the 119894th-
stage drive train 119894 = 1 21198681119895 Polar mass moment of inertia of 119895th
member for 1st-stage drive train 119895 =c s p1 p2 p1119873
1198721p Mass of 1st-stage planetary gear
1198681ce = 119868
1c + 11198731198721p1199032
1c1198682119895 Polar mass moment of inertia of 119895th
member for 2nd-stage drive train 119895 =c s p1 p2 p2119873
1198722p Mass of 2nd-stage planetary gear
1198682ce = 119868
2c + 21198731198722p1199032
2c119903119894b119895 Gear base radii of 119895th member for 119894th-
stage drive train 119894 = 1 2 3 119895 =
s r p1 p2 p119899 g1 g2
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
6 The Scientific World Journal
1198982p22p2 + 1198652sp2 + 1198652rp2 = 0
1198982p21198732p2119873 + 1198652sp2119873 + 1198652rp2119873 = 0
1198982r2r minus
2119873
sum
119895=1
1198652rp119895 minus
1198701c2r1199032br
(
1199061c1199031cminus
1199062r1199032br)
minus
1198621c2r1199032br
(
1c1199031cminus
2r1199032br) = 0
1198983g13g1 + 1198653g1g2 minus
1198702s3g1
1199033bg1
(
1199062s1199032bs
minus
1199063g1
1199033bg1
)
minus
1198622s3g1
1199033bg1
(
2s1199032bs
minus
3g1
1199033bg1
) = 0
1198983g23g2 + 1198653g1g2 +
119879119900
1199033bg2
= 0
(27)
The equations of the dynamic model are given in thematrix form as
119872 + 119862 + 119870119906 = 119865 (28)
where the displacement vector the mass matrix the dampingmatrix the stiffness matrix and the load vector are givenrespectively as
119906 = [1199061c 1199061s 1199061p1 1199061p2 1199061p3 1199062c 1199062s 1199062p1 1199062p2 1199062p3 1199062r 1199063g1 1199063g2]
119879
119872 = diag (1198981c 1198981s 1198981p1 1198981p2 1198981p3 1198982c 1198982s 1198982p1 1198982p2 1198982p3 1198982r 1198983g1 1198983g2)
119862 =
[
[
[
[
[
[
[
[
[
[
[
[
1198621c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198621sp119895 + cos1205722
1r
3
sum
119895=1
1198621rp119895 minus cos1205721s3
sum
119895=1
1198621sp119895 1198621rp1 cos1205721r minus 1198621sp1 cos1205721s 1198621rp2 cos1205721r minus 1198621sp2 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198621sp1198951198621s2c1199032
1bs+
3
sum
119895=1
1198621sp119895 1198621sp1 sdot sdot sdot
1198621sp1 + 1198621rp1 0 sdot sdot sdot
1198621sp2 + 1198621rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119870 =
[
[
[
[
[
[
[
[
[
[
[
[
1198701c2r1199032
1c+ cos1205722
1s
3
sum
119895=1
1198701sp119895 + cos1205722
1r
3
sum
119895=1
1198701rp119895 minus cos1205721s3
sum
119895=1
1198701sp119895 1198701rp1 cos1205721r minus 1198701sp1 cos1205721s 1198701rp2 cos1205721r minus 1198701s1199012 cos1205721s sdot sdot sdot
minus cos1205721s3
sum
119895=1
1198701sp1198951198701s2c1199032
1bs+
3
sum
119895=1
1198701sp119895 1198701sp1 sdot sdot sdot
1198701sp1 + 1198701rp1 0 sdot sdot sdot
1198701sp2 + 1198701rp2 sdot sdot sdot
symmetric d
]
]
]
]
]
]
]
]
]
]
]
]
119865 =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879119894
1199031cminus cos120572
1s(3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895) + cos120572
1r(3
sum
119895=1
1198621rp1198951119903p119895 +
3
sum
119895=1
1198701rp1198951198641rp119895)
3
sum
119895=1
1198621sp1198951sp119895 +
3
sum
119895=1
1198701sp1198951198641sp119895
1198621sp11119904p1 + 1198701sp11198641sp1 + 1198621rp11rp1 + 1198701rp11198641rp1
1198621sp21sp2 + 1198701sp21198641sp2 + 1198621rp21rp2 + 1198701rp21198641rp2
1198621sp31sp3 + 1198701sp31198641sp3 + 1198621rp31rp3 + 1198701rp31198641rp3
cos1205722r(3
sum
119895=1
1198702rp1198951198642rp119895 +
3
sum
119895=1
1198622rp1198952rp119895) minus cos120572
2s(3
sum
119895=1
1198702sp1198951198642sp119895 +
3
sum
119895=1
1198622sp1198952sp119895)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(29)
The Scientific World Journal 7
4 Load Sharing Characteristic of Load-SplitMultiple-Stage Planetary Gear Train
41 Calculation of Load Sharing Coefficient Use numericalintegration method for solving the dynamic equation (28) ofthe system obtain the responses to displacement and velocityof the system and substitute the responses into (22)ndash(25) andthen obtain the engaging forces 119865
1sp119895 1198651rp119895 1198652sp119895 and 1198652rp119895Make 119863
1sp119895119896s and 1198631sp119895119896r respectively represent the load
sharing coefficients of the internal and external meshing ofall gear pairs of the first-stage planetary gear train and119863
2sp119894119896sand119863
2sp119894119896r as those of the internal and external meshing of allgear pairs of the second-stage planetary gear train then loadsharing coefficients are expressed as
1198631sp119895119896s =
1119873(1198651sp119895119896s)max
sum1119873
119895=1(1198651sp119895119896s)max
1198631rp119895119896r =
1119873(1198651rp119895119896r)max
sum1119873
119895=1(1198651rp119895119896r)max
1198632sp119894119896s =
2119873(1198652sp119895119896s)max
sum2119873
119894=1(1198652sp119895119896s)max
1198632rp119894119896r =
2119873(1198652rp119895119896r)max
sum2119873
119894=1(1198652rp119895119896r)max
(30)
where 119896s 119896r are meshing cycle numbers for internal andexternal meshing of the planetary gear pair
When 1198891sp119895 and 1198891rp119895 are used to stand for load sharing
coefficient of internal and externalmeshing of each first-stagegear and 119889
2sp119895 and 1198892rp119895 for that of each second-stage gearin system period respectively the expression can be given asfollows
1198891sp119895 =
100381610038161003816100381610038161198631sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198891rp119895 =
100381610038161003816100381610038161198631rp119895119896r minus 1
10038161003816100381610038161003816max + 1
1198892sp119895 =
100381610038161003816100381610038161198632sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198892rp119895 =
100381610038161003816100381610038161198632rp119895119896r minus 1
10038161003816100381610038161003816max + 1
(31)
The paper analyzes the transmission system as shown inFigure 4 The basic parameters of the transmission systemare shown in Tables 1 and 2 and other parameters can bedetermined by [21] Substitute the relevant parameters of thesystem into (28) for solution Use (30) and (31) to obtain theload sharing coefficients of the transmission system
42 Influence of Mesh Error on Load Sharing Coefficient ofthe System Load sharing property of planetary gear trainis significantly affected by manufacturing error installationerror and eccentric error which cannot be neglected inplanetary gear train Considering systemrsquos complexity it is
Table 1 Primary parameters of planetary gear train
Parameter Carrier Ring Sun gear Planetarygear
Pitch radius 1199031(mm) 468 726 210 258
Base circle 1199031119887(mm) mdash 68222 19734 24244
Mass1198721(kg) 204277 41034 34491 38839
Moment of inertia 1198691
(kgsdotm2) 46255 22657 762 1625
Pressure angle 1205721(∘) mdash 20 20 20
Pitch radius 1199032(mm)
1199032mm 345 550 140 205
Base circle 1199032119887(mm) mdash 51683 13156 19264
Mass1198722(kg) 121249 8056 13140 17654
Moment of inertia 1198692
(kgsdotm2) 15175 2565 129 512
Pressure angle 1205722(∘) mdash 20 20 20
Table 2 Primary parameters of parallel-shaft gears
Parameter Pinion gear Driven gearPitch radius 119903
3(mm) 292 100
Base circle 1199033119887(mm) 27439 9397
Mass1198723(kg) 20822 2414
Moment of inertia 1198693(kgsdotm2) 887 012
Pressure angle 1205723(∘) 20 20
assumed that equivalent mesh error of each stage planetarygear at the direction of meshing line is equal and values of10 20 30 40 and 50 120583m are given respectively Load sharingproperties of multiple-stage gear train under these fiveconditions are studied Relationships between load sharingcoefficient curves of internal and external meshing of first-stage and second-stage which are calculated according to(31) are drawn in Figure 6 with different mesh errors
Results below can be concluded according to Figure 6
(1) Each load-sharing coefficient increases with increas-ing mesh error
(2) Load sharing coefficient of internal-meshing is dif-ferent from that of external-meshing under differ-ent mesh errors Maximum external-meshing andinternal-meshing load sharing coefficients of first-stage planetary gear are 1579 and 1645 respectivelywhile those of second-stage planetary gear are 1630and 1665 respectively
(3) Compared to the differences in change rate of eachload sharing coefficient of second-stage planetarygear that of first-stage planetary gear is more evidentThe maximum difference in change rate of first-stageplanetary gear is 010150 120583m while that of second-stage planetary gear is only 000350 120583m
43 Influence of Revolution Speed on Load Sharing CoefficientTo analyze the influence of revolution speed of the first-stageplanetary gear on load sharing coefficient the revolution
8 The Scientific World Journal
10 20 30 40 50155
16
165
17
175
d1sp1d1sp2d1sp3
Gear error E1sp (120583m)
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
10 20 30 40 50155
16
165
17
175
d1rp1d1rp2d1rp3
Gear error E1rp (120583m)
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2sp1d2sp2d2sp3
Gear error E2sp (120583m)
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2rp1d2rp2d2rp3
Gear error E2rp (120583m)
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 6 Load-sharing coefficient curves of each planetary gear with different mesh errors
speed is set as 5 rmin 10 rmin 15 rmin 20 rmin and25 rmin respectively Equation (31) is used to calculatethe load sharing coefficient under different conditions andcurves are obtained in Figure 7
Influence of revolution speed on load-sharing coefficientcan be concluded below according to Figure 7
(1) Each load sharing coefficient increases with raisingthe revolution speed which indicates that load shar-ing capacity of planetary gear train is weakened andvibration is aggravated with increasing revolutionspeed
(2) At the variation interval of revolution speed thechange rate difference of load-sharing coefficientbetween internal and external meshing of first-stageplanetary gear train is significantly different thoseof first-stage planetary gears 1 2 and 3 are 177084 and 149 respectively Similar result can beconcluded in second-stage planetary gear train and
change rate differences of 147 271 and 276 ofsecond-stage planetary gears 1 2 and 3 are figuredout respectively
5 Conclusion
(1) The dynamic model is built to account for thedynamic behavior of multiple-stage planetary geartrain used in wind driven generator The model canprovide useful guideline for the dynamic design ofthe multiple-stage planetary gear train of wind drivengenerator
(2) Each load-sharing coefficient of the first-stage plan-etary gear varies more than that of the second-stageplanetary gear At the same mesh error second-stage internal-meshing load sharing coefficient is thelargest the first-stage internal-meshing load sharingcoefficient is the second largest and the first-stage
The Scientific World Journal 9
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1sp1d1sp2d1sp3
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1rp1d1rp2d1rp3
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2sp1d2sp2d2sp3
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2rp1d2rp2d2rp3
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 7 Load sharing coefficient curves of each planetary at different revolution speeds
external-meshing load sharing coefficient is the min-imum
(3) Load sharing property is weakened and transmissionsystemrsquos vibration is aggravated with increasing rev-olution speed At each interval of revolution speedinternal and external meshing load sharing coeffi-cients of the second-stage planetary gear train varymore than those of the first-stage planetary gear train
Nomenclature120579119894 Angular displacement of 119894th member(119894 = s p119899 r 119899 = 1 2 3)
119903b119894 Gear base radii 119894 = s p119899 r 119899 = 1 2 3119903c Radius of the circle passing through planet
centers119903119894c 119894th-stage radii of the circle passing
through planet centers 119894 = 1 2
120572s Sun-planet engaging angle120572r Ring-planet engaging angle119894119873 Total number of planet sets for the 119894th-
stage drive train 119894 = 1 21198681119895 Polar mass moment of inertia of 119895th
member for 1st-stage drive train 119895 =c s p1 p2 p1119873
1198721p Mass of 1st-stage planetary gear
1198681ce = 119868
1c + 11198731198721p1199032
1c1198682119895 Polar mass moment of inertia of 119895th
member for 2nd-stage drive train 119895 =c s p1 p2 p2119873
1198722p Mass of 2nd-stage planetary gear
1198682ce = 119868
2c + 21198731198722p1199032
2c119903119894b119895 Gear base radii of 119895th member for 119894th-
stage drive train 119894 = 1 2 3 119895 =
s r p1 p2 p119899 g1 g2
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
The Scientific World Journal 7
4 Load Sharing Characteristic of Load-SplitMultiple-Stage Planetary Gear Train
41 Calculation of Load Sharing Coefficient Use numericalintegration method for solving the dynamic equation (28) ofthe system obtain the responses to displacement and velocityof the system and substitute the responses into (22)ndash(25) andthen obtain the engaging forces 119865
1sp119895 1198651rp119895 1198652sp119895 and 1198652rp119895Make 119863
1sp119895119896s and 1198631sp119895119896r respectively represent the load
sharing coefficients of the internal and external meshing ofall gear pairs of the first-stage planetary gear train and119863
2sp119894119896sand119863
2sp119894119896r as those of the internal and external meshing of allgear pairs of the second-stage planetary gear train then loadsharing coefficients are expressed as
1198631sp119895119896s =
1119873(1198651sp119895119896s)max
sum1119873
119895=1(1198651sp119895119896s)max
1198631rp119895119896r =
1119873(1198651rp119895119896r)max
sum1119873
119895=1(1198651rp119895119896r)max
1198632sp119894119896s =
2119873(1198652sp119895119896s)max
sum2119873
119894=1(1198652sp119895119896s)max
1198632rp119894119896r =
2119873(1198652rp119895119896r)max
sum2119873
119894=1(1198652rp119895119896r)max
(30)
where 119896s 119896r are meshing cycle numbers for internal andexternal meshing of the planetary gear pair
When 1198891sp119895 and 1198891rp119895 are used to stand for load sharing
coefficient of internal and externalmeshing of each first-stagegear and 119889
2sp119895 and 1198892rp119895 for that of each second-stage gearin system period respectively the expression can be given asfollows
1198891sp119895 =
100381610038161003816100381610038161198631sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198891rp119895 =
100381610038161003816100381610038161198631rp119895119896r minus 1
10038161003816100381610038161003816max + 1
1198892sp119895 =
100381610038161003816100381610038161198632sp119895119896s minus 1
10038161003816100381610038161003816max + 1
1198892rp119895 =
100381610038161003816100381610038161198632rp119895119896r minus 1
10038161003816100381610038161003816max + 1
(31)
The paper analyzes the transmission system as shown inFigure 4 The basic parameters of the transmission systemare shown in Tables 1 and 2 and other parameters can bedetermined by [21] Substitute the relevant parameters of thesystem into (28) for solution Use (30) and (31) to obtain theload sharing coefficients of the transmission system
42 Influence of Mesh Error on Load Sharing Coefficient ofthe System Load sharing property of planetary gear trainis significantly affected by manufacturing error installationerror and eccentric error which cannot be neglected inplanetary gear train Considering systemrsquos complexity it is
Table 1 Primary parameters of planetary gear train
Parameter Carrier Ring Sun gear Planetarygear
Pitch radius 1199031(mm) 468 726 210 258
Base circle 1199031119887(mm) mdash 68222 19734 24244
Mass1198721(kg) 204277 41034 34491 38839
Moment of inertia 1198691
(kgsdotm2) 46255 22657 762 1625
Pressure angle 1205721(∘) mdash 20 20 20
Pitch radius 1199032(mm)
1199032mm 345 550 140 205
Base circle 1199032119887(mm) mdash 51683 13156 19264
Mass1198722(kg) 121249 8056 13140 17654
Moment of inertia 1198692
(kgsdotm2) 15175 2565 129 512
Pressure angle 1205722(∘) mdash 20 20 20
Table 2 Primary parameters of parallel-shaft gears
Parameter Pinion gear Driven gearPitch radius 119903
3(mm) 292 100
Base circle 1199033119887(mm) 27439 9397
Mass1198723(kg) 20822 2414
Moment of inertia 1198693(kgsdotm2) 887 012
Pressure angle 1205723(∘) 20 20
assumed that equivalent mesh error of each stage planetarygear at the direction of meshing line is equal and values of10 20 30 40 and 50 120583m are given respectively Load sharingproperties of multiple-stage gear train under these fiveconditions are studied Relationships between load sharingcoefficient curves of internal and external meshing of first-stage and second-stage which are calculated according to(31) are drawn in Figure 6 with different mesh errors
Results below can be concluded according to Figure 6
(1) Each load-sharing coefficient increases with increas-ing mesh error
(2) Load sharing coefficient of internal-meshing is dif-ferent from that of external-meshing under differ-ent mesh errors Maximum external-meshing andinternal-meshing load sharing coefficients of first-stage planetary gear are 1579 and 1645 respectivelywhile those of second-stage planetary gear are 1630and 1665 respectively
(3) Compared to the differences in change rate of eachload sharing coefficient of second-stage planetarygear that of first-stage planetary gear is more evidentThe maximum difference in change rate of first-stageplanetary gear is 010150 120583m while that of second-stage planetary gear is only 000350 120583m
43 Influence of Revolution Speed on Load Sharing CoefficientTo analyze the influence of revolution speed of the first-stageplanetary gear on load sharing coefficient the revolution
8 The Scientific World Journal
10 20 30 40 50155
16
165
17
175
d1sp1d1sp2d1sp3
Gear error E1sp (120583m)
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
10 20 30 40 50155
16
165
17
175
d1rp1d1rp2d1rp3
Gear error E1rp (120583m)
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2sp1d2sp2d2sp3
Gear error E2sp (120583m)
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2rp1d2rp2d2rp3
Gear error E2rp (120583m)
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 6 Load-sharing coefficient curves of each planetary gear with different mesh errors
speed is set as 5 rmin 10 rmin 15 rmin 20 rmin and25 rmin respectively Equation (31) is used to calculatethe load sharing coefficient under different conditions andcurves are obtained in Figure 7
Influence of revolution speed on load-sharing coefficientcan be concluded below according to Figure 7
(1) Each load sharing coefficient increases with raisingthe revolution speed which indicates that load shar-ing capacity of planetary gear train is weakened andvibration is aggravated with increasing revolutionspeed
(2) At the variation interval of revolution speed thechange rate difference of load-sharing coefficientbetween internal and external meshing of first-stageplanetary gear train is significantly different thoseof first-stage planetary gears 1 2 and 3 are 177084 and 149 respectively Similar result can beconcluded in second-stage planetary gear train and
change rate differences of 147 271 and 276 ofsecond-stage planetary gears 1 2 and 3 are figuredout respectively
5 Conclusion
(1) The dynamic model is built to account for thedynamic behavior of multiple-stage planetary geartrain used in wind driven generator The model canprovide useful guideline for the dynamic design ofthe multiple-stage planetary gear train of wind drivengenerator
(2) Each load-sharing coefficient of the first-stage plan-etary gear varies more than that of the second-stageplanetary gear At the same mesh error second-stage internal-meshing load sharing coefficient is thelargest the first-stage internal-meshing load sharingcoefficient is the second largest and the first-stage
The Scientific World Journal 9
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1sp1d1sp2d1sp3
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1rp1d1rp2d1rp3
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2sp1d2sp2d2sp3
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2rp1d2rp2d2rp3
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 7 Load sharing coefficient curves of each planetary at different revolution speeds
external-meshing load sharing coefficient is the min-imum
(3) Load sharing property is weakened and transmissionsystemrsquos vibration is aggravated with increasing rev-olution speed At each interval of revolution speedinternal and external meshing load sharing coeffi-cients of the second-stage planetary gear train varymore than those of the first-stage planetary gear train
Nomenclature120579119894 Angular displacement of 119894th member(119894 = s p119899 r 119899 = 1 2 3)
119903b119894 Gear base radii 119894 = s p119899 r 119899 = 1 2 3119903c Radius of the circle passing through planet
centers119903119894c 119894th-stage radii of the circle passing
through planet centers 119894 = 1 2
120572s Sun-planet engaging angle120572r Ring-planet engaging angle119894119873 Total number of planet sets for the 119894th-
stage drive train 119894 = 1 21198681119895 Polar mass moment of inertia of 119895th
member for 1st-stage drive train 119895 =c s p1 p2 p1119873
1198721p Mass of 1st-stage planetary gear
1198681ce = 119868
1c + 11198731198721p1199032
1c1198682119895 Polar mass moment of inertia of 119895th
member for 2nd-stage drive train 119895 =c s p1 p2 p2119873
1198722p Mass of 2nd-stage planetary gear
1198682ce = 119868
2c + 21198731198722p1199032
2c119903119894b119895 Gear base radii of 119895th member for 119894th-
stage drive train 119894 = 1 2 3 119895 =
s r p1 p2 p119899 g1 g2
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
8 The Scientific World Journal
10 20 30 40 50155
16
165
17
175
d1sp1d1sp2d1sp3
Gear error E1sp (120583m)
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
10 20 30 40 50155
16
165
17
175
d1rp1d1rp2d1rp3
Gear error E1rp (120583m)
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2sp1d2sp2d2sp3
Gear error E2sp (120583m)
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
10 20 30 40 50
155
16
15
165
17
d2rp1d2rp2d2rp3
Gear error E2rp (120583m)
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 6 Load-sharing coefficient curves of each planetary gear with different mesh errors
speed is set as 5 rmin 10 rmin 15 rmin 20 rmin and25 rmin respectively Equation (31) is used to calculatethe load sharing coefficient under different conditions andcurves are obtained in Figure 7
Influence of revolution speed on load-sharing coefficientcan be concluded below according to Figure 7
(1) Each load sharing coefficient increases with raisingthe revolution speed which indicates that load shar-ing capacity of planetary gear train is weakened andvibration is aggravated with increasing revolutionspeed
(2) At the variation interval of revolution speed thechange rate difference of load-sharing coefficientbetween internal and external meshing of first-stageplanetary gear train is significantly different thoseof first-stage planetary gears 1 2 and 3 are 177084 and 149 respectively Similar result can beconcluded in second-stage planetary gear train and
change rate differences of 147 271 and 276 ofsecond-stage planetary gears 1 2 and 3 are figuredout respectively
5 Conclusion
(1) The dynamic model is built to account for thedynamic behavior of multiple-stage planetary geartrain used in wind driven generator The model canprovide useful guideline for the dynamic design ofthe multiple-stage planetary gear train of wind drivengenerator
(2) Each load-sharing coefficient of the first-stage plan-etary gear varies more than that of the second-stageplanetary gear At the same mesh error second-stage internal-meshing load sharing coefficient is thelargest the first-stage internal-meshing load sharingcoefficient is the second largest and the first-stage
The Scientific World Journal 9
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1sp1d1sp2d1sp3
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1rp1d1rp2d1rp3
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2sp1d2sp2d2sp3
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2rp1d2rp2d2rp3
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 7 Load sharing coefficient curves of each planetary at different revolution speeds
external-meshing load sharing coefficient is the min-imum
(3) Load sharing property is weakened and transmissionsystemrsquos vibration is aggravated with increasing rev-olution speed At each interval of revolution speedinternal and external meshing load sharing coeffi-cients of the second-stage planetary gear train varymore than those of the first-stage planetary gear train
Nomenclature120579119894 Angular displacement of 119894th member(119894 = s p119899 r 119899 = 1 2 3)
119903b119894 Gear base radii 119894 = s p119899 r 119899 = 1 2 3119903c Radius of the circle passing through planet
centers119903119894c 119894th-stage radii of the circle passing
through planet centers 119894 = 1 2
120572s Sun-planet engaging angle120572r Ring-planet engaging angle119894119873 Total number of planet sets for the 119894th-
stage drive train 119894 = 1 21198681119895 Polar mass moment of inertia of 119895th
member for 1st-stage drive train 119895 =c s p1 p2 p1119873
1198721p Mass of 1st-stage planetary gear
1198681ce = 119868
1c + 11198731198721p1199032
1c1198682119895 Polar mass moment of inertia of 119895th
member for 2nd-stage drive train 119895 =c s p1 p2 p2119873
1198722p Mass of 2nd-stage planetary gear
1198682ce = 119868
2c + 21198731198722p1199032
2c119903119894b119895 Gear base radii of 119895th member for 119894th-
stage drive train 119894 = 1 2 3 119895 =
s r p1 p2 p119899 g1 g2
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
The Scientific World Journal 9
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1sp1d1sp2d1sp3
Load
shar
ing
coeffi
cien
td1s
pj
(a) Each external-meshing first-stage planetary gear
5 10 15 20 25158
16
162
164
166
168
17
172
174
Revolution speed (rmin)
d1rp1d1rp2d1rp3
Load
shar
ing
coeffi
cien
td1r
pj
(b) Each internal-meshing first-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2sp1d2sp2d2sp3
Load
shar
ing
coeffi
cien
td2s
pj
(c) Each external-meshing second-stage planetary gear
5 10 15 20 25154
156
158
16
162
164
166
168
17
Revolution speed (rmin)
d2rp1d2rp2d2rp3
Load
shar
ing
coeffi
cien
td2r
pj
(d) Each internal-meshing second-stage planetary gear
Figure 7 Load sharing coefficient curves of each planetary at different revolution speeds
external-meshing load sharing coefficient is the min-imum
(3) Load sharing property is weakened and transmissionsystemrsquos vibration is aggravated with increasing rev-olution speed At each interval of revolution speedinternal and external meshing load sharing coeffi-cients of the second-stage planetary gear train varymore than those of the first-stage planetary gear train
Nomenclature120579119894 Angular displacement of 119894th member(119894 = s p119899 r 119899 = 1 2 3)
119903b119894 Gear base radii 119894 = s p119899 r 119899 = 1 2 3119903c Radius of the circle passing through planet
centers119903119894c 119894th-stage radii of the circle passing
through planet centers 119894 = 1 2
120572s Sun-planet engaging angle120572r Ring-planet engaging angle119894119873 Total number of planet sets for the 119894th-
stage drive train 119894 = 1 21198681119895 Polar mass moment of inertia of 119895th
member for 1st-stage drive train 119895 =c s p1 p2 p1119873
1198721p Mass of 1st-stage planetary gear
1198681ce = 119868
1c + 11198731198721p1199032
1c1198682119895 Polar mass moment of inertia of 119895th
member for 2nd-stage drive train 119895 =c s p1 p2 p2119873
1198722p Mass of 2nd-stage planetary gear
1198682ce = 119868
2c + 21198731198722p1199032
2c119903119894b119895 Gear base radii of 119895th member for 119894th-
stage drive train 119894 = 1 2 3 119895 =
s r p1 p2 p119899 g1 g2
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
10 The Scientific World Journal
120572119894s Sun-planet engaging angle for 119894th-
stage drive train 119894 = 1 2120572119894r Ring-planet engaging angle for 119894th-
stage drive train 119894 = 1 2119864sp Sun-planet mesh error119864rp Ring-planet mesh error119870sp Sun-planet mesh stiffness119870rp Ring-planet mesh stiffness119862sp Sun-planet mesh damping coefficient119862rp Ring-planetmesh damping coefficient120579119894119895 Angular displacement of 119895th member
for 119894th-stage drive train 119894 = 1 2 3 119895 =s r c p1 p2 p119899 g1 g2 119899 = 1 2 3
119864119894sp119895 Sun-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
119864119894rp119895 Ring-planet mesh error of 119895th planet
gear for 119894th-stage drive train 119894 = 1 2119895 = 1 2 3
1198643g1g2 Mesh error of parallel-shaft gears119870119894sp119895 Sun-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
119870119894rp119895 Ring-planet mesh stiffness of 119895th
planet gear for 119894th-stage drive train119894 = 1 2 119895 = 1 2 3
1198703g1g2 Mesh stiffness of parallel-shaft gears
1198701s2c Torsional stiffness associated with 1st-
stage sun and 2nd-stage carrier1198701c2r Torsional stiffness associated with 1st-
stage carrier and 2nd-stage ring1198702s3g1 Torsional stiffness associated with
2nd-stage sun and 3rd-stage gear119862119894sp119895 Sun-planet mesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
119862119894rp119895 Ring-planetmesh damping coefficient
of 119895th planet gear for 119894th-stage drivetrain 119894 = 1 2 119895 = 1 2 3
1198623g1g2 Mesh damping coefficient of parallel-
shaft gears1198621s2c Torsional damping coefficient associ-
ated with 1st-stage sun and 2nd-stagecarrier
1198621c2r Torsional damping coefficient associ-
ated with 1st-stage carrier and 2nd-stage ring
1198622s3g1 Torsional damping coefficient associ-
ated with 2nd-stage sun and 3rd-stagegear
119879119894 Input torque
119879119900 Output torque
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theChinese National Science Foundation (no 51175299) theShandong Provincial Natural Science Foundation China (noZR2010EM012) the Independent Innovation Foundation ofShandong University (IIFSDU2012TS044) and the GraduateIndependent Innovation Foundation of ShandongUniversityGIIFSDU (no yzc10117)
References
[1] Z D Fang Y W Shen and Z D Huang ldquoDynamic charac-teristics of 2K-H planetary gearingrdquo Journal of NorthwesternPolytechnical University vol 10 no 4 pp 361ndash371 1990
[2] A Kahraman ldquoPlanetary gear train dynamicsrdquo Journal ofMechanical Design Transactions of the ASME vol 116 no 3 pp713ndash720 1994
[3] R Hbaieb F Chaari T Fakhfakh and M Haddar ldquoDynamicstability of a planetary gear train under the influence of variablemeshing stiffnessesrdquo Proceedings of the Institution ofMechanicalEngineers D vol 220 no 12 pp 1711ndash1725 2006
[4] Y Guo and R G Parker ldquoPurely rotational model and vibrationmodes of compound planetary gearsrdquoMechanism and MachineTheory vol 45 no 3 pp 365ndash377 2010
[5] S-Y Wang Y-M Song Z-G Shen C Zhang T-Q Yangand W-D Xu ldquoResearch on natural characteristics and lociveering of planetary gear transmissionsrdquo Journal of VibrationEngineering vol 18 no 4 pp 412ndash417 2005
[6] Z F Ma K Liu and Y H Cui ldquoAnalysis of the torsional char-acteristics of planetary gear trains of an increasing gearboxrdquoMechanical Science and Technology For Aerospace Engineeringvol 29 no 6 pp 788ndash791 2010
[7] J Lin and R G Parker ldquoAnalytical characterization of theunique properties of planetary gear free vibrationrdquo Journal ofVibration and Acoustics Transactions of the ASME vol 121 no3 pp 316ndash321 1999
[8] J Lin and R G Parker ldquoStructured vibration characteristics ofplanetary gearswith unequally spaced planetsrdquo Journal of Soundand Vibration vol 233 no 5 pp 921ndash928 2000
[9] S Dhouib R Hbaieb F Chaari M S Abbes T Fakhfakhand M Haddar ldquoFree vibration characteristics of compoundplanetary gear train setsrdquo Proceedings of the Institution ofMechanical Engineers C vol 222 no 8 pp 1389ndash1401 2008
[10] S J Wu H Ren and E Y Zhu ldquoResearch advances fordynamics of planetary gear trainsrdquo Engineering Journal ofWuhan University vol 43 no 3 pp 398ndash403 2010
[11] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal C vol 47 no3 pp 908ndash915 2004
[12] A Kahraman ldquoNatural modes of planetary gear trainsrdquo Journalof Sound and Vibration vol 173 no 1 pp 125ndash130 1994
[13] A Kahraman ldquoFree torsional vibration characteristics of com-pound planetary gear setsrdquo Mechanism and Machine Theoryvol 36 no 8 pp 953ndash971 2001
[14] J F Du Z D Fang B B Wang and H Dong ldquoStudy on loadsharing behavior of planetary gear train based on deformationcompatibilityrdquo Journal of Aerospace Power vol 27 no 5 pp1166ndash1171 2012
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
The Scientific World Journal 11
[15] X Gu and P A Velex ldquodynamic model to study the influence ofplanet position errors in planetary gearsrdquo Journal of Sound andVibration vol 331 pp 4554ndash4574 2012
[16] A Singh ldquoEpicyclic load sharingmap development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[17] H Ligata A Kahraman and A Singh ldquoAn experimental studyof the influence of manufacturing errors on the planetary gearstresses and planet load sharingrdquo Journal of Mechanical DesignTransactions of the ASME vol 130 no 4 Article ID 0417012008
[18] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
[19] J Lu R Zhu and G Jin ldquoAnalysis of dynamic load sharingbehavior in planetary gearingrdquo Journal of Mechanical Engineer-ing vol 45 no 5 pp 85ndash90 2009
[20] F-M Ye R-P Zhu andH-Y Bao ldquoStatic load sharing behaviorin NGW planetary gear train with unequal modulus andpressure anglesrdquo Journal of Central South University vol 42 no7 pp 1960ndash1966 2011
[21] R F Li Wang and J J Vibration Shock and Noise of GearDynamics Science Press Beijng China 1997
