Research Article Load Sharing Multiobjective Optimization...
Transcript of Research Article Load Sharing Multiobjective Optimization...
Research ArticleLoad Sharing Multiobjective Optimization Design ofa Split Torque Helicopter Transmission
Chenxi Fu1 Ning Zhao1 and Yongzhi Zhao2
1School of Mechanical Engineering Northwestern Polytechnical University Xirsquoan 710072 China2Systems EngineeringDivision of ChinaAcademy of LaunchVehicle Technology ChinaAerospace Science andTechnologyCorporationBeijing China
Correspondence should be addressed to Chenxi Fu franten126com
Received 2 April 2015 Accepted 18 May 2015
Academic Editor Dapeng P Du
Copyright copy 2015 Chenxi Fu 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
Split torque designs can offer significant advantages over the traditional planetary designs for helicopter transmissions However ithas two unique properties gap and phase differences which result in the risk of unequal load sharing Various methods have beenproposed to eliminate the effect of gap and promote load sharing to a certain extent In this paper system design parameters willbe optimized to change the phase difference thereby further improving load sharing A nonlinear dynamic model is establishedto measure the load sharing with dynamic mesh forces quantitatively Afterwards a multiobjective optimization of a referencesplit torque design is conducted with the promoting of load sharing property lightweight and safety considered as the objectivesThe load sharing property which is measured by load sharing coefficient is evaluated under multiple operating conditions withdynamic analysis method To solve the multiobjective model with NSGA-II an improvement is done to overcome the problem oftime consuming Finally a satisfied optimal solution is picked up as the final design from the Pareto optimal front which achievesimprovements in all the three objectives compared with the reference design
1 Introduction
The drive system of a rotorcraft must meet especiallydemanding requirements of high reduction ratio high safetyand reliability lightweight and little vibration White [1]proposed a promising alternative design to the commonplanetary transmissions for helicopters known as the splittorque or split path arrangement which offers two parallelpaths for transmitting torque from the engine to the rotorKish [2] and Krantz [3 4] pointed out that although asplit torque design can offer significant advantages over thecommonly used planetary design the split torque design isconsidered even risk there might have been unequal torquesin the two parallel paths which cause excessive wear in oneof the paths and renders the split torque system ineffectiveTherefore the main problem in the design of split torquetransmission lies in how to ensure the torque splitting equallybetween different paths
Actually the problem of load sharing exists in almost alltypes of multipath transmission system such as split torque
and planetary designs There is a locked loop of gearingin a multipath design and each of the gear meshes in theloop will be engaged at the same time only if the gears areassembled with the required relative relationship which isconsidered a particular tooth timing relationship Howeverdue to the manufacturing and assembly errors the particulartooth timing relationship can not precisely be met in a realmultipath transmission system Therefore there might be agap (or several gaps) at one (or several) of the gear meshlocations under a nominal light load Usually this kind of gapis considered the main cause of unequaled load in multipathtransmission systems [3]
To analyze the load sharing property of amultipath trans-mission system a lot of research works have been conductedThe main research contents and methods include definingthe load sharing coefficient based on mesh forces which areevaluated through dynamic or static methods tomeasure theload sharing property of a multipath transmission systemand then effects of different factors such as errors andoperating conditions can be studied Hayashi et al [5] put
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 381010 15 pageshttpdxdoiorg1011552015381010
2 Mathematical Problems in Engineering
forward a method to measure the load sharing of planetarytransmissions and concluded that the dynamic load sharingdiffers greatly from static condition Therefore the dynamicanalysismethod is widely used because it reflects the real loadsharing property under operating condition and it is naturallyadopted in this paper Krantz [3] firstly studied the dynamicsof a split torque transmission system and concluded that theloads and motions of the two power paths differ although thesystem has symmetric geometry Kahraman [6] investigatedthe load sharing property of a planetary transmission systemwith a nonlinear dynamic model established which takesmanufacturing and assembly errors into consideration andfinds that the operating conditions also having great effectson load sharing Guo et al [7] studied the influences of relatedfactors on load sharing property of thewind turbine planetarygears based on its characteristics Kahraman [6] derived therelationship between dynamic load sharing coefficient andstatic load sharing coefficient based on dynamic methodwhich makes static analyzing meaningful [8] Bodas andKahraman [9] and Singh [10] studied the influences of man-ufacturing and assembly errors on load sharing property ofplanetary transmissions with 2D and 3D static contact mod-els adopted respectively Afterwards experimental studies[11 12] have also been conducted Ligata et al [13] establisheda discrete model of planetary transmissions to study its loadsharing property Then Singh [14 15] investigated the modeldeeply and obtains multiple load sharing coefficients undervaries of manufacturing errors and torques by defining theload sharing map
Benefiting from the centrosymmetry of design the plan-etary multipath transmission systems are always in capacityof automatical load sharing which offers an advantage overthe split torque design To promote the load sharing propertyof split torque transmission system somemethods have beenproposed to compensate for or minimize the effect of the gapincluding floating gears [16] quill shafts [2] and ldquoclockinganglerdquo [4 17] The floating gears arrangement permits theinput pinion to float until gear loads are balanced betweenthe two paths There are two kinds of quill shafts loadsharing devices the conventional quill shafts and the onebased on elastomeric elements The conventional quill shaftsassemble intermediate shafts with some torsional flexibilityso as to minimize the difference in torque split betweenpaths whereas the latter one add some materials with alower elastic modulus in the compound shafts to achieve thesame purpose The ldquoclocking anglerdquo method considers theldquoclocking anglerdquo as a design parameter to adjust and optimizethe load sharing and the ldquoclocking anglerdquo is adjusted byvarying the thicknesses of shim packs that axially positionedthe compound shafts
Further research of Krantz [3] indicated that even thoughthe manufacturing and assembly errors of a split torquetransmission are precisely controlled and the gap is elim-inated completely there still unequaled loads in the twopaths under real operating conditions Actually the uniquephase difference property of the split torque transmissionsystem results in desynchrony between the two paths whichcauses significant effects on load sharing However the phasedifferences are not independent design parameters which
are connected to the system geometry parametersThereforethe load sharing property of a split torque transmissioncan be promoted by adjusting the phase differences throughoptimizing the system geometry parameters Since it reflectsthe real load sharing of the split torque transmission underdynamic condition the load sharing coefficient used in thispaper is evaluated by solving the nonlinear dynamic model
Despite of the load sharing property the drive systemof a rotorcraft must also meet the demanding requirementsof lightweight and high safety [2 3] Savsani et al [18] andThompson et al [19] reduced the mass of a gear pair anda multistage gear system respectively through optimizingthe gear parameters Kumar et al [20] optimized a singlepair of gear transmission with promoting of load capacityof gears considered as the objective Based on the aboveconsiderations a multiobjective optimization design of asplit torque transmission system is conducted with thepromoting of load sharing property lightweight and safetyconsidered as the objectivesThe load sharing property whichis measured by load sharing coefficient is evaluated undermultiple operating conditionswith dynamic analysismethod
Deb et al [21] proposed the improved nondominatedsorting genetic algorithm (NSGA-II) based onmultiobjectiveevolutionary algorithm formultiobjective optimization prob-lems which has been proved a simple and effective method[22] In this paper NSGA-II is adopted to solve the multiob-jective optimizationmodel However there are large numbersof nonlinear dynamic equations to be solved under multipleoperating conditions when evaluating the fitness and thesolving time is not acceptable in engineering Therefore animprovement has been done toNSGA-II to solve the problemof time consuming prediction strategy is used in the fitnessevaluation step so as to avoid the evaluation of load sharingproperty which is computationally very expensive
2 Nonlinear Dynamics Model of a SplitTorque Transmission System
21 Modeling the Gear Mesh Regarding the spur gears asspecial helical gears with helix angle of 0 degree the general3D pinion-gear meshing model can be established as shownin Figure 1 Assuming that the geometry is not affected bydeflections (small displacements hypothesis) and providedthat mesh elasticity can be transferred onto the base planea rigid-body approach can be employed The pinion and thegear can therefore be assimilated to two rigid cylinders with4 degrees of freedom each which are connected by a stiffnesselement (or a distribution of stiffness elements) and a dampelement From a physical point of view the 8 degrees offreedomof a pair represent the generalized displacements of 3translational degrees (along axes 119909 119910 and 119911) and 1 rotationaldegree (around axis 119911) each Here the angle 120595 between axis119909 and the center line of the gear pair 997888997888997888997888rarr
119874119901119874119892 is defined asdirection angle
In this model flank deviations are taken into considera-tion Conventionally flank deviations relative to the perfectgeometry are positively defined in the direction of the outernormal and they are supposed to be small enough so that
Mathematical Problems in Engineering 3
Theoretical line of contact
y
xz
Op120595
120596p
Mlowast
M
120596g
Ogefp(M) gt 0
efg(M) lt 0
M
119847g
119847pPinion
Gear
Figure 1 Model of gear contact
tooth contacts remain on theoretical base planes Then thedeviation of a gear pair can be defined as sum of flankdeviations of the pinion and gear Each theoretical contactline on the base plane is discretized in elementary cellscentered in one point whose deviation is defined as thenormal distance on the base plane between a point of thepinion and a point of the gear that would be in contactfor perfect geometries [23] In fact due to the influence ofdeviations contact and deflection do not occur at all the pointin the theoretical contact line The real contact status of thegear pair can be obtained only when the contact status of allthe point in all the contact lines is solved
As Figure 1 illustrates when a gear pair is nominallyengaged contact occurs at one certain point 119872
lowast which hasthe maximum deviation namely 119890(119872
lowast) Select any point 119872
in the theoretical contact line different from 119872lowast and name
the flank deviations of 119872 in the pinion and gear as 119890119891119901(119872)
and 119890119891119892(119872) respectively The flank deviation is positive foran excess of material and negative when some material isremoved from the ideal geometryTherefore the deviation inpoint 119872 can be expressed as
119890 (119872) = 119890119891119901 (119872) + 119890119891119892 (119872) (1)
The gear model is shown in Figure 1 and pinion andgear are assimilated to rigid cylinders with four-degrees-of-freedom connected by series of stiffness and damp Thegeneralized displacements can be expressed as
q = 119909119901 119910119901 119911119901 120579119901 119909119892 119910119892 119911119892 120579119892
119879
(2)
where 119909119901 119910119901 119911119901 are translational displacements of the pinionalong axes 119909 119910 119911 respectively 120579119901 is rotational displacementof the pinion around axis 119911 119909119892 119910119892 119911119892 are translationaldisplacements of the gear along axes 119909 119910 119911 respectively 120579119892
is rotational displacement of the gear around axis 119911Since generalized displacements are small quantities it
can therefore be represented by infinitesimal translationsand rotations The perturbation q on pinion and gear gives
a normal approach 120575(119872) on the base plane relative to rigidbody positions
120575 (119872) = 120581119879q =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
cos120573119887 sin (120572 minus 120595)
cos120573119887 cos (120572 minus 120595)
sin120573119887
119877119887119901 cos120573119887
minus cos120573119887 sin (120572 minus 120595)
minus cos120573119887 cos (120572 minus 120595)
minus sin120573119887
119877119887119892 cos120573119887
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
119879
sdot
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119909119901
119910119901
119911119901
120579119901
119909119892
119910119892
119911119892
120579119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(3)
where 120581 is a projective vector depending on gear geometrywhich projects displacements q onto the base plane 120573119887 is thebase helix angle (the value is 0 to spur gears) 120572 is the pressureangle 119877119887119901 119877119887119892 are the base radius of pinion and gear
Contact and deflection in 119872 occur only if the normalapproach 120575(119872) is larger than the initial deviation 120575119890(119872) andin such case the deflection Δ(119872) can be evaluated as
Δ (119872) = 120575 (119872) minus 120575119890 (119872) (4)
where 120575119890(119872) = 119890(119872lowast) minus 119890(119872) 119890(119872
lowast) 119890(119872) is the deviation
in 119872lowast 119872 Else if the normal approach 120575(119872) is less than the
initial deviation 120575119890(119872) then the deflection Δ(119872) = 0The elemental mesh force transmitted from the pinion
onto the gear at one point of contact 119872 is
119889119865 (119872) = 119896 (119872) 119889119897 sdot Δ (119872) (5)
where 119896(119872) is the mesh stiffness at point 119872 per unit ofcontact length which is calculated according to ISO6336 and119889119897 is the elemental contact length
The total mesh force 119865119897 can be deducted by integratingover the time-varying and deflection dependent contactlength
119865119897 = int
119897
119896 (119872) Δ (119872) 119889119897
= int
119897
119896 (119872) 119889119897 sdot 120581119879qminus int
119897
119896 (119872) 120575119890 (119872) 119889119897
(6)
4 Mathematical Problems in Engineering
Outputshaft
First reductionstageEngine
input
High torquereduction stage
Figure 2 Full arrangement of a split torque transmission system
22 Gear Train Arrangement The full arrangement of a splittorque or split-path transmissions [3] is depicted in Figure 2which has two stages (1) One is first reduction stageThe firststage is where torque is split between the input pinion andthe two output gears Usually helical gears are used (2) Thesecond stage is high torque reduction stage The output shaftis driven by a gear which is driven simultaneously by two spurpinions each coaxial to the gear in the first reduction stage
As shown in Figure 2 the input pinion meshes with twogears offering two paths to transfer power to the output geartherefore the whole system is divided into two pathsThe twopower paths are identified as 119871 and 119877 with 119877 to the right of 119871The first-stage gear and second-stage pinion combination arecollectively called the compound gear The compound gearand gear shaft combination are called the compound shaft
As shown in Figure 3 a right-hand Cartesian coordinatesystem is established such that the 119911-axis is coincident withthe output gear shaft the positive 119910-axis extends from theinput gear center to the output pinion center and the inputgear drives clockwise The first-stage pinion gear (119871) andgear (119877) are marked with gears 1sim3 the second-stage pinion(119871) pinion (119877) and gear aremarkedwith gears 4sim6The inputshaft compound shaft (119871) compound shaft (119877) and outputshaft are marked with axes 1sim4 and the center of which ismarked with 1198741 1198742 1198743 1198744 Define the angle between
997888997888997888997888rarr
11987411198742
and 997888997888997888997888rarr
11987411198743 as shaft angle Φ1 and the angle between 997888997888997888997888rarr
11987441198742 and997888997888997888997888rarr
11987441198743 as shaft angle Φ2A unique property of a split torque transmission is the
phase relationships of the meshes The input pinion drivestwo gears simultaneously the length of the arc along thepitch circle joining the two pitch points 119886 119887 (the length ofarc 119886119887 in Figure 3) is probably not an integer multiple of thecircular pitch and then the two meshes will not pass throughthe pitch point at the same instant of time which results inthe desynchrony of the two paths It is the same with thesecond stage In order to describe the unique property of
Tout
L
c
O24
2y
xO
a b
d
Tin
Φ1
O1
O3 5
1
3
6
R
O4
Φ2
Figure 3 Coordinates of a split torque transmission system
a split torque transmission the concept of phase difference ofa stage is defined as quotient of the length of arc joining thetwo pitch points and the circular pitchThe phase differencesof the two stages are expressed as
Δ1205931 =
1006704119886119887
1199011199051=
1198891Φ121205871198981199051
=
1198851Φ12120587
Δ1205932 =
1006704119888119889
1199011199052=
1198896Φ221205871198981199052
=
1198856Φ22120587
(7)
where Δ12059312 is the phase difference of the first and secondstages 1006704
119886119887 1006704119888119889 are the length of arcs 119886119887 119888119889 1199011199051 1199011199052 are the
circular pitch of the first and second stages 1198981199051 1198981199052 are thetransversemodule of the first and second stages119889119894 is the pitchdiameter of gear 119894 119885119894 is the tooth number of gear 119894
There are 6 gears with 4 meshing pairs in a split torquetransmission system Each gear can be assimilated to a rigidcylinder with 4 degrees of freedom Therefore there are 24degrees of freedom in totalThe generalized displacements ofthe whole system can be expressed as
x = 1199091 1199101 1199111 1205791 1199092 1199102 1199112 1205792 1199096 1199106 1199116 1205796119879
(8)
where the subscripts 1sim6 correspond to gears 1sim6
23 Equations of Dynamics The general form of dynamicequations of a gear transmission system is
Mx +Cx +Kx = F (9)
whereM is the generalizedmassmatrixC is the dampmatrixK is the stiffness matrix and F is the load vector
With regard to the split torque transmission system thegeneral model can be specified to
Mx +Cx + (K119898 +K119887 +K119888) x = F0 + F (119890) (10)
where K119898 is the mesh stiffness term K119887 is the supportingstiffness term K119888 is the coupling stiffness term F0 is theconstant torques term and F(119890) is the additional force termcaused by gear deviations
Here we give the principle to set up all the matrices andvector in the model
Mathematical Problems in Engineering 5
231 Mass Matrix M Mass matrix M is a 24 times 24 diagonalmatrix which is expressed as
M = diag (1198981 1198981 1198981 1198681 1198982 1198982 1198982 1198682 1198986 1198986
1198986 1198686)
(11)
where 119898119894 119894 = 1 sim 6 is the mass of gears 1sim6 119868119894 119894 = 1 sim 6 isthe rotational inertia around 119911-axis
232 Stiffness Matrix K The stiffness matrix K consists ofmesh stiffness K119898 supporting stiffness K119887 and couplingstiffness K119888 and it is also a 24 times 24 symmetric matrix
K = K119898 +K119887 +K119888 (12)
(i) Meshing Stiffness Matrix K119898The mesh stiffness matrix ofthewhole systemK119898 comes frommesh stiffness of all the gearpairs it can therefore be obtained by assembling the entiremesh stiffness submatrix together
K119898 =
4sum
119899=1R119879119894119895K119898119894119895R119894119895 119894 119895 = 1 2 1 3 4 6 5 6 (13)
where K119898119894119895 = int119897119896119894119895(119872)119889119897 sdot 120581119894119895120581
119879
119894119895is the nonlinear and time-
dependent mesh stiffness submatrix of gear pair 119894 119895 with 120581119894119895the projective vector of gear pair 119894 and 119895
R119894119895
=
column 119894 column 119895
[
04times4 sdot sdot sdot R1119894 sdot sdot sdot 04times4 sdot sdot sdot 04times404times4 sdot sdot sdot 04times4 sdot sdot sdot R2119895 sdot sdot sdot 04times4
]
2times6
(14)
is assemble matrix with R1119894 = R2119895 = diag(1 1 1 1)
(ii) Supporting Stiffness Matrix K119887 The supporting stiffnessmatrixK119887 comes from all the gearsrsquo supporting stiffness in allthe translational motion freedoms
K119887 = diag (1198961199091 1198961199101 1198961199111 0 1198961199092 1198961199102 1198961199112 0 1198961199096 1198961199106
1198961199116 0)
(15)
where 119896119909119894 119896119910119894 119896119911119894 119894 = 1 sim 6 is the supporting stiffness of gear119894 in 119909 119910 119911 directions
(iii) Coupling Stiffness Matrix K119888 The coupling stiffnessmatrixK119888 of thewhole system comes from the two compoundshafts and it can be obtained by assembling the two couplingstiffness submatrix together
K119888 =2
sum
119899=1R119879119894119895K119888119894119895R119894119895 119894 119895 = 2 4 3 5 (16)
where
K119888119894119895
=
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119896119887119894119895 0 0 0 minus119896119887119894119895 0 0 0119896119887119894119895 0 0 0 minus119896119887119894119895 0 0
119896119886119894119895 0 0 0 minus119896119886119894119895 0119896119904119894119895 0 0 0 minus119896119904119894119895
119896119887119894119895 0 0 0119896119887119894119895 0 0
sym 119896119886119894119895 0119896119904119894119895
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
119894 119895 = 2 4 3 5
(17)
is the coupling stiffness submatrix of compound gear 119894 119895 with119896119904119894119895 119896119887119894119895 119896119886119894119895 the torsional stiffness bending stiffness axialstiffness between the compound gears 119894 and 119895
233 Damp Matrix C Here Rayleighrsquos damping is adopted
C = 119886M+ 119887K (18)
with 119886 119887 two constants to be adjusted from experimentalresults and experience
234 Load Vector F The load vector F consists of torquesvector F0 produced by input and output torques and addi-tional force vector F(119890) caused by gear deviations
F = F0 + F (119890) (19)
The torques vector F0 is expressed as
F0 = [0 0 0 119879in 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 119879out]119879
(20)
The additional force vector F(119890) caused by gear deviationsis given by
F (119890) =
4sum
119899=1R119879119894119895
int
119897
119896119894119895 (119872) 120575119890119894119895 (119872) 119889119871 sdot 120581119894119895 119894 119895
= 1 2 1 3 4 6 5 6
(21)
3 Case Study and Load Sharing Property
The load sharing properties of a split torque transmissionare how equivalent the load allocated between the two pathswhich can bemeasured by load sharing coefficient Accordingto [9] the load sharing coefficient 119896119897119904 is expressed as
119896119897119904 =
max (119865119871 119865119877)
1198650 (22)
where 119865119871 119865119877 are the average mesh forces of the first stage inleft and right paths1198650 is the staticmesh force of the first stage
6 Mathematical Problems in Engineering
Theoretical flank
Cumulative pitch error
Profile error
10
0
minus10
minus20
minus30
00
0
05 05
1
1
Root
Tip
Flan
k de
viat
ion
(120583m
)
Normalized profile length Normali
zed fa
ce width
A
A
Figure 4 An example of flank deviation surface for a tooth
Table 1 The geometry parameters of the reference design
Parameters First stage Second stagePiniongear Piniongear
Tooth number 32124 27176Modulemm 1590 254Pressure angle(∘) 20 25Helix angle(∘) 6 0Tooth widthmm 44453810 66045994Shaft angle(∘) 1225 502Phase difference 10889 24542
The larger load sharing coefficient 119896119897119904 the worth load sharingproperties
It is mentioned in Section 1 that there are two reasonswhich cause the unequal torques in the split torque transmis-sion system (1) the gap at one of the four gear mesh locationscaused bymanufacturing and assembly errors which directlyresults in the difference of deformations between the twopaths (2) the unique phase difference properties of the splittorque transmission system which result in the desynchronybetween the two paths Since various split torque load sharingmethods have been proposed to compensate for or minimizethe gap the effect of this gap caused by manufacturing andassembly errors will be ignored in this paperThemain factorwhich causes the unequal torques studied here is the phasedifference only
According to [3] a split torque transmission design forhelicopter is introduced as the reference design The loadsharing properties of the reference design will be studiedfirst and then an optimization will be conducted to promoteits load sharing The geometry parameters of the referencedesign have been listed in Table 1 other parameters are asfollows input power 119875in is 37285 kW input speed 119899in is8780 rmin center distance119867 between input and output shaftis 29345mm and material of gears is SAE 9310 An exampleof flank deviation surface for a teeth is shown in Figure 4 andthe flank deviation is considered the sum of cumulative pitcherror and profile error where the cumulative pitch error is
34 35 36 37 38 39 40
Dimensionless time
25
2
15
1
05
0
Mes
h fo
rce (
N)
times105
FLFRF0
Figure 5 Mesh force curves of the first stage in both paths at givenpower and speed
simulated by normally distributed constant and profile erroris simulated by sine surface The flank deviation of differentteeth is simulated differently and the periodicity of the gearpair is taken into consideration
The dynamic equations of the reference design systemare established according to the previous section and ode45order is adopted to solve the functions with MATLABsoftware The curves of mesh force and mesh stiffness of thefirst stage in both paths are evaluated at given power andspeed and parts of the curves are shown in Figures 5 and 6
As Figure 5 illustrates 119865119871 119865119877 are the mesh forces of leftand right paths of the first stage and 1198650 is the static meshforce of the first stage It can be seen from the figure that themesh force of the left path differs significantly from the rightone and the evaluated load sharing coefficient 119896119897119904 has reached1268
In Figure 6 119896119871 119896119877 are the mesh stiffness of left and rightpaths of the first stage and inside the dashed rectangle boxis the errorless mesh stiffness curve over one mesh periodcalculated by ISO6336 It can be seen that the actual mesh
Mathematical Problems in Engineering 7
34 35 36 37 38 39 40
Dimensionless time
Mes
h sti
ffnes
s (N
m)
times108
12
10
8
6
4
2
0
11
1
090 05 1
times109
kLkR
Figure 6 Mesh stiffness curves of the first stage in both paths atgiven power and speed
400
24
22
2
18
16
14
12
1Load
shar
ing
coeffi
cien
t
1400012000
100008000
6000Speed (rmin) 200
300
500600
Power (kW)
Load decreasesX 2237Y 8965Z 2136
X 2237Y 5268
Y 5268
Z 1304
Z 1062
X 522
X 522
Y 5638Z 1058
X 4435
Y 12292
Y 12292
Z 1211
X 2237
Z 2106
Figure 7 ldquoLoad sharing maprdquo of the reference design
stiffness of both paths decrease sometimes compared to theerrorless mesh stiffness It means the gear tooth of bothpaths does not always fully contact over the whole theoreticalcontact line Therefore contact length and mesh stiffnesscannot be predicted before the nonlinear model is solvedBesides the phase difference of mesh stiffness can be foundeasily in the figure
The load sharing coefficient 119896119897119904 under different powerand speed is evaluated as is illustrated in Figure 7 It canbe seen that the ldquoload sharing maprdquo (curved surface of loadsharing coefficient 119896119897119904 under different operating conditions)is complicated which comes from the nonlinearity of thesplit torque transmission system The evaluated load sharingcoefficient 119896119897119904 varies from 1058 to 2136 with input powervarying from 2237 kW to 522 kW and speed varying from5268 rmin to 12292 rmin The root mean square (RMS) ofload sharing coefficient 119896119897119904 is 1391 From a global point ofview the load sharing coefficient 119896119897119904 increases with the speedincreasing and power decreasing which correspond with [6]The mesh load of the gear pair is so little under the conditionof high speed and light power that the elastic deformationis not large enough to offset the initial deviations at all thecontact points However it is noteworthy that the law of
load sharing and operating conditions proposed here is notstrictly correct and there might be some counterexamplesIt is because the fact that the actual length of contact linedepends much on gear deviation under light load whichincreases the nonlinearity of system dynamics
4 Mathematical Model of Optimization
41 Objective Function According to the previous sectionthe load sharing coefficient of a split torque transmissionsystem changes greatly with different operating conditionsand the law of changing is complicated Therefore inorder to obtain better load sharing from a system pointof view multiple operating conditions have to be takeninto consideration Considering the average case the firstobjective function is promoted by minimizing the root meansquare of load sharing coefficient under a wide range ofoperating conditions (possible operating conditions) inputpower varying from 2237 kW (60 of 119875in) to 522 kW (140of 119875in) and input speed varying from 5268 rmin (60 of119899in) to 12292 rmin (140 of 119899in) When designing a geartransmission light weight and safety are always importantdesign targets Safety is always measured by safety factorsof contact fatigue strength and bending fatigue strength[24] Therefore the second and the third objective functionscan be promoted by minimizing the total system mass andmaximizing the total safety factors
To sum up the whole objective functions are expressed as
min 1198841 = 119896119897119904RMS
min 1198842 =
6sum
119894=1119872119892119894
min 1198843 = minus sum 119878 = minus (1198781198671 + 1198781198672 + 1198781198651 + 1198781198652)
(23)
where 119896119897119904RMS is the RMS of load sharing coefficient (L-S-CRMS) under the possible operating conditions 119872119892119894 is themass of gear 119894 1198781198671 1198781198672 are the safety factor of contact fatiguestrength of first and second stages and 1198781198651 1198781198652 are the safetyfactor of bending fatigue strength of first and second stagesHere only the mass of gears is considered in the total mass ofthe system
The safety factors of gear pairs can be evaluated accordingto ISO6336 it will not be discussed in detail here Howeverthe calculation of gear mass is a problem for the methodto calculate gear mass is associated with its wheel structureGenerally there are three types of wheel structure solid typepanel type and spoke type The reason to adopt differenttypes of wheel structure is reducing weight as the gear getslarger the more percentage of mass is removed from thewheel Here the light weight coefficient 120578 is introduced tomeasure the extent of light weight and then the gear mass119872119892119894 can be calculated by
119872119892119894 = 120578119894119872lowast
119892119894119894 = 1 2 6 (24)
where 119872lowast
119892119894= 120588(1205874)119887119894119889119894
2 is the solid mass of gear 119894 with 120588the material density 119887119894 the tooth width of gear 119894 and 119889119894 thepitch diameter of gear 119894
8 Mathematical Problems in Engineering
The introduction of light weight coefficient 120578 unifies thedifferent methods to calculate gear mass under differentwheel structures and the difference of three types of wheelstructure is presented by varying the value of 120578 The type ofwheel structure is decided by the tip diameter 119889119886 so the valueof 120578 is directly related to the tip diameter 120578 = 120578(119889119886) Accordingto wheel structure design criteria 3D parameterized modelof a spur gear is created in CATIA V5 system A series of gearmodels are created by varying the tip diameter 119889119886 in a widerange and the masses of them are measured in CATIA V5system and then the values of 120578 for gears with different tipdiameters can be calculated In the process of optimizationthe value of 120578 for a gear is obtained by interpolating accordingto its tip diameter
The values of the three objective functions of the referencedesign are evaluated the root mean square of load sharingcoefficient 119896119897119904RMS is 1391 the system mass is 40910 kg andthe total safety factors is 8966
42 Designing Variables There are a lot of designing param-eters in a split torque transmission system some of themare independent while others are not Picking up appropriateparameters as the designing variables is the prerequisite foroptimization design The special arrangement of the splittorque transmission leads to the special mounting conditionthe proportioning of gear tooth has definite interrelationwiththe two shaft angles [2] Once the proportioning of gear toothand the two shaft angles are determined the center distancesof the two stages are determined at the same time which arerestricted to the center distance between input and outputshaft Therefore the modules of the two stages can hardly bethe standard value In other words in order to guarantee thecorrect arrangement of a split torque transmission system thestandard of modules has to be sacrificed
Based on the above considerations the designing vari-ables selected here includes the gear ratio of the first stage 1198941the pinion tooth number of the first and second stages11988511198854the helix angle of the first stage 1205731198871 and the two shaft anglesΦ1 Φ2 as expressed in (25)
X = 1198941 1198851 1198854 1205731198871 Φ1 Φ2119879
(25)
Other parameters can be evaluated by
1198942 =
11989401198941
1198852 = round (1198941 sdot 1198851)
1198856 = round (1198942 sdot 1198854)
1198981 = 2119867
cos12057311988711198851 + 1198852
sdot
sin (Φ22)
sin (Φ12 + Φ22)
1198982 = 2119867
11198854 + 1198856
sdot
sin (Φ12)
sin (Φ12 + Φ22)
(26)
where 1198940 is the total gear ratio 1198942 is the gear ratio of thesecond stage 1198852 1198856 are the gear tooth numbers of the firstand second stages 1198981 1198982 are the module of the first and
second stages and119867 is the center distance between input andoutput shaftsThe function round(sdot)heremeans round sdot to thenearest integer
43 Constraints
431 Boundary Constraints The design variables meet thefollowing constraints
1198941min le 1198941 le 1198941max
1198851min le 1198851 le 1198851max
1198854min le 1198854 le 1198854max
1205731198871min le 1205731198871 le 1205731198871max
Φ1min le Φ1 le Φ1max
Φ2min le Φ2 le Φ2max
(27)
where 119894111988511198854 1205731198871Φ1Φ2 with the subscripts min andmaxare the boundaries of design variables which are determinedempirically according to the initial design
432 Performance Constraints (i) Contact and bendingfatigue strengths should be below the allowable values
12059011986712 le [12059011986712]
12059011986512 le [12059011986512]
(28)
where 12059011986712 is the contact stress of the first and second stages[12059011986712] is the allowable contact stress of the first and secondstages12059011986512 is the bending stress of the first and second stagesand [12059011986512] is the allowable bending stress of the first andsecond stages
(ii) First stage gears should not interfere with each other
1198981 (1198852 + 2ℎlowast
119886) lt
1198981 (1198851 + 1198852)
cos1205731198871sdot sin(
Φ12
) (29)
where ℎlowast
119886is the addendum factor of the first stage gear
(iii) Safety margin of each stage should be balanced
radic14
4sum
119894=1(Δ119878119894 minus Δ119878)
2lt [120576] (30)
where
Δ119878119894 =
119878119867119894 minus 119878119867min 119894 = 1 2
119878119865119894minus2 minus 119878119865min 119894 = 3 4(31)
is the safety margin with 119878119867min the minimum safety factorfor contact fatigue strength and 119878119865min the minimum safetyfactor for bending fatigue strength Δ119878 is the mean value ofsafety margin Δ119878 [120576] is the allowable upper limit of safetymargin standard deviation
Mathematical Problems in Engineering 9
Ωj
Ωi
j
i
rs
Figure 8 Fitness sharing area
5 Algorithm and Improvement
Classical optimization methods suggest converting the mul-tiobjective optimization problem to a single-objective opti-mization problem by weighted sum of all the objectiveswhich lead to disadvantage of subjectivity when determiningthe weights of objectives Whereas the improved nondomi-nated sorting genetic algorithm (NSGA-II) proposed by Debavoids this disadvantage In this paper NSGA-II is adopted tosolve the proposed multiobjective optimization model
However there are large numbers of nonlinear dynamicequations to be solved under multiple operating conditionswhen evaluating the fitness and the solving time can be toolong to accept Therefore an improved NSGA-II algorithm isbeing put forward to solve the problem of time consumingprediction strategy is used in the fitness evaluation step soas to avoid the evaluation of load sharing property which iscomputationally very expensiveThe key of the improvementis predicting instead of evaluating the real fitness
51The Fitness Prediction Strategy As is known to all NSGA-II is a population based evolutionary algorithm for multi-objective optimization problems and the population evolveswith the generation increasing In the improved NSGA-IIalgorithm each individual 119894 in the population has its fitnessvector
ftn (119894) = [119891tn1
(119894) 119891tn2
(119894) 119891tn119873
(119894)] (32)
and the fidelity 119877(119894) of the fitness vector where 119873 is thenumber of objectives The value of the fitness vector can beevaluated using the fitness functions or predicted throughthe values of other individuals If the fitness vector ftn(119894)
is evaluated using the original fitness functions the fidelity119877(119894) = 1 if ftn(119894) is estimated the fidelity 0 le 119877(119894) lt 1
As shown in Figure 8 for each individual 119894 specify itsfitness sharing radius 119903119904 The area in which the dimensionlessEuclidean distance between individual 119894 and any other oneis no greater than the fitness sharing radius is called fitnesssharing area for individual 119894 expressed as Ω(119894) Assume thereare119898 other individuals in the fitness sharing areaΩ(119894) whichcomposed a collection 119878 = 1199041 1199042 119904119898 And the evaluationmethod of ftn(119894) is as follows
Rs2 fs2Rs1 fs1
Rs3 fs3
Rs4 fs4
Rs5 fs5
rs 1205962 1205961
1205963
1205964
1205965
Evaluating
Predictingsumi
120596ifsiRlowast
Figure 9 Fitness prediction model
Firstly evaluate the fidelity 119877(119894) of individual 119894
119877 (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot 119877 (119904119895) (33)
where 119904119895 is an individual including fitness sharing area Ω(119894)119877(119904119895) is the fidelity of 119904119895 120596(119904119895 119894) is the weight of 119904119895 donated byindividual 119894 Let dimensionless Euclidean distance betweenindividuals 1199041 1199042 119904119898 and individual 119894 be 1198891
119894
1198892119894
119889119898
119894
respectively then 120596(119904119895 119894) can be evaluated by
120596 (119904119895 119894) =
exp (minus120574 sdot 119889119895
119894
)
sum119898
119896=1 exp (minus120574 sdot 119889119896
119894
)
119895 = 1 2 119898 (34)
where 120574 is weight rescaling factorThe closer the individual isto individual 119894 the greater contribution of fidelity it makes
As depicted in Figure 9 if fidelity 119877(119894) is greater thana given threshold 119877
lowast then the fitness vector ftn(119894) can bepredicted as
ftn (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot ftn (119904119895) (35)
Else if119877(119894) is less than the threshold119877lowast then the fitness vector
ftn(119894) should be evaluated using its original fitness functionsTo take full advantage of the historical population whose
fitness has been evaluated it is necessary to establish adatabase of historical populations of coordinates fitness andfidelities As the population evolves the database will expandthe scale gradually In order to reduce space complexityand the amount of computation redundant data need to beeliminated after each generation The concept of individualredundancy is introduced to determine whether the data isredundant which is defined as
119868119903 (119894) =
119899
sum
119896=1Δ119909119896 (119894) (36)
where Δ119909119896(119894) is the coordinate difference (absolute value)of individual 119894 between the former individual and the latterone in the 119896th dimensional and 119899 is the number of design
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
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2 Mathematical Problems in Engineering
forward a method to measure the load sharing of planetarytransmissions and concluded that the dynamic load sharingdiffers greatly from static condition Therefore the dynamicanalysismethod is widely used because it reflects the real loadsharing property under operating condition and it is naturallyadopted in this paper Krantz [3] firstly studied the dynamicsof a split torque transmission system and concluded that theloads and motions of the two power paths differ although thesystem has symmetric geometry Kahraman [6] investigatedthe load sharing property of a planetary transmission systemwith a nonlinear dynamic model established which takesmanufacturing and assembly errors into consideration andfinds that the operating conditions also having great effectson load sharing Guo et al [7] studied the influences of relatedfactors on load sharing property of thewind turbine planetarygears based on its characteristics Kahraman [6] derived therelationship between dynamic load sharing coefficient andstatic load sharing coefficient based on dynamic methodwhich makes static analyzing meaningful [8] Bodas andKahraman [9] and Singh [10] studied the influences of man-ufacturing and assembly errors on load sharing property ofplanetary transmissions with 2D and 3D static contact mod-els adopted respectively Afterwards experimental studies[11 12] have also been conducted Ligata et al [13] establisheda discrete model of planetary transmissions to study its loadsharing property Then Singh [14 15] investigated the modeldeeply and obtains multiple load sharing coefficients undervaries of manufacturing errors and torques by defining theload sharing map
Benefiting from the centrosymmetry of design the plan-etary multipath transmission systems are always in capacityof automatical load sharing which offers an advantage overthe split torque design To promote the load sharing propertyof split torque transmission system somemethods have beenproposed to compensate for or minimize the effect of the gapincluding floating gears [16] quill shafts [2] and ldquoclockinganglerdquo [4 17] The floating gears arrangement permits theinput pinion to float until gear loads are balanced betweenthe two paths There are two kinds of quill shafts loadsharing devices the conventional quill shafts and the onebased on elastomeric elements The conventional quill shaftsassemble intermediate shafts with some torsional flexibilityso as to minimize the difference in torque split betweenpaths whereas the latter one add some materials with alower elastic modulus in the compound shafts to achieve thesame purpose The ldquoclocking anglerdquo method considers theldquoclocking anglerdquo as a design parameter to adjust and optimizethe load sharing and the ldquoclocking anglerdquo is adjusted byvarying the thicknesses of shim packs that axially positionedthe compound shafts
Further research of Krantz [3] indicated that even thoughthe manufacturing and assembly errors of a split torquetransmission are precisely controlled and the gap is elim-inated completely there still unequaled loads in the twopaths under real operating conditions Actually the uniquephase difference property of the split torque transmissionsystem results in desynchrony between the two paths whichcauses significant effects on load sharing However the phasedifferences are not independent design parameters which
are connected to the system geometry parametersThereforethe load sharing property of a split torque transmissioncan be promoted by adjusting the phase differences throughoptimizing the system geometry parameters Since it reflectsthe real load sharing of the split torque transmission underdynamic condition the load sharing coefficient used in thispaper is evaluated by solving the nonlinear dynamic model
Despite of the load sharing property the drive systemof a rotorcraft must also meet the demanding requirementsof lightweight and high safety [2 3] Savsani et al [18] andThompson et al [19] reduced the mass of a gear pair anda multistage gear system respectively through optimizingthe gear parameters Kumar et al [20] optimized a singlepair of gear transmission with promoting of load capacityof gears considered as the objective Based on the aboveconsiderations a multiobjective optimization design of asplit torque transmission system is conducted with thepromoting of load sharing property lightweight and safetyconsidered as the objectivesThe load sharing property whichis measured by load sharing coefficient is evaluated undermultiple operating conditionswith dynamic analysismethod
Deb et al [21] proposed the improved nondominatedsorting genetic algorithm (NSGA-II) based onmultiobjectiveevolutionary algorithm formultiobjective optimization prob-lems which has been proved a simple and effective method[22] In this paper NSGA-II is adopted to solve the multiob-jective optimizationmodel However there are large numbersof nonlinear dynamic equations to be solved under multipleoperating conditions when evaluating the fitness and thesolving time is not acceptable in engineering Therefore animprovement has been done toNSGA-II to solve the problemof time consuming prediction strategy is used in the fitnessevaluation step so as to avoid the evaluation of load sharingproperty which is computationally very expensive
2 Nonlinear Dynamics Model of a SplitTorque Transmission System
21 Modeling the Gear Mesh Regarding the spur gears asspecial helical gears with helix angle of 0 degree the general3D pinion-gear meshing model can be established as shownin Figure 1 Assuming that the geometry is not affected bydeflections (small displacements hypothesis) and providedthat mesh elasticity can be transferred onto the base planea rigid-body approach can be employed The pinion and thegear can therefore be assimilated to two rigid cylinders with4 degrees of freedom each which are connected by a stiffnesselement (or a distribution of stiffness elements) and a dampelement From a physical point of view the 8 degrees offreedomof a pair represent the generalized displacements of 3translational degrees (along axes 119909 119910 and 119911) and 1 rotationaldegree (around axis 119911) each Here the angle 120595 between axis119909 and the center line of the gear pair 997888997888997888997888rarr
119874119901119874119892 is defined asdirection angle
In this model flank deviations are taken into considera-tion Conventionally flank deviations relative to the perfectgeometry are positively defined in the direction of the outernormal and they are supposed to be small enough so that
Mathematical Problems in Engineering 3
Theoretical line of contact
y
xz
Op120595
120596p
Mlowast
M
120596g
Ogefp(M) gt 0
efg(M) lt 0
M
119847g
119847pPinion
Gear
Figure 1 Model of gear contact
tooth contacts remain on theoretical base planes Then thedeviation of a gear pair can be defined as sum of flankdeviations of the pinion and gear Each theoretical contactline on the base plane is discretized in elementary cellscentered in one point whose deviation is defined as thenormal distance on the base plane between a point of thepinion and a point of the gear that would be in contactfor perfect geometries [23] In fact due to the influence ofdeviations contact and deflection do not occur at all the pointin the theoretical contact line The real contact status of thegear pair can be obtained only when the contact status of allthe point in all the contact lines is solved
As Figure 1 illustrates when a gear pair is nominallyengaged contact occurs at one certain point 119872
lowast which hasthe maximum deviation namely 119890(119872
lowast) Select any point 119872
in the theoretical contact line different from 119872lowast and name
the flank deviations of 119872 in the pinion and gear as 119890119891119901(119872)
and 119890119891119892(119872) respectively The flank deviation is positive foran excess of material and negative when some material isremoved from the ideal geometryTherefore the deviation inpoint 119872 can be expressed as
119890 (119872) = 119890119891119901 (119872) + 119890119891119892 (119872) (1)
The gear model is shown in Figure 1 and pinion andgear are assimilated to rigid cylinders with four-degrees-of-freedom connected by series of stiffness and damp Thegeneralized displacements can be expressed as
q = 119909119901 119910119901 119911119901 120579119901 119909119892 119910119892 119911119892 120579119892
119879
(2)
where 119909119901 119910119901 119911119901 are translational displacements of the pinionalong axes 119909 119910 119911 respectively 120579119901 is rotational displacementof the pinion around axis 119911 119909119892 119910119892 119911119892 are translationaldisplacements of the gear along axes 119909 119910 119911 respectively 120579119892
is rotational displacement of the gear around axis 119911Since generalized displacements are small quantities it
can therefore be represented by infinitesimal translationsand rotations The perturbation q on pinion and gear gives
a normal approach 120575(119872) on the base plane relative to rigidbody positions
120575 (119872) = 120581119879q =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
cos120573119887 sin (120572 minus 120595)
cos120573119887 cos (120572 minus 120595)
sin120573119887
119877119887119901 cos120573119887
minus cos120573119887 sin (120572 minus 120595)
minus cos120573119887 cos (120572 minus 120595)
minus sin120573119887
119877119887119892 cos120573119887
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
119879
sdot
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119909119901
119910119901
119911119901
120579119901
119909119892
119910119892
119911119892
120579119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(3)
where 120581 is a projective vector depending on gear geometrywhich projects displacements q onto the base plane 120573119887 is thebase helix angle (the value is 0 to spur gears) 120572 is the pressureangle 119877119887119901 119877119887119892 are the base radius of pinion and gear
Contact and deflection in 119872 occur only if the normalapproach 120575(119872) is larger than the initial deviation 120575119890(119872) andin such case the deflection Δ(119872) can be evaluated as
Δ (119872) = 120575 (119872) minus 120575119890 (119872) (4)
where 120575119890(119872) = 119890(119872lowast) minus 119890(119872) 119890(119872
lowast) 119890(119872) is the deviation
in 119872lowast 119872 Else if the normal approach 120575(119872) is less than the
initial deviation 120575119890(119872) then the deflection Δ(119872) = 0The elemental mesh force transmitted from the pinion
onto the gear at one point of contact 119872 is
119889119865 (119872) = 119896 (119872) 119889119897 sdot Δ (119872) (5)
where 119896(119872) is the mesh stiffness at point 119872 per unit ofcontact length which is calculated according to ISO6336 and119889119897 is the elemental contact length
The total mesh force 119865119897 can be deducted by integratingover the time-varying and deflection dependent contactlength
119865119897 = int
119897
119896 (119872) Δ (119872) 119889119897
= int
119897
119896 (119872) 119889119897 sdot 120581119879qminus int
119897
119896 (119872) 120575119890 (119872) 119889119897
(6)
4 Mathematical Problems in Engineering
Outputshaft
First reductionstageEngine
input
High torquereduction stage
Figure 2 Full arrangement of a split torque transmission system
22 Gear Train Arrangement The full arrangement of a splittorque or split-path transmissions [3] is depicted in Figure 2which has two stages (1) One is first reduction stageThe firststage is where torque is split between the input pinion andthe two output gears Usually helical gears are used (2) Thesecond stage is high torque reduction stage The output shaftis driven by a gear which is driven simultaneously by two spurpinions each coaxial to the gear in the first reduction stage
As shown in Figure 2 the input pinion meshes with twogears offering two paths to transfer power to the output geartherefore the whole system is divided into two pathsThe twopower paths are identified as 119871 and 119877 with 119877 to the right of 119871The first-stage gear and second-stage pinion combination arecollectively called the compound gear The compound gearand gear shaft combination are called the compound shaft
As shown in Figure 3 a right-hand Cartesian coordinatesystem is established such that the 119911-axis is coincident withthe output gear shaft the positive 119910-axis extends from theinput gear center to the output pinion center and the inputgear drives clockwise The first-stage pinion gear (119871) andgear (119877) are marked with gears 1sim3 the second-stage pinion(119871) pinion (119877) and gear aremarkedwith gears 4sim6The inputshaft compound shaft (119871) compound shaft (119877) and outputshaft are marked with axes 1sim4 and the center of which ismarked with 1198741 1198742 1198743 1198744 Define the angle between
997888997888997888997888rarr
11987411198742
and 997888997888997888997888rarr
11987411198743 as shaft angle Φ1 and the angle between 997888997888997888997888rarr
11987441198742 and997888997888997888997888rarr
11987441198743 as shaft angle Φ2A unique property of a split torque transmission is the
phase relationships of the meshes The input pinion drivestwo gears simultaneously the length of the arc along thepitch circle joining the two pitch points 119886 119887 (the length ofarc 119886119887 in Figure 3) is probably not an integer multiple of thecircular pitch and then the two meshes will not pass throughthe pitch point at the same instant of time which results inthe desynchrony of the two paths It is the same with thesecond stage In order to describe the unique property of
Tout
L
c
O24
2y
xO
a b
d
Tin
Φ1
O1
O3 5
1
3
6
R
O4
Φ2
Figure 3 Coordinates of a split torque transmission system
a split torque transmission the concept of phase difference ofa stage is defined as quotient of the length of arc joining thetwo pitch points and the circular pitchThe phase differencesof the two stages are expressed as
Δ1205931 =
1006704119886119887
1199011199051=
1198891Φ121205871198981199051
=
1198851Φ12120587
Δ1205932 =
1006704119888119889
1199011199052=
1198896Φ221205871198981199052
=
1198856Φ22120587
(7)
where Δ12059312 is the phase difference of the first and secondstages 1006704
119886119887 1006704119888119889 are the length of arcs 119886119887 119888119889 1199011199051 1199011199052 are the
circular pitch of the first and second stages 1198981199051 1198981199052 are thetransversemodule of the first and second stages119889119894 is the pitchdiameter of gear 119894 119885119894 is the tooth number of gear 119894
There are 6 gears with 4 meshing pairs in a split torquetransmission system Each gear can be assimilated to a rigidcylinder with 4 degrees of freedom Therefore there are 24degrees of freedom in totalThe generalized displacements ofthe whole system can be expressed as
x = 1199091 1199101 1199111 1205791 1199092 1199102 1199112 1205792 1199096 1199106 1199116 1205796119879
(8)
where the subscripts 1sim6 correspond to gears 1sim6
23 Equations of Dynamics The general form of dynamicequations of a gear transmission system is
Mx +Cx +Kx = F (9)
whereM is the generalizedmassmatrixC is the dampmatrixK is the stiffness matrix and F is the load vector
With regard to the split torque transmission system thegeneral model can be specified to
Mx +Cx + (K119898 +K119887 +K119888) x = F0 + F (119890) (10)
where K119898 is the mesh stiffness term K119887 is the supportingstiffness term K119888 is the coupling stiffness term F0 is theconstant torques term and F(119890) is the additional force termcaused by gear deviations
Here we give the principle to set up all the matrices andvector in the model
Mathematical Problems in Engineering 5
231 Mass Matrix M Mass matrix M is a 24 times 24 diagonalmatrix which is expressed as
M = diag (1198981 1198981 1198981 1198681 1198982 1198982 1198982 1198682 1198986 1198986
1198986 1198686)
(11)
where 119898119894 119894 = 1 sim 6 is the mass of gears 1sim6 119868119894 119894 = 1 sim 6 isthe rotational inertia around 119911-axis
232 Stiffness Matrix K The stiffness matrix K consists ofmesh stiffness K119898 supporting stiffness K119887 and couplingstiffness K119888 and it is also a 24 times 24 symmetric matrix
K = K119898 +K119887 +K119888 (12)
(i) Meshing Stiffness Matrix K119898The mesh stiffness matrix ofthewhole systemK119898 comes frommesh stiffness of all the gearpairs it can therefore be obtained by assembling the entiremesh stiffness submatrix together
K119898 =
4sum
119899=1R119879119894119895K119898119894119895R119894119895 119894 119895 = 1 2 1 3 4 6 5 6 (13)
where K119898119894119895 = int119897119896119894119895(119872)119889119897 sdot 120581119894119895120581
119879
119894119895is the nonlinear and time-
dependent mesh stiffness submatrix of gear pair 119894 119895 with 120581119894119895the projective vector of gear pair 119894 and 119895
R119894119895
=
column 119894 column 119895
[
04times4 sdot sdot sdot R1119894 sdot sdot sdot 04times4 sdot sdot sdot 04times404times4 sdot sdot sdot 04times4 sdot sdot sdot R2119895 sdot sdot sdot 04times4
]
2times6
(14)
is assemble matrix with R1119894 = R2119895 = diag(1 1 1 1)
(ii) Supporting Stiffness Matrix K119887 The supporting stiffnessmatrixK119887 comes from all the gearsrsquo supporting stiffness in allthe translational motion freedoms
K119887 = diag (1198961199091 1198961199101 1198961199111 0 1198961199092 1198961199102 1198961199112 0 1198961199096 1198961199106
1198961199116 0)
(15)
where 119896119909119894 119896119910119894 119896119911119894 119894 = 1 sim 6 is the supporting stiffness of gear119894 in 119909 119910 119911 directions
(iii) Coupling Stiffness Matrix K119888 The coupling stiffnessmatrixK119888 of thewhole system comes from the two compoundshafts and it can be obtained by assembling the two couplingstiffness submatrix together
K119888 =2
sum
119899=1R119879119894119895K119888119894119895R119894119895 119894 119895 = 2 4 3 5 (16)
where
K119888119894119895
=
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119896119887119894119895 0 0 0 minus119896119887119894119895 0 0 0119896119887119894119895 0 0 0 minus119896119887119894119895 0 0
119896119886119894119895 0 0 0 minus119896119886119894119895 0119896119904119894119895 0 0 0 minus119896119904119894119895
119896119887119894119895 0 0 0119896119887119894119895 0 0
sym 119896119886119894119895 0119896119904119894119895
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
119894 119895 = 2 4 3 5
(17)
is the coupling stiffness submatrix of compound gear 119894 119895 with119896119904119894119895 119896119887119894119895 119896119886119894119895 the torsional stiffness bending stiffness axialstiffness between the compound gears 119894 and 119895
233 Damp Matrix C Here Rayleighrsquos damping is adopted
C = 119886M+ 119887K (18)
with 119886 119887 two constants to be adjusted from experimentalresults and experience
234 Load Vector F The load vector F consists of torquesvector F0 produced by input and output torques and addi-tional force vector F(119890) caused by gear deviations
F = F0 + F (119890) (19)
The torques vector F0 is expressed as
F0 = [0 0 0 119879in 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 119879out]119879
(20)
The additional force vector F(119890) caused by gear deviationsis given by
F (119890) =
4sum
119899=1R119879119894119895
int
119897
119896119894119895 (119872) 120575119890119894119895 (119872) 119889119871 sdot 120581119894119895 119894 119895
= 1 2 1 3 4 6 5 6
(21)
3 Case Study and Load Sharing Property
The load sharing properties of a split torque transmissionare how equivalent the load allocated between the two pathswhich can bemeasured by load sharing coefficient Accordingto [9] the load sharing coefficient 119896119897119904 is expressed as
119896119897119904 =
max (119865119871 119865119877)
1198650 (22)
where 119865119871 119865119877 are the average mesh forces of the first stage inleft and right paths1198650 is the staticmesh force of the first stage
6 Mathematical Problems in Engineering
Theoretical flank
Cumulative pitch error
Profile error
10
0
minus10
minus20
minus30
00
0
05 05
1
1
Root
Tip
Flan
k de
viat
ion
(120583m
)
Normalized profile length Normali
zed fa
ce width
A
A
Figure 4 An example of flank deviation surface for a tooth
Table 1 The geometry parameters of the reference design
Parameters First stage Second stagePiniongear Piniongear
Tooth number 32124 27176Modulemm 1590 254Pressure angle(∘) 20 25Helix angle(∘) 6 0Tooth widthmm 44453810 66045994Shaft angle(∘) 1225 502Phase difference 10889 24542
The larger load sharing coefficient 119896119897119904 the worth load sharingproperties
It is mentioned in Section 1 that there are two reasonswhich cause the unequal torques in the split torque transmis-sion system (1) the gap at one of the four gear mesh locationscaused bymanufacturing and assembly errors which directlyresults in the difference of deformations between the twopaths (2) the unique phase difference properties of the splittorque transmission system which result in the desynchronybetween the two paths Since various split torque load sharingmethods have been proposed to compensate for or minimizethe gap the effect of this gap caused by manufacturing andassembly errors will be ignored in this paperThemain factorwhich causes the unequal torques studied here is the phasedifference only
According to [3] a split torque transmission design forhelicopter is introduced as the reference design The loadsharing properties of the reference design will be studiedfirst and then an optimization will be conducted to promoteits load sharing The geometry parameters of the referencedesign have been listed in Table 1 other parameters are asfollows input power 119875in is 37285 kW input speed 119899in is8780 rmin center distance119867 between input and output shaftis 29345mm and material of gears is SAE 9310 An exampleof flank deviation surface for a teeth is shown in Figure 4 andthe flank deviation is considered the sum of cumulative pitcherror and profile error where the cumulative pitch error is
34 35 36 37 38 39 40
Dimensionless time
25
2
15
1
05
0
Mes
h fo
rce (
N)
times105
FLFRF0
Figure 5 Mesh force curves of the first stage in both paths at givenpower and speed
simulated by normally distributed constant and profile erroris simulated by sine surface The flank deviation of differentteeth is simulated differently and the periodicity of the gearpair is taken into consideration
The dynamic equations of the reference design systemare established according to the previous section and ode45order is adopted to solve the functions with MATLABsoftware The curves of mesh force and mesh stiffness of thefirst stage in both paths are evaluated at given power andspeed and parts of the curves are shown in Figures 5 and 6
As Figure 5 illustrates 119865119871 119865119877 are the mesh forces of leftand right paths of the first stage and 1198650 is the static meshforce of the first stage It can be seen from the figure that themesh force of the left path differs significantly from the rightone and the evaluated load sharing coefficient 119896119897119904 has reached1268
In Figure 6 119896119871 119896119877 are the mesh stiffness of left and rightpaths of the first stage and inside the dashed rectangle boxis the errorless mesh stiffness curve over one mesh periodcalculated by ISO6336 It can be seen that the actual mesh
Mathematical Problems in Engineering 7
34 35 36 37 38 39 40
Dimensionless time
Mes
h sti
ffnes
s (N
m)
times108
12
10
8
6
4
2
0
11
1
090 05 1
times109
kLkR
Figure 6 Mesh stiffness curves of the first stage in both paths atgiven power and speed
400
24
22
2
18
16
14
12
1Load
shar
ing
coeffi
cien
t
1400012000
100008000
6000Speed (rmin) 200
300
500600
Power (kW)
Load decreasesX 2237Y 8965Z 2136
X 2237Y 5268
Y 5268
Z 1304
Z 1062
X 522
X 522
Y 5638Z 1058
X 4435
Y 12292
Y 12292
Z 1211
X 2237
Z 2106
Figure 7 ldquoLoad sharing maprdquo of the reference design
stiffness of both paths decrease sometimes compared to theerrorless mesh stiffness It means the gear tooth of bothpaths does not always fully contact over the whole theoreticalcontact line Therefore contact length and mesh stiffnesscannot be predicted before the nonlinear model is solvedBesides the phase difference of mesh stiffness can be foundeasily in the figure
The load sharing coefficient 119896119897119904 under different powerand speed is evaluated as is illustrated in Figure 7 It canbe seen that the ldquoload sharing maprdquo (curved surface of loadsharing coefficient 119896119897119904 under different operating conditions)is complicated which comes from the nonlinearity of thesplit torque transmission system The evaluated load sharingcoefficient 119896119897119904 varies from 1058 to 2136 with input powervarying from 2237 kW to 522 kW and speed varying from5268 rmin to 12292 rmin The root mean square (RMS) ofload sharing coefficient 119896119897119904 is 1391 From a global point ofview the load sharing coefficient 119896119897119904 increases with the speedincreasing and power decreasing which correspond with [6]The mesh load of the gear pair is so little under the conditionof high speed and light power that the elastic deformationis not large enough to offset the initial deviations at all thecontact points However it is noteworthy that the law of
load sharing and operating conditions proposed here is notstrictly correct and there might be some counterexamplesIt is because the fact that the actual length of contact linedepends much on gear deviation under light load whichincreases the nonlinearity of system dynamics
4 Mathematical Model of Optimization
41 Objective Function According to the previous sectionthe load sharing coefficient of a split torque transmissionsystem changes greatly with different operating conditionsand the law of changing is complicated Therefore inorder to obtain better load sharing from a system pointof view multiple operating conditions have to be takeninto consideration Considering the average case the firstobjective function is promoted by minimizing the root meansquare of load sharing coefficient under a wide range ofoperating conditions (possible operating conditions) inputpower varying from 2237 kW (60 of 119875in) to 522 kW (140of 119875in) and input speed varying from 5268 rmin (60 of119899in) to 12292 rmin (140 of 119899in) When designing a geartransmission light weight and safety are always importantdesign targets Safety is always measured by safety factorsof contact fatigue strength and bending fatigue strength[24] Therefore the second and the third objective functionscan be promoted by minimizing the total system mass andmaximizing the total safety factors
To sum up the whole objective functions are expressed as
min 1198841 = 119896119897119904RMS
min 1198842 =
6sum
119894=1119872119892119894
min 1198843 = minus sum 119878 = minus (1198781198671 + 1198781198672 + 1198781198651 + 1198781198652)
(23)
where 119896119897119904RMS is the RMS of load sharing coefficient (L-S-CRMS) under the possible operating conditions 119872119892119894 is themass of gear 119894 1198781198671 1198781198672 are the safety factor of contact fatiguestrength of first and second stages and 1198781198651 1198781198652 are the safetyfactor of bending fatigue strength of first and second stagesHere only the mass of gears is considered in the total mass ofthe system
The safety factors of gear pairs can be evaluated accordingto ISO6336 it will not be discussed in detail here Howeverthe calculation of gear mass is a problem for the methodto calculate gear mass is associated with its wheel structureGenerally there are three types of wheel structure solid typepanel type and spoke type The reason to adopt differenttypes of wheel structure is reducing weight as the gear getslarger the more percentage of mass is removed from thewheel Here the light weight coefficient 120578 is introduced tomeasure the extent of light weight and then the gear mass119872119892119894 can be calculated by
119872119892119894 = 120578119894119872lowast
119892119894119894 = 1 2 6 (24)
where 119872lowast
119892119894= 120588(1205874)119887119894119889119894
2 is the solid mass of gear 119894 with 120588the material density 119887119894 the tooth width of gear 119894 and 119889119894 thepitch diameter of gear 119894
8 Mathematical Problems in Engineering
The introduction of light weight coefficient 120578 unifies thedifferent methods to calculate gear mass under differentwheel structures and the difference of three types of wheelstructure is presented by varying the value of 120578 The type ofwheel structure is decided by the tip diameter 119889119886 so the valueof 120578 is directly related to the tip diameter 120578 = 120578(119889119886) Accordingto wheel structure design criteria 3D parameterized modelof a spur gear is created in CATIA V5 system A series of gearmodels are created by varying the tip diameter 119889119886 in a widerange and the masses of them are measured in CATIA V5system and then the values of 120578 for gears with different tipdiameters can be calculated In the process of optimizationthe value of 120578 for a gear is obtained by interpolating accordingto its tip diameter
The values of the three objective functions of the referencedesign are evaluated the root mean square of load sharingcoefficient 119896119897119904RMS is 1391 the system mass is 40910 kg andthe total safety factors is 8966
42 Designing Variables There are a lot of designing param-eters in a split torque transmission system some of themare independent while others are not Picking up appropriateparameters as the designing variables is the prerequisite foroptimization design The special arrangement of the splittorque transmission leads to the special mounting conditionthe proportioning of gear tooth has definite interrelationwiththe two shaft angles [2] Once the proportioning of gear toothand the two shaft angles are determined the center distancesof the two stages are determined at the same time which arerestricted to the center distance between input and outputshaft Therefore the modules of the two stages can hardly bethe standard value In other words in order to guarantee thecorrect arrangement of a split torque transmission system thestandard of modules has to be sacrificed
Based on the above considerations the designing vari-ables selected here includes the gear ratio of the first stage 1198941the pinion tooth number of the first and second stages11988511198854the helix angle of the first stage 1205731198871 and the two shaft anglesΦ1 Φ2 as expressed in (25)
X = 1198941 1198851 1198854 1205731198871 Φ1 Φ2119879
(25)
Other parameters can be evaluated by
1198942 =
11989401198941
1198852 = round (1198941 sdot 1198851)
1198856 = round (1198942 sdot 1198854)
1198981 = 2119867
cos12057311988711198851 + 1198852
sdot
sin (Φ22)
sin (Φ12 + Φ22)
1198982 = 2119867
11198854 + 1198856
sdot
sin (Φ12)
sin (Φ12 + Φ22)
(26)
where 1198940 is the total gear ratio 1198942 is the gear ratio of thesecond stage 1198852 1198856 are the gear tooth numbers of the firstand second stages 1198981 1198982 are the module of the first and
second stages and119867 is the center distance between input andoutput shaftsThe function round(sdot)heremeans round sdot to thenearest integer
43 Constraints
431 Boundary Constraints The design variables meet thefollowing constraints
1198941min le 1198941 le 1198941max
1198851min le 1198851 le 1198851max
1198854min le 1198854 le 1198854max
1205731198871min le 1205731198871 le 1205731198871max
Φ1min le Φ1 le Φ1max
Φ2min le Φ2 le Φ2max
(27)
where 119894111988511198854 1205731198871Φ1Φ2 with the subscripts min andmaxare the boundaries of design variables which are determinedempirically according to the initial design
432 Performance Constraints (i) Contact and bendingfatigue strengths should be below the allowable values
12059011986712 le [12059011986712]
12059011986512 le [12059011986512]
(28)
where 12059011986712 is the contact stress of the first and second stages[12059011986712] is the allowable contact stress of the first and secondstages12059011986512 is the bending stress of the first and second stagesand [12059011986512] is the allowable bending stress of the first andsecond stages
(ii) First stage gears should not interfere with each other
1198981 (1198852 + 2ℎlowast
119886) lt
1198981 (1198851 + 1198852)
cos1205731198871sdot sin(
Φ12
) (29)
where ℎlowast
119886is the addendum factor of the first stage gear
(iii) Safety margin of each stage should be balanced
radic14
4sum
119894=1(Δ119878119894 minus Δ119878)
2lt [120576] (30)
where
Δ119878119894 =
119878119867119894 minus 119878119867min 119894 = 1 2
119878119865119894minus2 minus 119878119865min 119894 = 3 4(31)
is the safety margin with 119878119867min the minimum safety factorfor contact fatigue strength and 119878119865min the minimum safetyfactor for bending fatigue strength Δ119878 is the mean value ofsafety margin Δ119878 [120576] is the allowable upper limit of safetymargin standard deviation
Mathematical Problems in Engineering 9
Ωj
Ωi
j
i
rs
Figure 8 Fitness sharing area
5 Algorithm and Improvement
Classical optimization methods suggest converting the mul-tiobjective optimization problem to a single-objective opti-mization problem by weighted sum of all the objectiveswhich lead to disadvantage of subjectivity when determiningthe weights of objectives Whereas the improved nondomi-nated sorting genetic algorithm (NSGA-II) proposed by Debavoids this disadvantage In this paper NSGA-II is adopted tosolve the proposed multiobjective optimization model
However there are large numbers of nonlinear dynamicequations to be solved under multiple operating conditionswhen evaluating the fitness and the solving time can be toolong to accept Therefore an improved NSGA-II algorithm isbeing put forward to solve the problem of time consumingprediction strategy is used in the fitness evaluation step soas to avoid the evaluation of load sharing property which iscomputationally very expensiveThe key of the improvementis predicting instead of evaluating the real fitness
51The Fitness Prediction Strategy As is known to all NSGA-II is a population based evolutionary algorithm for multi-objective optimization problems and the population evolveswith the generation increasing In the improved NSGA-IIalgorithm each individual 119894 in the population has its fitnessvector
ftn (119894) = [119891tn1
(119894) 119891tn2
(119894) 119891tn119873
(119894)] (32)
and the fidelity 119877(119894) of the fitness vector where 119873 is thenumber of objectives The value of the fitness vector can beevaluated using the fitness functions or predicted throughthe values of other individuals If the fitness vector ftn(119894)
is evaluated using the original fitness functions the fidelity119877(119894) = 1 if ftn(119894) is estimated the fidelity 0 le 119877(119894) lt 1
As shown in Figure 8 for each individual 119894 specify itsfitness sharing radius 119903119904 The area in which the dimensionlessEuclidean distance between individual 119894 and any other oneis no greater than the fitness sharing radius is called fitnesssharing area for individual 119894 expressed as Ω(119894) Assume thereare119898 other individuals in the fitness sharing areaΩ(119894) whichcomposed a collection 119878 = 1199041 1199042 119904119898 And the evaluationmethod of ftn(119894) is as follows
Rs2 fs2Rs1 fs1
Rs3 fs3
Rs4 fs4
Rs5 fs5
rs 1205962 1205961
1205963
1205964
1205965
Evaluating
Predictingsumi
120596ifsiRlowast
Figure 9 Fitness prediction model
Firstly evaluate the fidelity 119877(119894) of individual 119894
119877 (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot 119877 (119904119895) (33)
where 119904119895 is an individual including fitness sharing area Ω(119894)119877(119904119895) is the fidelity of 119904119895 120596(119904119895 119894) is the weight of 119904119895 donated byindividual 119894 Let dimensionless Euclidean distance betweenindividuals 1199041 1199042 119904119898 and individual 119894 be 1198891
119894
1198892119894
119889119898
119894
respectively then 120596(119904119895 119894) can be evaluated by
120596 (119904119895 119894) =
exp (minus120574 sdot 119889119895
119894
)
sum119898
119896=1 exp (minus120574 sdot 119889119896
119894
)
119895 = 1 2 119898 (34)
where 120574 is weight rescaling factorThe closer the individual isto individual 119894 the greater contribution of fidelity it makes
As depicted in Figure 9 if fidelity 119877(119894) is greater thana given threshold 119877
lowast then the fitness vector ftn(119894) can bepredicted as
ftn (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot ftn (119904119895) (35)
Else if119877(119894) is less than the threshold119877lowast then the fitness vector
ftn(119894) should be evaluated using its original fitness functionsTo take full advantage of the historical population whose
fitness has been evaluated it is necessary to establish adatabase of historical populations of coordinates fitness andfidelities As the population evolves the database will expandthe scale gradually In order to reduce space complexityand the amount of computation redundant data need to beeliminated after each generation The concept of individualredundancy is introduced to determine whether the data isredundant which is defined as
119868119903 (119894) =
119899
sum
119896=1Δ119909119896 (119894) (36)
where Δ119909119896(119894) is the coordinate difference (absolute value)of individual 119894 between the former individual and the latterone in the 119896th dimensional and 119899 is the number of design
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Theoretical line of contact
y
xz
Op120595
120596p
Mlowast
M
120596g
Ogefp(M) gt 0
efg(M) lt 0
M
119847g
119847pPinion
Gear
Figure 1 Model of gear contact
tooth contacts remain on theoretical base planes Then thedeviation of a gear pair can be defined as sum of flankdeviations of the pinion and gear Each theoretical contactline on the base plane is discretized in elementary cellscentered in one point whose deviation is defined as thenormal distance on the base plane between a point of thepinion and a point of the gear that would be in contactfor perfect geometries [23] In fact due to the influence ofdeviations contact and deflection do not occur at all the pointin the theoretical contact line The real contact status of thegear pair can be obtained only when the contact status of allthe point in all the contact lines is solved
As Figure 1 illustrates when a gear pair is nominallyengaged contact occurs at one certain point 119872
lowast which hasthe maximum deviation namely 119890(119872
lowast) Select any point 119872
in the theoretical contact line different from 119872lowast and name
the flank deviations of 119872 in the pinion and gear as 119890119891119901(119872)
and 119890119891119892(119872) respectively The flank deviation is positive foran excess of material and negative when some material isremoved from the ideal geometryTherefore the deviation inpoint 119872 can be expressed as
119890 (119872) = 119890119891119901 (119872) + 119890119891119892 (119872) (1)
The gear model is shown in Figure 1 and pinion andgear are assimilated to rigid cylinders with four-degrees-of-freedom connected by series of stiffness and damp Thegeneralized displacements can be expressed as
q = 119909119901 119910119901 119911119901 120579119901 119909119892 119910119892 119911119892 120579119892
119879
(2)
where 119909119901 119910119901 119911119901 are translational displacements of the pinionalong axes 119909 119910 119911 respectively 120579119901 is rotational displacementof the pinion around axis 119911 119909119892 119910119892 119911119892 are translationaldisplacements of the gear along axes 119909 119910 119911 respectively 120579119892
is rotational displacement of the gear around axis 119911Since generalized displacements are small quantities it
can therefore be represented by infinitesimal translationsand rotations The perturbation q on pinion and gear gives
a normal approach 120575(119872) on the base plane relative to rigidbody positions
120575 (119872) = 120581119879q =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
cos120573119887 sin (120572 minus 120595)
cos120573119887 cos (120572 minus 120595)
sin120573119887
119877119887119901 cos120573119887
minus cos120573119887 sin (120572 minus 120595)
minus cos120573119887 cos (120572 minus 120595)
minus sin120573119887
119877119887119892 cos120573119887
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
119879
sdot
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119909119901
119910119901
119911119901
120579119901
119909119892
119910119892
119911119892
120579119892
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(3)
where 120581 is a projective vector depending on gear geometrywhich projects displacements q onto the base plane 120573119887 is thebase helix angle (the value is 0 to spur gears) 120572 is the pressureangle 119877119887119901 119877119887119892 are the base radius of pinion and gear
Contact and deflection in 119872 occur only if the normalapproach 120575(119872) is larger than the initial deviation 120575119890(119872) andin such case the deflection Δ(119872) can be evaluated as
Δ (119872) = 120575 (119872) minus 120575119890 (119872) (4)
where 120575119890(119872) = 119890(119872lowast) minus 119890(119872) 119890(119872
lowast) 119890(119872) is the deviation
in 119872lowast 119872 Else if the normal approach 120575(119872) is less than the
initial deviation 120575119890(119872) then the deflection Δ(119872) = 0The elemental mesh force transmitted from the pinion
onto the gear at one point of contact 119872 is
119889119865 (119872) = 119896 (119872) 119889119897 sdot Δ (119872) (5)
where 119896(119872) is the mesh stiffness at point 119872 per unit ofcontact length which is calculated according to ISO6336 and119889119897 is the elemental contact length
The total mesh force 119865119897 can be deducted by integratingover the time-varying and deflection dependent contactlength
119865119897 = int
119897
119896 (119872) Δ (119872) 119889119897
= int
119897
119896 (119872) 119889119897 sdot 120581119879qminus int
119897
119896 (119872) 120575119890 (119872) 119889119897
(6)
4 Mathematical Problems in Engineering
Outputshaft
First reductionstageEngine
input
High torquereduction stage
Figure 2 Full arrangement of a split torque transmission system
22 Gear Train Arrangement The full arrangement of a splittorque or split-path transmissions [3] is depicted in Figure 2which has two stages (1) One is first reduction stageThe firststage is where torque is split between the input pinion andthe two output gears Usually helical gears are used (2) Thesecond stage is high torque reduction stage The output shaftis driven by a gear which is driven simultaneously by two spurpinions each coaxial to the gear in the first reduction stage
As shown in Figure 2 the input pinion meshes with twogears offering two paths to transfer power to the output geartherefore the whole system is divided into two pathsThe twopower paths are identified as 119871 and 119877 with 119877 to the right of 119871The first-stage gear and second-stage pinion combination arecollectively called the compound gear The compound gearand gear shaft combination are called the compound shaft
As shown in Figure 3 a right-hand Cartesian coordinatesystem is established such that the 119911-axis is coincident withthe output gear shaft the positive 119910-axis extends from theinput gear center to the output pinion center and the inputgear drives clockwise The first-stage pinion gear (119871) andgear (119877) are marked with gears 1sim3 the second-stage pinion(119871) pinion (119877) and gear aremarkedwith gears 4sim6The inputshaft compound shaft (119871) compound shaft (119877) and outputshaft are marked with axes 1sim4 and the center of which ismarked with 1198741 1198742 1198743 1198744 Define the angle between
997888997888997888997888rarr
11987411198742
and 997888997888997888997888rarr
11987411198743 as shaft angle Φ1 and the angle between 997888997888997888997888rarr
11987441198742 and997888997888997888997888rarr
11987441198743 as shaft angle Φ2A unique property of a split torque transmission is the
phase relationships of the meshes The input pinion drivestwo gears simultaneously the length of the arc along thepitch circle joining the two pitch points 119886 119887 (the length ofarc 119886119887 in Figure 3) is probably not an integer multiple of thecircular pitch and then the two meshes will not pass throughthe pitch point at the same instant of time which results inthe desynchrony of the two paths It is the same with thesecond stage In order to describe the unique property of
Tout
L
c
O24
2y
xO
a b
d
Tin
Φ1
O1
O3 5
1
3
6
R
O4
Φ2
Figure 3 Coordinates of a split torque transmission system
a split torque transmission the concept of phase difference ofa stage is defined as quotient of the length of arc joining thetwo pitch points and the circular pitchThe phase differencesof the two stages are expressed as
Δ1205931 =
1006704119886119887
1199011199051=
1198891Φ121205871198981199051
=
1198851Φ12120587
Δ1205932 =
1006704119888119889
1199011199052=
1198896Φ221205871198981199052
=
1198856Φ22120587
(7)
where Δ12059312 is the phase difference of the first and secondstages 1006704
119886119887 1006704119888119889 are the length of arcs 119886119887 119888119889 1199011199051 1199011199052 are the
circular pitch of the first and second stages 1198981199051 1198981199052 are thetransversemodule of the first and second stages119889119894 is the pitchdiameter of gear 119894 119885119894 is the tooth number of gear 119894
There are 6 gears with 4 meshing pairs in a split torquetransmission system Each gear can be assimilated to a rigidcylinder with 4 degrees of freedom Therefore there are 24degrees of freedom in totalThe generalized displacements ofthe whole system can be expressed as
x = 1199091 1199101 1199111 1205791 1199092 1199102 1199112 1205792 1199096 1199106 1199116 1205796119879
(8)
where the subscripts 1sim6 correspond to gears 1sim6
23 Equations of Dynamics The general form of dynamicequations of a gear transmission system is
Mx +Cx +Kx = F (9)
whereM is the generalizedmassmatrixC is the dampmatrixK is the stiffness matrix and F is the load vector
With regard to the split torque transmission system thegeneral model can be specified to
Mx +Cx + (K119898 +K119887 +K119888) x = F0 + F (119890) (10)
where K119898 is the mesh stiffness term K119887 is the supportingstiffness term K119888 is the coupling stiffness term F0 is theconstant torques term and F(119890) is the additional force termcaused by gear deviations
Here we give the principle to set up all the matrices andvector in the model
Mathematical Problems in Engineering 5
231 Mass Matrix M Mass matrix M is a 24 times 24 diagonalmatrix which is expressed as
M = diag (1198981 1198981 1198981 1198681 1198982 1198982 1198982 1198682 1198986 1198986
1198986 1198686)
(11)
where 119898119894 119894 = 1 sim 6 is the mass of gears 1sim6 119868119894 119894 = 1 sim 6 isthe rotational inertia around 119911-axis
232 Stiffness Matrix K The stiffness matrix K consists ofmesh stiffness K119898 supporting stiffness K119887 and couplingstiffness K119888 and it is also a 24 times 24 symmetric matrix
K = K119898 +K119887 +K119888 (12)
(i) Meshing Stiffness Matrix K119898The mesh stiffness matrix ofthewhole systemK119898 comes frommesh stiffness of all the gearpairs it can therefore be obtained by assembling the entiremesh stiffness submatrix together
K119898 =
4sum
119899=1R119879119894119895K119898119894119895R119894119895 119894 119895 = 1 2 1 3 4 6 5 6 (13)
where K119898119894119895 = int119897119896119894119895(119872)119889119897 sdot 120581119894119895120581
119879
119894119895is the nonlinear and time-
dependent mesh stiffness submatrix of gear pair 119894 119895 with 120581119894119895the projective vector of gear pair 119894 and 119895
R119894119895
=
column 119894 column 119895
[
04times4 sdot sdot sdot R1119894 sdot sdot sdot 04times4 sdot sdot sdot 04times404times4 sdot sdot sdot 04times4 sdot sdot sdot R2119895 sdot sdot sdot 04times4
]
2times6
(14)
is assemble matrix with R1119894 = R2119895 = diag(1 1 1 1)
(ii) Supporting Stiffness Matrix K119887 The supporting stiffnessmatrixK119887 comes from all the gearsrsquo supporting stiffness in allthe translational motion freedoms
K119887 = diag (1198961199091 1198961199101 1198961199111 0 1198961199092 1198961199102 1198961199112 0 1198961199096 1198961199106
1198961199116 0)
(15)
where 119896119909119894 119896119910119894 119896119911119894 119894 = 1 sim 6 is the supporting stiffness of gear119894 in 119909 119910 119911 directions
(iii) Coupling Stiffness Matrix K119888 The coupling stiffnessmatrixK119888 of thewhole system comes from the two compoundshafts and it can be obtained by assembling the two couplingstiffness submatrix together
K119888 =2
sum
119899=1R119879119894119895K119888119894119895R119894119895 119894 119895 = 2 4 3 5 (16)
where
K119888119894119895
=
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119896119887119894119895 0 0 0 minus119896119887119894119895 0 0 0119896119887119894119895 0 0 0 minus119896119887119894119895 0 0
119896119886119894119895 0 0 0 minus119896119886119894119895 0119896119904119894119895 0 0 0 minus119896119904119894119895
119896119887119894119895 0 0 0119896119887119894119895 0 0
sym 119896119886119894119895 0119896119904119894119895
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
119894 119895 = 2 4 3 5
(17)
is the coupling stiffness submatrix of compound gear 119894 119895 with119896119904119894119895 119896119887119894119895 119896119886119894119895 the torsional stiffness bending stiffness axialstiffness between the compound gears 119894 and 119895
233 Damp Matrix C Here Rayleighrsquos damping is adopted
C = 119886M+ 119887K (18)
with 119886 119887 two constants to be adjusted from experimentalresults and experience
234 Load Vector F The load vector F consists of torquesvector F0 produced by input and output torques and addi-tional force vector F(119890) caused by gear deviations
F = F0 + F (119890) (19)
The torques vector F0 is expressed as
F0 = [0 0 0 119879in 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 119879out]119879
(20)
The additional force vector F(119890) caused by gear deviationsis given by
F (119890) =
4sum
119899=1R119879119894119895
int
119897
119896119894119895 (119872) 120575119890119894119895 (119872) 119889119871 sdot 120581119894119895 119894 119895
= 1 2 1 3 4 6 5 6
(21)
3 Case Study and Load Sharing Property
The load sharing properties of a split torque transmissionare how equivalent the load allocated between the two pathswhich can bemeasured by load sharing coefficient Accordingto [9] the load sharing coefficient 119896119897119904 is expressed as
119896119897119904 =
max (119865119871 119865119877)
1198650 (22)
where 119865119871 119865119877 are the average mesh forces of the first stage inleft and right paths1198650 is the staticmesh force of the first stage
6 Mathematical Problems in Engineering
Theoretical flank
Cumulative pitch error
Profile error
10
0
minus10
minus20
minus30
00
0
05 05
1
1
Root
Tip
Flan
k de
viat
ion
(120583m
)
Normalized profile length Normali
zed fa
ce width
A
A
Figure 4 An example of flank deviation surface for a tooth
Table 1 The geometry parameters of the reference design
Parameters First stage Second stagePiniongear Piniongear
Tooth number 32124 27176Modulemm 1590 254Pressure angle(∘) 20 25Helix angle(∘) 6 0Tooth widthmm 44453810 66045994Shaft angle(∘) 1225 502Phase difference 10889 24542
The larger load sharing coefficient 119896119897119904 the worth load sharingproperties
It is mentioned in Section 1 that there are two reasonswhich cause the unequal torques in the split torque transmis-sion system (1) the gap at one of the four gear mesh locationscaused bymanufacturing and assembly errors which directlyresults in the difference of deformations between the twopaths (2) the unique phase difference properties of the splittorque transmission system which result in the desynchronybetween the two paths Since various split torque load sharingmethods have been proposed to compensate for or minimizethe gap the effect of this gap caused by manufacturing andassembly errors will be ignored in this paperThemain factorwhich causes the unequal torques studied here is the phasedifference only
According to [3] a split torque transmission design forhelicopter is introduced as the reference design The loadsharing properties of the reference design will be studiedfirst and then an optimization will be conducted to promoteits load sharing The geometry parameters of the referencedesign have been listed in Table 1 other parameters are asfollows input power 119875in is 37285 kW input speed 119899in is8780 rmin center distance119867 between input and output shaftis 29345mm and material of gears is SAE 9310 An exampleof flank deviation surface for a teeth is shown in Figure 4 andthe flank deviation is considered the sum of cumulative pitcherror and profile error where the cumulative pitch error is
34 35 36 37 38 39 40
Dimensionless time
25
2
15
1
05
0
Mes
h fo
rce (
N)
times105
FLFRF0
Figure 5 Mesh force curves of the first stage in both paths at givenpower and speed
simulated by normally distributed constant and profile erroris simulated by sine surface The flank deviation of differentteeth is simulated differently and the periodicity of the gearpair is taken into consideration
The dynamic equations of the reference design systemare established according to the previous section and ode45order is adopted to solve the functions with MATLABsoftware The curves of mesh force and mesh stiffness of thefirst stage in both paths are evaluated at given power andspeed and parts of the curves are shown in Figures 5 and 6
As Figure 5 illustrates 119865119871 119865119877 are the mesh forces of leftand right paths of the first stage and 1198650 is the static meshforce of the first stage It can be seen from the figure that themesh force of the left path differs significantly from the rightone and the evaluated load sharing coefficient 119896119897119904 has reached1268
In Figure 6 119896119871 119896119877 are the mesh stiffness of left and rightpaths of the first stage and inside the dashed rectangle boxis the errorless mesh stiffness curve over one mesh periodcalculated by ISO6336 It can be seen that the actual mesh
Mathematical Problems in Engineering 7
34 35 36 37 38 39 40
Dimensionless time
Mes
h sti
ffnes
s (N
m)
times108
12
10
8
6
4
2
0
11
1
090 05 1
times109
kLkR
Figure 6 Mesh stiffness curves of the first stage in both paths atgiven power and speed
400
24
22
2
18
16
14
12
1Load
shar
ing
coeffi
cien
t
1400012000
100008000
6000Speed (rmin) 200
300
500600
Power (kW)
Load decreasesX 2237Y 8965Z 2136
X 2237Y 5268
Y 5268
Z 1304
Z 1062
X 522
X 522
Y 5638Z 1058
X 4435
Y 12292
Y 12292
Z 1211
X 2237
Z 2106
Figure 7 ldquoLoad sharing maprdquo of the reference design
stiffness of both paths decrease sometimes compared to theerrorless mesh stiffness It means the gear tooth of bothpaths does not always fully contact over the whole theoreticalcontact line Therefore contact length and mesh stiffnesscannot be predicted before the nonlinear model is solvedBesides the phase difference of mesh stiffness can be foundeasily in the figure
The load sharing coefficient 119896119897119904 under different powerand speed is evaluated as is illustrated in Figure 7 It canbe seen that the ldquoload sharing maprdquo (curved surface of loadsharing coefficient 119896119897119904 under different operating conditions)is complicated which comes from the nonlinearity of thesplit torque transmission system The evaluated load sharingcoefficient 119896119897119904 varies from 1058 to 2136 with input powervarying from 2237 kW to 522 kW and speed varying from5268 rmin to 12292 rmin The root mean square (RMS) ofload sharing coefficient 119896119897119904 is 1391 From a global point ofview the load sharing coefficient 119896119897119904 increases with the speedincreasing and power decreasing which correspond with [6]The mesh load of the gear pair is so little under the conditionof high speed and light power that the elastic deformationis not large enough to offset the initial deviations at all thecontact points However it is noteworthy that the law of
load sharing and operating conditions proposed here is notstrictly correct and there might be some counterexamplesIt is because the fact that the actual length of contact linedepends much on gear deviation under light load whichincreases the nonlinearity of system dynamics
4 Mathematical Model of Optimization
41 Objective Function According to the previous sectionthe load sharing coefficient of a split torque transmissionsystem changes greatly with different operating conditionsand the law of changing is complicated Therefore inorder to obtain better load sharing from a system pointof view multiple operating conditions have to be takeninto consideration Considering the average case the firstobjective function is promoted by minimizing the root meansquare of load sharing coefficient under a wide range ofoperating conditions (possible operating conditions) inputpower varying from 2237 kW (60 of 119875in) to 522 kW (140of 119875in) and input speed varying from 5268 rmin (60 of119899in) to 12292 rmin (140 of 119899in) When designing a geartransmission light weight and safety are always importantdesign targets Safety is always measured by safety factorsof contact fatigue strength and bending fatigue strength[24] Therefore the second and the third objective functionscan be promoted by minimizing the total system mass andmaximizing the total safety factors
To sum up the whole objective functions are expressed as
min 1198841 = 119896119897119904RMS
min 1198842 =
6sum
119894=1119872119892119894
min 1198843 = minus sum 119878 = minus (1198781198671 + 1198781198672 + 1198781198651 + 1198781198652)
(23)
where 119896119897119904RMS is the RMS of load sharing coefficient (L-S-CRMS) under the possible operating conditions 119872119892119894 is themass of gear 119894 1198781198671 1198781198672 are the safety factor of contact fatiguestrength of first and second stages and 1198781198651 1198781198652 are the safetyfactor of bending fatigue strength of first and second stagesHere only the mass of gears is considered in the total mass ofthe system
The safety factors of gear pairs can be evaluated accordingto ISO6336 it will not be discussed in detail here Howeverthe calculation of gear mass is a problem for the methodto calculate gear mass is associated with its wheel structureGenerally there are three types of wheel structure solid typepanel type and spoke type The reason to adopt differenttypes of wheel structure is reducing weight as the gear getslarger the more percentage of mass is removed from thewheel Here the light weight coefficient 120578 is introduced tomeasure the extent of light weight and then the gear mass119872119892119894 can be calculated by
119872119892119894 = 120578119894119872lowast
119892119894119894 = 1 2 6 (24)
where 119872lowast
119892119894= 120588(1205874)119887119894119889119894
2 is the solid mass of gear 119894 with 120588the material density 119887119894 the tooth width of gear 119894 and 119889119894 thepitch diameter of gear 119894
8 Mathematical Problems in Engineering
The introduction of light weight coefficient 120578 unifies thedifferent methods to calculate gear mass under differentwheel structures and the difference of three types of wheelstructure is presented by varying the value of 120578 The type ofwheel structure is decided by the tip diameter 119889119886 so the valueof 120578 is directly related to the tip diameter 120578 = 120578(119889119886) Accordingto wheel structure design criteria 3D parameterized modelof a spur gear is created in CATIA V5 system A series of gearmodels are created by varying the tip diameter 119889119886 in a widerange and the masses of them are measured in CATIA V5system and then the values of 120578 for gears with different tipdiameters can be calculated In the process of optimizationthe value of 120578 for a gear is obtained by interpolating accordingto its tip diameter
The values of the three objective functions of the referencedesign are evaluated the root mean square of load sharingcoefficient 119896119897119904RMS is 1391 the system mass is 40910 kg andthe total safety factors is 8966
42 Designing Variables There are a lot of designing param-eters in a split torque transmission system some of themare independent while others are not Picking up appropriateparameters as the designing variables is the prerequisite foroptimization design The special arrangement of the splittorque transmission leads to the special mounting conditionthe proportioning of gear tooth has definite interrelationwiththe two shaft angles [2] Once the proportioning of gear toothand the two shaft angles are determined the center distancesof the two stages are determined at the same time which arerestricted to the center distance between input and outputshaft Therefore the modules of the two stages can hardly bethe standard value In other words in order to guarantee thecorrect arrangement of a split torque transmission system thestandard of modules has to be sacrificed
Based on the above considerations the designing vari-ables selected here includes the gear ratio of the first stage 1198941the pinion tooth number of the first and second stages11988511198854the helix angle of the first stage 1205731198871 and the two shaft anglesΦ1 Φ2 as expressed in (25)
X = 1198941 1198851 1198854 1205731198871 Φ1 Φ2119879
(25)
Other parameters can be evaluated by
1198942 =
11989401198941
1198852 = round (1198941 sdot 1198851)
1198856 = round (1198942 sdot 1198854)
1198981 = 2119867
cos12057311988711198851 + 1198852
sdot
sin (Φ22)
sin (Φ12 + Φ22)
1198982 = 2119867
11198854 + 1198856
sdot
sin (Φ12)
sin (Φ12 + Φ22)
(26)
where 1198940 is the total gear ratio 1198942 is the gear ratio of thesecond stage 1198852 1198856 are the gear tooth numbers of the firstand second stages 1198981 1198982 are the module of the first and
second stages and119867 is the center distance between input andoutput shaftsThe function round(sdot)heremeans round sdot to thenearest integer
43 Constraints
431 Boundary Constraints The design variables meet thefollowing constraints
1198941min le 1198941 le 1198941max
1198851min le 1198851 le 1198851max
1198854min le 1198854 le 1198854max
1205731198871min le 1205731198871 le 1205731198871max
Φ1min le Φ1 le Φ1max
Φ2min le Φ2 le Φ2max
(27)
where 119894111988511198854 1205731198871Φ1Φ2 with the subscripts min andmaxare the boundaries of design variables which are determinedempirically according to the initial design
432 Performance Constraints (i) Contact and bendingfatigue strengths should be below the allowable values
12059011986712 le [12059011986712]
12059011986512 le [12059011986512]
(28)
where 12059011986712 is the contact stress of the first and second stages[12059011986712] is the allowable contact stress of the first and secondstages12059011986512 is the bending stress of the first and second stagesand [12059011986512] is the allowable bending stress of the first andsecond stages
(ii) First stage gears should not interfere with each other
1198981 (1198852 + 2ℎlowast
119886) lt
1198981 (1198851 + 1198852)
cos1205731198871sdot sin(
Φ12
) (29)
where ℎlowast
119886is the addendum factor of the first stage gear
(iii) Safety margin of each stage should be balanced
radic14
4sum
119894=1(Δ119878119894 minus Δ119878)
2lt [120576] (30)
where
Δ119878119894 =
119878119867119894 minus 119878119867min 119894 = 1 2
119878119865119894minus2 minus 119878119865min 119894 = 3 4(31)
is the safety margin with 119878119867min the minimum safety factorfor contact fatigue strength and 119878119865min the minimum safetyfactor for bending fatigue strength Δ119878 is the mean value ofsafety margin Δ119878 [120576] is the allowable upper limit of safetymargin standard deviation
Mathematical Problems in Engineering 9
Ωj
Ωi
j
i
rs
Figure 8 Fitness sharing area
5 Algorithm and Improvement
Classical optimization methods suggest converting the mul-tiobjective optimization problem to a single-objective opti-mization problem by weighted sum of all the objectiveswhich lead to disadvantage of subjectivity when determiningthe weights of objectives Whereas the improved nondomi-nated sorting genetic algorithm (NSGA-II) proposed by Debavoids this disadvantage In this paper NSGA-II is adopted tosolve the proposed multiobjective optimization model
However there are large numbers of nonlinear dynamicequations to be solved under multiple operating conditionswhen evaluating the fitness and the solving time can be toolong to accept Therefore an improved NSGA-II algorithm isbeing put forward to solve the problem of time consumingprediction strategy is used in the fitness evaluation step soas to avoid the evaluation of load sharing property which iscomputationally very expensiveThe key of the improvementis predicting instead of evaluating the real fitness
51The Fitness Prediction Strategy As is known to all NSGA-II is a population based evolutionary algorithm for multi-objective optimization problems and the population evolveswith the generation increasing In the improved NSGA-IIalgorithm each individual 119894 in the population has its fitnessvector
ftn (119894) = [119891tn1
(119894) 119891tn2
(119894) 119891tn119873
(119894)] (32)
and the fidelity 119877(119894) of the fitness vector where 119873 is thenumber of objectives The value of the fitness vector can beevaluated using the fitness functions or predicted throughthe values of other individuals If the fitness vector ftn(119894)
is evaluated using the original fitness functions the fidelity119877(119894) = 1 if ftn(119894) is estimated the fidelity 0 le 119877(119894) lt 1
As shown in Figure 8 for each individual 119894 specify itsfitness sharing radius 119903119904 The area in which the dimensionlessEuclidean distance between individual 119894 and any other oneis no greater than the fitness sharing radius is called fitnesssharing area for individual 119894 expressed as Ω(119894) Assume thereare119898 other individuals in the fitness sharing areaΩ(119894) whichcomposed a collection 119878 = 1199041 1199042 119904119898 And the evaluationmethod of ftn(119894) is as follows
Rs2 fs2Rs1 fs1
Rs3 fs3
Rs4 fs4
Rs5 fs5
rs 1205962 1205961
1205963
1205964
1205965
Evaluating
Predictingsumi
120596ifsiRlowast
Figure 9 Fitness prediction model
Firstly evaluate the fidelity 119877(119894) of individual 119894
119877 (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot 119877 (119904119895) (33)
where 119904119895 is an individual including fitness sharing area Ω(119894)119877(119904119895) is the fidelity of 119904119895 120596(119904119895 119894) is the weight of 119904119895 donated byindividual 119894 Let dimensionless Euclidean distance betweenindividuals 1199041 1199042 119904119898 and individual 119894 be 1198891
119894
1198892119894
119889119898
119894
respectively then 120596(119904119895 119894) can be evaluated by
120596 (119904119895 119894) =
exp (minus120574 sdot 119889119895
119894
)
sum119898
119896=1 exp (minus120574 sdot 119889119896
119894
)
119895 = 1 2 119898 (34)
where 120574 is weight rescaling factorThe closer the individual isto individual 119894 the greater contribution of fidelity it makes
As depicted in Figure 9 if fidelity 119877(119894) is greater thana given threshold 119877
lowast then the fitness vector ftn(119894) can bepredicted as
ftn (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot ftn (119904119895) (35)
Else if119877(119894) is less than the threshold119877lowast then the fitness vector
ftn(119894) should be evaluated using its original fitness functionsTo take full advantage of the historical population whose
fitness has been evaluated it is necessary to establish adatabase of historical populations of coordinates fitness andfidelities As the population evolves the database will expandthe scale gradually In order to reduce space complexityand the amount of computation redundant data need to beeliminated after each generation The concept of individualredundancy is introduced to determine whether the data isredundant which is defined as
119868119903 (119894) =
119899
sum
119896=1Δ119909119896 (119894) (36)
where Δ119909119896(119894) is the coordinate difference (absolute value)of individual 119894 between the former individual and the latterone in the 119896th dimensional and 119899 is the number of design
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Outputshaft
First reductionstageEngine
input
High torquereduction stage
Figure 2 Full arrangement of a split torque transmission system
22 Gear Train Arrangement The full arrangement of a splittorque or split-path transmissions [3] is depicted in Figure 2which has two stages (1) One is first reduction stageThe firststage is where torque is split between the input pinion andthe two output gears Usually helical gears are used (2) Thesecond stage is high torque reduction stage The output shaftis driven by a gear which is driven simultaneously by two spurpinions each coaxial to the gear in the first reduction stage
As shown in Figure 2 the input pinion meshes with twogears offering two paths to transfer power to the output geartherefore the whole system is divided into two pathsThe twopower paths are identified as 119871 and 119877 with 119877 to the right of 119871The first-stage gear and second-stage pinion combination arecollectively called the compound gear The compound gearand gear shaft combination are called the compound shaft
As shown in Figure 3 a right-hand Cartesian coordinatesystem is established such that the 119911-axis is coincident withthe output gear shaft the positive 119910-axis extends from theinput gear center to the output pinion center and the inputgear drives clockwise The first-stage pinion gear (119871) andgear (119877) are marked with gears 1sim3 the second-stage pinion(119871) pinion (119877) and gear aremarkedwith gears 4sim6The inputshaft compound shaft (119871) compound shaft (119877) and outputshaft are marked with axes 1sim4 and the center of which ismarked with 1198741 1198742 1198743 1198744 Define the angle between
997888997888997888997888rarr
11987411198742
and 997888997888997888997888rarr
11987411198743 as shaft angle Φ1 and the angle between 997888997888997888997888rarr
11987441198742 and997888997888997888997888rarr
11987441198743 as shaft angle Φ2A unique property of a split torque transmission is the
phase relationships of the meshes The input pinion drivestwo gears simultaneously the length of the arc along thepitch circle joining the two pitch points 119886 119887 (the length ofarc 119886119887 in Figure 3) is probably not an integer multiple of thecircular pitch and then the two meshes will not pass throughthe pitch point at the same instant of time which results inthe desynchrony of the two paths It is the same with thesecond stage In order to describe the unique property of
Tout
L
c
O24
2y
xO
a b
d
Tin
Φ1
O1
O3 5
1
3
6
R
O4
Φ2
Figure 3 Coordinates of a split torque transmission system
a split torque transmission the concept of phase difference ofa stage is defined as quotient of the length of arc joining thetwo pitch points and the circular pitchThe phase differencesof the two stages are expressed as
Δ1205931 =
1006704119886119887
1199011199051=
1198891Φ121205871198981199051
=
1198851Φ12120587
Δ1205932 =
1006704119888119889
1199011199052=
1198896Φ221205871198981199052
=
1198856Φ22120587
(7)
where Δ12059312 is the phase difference of the first and secondstages 1006704
119886119887 1006704119888119889 are the length of arcs 119886119887 119888119889 1199011199051 1199011199052 are the
circular pitch of the first and second stages 1198981199051 1198981199052 are thetransversemodule of the first and second stages119889119894 is the pitchdiameter of gear 119894 119885119894 is the tooth number of gear 119894
There are 6 gears with 4 meshing pairs in a split torquetransmission system Each gear can be assimilated to a rigidcylinder with 4 degrees of freedom Therefore there are 24degrees of freedom in totalThe generalized displacements ofthe whole system can be expressed as
x = 1199091 1199101 1199111 1205791 1199092 1199102 1199112 1205792 1199096 1199106 1199116 1205796119879
(8)
where the subscripts 1sim6 correspond to gears 1sim6
23 Equations of Dynamics The general form of dynamicequations of a gear transmission system is
Mx +Cx +Kx = F (9)
whereM is the generalizedmassmatrixC is the dampmatrixK is the stiffness matrix and F is the load vector
With regard to the split torque transmission system thegeneral model can be specified to
Mx +Cx + (K119898 +K119887 +K119888) x = F0 + F (119890) (10)
where K119898 is the mesh stiffness term K119887 is the supportingstiffness term K119888 is the coupling stiffness term F0 is theconstant torques term and F(119890) is the additional force termcaused by gear deviations
Here we give the principle to set up all the matrices andvector in the model
Mathematical Problems in Engineering 5
231 Mass Matrix M Mass matrix M is a 24 times 24 diagonalmatrix which is expressed as
M = diag (1198981 1198981 1198981 1198681 1198982 1198982 1198982 1198682 1198986 1198986
1198986 1198686)
(11)
where 119898119894 119894 = 1 sim 6 is the mass of gears 1sim6 119868119894 119894 = 1 sim 6 isthe rotational inertia around 119911-axis
232 Stiffness Matrix K The stiffness matrix K consists ofmesh stiffness K119898 supporting stiffness K119887 and couplingstiffness K119888 and it is also a 24 times 24 symmetric matrix
K = K119898 +K119887 +K119888 (12)
(i) Meshing Stiffness Matrix K119898The mesh stiffness matrix ofthewhole systemK119898 comes frommesh stiffness of all the gearpairs it can therefore be obtained by assembling the entiremesh stiffness submatrix together
K119898 =
4sum
119899=1R119879119894119895K119898119894119895R119894119895 119894 119895 = 1 2 1 3 4 6 5 6 (13)
where K119898119894119895 = int119897119896119894119895(119872)119889119897 sdot 120581119894119895120581
119879
119894119895is the nonlinear and time-
dependent mesh stiffness submatrix of gear pair 119894 119895 with 120581119894119895the projective vector of gear pair 119894 and 119895
R119894119895
=
column 119894 column 119895
[
04times4 sdot sdot sdot R1119894 sdot sdot sdot 04times4 sdot sdot sdot 04times404times4 sdot sdot sdot 04times4 sdot sdot sdot R2119895 sdot sdot sdot 04times4
]
2times6
(14)
is assemble matrix with R1119894 = R2119895 = diag(1 1 1 1)
(ii) Supporting Stiffness Matrix K119887 The supporting stiffnessmatrixK119887 comes from all the gearsrsquo supporting stiffness in allthe translational motion freedoms
K119887 = diag (1198961199091 1198961199101 1198961199111 0 1198961199092 1198961199102 1198961199112 0 1198961199096 1198961199106
1198961199116 0)
(15)
where 119896119909119894 119896119910119894 119896119911119894 119894 = 1 sim 6 is the supporting stiffness of gear119894 in 119909 119910 119911 directions
(iii) Coupling Stiffness Matrix K119888 The coupling stiffnessmatrixK119888 of thewhole system comes from the two compoundshafts and it can be obtained by assembling the two couplingstiffness submatrix together
K119888 =2
sum
119899=1R119879119894119895K119888119894119895R119894119895 119894 119895 = 2 4 3 5 (16)
where
K119888119894119895
=
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119896119887119894119895 0 0 0 minus119896119887119894119895 0 0 0119896119887119894119895 0 0 0 minus119896119887119894119895 0 0
119896119886119894119895 0 0 0 minus119896119886119894119895 0119896119904119894119895 0 0 0 minus119896119904119894119895
119896119887119894119895 0 0 0119896119887119894119895 0 0
sym 119896119886119894119895 0119896119904119894119895
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
119894 119895 = 2 4 3 5
(17)
is the coupling stiffness submatrix of compound gear 119894 119895 with119896119904119894119895 119896119887119894119895 119896119886119894119895 the torsional stiffness bending stiffness axialstiffness between the compound gears 119894 and 119895
233 Damp Matrix C Here Rayleighrsquos damping is adopted
C = 119886M+ 119887K (18)
with 119886 119887 two constants to be adjusted from experimentalresults and experience
234 Load Vector F The load vector F consists of torquesvector F0 produced by input and output torques and addi-tional force vector F(119890) caused by gear deviations
F = F0 + F (119890) (19)
The torques vector F0 is expressed as
F0 = [0 0 0 119879in 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 119879out]119879
(20)
The additional force vector F(119890) caused by gear deviationsis given by
F (119890) =
4sum
119899=1R119879119894119895
int
119897
119896119894119895 (119872) 120575119890119894119895 (119872) 119889119871 sdot 120581119894119895 119894 119895
= 1 2 1 3 4 6 5 6
(21)
3 Case Study and Load Sharing Property
The load sharing properties of a split torque transmissionare how equivalent the load allocated between the two pathswhich can bemeasured by load sharing coefficient Accordingto [9] the load sharing coefficient 119896119897119904 is expressed as
119896119897119904 =
max (119865119871 119865119877)
1198650 (22)
where 119865119871 119865119877 are the average mesh forces of the first stage inleft and right paths1198650 is the staticmesh force of the first stage
6 Mathematical Problems in Engineering
Theoretical flank
Cumulative pitch error
Profile error
10
0
minus10
minus20
minus30
00
0
05 05
1
1
Root
Tip
Flan
k de
viat
ion
(120583m
)
Normalized profile length Normali
zed fa
ce width
A
A
Figure 4 An example of flank deviation surface for a tooth
Table 1 The geometry parameters of the reference design
Parameters First stage Second stagePiniongear Piniongear
Tooth number 32124 27176Modulemm 1590 254Pressure angle(∘) 20 25Helix angle(∘) 6 0Tooth widthmm 44453810 66045994Shaft angle(∘) 1225 502Phase difference 10889 24542
The larger load sharing coefficient 119896119897119904 the worth load sharingproperties
It is mentioned in Section 1 that there are two reasonswhich cause the unequal torques in the split torque transmis-sion system (1) the gap at one of the four gear mesh locationscaused bymanufacturing and assembly errors which directlyresults in the difference of deformations between the twopaths (2) the unique phase difference properties of the splittorque transmission system which result in the desynchronybetween the two paths Since various split torque load sharingmethods have been proposed to compensate for or minimizethe gap the effect of this gap caused by manufacturing andassembly errors will be ignored in this paperThemain factorwhich causes the unequal torques studied here is the phasedifference only
According to [3] a split torque transmission design forhelicopter is introduced as the reference design The loadsharing properties of the reference design will be studiedfirst and then an optimization will be conducted to promoteits load sharing The geometry parameters of the referencedesign have been listed in Table 1 other parameters are asfollows input power 119875in is 37285 kW input speed 119899in is8780 rmin center distance119867 between input and output shaftis 29345mm and material of gears is SAE 9310 An exampleof flank deviation surface for a teeth is shown in Figure 4 andthe flank deviation is considered the sum of cumulative pitcherror and profile error where the cumulative pitch error is
34 35 36 37 38 39 40
Dimensionless time
25
2
15
1
05
0
Mes
h fo
rce (
N)
times105
FLFRF0
Figure 5 Mesh force curves of the first stage in both paths at givenpower and speed
simulated by normally distributed constant and profile erroris simulated by sine surface The flank deviation of differentteeth is simulated differently and the periodicity of the gearpair is taken into consideration
The dynamic equations of the reference design systemare established according to the previous section and ode45order is adopted to solve the functions with MATLABsoftware The curves of mesh force and mesh stiffness of thefirst stage in both paths are evaluated at given power andspeed and parts of the curves are shown in Figures 5 and 6
As Figure 5 illustrates 119865119871 119865119877 are the mesh forces of leftand right paths of the first stage and 1198650 is the static meshforce of the first stage It can be seen from the figure that themesh force of the left path differs significantly from the rightone and the evaluated load sharing coefficient 119896119897119904 has reached1268
In Figure 6 119896119871 119896119877 are the mesh stiffness of left and rightpaths of the first stage and inside the dashed rectangle boxis the errorless mesh stiffness curve over one mesh periodcalculated by ISO6336 It can be seen that the actual mesh
Mathematical Problems in Engineering 7
34 35 36 37 38 39 40
Dimensionless time
Mes
h sti
ffnes
s (N
m)
times108
12
10
8
6
4
2
0
11
1
090 05 1
times109
kLkR
Figure 6 Mesh stiffness curves of the first stage in both paths atgiven power and speed
400
24
22
2
18
16
14
12
1Load
shar
ing
coeffi
cien
t
1400012000
100008000
6000Speed (rmin) 200
300
500600
Power (kW)
Load decreasesX 2237Y 8965Z 2136
X 2237Y 5268
Y 5268
Z 1304
Z 1062
X 522
X 522
Y 5638Z 1058
X 4435
Y 12292
Y 12292
Z 1211
X 2237
Z 2106
Figure 7 ldquoLoad sharing maprdquo of the reference design
stiffness of both paths decrease sometimes compared to theerrorless mesh stiffness It means the gear tooth of bothpaths does not always fully contact over the whole theoreticalcontact line Therefore contact length and mesh stiffnesscannot be predicted before the nonlinear model is solvedBesides the phase difference of mesh stiffness can be foundeasily in the figure
The load sharing coefficient 119896119897119904 under different powerand speed is evaluated as is illustrated in Figure 7 It canbe seen that the ldquoload sharing maprdquo (curved surface of loadsharing coefficient 119896119897119904 under different operating conditions)is complicated which comes from the nonlinearity of thesplit torque transmission system The evaluated load sharingcoefficient 119896119897119904 varies from 1058 to 2136 with input powervarying from 2237 kW to 522 kW and speed varying from5268 rmin to 12292 rmin The root mean square (RMS) ofload sharing coefficient 119896119897119904 is 1391 From a global point ofview the load sharing coefficient 119896119897119904 increases with the speedincreasing and power decreasing which correspond with [6]The mesh load of the gear pair is so little under the conditionof high speed and light power that the elastic deformationis not large enough to offset the initial deviations at all thecontact points However it is noteworthy that the law of
load sharing and operating conditions proposed here is notstrictly correct and there might be some counterexamplesIt is because the fact that the actual length of contact linedepends much on gear deviation under light load whichincreases the nonlinearity of system dynamics
4 Mathematical Model of Optimization
41 Objective Function According to the previous sectionthe load sharing coefficient of a split torque transmissionsystem changes greatly with different operating conditionsand the law of changing is complicated Therefore inorder to obtain better load sharing from a system pointof view multiple operating conditions have to be takeninto consideration Considering the average case the firstobjective function is promoted by minimizing the root meansquare of load sharing coefficient under a wide range ofoperating conditions (possible operating conditions) inputpower varying from 2237 kW (60 of 119875in) to 522 kW (140of 119875in) and input speed varying from 5268 rmin (60 of119899in) to 12292 rmin (140 of 119899in) When designing a geartransmission light weight and safety are always importantdesign targets Safety is always measured by safety factorsof contact fatigue strength and bending fatigue strength[24] Therefore the second and the third objective functionscan be promoted by minimizing the total system mass andmaximizing the total safety factors
To sum up the whole objective functions are expressed as
min 1198841 = 119896119897119904RMS
min 1198842 =
6sum
119894=1119872119892119894
min 1198843 = minus sum 119878 = minus (1198781198671 + 1198781198672 + 1198781198651 + 1198781198652)
(23)
where 119896119897119904RMS is the RMS of load sharing coefficient (L-S-CRMS) under the possible operating conditions 119872119892119894 is themass of gear 119894 1198781198671 1198781198672 are the safety factor of contact fatiguestrength of first and second stages and 1198781198651 1198781198652 are the safetyfactor of bending fatigue strength of first and second stagesHere only the mass of gears is considered in the total mass ofthe system
The safety factors of gear pairs can be evaluated accordingto ISO6336 it will not be discussed in detail here Howeverthe calculation of gear mass is a problem for the methodto calculate gear mass is associated with its wheel structureGenerally there are three types of wheel structure solid typepanel type and spoke type The reason to adopt differenttypes of wheel structure is reducing weight as the gear getslarger the more percentage of mass is removed from thewheel Here the light weight coefficient 120578 is introduced tomeasure the extent of light weight and then the gear mass119872119892119894 can be calculated by
119872119892119894 = 120578119894119872lowast
119892119894119894 = 1 2 6 (24)
where 119872lowast
119892119894= 120588(1205874)119887119894119889119894
2 is the solid mass of gear 119894 with 120588the material density 119887119894 the tooth width of gear 119894 and 119889119894 thepitch diameter of gear 119894
8 Mathematical Problems in Engineering
The introduction of light weight coefficient 120578 unifies thedifferent methods to calculate gear mass under differentwheel structures and the difference of three types of wheelstructure is presented by varying the value of 120578 The type ofwheel structure is decided by the tip diameter 119889119886 so the valueof 120578 is directly related to the tip diameter 120578 = 120578(119889119886) Accordingto wheel structure design criteria 3D parameterized modelof a spur gear is created in CATIA V5 system A series of gearmodels are created by varying the tip diameter 119889119886 in a widerange and the masses of them are measured in CATIA V5system and then the values of 120578 for gears with different tipdiameters can be calculated In the process of optimizationthe value of 120578 for a gear is obtained by interpolating accordingto its tip diameter
The values of the three objective functions of the referencedesign are evaluated the root mean square of load sharingcoefficient 119896119897119904RMS is 1391 the system mass is 40910 kg andthe total safety factors is 8966
42 Designing Variables There are a lot of designing param-eters in a split torque transmission system some of themare independent while others are not Picking up appropriateparameters as the designing variables is the prerequisite foroptimization design The special arrangement of the splittorque transmission leads to the special mounting conditionthe proportioning of gear tooth has definite interrelationwiththe two shaft angles [2] Once the proportioning of gear toothand the two shaft angles are determined the center distancesof the two stages are determined at the same time which arerestricted to the center distance between input and outputshaft Therefore the modules of the two stages can hardly bethe standard value In other words in order to guarantee thecorrect arrangement of a split torque transmission system thestandard of modules has to be sacrificed
Based on the above considerations the designing vari-ables selected here includes the gear ratio of the first stage 1198941the pinion tooth number of the first and second stages11988511198854the helix angle of the first stage 1205731198871 and the two shaft anglesΦ1 Φ2 as expressed in (25)
X = 1198941 1198851 1198854 1205731198871 Φ1 Φ2119879
(25)
Other parameters can be evaluated by
1198942 =
11989401198941
1198852 = round (1198941 sdot 1198851)
1198856 = round (1198942 sdot 1198854)
1198981 = 2119867
cos12057311988711198851 + 1198852
sdot
sin (Φ22)
sin (Φ12 + Φ22)
1198982 = 2119867
11198854 + 1198856
sdot
sin (Φ12)
sin (Φ12 + Φ22)
(26)
where 1198940 is the total gear ratio 1198942 is the gear ratio of thesecond stage 1198852 1198856 are the gear tooth numbers of the firstand second stages 1198981 1198982 are the module of the first and
second stages and119867 is the center distance between input andoutput shaftsThe function round(sdot)heremeans round sdot to thenearest integer
43 Constraints
431 Boundary Constraints The design variables meet thefollowing constraints
1198941min le 1198941 le 1198941max
1198851min le 1198851 le 1198851max
1198854min le 1198854 le 1198854max
1205731198871min le 1205731198871 le 1205731198871max
Φ1min le Φ1 le Φ1max
Φ2min le Φ2 le Φ2max
(27)
where 119894111988511198854 1205731198871Φ1Φ2 with the subscripts min andmaxare the boundaries of design variables which are determinedempirically according to the initial design
432 Performance Constraints (i) Contact and bendingfatigue strengths should be below the allowable values
12059011986712 le [12059011986712]
12059011986512 le [12059011986512]
(28)
where 12059011986712 is the contact stress of the first and second stages[12059011986712] is the allowable contact stress of the first and secondstages12059011986512 is the bending stress of the first and second stagesand [12059011986512] is the allowable bending stress of the first andsecond stages
(ii) First stage gears should not interfere with each other
1198981 (1198852 + 2ℎlowast
119886) lt
1198981 (1198851 + 1198852)
cos1205731198871sdot sin(
Φ12
) (29)
where ℎlowast
119886is the addendum factor of the first stage gear
(iii) Safety margin of each stage should be balanced
radic14
4sum
119894=1(Δ119878119894 minus Δ119878)
2lt [120576] (30)
where
Δ119878119894 =
119878119867119894 minus 119878119867min 119894 = 1 2
119878119865119894minus2 minus 119878119865min 119894 = 3 4(31)
is the safety margin with 119878119867min the minimum safety factorfor contact fatigue strength and 119878119865min the minimum safetyfactor for bending fatigue strength Δ119878 is the mean value ofsafety margin Δ119878 [120576] is the allowable upper limit of safetymargin standard deviation
Mathematical Problems in Engineering 9
Ωj
Ωi
j
i
rs
Figure 8 Fitness sharing area
5 Algorithm and Improvement
Classical optimization methods suggest converting the mul-tiobjective optimization problem to a single-objective opti-mization problem by weighted sum of all the objectiveswhich lead to disadvantage of subjectivity when determiningthe weights of objectives Whereas the improved nondomi-nated sorting genetic algorithm (NSGA-II) proposed by Debavoids this disadvantage In this paper NSGA-II is adopted tosolve the proposed multiobjective optimization model
However there are large numbers of nonlinear dynamicequations to be solved under multiple operating conditionswhen evaluating the fitness and the solving time can be toolong to accept Therefore an improved NSGA-II algorithm isbeing put forward to solve the problem of time consumingprediction strategy is used in the fitness evaluation step soas to avoid the evaluation of load sharing property which iscomputationally very expensiveThe key of the improvementis predicting instead of evaluating the real fitness
51The Fitness Prediction Strategy As is known to all NSGA-II is a population based evolutionary algorithm for multi-objective optimization problems and the population evolveswith the generation increasing In the improved NSGA-IIalgorithm each individual 119894 in the population has its fitnessvector
ftn (119894) = [119891tn1
(119894) 119891tn2
(119894) 119891tn119873
(119894)] (32)
and the fidelity 119877(119894) of the fitness vector where 119873 is thenumber of objectives The value of the fitness vector can beevaluated using the fitness functions or predicted throughthe values of other individuals If the fitness vector ftn(119894)
is evaluated using the original fitness functions the fidelity119877(119894) = 1 if ftn(119894) is estimated the fidelity 0 le 119877(119894) lt 1
As shown in Figure 8 for each individual 119894 specify itsfitness sharing radius 119903119904 The area in which the dimensionlessEuclidean distance between individual 119894 and any other oneis no greater than the fitness sharing radius is called fitnesssharing area for individual 119894 expressed as Ω(119894) Assume thereare119898 other individuals in the fitness sharing areaΩ(119894) whichcomposed a collection 119878 = 1199041 1199042 119904119898 And the evaluationmethod of ftn(119894) is as follows
Rs2 fs2Rs1 fs1
Rs3 fs3
Rs4 fs4
Rs5 fs5
rs 1205962 1205961
1205963
1205964
1205965
Evaluating
Predictingsumi
120596ifsiRlowast
Figure 9 Fitness prediction model
Firstly evaluate the fidelity 119877(119894) of individual 119894
119877 (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot 119877 (119904119895) (33)
where 119904119895 is an individual including fitness sharing area Ω(119894)119877(119904119895) is the fidelity of 119904119895 120596(119904119895 119894) is the weight of 119904119895 donated byindividual 119894 Let dimensionless Euclidean distance betweenindividuals 1199041 1199042 119904119898 and individual 119894 be 1198891
119894
1198892119894
119889119898
119894
respectively then 120596(119904119895 119894) can be evaluated by
120596 (119904119895 119894) =
exp (minus120574 sdot 119889119895
119894
)
sum119898
119896=1 exp (minus120574 sdot 119889119896
119894
)
119895 = 1 2 119898 (34)
where 120574 is weight rescaling factorThe closer the individual isto individual 119894 the greater contribution of fidelity it makes
As depicted in Figure 9 if fidelity 119877(119894) is greater thana given threshold 119877
lowast then the fitness vector ftn(119894) can bepredicted as
ftn (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot ftn (119904119895) (35)
Else if119877(119894) is less than the threshold119877lowast then the fitness vector
ftn(119894) should be evaluated using its original fitness functionsTo take full advantage of the historical population whose
fitness has been evaluated it is necessary to establish adatabase of historical populations of coordinates fitness andfidelities As the population evolves the database will expandthe scale gradually In order to reduce space complexityand the amount of computation redundant data need to beeliminated after each generation The concept of individualredundancy is introduced to determine whether the data isredundant which is defined as
119868119903 (119894) =
119899
sum
119896=1Δ119909119896 (119894) (36)
where Δ119909119896(119894) is the coordinate difference (absolute value)of individual 119894 between the former individual and the latterone in the 119896th dimensional and 119899 is the number of design
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
231 Mass Matrix M Mass matrix M is a 24 times 24 diagonalmatrix which is expressed as
M = diag (1198981 1198981 1198981 1198681 1198982 1198982 1198982 1198682 1198986 1198986
1198986 1198686)
(11)
where 119898119894 119894 = 1 sim 6 is the mass of gears 1sim6 119868119894 119894 = 1 sim 6 isthe rotational inertia around 119911-axis
232 Stiffness Matrix K The stiffness matrix K consists ofmesh stiffness K119898 supporting stiffness K119887 and couplingstiffness K119888 and it is also a 24 times 24 symmetric matrix
K = K119898 +K119887 +K119888 (12)
(i) Meshing Stiffness Matrix K119898The mesh stiffness matrix ofthewhole systemK119898 comes frommesh stiffness of all the gearpairs it can therefore be obtained by assembling the entiremesh stiffness submatrix together
K119898 =
4sum
119899=1R119879119894119895K119898119894119895R119894119895 119894 119895 = 1 2 1 3 4 6 5 6 (13)
where K119898119894119895 = int119897119896119894119895(119872)119889119897 sdot 120581119894119895120581
119879
119894119895is the nonlinear and time-
dependent mesh stiffness submatrix of gear pair 119894 119895 with 120581119894119895the projective vector of gear pair 119894 and 119895
R119894119895
=
column 119894 column 119895
[
04times4 sdot sdot sdot R1119894 sdot sdot sdot 04times4 sdot sdot sdot 04times404times4 sdot sdot sdot 04times4 sdot sdot sdot R2119895 sdot sdot sdot 04times4
]
2times6
(14)
is assemble matrix with R1119894 = R2119895 = diag(1 1 1 1)
(ii) Supporting Stiffness Matrix K119887 The supporting stiffnessmatrixK119887 comes from all the gearsrsquo supporting stiffness in allthe translational motion freedoms
K119887 = diag (1198961199091 1198961199101 1198961199111 0 1198961199092 1198961199102 1198961199112 0 1198961199096 1198961199106
1198961199116 0)
(15)
where 119896119909119894 119896119910119894 119896119911119894 119894 = 1 sim 6 is the supporting stiffness of gear119894 in 119909 119910 119911 directions
(iii) Coupling Stiffness Matrix K119888 The coupling stiffnessmatrixK119888 of thewhole system comes from the two compoundshafts and it can be obtained by assembling the two couplingstiffness submatrix together
K119888 =2
sum
119899=1R119879119894119895K119888119894119895R119894119895 119894 119895 = 2 4 3 5 (16)
where
K119888119894119895
=
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119896119887119894119895 0 0 0 minus119896119887119894119895 0 0 0119896119887119894119895 0 0 0 minus119896119887119894119895 0 0
119896119886119894119895 0 0 0 minus119896119886119894119895 0119896119904119894119895 0 0 0 minus119896119904119894119895
119896119887119894119895 0 0 0119896119887119894119895 0 0
sym 119896119886119894119895 0119896119904119894119895
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
119894 119895 = 2 4 3 5
(17)
is the coupling stiffness submatrix of compound gear 119894 119895 with119896119904119894119895 119896119887119894119895 119896119886119894119895 the torsional stiffness bending stiffness axialstiffness between the compound gears 119894 and 119895
233 Damp Matrix C Here Rayleighrsquos damping is adopted
C = 119886M+ 119887K (18)
with 119886 119887 two constants to be adjusted from experimentalresults and experience
234 Load Vector F The load vector F consists of torquesvector F0 produced by input and output torques and addi-tional force vector F(119890) caused by gear deviations
F = F0 + F (119890) (19)
The torques vector F0 is expressed as
F0 = [0 0 0 119879in 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 119879out]119879
(20)
The additional force vector F(119890) caused by gear deviationsis given by
F (119890) =
4sum
119899=1R119879119894119895
int
119897
119896119894119895 (119872) 120575119890119894119895 (119872) 119889119871 sdot 120581119894119895 119894 119895
= 1 2 1 3 4 6 5 6
(21)
3 Case Study and Load Sharing Property
The load sharing properties of a split torque transmissionare how equivalent the load allocated between the two pathswhich can bemeasured by load sharing coefficient Accordingto [9] the load sharing coefficient 119896119897119904 is expressed as
119896119897119904 =
max (119865119871 119865119877)
1198650 (22)
where 119865119871 119865119877 are the average mesh forces of the first stage inleft and right paths1198650 is the staticmesh force of the first stage
6 Mathematical Problems in Engineering
Theoretical flank
Cumulative pitch error
Profile error
10
0
minus10
minus20
minus30
00
0
05 05
1
1
Root
Tip
Flan
k de
viat
ion
(120583m
)
Normalized profile length Normali
zed fa
ce width
A
A
Figure 4 An example of flank deviation surface for a tooth
Table 1 The geometry parameters of the reference design
Parameters First stage Second stagePiniongear Piniongear
Tooth number 32124 27176Modulemm 1590 254Pressure angle(∘) 20 25Helix angle(∘) 6 0Tooth widthmm 44453810 66045994Shaft angle(∘) 1225 502Phase difference 10889 24542
The larger load sharing coefficient 119896119897119904 the worth load sharingproperties
It is mentioned in Section 1 that there are two reasonswhich cause the unequal torques in the split torque transmis-sion system (1) the gap at one of the four gear mesh locationscaused bymanufacturing and assembly errors which directlyresults in the difference of deformations between the twopaths (2) the unique phase difference properties of the splittorque transmission system which result in the desynchronybetween the two paths Since various split torque load sharingmethods have been proposed to compensate for or minimizethe gap the effect of this gap caused by manufacturing andassembly errors will be ignored in this paperThemain factorwhich causes the unequal torques studied here is the phasedifference only
According to [3] a split torque transmission design forhelicopter is introduced as the reference design The loadsharing properties of the reference design will be studiedfirst and then an optimization will be conducted to promoteits load sharing The geometry parameters of the referencedesign have been listed in Table 1 other parameters are asfollows input power 119875in is 37285 kW input speed 119899in is8780 rmin center distance119867 between input and output shaftis 29345mm and material of gears is SAE 9310 An exampleof flank deviation surface for a teeth is shown in Figure 4 andthe flank deviation is considered the sum of cumulative pitcherror and profile error where the cumulative pitch error is
34 35 36 37 38 39 40
Dimensionless time
25
2
15
1
05
0
Mes
h fo
rce (
N)
times105
FLFRF0
Figure 5 Mesh force curves of the first stage in both paths at givenpower and speed
simulated by normally distributed constant and profile erroris simulated by sine surface The flank deviation of differentteeth is simulated differently and the periodicity of the gearpair is taken into consideration
The dynamic equations of the reference design systemare established according to the previous section and ode45order is adopted to solve the functions with MATLABsoftware The curves of mesh force and mesh stiffness of thefirst stage in both paths are evaluated at given power andspeed and parts of the curves are shown in Figures 5 and 6
As Figure 5 illustrates 119865119871 119865119877 are the mesh forces of leftand right paths of the first stage and 1198650 is the static meshforce of the first stage It can be seen from the figure that themesh force of the left path differs significantly from the rightone and the evaluated load sharing coefficient 119896119897119904 has reached1268
In Figure 6 119896119871 119896119877 are the mesh stiffness of left and rightpaths of the first stage and inside the dashed rectangle boxis the errorless mesh stiffness curve over one mesh periodcalculated by ISO6336 It can be seen that the actual mesh
Mathematical Problems in Engineering 7
34 35 36 37 38 39 40
Dimensionless time
Mes
h sti
ffnes
s (N
m)
times108
12
10
8
6
4
2
0
11
1
090 05 1
times109
kLkR
Figure 6 Mesh stiffness curves of the first stage in both paths atgiven power and speed
400
24
22
2
18
16
14
12
1Load
shar
ing
coeffi
cien
t
1400012000
100008000
6000Speed (rmin) 200
300
500600
Power (kW)
Load decreasesX 2237Y 8965Z 2136
X 2237Y 5268
Y 5268
Z 1304
Z 1062
X 522
X 522
Y 5638Z 1058
X 4435
Y 12292
Y 12292
Z 1211
X 2237
Z 2106
Figure 7 ldquoLoad sharing maprdquo of the reference design
stiffness of both paths decrease sometimes compared to theerrorless mesh stiffness It means the gear tooth of bothpaths does not always fully contact over the whole theoreticalcontact line Therefore contact length and mesh stiffnesscannot be predicted before the nonlinear model is solvedBesides the phase difference of mesh stiffness can be foundeasily in the figure
The load sharing coefficient 119896119897119904 under different powerand speed is evaluated as is illustrated in Figure 7 It canbe seen that the ldquoload sharing maprdquo (curved surface of loadsharing coefficient 119896119897119904 under different operating conditions)is complicated which comes from the nonlinearity of thesplit torque transmission system The evaluated load sharingcoefficient 119896119897119904 varies from 1058 to 2136 with input powervarying from 2237 kW to 522 kW and speed varying from5268 rmin to 12292 rmin The root mean square (RMS) ofload sharing coefficient 119896119897119904 is 1391 From a global point ofview the load sharing coefficient 119896119897119904 increases with the speedincreasing and power decreasing which correspond with [6]The mesh load of the gear pair is so little under the conditionof high speed and light power that the elastic deformationis not large enough to offset the initial deviations at all thecontact points However it is noteworthy that the law of
load sharing and operating conditions proposed here is notstrictly correct and there might be some counterexamplesIt is because the fact that the actual length of contact linedepends much on gear deviation under light load whichincreases the nonlinearity of system dynamics
4 Mathematical Model of Optimization
41 Objective Function According to the previous sectionthe load sharing coefficient of a split torque transmissionsystem changes greatly with different operating conditionsand the law of changing is complicated Therefore inorder to obtain better load sharing from a system pointof view multiple operating conditions have to be takeninto consideration Considering the average case the firstobjective function is promoted by minimizing the root meansquare of load sharing coefficient under a wide range ofoperating conditions (possible operating conditions) inputpower varying from 2237 kW (60 of 119875in) to 522 kW (140of 119875in) and input speed varying from 5268 rmin (60 of119899in) to 12292 rmin (140 of 119899in) When designing a geartransmission light weight and safety are always importantdesign targets Safety is always measured by safety factorsof contact fatigue strength and bending fatigue strength[24] Therefore the second and the third objective functionscan be promoted by minimizing the total system mass andmaximizing the total safety factors
To sum up the whole objective functions are expressed as
min 1198841 = 119896119897119904RMS
min 1198842 =
6sum
119894=1119872119892119894
min 1198843 = minus sum 119878 = minus (1198781198671 + 1198781198672 + 1198781198651 + 1198781198652)
(23)
where 119896119897119904RMS is the RMS of load sharing coefficient (L-S-CRMS) under the possible operating conditions 119872119892119894 is themass of gear 119894 1198781198671 1198781198672 are the safety factor of contact fatiguestrength of first and second stages and 1198781198651 1198781198652 are the safetyfactor of bending fatigue strength of first and second stagesHere only the mass of gears is considered in the total mass ofthe system
The safety factors of gear pairs can be evaluated accordingto ISO6336 it will not be discussed in detail here Howeverthe calculation of gear mass is a problem for the methodto calculate gear mass is associated with its wheel structureGenerally there are three types of wheel structure solid typepanel type and spoke type The reason to adopt differenttypes of wheel structure is reducing weight as the gear getslarger the more percentage of mass is removed from thewheel Here the light weight coefficient 120578 is introduced tomeasure the extent of light weight and then the gear mass119872119892119894 can be calculated by
119872119892119894 = 120578119894119872lowast
119892119894119894 = 1 2 6 (24)
where 119872lowast
119892119894= 120588(1205874)119887119894119889119894
2 is the solid mass of gear 119894 with 120588the material density 119887119894 the tooth width of gear 119894 and 119889119894 thepitch diameter of gear 119894
8 Mathematical Problems in Engineering
The introduction of light weight coefficient 120578 unifies thedifferent methods to calculate gear mass under differentwheel structures and the difference of three types of wheelstructure is presented by varying the value of 120578 The type ofwheel structure is decided by the tip diameter 119889119886 so the valueof 120578 is directly related to the tip diameter 120578 = 120578(119889119886) Accordingto wheel structure design criteria 3D parameterized modelof a spur gear is created in CATIA V5 system A series of gearmodels are created by varying the tip diameter 119889119886 in a widerange and the masses of them are measured in CATIA V5system and then the values of 120578 for gears with different tipdiameters can be calculated In the process of optimizationthe value of 120578 for a gear is obtained by interpolating accordingto its tip diameter
The values of the three objective functions of the referencedesign are evaluated the root mean square of load sharingcoefficient 119896119897119904RMS is 1391 the system mass is 40910 kg andthe total safety factors is 8966
42 Designing Variables There are a lot of designing param-eters in a split torque transmission system some of themare independent while others are not Picking up appropriateparameters as the designing variables is the prerequisite foroptimization design The special arrangement of the splittorque transmission leads to the special mounting conditionthe proportioning of gear tooth has definite interrelationwiththe two shaft angles [2] Once the proportioning of gear toothand the two shaft angles are determined the center distancesof the two stages are determined at the same time which arerestricted to the center distance between input and outputshaft Therefore the modules of the two stages can hardly bethe standard value In other words in order to guarantee thecorrect arrangement of a split torque transmission system thestandard of modules has to be sacrificed
Based on the above considerations the designing vari-ables selected here includes the gear ratio of the first stage 1198941the pinion tooth number of the first and second stages11988511198854the helix angle of the first stage 1205731198871 and the two shaft anglesΦ1 Φ2 as expressed in (25)
X = 1198941 1198851 1198854 1205731198871 Φ1 Φ2119879
(25)
Other parameters can be evaluated by
1198942 =
11989401198941
1198852 = round (1198941 sdot 1198851)
1198856 = round (1198942 sdot 1198854)
1198981 = 2119867
cos12057311988711198851 + 1198852
sdot
sin (Φ22)
sin (Φ12 + Φ22)
1198982 = 2119867
11198854 + 1198856
sdot
sin (Φ12)
sin (Φ12 + Φ22)
(26)
where 1198940 is the total gear ratio 1198942 is the gear ratio of thesecond stage 1198852 1198856 are the gear tooth numbers of the firstand second stages 1198981 1198982 are the module of the first and
second stages and119867 is the center distance between input andoutput shaftsThe function round(sdot)heremeans round sdot to thenearest integer
43 Constraints
431 Boundary Constraints The design variables meet thefollowing constraints
1198941min le 1198941 le 1198941max
1198851min le 1198851 le 1198851max
1198854min le 1198854 le 1198854max
1205731198871min le 1205731198871 le 1205731198871max
Φ1min le Φ1 le Φ1max
Φ2min le Φ2 le Φ2max
(27)
where 119894111988511198854 1205731198871Φ1Φ2 with the subscripts min andmaxare the boundaries of design variables which are determinedempirically according to the initial design
432 Performance Constraints (i) Contact and bendingfatigue strengths should be below the allowable values
12059011986712 le [12059011986712]
12059011986512 le [12059011986512]
(28)
where 12059011986712 is the contact stress of the first and second stages[12059011986712] is the allowable contact stress of the first and secondstages12059011986512 is the bending stress of the first and second stagesand [12059011986512] is the allowable bending stress of the first andsecond stages
(ii) First stage gears should not interfere with each other
1198981 (1198852 + 2ℎlowast
119886) lt
1198981 (1198851 + 1198852)
cos1205731198871sdot sin(
Φ12
) (29)
where ℎlowast
119886is the addendum factor of the first stage gear
(iii) Safety margin of each stage should be balanced
radic14
4sum
119894=1(Δ119878119894 minus Δ119878)
2lt [120576] (30)
where
Δ119878119894 =
119878119867119894 minus 119878119867min 119894 = 1 2
119878119865119894minus2 minus 119878119865min 119894 = 3 4(31)
is the safety margin with 119878119867min the minimum safety factorfor contact fatigue strength and 119878119865min the minimum safetyfactor for bending fatigue strength Δ119878 is the mean value ofsafety margin Δ119878 [120576] is the allowable upper limit of safetymargin standard deviation
Mathematical Problems in Engineering 9
Ωj
Ωi
j
i
rs
Figure 8 Fitness sharing area
5 Algorithm and Improvement
Classical optimization methods suggest converting the mul-tiobjective optimization problem to a single-objective opti-mization problem by weighted sum of all the objectiveswhich lead to disadvantage of subjectivity when determiningthe weights of objectives Whereas the improved nondomi-nated sorting genetic algorithm (NSGA-II) proposed by Debavoids this disadvantage In this paper NSGA-II is adopted tosolve the proposed multiobjective optimization model
However there are large numbers of nonlinear dynamicequations to be solved under multiple operating conditionswhen evaluating the fitness and the solving time can be toolong to accept Therefore an improved NSGA-II algorithm isbeing put forward to solve the problem of time consumingprediction strategy is used in the fitness evaluation step soas to avoid the evaluation of load sharing property which iscomputationally very expensiveThe key of the improvementis predicting instead of evaluating the real fitness
51The Fitness Prediction Strategy As is known to all NSGA-II is a population based evolutionary algorithm for multi-objective optimization problems and the population evolveswith the generation increasing In the improved NSGA-IIalgorithm each individual 119894 in the population has its fitnessvector
ftn (119894) = [119891tn1
(119894) 119891tn2
(119894) 119891tn119873
(119894)] (32)
and the fidelity 119877(119894) of the fitness vector where 119873 is thenumber of objectives The value of the fitness vector can beevaluated using the fitness functions or predicted throughthe values of other individuals If the fitness vector ftn(119894)
is evaluated using the original fitness functions the fidelity119877(119894) = 1 if ftn(119894) is estimated the fidelity 0 le 119877(119894) lt 1
As shown in Figure 8 for each individual 119894 specify itsfitness sharing radius 119903119904 The area in which the dimensionlessEuclidean distance between individual 119894 and any other oneis no greater than the fitness sharing radius is called fitnesssharing area for individual 119894 expressed as Ω(119894) Assume thereare119898 other individuals in the fitness sharing areaΩ(119894) whichcomposed a collection 119878 = 1199041 1199042 119904119898 And the evaluationmethod of ftn(119894) is as follows
Rs2 fs2Rs1 fs1
Rs3 fs3
Rs4 fs4
Rs5 fs5
rs 1205962 1205961
1205963
1205964
1205965
Evaluating
Predictingsumi
120596ifsiRlowast
Figure 9 Fitness prediction model
Firstly evaluate the fidelity 119877(119894) of individual 119894
119877 (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot 119877 (119904119895) (33)
where 119904119895 is an individual including fitness sharing area Ω(119894)119877(119904119895) is the fidelity of 119904119895 120596(119904119895 119894) is the weight of 119904119895 donated byindividual 119894 Let dimensionless Euclidean distance betweenindividuals 1199041 1199042 119904119898 and individual 119894 be 1198891
119894
1198892119894
119889119898
119894
respectively then 120596(119904119895 119894) can be evaluated by
120596 (119904119895 119894) =
exp (minus120574 sdot 119889119895
119894
)
sum119898
119896=1 exp (minus120574 sdot 119889119896
119894
)
119895 = 1 2 119898 (34)
where 120574 is weight rescaling factorThe closer the individual isto individual 119894 the greater contribution of fidelity it makes
As depicted in Figure 9 if fidelity 119877(119894) is greater thana given threshold 119877
lowast then the fitness vector ftn(119894) can bepredicted as
ftn (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot ftn (119904119895) (35)
Else if119877(119894) is less than the threshold119877lowast then the fitness vector
ftn(119894) should be evaluated using its original fitness functionsTo take full advantage of the historical population whose
fitness has been evaluated it is necessary to establish adatabase of historical populations of coordinates fitness andfidelities As the population evolves the database will expandthe scale gradually In order to reduce space complexityand the amount of computation redundant data need to beeliminated after each generation The concept of individualredundancy is introduced to determine whether the data isredundant which is defined as
119868119903 (119894) =
119899
sum
119896=1Δ119909119896 (119894) (36)
where Δ119909119896(119894) is the coordinate difference (absolute value)of individual 119894 between the former individual and the latterone in the 119896th dimensional and 119899 is the number of design
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
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Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Theoretical flank
Cumulative pitch error
Profile error
10
0
minus10
minus20
minus30
00
0
05 05
1
1
Root
Tip
Flan
k de
viat
ion
(120583m
)
Normalized profile length Normali
zed fa
ce width
A
A
Figure 4 An example of flank deviation surface for a tooth
Table 1 The geometry parameters of the reference design
Parameters First stage Second stagePiniongear Piniongear
Tooth number 32124 27176Modulemm 1590 254Pressure angle(∘) 20 25Helix angle(∘) 6 0Tooth widthmm 44453810 66045994Shaft angle(∘) 1225 502Phase difference 10889 24542
The larger load sharing coefficient 119896119897119904 the worth load sharingproperties
It is mentioned in Section 1 that there are two reasonswhich cause the unequal torques in the split torque transmis-sion system (1) the gap at one of the four gear mesh locationscaused bymanufacturing and assembly errors which directlyresults in the difference of deformations between the twopaths (2) the unique phase difference properties of the splittorque transmission system which result in the desynchronybetween the two paths Since various split torque load sharingmethods have been proposed to compensate for or minimizethe gap the effect of this gap caused by manufacturing andassembly errors will be ignored in this paperThemain factorwhich causes the unequal torques studied here is the phasedifference only
According to [3] a split torque transmission design forhelicopter is introduced as the reference design The loadsharing properties of the reference design will be studiedfirst and then an optimization will be conducted to promoteits load sharing The geometry parameters of the referencedesign have been listed in Table 1 other parameters are asfollows input power 119875in is 37285 kW input speed 119899in is8780 rmin center distance119867 between input and output shaftis 29345mm and material of gears is SAE 9310 An exampleof flank deviation surface for a teeth is shown in Figure 4 andthe flank deviation is considered the sum of cumulative pitcherror and profile error where the cumulative pitch error is
34 35 36 37 38 39 40
Dimensionless time
25
2
15
1
05
0
Mes
h fo
rce (
N)
times105
FLFRF0
Figure 5 Mesh force curves of the first stage in both paths at givenpower and speed
simulated by normally distributed constant and profile erroris simulated by sine surface The flank deviation of differentteeth is simulated differently and the periodicity of the gearpair is taken into consideration
The dynamic equations of the reference design systemare established according to the previous section and ode45order is adopted to solve the functions with MATLABsoftware The curves of mesh force and mesh stiffness of thefirst stage in both paths are evaluated at given power andspeed and parts of the curves are shown in Figures 5 and 6
As Figure 5 illustrates 119865119871 119865119877 are the mesh forces of leftand right paths of the first stage and 1198650 is the static meshforce of the first stage It can be seen from the figure that themesh force of the left path differs significantly from the rightone and the evaluated load sharing coefficient 119896119897119904 has reached1268
In Figure 6 119896119871 119896119877 are the mesh stiffness of left and rightpaths of the first stage and inside the dashed rectangle boxis the errorless mesh stiffness curve over one mesh periodcalculated by ISO6336 It can be seen that the actual mesh
Mathematical Problems in Engineering 7
34 35 36 37 38 39 40
Dimensionless time
Mes
h sti
ffnes
s (N
m)
times108
12
10
8
6
4
2
0
11
1
090 05 1
times109
kLkR
Figure 6 Mesh stiffness curves of the first stage in both paths atgiven power and speed
400
24
22
2
18
16
14
12
1Load
shar
ing
coeffi
cien
t
1400012000
100008000
6000Speed (rmin) 200
300
500600
Power (kW)
Load decreasesX 2237Y 8965Z 2136
X 2237Y 5268
Y 5268
Z 1304
Z 1062
X 522
X 522
Y 5638Z 1058
X 4435
Y 12292
Y 12292
Z 1211
X 2237
Z 2106
Figure 7 ldquoLoad sharing maprdquo of the reference design
stiffness of both paths decrease sometimes compared to theerrorless mesh stiffness It means the gear tooth of bothpaths does not always fully contact over the whole theoreticalcontact line Therefore contact length and mesh stiffnesscannot be predicted before the nonlinear model is solvedBesides the phase difference of mesh stiffness can be foundeasily in the figure
The load sharing coefficient 119896119897119904 under different powerand speed is evaluated as is illustrated in Figure 7 It canbe seen that the ldquoload sharing maprdquo (curved surface of loadsharing coefficient 119896119897119904 under different operating conditions)is complicated which comes from the nonlinearity of thesplit torque transmission system The evaluated load sharingcoefficient 119896119897119904 varies from 1058 to 2136 with input powervarying from 2237 kW to 522 kW and speed varying from5268 rmin to 12292 rmin The root mean square (RMS) ofload sharing coefficient 119896119897119904 is 1391 From a global point ofview the load sharing coefficient 119896119897119904 increases with the speedincreasing and power decreasing which correspond with [6]The mesh load of the gear pair is so little under the conditionof high speed and light power that the elastic deformationis not large enough to offset the initial deviations at all thecontact points However it is noteworthy that the law of
load sharing and operating conditions proposed here is notstrictly correct and there might be some counterexamplesIt is because the fact that the actual length of contact linedepends much on gear deviation under light load whichincreases the nonlinearity of system dynamics
4 Mathematical Model of Optimization
41 Objective Function According to the previous sectionthe load sharing coefficient of a split torque transmissionsystem changes greatly with different operating conditionsand the law of changing is complicated Therefore inorder to obtain better load sharing from a system pointof view multiple operating conditions have to be takeninto consideration Considering the average case the firstobjective function is promoted by minimizing the root meansquare of load sharing coefficient under a wide range ofoperating conditions (possible operating conditions) inputpower varying from 2237 kW (60 of 119875in) to 522 kW (140of 119875in) and input speed varying from 5268 rmin (60 of119899in) to 12292 rmin (140 of 119899in) When designing a geartransmission light weight and safety are always importantdesign targets Safety is always measured by safety factorsof contact fatigue strength and bending fatigue strength[24] Therefore the second and the third objective functionscan be promoted by minimizing the total system mass andmaximizing the total safety factors
To sum up the whole objective functions are expressed as
min 1198841 = 119896119897119904RMS
min 1198842 =
6sum
119894=1119872119892119894
min 1198843 = minus sum 119878 = minus (1198781198671 + 1198781198672 + 1198781198651 + 1198781198652)
(23)
where 119896119897119904RMS is the RMS of load sharing coefficient (L-S-CRMS) under the possible operating conditions 119872119892119894 is themass of gear 119894 1198781198671 1198781198672 are the safety factor of contact fatiguestrength of first and second stages and 1198781198651 1198781198652 are the safetyfactor of bending fatigue strength of first and second stagesHere only the mass of gears is considered in the total mass ofthe system
The safety factors of gear pairs can be evaluated accordingto ISO6336 it will not be discussed in detail here Howeverthe calculation of gear mass is a problem for the methodto calculate gear mass is associated with its wheel structureGenerally there are three types of wheel structure solid typepanel type and spoke type The reason to adopt differenttypes of wheel structure is reducing weight as the gear getslarger the more percentage of mass is removed from thewheel Here the light weight coefficient 120578 is introduced tomeasure the extent of light weight and then the gear mass119872119892119894 can be calculated by
119872119892119894 = 120578119894119872lowast
119892119894119894 = 1 2 6 (24)
where 119872lowast
119892119894= 120588(1205874)119887119894119889119894
2 is the solid mass of gear 119894 with 120588the material density 119887119894 the tooth width of gear 119894 and 119889119894 thepitch diameter of gear 119894
8 Mathematical Problems in Engineering
The introduction of light weight coefficient 120578 unifies thedifferent methods to calculate gear mass under differentwheel structures and the difference of three types of wheelstructure is presented by varying the value of 120578 The type ofwheel structure is decided by the tip diameter 119889119886 so the valueof 120578 is directly related to the tip diameter 120578 = 120578(119889119886) Accordingto wheel structure design criteria 3D parameterized modelof a spur gear is created in CATIA V5 system A series of gearmodels are created by varying the tip diameter 119889119886 in a widerange and the masses of them are measured in CATIA V5system and then the values of 120578 for gears with different tipdiameters can be calculated In the process of optimizationthe value of 120578 for a gear is obtained by interpolating accordingto its tip diameter
The values of the three objective functions of the referencedesign are evaluated the root mean square of load sharingcoefficient 119896119897119904RMS is 1391 the system mass is 40910 kg andthe total safety factors is 8966
42 Designing Variables There are a lot of designing param-eters in a split torque transmission system some of themare independent while others are not Picking up appropriateparameters as the designing variables is the prerequisite foroptimization design The special arrangement of the splittorque transmission leads to the special mounting conditionthe proportioning of gear tooth has definite interrelationwiththe two shaft angles [2] Once the proportioning of gear toothand the two shaft angles are determined the center distancesof the two stages are determined at the same time which arerestricted to the center distance between input and outputshaft Therefore the modules of the two stages can hardly bethe standard value In other words in order to guarantee thecorrect arrangement of a split torque transmission system thestandard of modules has to be sacrificed
Based on the above considerations the designing vari-ables selected here includes the gear ratio of the first stage 1198941the pinion tooth number of the first and second stages11988511198854the helix angle of the first stage 1205731198871 and the two shaft anglesΦ1 Φ2 as expressed in (25)
X = 1198941 1198851 1198854 1205731198871 Φ1 Φ2119879
(25)
Other parameters can be evaluated by
1198942 =
11989401198941
1198852 = round (1198941 sdot 1198851)
1198856 = round (1198942 sdot 1198854)
1198981 = 2119867
cos12057311988711198851 + 1198852
sdot
sin (Φ22)
sin (Φ12 + Φ22)
1198982 = 2119867
11198854 + 1198856
sdot
sin (Φ12)
sin (Φ12 + Φ22)
(26)
where 1198940 is the total gear ratio 1198942 is the gear ratio of thesecond stage 1198852 1198856 are the gear tooth numbers of the firstand second stages 1198981 1198982 are the module of the first and
second stages and119867 is the center distance between input andoutput shaftsThe function round(sdot)heremeans round sdot to thenearest integer
43 Constraints
431 Boundary Constraints The design variables meet thefollowing constraints
1198941min le 1198941 le 1198941max
1198851min le 1198851 le 1198851max
1198854min le 1198854 le 1198854max
1205731198871min le 1205731198871 le 1205731198871max
Φ1min le Φ1 le Φ1max
Φ2min le Φ2 le Φ2max
(27)
where 119894111988511198854 1205731198871Φ1Φ2 with the subscripts min andmaxare the boundaries of design variables which are determinedempirically according to the initial design
432 Performance Constraints (i) Contact and bendingfatigue strengths should be below the allowable values
12059011986712 le [12059011986712]
12059011986512 le [12059011986512]
(28)
where 12059011986712 is the contact stress of the first and second stages[12059011986712] is the allowable contact stress of the first and secondstages12059011986512 is the bending stress of the first and second stagesand [12059011986512] is the allowable bending stress of the first andsecond stages
(ii) First stage gears should not interfere with each other
1198981 (1198852 + 2ℎlowast
119886) lt
1198981 (1198851 + 1198852)
cos1205731198871sdot sin(
Φ12
) (29)
where ℎlowast
119886is the addendum factor of the first stage gear
(iii) Safety margin of each stage should be balanced
radic14
4sum
119894=1(Δ119878119894 minus Δ119878)
2lt [120576] (30)
where
Δ119878119894 =
119878119867119894 minus 119878119867min 119894 = 1 2
119878119865119894minus2 minus 119878119865min 119894 = 3 4(31)
is the safety margin with 119878119867min the minimum safety factorfor contact fatigue strength and 119878119865min the minimum safetyfactor for bending fatigue strength Δ119878 is the mean value ofsafety margin Δ119878 [120576] is the allowable upper limit of safetymargin standard deviation
Mathematical Problems in Engineering 9
Ωj
Ωi
j
i
rs
Figure 8 Fitness sharing area
5 Algorithm and Improvement
Classical optimization methods suggest converting the mul-tiobjective optimization problem to a single-objective opti-mization problem by weighted sum of all the objectiveswhich lead to disadvantage of subjectivity when determiningthe weights of objectives Whereas the improved nondomi-nated sorting genetic algorithm (NSGA-II) proposed by Debavoids this disadvantage In this paper NSGA-II is adopted tosolve the proposed multiobjective optimization model
However there are large numbers of nonlinear dynamicequations to be solved under multiple operating conditionswhen evaluating the fitness and the solving time can be toolong to accept Therefore an improved NSGA-II algorithm isbeing put forward to solve the problem of time consumingprediction strategy is used in the fitness evaluation step soas to avoid the evaluation of load sharing property which iscomputationally very expensiveThe key of the improvementis predicting instead of evaluating the real fitness
51The Fitness Prediction Strategy As is known to all NSGA-II is a population based evolutionary algorithm for multi-objective optimization problems and the population evolveswith the generation increasing In the improved NSGA-IIalgorithm each individual 119894 in the population has its fitnessvector
ftn (119894) = [119891tn1
(119894) 119891tn2
(119894) 119891tn119873
(119894)] (32)
and the fidelity 119877(119894) of the fitness vector where 119873 is thenumber of objectives The value of the fitness vector can beevaluated using the fitness functions or predicted throughthe values of other individuals If the fitness vector ftn(119894)
is evaluated using the original fitness functions the fidelity119877(119894) = 1 if ftn(119894) is estimated the fidelity 0 le 119877(119894) lt 1
As shown in Figure 8 for each individual 119894 specify itsfitness sharing radius 119903119904 The area in which the dimensionlessEuclidean distance between individual 119894 and any other oneis no greater than the fitness sharing radius is called fitnesssharing area for individual 119894 expressed as Ω(119894) Assume thereare119898 other individuals in the fitness sharing areaΩ(119894) whichcomposed a collection 119878 = 1199041 1199042 119904119898 And the evaluationmethod of ftn(119894) is as follows
Rs2 fs2Rs1 fs1
Rs3 fs3
Rs4 fs4
Rs5 fs5
rs 1205962 1205961
1205963
1205964
1205965
Evaluating
Predictingsumi
120596ifsiRlowast
Figure 9 Fitness prediction model
Firstly evaluate the fidelity 119877(119894) of individual 119894
119877 (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot 119877 (119904119895) (33)
where 119904119895 is an individual including fitness sharing area Ω(119894)119877(119904119895) is the fidelity of 119904119895 120596(119904119895 119894) is the weight of 119904119895 donated byindividual 119894 Let dimensionless Euclidean distance betweenindividuals 1199041 1199042 119904119898 and individual 119894 be 1198891
119894
1198892119894
119889119898
119894
respectively then 120596(119904119895 119894) can be evaluated by
120596 (119904119895 119894) =
exp (minus120574 sdot 119889119895
119894
)
sum119898
119896=1 exp (minus120574 sdot 119889119896
119894
)
119895 = 1 2 119898 (34)
where 120574 is weight rescaling factorThe closer the individual isto individual 119894 the greater contribution of fidelity it makes
As depicted in Figure 9 if fidelity 119877(119894) is greater thana given threshold 119877
lowast then the fitness vector ftn(119894) can bepredicted as
ftn (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot ftn (119904119895) (35)
Else if119877(119894) is less than the threshold119877lowast then the fitness vector
ftn(119894) should be evaluated using its original fitness functionsTo take full advantage of the historical population whose
fitness has been evaluated it is necessary to establish adatabase of historical populations of coordinates fitness andfidelities As the population evolves the database will expandthe scale gradually In order to reduce space complexityand the amount of computation redundant data need to beeliminated after each generation The concept of individualredundancy is introduced to determine whether the data isredundant which is defined as
119868119903 (119894) =
119899
sum
119896=1Δ119909119896 (119894) (36)
where Δ119909119896(119894) is the coordinate difference (absolute value)of individual 119894 between the former individual and the latterone in the 119896th dimensional and 119899 is the number of design
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
34 35 36 37 38 39 40
Dimensionless time
Mes
h sti
ffnes
s (N
m)
times108
12
10
8
6
4
2
0
11
1
090 05 1
times109
kLkR
Figure 6 Mesh stiffness curves of the first stage in both paths atgiven power and speed
400
24
22
2
18
16
14
12
1Load
shar
ing
coeffi
cien
t
1400012000
100008000
6000Speed (rmin) 200
300
500600
Power (kW)
Load decreasesX 2237Y 8965Z 2136
X 2237Y 5268
Y 5268
Z 1304
Z 1062
X 522
X 522
Y 5638Z 1058
X 4435
Y 12292
Y 12292
Z 1211
X 2237
Z 2106
Figure 7 ldquoLoad sharing maprdquo of the reference design
stiffness of both paths decrease sometimes compared to theerrorless mesh stiffness It means the gear tooth of bothpaths does not always fully contact over the whole theoreticalcontact line Therefore contact length and mesh stiffnesscannot be predicted before the nonlinear model is solvedBesides the phase difference of mesh stiffness can be foundeasily in the figure
The load sharing coefficient 119896119897119904 under different powerand speed is evaluated as is illustrated in Figure 7 It canbe seen that the ldquoload sharing maprdquo (curved surface of loadsharing coefficient 119896119897119904 under different operating conditions)is complicated which comes from the nonlinearity of thesplit torque transmission system The evaluated load sharingcoefficient 119896119897119904 varies from 1058 to 2136 with input powervarying from 2237 kW to 522 kW and speed varying from5268 rmin to 12292 rmin The root mean square (RMS) ofload sharing coefficient 119896119897119904 is 1391 From a global point ofview the load sharing coefficient 119896119897119904 increases with the speedincreasing and power decreasing which correspond with [6]The mesh load of the gear pair is so little under the conditionof high speed and light power that the elastic deformationis not large enough to offset the initial deviations at all thecontact points However it is noteworthy that the law of
load sharing and operating conditions proposed here is notstrictly correct and there might be some counterexamplesIt is because the fact that the actual length of contact linedepends much on gear deviation under light load whichincreases the nonlinearity of system dynamics
4 Mathematical Model of Optimization
41 Objective Function According to the previous sectionthe load sharing coefficient of a split torque transmissionsystem changes greatly with different operating conditionsand the law of changing is complicated Therefore inorder to obtain better load sharing from a system pointof view multiple operating conditions have to be takeninto consideration Considering the average case the firstobjective function is promoted by minimizing the root meansquare of load sharing coefficient under a wide range ofoperating conditions (possible operating conditions) inputpower varying from 2237 kW (60 of 119875in) to 522 kW (140of 119875in) and input speed varying from 5268 rmin (60 of119899in) to 12292 rmin (140 of 119899in) When designing a geartransmission light weight and safety are always importantdesign targets Safety is always measured by safety factorsof contact fatigue strength and bending fatigue strength[24] Therefore the second and the third objective functionscan be promoted by minimizing the total system mass andmaximizing the total safety factors
To sum up the whole objective functions are expressed as
min 1198841 = 119896119897119904RMS
min 1198842 =
6sum
119894=1119872119892119894
min 1198843 = minus sum 119878 = minus (1198781198671 + 1198781198672 + 1198781198651 + 1198781198652)
(23)
where 119896119897119904RMS is the RMS of load sharing coefficient (L-S-CRMS) under the possible operating conditions 119872119892119894 is themass of gear 119894 1198781198671 1198781198672 are the safety factor of contact fatiguestrength of first and second stages and 1198781198651 1198781198652 are the safetyfactor of bending fatigue strength of first and second stagesHere only the mass of gears is considered in the total mass ofthe system
The safety factors of gear pairs can be evaluated accordingto ISO6336 it will not be discussed in detail here Howeverthe calculation of gear mass is a problem for the methodto calculate gear mass is associated with its wheel structureGenerally there are three types of wheel structure solid typepanel type and spoke type The reason to adopt differenttypes of wheel structure is reducing weight as the gear getslarger the more percentage of mass is removed from thewheel Here the light weight coefficient 120578 is introduced tomeasure the extent of light weight and then the gear mass119872119892119894 can be calculated by
119872119892119894 = 120578119894119872lowast
119892119894119894 = 1 2 6 (24)
where 119872lowast
119892119894= 120588(1205874)119887119894119889119894
2 is the solid mass of gear 119894 with 120588the material density 119887119894 the tooth width of gear 119894 and 119889119894 thepitch diameter of gear 119894
8 Mathematical Problems in Engineering
The introduction of light weight coefficient 120578 unifies thedifferent methods to calculate gear mass under differentwheel structures and the difference of three types of wheelstructure is presented by varying the value of 120578 The type ofwheel structure is decided by the tip diameter 119889119886 so the valueof 120578 is directly related to the tip diameter 120578 = 120578(119889119886) Accordingto wheel structure design criteria 3D parameterized modelof a spur gear is created in CATIA V5 system A series of gearmodels are created by varying the tip diameter 119889119886 in a widerange and the masses of them are measured in CATIA V5system and then the values of 120578 for gears with different tipdiameters can be calculated In the process of optimizationthe value of 120578 for a gear is obtained by interpolating accordingto its tip diameter
The values of the three objective functions of the referencedesign are evaluated the root mean square of load sharingcoefficient 119896119897119904RMS is 1391 the system mass is 40910 kg andthe total safety factors is 8966
42 Designing Variables There are a lot of designing param-eters in a split torque transmission system some of themare independent while others are not Picking up appropriateparameters as the designing variables is the prerequisite foroptimization design The special arrangement of the splittorque transmission leads to the special mounting conditionthe proportioning of gear tooth has definite interrelationwiththe two shaft angles [2] Once the proportioning of gear toothand the two shaft angles are determined the center distancesof the two stages are determined at the same time which arerestricted to the center distance between input and outputshaft Therefore the modules of the two stages can hardly bethe standard value In other words in order to guarantee thecorrect arrangement of a split torque transmission system thestandard of modules has to be sacrificed
Based on the above considerations the designing vari-ables selected here includes the gear ratio of the first stage 1198941the pinion tooth number of the first and second stages11988511198854the helix angle of the first stage 1205731198871 and the two shaft anglesΦ1 Φ2 as expressed in (25)
X = 1198941 1198851 1198854 1205731198871 Φ1 Φ2119879
(25)
Other parameters can be evaluated by
1198942 =
11989401198941
1198852 = round (1198941 sdot 1198851)
1198856 = round (1198942 sdot 1198854)
1198981 = 2119867
cos12057311988711198851 + 1198852
sdot
sin (Φ22)
sin (Φ12 + Φ22)
1198982 = 2119867
11198854 + 1198856
sdot
sin (Φ12)
sin (Φ12 + Φ22)
(26)
where 1198940 is the total gear ratio 1198942 is the gear ratio of thesecond stage 1198852 1198856 are the gear tooth numbers of the firstand second stages 1198981 1198982 are the module of the first and
second stages and119867 is the center distance between input andoutput shaftsThe function round(sdot)heremeans round sdot to thenearest integer
43 Constraints
431 Boundary Constraints The design variables meet thefollowing constraints
1198941min le 1198941 le 1198941max
1198851min le 1198851 le 1198851max
1198854min le 1198854 le 1198854max
1205731198871min le 1205731198871 le 1205731198871max
Φ1min le Φ1 le Φ1max
Φ2min le Φ2 le Φ2max
(27)
where 119894111988511198854 1205731198871Φ1Φ2 with the subscripts min andmaxare the boundaries of design variables which are determinedempirically according to the initial design
432 Performance Constraints (i) Contact and bendingfatigue strengths should be below the allowable values
12059011986712 le [12059011986712]
12059011986512 le [12059011986512]
(28)
where 12059011986712 is the contact stress of the first and second stages[12059011986712] is the allowable contact stress of the first and secondstages12059011986512 is the bending stress of the first and second stagesand [12059011986512] is the allowable bending stress of the first andsecond stages
(ii) First stage gears should not interfere with each other
1198981 (1198852 + 2ℎlowast
119886) lt
1198981 (1198851 + 1198852)
cos1205731198871sdot sin(
Φ12
) (29)
where ℎlowast
119886is the addendum factor of the first stage gear
(iii) Safety margin of each stage should be balanced
radic14
4sum
119894=1(Δ119878119894 minus Δ119878)
2lt [120576] (30)
where
Δ119878119894 =
119878119867119894 minus 119878119867min 119894 = 1 2
119878119865119894minus2 minus 119878119865min 119894 = 3 4(31)
is the safety margin with 119878119867min the minimum safety factorfor contact fatigue strength and 119878119865min the minimum safetyfactor for bending fatigue strength Δ119878 is the mean value ofsafety margin Δ119878 [120576] is the allowable upper limit of safetymargin standard deviation
Mathematical Problems in Engineering 9
Ωj
Ωi
j
i
rs
Figure 8 Fitness sharing area
5 Algorithm and Improvement
Classical optimization methods suggest converting the mul-tiobjective optimization problem to a single-objective opti-mization problem by weighted sum of all the objectiveswhich lead to disadvantage of subjectivity when determiningthe weights of objectives Whereas the improved nondomi-nated sorting genetic algorithm (NSGA-II) proposed by Debavoids this disadvantage In this paper NSGA-II is adopted tosolve the proposed multiobjective optimization model
However there are large numbers of nonlinear dynamicequations to be solved under multiple operating conditionswhen evaluating the fitness and the solving time can be toolong to accept Therefore an improved NSGA-II algorithm isbeing put forward to solve the problem of time consumingprediction strategy is used in the fitness evaluation step soas to avoid the evaluation of load sharing property which iscomputationally very expensiveThe key of the improvementis predicting instead of evaluating the real fitness
51The Fitness Prediction Strategy As is known to all NSGA-II is a population based evolutionary algorithm for multi-objective optimization problems and the population evolveswith the generation increasing In the improved NSGA-IIalgorithm each individual 119894 in the population has its fitnessvector
ftn (119894) = [119891tn1
(119894) 119891tn2
(119894) 119891tn119873
(119894)] (32)
and the fidelity 119877(119894) of the fitness vector where 119873 is thenumber of objectives The value of the fitness vector can beevaluated using the fitness functions or predicted throughthe values of other individuals If the fitness vector ftn(119894)
is evaluated using the original fitness functions the fidelity119877(119894) = 1 if ftn(119894) is estimated the fidelity 0 le 119877(119894) lt 1
As shown in Figure 8 for each individual 119894 specify itsfitness sharing radius 119903119904 The area in which the dimensionlessEuclidean distance between individual 119894 and any other oneis no greater than the fitness sharing radius is called fitnesssharing area for individual 119894 expressed as Ω(119894) Assume thereare119898 other individuals in the fitness sharing areaΩ(119894) whichcomposed a collection 119878 = 1199041 1199042 119904119898 And the evaluationmethod of ftn(119894) is as follows
Rs2 fs2Rs1 fs1
Rs3 fs3
Rs4 fs4
Rs5 fs5
rs 1205962 1205961
1205963
1205964
1205965
Evaluating
Predictingsumi
120596ifsiRlowast
Figure 9 Fitness prediction model
Firstly evaluate the fidelity 119877(119894) of individual 119894
119877 (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot 119877 (119904119895) (33)
where 119904119895 is an individual including fitness sharing area Ω(119894)119877(119904119895) is the fidelity of 119904119895 120596(119904119895 119894) is the weight of 119904119895 donated byindividual 119894 Let dimensionless Euclidean distance betweenindividuals 1199041 1199042 119904119898 and individual 119894 be 1198891
119894
1198892119894
119889119898
119894
respectively then 120596(119904119895 119894) can be evaluated by
120596 (119904119895 119894) =
exp (minus120574 sdot 119889119895
119894
)
sum119898
119896=1 exp (minus120574 sdot 119889119896
119894
)
119895 = 1 2 119898 (34)
where 120574 is weight rescaling factorThe closer the individual isto individual 119894 the greater contribution of fidelity it makes
As depicted in Figure 9 if fidelity 119877(119894) is greater thana given threshold 119877
lowast then the fitness vector ftn(119894) can bepredicted as
ftn (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot ftn (119904119895) (35)
Else if119877(119894) is less than the threshold119877lowast then the fitness vector
ftn(119894) should be evaluated using its original fitness functionsTo take full advantage of the historical population whose
fitness has been evaluated it is necessary to establish adatabase of historical populations of coordinates fitness andfidelities As the population evolves the database will expandthe scale gradually In order to reduce space complexityand the amount of computation redundant data need to beeliminated after each generation The concept of individualredundancy is introduced to determine whether the data isredundant which is defined as
119868119903 (119894) =
119899
sum
119896=1Δ119909119896 (119894) (36)
where Δ119909119896(119894) is the coordinate difference (absolute value)of individual 119894 between the former individual and the latterone in the 119896th dimensional and 119899 is the number of design
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
The introduction of light weight coefficient 120578 unifies thedifferent methods to calculate gear mass under differentwheel structures and the difference of three types of wheelstructure is presented by varying the value of 120578 The type ofwheel structure is decided by the tip diameter 119889119886 so the valueof 120578 is directly related to the tip diameter 120578 = 120578(119889119886) Accordingto wheel structure design criteria 3D parameterized modelof a spur gear is created in CATIA V5 system A series of gearmodels are created by varying the tip diameter 119889119886 in a widerange and the masses of them are measured in CATIA V5system and then the values of 120578 for gears with different tipdiameters can be calculated In the process of optimizationthe value of 120578 for a gear is obtained by interpolating accordingto its tip diameter
The values of the three objective functions of the referencedesign are evaluated the root mean square of load sharingcoefficient 119896119897119904RMS is 1391 the system mass is 40910 kg andthe total safety factors is 8966
42 Designing Variables There are a lot of designing param-eters in a split torque transmission system some of themare independent while others are not Picking up appropriateparameters as the designing variables is the prerequisite foroptimization design The special arrangement of the splittorque transmission leads to the special mounting conditionthe proportioning of gear tooth has definite interrelationwiththe two shaft angles [2] Once the proportioning of gear toothand the two shaft angles are determined the center distancesof the two stages are determined at the same time which arerestricted to the center distance between input and outputshaft Therefore the modules of the two stages can hardly bethe standard value In other words in order to guarantee thecorrect arrangement of a split torque transmission system thestandard of modules has to be sacrificed
Based on the above considerations the designing vari-ables selected here includes the gear ratio of the first stage 1198941the pinion tooth number of the first and second stages11988511198854the helix angle of the first stage 1205731198871 and the two shaft anglesΦ1 Φ2 as expressed in (25)
X = 1198941 1198851 1198854 1205731198871 Φ1 Φ2119879
(25)
Other parameters can be evaluated by
1198942 =
11989401198941
1198852 = round (1198941 sdot 1198851)
1198856 = round (1198942 sdot 1198854)
1198981 = 2119867
cos12057311988711198851 + 1198852
sdot
sin (Φ22)
sin (Φ12 + Φ22)
1198982 = 2119867
11198854 + 1198856
sdot
sin (Φ12)
sin (Φ12 + Φ22)
(26)
where 1198940 is the total gear ratio 1198942 is the gear ratio of thesecond stage 1198852 1198856 are the gear tooth numbers of the firstand second stages 1198981 1198982 are the module of the first and
second stages and119867 is the center distance between input andoutput shaftsThe function round(sdot)heremeans round sdot to thenearest integer
43 Constraints
431 Boundary Constraints The design variables meet thefollowing constraints
1198941min le 1198941 le 1198941max
1198851min le 1198851 le 1198851max
1198854min le 1198854 le 1198854max
1205731198871min le 1205731198871 le 1205731198871max
Φ1min le Φ1 le Φ1max
Φ2min le Φ2 le Φ2max
(27)
where 119894111988511198854 1205731198871Φ1Φ2 with the subscripts min andmaxare the boundaries of design variables which are determinedempirically according to the initial design
432 Performance Constraints (i) Contact and bendingfatigue strengths should be below the allowable values
12059011986712 le [12059011986712]
12059011986512 le [12059011986512]
(28)
where 12059011986712 is the contact stress of the first and second stages[12059011986712] is the allowable contact stress of the first and secondstages12059011986512 is the bending stress of the first and second stagesand [12059011986512] is the allowable bending stress of the first andsecond stages
(ii) First stage gears should not interfere with each other
1198981 (1198852 + 2ℎlowast
119886) lt
1198981 (1198851 + 1198852)
cos1205731198871sdot sin(
Φ12
) (29)
where ℎlowast
119886is the addendum factor of the first stage gear
(iii) Safety margin of each stage should be balanced
radic14
4sum
119894=1(Δ119878119894 minus Δ119878)
2lt [120576] (30)
where
Δ119878119894 =
119878119867119894 minus 119878119867min 119894 = 1 2
119878119865119894minus2 minus 119878119865min 119894 = 3 4(31)
is the safety margin with 119878119867min the minimum safety factorfor contact fatigue strength and 119878119865min the minimum safetyfactor for bending fatigue strength Δ119878 is the mean value ofsafety margin Δ119878 [120576] is the allowable upper limit of safetymargin standard deviation
Mathematical Problems in Engineering 9
Ωj
Ωi
j
i
rs
Figure 8 Fitness sharing area
5 Algorithm and Improvement
Classical optimization methods suggest converting the mul-tiobjective optimization problem to a single-objective opti-mization problem by weighted sum of all the objectiveswhich lead to disadvantage of subjectivity when determiningthe weights of objectives Whereas the improved nondomi-nated sorting genetic algorithm (NSGA-II) proposed by Debavoids this disadvantage In this paper NSGA-II is adopted tosolve the proposed multiobjective optimization model
However there are large numbers of nonlinear dynamicequations to be solved under multiple operating conditionswhen evaluating the fitness and the solving time can be toolong to accept Therefore an improved NSGA-II algorithm isbeing put forward to solve the problem of time consumingprediction strategy is used in the fitness evaluation step soas to avoid the evaluation of load sharing property which iscomputationally very expensiveThe key of the improvementis predicting instead of evaluating the real fitness
51The Fitness Prediction Strategy As is known to all NSGA-II is a population based evolutionary algorithm for multi-objective optimization problems and the population evolveswith the generation increasing In the improved NSGA-IIalgorithm each individual 119894 in the population has its fitnessvector
ftn (119894) = [119891tn1
(119894) 119891tn2
(119894) 119891tn119873
(119894)] (32)
and the fidelity 119877(119894) of the fitness vector where 119873 is thenumber of objectives The value of the fitness vector can beevaluated using the fitness functions or predicted throughthe values of other individuals If the fitness vector ftn(119894)
is evaluated using the original fitness functions the fidelity119877(119894) = 1 if ftn(119894) is estimated the fidelity 0 le 119877(119894) lt 1
As shown in Figure 8 for each individual 119894 specify itsfitness sharing radius 119903119904 The area in which the dimensionlessEuclidean distance between individual 119894 and any other oneis no greater than the fitness sharing radius is called fitnesssharing area for individual 119894 expressed as Ω(119894) Assume thereare119898 other individuals in the fitness sharing areaΩ(119894) whichcomposed a collection 119878 = 1199041 1199042 119904119898 And the evaluationmethod of ftn(119894) is as follows
Rs2 fs2Rs1 fs1
Rs3 fs3
Rs4 fs4
Rs5 fs5
rs 1205962 1205961
1205963
1205964
1205965
Evaluating
Predictingsumi
120596ifsiRlowast
Figure 9 Fitness prediction model
Firstly evaluate the fidelity 119877(119894) of individual 119894
119877 (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot 119877 (119904119895) (33)
where 119904119895 is an individual including fitness sharing area Ω(119894)119877(119904119895) is the fidelity of 119904119895 120596(119904119895 119894) is the weight of 119904119895 donated byindividual 119894 Let dimensionless Euclidean distance betweenindividuals 1199041 1199042 119904119898 and individual 119894 be 1198891
119894
1198892119894
119889119898
119894
respectively then 120596(119904119895 119894) can be evaluated by
120596 (119904119895 119894) =
exp (minus120574 sdot 119889119895
119894
)
sum119898
119896=1 exp (minus120574 sdot 119889119896
119894
)
119895 = 1 2 119898 (34)
where 120574 is weight rescaling factorThe closer the individual isto individual 119894 the greater contribution of fidelity it makes
As depicted in Figure 9 if fidelity 119877(119894) is greater thana given threshold 119877
lowast then the fitness vector ftn(119894) can bepredicted as
ftn (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot ftn (119904119895) (35)
Else if119877(119894) is less than the threshold119877lowast then the fitness vector
ftn(119894) should be evaluated using its original fitness functionsTo take full advantage of the historical population whose
fitness has been evaluated it is necessary to establish adatabase of historical populations of coordinates fitness andfidelities As the population evolves the database will expandthe scale gradually In order to reduce space complexityand the amount of computation redundant data need to beeliminated after each generation The concept of individualredundancy is introduced to determine whether the data isredundant which is defined as
119868119903 (119894) =
119899
sum
119896=1Δ119909119896 (119894) (36)
where Δ119909119896(119894) is the coordinate difference (absolute value)of individual 119894 between the former individual and the latterone in the 119896th dimensional and 119899 is the number of design
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Ωj
Ωi
j
i
rs
Figure 8 Fitness sharing area
5 Algorithm and Improvement
Classical optimization methods suggest converting the mul-tiobjective optimization problem to a single-objective opti-mization problem by weighted sum of all the objectiveswhich lead to disadvantage of subjectivity when determiningthe weights of objectives Whereas the improved nondomi-nated sorting genetic algorithm (NSGA-II) proposed by Debavoids this disadvantage In this paper NSGA-II is adopted tosolve the proposed multiobjective optimization model
However there are large numbers of nonlinear dynamicequations to be solved under multiple operating conditionswhen evaluating the fitness and the solving time can be toolong to accept Therefore an improved NSGA-II algorithm isbeing put forward to solve the problem of time consumingprediction strategy is used in the fitness evaluation step soas to avoid the evaluation of load sharing property which iscomputationally very expensiveThe key of the improvementis predicting instead of evaluating the real fitness
51The Fitness Prediction Strategy As is known to all NSGA-II is a population based evolutionary algorithm for multi-objective optimization problems and the population evolveswith the generation increasing In the improved NSGA-IIalgorithm each individual 119894 in the population has its fitnessvector
ftn (119894) = [119891tn1
(119894) 119891tn2
(119894) 119891tn119873
(119894)] (32)
and the fidelity 119877(119894) of the fitness vector where 119873 is thenumber of objectives The value of the fitness vector can beevaluated using the fitness functions or predicted throughthe values of other individuals If the fitness vector ftn(119894)
is evaluated using the original fitness functions the fidelity119877(119894) = 1 if ftn(119894) is estimated the fidelity 0 le 119877(119894) lt 1
As shown in Figure 8 for each individual 119894 specify itsfitness sharing radius 119903119904 The area in which the dimensionlessEuclidean distance between individual 119894 and any other oneis no greater than the fitness sharing radius is called fitnesssharing area for individual 119894 expressed as Ω(119894) Assume thereare119898 other individuals in the fitness sharing areaΩ(119894) whichcomposed a collection 119878 = 1199041 1199042 119904119898 And the evaluationmethod of ftn(119894) is as follows
Rs2 fs2Rs1 fs1
Rs3 fs3
Rs4 fs4
Rs5 fs5
rs 1205962 1205961
1205963
1205964
1205965
Evaluating
Predictingsumi
120596ifsiRlowast
Figure 9 Fitness prediction model
Firstly evaluate the fidelity 119877(119894) of individual 119894
119877 (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot 119877 (119904119895) (33)
where 119904119895 is an individual including fitness sharing area Ω(119894)119877(119904119895) is the fidelity of 119904119895 120596(119904119895 119894) is the weight of 119904119895 donated byindividual 119894 Let dimensionless Euclidean distance betweenindividuals 1199041 1199042 119904119898 and individual 119894 be 1198891
119894
1198892119894
119889119898
119894
respectively then 120596(119904119895 119894) can be evaluated by
120596 (119904119895 119894) =
exp (minus120574 sdot 119889119895
119894
)
sum119898
119896=1 exp (minus120574 sdot 119889119896
119894
)
119895 = 1 2 119898 (34)
where 120574 is weight rescaling factorThe closer the individual isto individual 119894 the greater contribution of fidelity it makes
As depicted in Figure 9 if fidelity 119877(119894) is greater thana given threshold 119877
lowast then the fitness vector ftn(119894) can bepredicted as
ftn (119894) =
119898
sum
119895=1120596 (119904119895 119894) sdot ftn (119904119895) (35)
Else if119877(119894) is less than the threshold119877lowast then the fitness vector
ftn(119894) should be evaluated using its original fitness functionsTo take full advantage of the historical population whose
fitness has been evaluated it is necessary to establish adatabase of historical populations of coordinates fitness andfidelities As the population evolves the database will expandthe scale gradually In order to reduce space complexityand the amount of computation redundant data need to beeliminated after each generation The concept of individualredundancy is introduced to determine whether the data isredundant which is defined as
119868119903 (119894) =
119899
sum
119896=1Δ119909119896 (119894) (36)
where Δ119909119896(119894) is the coordinate difference (absolute value)of individual 119894 between the former individual and the latterone in the 119896th dimensional and 119899 is the number of design
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
parameters If the redundancy value of an individual is lessthan a given threshold the individual is knocked out
In addition since not all of the individualsrsquo fitnessvectors are evaluated using its original fitness functionsthe predicted fitness vectors are not accurate Thus as thepopulation evolves gradually the fidelities of individuals withpredicted fitness vectors should decline gradually Assumethat individual 119894 is with the predicted fitness vectors let thefidelity of individual 119894 in generation 119905 be 119877(119894 119905) and then thefidelity in generation 119905 + 1 can be updated as
119877 (119894 119905 + 1) = 120573 sdot 119877 (119894 119905) (37)
where 120573 is fidelity drain factor with 0 lt 120573 lt 1 As the popu-lation evolves the fidelity drops below a given threshold 1198770and the individual should also be removed from the database
52 Algorithm Flow
Step 1 Initialize historical population database set the initialpopulation blank and set fitness vectors and fidelities to 0
Step 2 Find the fitness sharing area for each individual 119894and find the collection of individuals in the area from thedatabase
Step 3 Evaluate the fidelity 119877(119894) of individual 119894 and deter-mine whether 119877(119894) is greater than the threshold 119877
lowast If119877(119894) ge 119877
lowast predict fitness vector of individual 119894 according to(35) otherwise evaluate the fitness vector using its originalfunctions and set the fidelity 119877(119894) to 1
Step 4 Add individual 119894 to the database
Step 5 Update the database as follows (1) calculate redun-dancy for all individuals and eliminate all redundant indi-viduals (2) for all individuals with predicted fitness vectorsupdate its fidelity according to (37) and remove all theindividuals with low fidelity
53 Numerical Experiments There are two purposes of con-ducting numerical experiments on the improved NSGA-II(iNSGA-II) (1) testing convergence of the algorithm whichtests whether the algorithm can correctly guide the evolutionso that the Pareto optimal solution (Pareto front) of theoriginal problem can be obtained (2) testing the effectivenessof the algorithm namely testing what extent the algorithmcan reduce the amount of computation
Four benchmark problems are chosen from a numberof significant past studies in multiobjective optimizationarea Schafferrsquos study (SCH) [25] Fonseca and Flemingrsquosstudy (FON) [26] Polonirsquos study (POL) [27] and Kursawersquosstudy (KUR) [28] The benchmark problems are described asfollows
(1) SCH Problem (119899 = 1)
min 1198911 (119909) = 1199092
min 1198912 (119909) = (119909 minus 2)2
(38)
where the variable bound is [minus103 103] and the Paretooptimal front is convex
(2) FON Problem (119899 = 3)
min 1198911 (x) = 1minus exp(minus
3sum
119894=1(119909119894 minus
1radic3
)
2)
min 1198912 (x) = 1minus exp(minus
3sum
119894=1(119909119894 +
1radic3
)
2)
(39)
where the variable bounds are [minus4 4] and the Pareto optimalfront is nonconvex
(3) POL Problem (119899 = 2)
min 1198911 (x) = 1+ (1198601 minus 1198611)2
+ (1198602 minus 1198612)2
min 1198912 (x) = (1199091 + 3)2
+ (1199092 + 1)2
(40)
where
1198601 = 05 sin 1minus 2 cos 1+ sin 2minus 15 cos 2
1198602 = 15 sin 1minus cos 1+ 2 sin 2minus 05 cos 2
1198611 = 05 sin1199091 minus 2 cos1199091 + sin1199092 minus 15 cos1199092
1198612 = 15 sin1199091 minus cos1199091 + 2 sin1199092 minus 05 cos1199092
(41)
and the variable bounds are [minus120587 120587] and the Pareto optimalfront is nonconvex and disconnected
(4) KUR Problem (119899 = 3)
min 1198911 (x) =
119899minus1sum
119894=1(minus10 exp (minus02radic119909
2119894
+ 1199092119894+1))
min 1198912 (x) =
119899
sum
119894=1(1003816100381610038161003816119909119894
1003816100381610038161003816
08+ 5 sin119909
3119894)
(42)
where the variable bounds are [minus5 5] and the Pareto optimalfront is nonconvex and disconnected
The four benchmark problems are solved by iNSGA-IIwith MATLAB programming Binary-coding single-pointcrossover and bitwise mutation are used in the algorithmThe algorithm parameters are settled as follows populationsize is 100 evolution generation is 200 the crossover prob-ability is 09 the mutation probability is 01 threshold 119877
lowast=
06 and fidelity drain factor 120573 = 09 Each problem is tested20 times respectively
Pareto optimal fronts of the four benchmark problemswith iNSGA-II are illustrated in Figure 10 where (a) (b)(c) and (d) represent SCH problem FON problem POLproblem and KUR problem respectively It can be seen thatthe improvedNSGA-II algorithm (iNSGA-II) achieves Paretofronts correctly in the four benchmark problems
Percentage of real fitness vectors evaluated in each gen-eration of the four benchmark problems are representedin Figure 11 where (a) (b) (c) and (d) represent SCH
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
4
3
2
1
043210
f2
f1
(a) SCH (convex)
f2
f1
1080604020
1
08
06
04
02
0
(b) FON (nonconvex)
f2
f1
25
20
15
10
5
0
151050
(c) POL (nonconvex and disconnected)
f2
f1
2
0
minus2
minus4
minus6
minus8
minus10
minus12minus20 minus18 minus16 minus14
(d) KUR (nonconvex and disconnected)
Figure 10 Pareto optimal fronts of the four benchmark problems with iNSGA-II
Table 2 Effectiveness test results of the algorithm
Problems MaxNum EvalNum PercentageSCH 20000 71675 35838FON 20000 75605 37803POL 20000 73825 36913KUR 20000 721205 36060
problem FON problem POL problem and KUR problemrespectively Percentage of real fitness evaluated in eachgeneration substantially stabilized at 30 to 50 A certainpercentage of the individualsrsquo fitness vectors is evaluatedusing the original fitness functions in each generation so thatthe evolutionary direction can be guided correctly
The detailed testing results of effectiveness are illustratedin Table 2 where ldquoMaxNumrdquo represents themaximum num-ber of fitness vectors to be evaluated or predicted ldquoEvalNumrdquo
represents the average number of evaluated fitness vectorsand ldquoPercentagerdquo represents the percentage of ldquoMaxNumrdquoand ldquoEvalNumrdquo It can be known from Table 2 that iNSGA-IIcan reduce much computation amount of real fitness com-pared with traditional NSGA-II It means that when the realfitness evaluation is computationally very expensive usingiNSGA-II can save approximately 23 of the computing time
6 Results
The improved NSGA-II algorithm for solving the multi-objective optimization model is realized by MATLAB pro-gramming As is known to all there are a set of optimalsolutions (largely known as Pareto optimal solutions) ina multiobjective optimization problem instead of a singleoptimal solution The Pareto optimal solutions form a Paretooptimal front which has the property that one solution in thePareto optimal front cannot be said to be better than any of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(a) SCH
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(b) FON
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(c) POL
60
50
40
30
20
10
00 50 100 150 200
Generation
Eval
num
(d) KUR
Figure 11 Percentage of real fitness evaluated
the others [21] Therefore when the Pareto optimal front isobtained designers can choose any solution in it according todifferent requirements and favors The Pareto optimal frontfor two objective problems is always a continuous curvewhereas for three objective problems it is a curved surface
Pareto optimal front (blue points) for the three objec-tive problems of split torque transmission is illustrated inFigure 12 In order to observe the Pareto optimal frontconveniently spatial curved surface fitting for the Paretooptimal solution points is applied As shown in Figure 12coordinate 1198841 is the L-S-C RMS 119896119897119904RMS coordinate 1198842 isthe total mass sum 119872 of the system and coordinate 1198843 is theopposite number of total safety factors sum 119878
In order to choose an optimal design conveniently fromthe Pareto optimal front the curved surface of the frontis projected onto a 2D plane as shown in Figure 13 The119909-coordinate is the total mass of the system sum 119872 the 119910-coordinate is the opposite number of total safety factors sum 119878and the information of L-S-C RMS 119896119897119904RMS is expressed bycolor As Figure 13 illustrates the Pareto optimal front is
divided into 10 belt areas and L-S-C RMS 119896119897119904RMS of each areadecreases from area (a) to area (j) The distributions of valuesof the three objective functions in the Pareto optimal frontare as follows the L-S-C RMS 119896119897119904RMS varies from 1166 to1709 the total mass of the system sum 119872 varies from 33513 kgto 50448 kg and the total safety factors sum 119878 varies from7796 to 14450 As is mentioned above one solution in thePareto optimal front cannot be said to be better than any ofthe others therefore any solution can be chosen accordingto designersrsquo requirements and favors If the load sharingproperty is biased then areas (i) and (j) will be suitable if thelightweight is biased then areas (e) and (f) will be selected ifthe safety is biased then areas (a) and (b) will be the answerBesides designer can also judge and weigh among the threeobjectives and then make an optimal choice
To promise that the optimal design is better than thereferance design in all the three objectives all the unqualifiedsolutions are eliminated and then the left solutions can befurther divided based on RMS of load sharing coefficient AsFigure 14 illustrates there are still 8 optimal solutions which
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
18
16
14
12
1Y1
load
shar
ing
coeffi
cien
t
35
40
45
50
Y2 m
ass (kg)
minus14 minus13 minus12 minus11 minus10 minus9 minus8
Y3 safety factor
Figure 12 The Pareto optimal front
minus8
minus9
minus10
minus11
minus12
minus13
minus14
35 40 45 50
Safe
ty fa
ctor
Mass (kg)
16
15
14
13
12
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(i)
(h)
(j)
Figure 13 Projected Pareto optimal front
are better than the referance one in all the three objectivesFour alternative solutions 1sim4 are selected with 1 thesolution with minimum L-S-C RMS 2 the solution withmaximum total safety factor 3 the solution with minimumtotal mass of the system and 4 the optimal solution whichweighs among the three objectives The values of designparameters and objective functions of solutions 1sim4 arelisted in Table 3
Solution 3 is selected as the final optimal design throughbalancing with L-S-C RMS deceasing from 1391 to 1317 thesystem mass deceasing from 40910 kg to 36338 kg and thetotal safety factors increasing from 8966 to 9253 Figure 15illustrates the ldquoload sharingmaprdquo of solution 3 (the colorizedone) and it can be seen that the nonlinearity still exists in theoptimal design The load sharing coefficient 119896119897119904 of solution3 under possible operating conditions varies from 1032 to
Safe
ty fa
ctor
Mass (kg)
minus92
minus94
minus96
minus98
minus10
minus102
minus104
37 38 39 40
138
137
136
135
134
133
132
131
13
1292
1
4
3
Figure 14 Further divided optimal solutions
200 250 300 350 400 450 500 5506000
1000014000
1
15
2
25
Y 12292X 2237
Z 2023
Y 5268X 2237
Z 1125
X 522Y 12292Z 1057 X 522
Y 7486Z 1032
X 522Y 5268Z 1039
Power (kW)Speed (rmin)
Load
shar
ing
coeffi
cien
t
The optimaldesign
The reference design
Figure 15 Load sharing map of 3 and the reference design
2023 which under the normal work condition is 1083 Com-paredwith the ldquoload sharingmaprdquo of the reference design (thegray translucent one) the optimal design achieves better loadsharing property under wide range of operating conditions
7 Conclusions
Gap and phase differences are two of the mainly causes ofunequaled torques in a split torque transmission system Var-iousmethods have been proposed to compensate for ormini-mize the effect of gap promoting the load sharing property toa certain extent In this paper system design parameters wereoptimized to change the phase difference thereby furtherimproving the load sharing property Therefore a multiob-jective optimization design of a split torque transmissionsystem was conducted with the promoting of load sharingproperty lightweight and safety considered as the objectivesThe load sharing property which was measured by loadsharing coefficient was evaluated under multiple operatingconditions with dynamic analysis method An improvedNSGA-II algorithm was adopted to solve the multiobjectivemodel and a satisfied optimal solution was picked up as thefinal optimal design from the Pareto optimal front
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
Table 3 The values of design parameters and objective functions of solutions 1sim4
Parameters 1 2 3 4First stage
Tooth number 11988511198852 28119 2691 26105 26105Module 1198981mm 1538 1861 1683 1694Gear ratio 1198941 425 35 4038 4038Helix angle 1205731198871(
∘) 18900 14296 18196 17402Second stage
Tooth number 11988541198856 22131 22159 22138 22138Module 1198982mm 3270 2981 3095 3119Gear ratio 1198942 5955 7227 6273 6273
Shaft angle Φ1Φ2 (∘) 11474147439 13364145018 11161545617 11358945899
Phase difference Δ1205931Δ1205932 165588411 192959145 166987559 168397709L-S-C RMS 119896lsRMS 1287 1381 1317 1318Total mass sum 119872kg 39066 40516 36338 37088Total safety factor sum 119878 9279 10582 9253 9367
The main contributions of this paper include the follow-ing
(1) Anewmethod for promoting load sharing property ofa split torque transmission system is proposed whichchanges the phase difference by optimizing the systemdesign parametersThis method is proved effective bythe results with the reference design and can be usedwith other load sharing methods together
(2) The concept of ldquoload sharing maprdquo is put forwardto measure the load sharing property under multipleoperating conditions which allows us to find anoptimized design over a wide range of operatingconditions
(3) An improved NSGA-II is proposed to overcome theproblem of time consuming during the process offitness evaluationwhich promotes the effectiveness ofoptimizationwith nearly 23 of the total computationsreduced
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] G White ldquoDesign study of a 375 kW helicopter transmissionwith split torque epicyclic and bevel drive stagesrdquo Proceedingsof the Institution of Mechanical Engineers Part C MechanicalEngineering Science vol 197 pp 213ndash224 1983
[2] J G Kish ldquoSikorsky aircraft advanced rotorcraft transmission(ART) programmdashfinal reportrdquo NASA CR-191079 NASA LewisResearch Center Cleveland Ohio USA 1993
[3] T L Krantz ldquoDynamics of a split torque helicopter trans-missionrdquo NASA TM-106410 NASA Lewis Research CenterCleveland Ohio USA 1994
[4] T L Krantz ldquoAmethod to analyze and optimize the load sharingof split path transmissionsrdquo NASA TM-107201 NASA LewisResearch Center Cleveland Ohio USA 1996
[5] T Hayashi X Y Li I Hayashi K Endo and W WatanabeldquoMeasurement and some discussions on dynamic load sharingin planetary gearsrdquo Bulletin of JSME vol 29 no 253 pp 2290ndash2297 1986
[6] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994
[7] Y Guo J Keller and W LaCava ldquoCombined effects of gravitybending moment bearing clearance and input torque on windturbine planetary gear load sharingrdquo in Proceedings of theAGMA Fall Technical Meeting NRELCP-5000-55968 Dear-born Mich USA October 2012
[8] A Kahraman ldquoStatic load sharing characteristics of transmis-sion planetary gear sets model and experimentrdquo SAE TechnicalPaper 1999-01-1050 1999
[9] A Bodas and A Kahraman ldquoInfluence of carrier and gearmanufacturing errors on the static load sharing behavior ofplanetary gear setsrdquo JSME International Journal Series CMechanical Systems Machine Elements amp Manufacturing vol47 no 3 pp 908ndash915 2004
[10] A Singh ldquoApplication of a system level model to study theplanetary load sharing behaviorrdquo Transactions of the ASMEJournal of Mechanical Design vol 127 no 3 pp 469ndash476 2005
[11] 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 Designvol 130 no 4 Article ID 041701 2008
[12] A Singh A Kahraman and H Ligata ldquoInternal gear strainsand load sharing in planetary transmissions model and experi-mentsrdquoTransactions of the ASME Journal ofMechanical Designvol 130 no 7 Article ID 072602 10 pages 2008
[13] H Ligata A Kahraman and A Singh ldquoA closed-form planetload sharing formulation for planetary gear sets using a trans-lational analogyrdquo Journal of Mechanical Design vol 131 no 2Article ID 021007 2009
[14] A Singh ldquoLoad sharing behavior in epicyclic gears physi-cal explanation and generalized formulationrdquo Mechanism andMachine Theory vol 45 no 3 pp 511ndash530 2010
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
[15] A Singh ldquoEpicyclic load sharing map-development and valida-tionrdquo Mechanism and Machine Theory vol 46 no 5 pp 632ndash646 2011
[16] A Segade-Robleda J A Vilan-Vilan M Lopez-Lago et alldquoSplit torque gearboxes requirements performance and appli-cationsrdquo inMechanical Engineering M Gokcek Ed pp 56ndash74In Tech Rijeka Croatia 2012
[17] T L Krantz and I R Delgado ldquoExperimental study of split-pathtransmission load sharingrdquo NASA TM-107202 NASA LewisResearch Center Cleveland Ohio USA 1996
[18] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010
[19] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000
[20] V S Kumar D V Muni and G Muthuveerappan ldquoOptimiza-tion of asymmetric spur gear drives to improve the bendingload capacityrdquo Mechanism and Machine Theory vol 43 no 7pp 829ndash858 2008
[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multi-objective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002
[22] K Deb and S Karthik ldquoDynamic multi-objective optimizationand decision-making using modified NSGA-II a case studyon hydro-thermal power schedulingrdquo in Evolutionary Multi-Criterion Optimization S Obayashi K Deb C Poloni THiroyasu and T Murata Eds vol 4403 of Lecture Notes inComputer Science pp 803ndash817 Springer Berlin Germany 2007
[23] P Velex and M Maatar ldquoA mathematical model for analyzingthe influence of shape deviations and mounting errors on geardynamic behaviourrdquo Journal of Sound andVibration vol 191 no5 pp 629ndash660 1996
[24] N Marjanovic B Isailovic V Marjanovic Z Milojevic MBlagojevic andM Bojic ldquoA practical approach to the optimiza-tion of gear trains with spur gearsrdquo Mechanism and MachineTheory vol 53 pp 1ndash16 2012
[25] J D Schaffer ldquoMultiple objective optimization with vector eval-uated genetic algorithmsrdquo in Proceedings of the 1st InternationalConference on Genetic Algorithms pp 93ndash100 Pittsburgh PaUSA July 1985
[26] C M Fonseca and P J Fleming ldquoMultiobjective optimizationandmultiple constraint handling with evolutionary algorithmsII Application examplerdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 28 no 1 pp38ndash47 1998
[27] C Poloni ldquoHybrid GA for multi objective aerodynamic shapeoptimisationrdquo in Genetic Algorithms in Engineering And Com-puter Science GWinter J Periaux andMGalan Eds pp 397ndash414 John Wiley amp Sons New York NY USA 1996
[28] F Kursawe ldquoA variant of evolution strategies for vector opti-mizationrdquo in Parallel Problem Solving from Nature H P Schwe-fel and R Manner Eds vol 496 of Lecture Notes in ComputerScience pp 193ndash197 Springer Berlin Germany 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of