research article multiobjective optimization of steering...

Research ArticleMultiobjective Optimization of Steering Mechanism for RotarySteering System Using Modified NSGA-II and Fuzzy Set Theory

Hongtao Li1 Wentie Niu1 Shengli Fu2 and Dawei Zhang1

1Key Laboratory of MechanismTheory and Equipment Design of Ministry of Education Tianjin University Tianjin 300072 China2CNPC Bohai Drilling Engineering Company Limited Tianjin 300457 China

Correspondence should be addressed to Wentie Niu niuwentietjueducn

Received 28 January 2015 Accepted 11 May 2015

Academic Editor Mitsuhiro Okayasu

Copyright copy 2015 Hongtao Li 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

Due to the complicated design process of gear train optimization is a significant approach to improve design efficiencyHowever thedesign of gear train is a complex multiobjective optimization with mixed continuous-discrete variables under numerous nonlinearconstraints and conventional optimization algorithms are not suitable to deal with such optimization problems In this paperbased on the established dynamic model of steering mechanism for rotary steering system the key component of which is aplanetary gear setwith teeth number difference the optimization problemof steeringmechanism is formulated to achieveminimumdynamic responses and outer diameter by optimizing structural parameters under geometric kinematic and strength constraintsAnoptimization procedure based onmodifiedNSGA-II by incorporating dynamic crowding distance strategies and fuzzy set theoryis applied to the multiobjective optimization For comparative purpose NSGA-II is also employed to obtain Pareto optimal set anddynamic responses of original and optimized designs are compared The results show the optimized design has better dynamicresponses with minimum outer diameter and the response decay decreases faster The optimization procedure is feasible to thedesign of gear train and this study can provide guidance for designer at the preliminary design phase of mechanical structures withgear train

1 Introduction

Gear trains are widely used in mechanical engineering foradvantages of compact structure high reliability and largepower transmission However the design of gear trains is acomplex process and the traditional design process of geartrains depends on the designerrsquos intuition experience andskills which is not satisfactory to the increasing demandsfor compactness efficiency and reliability in engineeringapplication Therefore the optimization for gear trains hasbeen a necessary process to solve the above problems at thepreliminary design phase of gear trains and many differentoptimization techniques have been reported in the literatureson gear trains

The sequential quadratic programming (SQP) methodwas employed respectively by Bozca [1] and Huang et al [2]to obtain a light-weight-gearbox structure by optimizing thegeometric parameters of the gearbox Chong et al described

a method for reduction of geometrical volume and meshingvibration of cylindrical gear pairs while satisfying strengthand geometric constraints using a goal programming for-mulation [3] Based on the random search method Zarefarand Muthukrishnan investigated the optimization of helicalgear design [4] Ciavarella and Demelio investigated theoptimization of stress concentration specific sliding andfatigue life of gears with numerical methods [5] Huang etal developed an interactive physical programming in orderto optimize a three-stage spur gear reduction unit [6] ARandom-Simplex optimization algorithm was developed byFaggioni et al for gear vibration reduction bymeans of profilemodifications [7] Based onmin-maxmethod combinedwitha direct search technique Abuid and Ameen had done theoptimization problem containing seven objective functionsgear volume center distance and five dynamic factors ofshafts and gears [8] Thompson et al optimized minimum

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 875872 13 pageshttpdxdoiorg1011552015875872

2 Mathematical Problems in Engineering

volume and surface fatigue of multistage spur gear reductionunits by employing quasi-Newton method [9]

The mentioned optimization algorithms above are cata-loged as conventional optimization techniques Though theyare efficient for some optimization problems in applicationdifficulties still exist in tackling some special problems withnoncontinuous variables complex constraints and stronglynonlinear objectives Therefore some modern optimizationmethods such as genetic algorithm (GA) have been proposedto solve such problems in gear trains By using the geneticalgorithmMendi et al investigated optimization of themod-ulus of spur gears the diameters of shafts and rolling bearing[10] Chong and Lee presented a design method to optimizethe volume of two stage gear trains by using the geneticalgorithm which shows that the genetic algorithm is betterthan other conventional algorithms for solving the discreteinteger variable and continuous problems [11] Gologlu andZeyveli introduced an automated preliminary design of geardrives by minimizing volume of gear trains using a geneticalgorithm and static and dynamic penalty functions werepresented to the objective function for handling the designconstraints [12] Marcelin conducted optimum design ofgears by GAs and penalty selection methods [13] Buiga andTudose investigated mass minimization design of a two-stage coaxial helical speed reducer with genetic algorithms[14] Ciglaric and Kidric conducted automatic dynamicoptimization of a gear pair by using genetic programmingalgorithm [15] Bonori et al performed profile modificationoptimization of spur gears by means of genetic algorithmswhich aimed to reduce vibration and noise [16 17]

However most real-world optimization problems involvemultiple objectives with mixed continuous-discrete variablesunder nonlinear constraints and the conventional optimiza-tion techniques are not developed for such optimizationproblems in mind Multiple objectives have to be reformu-lated as a single objective by weighted-sum approaches priorto optimization in practice and the optimization programwill be performed repeatedly by changing weighting factorsto obtain Pareto optimal frontMeanwhile the nonlinear con-straints are handled by introducing penalty coefficients andthe inappropriate selection of penalty coefficients will leadto nonconvergence Therefore multiobjective evolutionaryalgorithms (MOEAs) as NSGA-II [18] have been developedto solve these problems Deb and Jain demonstrated theuse of NSGA-II in solving the original problem involvingmixed discrete and real-valued parameters and more thanone objective [19] Sanghvi et al investigated multiobjectiveoptimization of a two-stage helical gear train by usingNSGA-II [20]

Though NSGA-II is considered as a successful mul-tiobjective optimization algorithm some drawbacks havebeen discovered as lack of uniform diversity in obtainednondominated solutions To solve such problem Luo et alpresented dynamic crowding distance (DCD) [21] which canbe incorporated in NSGA-II as modified NSGA-II (MNSGA-II)This optimization algorithmhas never been applied in thedesign of gear trains

In this paperMNSGA-II is applied tomultiobjective opti-mization of a novel steering mechanism for rotary steering

M1

M2

1

23

3

2

6

104

5

7

8 9 11

12F

s

T1

T2

Figure 1 Schematic diagram of steering mechanism

system (RSS) the key component of which is a planetary gearset with teeth number difference (PGSTND) [22] This studyaims to minimize the dynamic responses and outer diam-eter of steering mechanism with structural parameters asdesign variables subject to geometric kinematic and strengthconstraints Based on the established dynamic model theoptimization problem is formulated and both MNSGA-II and NSGA-II are applied to the optimization problemMeanwhile in order to avoid human interference in selectionof the best solution from Pareto optimal solutions whenthe objective preferences are absent the best compromisesolution is obtained by fuzzy set theory (FST) and thedynamic responses of optimized and original designs arecompared and analyzed

The rest of this paper is organized as follows the problemformulation is established in Section 2 The optimizationprocedure based on MNSGA-II and FST is introduced andapplied to the multiobjective optimization of steering mech-anism for RSS in Section 3 Comparisons between resultsby MNSGA-II and NSGA-II are conducted and the bestcompromise solution is obtained by FST and compared withthe original design in Section 4 Finally main conclusions aredrawn in Section 5

2 Problem Formulation

The studied object is a steering mechanism for RSS the keycomponents of which are a specially designed PGSTND andtwo servo motors [22] The schematic diagram of steeringmechanism is illustrated in Figure 1

In Figure 1 M1 and M2 denote DC servo motors 1 and 2respectively 1 and 2 respectively denote driving and drivengears for rotation 3 denotes annulus gear 23 denotes aparallel gear consisting of gear 2 and gear 3 4 denotes planetgear 5 and 6 respectively denote driving and driven gearsfor revolution 7 denotes drill-string 8 denotes universaljoint 9 denotes spherical plain bearing 10 denotes drillingmandrel 11 denotes spherical roller bearing 12 denotes drillbit 119865 denotes external excitation 119878 denotes distance betweenrotating axis of gear 6 and eccentric axis of gear 4

In drilling process the steering mechanism is inclinedto fail due to the dynamic load which is related not onlyto operation conditions but also to the structural dynamic

Mathematical Problems in Engineering 3

x5

x4

x6x23

y5

1205795

y6

y4

y23

1205796

1205794

12057923

Gear 5

C5y

K5xK5y

C56

C34

C12

x1

y1 Gear 1

Gear 6

Gear 23

Gear 4

C5x

1205791

C1y

C4y

K1x K6x

K1y

K4y

C6y K6y

C1x C6xK12

K56

K34

C23y K23y

C23x

C4x

K4x

K23xO1

O4O5O23(O6)

Figure 2 Dynamic model of steering mechanism

characteristics of steering mechanism Therefore it is of sig-nificance to optimize the dynamic characteristics of steeringmechanism for high reliability and long lifetime Meanwhilein consideration of the requirements for borehole size andannulus the outer diameter of steeringmechanism should beas small as possible In this section based on the establisheddynamic model of steering mechanism the optimizationproblem is formulated for minimum dynamic responsesand outer diameter of steering mechanism with structuralparameters as design variables under geometric kinematicand strength constraints

21 Dynamic Modeling

211 Dynamic Model The lumped-parameter method isemployed to establish the equivalent dynamic model ofsteering mechanism as shown in Figure 2 Each componenthas 3 degrees of freedom consisting of one angular dis-placement and two transverse displacements Gear meshinginteraction is modeled with time-varying meshing stiffnessand damping The backlash and dynamic transmission errorare not considered in this study The deformations of allbearings under load are represented by stiffness and dampingbetween bodies and their housings in both119909 and119910 directions

212 Equations of Motion (EOMs) By taking one angulardisplacement and two transverse displacements into consid-eration EOMs of each gear are respectively derived by usingLagrange function as follows

11986811205791 + (1198621212 +11987012 (119905) 11990912) 1199031198871 = 1198791

11989811 +11986211199091 +11987011199091199091

+ (1198621212 +11987012 (119905) 11990912) sin120572 = 0

1198981 1199101 +1198621119910 1199101 +11987011199101199101

+ (1198621212 +11987012 (119905) 11990912) cos120572 = 0

1198682312057923 minus (1198621212 +11987012 (119905) 11990912) 1199031198872

+ (1198623434 +11987034 (119905) 11990934) 1199031198873 = 0

1198982323 +1198622311990923 +1198702311990911990923

minus (1198621212 +11987012 (119905) 11990912) sin120572

+ (1198623434 +11987034 (119905) 11990934) sin (120579 minus 12057234) = 0

11989823 11991023 +11986223119910 11991023 +1198702311991011991023

minus (1198621212 +11987012 (119905) 11990912) cos120572

+ (1198623434 +11987034 (119905) 11990934) cos (120579 minus 12057234) = 0

11986841205794 minus (1198623434 +11987034 (119905) 11990934) 1199031198874 = minus119865119878

11989844 +11986241199094 +11987041199091199094

+ (1198623434 +11987034 (119905) 11990934) sin12057234 = 0

1198984 1199104 +1198624119910 1199104 +11987041199101199104

minus (1198623434 +11987034 (119905) 11990934) cos12057234 = 0

11986851205795 + (1198625656 +11987056 (119905) 11990956) 1199031198875 = 1198792

11989855 +11986251199095 +11987051199091199095

+ (1198625656 +11987056 (119905) 11990956) sin120572 = 0

1198985 1199105 +1198625119910 1199105 +11987051199101199105

+ (1198625656 +11987056 (119905) 11990956) cos120572 = 0


11986861205796 minus (1198625656 +11987056 (119905) 11990956) 1199031198876

+ (1198623434 +11987034 (119905) 11990934) (1199031198873 minus 1199031198874) = 0

11989866 +11986261199096 +11987061199091199096 minus (1198625656 +11987056 (119905) 11990956) sin120572

minus (1198623434 +11987034 (119905) 11990934) sin (120579 minus 12057234) = 0

1198986 1199106 +1198626119910 1199106 +11987061199101199106 minus (1198625656 +11987056 (119905) 11990956) cos120572

minus (1198623434 +11987034 (119905) 11990934) cos (120579 minus 12057234) = 0(1)

where 11990912 11990934 and 11990956 respectively denote the relativedisplacements of gear 1 and gear 2 gear 3 and gear 4 gear5 and gear 6 The expressions are as follows

11990912 = 11990311988711205791 minus 119903119887212057923 + (1199091 minus11990923) sin120572

+ (1199101 minus11991023) cos120572

11990934 = 1199031198873 (12057923 + 1205796) minus 1199031198874 (1205794 + 1205796)

+ (11990923 minus1199094 minus1199096) sin (120579 minus 12057234)

+ (11991023 minus1199104 minus1199106) cos (120579 minus 12057234)

11990956 = 11990311988751205795 minus 11990311988761205796 + (1199095 minus1199096) sin120572+ (1199105 minus1199106) cos120572

(2)

In (1)-(2) 119898119894 denotes mass of gear 119894 119909119894 denotes displace-ment of gear 119894 in 119909 direction 119910119894 denotes displacement ofgear 119894 in 119910 direction 119868119894 denotes inertia of gear 119894 120572 denotespressure angle of each gear 12057234 denotes meshing angle ofgear 3 and gear 4 119870119894119895(119905) and 119862119894119895 respectively denote time-varying meshing stiffness and damping between gear 119894 andgear 119895 which will be given in Section 213 119862119894119909 and 119862119894119910respectively denote the supporting dampings of gear 119894 in 119909

and 119910 directions119870119894119909 and119870119894119910 denote the supporting stiffnessof gear 119894 in 119909 and 119910 directions respectively 1198791 and 1198792 denoteinput torques of M1 and M2 119865 denotes external excitation

By substituting (2) into (1) the global EOMs of system aretransformed in matrix form as

MX+CX+K (119905)X = P (3)

where C denotes damping matrix K(119905) denotes stiffnessmatrix P denotes excitation vector M and X respectivelyrepresent mass matrix and displacement vector which aregiven as

M = diag [1198681 1198981 1198981 11986823 11989823 11989823 1198684 1198984 1198984 1198685 1198985

1198985 1198686 1198986 1198986]

X = [1205791 1199091 1199101 12057923 11990923 11991023 1205794 1199094 1199104 1205795 1199095 1199105 1205796

1199096 1199106]119879

(4)

213 System Excitation

(1) Meshing StiffnessThemeshing stiffness will fluctuate withtime due to the periodic change of gear meshing Meanwhile

the deformations of gear teeth are different from dedendumto addendum in meshing which also causes the variation ofmeshing stiffnessThemeshing stiffness can be formulated bysine expression as [23]

119870 (119905) = 119870119898 +119870119886 sin (120596119898119905 + 120591) (5)

where119870119898 and119870119886 respectively represent the average value ofmesh stiffness and the amplitude of variable mesh stiffness120596119898 is meshing frequency and the expression is 120596119898 = 12058711989911991130119899 119911 and 120591 denote working speed teeth number andmeshingphase angle respectively

(2) Meshing DampingThe expression of meshing damping is[23]

11986212 = 2120585radic11987011989811989811198982

1198981 + 1198982

(6)

where 120585 1198981 and 1198982 are respectively meshing dampingcoefficient mass of driving gear and driven gear

22 Objective Functions The structural damage mainlyresults from dynamic characteristics of steering mechanismand the maximum transverse acceleration of each gear ischosen as the optimization objective

119891119894119909min = 119886119894119909

119891119894119910min = 119886119894119910

(119894 = 1 23 4 5 6)

(7)

where 119886119894119909 = 119894 is the maximum acceleration of gear 119894 in 119909

direction 119886119894119910 = 119910119894 is the maximum acceleration of gear 119894 in119910 direction 119894 and 119910119894 can be obtained by solving (3) usingNewmark-120573method

Besides in consideration of borehole size (the studiedcase is 12 1410158401015840) and annulus the outer diameter must beminimized for more drilling requirements on the conditionof good dynamic performances

119891119863min = 119863 (8)

where119863 is the outer diameter of steering mechanism

23 Design Variables Parameters with significant impacton objective functions are chosen as the design variablesHowever increasing design variables lead to more time-consuming iterations so it is necessary to reduce designvariables properly The aim of this study is to optimize thedynamic characteristics and dimension of steering mecha-nism which depend on the system parameters as mass stiff-ness damping inertia and external excitation according to(1) Meanwhile these system parameters are determinate bythe structural parameters of steering mechanism in essenceTherefore teeth number of gear 119894 119911119894 (119894 = 1 6) modulusof gears 119894 and 119895 119898119894119895 (119894119895 = 12 34 56) width of gears 119894 and119895 119887119894119895 (119894119895 = 12 34 56) diameter of mandrel 119889 locationparameter of steering point 1198971 and location parameter of


Table 1 Design variables

Items 1199111 1199112 1199113 1199114 1199115 119911611989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119889

(mm)1198971(m)

1198972(m)

Lower 17 20 25 21 17 20 15 3 15 30 30 30 80 03 02Upper 32 51 56 52 32 51 4 5 4 70 70 70 120 062 042

PO

l1 l2

dM Db34

b12 b56

Figure 3 Structure of steering mechanism for RSS

Table 2 Operating parameters

Parameters 1198791 (Nsdotm) 1198792 (Nsdotm) 119865 (KN)Values 50 50 20

spherical roller bearing 1198972 are chosen as design variablesand the lower and upper limits of variables are listed inTable 1 based onmechanical design criterion and engineeringexperiences

Dimensions of design variables in optimization formula-tion are illustrated in Figure 3

Operating parameters of steeringmechanism are listed inTable 2

24 Constraints

(1) Number of Teeth Due to the application of PGSTNDthe teeth number difference of inner meshing pair should belimited in a certain range according to the mechanical designcriterion and all teeth numbers must be integers

1 le 1199113 minus 1199114 le 4

119911119894 = Integer (119894 = 1 6) (9)

(2) Contact Ratio In order to ensure the continuity andstability of gear transmission the contact ratio of eachmeshing pair is limited as

11 le 120576119894119895 le 22 (10)

where 120576119894119895 (119894119895 = 12 34 56) denotes the contact ratio of gear119894-gear 119895 pair

(3) Tooth Thickness at Tip Cylinder To ensure the strengthof gear teeth at tip cylinder tooth thickness at tip cylindershould be constrained during machining process [24]

04119898119905119894 minus 119878119886119894 le 0 (11)

where 119878119886119894 (119894 = 1 6) is the tooth thickness of gear 119894 at tipcylinder

(4) Profile Modification Coefficient PGSTND is applied insteering mechanism and profile modification coefficientsshould be limited to avoid interference of gear 3 and gear 4

minus 05 le 1199091198993 le 05

minus 05 le 1199091198994 le 05(12)

where 1199091198993 and 1199091198994 are respectively profile modificationcoefficients of gear 3 and gear 4

(5) Gear Strength Since gear failures usually exist in formsof crack and pitting corrosion it is essential to check thecontacting strength and bending strength of gear teeth asfollows [25]

Contact strength

120590119867 =1070119886

radic(119894 + 1)3119870119889119879

119887119894le [120590119867]

(13)

where 120590119867 denote maximum contact strength 119886 is centredistance of meshing pair 119894 denotes transmission ratio ofmeshing pair 119870119889 is dynamic load coefficient (119870119889 is equal to13) 119879 is torque of pinion in meshing pair [120590119867] is allowablecontact stress

Bending strength

120590119865 =211987011988911987911198871198891119898119910119865

le [120590119865] (14)

where 120590119865 denotes maximum bending strength 1198891 denotespitch diameter of pinion in a meshing pair [120590119865] denotesallowable bending stress 119910119865 denotes tooth form factor

(6) Outer Diameter Based on the borehole size and annulusrequirements the outer diameter of steering mechanism isconstrained as

119863 le 280mm (15)


Start

End

Composite dominatedsorting (nondominated

sorting aheadconstrained-dominated

sorting behind)

Population of unfeasible

solutions

Constrained-dominated

sorting

Setup algorithm parameters

Initialize population

Terminate

Update population

No

Yes

Pareto optimal solutions

Fuzzy set theory

Best compromise solution

Dominated sorting

Feasiblesolution

Yes

No No

Selection

Crossover

Mutation

Offspring population

Population of feasible solutions

Nondominated sorting by

incorporating DCD strategies

Yes

Figure 4 Flowchart of optimization procedure based on MNSGA-II and FST

(7) Alternating Stress ofMandrelThemandrel suffers alternat-ing stress during operation and themaximum bending stressshould be limited to avoid destruction

120590max le [120590] (16)

where 120590max is maximum bending stress and [120590] is allowablebending stress

(8) Torque Constraint To meet the requirement for cuttingforce the enlarged input torque must be enough to conquerthe load torque

1198941198791 =0511987811986511989721198971

(17)

(9) Build-Up Rate To meet the requirements of build-up ratethat 119896 is equal to 8∘30m and build-up rate is constrained bythe expression in [22]

107120573 = 8 (18)

where 120573 = arctan(21198781198971) is the steering angle of steeringmechanism

25 Optimization Problem Formulation Based on the aboveanalysis the optimization problem of steering mechanismis a multiobjective optimization with mixed continuous-discrete variables under nonlinear constraints that is theminimizations of dynamic responses and outer diameter of

steering mechanism are investigated by selecting the struc-tural parameters of PGSTND and mandrel subject to somestructural and stress constraints In general the optimizationproblem can be formulated as follows

Minimize lfloor119891119894119909min (119909) 119891119894119910min (119909) 119891119863 (119909)rfloor

(119894 = 1 23 4 5 6)

Subject to ℎV (x) = 0

119892119906 (x) le 0

(19)

where ℎV(x) is equal constraint function 119892119906(x) is unequalconstraint function V 119906 are respectively the numbers ofequal and unequal constraint functions x is a set of designvariables and given byx

= [1199111 1199112 1199113 1199114 1199115 1199116 11989812 11989834 11989856 11988712 11988734 11988756 119889 1198971 1198972]119879

(20)

The optimization procedure based on MNSGA-II andFST is applied to solve this optimization problem andthe optimization procedure and results will be respectivelydemonstrated in Sections 3 and 4

3 Optimization Procedure Based onMNSGA-II and FST

31 Optimization Flowchart The optimization procedurebased on MNSGA-II and FST is shown in Figure 4 The core


Table 3 Algorithm parameters

Parameters Populationsize

Number ofgeneration

Crossoverprobability

Mutationrate

Values 300 150 09 01

of optimization is constrained-dominated sorting algorithmMeanwhile the DCD strategies are incorporated to improvethe uniformity of Pareto front

In traditional design the best solution in Pareto optimalsolutions is usually selected based on decision makerrsquos expe-rience and skills which is subjected to human preference Todeal with the drawbacks FST is applied to obtain the bestcomprise solution from Pareto optimal solutions in whichempirical design is replaced by theoretical design

The detailed introductions of MNSGA-II and FST will bemade in Sections 32 and 33

32 Introduction of MNSGA-II

321 Algorithm Initialization and Genetic Operators InMNSGA-II the algorithm parameters include populationsize number of generation crossover probability and muta-tion rate which can be determinate by usual methods Basedon the optimization problem formulation in Section 2 thealgorithm parameters are listed in Table 3 Meanwhile binaryencoding is used to initialize population with discrete andcontinuous variables [26]

Selection is the first genetic operator which guaranteesthat individuals with excellent genes are selected from parentpopulation and binary tournament selection is chosen forcalculation in MNSGA-II Crossover and mutation are thegenetic operators to maintain the diversity of population byproducing offspring individuals and uniform crossover andsingle-point mutation are respectively applied for mutationand mutation in this study [19]

322 Nondominated Sorting by Incorporating DCD StrategiesNondominated sorting is still the core ofMNSGA-II to deter-minate the distribution of Pareto optimal solutions Owingto the particularity of nonlinear constraints in this studythe constrained-dominated sorting [18] and the modifiednondominated sorting by incorporating DCD strategies areused for the optimization

The modified nondominated sorting by incorporatingDCD strategies is as follows

In conventional NSGA-II approach the solutions in thesame rank are sorted based on crowding distance in nondom-inated sorting and the crowding distance is calculated as

CD119894 =1119903

119903

sum

119896=1

10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816 (21)

where CD119894 is crowding distance of the 119894th solution 119903 is thenumber of objectives 119891119896

119894is the 119896th objective value of the 119894th

solutionHowever nondominated sorting using the above crowd-

ing distance has drawback of weak uniform diversity in

obtained Pareto optimal solutions DCD strategies are incor-porated in currently used NSGA-II to deal with the men-tioned problem and DCD of the 119894th solution is expressed as[21]

DCD119894 =CD119894

log (1119881119894) (22)

where DCD119894 denotes the dynamic crowding distance of the119894th solution and 119881119894 is expressed as

119881119894 =1119903

119903

sum

119896=1(10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816minusCD119894)

2 (23)

119881119894 is the variance of CDs of individuals which areneighbors of the 119894th solution and it gives some informationabout the difference degree of CD in different objectives

In addition the nondominated sorting algorithm ischanged due to the incorporation of DCD strategies Supposepopulation size is 119873 the 119894th generation of nondominatedsorting set is 119876(119905) and the size of 119876(119905) is 119872 then 119872 minus 119873

solutions are wiped off and procedures are performed asfollows [21]

Step 1 If |119876(119905)| le 119873 go to Step 5 else keep on

Step 2 Calculate all solutionsrsquo DCD in the 119876(119905) based on(22)

Step 3 Sort 119876(119905) based on solutionsrsquo DCD

Step 4 Wipe off a solution which has the lowest DCD in the119876(119905)

Step 5 If |119876(119905)| le 119873 stop population maintenance else goStep 2 and keep on

It can be seen that one solution is wiped off every timeand all solutionsrsquo DCD in the 119876(119905) will be recalculatedTherefore the diversity of modified nondominated sortingcan bemaintained and a Pareto front with high uniformity isalso obtained

33 Fuzzy Set Theory For Pareto optimal solutions with119873obj objectives and 119872 solutions a membership function 120583119894denotes the 119894th objective function of a solution in Paretooptimal solutions which is defined as [27]

120583119894 =

1 119865119894 le 119865min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

le 119865119894 le 119865max119894

0 119865119894 ge 119865max119894

(24)

where 119865max119894

and 119865min119894

respectively denote the maximumand minimum values of the 119894th objective function Foreach nondominated solution 119896 the normalized membershipfunction 120583119896 is expressed as

120583119896=

sum119873obj119894=1 120583119896

119894

sum119872

119895=1sum119873obj119894=1 120583119895

119894

(25)


Table 4 Comparisons between original and optimized designs

Items 1199111

1199112

1199113

1199114

1199115

1199116

11989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119863

(mm)1198971(m)

1198972(m)

Original 17 38 33 30 17 38 35 35 35 50 50 50 80 0601 0367Optimized 18 43 37 35 22 37 3 45 3 45 63 47 81 0564 0297

Table 5 Comparisons of dynamic responses and outer diameter in original and optimized designs

Items 1198861119909

(ms2)1198861119910

(ms2)11988623119909

(ms2)11988623119910

(ms2)1198864119909

(ms2)1198864119910

(ms2)1198865119909

(ms2)1198865119910

(ms2)1198866119909

(ms2)1198866119910

(ms2)119863

(m)Original 151 414 094 263 025 055 151 415 1329 2108 0258Optimized 143 392 056 163 023 052 141 389 100 162 0240Difference () 541 540 4010 3796 602 538 628 627 2476 2314 698

In (25) larger 120583119896 indicates better compromise solu-

tion Therefore a priority list of nondominated solutions isobtained by descending sort of 120583119896 and it is beneficial fordecision maker to choose the best compromise solution inPareto optimal solutions

4 Results and Discussions

41 Comparison between Optimized Results In order toevaluate the performance of MNSGA-II for optimizationproblemof steeringmechanism bothMNSGA-II andNSGA-II are used to solve the optimization and Pareto fronts areillustrated in Figure 5

As shown in Figure 5(a) the competitive relationshipbetween maximum accelerations of gear 1 in 119909 and 119910 direc-tions is linear and the distribution of Pareto optimal solutionsby MNSGA-II is more uniform than that by NSGA-II Thesame phenomena also can be discovered in the competitiverelationships between maximum accelerations of gear 4 andgear 5 in 119909 and 119910 directions as shown in Figures 5(c) and 5(d)The competitive relationship between maximum accelera-tions of gear 23 in 119909 and119910 directions is approximatively linearas shown in Figure 5(b)Meanwhile the distribution of Paretooptimal solutions byMNSGA-II ismore uniform than that byNSGA-IIThe same phenomena also can be discovered in thecompetitive relationship between maximum accelerations ofgear 6 in 119909 and 119910 directions as shown in Figure 5(e)

Based on the above analysis MNSGA-II can avoid phe-nomena that some parts of the Pareto fronts are too crowdedand some parts are sparseness The uniformity of Paretofronts is improved which indicates that the MNSGA-II isfeasible for optimization problem of steering mechanism

42 Pareto Optimal Solutions In order to reveal the variationtrends of all objectives directly the Pareto optimal solutionsobtained by MNSGA-II are shown in Figure 6 in whicheach vertical axis represents an objective and each brokenline represents a solution As illustrated in Figure 6 themaximum accelerations of each gear in 119909 and 119910 directionsappear in linear or approximately linear relationship asdepicted in Section 41 However the competitive relationshipbetween maximum acceleration of gear 1 in 119910 direction and

that of gear 23 in 119909 direction is irregular and the samephenomena can be reflected in maximum accelerations ofgear 23 in 119910 direction and gear 4 in 119909 direction gear 4in 119910 direction and gear 5 in 119909 direction and gear 5 in 119910

direction and gear 6 in 119909 direction which indicate that theeffect of structural parameters on the dynamics of steeringmechanism is complexTherefore it is significant to optimizethe dynamics of steering mechanism Meanwhile due tothe conflicts between objectives it is difficult for designerto select the best solution from Pareto optimal solutionsand it is also subjected to human preference In this studyFST in Section 33 is applied to select the best compromisesolution fromPareto optimal solutions which avoid the effectof human behaviors

The best compromise solution obtained by FST is shownby green broken line in Figure 6 and the red broken linerepresents the dynamic characteristics of original designThecomparisons between original and optimized designs areshown in Table 4

Based on the analysis of dynamic characteristics shownin Figure 6 the comparisons of maximum accelerations andouter diameter of original and optimized designs are shownin Table 5 The maximum accelerations of all gears decreasein different extent and the maximum accelerations of gear 23decrease from 094ms2 to 056ms2 by 401 in 119909 directionand from 263ms2 to 163ms2 by 3796 in 119910 direction Inaddition the outer diameter of steeringmechanism decreasesfrom 0258m to 024m by 698 Optimization results indi-cate that not only are the dynamic characteristics improvedbut also the outer diameter is decreasedwhich canmeetmorerequirements on borehole size

43 Comparisons between Responses in Original and Opti-mized Designs To reflect the response variance of each gearin original and optimized designs the response curves ofeach gear are depicted in Figures 7ndash11 in which (a) and (b)respectively represent responses in 119909 and 119910 directions Theacceleration amplitudes in 119909 and 119910 directions decrease indifferent degrees as shown in Figures 7ndash11 and the responsedecays get faster as shown in Figures 8 9 and 11 whichindicate that the responses are improved after optimizationand the optimization procedure is effective for the steeringmechanism


05 1 15 2 251

2

3

4

5

6a 1

y(m

s2)

a1x (ms2)

(a)

01 02 03 04 05 06 0705

1

15

2

a 23y

(ms2)

a23x (ms2)

(b)

0 05 1 15 20

1

2

3

4

5

a 4y

(ms2)

a4x (ms2)

(c)

05 1 15 2 251

2

3

4

5

6

a 5y

(ms2)

a5x (ms2)

(d)

0 05 1 15 20

05

1

15

2

25

3

a 6y

(ms2)

a6x (ms2)

(e)

Figure 5 Pareto fronts for different objectives (998779 NSGA-II I MNSGA-II)

5 Conclusions

The design of gear train is a complex multiobjective opti-mization with mixed continuous-discrete variables undernumerous nonlinear constraints In this paper based onthe dynamic model of steering mechanism in which thekey component is a PGSTND the optimization problem

of steering mechanism is investigated by the optimizationprocedure based onMNSGA-II and FST and conclusions canbe drawn as follows

(1) For multiobjective optimization of steering mecha-nism for RSS MNSGA-II can improve uniformity ofPareto fronts compared with NSGA-II


Pareto optimal solutionsOriginal designBest compromise solution

a1ya1x a6ya6xa5ya5xa4ya4xa23ya23x D200

186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020

Figure 6 Pareto optimal solutions for all objectives (units 119886 (ms2)119863 (m))

3 35 4 45 5

0

1

2

3

Time (s)

Original designOptimized design

minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)

Figure 7 Responses of gear 1 in original and optimized designs

3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)



(2) The best compromise solution in Pareto optimal solu-tions is obtained by FST to avoid human preferenceCompared with original design the optimized designhas better dynamic responses with minimum outerdiameter which indicate FST is beneficial for decisionmaker

(3) The comparisons between response curves of originaland optimized designs demonstrate that the responseamplitudes become smaller and decay gets faster afteroptimization which indicate that the optimizationprocedure is effective for the optimization of steeringmechanism

In conclusion the proposed optimization procedurebased on MNSGA-II and FST is feasible to solve the multi-objective optimization of gear train with mixed continuous-discrete variables under nonlinear constraints In additionthe procedure is flexible and can be extended to optimizationsof other more complex mechanical structures

Nomenclature

119878 Distance between rotating axis ofgear 6 and eccentric axis of gear 4

120579119894 Angular displacement of gear 119894119909119894 Displacement of gear 119894 in 119909 direction119910119894 Displacement of gear 119894 in 119910 direction119909119894119895 Relative displacement of gears 119894 and 1198951199091198993 1199091198994 Profile modification coefficients of

gears 3 and 4119903119887119894 Radius of base circle of gear 119894119870119894119895(119905) Time-varying meshing stiffness

between gears 119894 and 119895119862119894119895 Damping between gears 119894 and 119895119870119894119909119870119894119910 Stiffness of gear 119894 in 119909 and 119910

directions119862119894119909 119862119894119910 Damping of gear 119894 in 119909 and 119910

directions119879119894 Torque of DC servo motor 119894119865 External excitationM Mass matrixK Stiffness matrixC Damping matrixX Displacement vectorP Excitation vector119898119894 Mass of gear 119894120596119898 Meshing frequency120572 Pressure angle12057234 Meshing angle of gears 3 and 4119868119894 Inertia of gear 119894119905 Time119911119894 Teeth number of gear 119894120576119894119895 Contact ratio of gears 119894 and 119895119878119886119894 Thickness of gear 119894 at tip cylinder120591 Meshing phase angle120577 Meshing damping coefficient120578 Shaft damping coefficient119886119894119909 119886119894119910 Acceleration of gear 119894 in 119909 direction

119863 Outer diameter of steeringmechanism

119887119894119895 Width of gears 119894 and 119895119898119894119895 Modulus of gears 119894 and 1198951198971 1198972 Location parameters of steering point

and spherical roller bearing119889 Outer diameter of mandrel120590119867 Maximum contact strength[120590119867] Allowable contact stress119886 Centre distance of meshing pair119894 Transmission ratio of meshing pair119870119889 Dynamic load coefficient120590119865 Maximum bending strength[120590119865] Allowable bending stress119910119865 Tooth form factor120590max Maximum bending stress of mandrel[120590] Allowable bending stress of mandrel119896 Build-up rate120573 Steering angle of steering mechanism

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was sponsored by the Key Technologies RampDProgram of Tianjin under Grant no 11ZCKFGX03500

References

[1] M Bozca ldquoTorsional vibration model based optimization ofgearbox geometric design parameters to reduce rattle noise inan automotive transmissionrdquo Mechanism and Machine Theoryvol 45 no 11 pp 1583ndash1598 2010

[2] W Huang L Fu X Liu Z Wen and L Zhao ldquoThe struc-tural optimization of gearbox based on sequential quadraticprogramming methodrdquo in Proceedings of the 2nd InternationalConference on Intelligent Computing Technology and Automa-tion (ICICTA rsquo09) pp 356ndash359 Hunan China October 2009

[3] T H Chong I Bae and A Kubo ldquoMultiobjective optimaldesign of cylindrical gear pairs for the reduction of gear sizeand meshing vibrationrdquo JSME International Journal Series CMechanical Systems Machine Elements and Manufacturing vol44 no 1 pp 291ndash298 2001

[4] H Zarefar and S N Muthukrishnan ldquoComputer-aided opti-mal design via modified adaptive random-search algorithmrdquoComputer-Aided Design vol 25 no 4 pp 240ndash248 1993

[5] M Ciavarella and G Demelio ldquoNumerical methods for theoptimisation of specific sliding stress concentration and fatiguelife of gearsrdquo International Journal of Fatigue vol 21 no 5 pp465ndash474 1999

[6] H-Z Huang Z-G Tian and M J Zuo ldquoMultiobjectiveoptimization of three-stage spur gear reduction units usinginteractive physical programmingrdquo Journal of Mechanical Sci-ence and Technology vol 19 no 5 pp 1080ndash1086 2005

[7] M Faggioni F S Samani G Bertacchi and F PellicanoldquoDynamic optimization of spur gearsrdquoMechanism andMachineTheory vol 46 no 4 pp 544ndash557 2011


[8] B A Abuid and Y M Ameen ldquoProcedure for optimumdesign of a two-stage spur gear systemrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 46 no 4 pp 1582ndash1590 2003

[9] 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

[10] F Mendi T Baskal K Boran and F E Boran ldquoOptimizationof module shaft diameter and rolling bearing for spur gearthrough genetic algorithmrdquo Expert Systems with Applicationsvol 37 no 12 pp 8058ndash8064 2010

[11] T H Chong and J S Lee ldquoA design method of gear trains usinga genetic algorithmrdquo International Journal of the Korean Societyof Precision Engineering vol 1 no 1 pp 62ndash70 2000

[12] C Gologlu and M Zeyveli ldquoA genetic approach to automatepreliminary design of gear drivesrdquo Computers amp IndustrialEngineering vol 57 no 3 pp 1043ndash1051 2009

[13] J L Marcelin ldquoGenetic optimisation of gearsrdquo InternationalJournal of Advanced Manufacturing Technology vol 17 no 12pp 910ndash915 2001

[14] O Buiga and L Tudose ldquoOptimal mass minimization designof a two-stage coaxial helical speed reducer with GeneticAlgorithmsrdquo Advances in Engineering Software vol 68 pp 25ndash32 2014

[15] I Ciglaric and A Kidric ldquoComputer-aided derivation of theoptimal mathematical models to study gear-pair dynamic byusing genetic programmingrdquo Structural and MultidisciplinaryOptimization vol 32 no 2 pp 153ndash160 2006

[16] G Bonori M Barbieri and F Pellicano ldquoOptimum profilemodifications of spur gears by means of genetic algorithmsrdquoJournal of Sound and Vibration vol 313 no 3ndash5 pp 603ndash6162008

[17] M Barbieri G Bonori and F Pellicano ldquoCorrigendum tooptimumprofilemodifications of spur gears bymeans of geneticalgorithmsrdquo Journal of Sound and Vibration vol 331 pp 4825ndash4829 2012

[18] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[19] K Deb and S Jain ldquoMulti-speed gearbox design using multi-objective evolutionary algorithmsrdquo Transactions of the ASMEJournal of Mechanical Design vol 125 no 3 pp 609ndash619 2003

[20] R C Sanghvi A S Vashi H P Patolia and R G JivanildquoMulti-objective optimization of two-stage helical gear trainusing NSGA-IIrdquo Journal of Optimization vol 2014 Article ID670297 8 pages 2014

[21] B Luo J Zheng J Xie and J Wu ldquoDynamic crowdingdistancemdasha new diversity maintenance strategy for MOEAsrdquoin Proceedings of the 4th International Conference on NaturalComputation (ICNC rsquo08) pp 580ndash585 IEEE Jinan ChinaOctober 2008

[22] Y Z Li W T Niu H T Li Z J Luo and L N Wang ldquoStudyon a new Steerablemechanism for point-the-bit rotary steerablesystemrdquo Advance in Mechanical Engineering vol 6 Article ID923178 14 pages 2014

[23] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[24] J Wei C Lv W Sun X Li and Y Wang ldquoA study onoptimum design method of gear transmission system for windturbinerdquo International Journal of Precision Engineering andManufacturing vol 14 no 5 pp 767ndash778 2013

[25] G J Yue C J Bao and S F Dong ldquoOptimization design of spurgear transmissionrdquoCoal Technology vol 31 no 4 pp 16ndash17 2012(Chinese)

[26] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons Toronto Canada 1997

[27] T Ray K Tai and K C Seow ldquoMultiobjective design optimiza-tion by an evolutionary algorithmrdquo Engineering Optimizationvol 33 no 4 pp 399ndash424 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


volume and surface fatigue of multistage spur gear reductionunits by employing quasi-Newton method [9]

The mentioned optimization algorithms above are cata-loged as conventional optimization techniques Though theyare efficient for some optimization problems in applicationdifficulties still exist in tackling some special problems withnoncontinuous variables complex constraints and stronglynonlinear objectives Therefore some modern optimizationmethods such as genetic algorithm (GA) have been proposedto solve such problems in gear trains By using the geneticalgorithmMendi et al investigated optimization of themod-ulus of spur gears the diameters of shafts and rolling bearing[10] Chong and Lee presented a design method to optimizethe volume of two stage gear trains by using the geneticalgorithm which shows that the genetic algorithm is betterthan other conventional algorithms for solving the discreteinteger variable and continuous problems [11] Gologlu andZeyveli introduced an automated preliminary design of geardrives by minimizing volume of gear trains using a geneticalgorithm and static and dynamic penalty functions werepresented to the objective function for handling the designconstraints [12] Marcelin conducted optimum design ofgears by GAs and penalty selection methods [13] Buiga andTudose investigated mass minimization design of a two-stage coaxial helical speed reducer with genetic algorithms[14] Ciglaric and Kidric conducted automatic dynamicoptimization of a gear pair by using genetic programmingalgorithm [15] Bonori et al performed profile modificationoptimization of spur gears by means of genetic algorithmswhich aimed to reduce vibration and noise [16 17]

However most real-world optimization problems involvemultiple objectives with mixed continuous-discrete variablesunder nonlinear constraints and the conventional optimiza-tion techniques are not developed for such optimizationproblems in mind Multiple objectives have to be reformu-lated as a single objective by weighted-sum approaches priorto optimization in practice and the optimization programwill be performed repeatedly by changing weighting factorsto obtain Pareto optimal frontMeanwhile the nonlinear con-straints are handled by introducing penalty coefficients andthe inappropriate selection of penalty coefficients will leadto nonconvergence Therefore multiobjective evolutionaryalgorithms (MOEAs) as NSGA-II [18] have been developedto solve these problems Deb and Jain demonstrated theuse of NSGA-II in solving the original problem involvingmixed discrete and real-valued parameters and more thanone objective [19] Sanghvi et al investigated multiobjectiveoptimization of a two-stage helical gear train by usingNSGA-II [20]

Though NSGA-II is considered as a successful mul-tiobjective optimization algorithm some drawbacks havebeen discovered as lack of uniform diversity in obtainednondominated solutions To solve such problem Luo et alpresented dynamic crowding distance (DCD) [21] which canbe incorporated in NSGA-II as modified NSGA-II (MNSGA-II)This optimization algorithmhas never been applied in thedesign of gear trains

In this paperMNSGA-II is applied tomultiobjective opti-mization of a novel steering mechanism for rotary steering

M1

M2

1

23

3

2

6

104

5

7

8 9 11

12F

s

T1

T2

Figure 1 Schematic diagram of steering mechanism

system (RSS) the key component of which is a planetary gearset with teeth number difference (PGSTND) [22] This studyaims to minimize the dynamic responses and outer diam-eter of steering mechanism with structural parameters asdesign variables subject to geometric kinematic and strengthconstraints Based on the established dynamic model theoptimization problem is formulated and both MNSGA-II and NSGA-II are applied to the optimization problemMeanwhile in order to avoid human interference in selectionof the best solution from Pareto optimal solutions whenthe objective preferences are absent the best compromisesolution is obtained by fuzzy set theory (FST) and thedynamic responses of optimized and original designs arecompared and analyzed

The rest of this paper is organized as follows the problemformulation is established in Section 2 The optimizationprocedure based on MNSGA-II and FST is introduced andapplied to the multiobjective optimization of steering mech-anism for RSS in Section 3 Comparisons between resultsby MNSGA-II and NSGA-II are conducted and the bestcompromise solution is obtained by FST and compared withthe original design in Section 4 Finally main conclusions aredrawn in Section 5

2 Problem Formulation

The studied object is a steering mechanism for RSS the keycomponents of which are a specially designed PGSTND andtwo servo motors [22] The schematic diagram of steeringmechanism is illustrated in Figure 1

In Figure 1 M1 and M2 denote DC servo motors 1 and 2respectively 1 and 2 respectively denote driving and drivengears for rotation 3 denotes annulus gear 23 denotes aparallel gear consisting of gear 2 and gear 3 4 denotes planetgear 5 and 6 respectively denote driving and driven gearsfor revolution 7 denotes drill-string 8 denotes universaljoint 9 denotes spherical plain bearing 10 denotes drillingmandrel 11 denotes spherical roller bearing 12 denotes drillbit 119865 denotes external excitation 119878 denotes distance betweenrotating axis of gear 6 and eccentric axis of gear 4

In drilling process the steering mechanism is inclinedto fail due to the dynamic load which is related not onlyto operation conditions but also to the structural dynamic


x5

x4

x6x23

y5

1205795

y6

y4

y23

1205796

1205794

12057923

Gear 5

C5y

K5xK5y

C56

C34

C12

x1

y1 Gear 1

Gear 6

Gear 23

Gear 4

C5x

1205791

C1y

C4y

K1x K6x

K1y

K4y

C6y K6y

C1x C6xK12

K56

K34

C23y K23y

C23x

C4x

K4x

K23xO1

O4O5O23(O6)



21 Dynamic Modeling



11986811205791 + (1198621212 +11987012 (119905) 11990912) 1199031198871 = 1198791

11989811 +11986211199091 +11987011199091199091

+ (1198621212 +11987012 (119905) 11990912) sin120572 = 0

1198981 1199101 +1198621119910 1199101 +11987011199101199101

+ (1198621212 +11987012 (119905) 11990912) cos120572 = 0

1198682312057923 minus (1198621212 +11987012 (119905) 11990912) 1199031198872

+ (1198623434 +11987034 (119905) 11990934) 1199031198873 = 0

1198982323 +1198622311990923 +1198702311990911990923

minus (1198621212 +11987012 (119905) 11990912) sin120572

+ (1198623434 +11987034 (119905) 11990934) sin (120579 minus 12057234) = 0

11989823 11991023 +11986223119910 11991023 +1198702311991011991023

minus (1198621212 +11987012 (119905) 11990912) cos120572

+ (1198623434 +11987034 (119905) 11990934) cos (120579 minus 12057234) = 0

11986841205794 minus (1198623434 +11987034 (119905) 11990934) 1199031198874 = minus119865119878

11989844 +11986241199094 +11987041199091199094

+ (1198623434 +11987034 (119905) 11990934) sin12057234 = 0

1198984 1199104 +1198624119910 1199104 +11987041199101199104

minus (1198623434 +11987034 (119905) 11990934) cos12057234 = 0

11986851205795 + (1198625656 +11987056 (119905) 11990956) 1199031198875 = 1198792

11989855 +11986251199095 +11987051199091199095

+ (1198625656 +11987056 (119905) 11990956) sin120572 = 0

1198985 1199105 +1198625119910 1199105 +11987051199101199105

+ (1198625656 +11987056 (119905) 11990956) cos120572 = 0


11986861205796 minus (1198625656 +11987056 (119905) 11990956) 1199031198876

+ (1198623434 +11987034 (119905) 11990934) (1199031198873 minus 1199031198874) = 0

11989866 +11986261199096 +11987061199091199096 minus (1198625656 +11987056 (119905) 11990956) sin120572

minus (1198623434 +11987034 (119905) 11990934) sin (120579 minus 12057234) = 0

1198986 1199106 +1198626119910 1199106 +11987061199101199106 minus (1198625656 +11987056 (119905) 11990956) cos120572

minus (1198623434 +11987034 (119905) 11990934) cos (120579 minus 12057234) = 0(1)


11990912 = 11990311988711205791 minus 119903119887212057923 + (1199091 minus11990923) sin120572

+ (1199101 minus11991023) cos120572

11990934 = 1199031198873 (12057923 + 1205796) minus 1199031198874 (1205794 + 1205796)




(2)




MX+CX+K (119905)X = P (3)


M = diag [1198681 1198981 1198981 11986823 11989823 11989823 1198684 1198984 1198984 1198685 1198985

1198985 1198686 1198986 1198986]

X = [1205791 1199091 1199101 12057923 11990923 11991023 1205794 1199094 1199104 1205795 1199095 1199105 1205796

1199096 1199106]119879

(4)




119870 (119905) = 119870119898 +119870119886 sin (120596119898119905 + 120591) (5)



11986212 = 2120585radic11987011989811989811198982

1198981 + 1198982

(6)



119891119894119909min = 119886119894119909

119891119894119910min = 119886119894119910

(119894 = 1 23 4 5 6)

(7)




119891119863min = 119863 (8)





Items 1199111 1199112 1199113 1199114 1199115 119911611989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119889

(mm)1198971(m)

1198972(m)

Lower 17 20 25 21 17 20 15 3 15 30 30 30 80 03 02Upper 32 51 56 52 32 51 4 5 4 70 70 70 120 062 042

PO

l1 l2

dM Db34

b12 b56







24 Constraints


1 le 1199113 minus 1199114 le 4

119911119894 = Integer (119894 = 1 6) (9)


11 le 120576119894119895 le 22 (10)



04119898119905119894 minus 119878119886119894 le 0 (11)



minus 05 le 1199091198993 le 05

minus 05 le 1199091198994 le 05(12)



Contact strength

120590119867 =1070119886

radic(119894 + 1)3119870119889119879

119887119894le [120590119867]

(13)


Bending strength

120590119865 =211987011988911987911198871198891119898119910119865

le [120590119865] (14)



119863 le 280mm (15)


Start

End



sorting behind)


solutions


sorting



Terminate

Update population

No

Yes


Fuzzy set theory


Dominated sorting

Feasiblesolution

Yes

No No

Selection

Crossover

Mutation





Yes



120590max le [120590] (16)



1198941198791 =0511987811986511989721198971

(17)


107120573 = 8 (18)





(119894 = 1 23 4 5 6)


119892119906 (x) le 0

(19)


= [1199111 1199112 1199113 1199114 1199115 1199116 11989812 11989834 11989856 11988712 11988734 11988756 119889 1198971 1198972]119879

(20)







Number ofgeneration


Mutationrate

Values 300 150 09 01










CD119894 =1119903

119903

sum

119896=1

10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816 (21)






DCD119894 =CD119894

log (1119881119894) (22)


119881119894 =1119903

119903

sum

119896=1(10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816minusCD119894)

2 (23)











120583119894 =

1 119865119894 le 119865min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

le 119865119894 le 119865max119894

0 119865119894 ge 119865max119894

(24)

where 119865max119894

and 119865min119894


120583119896=

sum119873obj119894=1 120583119896

119894

sum119872

119895=1sum119873obj119894=1 120583119895

119894

(25)



Items 1199111

1199112

1199113

1199114

1199115

1199116

11989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119863

(mm)1198971(m)

1198972(m)



Items 1198861119909

(ms2)1198861119910

(ms2)11988623119909

(ms2)11988623119910

(ms2)1198864119909

(ms2)1198864119910

(ms2)1198865119909

(ms2)1198865119910

(ms2)1198866119909

(ms2)1198866119910

(ms2)119863















05 1 15 2 251

2

3

4

5

6a 1

y(m

s2)

a1x (ms2)

(a)

01 02 03 04 05 06 0705

1

15

2

a 23y

(ms2)

a23x (ms2)

(b)

0 05 1 15 20

1

2

3

4

5

a 4y

(ms2)

a4x (ms2)

(c)

05 1 15 2 251

2

3

4

5

6

a 5y

(ms2)

a5x (ms2)

(d)

0 05 1 15 20

05

1

15

2

25

3

a 6y

(ms2)

a6x (ms2)

(e)


5 Conclusions







186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)


3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x5

x4

x6x23

y5

1205795

y6

y4

y23

1205796

1205794

12057923

Gear 5

C5y

K5xK5y

C56

C34

C12

x1

y1 Gear 1

Gear 6

Gear 23

Gear 4

C5x

1205791

C1y

C4y

K1x K6x

K1y

K4y

C6y K6y

C1x C6xK12

K56

K34

C23y K23y

C23x

C4x

K4x

K23xO1

O4O5O23(O6)



21 Dynamic Modeling



11986811205791 + (1198621212 +11987012 (119905) 11990912) 1199031198871 = 1198791

11989811 +11986211199091 +11987011199091199091

+ (1198621212 +11987012 (119905) 11990912) sin120572 = 0

1198981 1199101 +1198621119910 1199101 +11987011199101199101

+ (1198621212 +11987012 (119905) 11990912) cos120572 = 0

1198682312057923 minus (1198621212 +11987012 (119905) 11990912) 1199031198872

+ (1198623434 +11987034 (119905) 11990934) 1199031198873 = 0

1198982323 +1198622311990923 +1198702311990911990923

minus (1198621212 +11987012 (119905) 11990912) sin120572

+ (1198623434 +11987034 (119905) 11990934) sin (120579 minus 12057234) = 0

11989823 11991023 +11986223119910 11991023 +1198702311991011991023

minus (1198621212 +11987012 (119905) 11990912) cos120572

+ (1198623434 +11987034 (119905) 11990934) cos (120579 minus 12057234) = 0

11986841205794 minus (1198623434 +11987034 (119905) 11990934) 1199031198874 = minus119865119878

11989844 +11986241199094 +11987041199091199094

+ (1198623434 +11987034 (119905) 11990934) sin12057234 = 0

1198984 1199104 +1198624119910 1199104 +11987041199101199104

minus (1198623434 +11987034 (119905) 11990934) cos12057234 = 0

11986851205795 + (1198625656 +11987056 (119905) 11990956) 1199031198875 = 1198792

11989855 +11986251199095 +11987051199091199095

+ (1198625656 +11987056 (119905) 11990956) sin120572 = 0

1198985 1199105 +1198625119910 1199105 +11987051199101199105

+ (1198625656 +11987056 (119905) 11990956) cos120572 = 0


11986861205796 minus (1198625656 +11987056 (119905) 11990956) 1199031198876

+ (1198623434 +11987034 (119905) 11990934) (1199031198873 minus 1199031198874) = 0

11989866 +11986261199096 +11987061199091199096 minus (1198625656 +11987056 (119905) 11990956) sin120572

minus (1198623434 +11987034 (119905) 11990934) sin (120579 minus 12057234) = 0

1198986 1199106 +1198626119910 1199106 +11987061199101199106 minus (1198625656 +11987056 (119905) 11990956) cos120572

minus (1198623434 +11987034 (119905) 11990934) cos (120579 minus 12057234) = 0(1)


11990912 = 11990311988711205791 minus 119903119887212057923 + (1199091 minus11990923) sin120572

+ (1199101 minus11991023) cos120572

11990934 = 1199031198873 (12057923 + 1205796) minus 1199031198874 (1205794 + 1205796)




(2)




MX+CX+K (119905)X = P (3)


M = diag [1198681 1198981 1198981 11986823 11989823 11989823 1198684 1198984 1198984 1198685 1198985

1198985 1198686 1198986 1198986]

X = [1205791 1199091 1199101 12057923 11990923 11991023 1205794 1199094 1199104 1205795 1199095 1199105 1205796

1199096 1199106]119879

(4)




119870 (119905) = 119870119898 +119870119886 sin (120596119898119905 + 120591) (5)



11986212 = 2120585radic11987011989811989811198982

1198981 + 1198982

(6)



119891119894119909min = 119886119894119909

119891119894119910min = 119886119894119910

(119894 = 1 23 4 5 6)

(7)




119891119863min = 119863 (8)





Items 1199111 1199112 1199113 1199114 1199115 119911611989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119889

(mm)1198971(m)

1198972(m)

Lower 17 20 25 21 17 20 15 3 15 30 30 30 80 03 02Upper 32 51 56 52 32 51 4 5 4 70 70 70 120 062 042

PO

l1 l2

dM Db34

b12 b56







24 Constraints


1 le 1199113 minus 1199114 le 4

119911119894 = Integer (119894 = 1 6) (9)


11 le 120576119894119895 le 22 (10)



04119898119905119894 minus 119878119886119894 le 0 (11)



minus 05 le 1199091198993 le 05

minus 05 le 1199091198994 le 05(12)



Contact strength

120590119867 =1070119886

radic(119894 + 1)3119870119889119879

119887119894le [120590119867]

(13)


Bending strength

120590119865 =211987011988911987911198871198891119898119910119865

le [120590119865] (14)



119863 le 280mm (15)


Start

End



sorting behind)


solutions


sorting



Terminate

Update population

No

Yes


Fuzzy set theory


Dominated sorting

Feasiblesolution

Yes

No No

Selection

Crossover

Mutation





Yes



120590max le [120590] (16)



1198941198791 =0511987811986511989721198971

(17)


107120573 = 8 (18)





(119894 = 1 23 4 5 6)


119892119906 (x) le 0

(19)


= [1199111 1199112 1199113 1199114 1199115 1199116 11989812 11989834 11989856 11988712 11988734 11988756 119889 1198971 1198972]119879

(20)







Number ofgeneration


Mutationrate

Values 300 150 09 01










CD119894 =1119903

119903

sum

119896=1

10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816 (21)






DCD119894 =CD119894

log (1119881119894) (22)


119881119894 =1119903

119903

sum

119896=1(10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816minusCD119894)

2 (23)











120583119894 =

1 119865119894 le 119865min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

le 119865119894 le 119865max119894

0 119865119894 ge 119865max119894

(24)

where 119865max119894

and 119865min119894


120583119896=

sum119873obj119894=1 120583119896

119894

sum119872

119895=1sum119873obj119894=1 120583119895

119894

(25)



Items 1199111

1199112

1199113

1199114

1199115

1199116

11989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119863

(mm)1198971(m)

1198972(m)



Items 1198861119909

(ms2)1198861119910

(ms2)11988623119909

(ms2)11988623119910

(ms2)1198864119909

(ms2)1198864119910

(ms2)1198865119909

(ms2)1198865119910

(ms2)1198866119909

(ms2)1198866119910

(ms2)119863















05 1 15 2 251

2

3

4

5

6a 1

y(m

s2)

a1x (ms2)

(a)

01 02 03 04 05 06 0705

1

15

2

a 23y

(ms2)

a23x (ms2)

(b)

0 05 1 15 20

1

2

3

4

5

a 4y

(ms2)

a4x (ms2)

(c)

05 1 15 2 251

2

3

4

5

6

a 5y

(ms2)

a5x (ms2)

(d)

0 05 1 15 20

05

1

15

2

25

3

a 6y

(ms2)

a6x (ms2)

(e)


5 Conclusions







186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)


3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










11986861205796 minus (1198625656 +11987056 (119905) 11990956) 1199031198876

+ (1198623434 +11987034 (119905) 11990934) (1199031198873 minus 1199031198874) = 0

11989866 +11986261199096 +11987061199091199096 minus (1198625656 +11987056 (119905) 11990956) sin120572

minus (1198623434 +11987034 (119905) 11990934) sin (120579 minus 12057234) = 0

1198986 1199106 +1198626119910 1199106 +11987061199101199106 minus (1198625656 +11987056 (119905) 11990956) cos120572

minus (1198623434 +11987034 (119905) 11990934) cos (120579 minus 12057234) = 0(1)


11990912 = 11990311988711205791 minus 119903119887212057923 + (1199091 minus11990923) sin120572

+ (1199101 minus11991023) cos120572

11990934 = 1199031198873 (12057923 + 1205796) minus 1199031198874 (1205794 + 1205796)




(2)




MX+CX+K (119905)X = P (3)


M = diag [1198681 1198981 1198981 11986823 11989823 11989823 1198684 1198984 1198984 1198685 1198985

1198985 1198686 1198986 1198986]

X = [1205791 1199091 1199101 12057923 11990923 11991023 1205794 1199094 1199104 1205795 1199095 1199105 1205796

1199096 1199106]119879

(4)




119870 (119905) = 119870119898 +119870119886 sin (120596119898119905 + 120591) (5)



11986212 = 2120585radic11987011989811989811198982

1198981 + 1198982

(6)



119891119894119909min = 119886119894119909

119891119894119910min = 119886119894119910

(119894 = 1 23 4 5 6)

(7)




119891119863min = 119863 (8)





Items 1199111 1199112 1199113 1199114 1199115 119911611989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119889

(mm)1198971(m)

1198972(m)

Lower 17 20 25 21 17 20 15 3 15 30 30 30 80 03 02Upper 32 51 56 52 32 51 4 5 4 70 70 70 120 062 042

PO

l1 l2

dM Db34

b12 b56







24 Constraints


1 le 1199113 minus 1199114 le 4

119911119894 = Integer (119894 = 1 6) (9)


11 le 120576119894119895 le 22 (10)



04119898119905119894 minus 119878119886119894 le 0 (11)



minus 05 le 1199091198993 le 05

minus 05 le 1199091198994 le 05(12)



Contact strength

120590119867 =1070119886

radic(119894 + 1)3119870119889119879

119887119894le [120590119867]

(13)


Bending strength

120590119865 =211987011988911987911198871198891119898119910119865

le [120590119865] (14)



119863 le 280mm (15)


Start

End



sorting behind)


solutions


sorting



Terminate

Update population

No

Yes


Fuzzy set theory


Dominated sorting

Feasiblesolution

Yes

No No

Selection

Crossover

Mutation





Yes



120590max le [120590] (16)



1198941198791 =0511987811986511989721198971

(17)


107120573 = 8 (18)





(119894 = 1 23 4 5 6)


119892119906 (x) le 0

(19)


= [1199111 1199112 1199113 1199114 1199115 1199116 11989812 11989834 11989856 11988712 11988734 11988756 119889 1198971 1198972]119879

(20)







Number ofgeneration


Mutationrate

Values 300 150 09 01










CD119894 =1119903

119903

sum

119896=1

10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816 (21)






DCD119894 =CD119894

log (1119881119894) (22)


119881119894 =1119903

119903

sum

119896=1(10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816minusCD119894)

2 (23)











120583119894 =

1 119865119894 le 119865min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

le 119865119894 le 119865max119894

0 119865119894 ge 119865max119894

(24)

where 119865max119894

and 119865min119894


120583119896=

sum119873obj119894=1 120583119896

119894

sum119872

119895=1sum119873obj119894=1 120583119895

119894

(25)



Items 1199111

1199112

1199113

1199114

1199115

1199116

11989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119863

(mm)1198971(m)

1198972(m)



Items 1198861119909

(ms2)1198861119910

(ms2)11988623119909

(ms2)11988623119910

(ms2)1198864119909

(ms2)1198864119910

(ms2)1198865119909

(ms2)1198865119910

(ms2)1198866119909

(ms2)1198866119910

(ms2)119863















05 1 15 2 251

2

3

4

5

6a 1

y(m

s2)

a1x (ms2)

(a)

01 02 03 04 05 06 0705

1

15

2

a 23y

(ms2)

a23x (ms2)

(b)

0 05 1 15 20

1

2

3

4

5

a 4y

(ms2)

a4x (ms2)

(c)

05 1 15 2 251

2

3

4

5

6

a 5y

(ms2)

a5x (ms2)

(d)

0 05 1 15 20

05

1

15

2

25

3

a 6y

(ms2)

a6x (ms2)

(e)


5 Conclusions







186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)


3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Items 1199111 1199112 1199113 1199114 1199115 119911611989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119889

(mm)1198971(m)

1198972(m)

Lower 17 20 25 21 17 20 15 3 15 30 30 30 80 03 02Upper 32 51 56 52 32 51 4 5 4 70 70 70 120 062 042

PO

l1 l2

dM Db34

b12 b56







24 Constraints


1 le 1199113 minus 1199114 le 4

119911119894 = Integer (119894 = 1 6) (9)


11 le 120576119894119895 le 22 (10)



04119898119905119894 minus 119878119886119894 le 0 (11)



minus 05 le 1199091198993 le 05

minus 05 le 1199091198994 le 05(12)



Contact strength

120590119867 =1070119886

radic(119894 + 1)3119870119889119879

119887119894le [120590119867]

(13)


Bending strength

120590119865 =211987011988911987911198871198891119898119910119865

le [120590119865] (14)



119863 le 280mm (15)


Start

End



sorting behind)


solutions


sorting



Terminate

Update population

No

Yes


Fuzzy set theory


Dominated sorting

Feasiblesolution

Yes

No No

Selection

Crossover

Mutation





Yes



120590max le [120590] (16)



1198941198791 =0511987811986511989721198971

(17)


107120573 = 8 (18)





(119894 = 1 23 4 5 6)


119892119906 (x) le 0

(19)


= [1199111 1199112 1199113 1199114 1199115 1199116 11989812 11989834 11989856 11988712 11988734 11988756 119889 1198971 1198972]119879

(20)







Number ofgeneration


Mutationrate

Values 300 150 09 01










CD119894 =1119903

119903

sum

119896=1

10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816 (21)






DCD119894 =CD119894

log (1119881119894) (22)


119881119894 =1119903

119903

sum

119896=1(10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816minusCD119894)

2 (23)











120583119894 =

1 119865119894 le 119865min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

le 119865119894 le 119865max119894

0 119865119894 ge 119865max119894

(24)

where 119865max119894

and 119865min119894


120583119896=

sum119873obj119894=1 120583119896

119894

sum119872

119895=1sum119873obj119894=1 120583119895

119894

(25)



Items 1199111

1199112

1199113

1199114

1199115

1199116

11989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119863

(mm)1198971(m)

1198972(m)



Items 1198861119909

(ms2)1198861119910

(ms2)11988623119909

(ms2)11988623119910

(ms2)1198864119909

(ms2)1198864119910

(ms2)1198865119909

(ms2)1198865119910

(ms2)1198866119909

(ms2)1198866119910

(ms2)119863















05 1 15 2 251

2

3

4

5

6a 1

y(m

s2)

a1x (ms2)

(a)

01 02 03 04 05 06 0705

1

15

2

a 23y

(ms2)

a23x (ms2)

(b)

0 05 1 15 20

1

2

3

4

5

a 4y

(ms2)

a4x (ms2)

(c)

05 1 15 2 251

2

3

4

5

6

a 5y

(ms2)

a5x (ms2)

(d)

0 05 1 15 20

05

1

15

2

25

3

a 6y

(ms2)

a6x (ms2)

(e)


5 Conclusions







186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)


3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

End



sorting behind)


solutions


sorting



Terminate

Update population

No

Yes


Fuzzy set theory


Dominated sorting

Feasiblesolution

Yes

No No

Selection

Crossover

Mutation





Yes



120590max le [120590] (16)



1198941198791 =0511987811986511989721198971

(17)


107120573 = 8 (18)





(119894 = 1 23 4 5 6)


119892119906 (x) le 0

(19)


= [1199111 1199112 1199113 1199114 1199115 1199116 11989812 11989834 11989856 11988712 11988734 11988756 119889 1198971 1198972]119879

(20)







Number ofgeneration


Mutationrate

Values 300 150 09 01










CD119894 =1119903

119903

sum

119896=1

10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816 (21)






DCD119894 =CD119894

log (1119881119894) (22)


119881119894 =1119903

119903

sum

119896=1(10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816minusCD119894)

2 (23)











120583119894 =

1 119865119894 le 119865min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

le 119865119894 le 119865max119894

0 119865119894 ge 119865max119894

(24)

where 119865max119894

and 119865min119894


120583119896=

sum119873obj119894=1 120583119896

119894

sum119872

119895=1sum119873obj119894=1 120583119895

119894

(25)



Items 1199111

1199112

1199113

1199114

1199115

1199116

11989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119863

(mm)1198971(m)

1198972(m)



Items 1198861119909

(ms2)1198861119910

(ms2)11988623119909

(ms2)11988623119910

(ms2)1198864119909

(ms2)1198864119910

(ms2)1198865119909

(ms2)1198865119910

(ms2)1198866119909

(ms2)1198866119910

(ms2)119863















05 1 15 2 251

2

3

4

5

6a 1

y(m

s2)

a1x (ms2)

(a)

01 02 03 04 05 06 0705

1

15

2

a 23y

(ms2)

a23x (ms2)

(b)

0 05 1 15 20

1

2

3

4

5

a 4y

(ms2)

a4x (ms2)

(c)

05 1 15 2 251

2

3

4

5

6

a 5y

(ms2)

a5x (ms2)

(d)

0 05 1 15 20

05

1

15

2

25

3

a 6y

(ms2)

a6x (ms2)

(e)


5 Conclusions







186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)


3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Number ofgeneration


Mutationrate

Values 300 150 09 01










CD119894 =1119903

119903

sum

119896=1

10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816 (21)






DCD119894 =CD119894

log (1119881119894) (22)


119881119894 =1119903

119903

sum

119896=1(10038161003816100381610038161003816119891119896

119894+1 minus119891119896

119894minus110038161003816100381610038161003816minusCD119894)

2 (23)











120583119894 =

1 119865119894 le 119865min119894

119865max119894

minus 119865119894

119865max119894

minus 119865min119894

119865min119894

le 119865119894 le 119865max119894

0 119865119894 ge 119865max119894

(24)

where 119865max119894

and 119865min119894


120583119896=

sum119873obj119894=1 120583119896

119894

sum119872

119895=1sum119873obj119894=1 120583119895

119894

(25)



Items 1199111

1199112

1199113

1199114

1199115

1199116

11989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119863

(mm)1198971(m)

1198972(m)



Items 1198861119909

(ms2)1198861119910

(ms2)11988623119909

(ms2)11988623119910

(ms2)1198864119909

(ms2)1198864119910

(ms2)1198865119909

(ms2)1198865119910

(ms2)1198866119909

(ms2)1198866119910

(ms2)119863















05 1 15 2 251

2

3

4

5

6a 1

y(m

s2)

a1x (ms2)

(a)

01 02 03 04 05 06 0705

1

15

2

a 23y

(ms2)

a23x (ms2)

(b)

0 05 1 15 20

1

2

3

4

5

a 4y

(ms2)

a4x (ms2)

(c)

05 1 15 2 251

2

3

4

5

6

a 5y

(ms2)

a5x (ms2)

(d)

0 05 1 15 20

05

1

15

2

25

3

a 6y

(ms2)

a6x (ms2)

(e)


5 Conclusions







186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)


3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Items 1199111

1199112

1199113

1199114

1199115

1199116

11989812(mm)

11989834(mm)

11989856(mm)

11988712(mm)

11988734(mm)

11988756(mm)

119863

(mm)1198971(m)

1198972(m)



Items 1198861119909

(ms2)1198861119910

(ms2)11988623119909

(ms2)11988623119910

(ms2)1198864119909

(ms2)1198864119910

(ms2)1198865119909

(ms2)1198865119910

(ms2)1198866119909

(ms2)1198866119910

(ms2)119863















05 1 15 2 251

2

3

4

5

6a 1

y(m

s2)

a1x (ms2)

(a)

01 02 03 04 05 06 0705

1

15

2

a 23y

(ms2)

a23x (ms2)

(b)

0 05 1 15 20

1

2

3

4

5

a 4y

(ms2)

a4x (ms2)

(c)

05 1 15 2 251

2

3

4

5

6

a 5y

(ms2)

a5x (ms2)

(d)

0 05 1 15 20

05

1

15

2

25

3

a 6y

(ms2)

a6x (ms2)

(e)


5 Conclusions







186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)


3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










05 1 15 2 251

2

3

4

5

6a 1

y(m

s2)

a1x (ms2)

(a)

01 02 03 04 05 06 0705

1

15

2

a 23y

(ms2)

a23x (ms2)

(b)

0 05 1 15 20

1

2

3

4

5

a 4y

(ms2)

a4x (ms2)

(c)

05 1 15 2 251

2

3

4

5

6

a 5y

(ms2)

a5x (ms2)

(d)

0 05 1 15 20

05

1

15

2

25

3

a 6y

(ms2)

a6x (ms2)

(e)


5 Conclusions







186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)


3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












186

172

158

144

130

55

51

47

43

39

35

121

104

087

070

053

036

272

236

200

164

128

092

050

043

036

029

022

015

100

087

074

061

048

035

200

184

168

152

136

120

543

420

397

374

351

328

181

146

111

076

041

006

312

255

198

141

084

027

028

0264

0248

0232

0216

020


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 1x

(ms2)

(a)


3 35 4 45 5

0

4

8

Time (s)

minus8

minus4

a 1y

(ms2)

(b)


3 35 4 45 5

0

05

1

15

Time (s)


minus05

minus15

minus1

a 23x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 23y

(ms2)

(b)



3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3 35 4 45 5

0

02

04

Time (s)


minus02

minus04

a 4x

(ms2)

(a)


3 35 4 45 5

0

05

1

Time (s)

minus05

minus1

a 4y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 5x

(ms2)

(a)

3 35 4 45 5

0

4

8

Time (s)


minus4

minus8

a 5y

(ms2)

(b)


3 35 4 45 5

0

1

2

3

Time (s)


minus1

minus2

minus3

a 6x

(ms2)

(a)

3 35 4 45 5

0

2

4

Time (s)


minus2

minus4

a 6y

(ms2)

(b)






Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Nomenclature












Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article multiobjective optimization of steering...

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