optimization of rail profiles to improve vehicle running...

Research ArticleOptimization of Rail Profiles to Improve Vehicle RunningStability in Switch Panel of High-Speed Railway Turnouts

Ping Wang12 Xiaochuan Ma12 Jian Wang12 Jingmang Xu12 and Rong Chen12

1MOE Key Laboratory of High-Speed Railway Engineering Southwest Jiaotong University Chengdu 610031 China2School of Civil Engineering Southwest Jiaotong University Chengdu 610031 China

Correspondence should be addressed to Jingmang Xu mang080887163com

Received 31 October 2016 Accepted 23 January 2017 Published 12 February 2017

Academic Editor Gen Q Xu

Copyright copy 2017 Ping Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Amethod for optimizing rail profiles to improve vehicle running stability in switch panel of high-speed railway turnouts is proposedin this paper The stock rail profiles are optimized to decrease the rolling radii difference (RRD) Such characteristics are definedthrough given rail profiles and the target rolling radii difference is defined as a function of lateral displacements of wheel setThe improved sequential quadratic programming (SQP) method is used to generate a sequence of improving profiles leading tothe optimum one The wheel-rail contact geometry and train-turnout dynamic interaction of the optimized profiles and thoseof nominal profiles are calculated for comparison Without lateral displacement of wheel set the maximum RRD in relation to anominal profile will be kept within 05mmndash1mm while that in relation to an optimized profile will be kept within 03mmndash05mmFor the facing and trailing move of vehicle passing the switch panel in the through route the lateral wheel-rail contact force isdecreased by 340 and 299 respectively the lateral acceleration of car body is decreased by 419 and 407 respectively andthe optimized profile will not greatly influence the vertical wheel-rail contact forceThe proposed method works efficiently and theresults prove to be quite reasonable

1 Introduction

Railway turnouts are essential components in railway appli-cations as they can provide flexibility for traffic operationsenabling vehicles to change among the tracks They are madeup of a switch panel a crossing panel and a closure panel[1] To realise wheel transfer from a stock rail to a switchrail switch rail profiles are designed with variation along theswitch panel and combined with the stock rail The variationin rail profiles changes boundary conditions of wheel-railcontact resulting in a contact patch that is no longer ellipticalIn railway turnouts the combination of a switch rail anda stock rail to bear wheel loads together makes multipointcontact more common Situations of nominal wheel-rail con-tact are disturbed when a vehicle passes through a turnoutpotentially resulting in severe impact loads Material lossrolling contact fatigue and accumulated plastic deformationsare common damage mechanisms that further disturb nomi-nal contact conditions [2] Due to this phenomenon railwayturnouts account for the largest contribution of reported

faults and require more maintenance than other componentsof railway networks

The design of rail and wheel profiles is a fundamentalproblem and various approaches were developed to obtaina satisfactory combination of wheel and rail Typically arail or wheel profile was designed using the trial-and-errorapproach but during the last few decades a number of effortshave been made using numerical methods in the designprocessThemain idea in optimizing rail andwheel profiles ofordinary tracks is based on wheel-rail contact characteristicssuch as rolling radii difference (RRD) of wheel set Shevtsovet al [3] established models to design an optimal wheelprofile by minimizing the difference between the target andthe actual RRD function and three approaches for choosinga target RRD function were suggested they also used thesame method to design a wheel profile by considering wrrolling contact fatigue (RCF) and wear the paper showsthat the RRD of wheel set on the straight line should beas low as possible in order to ensure the lateral runningstability of the wheel set [4] Jahed et al [5] proposed a

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 2856030 13 pageshttpsdoiorg10115520172856030

2 Mathematical Problems in Engineering

computationally efficient optimization algorithm that workswith five design variables using a method targeting the RRDfunction and they put forward a cubic spline interpolationmethod to ensure the convexity and monotony of wheelprofiles Markine et al [6] designed an optimal wheel pro-file based on geometrical wheel-rail contact characteristicssuch as RRD They chose an optimization method calledmultipoint approximations based on response surface fitting(MARS) as a numerical technique Other efforts have beenmade based on different optimal targets Polach [7] studiedthe interrelationship between the nominal contact angleconformity and equivalent conicity and this relationship wasconsidered in the method for wheel profile design Shen etal [8] developed a method for generating new wheel profilesusing the contact angle function between wheels and railsCui et al [9] used a conventional optimization procedureto obtain wheel profiles that placed the weighted wr gap asthe objective function and they put forward a new wheelprofile design method to reduce the hollow wear by seekingan optimization match of the wheel profiles the vehiclersquossuspension systems and the wear behaviour of wheels inservice [10] Smallwood et al [11] suggested a method tomodify transverse rail profiles in order to reduce contactstress and used theoretical methods to investigate the effectof profile changes on contact stress and conicity Novales etal [12] proposed a new general methodology for improvingwheel profiles in relation to certain physical phenomenathat arise during the running of the vehicle over tracksand the methodology was based on the genetic algorithmtechnique Ignesti et al [13] presented an innovative wheelprofile optimization procedure specifically designed with theaim of improving the wear and stability of vehicles Lin et al[14] proposed a multiobjective optimization model of wheelprofile to reduce the wheel wear of electric multiple unitsFor the curved track some optimum design methods forasymmetric profiles are often used to reduce the wear ofwheel flange and rail side Choi et al [15 16] proposed anoptimum design procedure for asymmetric railhead profilesby minimizing the wear and fatigue on curved tracks Zhai etal [17] put forward a designmethodology of rail asymmetric-grinding profiles based on the principle of a low dynamicinteraction of wheel-rail

The optimal studies of railway turnouts are less thanordinary track because the wr contact relationship of railwayturnouts is more complex At present the main optimal ideaof railway turnouts focuses on the geometry parameters suchas the height of switch rail the thickness of nose rail and thetrack gauge design of switch panel Palsson and Nielsen [18]proposed a methodology for the optimization of a prescribedtrack gauge variation in the switch panel of a railway turnoutthe aim of which is reducing rail profile degradation AlsoPalsson et al presented some methods for profile geometryoptimization of switch rail and nose rail by using a genetictype optimization algorithm the purpose of these efforts isto improve the dynamic interactions between wheel and railand minimize contact pressure and accumulated damage ofrails [19 20] To improve the dynamic behaviour of turnoutcrossings Wan et al [20] proposed a numerical optimizationmethod to minimize RCF damage and wear in the crossing

panel by varying the nose rail shape Nicklisch et al [21]represented the geometry of a designed track gauge variationin the switch panel in a parametric way to find the means ofimproving the switch panel design Bugarın and Garcıa Dıaz-De-Villegas [22] studied the problem of turnout optimizationwith respect to impact loads occurring during the negotiationof the main line proposing a gauge widening method in aswitch panel These optimal efforts can improve the dynamicbehaviour of railway turnouts to minimize RCF damage andwear

For the switch panel of railway turnout the variable cross-section rails will lead to the RRD of wheel set The RRDof wheel set is not always the same at different positionsalong the longitudinal direction of the railway line whichcan cause further lateral motion of the wheel set decreaseof vehicle running stability severe dynamic vehicle-turnoutinteraction and damage to wheels and rails In this paperan optimal method in switch panel of high-speed railwayturnouts is present based on the RRD function and theoptimal principle is tominimize the rolling radii difference ofwheel set as far as possible Based on a standard designCN60-1100-118 turnout (curve radius 1100m turnout angle 118)the rail profiles are optimized by this method The wheel-railcontact geometry and dynamic interaction between vehicleand turnout of the optimized profiles and those of nominalprofiles are calculated for comparison

2 Optimization of Rail Profiles

21 Rolling Radii Difference in Switch Panel For switchpanel of railway turnout the variable cross-section rails willinevitably causeRRDofwheel set As shown in Figure 1 in theswitch panel of railway turnout in the through route withoutlateral displacement of wheel set the wheel-rail contactposition changes with the variable cross-section switch railsThe rolling radius of the wheel at both sides is differenttherefore the rolling distances of the wheel at both sides aredifferent at the same time and the wheel set moves underthe action of lateral creep force The lateral motion of wheelset will reduce the vehicle running stability at a high-speedlevel In Figure 1 119909119910119911 refers to the coordinate system of thecentreline the 119909-axis and 119910-axis refer to the longitudinaldirection and lateral direction of the railway line and 119911-axisrefers to the vertical direction

The varied profiles lead to differences in rolling radiusof wheel set The normal wheel-rail contact situations aredisturbed when wheels transfer between two rails in railwayturnouts (eg the stock rail and switch rail) sometimesresulting in severe impact loads Dynamic vehicle-turnoutinteraction between train and turnout is a time variantprocess and is more complex than plain line due to thelateral translations of wheel set and varied rail profilesthus having a significant effect on the dynamic interactionbetween train and turnout which may influence the vehiclerunning stability The rolling radii difference in switch panelcan be expressed as follows

Δ119903 = 1003816100381610038161003816119903119903 minus 1199031198971003816100381610038161003816 (1)

Mathematical Problems in Engineering 3

A B

Straight stock rail Straight switch rail and curved stock rail

Geometric centre of wheel set

Centreline of track

y

zx

rl = r0 rr = r0 + Δr

Figure 1 Rolling radii in switch panel of railway turnout Points A and B refer to the wheel-rail contact positions 119903119897 and 119903119903 are rolling radiiof the wheels at both sides r0 refers to the nominal rolling radius of the wheel and Δr is RRD of the wheel set

Section A Section BSection C

Straight stock rail

Centreline of track

Straight switch rail

Curved stock rail

Figure 2 Optimization plan of rail profiles in switch panel The red dotted lines refer to the path line of wheel-rail contact points

where 119903119903 is the rolling radius of the wheel on the side of theswitch rail and 119903119897 is the rolling radius of the wheel on theopposite side of the track

22 Optimization Plan As shown in Figure 2 the variablecross-section rails lead to RRD of the wheel set betweenSectionA and SectionB evenwithout the lateral displacementof wheel set therefore this region can be taken as theoptimization range Section A is the point of switch rail and isthe starting point of the variable cross-section rails Section Bis the end point of the variable cross-section rails Within theoptimization region the RRD of wheel set is not always thesame at different positions along the longitudinal directionof the railway line but reaches the peak value at Section Ctherefore Section C is selected as the control position on theoptimization planThe rail profiles of SectionA and Section Bare not optimized and the rail profile of the straight stock railat Section C is optimized using the method which describedin the subsequent sectionsThe rail profile between Section Aand Section C as well as that between Section C and Section Bcan be obtained according to the linear interpolationmethod

Section C is the position before the wheels transfer fromthe stock rail to the switch rail when vehicle passes theswitch panel with the facing move in the through route thecalculation method of the position is as follows

(1) Calculate the positions of wheel-rail contact pointsper 01m along the longitudinal direction of therailway line from Section A to Section B Since thelateral displacement of the wheel set is low whenvehicle passes the switch panel in the through routethe situation of wheel flanges clinging to rails isneglected therefore the lateral displacement of wheelset is selected from minus8mm to 8mm

(2) When all of wheel-rail contact points of a positionare located on the stock rail and at the next positionall or part of contact points are transferred from thestock rail to the switch rail then the position can beconsidered as Section C

Considering the influence law of variable cross-sectionrails of the switch panel on RRD of wheel set along thelongitudinal direction of the railway line the rail profile of thestraight stock rail at Section C can be selected as the object tobe optimized The rail profile between Section A and SectionC as well as that between Section B and Section C can beobtained according to the linear interpolation method Thisoptimization plan can help to reduce RRDs of wheel sets fordifferent positions of the switch panel as much as possible

23 OptimizationMethod Theoptimization region is locatedon the railhead of the straight stock rail For example in the


ℎ (mm)

(m

m)

Rail profileMoving pointsFixed points

Centreline of rail(ℎi i) 15 mm

A

B

Optimization region

Measuring point of track gauge

Figure 3 Optimization region of rail profile

optimized profile design of a straight stock rail (Figure 3)the optimization region of the rail profile is from A to BThe range from A to B relates to the region with possiblewheel-rail contact where Point A is the measuring point oftrack gauge and Point B is a position 15mm away from thecentreline of the railhead on the right side The position ofthese moving points (ℎ119894 V119894 119894 = 1 2 119899) can be variedin order to get the optimized profile Respectively ℎ119894 and V119894are the lateral and vertical coordinates of the moving pointsTo simplify modelling the abscissa of each moving point ℎ119894(119894 = 1 2 119899) is selected as a constant and the verticalcoordinates of each moving point V119894 are considered to bevariedMoreover V119894 is chosen as a variable in the optimizationproblem The rail profile can be expressed as 119891 (V119894)

When the lateral displacement of wheel set is 119910119896 thepositions of wheel-rail contact points at both sides can beobtained based on the principle of the tracing line method[23] The position of wheel-rail contact point at the right sideis 119862119903119896 (119910119903119896 119911119903119896) and the position of wheel-rail contact pointat the left side is 119862119897119896 (119910119897119896 119911119897119896) The rolling radii of the wheelscan be calculated according to the positions of wheel-railcontact pointsThe RRD function of wheel set can be definedas

Δ119903119896 =1003816100381610038161003816119903119903119896 minus 119903119897119896

1003816100381610038161003816 (2)

where 119903119903119896 and 119903119897119896 are the rolling radii of the wheels at bothsides The rolling radii of the wheels relate to the positions ofwheel-rail contact points which are dependent on the lateralwheel set displacement and wheel-rail profiles so the RRDfunction of wheel set can also be defined as

Δ119903119896 = Δ119903119896 (119910119896 V1 V2 V119899) (3)

Considering the lateral wheel set displacement range theaverage RRD function of wheel set can be defined as

119878 = sum119898119896=1 Δ119903119896 (119910119896 V1 V2 V119899)

119898 (4)

where 119898 refers to the number of calculations for lateralwheel set displacement which can be obtained according tothe lateral wheel set displacement range and the calculationstep length Since the probability of lateral wheel set dis-placement varies the weight coefficient 119908 should be adoptedfor controlling the influence of different lateral displacementof wheel set on function 119878 Supposing that the probabilityof lateral wheel set displacement meets the law of normaldistribution (120583 1205902) then according to [9] the lateral wheelset displacement has the highest probability of being lessthan 4mm So in the normal distribution curve of lateralwheel set displacement the mathematical expectation 120583 is 0and the standard deviation 120590 is 4 The weight coefficient inrelation to different lateral wheel set displacement results canbe expressed as

119908 =

068 10038161003816100381610038161199101198941003816100381610038161003816 isin [0 4]

027 10038161003816100381610038161199101198941003816100381610038161003816 isin (4 8]

005 10038161003816100381610038161199101198941003816100381610038161003816 isin (8 12]

(5)

When the weight coefficient is considered the function 119878can be further expressed as

119878 =sum119898119896=1 119908119895Δ119903119896 (119910119896 V1 V2 V119899)

119898 (6)

Since the parameter 119910119896 is given in the calculation processthe function 119878 in (6) can be expressed as follows

119878 = 119878 (V1 V2 V119899) (7)

Equation (7) is used as the objective function to find theoptimized profile of the rail

In the process of finding the optimized profile it isnoted that the rail profile is a convex curve To ensure theauthenticity of the rail profile a constraint equation is usedas follows

119866119894 = V119894+1 minusV119894 + V119894+22 gt 0 119894 = 1 2 119899 minus 2 (8)

Considering the initial profile of the rail and the capacityof optimization the boundaries of variables V119894 are defined asfollows

119886119894 le V119894 le 119887119894 119894 = 1 2 119899 (9)

Equations (8) and (9) are used as the constrained func-tions to find the optimized profile of the rail

An algorithm is developed using the improved SQP(sequential quadratic programming) method combined withthe Quasi-Newton and BFGS (Broyden-Fletcher-Goldfarb-Shanno) method [24ndash28]The optimization problem accord-ing to (7)ndash(9) can be represented as follows

min 119878 = 119878 (V1 V2 V119899)

st 119866119894 gt 0 119886119894 le V119894 le 119887119894 119894 = 1 2 119899(10)


Input the position parameters wheelrail profilesinitial rail profile track gauge rail cant

Input other parameters weighting factors range andcalculating step of lateral wheel set displacement

Define the initial rail profile function

The RRD function of wheel set

The value of objective function

Adjusted the value of

Yes

No SQP

f(1 2 n)

Δrk120577(yk 1 2 n)

S120577(1 2 n)

Variables i

10038161003816100381610038161003816S120577 minus S120577minus110038161003816100381610038161003816 lt 120576

Output the value i get the optimized rail profile which defined by f(1 2 n)

Figure 4 Flow chart of the optimization algorithm

Table 1 The parameter values of switch panel

Parameters Value UnitTotal length of switch rail 21449 mVariable cross-section length of switch rail 10962 mTrack gauge 1435 mRail cant 140 mdash

This paper develops the relevant computer code in orderto solve the optimization of rail profiles Figure 4 shows theflow chart of the optimization algorithm

24 Optimization Results A case study is based on a standarddesign CN60-1100-118 turnout (curve radius 1100m turnoutangle 118) See Table 1 for the key parameters of switch panelfor other structural parameters see [29] which will not berepeated here As shown in Table 1 Section B is 10962m fromthe point of switch rail

Figure 5 shows the wheel-rail contact points distributionat four consecutive positions along the direction of therailway line The range of lateral wheel set displacement isfrom minus8mm to 8mm and the wheel tread is LMA When119909 = 54m all the wheel-rail contact points are located onthe stock rail and when 119909 = 55m part of contact points aretransferred from the stock rail to the switch rail According tothe calculation method Section C is 54m from the point ofswitch rail

The range of lateral wheel set displacement is fromminus12mm to 12mm and the calculated step length is adoptedas 05mm In the constraint condition of optimization theoptimized capacity of parameter V119894 is adopted as 05mmThe number of the moving points is 14 [5 9] The straight

stock rail profile at Section C is optimized according tothe method which is described in Section 23 For thecomparison between the rail profiles obtained before andafter the optimization of the straight stock rail side at SectionC see Figure 6 Furthermore the CPU time taken by theoptimization algorithm to solve the case in Figure 6 usinga conventional desktop with 32GHz processor is 62931 s Itshould also be noted that the optimization algorithm is codedas a Matlab function

3 Calculation of Wheel-RailContact Geometry

The calculation process for wheel-rail contact geometricparameters was developed based on the principle of thetrace line method [23] The wheel-rail contact geometricparameters with nominal profiles and optimized profiles werefirst calculated and then the influence of the optimized profileon the wheel-rail contact position and RRD was analysed inthis section

31 Contact Distribution of Section C The wheel-rail contactdistribution of Section C with a nominal profile is shown(Figure 7) In the calculation process the lateral displacementrange of the wheel set is from minus12mm to 12mm and thecalculation step is 05mmThe direction is positive when thewheel set moves to the switch rail

With the nominal profile Figure 7 indicates that when thelateral displacement of wheel set is 0 the 119910 coordinate of thewheel-rail contact point on straight stock rail is minus7529 Withthe lateral displacement of wheel set changes from minus4mmto 4mm the 119910 coordinate of the wheel-rail points on theleft wheel is changed from minus7423 to minus7570 and the possible


680 700 720 740 760 780 800 820

y (mm)

z(m

m)

Wheel profileRail profilesPairs of contact points

0

15

30

45

60

minus88

(a)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(b)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(c)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus8

8

(d)

Figure 5 Distribution of wheel-rail contact points at different longitudinal positions (a) 119909 = 53m (b) 119909 = 54m (c) 119909 = 55m and (d)119909 = 56m

wheel-rail contact width on the wheel is 147mmThe wheel-rail contact distribution of Section C with an optimizedprofile is schematically shown (Figure 8)

With the optimized profile when the lateral displacementof wheel set is 0 the 119910 coordinate of the wheel-rail con-tact point on straight stock rail is minus7591 With the lateraldisplacement of wheel set changes from minus4mm to 4mmthe 119910 coordinate of the wheel-rail points on the left wheelis changed from minus7543 to minus7620 and the possible wheel-rail contact width on the wheel is 77mm The optimizedprofile has little influence on the wheel-rail contact pointdistribution at the switch rail side (right side) The positionof wheel-rail contact points at the straight stock rail (left side)

moves to the outside of the rail and the movement value isaround 62mm When the lateral displacement of wheel setis kept within minus4mmndash4mm the possible wheel-rail contactwidth at the straight stock rail side with optimized profilewill be reduced from 147mm to 77mm The contact pointdistribution on the wheel is more centralized which to acertain extent is good for improving the running stability ofwheel set

32 Contact Point Position on Wheels As mentioned inSection 21 for the switch panel of railway turnout the changeof wheel-rail contact points at the switch rail side will cause


minus40 minus20 0 20 40

ℎ (mm)

0

5

10

15

20

25

(m

m)

Normal profileOptimized profile

Figure 6 Optimized rail profile for Section C

minus780 minus760 minus740 minus720 minus700

y (mm)

0

10

20

30

40

50

60

z(m

m)

12 8 4 0 minus4 minus8 minus12

(a)

0

10

20

30

40

50

60z

(mm

)

12 8 4 minus4minus8minus12

700 720 740 760 780 800 820

y (mm)

(b)

Figure 7 Contact distribution with nominal profile (a) Stock rail and (b) switch rails The red solid line represents the possible wheel-railcontact range on wheel with the lateral wheel set displacement from minus4mm to 4mm

RRDof wheel setThemore the contact points at both sides ofthe wheels deviate the lower the wheel set running stabilitywill become So in this paper the influence of optimizedprofile on the wheel set running stability can be evaluatedroughly through calculating the contact point distributionat both sides of the wheels along the longitudinal directionof the railway line without lateral displacement of wheel setFigure 9 shows the law of changing contact point positionson wheels along the longitudinal direction of the railwayline under the comparison between nominal and optimizedprofile

The optimized profiles have little influence on the wheel-rail contact points at the switch rail side while the con-tact points at the straight stock rail side are closer to theoutside of the wheels For nominal profiles the maximumdistance between the contact points of wheels at both sides is

239mmwhile for optimized profiles themaximumdistanceis 175mm decreasing by 268

33 Rolling Radii Difference As shown in Figure 10 the RRDof wheel set in a contour plot is shown as a function ofwheel set distance from the point of the switch rail and lateraldisplacement of the wheel set In the calculation process thelateral displacement range of the wheel set is from minus8mmto 8mm and the calculation step is 05mm In addition thelateral displacement is positive when the wheel set moves tothe switch rail

The figure illustrates that the RRD of wheel set decreaseswith the optimized profilesWhen the lateral displacement ofwheel set is changed from minus8mm to 8mm for the optimizedprofiles themaximumRRDof wheel set can be reduced from



y (mm)

0

10

20

30

40

50

60z

(mm

)


(a)

0

10

20

30

40

50

60

z(m

m)

700 720 740 760 780 800 820

y (mm)

12 8 4 minus4minus8 minus12

(b)

Figure 8 Contact distribution with optimized profile (a) Stock rail and (b) switch rails The red solid line represents the possible wheel-railcontact range on wheel with the lateral wheel set displacement from minus4mm to 4mm

0

10

20

30

40

Con

tact

loca

tion

on w

heel

(mm

)

0 3 6 9 12

Distance from front of switch (m)

Δmax = 239 mm

Left wheel the stock rail sideRight wheel the switch rail side

(a)

0

10

20

30

40C

onta

ct lo

catio

n on

whe

el (m

m)

0 3 6 9 12



Δmax = 175 mm

(b)

Figure 9 Distribution of wheel-rail contact points without lateral wheel set displacement (a) Nominal profile and (b) optimized profile Asthe value of the vertical axis increases the contact point will come closer to the outside of wheels

38mm to 31mm Without the lateral displacement of wheelset the maximum RRD in relation to a nominal profile willbe kept within 05mmndash1mm while that in relation to anoptimized profile will be kept within 03mmndash05mm

4 Dynamic Interactions ofVehicle and Turnout

41 Calculation Model The vehicle-turnout dynamic inter-action is simulated in the validated commercial softwareSimpack The calculation model includes two interactiveparts a three-dimensional multibody model of the Chinesehigh-speed vehicle CRH2 (Figure 11) and a space-dependentmodel of the railway turnout with a flexible track foundation(Figure 12) which are connected by a local model of the wrcontact [30]

The calculation model of the vehicle includes modellingof the car body suspension elements bogies and wheelsetsIn this paper a three-dimensional multibody model of thevehicle is implemented The car body bogie frames and thewheelsets are modelled as rigid bodies The parameter valuesof the vehicle model are listed below (Table 2) The simulatedturnout model is based on a standard design CN60-1100-118 (curve radius 1100m with a turnout angle 118) Thevaried rail profiles are realised by sampling several crosssections of the switch rail at certain positions along theswitch panel Furthermore the specific features of the switchpanel such as sudden changes of track curvature due to theabsence of transition curves cant deficiency and variationsin rail cross section are considered In the wr contactmodel the Hertz contact theory is applied for calculationof the normal contact force and the FASTSIM algorithm


minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


38

30

25

20

15

10

05

03

01

(a)

minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


31

30

25

20

15

10

05

03

01

(b)

Figure 10 Contour plot of RRD (mm) showing the lateral displacement of the wheel set versus the distance from the point of the switch rail(a) Nominal profile and (b) optimized profile

Table 2 Parameter values of vehicle model

Parameters Value UnitThe mass of car body 42400 kgThe rolling moment of inertia of car body 706e5 kgsdotm2The nodding moment of inertia of car body 227e6 kgsdotm2The yawing moment of inertia of car body 208e6 kgsdotm2Themass of bogie 3100 kgThe rolling moment of inertia of bogie 5045 kgsdotm2The nodding moment of inertia of bogie 2806 kgsdotm2The yawing moment of inertia of bogie 2247 kgsdotm2Themass of wheel set 1850 kgThe rolling moment of inertia of wheel set 717 kgsdotm2The yawing moment of inertia of wheel set 717 kgsdotm2Half of longitudinal stiffness of secondary suspension 145e5 NmHalf of lateral stiffness of secondary suspension 205e5 NmHalf of vertical stiffness of secondary suspension 148e5 NmHalf of longitudinal damping of secondary suspension 343e5 NsdotsmHalf of lateral damping of secondary suspension 245e4 NsdotsmHalf of vertical damping of secondary suspension 316e4 NsdotsmHalf of longitudinal stiffness of primary suspension 280e7 NmHalf of lateral stiffness of primary suspension 280e7 NmHalf of vertical stiffness of primary suspension 280e7 NmHalf of lateral damping of primary suspension 177e4 NsdotsmThe vertical distance between mass centre of vehicle body and secondary suspension 1100 mThe vertical distance between secondary suspension and mass centre of bogie 0100 mThe vertical distance between mass centre of bogie and primary suspension 0270 mHalf of lateral distance between primary suspensions 0813 mHalf of lateral distance between secondary suspensions 0978 mHalf of bogie fixed axles distance 1350 mHalf of distance between mass centres of car bodies 8750 mNominal rolling radius of wheel 0430 m


Figure 11 Multibody model of vehicle

is used to solve the tangential contact problem During thesimulation calculation a vehicle passing the switch panel inthe through route with facing and trailing move is simulatedrespectively

42 Dynamic Responses with Facing Move When a vehiclepasses the switch panel in the through route with facingmove the effects of the optimized profile on the dynamicinteraction between the vehicle and turnout are analysedand the simulation is carried out at the speed of 350 kmhfor the vehicle The calculation results include the verticaland lateral wheel-rail contact force at the side of the switchrail the lateral displacement of the wheel set and the lateralvibration acceleration of car body (the vibration accelerationis calculated at the centre of mass of the car body) (Figure 13)

Theoptimized profiles have a great influence on the lateraldynamic interaction between the vehicle and turnout In theswitch panel the maximum lateral contact force is 94 kN forthe nominal profile and is 62 kN for the optimized profiledecreasing by 340 The optimized profile will not greatlyinfluence the vertical wheel-rail contact forceThemaximumlateral displacement of wheel set is 49mm for the nominalprofile and is 31mm for the optimized profile decreasingby 367 The hunting motion amplitude of the wheel set inthe switch panel is decreased extensively The amplitude ofthe lateral acceleration of the car body is decreased by 419(from 031ms2 to 018ms2)

43 Dynamic Responses with Trailing Move When a vehiclepasses the switch panel in the through route with trailingmove the effects of the optimized profile on the dynamicinteraction between the vehicle and turnout are analysed andthe simulation is carried out at the speed of 350 kmh forthe vehicle The content of simulation calculation is the sameas that mentioned in Section 42 The calculation results areshown in Figure 14

In the switch panel the maximum lateral contact force is77 kN for the nominal profile and is 54 kN for the optimizedprofile decreasing by 299 The optimized profile will notgreatly influence the vertical wheel-rail contact force The

Ksz Csz

Ksy

Csy

Csz = 94000 Nsm Ksz = 10 lowast 108 NmCsy = 49000 Nsm Ksy = 50 lowast 107 Nm

Figure 12 Track foundation of flexibility

maximum lateral displacement of wheel set is 48mm forthe nominal profile and is 33mm for the optimized profiledecreasing by 313 The hunting motion amplitude of thewheel set in the switch panel is decreased extensively Theamplitude of the lateral acceleration of the car body isdecreased by 407 (from 027ms2 to 016ms2) Overallthe optimized profiles have a great influence on the lateraldynamic interaction between the vehicle and turnout and thelateral stability of the vehicle is improved obviously by theoptimized design

5 Conclusions

In this paper a method for optimization of rail profilesto improve vehicle stability in switch panel of high-speedrailway turnouts is presented the effects of optimized profileon the wr contact geometric parameters and dynamic inter-action of vehicle and turnout are simulated and both nominaland optimized profiles are taken as inputs for the simulationThe optimized profile strongly improves the lateral dynamicinteraction of vehicle and turnoutThe outcome of this papercould provide some theoretical guidance for the grinding ofswitch rail profiles and modify the dynamic performance ofthe vehicle thus increasing ride quality According to theresults the following can be summarized

(1) The optimized profile of Section C has little influenceon the wheel-rail contact point distribution at theswitch rail side The position of wheel-rail contactpoints at the straight stock rail moves to the outsideof the rail and the maximum movement value isaround 62mm When the lateral displacement ofwheel set is kept within minus4mmndash4mm the possiblewheel-rail contact width at the straight stock rail sidewith optimized profile will be reduced from 147mmto 77mmThe contact point distribution on thewheelis more centralized which is good for improving therunning stability of wheel set

(2) When the lateral displacement of wheel set is changedfrom minus8mm to 8mm for the optimized profiles themaximum RRD of wheel set can be reduced from38mm to 31mmWithout the lateral displacement ofwheel set themaximumRRD in relation to a nominalprofile will be kept within 05mmndash1mm while thatin relation to an optimized profile will be kept within


minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh

Nominal profileOptimized profile

(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)

Figure 13 The comparison of dynamic interaction with facing move (a) Lateral contact force (b) vertical contact force (c) lateraldisplacement of wheel set and (d) lateral acceleration of car body

03mmndash05mm The RRD of wheel set is decreasedwith the optimized profile

(3) When a vehicle passes the switch panel in the throughroute with facing move the maximum lateral contactforce is 94 kN for the nominal profile and is 62 kNfor the optimized profile decreasing by 340 Theoptimized profile will not greatly influence the ver-tical wheel-rail contact force The maximum lateraldisplacement of wheel set is 49mm for the nomi-nal profile and is 31mm for the optimized profiledecreasing by 367 The amplitude of the lateralacceleration of the car body is decreased by 419(from 031ms2 to 018ms2)

(4) When a vehicle passes the switch panel in the throughroutewith trailingmove themaximum lateral contactforce is 77 kN for the nominal profile and is 54 kNfor the optimized profile decreasing by 299 Theoptimized profile will not greatly influence the ver-tical wheel-rail contact force The maximum lateraldisplacement of wheel set is 48mm for the nomi-nal profile and is 33mm for the optimized profiledecreasing by 313 The amplitude of the lateralacceleration of the car body is decreased by 407(from 027ms2 to 016ms2)

The profile optimization of the railway turnout is a long-termand complicated process and different objectives should


minus10minus20 0 10 20


350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)

Figure 14 The comparison of dynamic interaction with trailing move (a) Lateral contact force (b) vertical contact force (c) lateraldisplacement of wheel set and (d) lateral acceleration of car body

be chosen under different conditions in the optimal designFor example the stability of the vehicle is the most importantin high-speed railway turnout therefore an optimal designmethod is presented in this paper based on the objective thatcan decrease the amplitude of hunting motion of the wheelset in the switch panel In the heavy-haul railway turnout thewear problem of the wheel and rail is the focus in optimizedresearch Since the speed of the vehicle when passing theturnout in the through route is much faster than divergingline therefore the requirement of vehicle running stability inthe through route is stricterThe optimization method in thispaper is presented to improve vehicle running stability in thethrough route the influence of optimized profiles on vehiclerunning stability in diverging line is not considered and itwill be considered in the future work if necessary

Competing Interests

The authors declared that there is no conflict of interests intheir submitted paper

Acknowledgments

Thepresent work has been supported by the National NaturalScience Foundation of China (51425804 and 51608459) theKey Project of the Chinarsquos High-Speed Railway United Fund(U1234201) and the Science andTechnology Project of ChinaRailwayCorporation (2015G006-C) and thanks are due to thesupport with the Doctoral Dissertation Cultivation Project ofSouthwest Jiaotong University (2015)


References

[1] E Kassa C Andersson and J C O Nielsen ldquoSimulation ofdynamic interaction between train and railway turnoutrdquoVehicleSystem Dynamics vol 44 no 3 pp 247ndash258 2006

[2] N Burgelman Z Li and R Dollevoet ldquoA new rolling contactmethod applied to conformal contact and the train-turnoutinteractionrdquoWear vol 321 pp 94ndash105 2014

[3] I Y Shevtsov V L Markine and C Esveld ldquoOptimal design ofwheel profile for railway vehiclesrdquo Wear vol 258 no 7-8 pp1022ndash1030 2005

[4] I Y Shevtsov V L Markine and C Esveld ldquoDesign of railwaywheel profile taking into account rolling contact fatigue andwearrdquoWear vol 265 no 9-10 pp 1273ndash1282 2008

[5] H Jahed B Farshi M A Eshraghi and A Nasr ldquoA numericaloptimization technique for design of wheel profilesrdquoWear vol264 no 1-2 pp 1ndash10 2008

[6] V L Markine I Y Shevtsov and C Esveld ldquoAn inverseshape design method for railway wheel profilesrdquo Structural andMultidisciplinary Optimization vol 33 no 3 pp 243ndash253 2007

[7] O Polach ldquoWheel profile design for target conicity and widetread wear spreadingrdquoWear vol 271 no 1-2 pp 195ndash202 2011

[8] G Shen J B Ayasse H Chollet and I Pratt ldquoA unique designmethod for wheel profiles by considering the contact anglefunctionrdquo Proceedings of the Institution ofMechanical EngineersPart F Journal of Rail and Rapid Transit vol 217 no 1 pp 25ndash30 2003

[9] D Cui L Li X Jin and X Li ldquoOptimal design of wheel profilesbased on weighed wheelrail gaprdquo Wear vol 271 no 1-2 pp218ndash226 2011

[10] D Cui H Wang L Li and X Jin ldquoOptimal design of wheelprofiles for high-speed trainsrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and Rapid Transitvol 229 no 3 pp 248ndash261 2015

[11] R Smallwood J C Sinclair and K J Sawley ldquoAn optimizationtechnique to minimize rail contact stressesrdquoWear vol 144 no1-2 pp 373ndash384 1991

[12] M Novales A Orro and M R Bugarın ldquoUse of a geneticalgorithm to optimize wheel profile geometryrdquo Proceedings ofthe Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 221 no 4 pp 467ndash476 2007

[13] M Ignesti A Innocenti L Marini E Meli and A RindildquoA numerical procedure for the wheel profile optimisation onrailway vehiclesrdquo Proceedings of the Institution of MechanicalEngineers Part J Journal of Engineering Tribology vol 228 no2 pp 206ndash222 2014

[14] F T Lin X Q Dong Y M Wang and C S Ni ldquoMultiobjec-tive optimization of CRH3 EMU wheel profilerdquo Advances inMechanical Engineering vol 7 no 1 Article ID 284043 pp 1ndash82015

[15] H-Y Choi D-H Lee C Y Song and J Lee ldquoOptimizationof rail profile to reduce wear on curved trackrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14 no4 pp 619ndash625 2013

[16] H-Y Choi D-H Lee and J Lee ldquoOptimization of a railwaywheel profile to minimize flange wear and surface fatiguerdquoWear vol 300 no 1-2 pp 225ndash233 2013

[17] W Zhai J Gao P Liu and K Wang ldquoReducing rail side wearon heavy-haul railway curves based on wheel-rail dynamicinteractionrdquo Vehicle System Dynamics vol 52 no 1 pp 440ndash454 2014

[18] B A Palsson and J C O Nielsen ldquoTrack gauge optimisationof railway switches using a genetic algorithmrdquo Vehicle SystemDynamics vol 50 no 1 pp 365ndash387 2012

[19] B A Palsson ldquoOptimisation of railway crossing geometryconsidering a representative set of wheel profilesrdquo VehicleSystem Dynamics vol 53 no 2 pp 274ndash301 2015

[20] C Wan V L Markine and I Y Shevtsov ldquoImprovement ofvehicle-turnout interaction by optimising the shape of crossingnoserdquo Vehicle System Dynamics vol 52 no 11 pp 1517ndash15402014

[21] D Nicklisch E Kassa J Nielsen M Ekh and S IwnickildquoGeometry and stiffness optimization for switches and cross-ings and simulation of material degradationrdquo Proceedings of theInstitution of Mechanical Engineers Part F Journal of Rail andRapid Transit vol 224 no 4 pp 279ndash292 2010

[22] M R Bugarın and J-M Garcıa Dıaz-De-Villegas ldquoImprove-ments in railway switchesrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and Rapid Transitvol 216 no 4 pp 275ndash286 2002

[23] Z Li Wheel-Rail Rolling Contact and Its Application to WearSimulation Delft University of Technology Delft The Nether-lands 2002

[24] P E Gill and D P Robinson ldquoA globally convergent stabilizedSQP methodrdquo SIAM Journal on Optimization vol 23 no 4 pp1983ndash2010 2013

[25] W Zhao D Song and B Liu ldquoError bounds and finite termi-nation for constrained optimization problemsrdquo MathematicalProblems in Engineering vol 2014 Article ID 158780 10 pages2014

[26] T-W Liu and D-H Li ldquoA cautious BFGS update for reducedHessian SQPrdquo Numerical Algorithms vol 44 no 1 pp 11ndash282007

[27] M Liu X Li and Q Wu ldquoA filter algorithm with inexact linesearchrdquoMathematical Problems in Engineering vol 2012 ArticleID 349178 20 pages 2012

[28] A F Izmailov and M V Solodov ldquoOn attraction of linearlyconstrained Lagrangian methods and of stabilized and quasi-Newton SQP methods to critical multipliersrdquo MathematicalProgramming vol 126 no 2 pp 231ndash257 2011

[29] P Wang J Xu K Xie and R Chen ldquoNumerical simulation ofrail profiles evolution in the switch panel of a railway turnoutrdquoWear pp 105ndash115 2016

[30] J Xu P Wang L Wang and R Chen ldquoEffects of profile wearon wheelndashrail contact conditions and dynamic interaction ofvehicle and turnoutrdquo Advances in Mechanical Engineering vol8 no 1 pp 1ndash14 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

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


computationally efficient optimization algorithm that workswith five design variables using a method targeting the RRDfunction and they put forward a cubic spline interpolationmethod to ensure the convexity and monotony of wheelprofiles Markine et al [6] designed an optimal wheel pro-file based on geometrical wheel-rail contact characteristicssuch as RRD They chose an optimization method calledmultipoint approximations based on response surface fitting(MARS) as a numerical technique Other efforts have beenmade based on different optimal targets Polach [7] studiedthe interrelationship between the nominal contact angleconformity and equivalent conicity and this relationship wasconsidered in the method for wheel profile design Shen etal [8] developed a method for generating new wheel profilesusing the contact angle function between wheels and railsCui et al [9] used a conventional optimization procedureto obtain wheel profiles that placed the weighted wr gap asthe objective function and they put forward a new wheelprofile design method to reduce the hollow wear by seekingan optimization match of the wheel profiles the vehiclersquossuspension systems and the wear behaviour of wheels inservice [10] Smallwood et al [11] suggested a method tomodify transverse rail profiles in order to reduce contactstress and used theoretical methods to investigate the effectof profile changes on contact stress and conicity Novales etal [12] proposed a new general methodology for improvingwheel profiles in relation to certain physical phenomenathat arise during the running of the vehicle over tracksand the methodology was based on the genetic algorithmtechnique Ignesti et al [13] presented an innovative wheelprofile optimization procedure specifically designed with theaim of improving the wear and stability of vehicles Lin et al[14] proposed a multiobjective optimization model of wheelprofile to reduce the wheel wear of electric multiple unitsFor the curved track some optimum design methods forasymmetric profiles are often used to reduce the wear ofwheel flange and rail side Choi et al [15 16] proposed anoptimum design procedure for asymmetric railhead profilesby minimizing the wear and fatigue on curved tracks Zhai etal [17] put forward a designmethodology of rail asymmetric-grinding profiles based on the principle of a low dynamicinteraction of wheel-rail

The optimal studies of railway turnouts are less thanordinary track because the wr contact relationship of railwayturnouts is more complex At present the main optimal ideaof railway turnouts focuses on the geometry parameters suchas the height of switch rail the thickness of nose rail and thetrack gauge design of switch panel Palsson and Nielsen [18]proposed a methodology for the optimization of a prescribedtrack gauge variation in the switch panel of a railway turnoutthe aim of which is reducing rail profile degradation AlsoPalsson et al presented some methods for profile geometryoptimization of switch rail and nose rail by using a genetictype optimization algorithm the purpose of these efforts isto improve the dynamic interactions between wheel and railand minimize contact pressure and accumulated damage ofrails [19 20] To improve the dynamic behaviour of turnoutcrossings Wan et al [20] proposed a numerical optimizationmethod to minimize RCF damage and wear in the crossing

panel by varying the nose rail shape Nicklisch et al [21]represented the geometry of a designed track gauge variationin the switch panel in a parametric way to find the means ofimproving the switch panel design Bugarın and Garcıa Dıaz-De-Villegas [22] studied the problem of turnout optimizationwith respect to impact loads occurring during the negotiationof the main line proposing a gauge widening method in aswitch panel These optimal efforts can improve the dynamicbehaviour of railway turnouts to minimize RCF damage andwear

For the switch panel of railway turnout the variable cross-section rails will lead to the RRD of wheel set The RRDof wheel set is not always the same at different positionsalong the longitudinal direction of the railway line whichcan cause further lateral motion of the wheel set decreaseof vehicle running stability severe dynamic vehicle-turnoutinteraction and damage to wheels and rails In this paperan optimal method in switch panel of high-speed railwayturnouts is present based on the RRD function and theoptimal principle is tominimize the rolling radii difference ofwheel set as far as possible Based on a standard designCN60-1100-118 turnout (curve radius 1100m turnout angle 118)the rail profiles are optimized by this method The wheel-railcontact geometry and dynamic interaction between vehicleand turnout of the optimized profiles and those of nominalprofiles are calculated for comparison

2 Optimization of Rail Profiles

21 Rolling Radii Difference in Switch Panel For switchpanel of railway turnout the variable cross-section rails willinevitably causeRRDofwheel set As shown in Figure 1 in theswitch panel of railway turnout in the through route withoutlateral displacement of wheel set the wheel-rail contactposition changes with the variable cross-section switch railsThe rolling radius of the wheel at both sides is differenttherefore the rolling distances of the wheel at both sides aredifferent at the same time and the wheel set moves underthe action of lateral creep force The lateral motion of wheelset will reduce the vehicle running stability at a high-speedlevel In Figure 1 119909119910119911 refers to the coordinate system of thecentreline the 119909-axis and 119910-axis refer to the longitudinaldirection and lateral direction of the railway line and 119911-axisrefers to the vertical direction

The varied profiles lead to differences in rolling radiusof wheel set The normal wheel-rail contact situations aredisturbed when wheels transfer between two rails in railwayturnouts (eg the stock rail and switch rail) sometimesresulting in severe impact loads Dynamic vehicle-turnoutinteraction between train and turnout is a time variantprocess and is more complex than plain line due to thelateral translations of wheel set and varied rail profilesthus having a significant effect on the dynamic interactionbetween train and turnout which may influence the vehiclerunning stability The rolling radii difference in switch panelcan be expressed as follows

Δ119903 = 1003816100381610038161003816119903119903 minus 1199031198971003816100381610038161003816 (1)


A B



Centreline of track

y

zx




Straight stock rail

Centreline of track


Curved stock rail










ℎ (mm)

(m

m)



A

B

Optimization region





Δ119903119896 =1003816100381610038161003816119903119903119896 minus 119903119897119896

1003816100381610038161003816 (2)


Δ119903119896 = Δ119903119896 (119910119896 V1 V2 V119899) (3)


119878 = sum119898119896=1 Δ119903119896 (119910119896 V1 V2 V119899)

119898 (4)


119908 =

068 10038161003816100381610038161199101198941003816100381610038161003816 isin [0 4]

027 10038161003816100381610038161199101198941003816100381610038161003816 isin (4 8]

005 10038161003816100381610038161199101198941003816100381610038161003816 isin (8 12]

(5)


119878 =sum119898119896=1 119908119895Δ119903119896 (119910119896 V1 V2 V119899)

119898 (6)


119878 = 119878 (V1 V2 V119899) (7)





119886119894 le V119894 le 119887119894 119894 = 1 2 119899 (9)



min 119878 = 119878 (V1 V2 V119899)

st 119866119894 gt 0 119886119894 le V119894 le 119887119894 119894 = 1 2 119899(10)








Yes

No SQP

f(1 2 n)

Δrk120577(yk 1 2 n)

S120577(1 2 n)

Variables i

10038161003816100381610038161003816S120577 minus S120577minus110038161003816100381610038161003816 lt 120576















680 700 720 740 760 780 800 820

y (mm)

z(m

m)


0

15

30

45

60

minus88

(a)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(b)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(c)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus8

8

(d)








ℎ (mm)

0

5

10

15

20

25

(m

m)




y (mm)

0

10

20

30

40

50

60

z(m

m)


(a)

0

10

20

30

40

50

60z

(mm

)


700 720 740 760 780 800 820

y (mm)

(b)









y (mm)

0

10

20

30

40

50

60z

(mm

)


(a)

0

10

20

30

40

50

60

z(m

m)

700 720 740 760 780 800 820

y (mm)


(b)


0

10

20

30

40

Con

tact

loca

tion

on w

heel

(mm

)

0 3 6 9 12


Δmax = 239 mm


(a)

0

10

20

30

40C

onta

ct lo

catio

n on

whe

el (m

m)

0 3 6 9 12



Δmax = 175 mm

(b)







minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


38

30

25

20

15

10

05

03

01

(a)

minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


31

30

25

20

15

10

05

03

01

(b)











Ksz Csz

Ksy

Csy




5 Conclusions





minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A B



Centreline of track

y

zx




Straight stock rail

Centreline of track


Curved stock rail










ℎ (mm)

(m

m)



A

B

Optimization region





Δ119903119896 =1003816100381610038161003816119903119903119896 minus 119903119897119896

1003816100381610038161003816 (2)


Δ119903119896 = Δ119903119896 (119910119896 V1 V2 V119899) (3)


119878 = sum119898119896=1 Δ119903119896 (119910119896 V1 V2 V119899)

119898 (4)


119908 =

068 10038161003816100381610038161199101198941003816100381610038161003816 isin [0 4]

027 10038161003816100381610038161199101198941003816100381610038161003816 isin (4 8]

005 10038161003816100381610038161199101198941003816100381610038161003816 isin (8 12]

(5)


119878 =sum119898119896=1 119908119895Δ119903119896 (119910119896 V1 V2 V119899)

119898 (6)


119878 = 119878 (V1 V2 V119899) (7)





119886119894 le V119894 le 119887119894 119894 = 1 2 119899 (9)



min 119878 = 119878 (V1 V2 V119899)

st 119866119894 gt 0 119886119894 le V119894 le 119887119894 119894 = 1 2 119899(10)








Yes

No SQP

f(1 2 n)

Δrk120577(yk 1 2 n)

S120577(1 2 n)

Variables i

10038161003816100381610038161003816S120577 minus S120577minus110038161003816100381610038161003816 lt 120576















680 700 720 740 760 780 800 820

y (mm)

z(m

m)


0

15

30

45

60

minus88

(a)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(b)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(c)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus8

8

(d)








ℎ (mm)

0

5

10

15

20

25

(m

m)




y (mm)

0

10

20

30

40

50

60

z(m

m)


(a)

0

10

20

30

40

50

60z

(mm

)


700 720 740 760 780 800 820

y (mm)

(b)









y (mm)

0

10

20

30

40

50

60z

(mm

)


(a)

0

10

20

30

40

50

60

z(m

m)

700 720 740 760 780 800 820

y (mm)


(b)


0

10

20

30

40

Con

tact

loca

tion

on w

heel

(mm

)

0 3 6 9 12


Δmax = 239 mm


(a)

0

10

20

30

40C

onta

ct lo

catio

n on

whe

el (m

m)

0 3 6 9 12



Δmax = 175 mm

(b)







minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


38

30

25

20

15

10

05

03

01

(a)

minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


31

30

25

20

15

10

05

03

01

(b)











Ksz Csz

Ksy

Csy




5 Conclusions





minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ℎ (mm)

(m

m)



A

B

Optimization region





Δ119903119896 =1003816100381610038161003816119903119903119896 minus 119903119897119896

1003816100381610038161003816 (2)


Δ119903119896 = Δ119903119896 (119910119896 V1 V2 V119899) (3)


119878 = sum119898119896=1 Δ119903119896 (119910119896 V1 V2 V119899)

119898 (4)


119908 =

068 10038161003816100381610038161199101198941003816100381610038161003816 isin [0 4]

027 10038161003816100381610038161199101198941003816100381610038161003816 isin (4 8]

005 10038161003816100381610038161199101198941003816100381610038161003816 isin (8 12]

(5)


119878 =sum119898119896=1 119908119895Δ119903119896 (119910119896 V1 V2 V119899)

119898 (6)


119878 = 119878 (V1 V2 V119899) (7)





119886119894 le V119894 le 119887119894 119894 = 1 2 119899 (9)



min 119878 = 119878 (V1 V2 V119899)

st 119866119894 gt 0 119886119894 le V119894 le 119887119894 119894 = 1 2 119899(10)








Yes

No SQP

f(1 2 n)

Δrk120577(yk 1 2 n)

S120577(1 2 n)

Variables i

10038161003816100381610038161003816S120577 minus S120577minus110038161003816100381610038161003816 lt 120576















680 700 720 740 760 780 800 820

y (mm)

z(m

m)


0

15

30

45

60

minus88

(a)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(b)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(c)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus8

8

(d)








ℎ (mm)

0

5

10

15

20

25

(m

m)




y (mm)

0

10

20

30

40

50

60

z(m

m)


(a)

0

10

20

30

40

50

60z

(mm

)


700 720 740 760 780 800 820

y (mm)

(b)









y (mm)

0

10

20

30

40

50

60z

(mm

)


(a)

0

10

20

30

40

50

60

z(m

m)

700 720 740 760 780 800 820

y (mm)


(b)


0

10

20

30

40

Con

tact

loca

tion

on w

heel

(mm

)

0 3 6 9 12


Δmax = 239 mm


(a)

0

10

20

30

40C

onta

ct lo

catio

n on

whe

el (m

m)

0 3 6 9 12



Δmax = 175 mm

(b)







minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


38

30

25

20

15

10

05

03

01

(a)

minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


31

30

25

20

15

10

05

03

01

(b)











Ksz Csz

Ksy

Csy




5 Conclusions





minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Yes

No SQP

f(1 2 n)

Δrk120577(yk 1 2 n)

S120577(1 2 n)

Variables i

10038161003816100381610038161003816S120577 minus S120577minus110038161003816100381610038161003816 lt 120576















680 700 720 740 760 780 800 820

y (mm)

z(m

m)


0

15

30

45

60

minus88

(a)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(b)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(c)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus8

8

(d)








ℎ (mm)

0

5

10

15

20

25

(m

m)




y (mm)

0

10

20

30

40

50

60

z(m

m)


(a)

0

10

20

30

40

50

60z

(mm

)


700 720 740 760 780 800 820

y (mm)

(b)









y (mm)

0

10

20

30

40

50

60z

(mm

)


(a)

0

10

20

30

40

50

60

z(m

m)

700 720 740 760 780 800 820

y (mm)


(b)


0

10

20

30

40

Con

tact

loca

tion

on w

heel

(mm

)

0 3 6 9 12


Δmax = 239 mm


(a)

0

10

20

30

40C

onta

ct lo

catio

n on

whe

el (m

m)

0 3 6 9 12



Δmax = 175 mm

(b)







minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


38

30

25

20

15

10

05

03

01

(a)

minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


31

30

25

20

15

10

05

03

01

(b)











Ksz Csz

Ksy

Csy




5 Conclusions





minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










680 700 720 740 760 780 800 820

y (mm)

z(m

m)


0

15

30

45

60

minus88

(a)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(b)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus88

(c)

680 700 720 740 760 780 800 820

y (mm)


z(m

m)

0

15

30

45

60

minus8

8

(d)








ℎ (mm)

0

5

10

15

20

25

(m

m)




y (mm)

0

10

20

30

40

50

60

z(m

m)


(a)

0

10

20

30

40

50

60z

(mm

)


700 720 740 760 780 800 820

y (mm)

(b)









y (mm)

0

10

20

30

40

50

60z

(mm

)


(a)

0

10

20

30

40

50

60

z(m

m)

700 720 740 760 780 800 820

y (mm)


(b)


0

10

20

30

40

Con

tact

loca

tion

on w

heel

(mm

)

0 3 6 9 12


Δmax = 239 mm


(a)

0

10

20

30

40C

onta

ct lo

catio

n on

whe

el (m

m)

0 3 6 9 12



Δmax = 175 mm

(b)







minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


38

30

25

20

15

10

05

03

01

(a)

minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


31

30

25

20

15

10

05

03

01

(b)











Ksz Csz

Ksy

Csy




5 Conclusions





minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ (mm)

0

5

10

15

20

25

(m

m)




y (mm)

0

10

20

30

40

50

60

z(m

m)


(a)

0

10

20

30

40

50

60z

(mm

)


700 720 740 760 780 800 820

y (mm)

(b)









y (mm)

0

10

20

30

40

50

60z

(mm

)


(a)

0

10

20

30

40

50

60

z(m

m)

700 720 740 760 780 800 820

y (mm)


(b)


0

10

20

30

40

Con

tact

loca

tion

on w

heel

(mm

)

0 3 6 9 12


Δmax = 239 mm


(a)

0

10

20

30

40C

onta

ct lo

catio

n on

whe

el (m

m)

0 3 6 9 12



Δmax = 175 mm

(b)







minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


38

30

25

20

15

10

05

03

01

(a)

minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


31

30

25

20

15

10

05

03

01

(b)











Ksz Csz

Ksy

Csy




5 Conclusions





minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











y (mm)

0

10

20

30

40

50

60z

(mm

)


(a)

0

10

20

30

40

50

60

z(m

m)

700 720 740 760 780 800 820

y (mm)


(b)


0

10

20

30

40

Con

tact

loca

tion

on w

heel

(mm

)

0 3 6 9 12


Δmax = 239 mm


(a)

0

10

20

30

40C

onta

ct lo

catio

n on

whe

el (m

m)

0 3 6 9 12



Δmax = 175 mm

(b)







minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


38

30

25

20

15

10

05

03

01

(a)

minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


31

30

25

20

15

10

05

03

01

(b)











Ksz Csz

Ksy

Csy




5 Conclusions





minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


38

30

25

20

15

10

05

03

01

(a)

minus8

minus4

0

4

8

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

0 2 4 6 8 10


31

30

25

20

15

10

05

03

01

(b)











Ksz Csz

Ksy

Csy




5 Conclusions





minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Ksz Csz

Ksy

Csy




5 Conclusions





minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus10 0 10 20 30


minus5

0

5

10La

tera

l for

ce (k

N)

350 kmh


(a)

minus10 0 10 20 30


350 kmh


60

66

72

78

84

Vert

ical

forc

e (kN

)

(b)

minus10 0 10 20 30


350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)

minus10 0 10 20 30


350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)









350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












350 kmh


minus5

0

5

10La

tera

l for

ce (k

N)

(a)



350 kmh


60

65

70

75

80

Vert

ical

forc

e (kN

)

(b)



350 kmh


minus3

0

3

6

Late

ral d

ispla

cem

ent o

f whe

el se

t (m

m)

(c)



350 kmh


minus04

minus02

00

02

04La

tera

l vib

ratio

n ac

cele

ratio

n (m

s2)

(d)



Competing Interests


Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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