The Dynamic Performance Analysis of a Low-Floor Tram Hydraulic Anti-Kink System Based on Multidisciplinary Collaboration

energies

Article The Dynamic Performance Analysis of a Low-Floor Tram Hydraulic Anti-Kink System Based on Multidisciplinary Collaboration

Xiaokang Liao 1 , Zili Chen 2,*, Yiping Jia 3 and Jianhui Lin 4

1 School of Electrical Engineering, Southwest Jiaotong University, Chengdu 611730, Sichuan, China; [email protected] 2 School of Mathematics, Southwest Jiaotong University, Chengdu 611730, Sichuan, China 3 Sichuan Aerospace Industry Group Co., Ltd., Chengdu 610100, Sichuan, China; [email protected] 4 State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610036, Sichuan, China; [email protected] * Correspondence: [email protected]; Tel.: +86-139-8005-0975

Received: 5 August 2020; Accepted: 18 August 2020; Published: 21 August 2020

Abstract: According to the basic principle of the hydraulic anti-kink system and flow continuity equation, this paper takes the low-floor tram as the research object and the four vehicles as the research carrier. Based on the correlation parameters between the vehicle subsystem and the hydraulic subsystem, a co-simulation platform of a low-floor tram with hydraulic an anti-kink system is built. The co-simulation results show that the anti-kink system can well maintain the relative yaw angle consistency between the vehicle body and bogie. The anti-kink system restrains the maximum yaw angle and excessive lateral displacement of the vehicle body effectively. The consistency between the experiment results and the simulation results shows the accuracy of the model. The co-simulation model of the low-floor tram with hydraulic anti-kink system can be used to research the dynamic performance when it passes through curve line.

Keywords: hydraulic anti-kink system; low-ﬂoor tram; co-simulation; dynamic performance

1. Introduction With the continuous expansion of the city boundary, the traffic problems inside the city are becoming increasingly serious. As an efficient means of transportation, urban rail transit has been paid more and more attention by urban decision makers. Among them, the low-floor tram has been widely used in many cities because of its economic and environmental protection, energy saving, and other characteristics [1,2]. The low-floor tram is composed of three motor vehicles Mc1, M, Mc2, and a trailer T. The overall structure of a low-floor tram is shown in Figure1. Mc1 and T form a unit and M and Mc2 form another unit. The two units are connected by a single hinge joint, and only two vehicles are allowed to yaw around the hinge center in the horizontal plane [3]. Because this kind of tram adopts the structure of one bogie for each vehicle, the yaw angle of the vehicle body relative to the bogie in the plane is larger when crossing the curve. When the train passes through the curve, the bogie and articulated device will bear a certain torque, which can cause the relative rotation of the vehicle body and the lateral force on the wheel flange to increase. The excessive torque may even cause the train derailment. The installation of a hydraulic anti-kink system can restrain the yaw angle of the vehicle body relative to the bogie and greatly reduce the lateral force on the wheel flange [4]. Therefore, it is essential to research the curving performance and running stability of the low-floor tram.

Energies 2020, 13, 4335; doi:10.3390/en13174335 www.mdpi.com/journal/energies Energies 2020, 13, 4335 2 of 19 Energies 2020, 13, x FOR PEER REVIEW 2 of 19

FigureFigure 1.1. OverallOverall structurestructure ofof low-ﬂoorlow-floor tram.tram.

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[8] established the models the mo of thedels control of the control subsystem, subsystem, air brake air subsystem, brake subsystem, electric brakeelectric subsystem, brake subsystem, and brake and execution brake execution subsystem subsystem by using by multidisciplinary using multidisciplinary collaborative collaborative analysis method.analysis Theirmethod. results Their provided results theoretical provided basis theo forretical the development basis for andthe design development optimization and of design brake system.optimization Li J. et of al. brake [9] established system. Li the J. damping et al. 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Wang two sets Y.Q. of theet al. hydraulic [10] designed anti-kink two system,sets of the which hydraulic can satisfy anti- thekink operation system, which state of can anti-kink satisfy systemthe operation in normal state mode, of anti-kink fault mode, system and in anti-kink normal mode, mode. fault They mode, took theand articulated anti-kink mode. Mc1 and They vehicle took the T as articulated their research Mc1 objects.and vehicle The T two as their sets ofresearch anti-kink objects. system The were two simulatedsets of anti-kink under normalsystem were operation simulated mode under by AMESim normal software, operation they mode obtained by AMESim the displacement software, they of vehicleobtained Mc1’s the maindisplacement control cylinder of vehicle and Mc1’s vehicle main T’s main control control cylinder cylinder and piston. vehicle Ding T’s W.S. main et al.control [11] basedcylinder on piston. the establishment Ding W.S. et of al. a corresponding[11] based on the mathematical establishment model of a andcorresponding analyzed the mathematical formation principlemodel and of combinedanalyzed the damping. formation The principle influence of of combined the design damping. parameters The of influence the buffer of valve the groupdesign onparameters the damping of the characteristics buffer valve wasgroup further on the analyzed. damping Their characteristics results showed was thatfurther the combinationanalyzed. Their of throttleresults showed valve and that pressure the combination limiting valve of throttle in the buvafflveer valveand pressure group forms limiting the valve damping in the characteristics buffer valve ofgroup multi forms segment the parabola damping combination characteristics with of multi multi inflection segment points. parabolaLiu X.combination et al. [12] established with multi a lateralinflection dynamic points. curve Liu X. negotiation et al. [12] established model of a suspension lateral dynamic rail vehiclecurve negotiation bogies based model on of the suspension coupling systemrail vehicle of rubber bogies tire based and on ground. the coupling They analyzed system of the rubber lateral tire and and yaw ground. motion They dynamics analyzed of wheelthe lateral set structureand yaw with motion rubber dynamics tire and groundof wheel coupling, set structur the displacemente with rubber and tire acceleration and ground response coupling, of bogie the weredisplacement calculated and when acceleration vehicle frame response and wheelsetof bogie passwere through calculated curve when section. vehicle Based frame on and the theorywheelset of vehiclepass through track coupling curve section. dynamics, BasedWang on the K.Y. theory et al. of[13 vehicle] simulated track the coupling performance dynamics, indexes Wang of wheelK.Y. et rail al. dynamic[13] simulated lateral the interaction performance between indexes locomotive of wheel andrail dynamic vehicle when lateral passing interaction through between different locomotive curve tracks,and vehicle including when passenger passing andthrough freight different trains passing curve tracks, through including mountain passenger railway with and a freight small radius trains curve,passing 160 through km/h passenger mountain train railway and with 200 kma small/h freight radius train curve, passing 160 km/h through passenger curve track train withand 200 diff erentkm/h radius.freight Zboinskitrain passing K. et through al. [14] researched curve track the with nonlinear different lateral radius. stability Zboinski of rail K. vehicles et al. [14] on researched the curve, and the proposednonlinear a lateral method stability to solve of the rail more vehicles complex on the rail curve, vehicle and model. proposed a method to solve the more complexComplex rail vehicle products model. have the intersection of multi-disciplinary information, and the manufacturing cycleComplex of physical products prototype ishave long the and theintersection cost is high. of Withmulti-disciplinary the development information, of computer hardwareand the andmanufacturing the technical cycle progress of physical in the fieldprototype of single is long subject and simulation,the cost is high. the product With the analysis development of virtual of prototypecomputer of hardware complex productsand the technical using multidisciplinary progress in th jointe field simulation of single technologysubject simulation, plays an increasinglythe product importantanalysis of role virtual in the prototype process of of product complex development. products using The multidisciplinary traditional empirical joint formula simulation and technology estimation plays an increasingly important role in the process of product development. The traditional empirical formula and estimation method are difficult to satisfy the design requirements of a low-floor tram Energies 2020, 13, 4335 3 of 19

Energies 2020, 13, x FOR PEER REVIEW 3 of 19 method are difficult to satisfy the design requirements of a low-floor tram when crossing the curve. whenThe simulation crossing the method curve. of The multidisciplinary simulation method collaborative of multidisciplinary simulation collaborative is used to design simulation and calculate is used tosuch design that aand comparison calculate issuch made that between a comparison experiment is made and between simulation. experiment This is an and effective simulation. way to This realize is anthe effective rational optimizationway to realize design the rational of low-floor optimizatio tram. Thisn design paper of took low-floor the low-floor tram. This tram paper as the took research the low-floorobject and tram the four as vehiclesthe research as the object research and carrier, the four based vehicles on the multidisciplinaryas the research carrier, collaborative based analysis on the multidisciplinarymethod and the correlation collaborative parameters analysis method among subsystems.and the correlation The simulation parameters platform among ofsubsystems. SIMPACK (AThe mechanical simulation dynamicsplatform software)of SIMPACK vehicle (A mechanical dynamic model, dynamics AMESim software) (A complex vehicle systemdynamic modeling model, AMESimand simulation (A complex software) system anti-kink modeling system and hydraulic simulation model software) and Simulink anti-kink model system are hydraulic built to research model andthe influenceSimulink ofmodel hydraulic are built anti-kink to research system the on influence the dynamic of hydraulic performance anti-kink of the system low-floor on the tram dynamic under performancecurve conditions. of the low-floor tram under curve conditions.

2. Model Model of of Hydraulic Hydraulic Anti-Kink System 2.1. Structure of Anti-Kink System 2.1. Structure of Anti-Kink System The hydraulic anti-kink system is installed between the vehicle body and the bogie. The hydraulic The hydraulic anti-kink system is installed between the vehicle body and the bogie. The cylinder, elastic stop, and lateral shock absorber are used to restrain the yaw motion of the vehicle body. hydraulic cylinder, elastic stop, and lateral shock absorber are used to restrain the yaw motion of the When the vehicle body moves laterally, the oil in the hydraulic pipeline flows in the high-pressure vehicle body. When the vehicle body moves laterally, the oil in the hydraulic pipeline flows in the chamber and low-pressure chamber of the hydraulic cylinder. The hydraulic oil flows through the high-pressure chamber and low-pressure chamber of the hydraulic cylinder. The hydraulic oil flows throttle valve to play a damping role. When the vehicle yaws its head, the hydraulic oil in the through the throttle valve to play a damping role. When the vehicle yaws its head, the hydraulic oil high-pressure chamber of the two hydraulic cylinders is controlled by the one-way valve, and flows to in the high-pressure chamber of the two hydraulic cylinders is controlled by the one-way valve, and the same pipeline. The buffer hydraulic cylinder reduces the oil pressure in the hydraulic pipeline. flows to the same pipeline. The buffer hydraulic cylinder reduces the oil pressure in the hydraulic When the front end of the vehicle enters the curve, the vehicle body yaws its head. Because of the buffer pipeline. When the front end of the vehicle enters the curve, the vehicle body yaws its head. Because hydraulic cylinder, the pressure in the hydraulic pipeline does not rise sharply, but increase slowly. At of the buffer hydraulic cylinder, the pressure in the hydraulic pipeline does not rise sharply, but this time, the anti- kink system allows the angle between the front and rear vehicle bodies and the increase slowly. At this time, the anti- kink system allows the angle between the front and rear vehicle corresponding bogies to have a certain difference, which plays the critical role of anti yaw damping. bodies and the corresponding bogies to have a certain difference, which plays the critical role of anti When the rear vehicle body enters the curve, the pressure maintained in the buffer cylinder pushes yaw damping. When the rear vehicle body enters the curve, the pressure maintained in the buffer the rear body to make the head yaw and play the role of auxiliary steering. Therefore, the parameter cylinder pushes the rear body to make the head yaw and play the role of auxiliary steering. Therefore, setting of buffer hydraulic cylinder is the key of hydraulic anti-kink system. The simplified schematic the parameter setting of buffer hydraulic cylinder is the key of hydraulic anti-kink system. The diagram of hydraulic anti-kink system is shown in Figure2. simplified schematic diagram of hydraulic anti-kink system is shown in Figure 2.

Figure 2. SimplifiedSimpliﬁed schematic diagram of hydraulic anti-kink system.

2.2. Mathematical Mathematical Model Model of Anti-Kink System

The anti-kink systemsystem cancan keep keep the the yaw yaw angle angle of of two two vehicle vehicle bodies bodiesγ1 and𝛾 andγ2 in 𝛾 a unit in a equalunit equal to their to theirrespective respective bogies. bogies. When When the two the angles two angles are not are equal, not equal, the anti-kink the anti-kink system system produces produces a reverse a reverse torque torque 𝑀 between the bogie and the vehicle body, forcing the two angles to be consistent. The equation of torque and yaw angle is satisfied [15]: Energies 2020, 13, 4335 4 of 19

Mγ between the bogie and the vehicle body, forcing the two angles to be consistent. The equation of Energiestorque 2020 and, 13 yaw, x FOR angle PEER is REVIEW satisﬁed [15]: 4 of 19

Mγ1 = Mγ2 = Kγ(γ1 γ2) (1) 𝑀 =−𝑀− =𝐾(𝛾 −𝛾− ) (1)

Here: M𝑀γ1 is is thethe torquetorque betweenbetween the front vehicle body and and its its bogie, bogie, M𝑀γ2 is is the torque between rear vehicle and its bogie, and K𝐾γ is is the the rotation rotation sti stiffnessffness between between vehicle vehicle bodies. bodies. When the two ends of the hydraulic cylinder areare respectively connected with the vehicle body and thethe bogie,bogie, the the relationship relationship between between the the yaw yaw angle angleγ and 𝛾 theand displacement the displacementY of the 𝑌 movingof the moving parts of partsthe hydraulic of the hydraulic cylinder cy islinder expressed is expressed as: as: Y = Hγ (2) 𝑌=𝐻𝛾 (2) Here: H is the longitudinal distance from the connection point of the anti-kink system and the Here: 𝐻 is the longitudinal distance from the connection point of the anti-kink system and the vehicle body to the center of the bogie. vehicle body to the center of the bogie. In order to establish the mathematical model of the hydraulic anti-kink system composed of a In order to establish the mathematical model of the hydraulic anti-kink system composed of a control hydraulic cylinder, buffer hydraulic cylinder, and bypass throttle valve, the simplified model of control hydraulic cylinder, buffer hydraulic cylinder, and bypass throttle valve, the simplified model the control hydraulic cylinder was analyzed. The simplified model of the hydraulic anti-kink system is of the control hydraulic cylinder was analyzed. The simplified model of the hydraulic anti-kink shown in Figure3. system is shown in Figure 3.

Figure 3. SimplifiedSimpliﬁed model of the hydraulic anti-kink system.

Assume thatthat thethe piston piston sectional sectional area area and and initial initial volume volume of each of hydrauliceach hydraulic cylinder cylinder are A0 andare V𝐴0. Theand sectional𝑉. The sectional area and area initial and volume initial volume of the combined of the combined hydraulic hydraulic cylinder cylinder in Figure in3 Figureare 2A 30 areand 22𝐴V0. andSuppose 2𝑉 . thatSuppose the controlthat the hydraulic control hydraulic cylinder 1cylind produceser 1 aproduces downward a downward displacement displacementY1 under the 𝑌 underaction oftheF 1action, and theof 𝐹 control, and the hydraulic control cylinder hydraulic 2 has cylinder an upward 2 has displacementan upward displacementY2 under the 𝑌 action under of pressure,the action and of pressure, the resistance and istheF2 resistance. The flow is continuity 𝐹. The flow equation continuity of control equation hydraulic of control cylinder hydraulic 1 can be cylinderexpressed 1 as:can be expressed as: dY1 2V0 2A0Y1 dP1 2A0 𝑑𝑌 QB1 = 2𝑉 −−2𝐴𝑌 𝑑𝑃 (3) 2𝐴dt −−𝑄 = βe dt (3) 𝑑𝑡 𝛽 𝑑𝑡 dY1 2V0 + 2A0Y1 dP2 QA1 2A0 𝑑𝑌= 2𝑉 +2𝐴𝑌 𝑑𝑃 (4) 𝑄−−2𝐴dt = βe dt (4) 𝑑𝑡 𝛽 𝑑𝑡 Here: QA1 is the inlet oil flow of chamber A, QB1 is the flow of oil from chamber B, P1 is the Here: 𝑄 is the inlet oil flow of chamber A, 𝑄 is the flow of oil from chamber B, 𝑃 is the pressure of control hydraulic cylinder chamber B, P2 is the pressure of control hydraulic cylinder pressure of control hydraulic cylinder chamber B, 𝑃 is the pressure of control hydraulic cylinder chamber A, and βe is the bulk modulus of hydraulic oil. chamber A, and 𝛽 is the bulk modulus of hydraulic oil. The differential equation of the moving parts of hydraulic cylinder 1 is expressed as: The differential equation of the moving parts of hydraulic cylinder 1 is expressed as: d2Y dY 𝑑 𝑌1 𝑑𝑌1 M𝑀k ==𝐹F1 −(𝑃(P1 −𝑃P2)A𝐴0 −𝛿δ (5)(5) dt𝑑𝑡2 − − − dt𝑑𝑡

Here: 𝑀 is the mass of the moving part of the hydraulic cylinder and 𝛿 is the motion damping coefficient of piston cylinder. According to the flow continuity formula, the flow equations of chamber A and chamber B of the buffer cylinder are expressed as: Energies 2020, 13, 4335 5 of 19

Here: Mk is the mass of the moving part of the hydraulic cylinder and δ is the motion damping coefficient of piston cylinder. According to the flow continuity formula, the flow equations of chamber A and chamber B of the buffer cylinder are expressed as:

dx V1 AHx dP2 AH QH2 = − (6) dt − βe dt

dx V1 + AHx dP1 QH1 AH = (7) − dt βe dt The diﬀerential equation of the moving parts of the buﬀer cylinder is expressed as:

d2x dx MH + δ + 2Ktx = AH(P P2) (8) dt2 dt 1 −

Here: MH is the mass of the moving part of the buffer cylinder, V1 is the initial volume of chamber A and chamber B of buffer hydraulic cylinder, and Kt is the spring stiffness. The flow continuity equation of control hydraulic cylinder 2 can be expressed as:

dY2 2V0 2A0Y2 dP2 2A0 QA2 = − (9) dt − βe dt

dY2 2V0 + 2A0Y2 dP1 QB2 2A0 = (10) − dt βe dt

Here: QA2 is the inlet oil flow of chamber A and QB2 is the flow of oil from chamber B. The differential equation of the moving parts of hydraulic cylinder 2 is expressed as:

2 d Y2 dY2 M = (P P2)A0 δ F2 (11) k dt2 1 − − dt − The ﬂow rate of the bypass throttle valve is expressed by the following formula: s 2 Qr = C A (P P ) (12) d 2 ρ 1 − 2

Here: Qr is the flow through the throttle valve, Cd is the throttle flow coefficient, ρ is the oil density, and A2 is the orifice area of bypass throttle valve. The flow continuity equation at node M is:

QB1 = Qr + QH1 + QB2 (13)

The ﬂow continuity equation at node N is:

QA1 = Qr + QH2 + QA2 (14)

2.3. AMESim Model of Anti-Kink System When the vehicle body yaws its head relative to the bogie, the oil ﬂows through the buﬀer valve group into or out of the hydraulic cylinder, which plays the role of shock absorption. According to the basic principle of hydraulic anti-kink system, the internal hydraulic simulation model is constructed by AMESim software, as shown in Figure4. Energies 2020, 13, 4335 6 of 19 Energies 2020, 13, x FOR PEER REVIEW 6 of 19 Energies 2020, 13, x FOR PEER REVIEW 6 of 19

Figure 4. AMESim hydraulic simulation model. FigureFigure 4. 4. AMESim AMESim hydraulic hydraulic simulation simulation model. model. The basic simulation parameters of anti-kink system are shown in Table1. TheThe basic basic simulation simulation parameters parameters of of an anti-kinkti-kink system system are are shown shown in in Table Table 1. 1. Table 1. Basic simulation parameters of anti-kink system. TableTable 1. 1. Basic Basic simulation simulation parameters parameters of of anti-kink anti-kink system. system. Name Value NameName ValueValue PistonPistonPiston diameter diameter of of of control control control cylinder cylinder cylinder (mm) (mm) (mm) 50 50 50 Piston rodrod diameter diameter of of control control cylinder cylinder (mm) (mm) 30 30 Piston rod diameter of control cylinder2 (mm) 30 Orifice area of throttle valve (mm ) 2 1.28 Orifice area of throttle valve (mm2 ) 1.28 Orifice areaOil density of throttle (kg/m valve3) (mm ) 1.28842 Oil density (kg/m3) 842 BulkOil modulus density of (kg/m oil (GPa)3) 842 1.46 BulkBulk modulus modulus of of oil oil (GPa) (GPa) 1.46 1.46 3. SIMPACK Model of Low-Floor Tram 3.3. SIMPACK SIMPACK Model Model of of Low-Floor Low-Floor Tram Tram The low-floor tram is different from the traditional railway vehicles. It uses the independent The low-floor tram is different from the traditional railway vehicles. It uses the independent rotatingThe low-floor wheel as thetram supporting is different motion from the device, tradit andional the railway four wheels vehicles. are independentIt uses the independent individuals. rotating wheel as the supporting motion device, and the four wheels are independent individuals. rotatingThe front wheel and as rear the wheels supporting on both motion sides device, of the and power the bogiefour wh areeels coupled are independent together through individuals. gears. The front and rear wheels on both sides of the power bogie are coupled together through gears. The TheThe front longitudinal and rear couplingwheels on is both simulated sides of by the the power gear transmissionbogie are coupled force elementtogether inthrough the whole gears. vehicle The longitudinal coupling is simulated by the gear transmission force element in the whole vehicle longitudinalmodeling. Referringcoupling is to simulated Figure1, theby SIMPACKthe gear transmission model of low-floor force element tram isin shownthe whole in Figure vehicle5. modeling. Referring to Figure 1, the SIMPACK model of low-floor tram is shown in Figure 5. The modeling.The position Referring of each to ball Figure is the 1, centroid the SIMPACK of the corresponding model of low-floor vehicle tram body. is shown in Figure 5. The positionposition of of each each ball ball is is the the centroid centroid of of the the corresponding corresponding vehicle vehicle body. body.

FigureFigureFigure 5. 5. 5. SIMPACK SIMPACKSIMPACK model model model of of of the the the low-floor low-floor low-ﬂoor tram. tram. tram.

TheTheThe basic basic basic simulation simulation simulation parameters parameters of of the the low-floor low-floor low-ﬂoor tram tram are are shown shown in in Table Table 2.2 2..

TableTable 2. 2. Basic Basic simulation simulation parameters parameters of of the the low-floor low-floor tram. tram.

DescriptionDescription Mc1Mc1 TT MM Mc2Mc2 VehicleVehicle body body mass mass (kg) (kg) 10,221 10,221 9233 9233 7793 7793 9983 9983 Energies 2020, 13, 4335 7 of 19

Table 2. Basic simulation parameters of the low-ﬂoor tram. EnergiesEnergies 20202020,, 1313,, xx FORFOR PEERPEER REVIEWREVIEW 77 ofof 1919 Description Mc1 T M Mc2 Unspring mass per wheelset (kg) 720 875 720 720 UnspringVehicle body mass mass per (kg) wheelset (kg) 10,221 720 9233 875 7793 720 9983 720 BogieBogieUnspring massmass (Not(Not mass include perinclude wheelset unspringunspring (kg) mass)mass) (kg)(kg) 720 3430 3430 875 1540 1540 720 3420 3420 3430 3430720 RollRoll inertiainertiaBogie mass ofof bogiebogie (Not include (Not(Not include unspringinclude mass)unspringunspring (kg) mass)mass) (kg·m(kg·m 343033)) 25002500 1540 790790 3420 25002500 343025002500 Roll inertia of bogie (Not include unspring mass) (kg m3) 2500 790 2500 2500 PitchPitch inertiainertia ofof bogiebogie (Not(Not includeinclude unspringunspring· mass)mass) (kg·m(kg·m33)) 10001000 500 500 10001000 10001000 Pitch inertia of bogie (Not include unspring mass) (kg m3) 1000 500 1000 1000 · 33 YawYawYaw inertia inertiainertia of of bogieof bogiebogie (Not (Not(Not include inin unspringcludeclude unspringunspring mass) (kg mass)mass)m3) (kg·m(kg·m3050)) 30503050 1200 12001200 3050 30503050 305030503050 ·

4.4.4. TheThe The Co-SimulationCo-Simulation Co-Simulation ofof of SIMPACK/AMESim/SimulinkSIMPACK/AMESim/Simulink SIMPACK/AMESim/Simulink InInIn orderorder order toto to betterbetter better analyzeanalyze analyze thethe the influenceinfluence influence ofof of thethe anti-kinkanti-kink system system on on vehivehi vehicleclecle drivingdriving performance performance underunderunder curvecurve curve crossingcrossing crossing conditions,conditions, conditions, thethe the vehiclevehicle vehicle dynamicdynamic dynamic modelmodel model basedbased based onon on SIMPACKSIMPACK SIMPACK andand and thethe hydraulichydraulic anti-kinkanti-kinkanti-kink systemsystem system modelmodel basedbased onon AMESim AMESim are are established.estaestablished.blished. ByBy By definingdefining defining thethe the interactioninteraction interfaceinterface betweenbetweenbetween differentdifferent different softwaresoftware simulationsimulation modelsmodels models in in SimulinkSimulink environment,environment, thethe the integrationintegration ofof of vehiclevehicle dynamicsdynamicsdynamics analysisanalysis analysis simulationsimulation simulation modelmodel model andand and thethe the anti-kinkanti-kink anti-kink systemsystem system hydraulihydrauli hydrauliccc simulationsimulation simulation modelmodel model isis is realized.realized. realized. TheTheThe schemescheme ofof thethe vehiclevehicle co-simulationco-simulation platformplatform isis establishedestablished as as shown shown in in Figure Figure 6 6.6..

Figure 6. Scheme of the vehicle co-simulation platform. FigureFigure 6. 6. SchemeScheme of of the the vehicle vehicle co-simulation co-simulation platform. platform.

5.5.5. InfluenceInfluence Inﬂuence ofof of thethe Anti-KinkAnti-Kink SystSyst Systememem onon VehicleVehicle Dynamic Dynamic Performance Performance InInIn anan an urbanurban urban railrail rail transittransit transit system,system, system, thethe the mostmost most commocommo commonnn smallsmall small curvecurve curve basicallybasically basically includesincludes includes S-shapedS-shaped S-shaped curvecurve curve lineslineslines andand and C-shapedC-shaped C-shaped curvecurve curve lines.lines. lines. Therefore,Therefore, Therefore, itit it isis is veryvery very importantimportant important toto to researchresearch research thethe the performanceperformance performance ofof of thethe the hydraulichydraulichydraulic anti-kinkanti-kink systemsystem whenwhen thethe curvecurve isis tootoo smsm small.all.all. AnAn An S-shapedS-shaped S-shaped curvecurve curve lineline line andand and aa a C-shapedC-shaped C-shaped curvecurve curve linelineline areare are shownshown shown inin in FigureFigure Figure 77.7..

FigureFigureFigure 7.7. 7. S-shapedS-shapedS-shaped curvecurve curve lineline line andand and C-shapedC-shaped C-shaped curvecurve curve line.line. line.

AccordingAccording toto thethe requirementsrequirements ofof codecode forfor designdesign ofof railwayrailway lineslines (GB(GB 50090-2006)50090-2006) andand generalgeneral technicaltechnical conditionsconditions forfor low-floorlow-floor tramtram vehiclesvehicles (CJ/T(CJ/T 417-2012),417-2012), thethe specificspecific parametersparameters ofof anan S-S- shapedshaped curvecurve lineline andand aa C-shapedC-shaped curvecurve lineline areare asas follows:follows: Energies 2020, 13, 4335 8 of 19

According to the requirements of code for design of railway lines (GB 50090-2006) and general technical conditions for low-ﬂoor tram vehicles (CJ/T 417-2012), the speciﬁc parameters of an S-shaped curve line and a C-shaped curve line are as follows: S-shaped curve line and C-shaped curve line: the length of straight line 1 and 3 is 50 m, and the length of straight line 2 is 10 m, the length of two circular curves is 10 m and the radius is 25 m.

5.1. Evaluation Index of Dynamic Performance The safety of vehicle operation mainly involves whether the vehicle will derail and overturn. Generally, derailment coeﬃcient, wheel load reduction rate, wheel–rail lateral force and other indicators are used to evaluate the safety of vehicle operation. At present, China’s vehicle departments mainly use derailment coeﬃcient and wheel load reduction rate.

5.1.1. Derailment Coefficient When the vehicle is running, the wheel derailment may be caused under the most unfavorable combination of line condition, operation condition, vehicle structural parameters, and loading. The “derailment coefficient” is used to evaluate the stability of wheel derailment, and the calculation formula is given by Nadal [16]: Q tgα µ = − (15) P 1 + µ tgα · Here: Q is the lateral force acting on the wheel, P is the vertical force acting on the wheel, µ is the friction coefficient at the flange, and α is the maximum flange contact angle. The International Union of Railways (UIC) stipulates that: Q/P 1.2. ≤ 5.1.2. Wheel Unloading Rate The vertical force and lateral force of the wheel are the main parameters affecting derailment. Combined with the definition of the derailment coefficient, the main reason for derailment in theory is that the lateral force is too large. However, in practical application, it is found that sometimes the lateral force is small, but the derailment phenomenon may occur when the wheel load is reduced. Therefore, the wheel load reduction rate is used as a supplement to jointly determine the derailment safety of vehicles. 1 1 W = (W + W ) ∆W = (W W ) (16) 2 1 2 2 2 − 1 Here: W is the average wheel weight of the unloaded side wheel and the loaded side wheel, ∆W is the load reduction variation of wheel load, W1 is the wheel weight of the unloaded side wheel, and W2 is the wheel weight of the loaded side wheel.

5.1.3. Wheel–Rail Lateral Force Excessive wheel–rail lateral force will lead to deformation or even damage of the line, so the maximum value of lateral force should be controlled. According to GB5599-85, the maximum allowable value of wheel–rail lateral force during curve passing is shown in formula (17) and formula (18). When the track spike is pulled out, the stress of the track spike is the limit of elastic limit:

Q 19 + 0.39Pst (17) ≤ When the track spike is pulled out, the stress of the track spike is the limit of yield limit:

Q 29 + 0.39Pst (18) ≤

Here: Q is the lateral force acting on the wheel and Pst is the wheel static load. Energies 2020, 13, 4335 9 of 19

The wheel rail lateral force shall not exceed the design load of elastic fastener. The lateral force limit of Shinkansen in Japan and Europe and America is usually 0.4 times of axle load. Energies 2020, 13, x FOR PEER REVIEW 9 of 19 Q 0.4(P + Pst ) (19) ≤ st1 2 5.1.4. Vehicle Running Stability Index Here: Pst1 is the static load of the left wheel and Pst2 is the static load of the right wheel. The evaluation indexes of vehicle running stability mainly include vehicle running stability 5.1.4. Vehicle Runningindex Stability and Index comfort index. These indexes can not only reflect the performance of vehicle system, but The evaluation indexesalso reflect of vehicle the physical running stabilityreaction mainlyof passengers include to vehicle vehicle running running stability quality. index and comfort index. These indexesThis paper can mainly not only introduces reﬂect the Sperling performance stability of vehicleindex, and system, evaluates but also vehicle running stability ℵ reﬂect the physical reactionthrough of passengers this index. toThe vehicle formula running of stationarity quality. index is shown as: This paper mainly introduces Sperling stability index, and evaluates vehicle running stability 𝑎 through this index. The formula of stationarity index ℵ=2.7is shown𝑍 as:𝑓𝐹(𝑓) = 0.896 𝐹(𝑓) (20) 𝑓 s Here: 𝑍 is the amplitude,q 𝑓 is the vibrationa3 frequency, 𝑎 is the vibration acceleration, and 𝐹(𝑓) is = 2.710 Z3 f 5F( f ) = 0.89610 F( f ) (20) the factor related to vibration frequency. Thef values of 𝐹(𝑓) are shown in Table 3.

Here: Z is the amplitude, f is the vibration frequency,Tablea is 3. the Frequency vibration correction acceleration, factor. and F( f ) is the factor related to vibration frequency. The values of F( f ) are shown in Table3. Vertical Vibration Lateral Vibration Table 3. Frequency0.5–5.9 Hz correction 𝐹(𝑓) factor.= 0.325𝑓 0.5–5.4 Hz 𝐹(𝑓) =0.8𝑓 5.9–20 Hz 𝐹(𝑓) = 400/𝑓 5.4–26 Hz 𝐹(𝑓) = 650/𝑓 Vertical Vibration>20 Hz 𝐹(𝑓 Lateral) =1 Vibration>26 Hz 𝐹(𝑓) =1 0.5–5.9 Hz F( f ) = 0.325 f 2 0.5–5.4 Hz F( f ) = 0.8 f 2 5.9–20 Hz The stationarityF( f ) = 400/ indexf 2 can 5.4–26only be Hz applied toF( fa) =single650/ fvibration2 of one frequency and one >20 Hzamplitude. ButF( fin) =the1 actual line, >the26 Hzvariation of vibrationF( f ) = 1frequency and amplitude with time is relatively complex. Therefore, before calculating the vehicle stability index, we analyze the frequency The stationarity indexspectrum can only of bethe applied measured to a single vehicle vibration vibration of one accele frequencyration andto obtain one amplitude. the amplitude value of each But in the actual line, thefrequency variation range, of vibration and then frequency calculate and the amplitude ride comfort with timeindex is relativelyℵ of each complex. band, and finally calculate the Therefore, before calculatingtotal stability the vehicle index stability of all bands. index, we analyze the frequency spectrum of the measured vehicle vibration acceleration to obtain the amplitude value of each frequency . range, and ℵ =ℵ +ℵ +⋯+ℵ (21) then calculate the ride comfort index t of each band, and finally calculate the total stability index of all bands. The vehicle stability levels specified in GB5599-85 are shown in Table 4. The running stability of 10 10 10 0.1 ordinary passenger( t = cars+ in China+ +shalln not) be lower than the secondary standard,(21) that is, the lateral 1 2 ··· and vertical stability indexes shall not exceed 2.75. The vehicle stability levels specified in GB5599-85 are shown in Table4. The running stability of ordinary passenger cars in China shall not be lower than the secondary standard, that is, the lateral Table 4. Passenger and Freight train stability level. and vertical stability indexes shall not exceed 2.75. Stability Index Stability Level Evaluation Result Table 4. Passenger and Freight train stability level. Passenger Train Freight Train Class A excellent <2.5 <3.5 Stability Index Stability Level EvaluationClass Result B good 2.5–2.75 3.5–4.0 Passenger Train Freight Train Class C qualified 2.75–3.0 4.0–4.25 Class A excellent <2.5 <3.5 Class B5.2. S-Shaped Curve good Line 2.5–2.75 3.5–4.0 Class C qualified 2.75–3.0 4.0–4.25 The change of angle between vehicle body and bogie with time for tram running on S-shaped 5.2. S-Shaped Curve Linecurve line with the anti-kink system is shown in Figure 8. The change of angle between vehicle body and bogie with time for tram running on S-shaped curve line with the anti-kink system is shown in Figure8. Energies 2020, 13, 4335 10 of 19 Energies 2020, 13, x FOR PEER REVIEW 10 of 19

(A) Angle between vehicle body Mc1 and bogie (B) Angle between vehicle body T and bogie

Figure 8. AngleAngle between between vehicle vehicle and and bogie bogie wi withth hydraulic anti-kink system.

When the vehicle withwith thethe anti-kinkanti-kink system system passes passes through through the the S-shaped S-shaped curve, curve, the the change change trend trend of ofthe the angle angle between between the the vehicle vehicle body body and itsand bogie its bo isgie diff iserent. different. The maximum The maximum angle betweenangle between the vehicle the vehiclebody and body the and bogie the can bogie reach can 2.42 reach◦, which 2.42°, is which located is in located the Mc1 in vehicle.the Mc1 Due vehicle. to the Due buff toer the hydraulic buffer hydrauliccylinder of cylinder the working of the hydraulic working anti-kinkhydraulic system,anti-kink the system, overall the stiff overallness is stiffness relatively is small, relatively allowing small, a allowingcertain di ffaerence certain in thedifference angle between in the theangle front between and rear the vehicle front bodies and rear and thevehicle corresponding bodies and bogies. the correspondingThere will be a certainbogies. diTherefference will between be a certain the anglesdifference between between the vehiclethe angles body between and the the bogie, vehicle and body there andis a delaythe bogie, in the and rotation there ofis thea delay front in and the rear rotation vehicle of the bodies front relative and re toar thevehicle corresponding bodies relative bogies. to the correspondingFigure9 shows bogies. the change of the maximum derailment coe fficient of each wheel group under the operatingFigure conditions. 9 shows the Before change and of after the the maximum installation dera ofilment anti-kink coefficient system, of the each derailment wheel group coefficient under of themost operating wheel groups conditions. changes Before slightly, and while after the the derailment installation coe ffiofcient anti-kink of the system, two front the wheel derailment groups coefficientchanges greatly. of most When wheel the groups vehicle changes passes slightly, through while the S-shaped the derailment curve coefficient at a faster speed,of the two the front most wheelfront-end groups bogie changes suffers greatly. a greater When lateral the impact. vehicle The pa maximumsses through derailment the S-shaped coeffi curvecient withoutat a faster anti-kink speed, thesystem most reaches front-end 1.45, thebogie maximum suffers derailmenta greater coelateffiralcient impact. of installing The maximum anti-kink systemderailment wheel coefficient set is 0.72. withoutFigure 10 anti-kink shows the system change reaches of wear 1.45, index. the After maximum the anti-kink derailment system coefficient is installed, of installing the change anti-kink range of systemwear index wheel is small, set is the0.72. wear Figure of front 10 wheelshowsset the is change large, and of thewear wear index. of rear After wheel the set anti-kink is small. system Figure 11is installed,shows the the variation change of range the maximum of wear index wheel is unloading small, the rate wear under of front operating wheel conditions.set is large, The and maximum the wear ofwheel rear unloadingwheel set is rate small. of the Figure vehicles 11 shows without the thevariation anti-kink of the system maximum reaches wheel 0.57, unloading close to the rate specified under operatinglimit, and theconditions. maximum The wheel maximum unloading wheel rate unloadin with theg anti-kink rate of the system vehicles is 0.28. without Figure the12 showsanti-kink the systemvariation reaches of the 0.57, maximum close to wheel–rail the specified lateral limit, force and under the maximum the operating wheel conditions. unloading The rate maximum with the anti-kink system is 0.28. Figure 12 shows the variation of the maximum wheel–rail lateral force under EnergiesEnergies 20202020,, , 13 13,, , x 4335xx FOR FORFOR PEER PEERPEER REVIEW REVIEWREVIEW 111111 of ofof 19 1919 thethe operatingoperating conditions.conditions. TheThe maximummaximum wheel–railwheel–rail latelateralral forceforce ofof thethe vehiclesvehicles withoutwithout thethe anti-kinkanti-kink wheel–rail lateral force of the vehicles without the anti-kink system reaches 82.8 KN, far exceeding the systemsystem reachesreaches 82.882.8 KN,KN, farfar exexceedingceeding thethe maximummaximum valuevalue alloweallowedd byby thethe standard.standard. TheThe maximummaximum maximum value allowed by the standard. The maximum wheel–rail transverse force of the vehicles wheel–railwheel–rail transversetransverse forceforce ofof thethe vehiclesvehicles withwith antianti-kink-kink systemsystem isis 38.238.2 KN,KN, whichwhich isis lowerlower thanthan thethe with anti-kink system is 38.2 KN, which is lower than the national standard value of 43 KN. Therefore, nationalnational standardstandard valuevalue ofof 4343 KN.KN. Therefore,Therefore, thethe anti-kinkanti-kink systemsystem cancan improveimprove thethe safetysafety ofof aa singlesingle the anti-kink system can improve the safety of a single tram passing through the S-shaped curve. tramtram passingpassing throughthrough thethe S-shapedS-shaped curve.curve.

FigureFigure 9. 9. DerailmentDerailment coefficient. coecoefficient.ﬃcient.

FigureFigureFigure 10. 10. WearWear index. index.

Figure 11. Wheel unloading rate. FigureFigure 11.11. WheelWheel unloading unloading rate. rate.rate.

Energies 2020, 13, 4335 12 of 19 vehicle. Because the vehicle T has no longitudinal coupling system, and the bogie lateral movement is larger than the other three vehicles, the yaw angle of T vehicle is larger and the anti-kink system force is larger. The maximum vertical stability index of vehicle body without the anti-kink system is 1.74, concentrated in M vehicle, the maximum vertical stability index of vehicle body with the anti-kink system is 1.26, concentrated in Mc1 vehicle. So the stability of vehicles with anti-kink system are better than those without anti-kink system. The amplitude and frequency of yaw head of vehicle body have inﬂuence on the stability index when the vehicle passes through an S-shaped curve. Energies 2020, 13, x FOR PEER REVIEW 12 of 19

Figure 12. Wheel–rail lateral force.

Figures 13 and 14 show the change of vehicle body stability index, the maximum lateral stability index of vehicle body without anti-kink system is 2.28,2.28, whichwhich isis concentratedconcentrated inin MM vehicle,vehicle, thethe maximum lateral stability index of vehicle body with anti-kink system is 1.74, which is also concentrated in M vehicle. Because the vehicle T has no longitudinal coupling system, and the bogie lateral movement is larger than the other three vehicles, the yaw angle of T vehicle is larger and the anti-kink system force is larger. The maximum vertical stability index of vehicle body without the anti-kink system is 1.74, concentrated in M vehicle, the maximum vertical stability index of vehicle body with the anti-kink system is 1.26, concentrated in Mc1 vehicle. So the stability of vehicles with anti-kink system are better than those without anti-kink system. The amplitude and frequency of yaw head of vehicle body have influence on the stability index when the vehicle passes through an S- shaped curve. Figure 12. Wheel–rail lateral force.

Figures 13 and 14 show the change of vehicle body stability index, the maximum lateral stability index of vehicle body without anti-kink system is 2.28, which is concentrated in M vehicle, the maximum lateral stability index of vehicle body with anti-kink system is 1.74, which is also concentrated in M vehicle. Because the vehicle T has no longitudinal coupling system, and the bogie lateral movement is larger than the other three vehicles, the yaw angle of T vehicle is larger and the anti-kink system force is larger. The maximum vertical stability index of vehicle body without the anti-kink system is 1.74, concentrated in M vehicle, the maximum vertical stability index of vehicle body with the anti-kink system is 1.26, concentrated in Mc1 vehicle. So the stability of vehicles with anti-kink system are better than those without anti-kink system. The amplitude and frequency of yaw head of vehicle body have influence on the stability index when the vehicle passes through an S- shaped curve.

FigureFigure 13. LateralLateral stability stability of of vehicle vehicle body. body.

Figure 13. Lateral stability of vehicle body. FigureFigure 14. 14. VerticalVertical stability stability of vehicle body.

Figure 14. Vertical stability of vehicle body. Energies 2020, 13, 4335 13 of 19 Energies 2020, 13, x FOR PEER REVIEW 13 of 19

5.3.5.3. C-Shaped C-Shaped Curve Curve Line Line TheThe change change of of angle angle between between vehicle vehicle body body and and bogie bogie with with time time for for tram tram running running on on a a C-shaped C-shaped curvecurve line line with with the the anti-kink anti-kink system system is shown in Figure 15.15.

(A) Angle between vehicle body Mc1 and bogie (B) Angle between vehicle body T and bogie

FigureFigure 15. 15. AngleAngle between between vehicle vehicle and and bogie bogie wi withth hydraulic hydraulic anti-kink anti-kink system. system.

WhenWhen the the vehicle vehicle without without the the anti-kink anti-kink system system pa passessses through through the the C-shaped C-shaped curve, curve, the the change change trendtrend of of the the angle angle between between the the vehicle vehicle body body and and it itss bogie is different. different. The The maximum angle angle between between thethe vehicle vehicle body body and and the the bogie bogie can can reach reach 2.39°, 2.39 which◦, which is located is located in the in Mc1 the vehicle. Mc1 vehicle. Due to Due the buffer to the hydraulicbuffer hydraulic cylinder cylinder of the working of the working hydraulic hydraulic anti-kink anti-kink system, system, the overall the overallstiffness sti isff nessrelatively is relatively small, allowingsmall, allowing a certain a certain difference difference in the in angle the angle between between the thefront front and and rear rear vehicle vehicle bodies bodies and and the the correspondingcorresponding bogies, bogies, there there will will be be a a certain certain differ differenceence between between the the angles angles between between the the vehicle vehicle body body andand the the bogie, bogie, and and there there is is a a delay delay in in the the rotation of of the front and re rearar vehicle bodies relative relative to to the the correspondingcorresponding bogies. bogies. FigureFigure 16 shows thethe changechange ofof the the maximum maximum derailment derailment coe coefficientfficient of of each each wheel wheel group group under under the theoperating operating conditions. conditions. When When the vehiclethe vehicle passes passes through through the C-shapedthe C-shaped curve curve at a fasterat a faster speed, speed, the most the mostfront-end front-end bogie bogie suffers suffers a greater a greater lateral lateral impact. impact. The The maximum maximum derailment derailment coe fficoefficientcient without without the theanti-kink anti-kink system system reaches reaches 1.48, 1.48, the maximum the maximum derailment derailment coeffi cientcoefficient of installing of installing the anti-kink the anti-kink system systemwheel set wheel is 0.82. set Figureis 0.82. 17 Figure shows 17 the shows change the of wearchange index. of wear After index. the anti-kink After the system anti-kink is installed, system the is installed,change range the change of wear range index of is wear small, index the wearis small, of front the wear wheel of set front is large, wheel and set theis large, wear and of rear thewheel wear ofset rear is small. wheel Figureset is small. 18 shows Figure the 18 variation shows the of variation the maximum of the wheelmaximum unloading wheel unloading rate under rate operating under operatingconditions. conditions. The maximum The wheelmaximum unloading wheel rateunloadin of theg vehicles rate of withoutthe vehicles the anti-kink without system the anti-kink reaches system reaches 0.58, close to the specified limit, and the maximum wheel unloading rate with the Energies 2020, 13, 4335 14 of 19

Energies 2020, 13, x FOR PEER REVIEW 14 of 19 0.58,Energies close 2020,,to 13, the, xx FORFOR speciﬁed PEERPEER REVIEWREVIEW limit, and the maximum wheel unloading rate with the anti-kink system14 of 19 is 0.29. Figure 19 shows the variation of the maximum wheel–rail lateral force under the operating anti-kinkanti-kink systemsystem isis 0.29.0.29. FigureFigure 1919 showsshows thethe varivariationation ofof thethe maximummaximum wheel–railwheel–rail laterallateral forceforce underunder conditions.anti-kink system The maximum is 0.29. Figure wheel–rail 19 shows lateral the forcevariation of the of vehicles the maximum without wheel–rail the anti-kink lateral system force reaches under thethe operatingoperating conditions.conditions. TheThe maximummaximum wheel–railwheel–rail latelateralral forceforce ofof thethe vehiclesvehicles withoutwithout thethe anti-kinkanti-kink 85.1the operating KN, far exceedingconditions. the The maximum maximum value wheel–rail allowed late byral theforce standard. of the vehicles The maximum without the wheel–rail anti-kink systemsystem reachesreaches 85.185.1 KN,KN, farfar exexceedingceeding thethe maximummaximum valuevalue alloweallowedd byby thethe standard.standard. TheThe maximummaximum transversesystem reaches force 85.1 of the KN, vehicles far exceeding with the the anti-kink maximum system value is 39.8 allowe KN,d whichby the isstandard. lower than The the maximum national wheel–railwheel–rail transversetransverse forceforce ofof thethe vehiclesvehicles withwith thethe anti-kinkanti-kink systemsystem isis 39.839.8 KN,KN, whichwhich isis lowerlower thanthan standardwheel–rail value transverse of 43 KN.force Therefore, of the vehicles the anti-kink with the anti-kink system can system improve is 39.8 the KN, safety which of ais single lower tramthan thethe nationalnational standardstandard valuevalue ofof 4343 KN.KN. Therefore,Therefore, thethe anti-kinkanti-kink systemsystem cancan improveimprove thethe safetysafety ofof aa passingthe national through standard the C-shaped value of curve. 43 KN. Therefore, the anti-kink system can improve the safety of a singlesingle tramtram passingpassing throthroughugh thethe C-shapedC-shaped curve.curve.

Figure 16. Derailment coefficient. FigureFigure 16.16. Derailment coecoefficient.ﬃcient.

Figure 17. Wear index. Figure 17. Wear index.index.

Figure 18. Wheel unloading rate. FigureFigure 18.18. Wheel unloading rate. Energies 2020, 13, 4335 15 of 19

Energies 2020, 13, x FOR PEER REVIEW 15 of 19

Figure 19. Wheel–rail lateral force.

Figures 20 and 2121 showshow thethe changechange ofof vehiclevehicle bodybody stabilitystability indexindex andand thethe maximummaximum laterallateral stability index index of of vehicle vehicle body body without without the the anti-kink anti-kink system system is 2.21, is 2.21,which which is concentrated is concentrated in M vehicle, in M vehicle,the maximum the maximum lateral lateralstability stability index indexof vehicle of vehicle body bodywith withthe anti-kink the anti-kink system system is 1.79, is 1.79, which which is isconcentrated concentrated in inMc2 Mc2 vehicle. vehicle. The The maximum maximum vertical vertical stability stability index index of vehicle of vehicle body body without without the anti- the anti-kinkkink system system is 1.62, is 1.62,concentrated concentrated in M invehicle, M vehicle, the maximum the maximum vertical vertical stability stability indexindex of vehicle of vehicle body bodywith the with anti-kink the anti-kink system system is 1.21 is concentrated 1.21 concentrated in Mc in2 vehicle. Mc2 vehicle. So the So stability the stability indexes indexes of vehicles of vehicles with withthe anti-kink the anti-kink system system are better are than better those than without those without the anti-kink the anti-kink system. The system. amplitude The amplitude and frequency and frequencyof yaw head of yawof vehicle head ofbody vehicle have body influence have inﬂuenceon the stability on the index stability when index the when vehicle the passes vehicle through passes throughthe C-shaped the C-shaped curve. curve.

Figure 20. Lateral stability of vehicle body.

Figure 21. Vertical stability of vehicle body. Figure 21. Vertical stability of vehicle body.

6. Experiment The displacement between each bogie and its correspondingcorresponding vehicle body was measured by two displacement sensors. Which were respectively installed at the edge of the vehicle body and the end of the frame to test the lateral displacement of the outermost end of the bogie relative to the vehicle Energies 2020, 13, 4335 16 of 19

Energies 2020, 13, x FOR PEER REVIEW 16 of 19 EnergiesofEnergies the 2020 frame 2020, 13, 13, tox, xFORtest FOR PEERthe PEER lateralREVIEW REVIEW displacement of the outermost end of the bogie relative to the vehicle1616 of of 19 19 body. The installation diagram is shown in Figure 22. The simpliﬁed dimensioning of displacement body. The installation diagram is shown in Figure 22. The simplified dimensioning of displacement body.measurementbody. The The installation installation is shown diagram diagram in Figure is is shown23 shown. in in Figure Figure 22. 22. The The simplified simplified dimens dimensioningioning of of displacement displacement measurement is shown in Figure 23. measurementmeasurement is is shown shown in in Figure Figure 23. 23.

Figure 22. Schematic diagram of sensor installation position. FigureFigureFigure 22. 22.22. Schematic SchematicSchematic diagram diagramdiagram of ofof sensor sensorsensor installation installationinstallation position. position.position.

Figure 23. Simplified dimensioning of displacement measurement. FigureFigureFigure 23. 23.23. Simplified SimpliﬁedSimplified dimensioning dimensioningdimensioning of ofof displacement displacementdisplacement measurement. measurement.measurement. Referring to the Figure 23, the yaw angle between vehicle body and bogie is expressed as: ReferringReferringReferring to toto the thethe Figure FigureFigure 23, 23 23, ,the thethe yaw yawyaw angle angleangle betw betweenbetweeneen vehicle vehicle vehicle body body body and and and bogie bogie bogie is is is expressed expressed expressed as: as: as: 180(𝐿 −𝐿 ) 180(𝐿180(𝐿−𝐿−𝐿) ) (22) 𝛾𝛾== 180(L1 L2) (22) γ𝛾== 𝜋𝐿𝜋𝐿− (22)(22) bogie π𝜋𝐿L Here:Here: 𝐿𝐿 andand 𝐿 𝐿 are are the the horizontal horizontal lateral lateral displacement displacement of of bogie bogie frame frame relative relative to to vehicle vehicle body body Here: 𝐿 and 𝐿 are the horizontal lateral displacement of bogie frame relative to vehicle body andand 𝐿𝐿Here:is is the the Ldistance distance1 and L2 between betweenare the horizontaltwo two horizontal horizontal lateral measuring measuring displacement points. points. of bogie frame relative to vehicle body and 𝐿 is the distance between two horizontal measuring points. and LTheis thefollowing distance is betweenthe angle two change horizontal of the measuringlow-floor tram points. Mc1 and T relative to their own bogies TheThe following following is is the the angle angle change change of of the the low-floor low-floor tram tram Mc1 Mc1 and and T T relative relative to to their their own own bogies bogies on theThe S-shaped following curve is the line, angle as shown change in of Figure the low-ﬂoor 24. tram Mc1 and T relative to their own bogies on onon the the S-shaped S-shaped curve curve line, line, as as shown shown in in Figure Figure 24. 24. the S-shaped curve line, as shown in Figure 24.

Mc1 T Mc1Mc1 T T Figure 24. Mc1 and T relative to their own bogies on the S-shaped curve line. FigureFigureFigure 24. 24.24. Mc1 Mc1Mc1 and andand T T Trelative relativerelative to toto their theirtheir own ownown bo bogiesbogiesgies on onon the thethe S-shaped S-shapedS-shaped curve curvecurve line. line.line. Due to the limitation of experimental conditions, there is no separate R = 25 m curve radius curve. DueDue to to the the limitation limitation of of experiment experimentalal conditions, conditions, there there is is no no separa separatete R R = =25 25 m m curve curve radius radius curve. curve. Therefore, a continuous curve with R = 25 m curve radius and C curve is selected, that is, after the Therefore,Therefore, a acontinuous continuous curve curve with with R R = =25 25 m m curve curve radius radius and and C C curve curve is is selected, selected, that that is, is, after after the the vehicle passes through the curve with R = 25 m small curve radius at the speed limit, and then passes vehiclevehicle passes passes through through the the curve curve with with R R = =25 25 m m smal small curvel curve radius radius at at the the speed speed limit, limit, and and then then passes passes Energies 2020, 13, 4335 17 of 19

Due to the limitation of experimental conditions, there is no separate R = 25 m curve radius curve. Therefore,Energies 2020, a 13 continuous, x FOR PEER curveREVIEW with R = 25 m curve radius and C curve is selected, that is, after17 of the 19 vehicle passes through the curve with R = 25 m small curve radius at the speed limit, and then passes the C curve track at the speed limit. The test results of the R = 25 m curve and the C-shaped curve are the C curve track at the speed limit. The test results of the R = 25 m curve and the C-shaped curve are displayed together as shown in Figure 25. displayed together as shown in Figure 25.

Mc1 T

FigureFigure 25.25. Mc1 and T relative to their own bogiesbogies onon thethe C-shapedC-shaped curvecurve line.line.

7.7. Discussion ComparedCompared withwith the the experimental experimental results results (Figure (Figure 24) 24) and and the simulationthe simulation results results (Figure (Figure8), the 8), curve the changescurve changes are consistent, are consistent, but the but peak the valuepeak value of the of yaw the angleyaw angle is di ﬀiserent. different. The The diﬀerence difference between between the experimentalthe experimental results results and and the simulationthe simulation results resu (S-shapedlts (S-shaped curve curve line) line) is shown is shown in Table in Table5. 5.

TableTable 5.5. DiDifferenceﬀerence betweenbetween thethe experimentalexperimental resultsresults andand thethe simulationsimulation results,results, (S-shaped(S-shaped curvecurve line).line).

LineLine Condition Condition Data Data Source Source Vehicle Vehicle Max Max (deg) Min Min (deg) (deg) Error Error SimulationSimulation Mc1Mc1 +2.42+2.42 −2.342.34 − 13.1%/1.3%13.1%/1.3% ExperimentExperiment Mc1Mc1 +2.14+2.14 −2.312.31 S-shapedS-shaped curve curve line line − SimulationSimulation T T +1.99+1.99 −2.202.20 − 5.2%/3.5%5.2%/3.5% ExperimentExperiment T T +2.10+2.10 −2.282.28 −

ComparedCompared with the the experimental experimental results results (Figure (Figure 25) 25 and) and the the simulation simulation results results (Figure (Figure 15), 15the), thecurve curve changes changes are consistent, are consistent, but the but peak the value peak valueof the yaw of the angle yaw is angle different. is di ﬀTheerent. difference The di betweenﬀerence betweenthe experimental the experimental results and results the simula and thetion simulation results is results shown is in shown Table in6. Table6.

TableTable 6.6.Di Differenceﬀerence between between the the experimental experimental results results and an thed simulationthe simulation results, results, (R = 25 (R m = and 25 m C-shaped and C- curveshaped line). curve line).

LineLine Condition Condition Data Data Source Source VehicleVehicle Max Max (deg) (deg) Min Min(deg) (deg) Error Error Simulation Mc1 +2.29 −2.33 Simulation Mc1 +2.29 2.33 2.1%/3.6% Experiment Mc1 +2.34 −2.25− 2.1%/3.6% R = 25 m Experiment Mc1 +2.34 2.25 R = 25 m Simulation T +2.25 −1.96− Simulation T +2.25 1.96 2.6%/5.8% Experiment T +2.31 −2.08− 2.6%/5.8% Experiment T +2.31 2.08 Simulation Mc1 +2.39 −2.32− Simulation Mc1 +2.39 2.32 16.6%/0.4% Experiment Mc1 +2.05 −2.33− 16.6%/0.4% C-shaped curve line Experiment Mc1 +2.05 2.33 C-shaped curve line Simulation T +1.92 −2.24− Simulation T +1.92 2.24 4.5%/1.8% Experiment T +2.01 −2.28− 4.5%/1.8% Experiment T +2.01 2.28 − Referring to Tables 5 and 6, when the vehicles pass the S-shaped curve line and the C-shaped curve line, the maximum yaw angle error of experimental results and simulation results is concentrated on Mc1 vehicle. Because the working conditions of Mc1 as a motor vehicle are much more complicated than that of T as a trailer when crossing different curves. The reasons for the Energies 2020, 13, 4335 18 of 19

Referring to Tables5 and6, when the vehicles pass the S-shaped curve line and the C-shaped curve line, the maximum yaw angle error of experimental results and simulation results is concentrated on Mc1 vehicle. Because the working conditions of Mc1 as a motor vehicle are much more complicated than that of T as a trailer when crossing diﬀerent curves. The reasons for the diﬀerence between the experimental results and the simulation results (S-shaped curve and C-shaped curve) may be as follows:

(1) The area of the throttle valve and the size of piston are not very accurate in the hydraulic anti-kink system. (2) When passing through a small curve, the wheel rail resistance is very large, and the actual running speed cannot be as constant as the simulation environment. (3) In the actual operation process, M, Mc1, and Mc2 vehicles have power, and the drive system has the ability of traction torque control. However, the simulation process relies on the No. 9 hinge to simulate the vehicle operation conditions, which is diﬀerent from the real situation.

After installing the anti-kink system, the front and rear vehicle bodies have internal relations in parameters. The original secondary suspension parameters and workshop hinge parameters are not necessarily the best parameters after installing the anti-kink system. The secondary suspension parameters and the hinge stiffness between the vehicle bodies can be further optimized, such as the spring stiffness of the buffer hydraulic cylinder or the sectional area of the bypass throttle valve, so as to make the anti-kink system to achieve better performance. Although the simulation has some defects, the consistency of the change curve shows the accuracy of the model.

Author Contributions: Methodology, all authors; Software, all authors; writing—original draft preparation, X.L., Z.C., and Y.J.; writing—review and editing, X.L., Z.C., Y.J., and J.L.; supervision, Z.C. and J.L.; project administration, X.L. and Y.J. All authors have read and agreed to the published version of the manuscript. Funding: The work was supported by National Natural Science Foundation of China (No. U1934203) and authors would like to thank the State Key Laboratory of Traction Power for providing office, equipment, and materials to this project. Conflicts of Interest: The authors declare no conflict of interest.

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