# Determination of Mapping Relation between Wheel Polygonalisation and Wheel/Rail Contact Force for Railway Freight Wagon Using Dynamic Simulation

Authors: Jian Mu; Jing Zeng; Qunsheng Wang; Hutang Sang
Journal: Shock and Vibration
 (2021)

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## Abstract

The polygonal wear around the wheel circumference could pose highly adverse influences on the wheel/rail interactions and thereby the performance of the vehicle system. In this study, the effects of wheel polygonalisation on the dynamic responses of a freight wagon are investigated through development and simulations of a comprehensive coupled vehicle-track dynamic model. The model integrates flexible ballasted track and wheelsets subsystem models so as to account for elastic deformations caused by impact loads induced by the wheel polygonalisation. Subsequently, the vehicles with low-order polygonal wear, whether in empty or loaded conditions, are simulated at different speeds considering different amplitudes and harmonic orders of the wheel polygonalisation and thus the mapping relation between wheel/rail impact force and wheel polygonalisation is obtained. The results reveal that the low-order wheel polygonalisation except 1st order and 3rd order can give rise to high-frequency impact loads at the wheel/rail interface and excite 1st-bend modes of the wheelset and “P2 resonance” leading to high-magnitude wheel/rail contact force at the corresponding speed.


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## Body

## 1. Introduction

Wheel polygonalisation is a form of wheel circumferential uneven wear, also referred to as wheel polygonalisation or wheel corrugation, which generally exists in vehicles [1]. The periodic circumferential wear has been associated with high magnitudes of high-frequency impact loads at the wheel/rail interface, which contribute to undesired dynamic responses and reduced fatigue lives of the vehicle-track substructures, especially with higher axle loads and high-speed operations [2].Owing to highly adverse effects of wheel polygonalisation, considerable efforts have been made to understand the mechanisms leading to periodic circumferential defects in railway wheels. Wheel polygonalisation was initially observed in European railways [3]. The studies suggested that wheel polygonalisation is strongly related to bending vibration of the wheelset, track properties, and dynamic unbalances in wheels. Nielsen et al. [4, 5] presented a comprehensive review of different types of wheel defects together with associated potential wear mechanisms and the resulting influences on dynamic responses of the vehicle. The study also proposed the wheel removal criterion in the presence of polygonalisation deformities. The influences of wheel out-of-roundness were also reviewed by Barke and Chiu [6] including the wheel/rail impacts, rail fatigue, rail joint deterioration, sleeper degradation, and noise generation. Liu and Zhai [7] investigated vertical dynamic wheel/rail interaction resulting from two types of out-of-round wheels at high speeds by a vertical vehicle-track coupled dynamics model. The results demonstrate the influence of the out-of-round wheel on vehicle system is mainly related to the wheelset vibration and derivative of wheel radial deviations can effectively reflect dynamic wheel/rail contact force.A few studies have shown that high-magnitude impact loads could excite different bending and torsional deformation modes of the wheelset [8], which could cause higher lateral slippage of the wheel and wheel/rail material excavation, and thereby enlargement of out-of-round (OOR) deformities. Jin and his research team [9, 10] experimentally investigated the effects and growth of polygonal wear of subway wheels and concluded that the ninth-order polygonal wear observed in wheel deformations was due to the first bending mode of the wheelset. The effects of wheel polygonalisation on the dynamic responses of a high-speed rail vehicle are investigated through development and simulations of a comprehensive coupled vehicle-track dynamic model by Wu [11]. The results suggested that the high-order wheel polygonalisation could give rise to high-frequency impact loads at the wheel/rail interface and excited some of the vibration modes of the wheelset leading to high-magnitude axle box acceleration and dynamic stress in the wheelset axle. The effects of the wheel polygonization on the fatigue damage of the gearbox housing of a high-speed train are investigated through a multibody system (MBS) railway vehicle model by Wu [12]. The results stated that the fatigue damage with a 20th-order polygonal wear is 63% larger than that without the polygonal wear on the wheel. Moreover, the torsional vibration of the gear transmission system [13] and wheel/rail noise [14] are also affected by wheel polygonization. However, almost all of the research focuses on single order wheel polygonalisation obtained in high-speed vehicles or subways.Wheel polygonalisation is common in freight wagon and its wear depth changes exponentially with time [15]. It is important to establish the mapping relationship detecting wheel polygonalisation by the trackside detection for freight wagon condition repair. In this study, a comprehensive coupled vehicle-track dynamic model of freight wagon is established so as to investigate the effects of low-order wheel polygonalisation on wheel/rail contact force. The model integrates flexible ballasted track and wheelset subsystem models so as to account for elastic deformations that may occur in the presence of repetitive impact loads induced by the wheel polygonalisation at the wheel/rail interface. The effects of amplitude and harmonic order of wheel polygonal wear are subsequently investigated and the corresponding mapping relations are obtained to detect polygonal wheel by trackside detection.


## 2. Establishment of Vehicle-Track Model

A comprehensive model of a top open wagon vehicle-track system is formulated through integration of models of flexible wheelsets, a ballasted track, and the freight wagon. Figure1 illustrates the simulation scheme comprising finite element (FE) models of the flexible wheelset and the ballasted track coupled with the open wagon model in the SIMPACK platform [11]. The FE model of the wheelset is used to determine its modal properties via the eigen analysis. The modal vectors are subsequently integrated to the wagon model using the finite element multibody systems (FEMBS) interface available in SIMPACK [11]. The resulting dynamic wheel/rail contact forces, the displacement of contact spot, and speed are obtained to serve as excitations to the ballasted track model to evaluate the deflection response of the track. The rail deflection response is subsequently integrated to the vehicle model using the SIMAT (SIMPACK-MATLAB) cosimulation interface, as shown in Figure 1, to study its dynamic responses in the presence of a wheel polygonal wear. In simulation, the track deflection is solved by a new fast numerical integration method [16] with the step size of 0.0001s. The component models’ formulations are described in the following subsections.Figure 1
Vehicle-track dynamic system model. FEMBS: finite element multibody systems.


### 2.1. Flexible Wheelset Model

The FE model of wheelset is discretized into 22066 Solid 185 elements, as shown in Figure1. The modal characteristics of wheelset are obtained through eigen analysis. The substructure method is introduced to deal with the wheelset FE model to speed up the dynamic calculation, and the model DOF is transformed into the structural model represented by the master DOF. The nodes DOF where the force is applied or output and the boundary conditions are calculated is defined as the master DOF and the other defined as the slave DOF. The substructure method is based on Guyan reduction method condensing a set of elements into a superelement by matrix transformation [17]. The dynamic equation of the FE model can be expressed as(1)Mu¨+Du˙+Ku=F,where M, K, and D are mass matrix, stiffness matrix, and damping matrix, respectively; u is displacement matrix; and F is load matrix. Assuming that there are m master DOF and s slave DOF, the matrix form of (1) can be expressed as(2)MmmMmsMsmMssu¨mmu¨ss+DmmDmsDsmDssu˙mmu˙ss+KmmKmsKsmKssummuss=Fmm0,where Kmm, Kms, Ksm, and Kss are m × m, m × m, s × m, and s × s stiffness matrices, respectively; M and D are the corresponding mass and damping matrices; and Fmm is the load vector acting on the master DOF. Considering that Guyan reduction theory is based on the static transformation of Ku = F [18], uss can be assumed as(3)uss=−Ks−1Ksmumm.Thus,(4)u=ummuss=I−Kss−1Ksmumm.The DOF conversion matrix can be defined as(5)T=I−Kss−1Ksm.Substituting (4) and (5) into equation (1), the motion equation of the finite element model with reduced degrees of freedom is(6)Mmm∗u¨mm+Dmm∗u˙mm+Kmm∗u=F∗,where(7)Μmm∗=TTMT,Dmm∗=TTDT,Kmm∗=TTKT,Fmm∗=TTF.The selected master DOF must retain the structural features of the components, as well as the master nodes such as hinge points and force action points. For this purpose, a total of 132 master nodes located on 13 cross sections of the wheelset along the axle direction are selected. Each of the wheel and axle end has 2 cross-sections. Additional 5 cross-sections are selected on the axle between the wheels. Considering that the wheel/rail impact force induced by wheel polygonal wear can excite some modes of the wheelset, a total of 3 vibration modes with frequency up to 229 Hz are considered to determine the wheelset response and 3 modes of the wheelset are shown in Figure2.Figure 2
Selected vibration modes of the wheelset.


### 2.2. The Ballasted Track Model

Figure3 illustrates the ballasted track is composed of rail, fasteners, sleepers, and ballast. The rail is represented by a finite length continuous Timoshenko beam discretely supported on ballast, which is represented by mass block connected by spring-damping element, via fasteners. The model is formulated for the 100m rail with the sleeper bay of 0.625 m, which is considered to represent characteristics of the infinite length rail with reasonable accuracy and computational efficiency [19].Figure 3
Schematic of ballasted track model: (a) main view and (b) side view.
(a)(b)(c)Both ends of the rail are assumed to be fixed, while each rail is subject to moving loads attributed to the lateral and vertical wheel/rail contact forces,Fwryj(t) and Fwrzj(t) (j = 1∼4). The lateral and vertical forces due to discrete supports are represented as Fsyi(t), Fszi(t) (i = 1∼Ns), where Ns is the number of supports. Figure 3(c) also illustrates the moments caused by the wheel/rail contact forces MGj(t) (j = 1∼4) and the support forces Msi(t) (i = 1∼Ns), which are taken about the center of the rail. xwj (j = 1∼4) and xsi(i = 1∼Ns) define the position of each wheel and the ith discrete support along the rail, respectively.The governing equations describing the verticalz and bending Ψy deflections of the rail subjected to vertical wheel/rail contact forces Fwrzj and support forces Fszi are obtained as [20](8)ρAr∂2zx,t∂t2+κzGAr∂ψyx,y∂x−∂2zx,t∂x2=−∑i=1NsFszitδx−xsi+∑j=1NwFwrzjtδx−xwj,ρIy∂2ψyx,t∂t2+κzGArψyx,t−∂zx,t∂x−EIy∂2ψyx,t∂x2=0,where ρ is the mass density, Ar is the rail cross-sectional area, G is the shear modulus, E is Young’s modulus, and Iy is the second moment of area of the rail cross-section about the y-axis. In the above equations, κz is the vertical shear coefficient, κz = 0.5329. δ(x) is the Dirac delta function, and Nw is the number of wheelset considered in the model.Similarly, the governing equations describing the lateraly and yaw Ψz deflections of the rail subjected to lateral wheel/rail contact forces Fwryj and support forces Fsyi are(9)ρAr∂2yx,t∂t2+κyGAr∂ψzx,tδx−∂2yx,t∂x2=−∑i=1NsFsyitδx−xsi+∑j=1NwFwryjtδx−xwj,ρIz∂2ψzx,t∂t2+κyGArψzx,t−∂yx,t∂x−EIz∂2ψzx,t∂x2=0,where Iz is the second moment of area of the rail cross section about the z-axis and κy is the lateral shear coefficient, κy = 0.4507.The equation of motion describing the torsional deflection Ψx of the rail subjected to the wheel/rail forces and support forces is formulated as(10)ρI0∂2ϕx,t∂t2−GK∂2ϕx,t∂t2=−∑i=1NsMsitδx−xsi+∑j=1NwMGjtδx−xwj,where I0 is the polar moment of inertia of the rail cross section and GK is the torsional stiffness coefficient.The above governing partial differential equations (7)∼(9) describing the deflections of the rail can be converted to a series of ordinary differential equations using the variable separation method in terms of generalized coordinates as(11)zx,t=∑k=1NzZkxqzkt,ψzx,t=∑k=1Nψzψzkxwzkt,yx,t=∑k=1NyYkxqykt,ψyx,t=∑k=1Nψyψykxwykt,ϕx,t=∑k=1NϕϕkxqTkt,where qyk(t), qzk(t), qTk(t), wzkt, and wykt are the vectors of corresponding modal coordinates, N in the above formulation relates to the number of modes considered for each coordinate, and Zk(x), Ψzk(x), Yk(x), Ψyk(x), and Φk(x) are the normalized shape functions of the rail. Using the condition that the rail is simply supported at two ends, the normalized shape functions are given as(12)Zkx=2ρIzlsinkπlx,ψzkx=2ρIylcoskπlx,Ykx=2ρIylsinkπlx,ψykx=2ρIzlcoskπlx,ϕkx=2ρI0lsinkπlx.Upon substituting the above relations in the governing equations of motion, equations (7)∼(9), and applying the normalized mode shape functions of the rails, the partial differential equations of the rail are converted to a set of ordinary differential equations, such that(13)g¨zkt+κzGArmkπl2qzkt−κzGArkπl1mρIywykt=−∑i=1NsFszitZkxsi+∑j=1NwFwrzjtZkxwj,w¨ykt+κzGArρIy+EIyρIykπl2wykt−κzGArkπl1mρIyqzkt=0,(14)g¨ykt+κyGArmkπl2qykt−κyGArkπl1mρIzwzkt=−∑i=1NsFsyitYkxsi+∑j=1NwFwryjtYkxwj,w¨zkt+κyGArρIz+EIzρIzkπl2wzkt−κyGArkπl1mρIzqykt=0,(15)g¨Tkt+GKρI0kπl2qTkt=−∑i=1NsMsitϕkxsi+∑j=1NwMGjtϕkxwj.The solutions of equations (13) ∼ (15) yield deflection responses of the rail, which are applied to the vehicle model to determine the normal contact forces, Fwrzi and Fwryi (i = 1∼4), using the Hertzian contact model available in the SIMPACK platform.The high-magnitude impact forces induced by the wheel defects are transmitted to the ballast through the fasteners, which causes vibration of the ballast. Meanwhile, the vibration of the ballast directly contributes to the rail deflections and thus the wheel/rail contact forces. As Figure3 shows, the fastening is treated as the spring-damper element with vertical stiffness Kfz, vertical damping coefficient Cfz, lateral stiffness Kfy, and lateral damping coefficient Cfz. The sleeper is modeled as a rigid body with mass ms and inertial moment Isx. The motions of the sleeper are described by vertical displacement zs and lateral displacement ys of the mass center of the sleeper and rotation angle Φs. The ballast is modeled as the equivalent mass block with mass mb and only the vertical motion zbL (zbR) is considered. Besides, the equivalent shear spring-damping elements Kr–Cr are set between track bed blocks to consider the shear interaction between left and right, front, and rear adjacent track beds. Parameters of ballasted track model are listed in Table 1.Table 1
Parameters of the ballasted track.
VariableNotationValueEYoung's modulus of rail2.06 × 1011 N/m2GShear modulus of rail7.92 × 1010 N/m2ρDensity of rail7850 kg/m2ACross-section area of rail7.745×10−3m2IySecond moments of area of rail cross section about they-axis5.240×10−6m2IzSecond moments of area of rail cross section about thez-axis3.217×10−5m4I0Solar moment of inertia of rail cross section3.741×10−5m4eDistance from load point to the central line of rail cross section0.023 maDistance from bottle surface to the torsional center of rail cross section0.08147 mbDistance from bottle surface left/right side to the central line of rail cross section0.075 mhRDistance from top surface to the torsional center of rail cross section0.095 mKfzEquivalent vertical stiffness of fastening2.8 × 107 N/mKfyEquivalent lateral stiffness of fastening3.0✕107 N/mCfzEquivalent vertical damping coefficient of fastening3.625 × 104 N/mCfyEquivalent lateral damping coefficient of fastening3.0 × 104 N/mmsMass of sleeper349 kgIsxMoment of inertia of sleeper about thex-axis199 kg m2mbMass of ballast equivalent mass block139.8 kgKfzEquivalent vertical stiffness of fastening2.8 × 107 N/mKfyEquivalent lateral stiffness of fastening3.0 × 107 N/mCfzEquivalent vertical damping coefficient of fastening3.6 × 104N·s/mCfyEquivalent lateral damping coefficient of fastening3.0 × 104N·s/mKsbyLateral equivalent stiffness between sleeper and ballast7.0 × 107 N/mCsbyLateral equivalent damping coefficient between sleeper and ballast6.0 × 104N·s/mKsbzVertical equivalent stiffness between sleeper and ballast7.0 × 107 N/mCsbzVertical equivalent damping coefficient between sleeper and ballast6.0 × 104N·s/mKbzVertical equivalent stiffness between ballast and ground6.7 × 107 N/mKbzVertical equivalent damping coefficient between ballast and ground3.1 × 104N·s/mKrVertical equivalent stiffness between ballasts7.8 × 107 N/mCrVertical equivalent damping coefficient between ballasts8.0 × 104N·s/mdBDistance between mass centers of ballast and sleeper0.6 m


### 2.3. Vehicle Model

The mathematical model coupling the lateral and vertical motions is considered in order to accurately simulate the operation performance of the Chinese open top wagon C80. The model consists of a rigid car body supported on two rigid bolsters via the center plates and side bearings, and four flexible wheelsets coupled to four side frames through the primary rubber suspensions. Meanwhile, two side frames and one bolster are connected by coil springs and friction wedge dampers. Each wheelset is considered as a rotating flexible body, whose deflection responses are obtained from modal superposition, shown in Section2.1.The freight wagon has friction elements such as center plate and side bearing, which will provide rotary friction torques between car body and bogies to ensure the vehicle hunting stability. The rotary friction torque of the center plate can be calculated by the following:(16)Tc=∫0Rc2πr2μcPcAcdr,=23μcPcRc,where Pc is the load on the center plate, Ac and Rc are the area and radius of the center plate, and μc is the friction coefficient of the center plate surface. The side bearing here is a double acting elastic side bearing, which can effectively prevent the excessive increase of the rotary friction torque as the gap reached and improve the curving performance of the freight wagon. The rotary friction torque of the side bearing can be calculated as follows:(17)Ts=Ps+kzΔzμsds,Δz≤Δc,Ps+kzΔcμsds,Δz>Δc,where Ps is side bearing preload, kz is the vertical stiffness of the side bearing, ∆c is side bearing gap with ∆c = 6 mm, μs is the friction coefficient of the side bearing surface, and ds is the distance between the side bearing and the bogie center. In addition, the Chinese type LM wheel profile and CHN60 rail profile are adopted for the simulation. The structural and suspension parameters of freight wagon model are set according to Table 2.Table 2
Freight wagon model parameters.
ParametersValuesCar body mass10829 kgSide frame mass497 kgBolster mass745 kgWheelset mass1171 kgCar body roll moment of inertia aroundx-axis6.932 × 104 kg·m2Car body pitch moment of inertia aroundy-axis1.837 × 105 kg·m2Car body yaw moment of inertia aroundz-axis2.127 × 105 kg·m2Side frame roll moment of inertia aroundx-axis20.735 kg·m2Side frame pitch moment of inertia aroundy-axis188.2 kg·m2Side frame yaw moment of inertia aroundz-axis172.5 kg·m2Bolster roll moment of inertia aroundx-axis259.7 kg·m2Bolster pitch moment of inertia aroundy-axis16.158 kg·m2Bolster yaw moment of inertia aroundz-axis258.08 kg·m2Wheelset roll moment of inertia aroundx-axis700 kg·m2Wheelset pitch moment of inertia aroundy-axis72 kg·m2Wheelset yaw moment of inertia aroundz-axis700 kg·m2Primary suspension vertical stiffness alongz-axis160 MN/mSecondary suspension vertical stiffness alongz-axis2.233 MN/mThe distance between the side bearing and the bogie center0.76 mRadius of friction moment of center plate0.125 mSide bearing preload19.8 kN


### 2.4. Wheel Polygonalisation

Wheel polygonalisation is described by variations in the wheel radius as a function of angular position of the wheelϕh. Figure 4(a) illustrates the OOR deformities measured on a wheel, which suggests nearly periodic variations or wheel polygonalisation. The measured wear profile is idealised by a periodic waveform superimposed on the wheel circumference, as seen in Figure 4(b). The coordinates of a point on the circumference or the contact point can thus be expressed as(18)x=Rw+ApsinNpϕhsinϕh,y=Rw+ApsinNpϕhcosϕh,ϕh∈0,2π,where x and y are the coordinates of a point on the wheel circumference, Ap denotes the amplitude of the polygonal wear, Np is the order of harmonic of the polygonal wear, and Rw is the nominal wheel radius. The radius of the wheel at a given point on the wheel circumference can be given by Rp=Rw+ApsinNpΦh.Figure 4
(a) Measured circumferential polygonal wear and (b) polygonal wear idealised by a harmonic waveform.
(a)(b)Through measurements of circumferential profiles of 99 different wheels employed in high-speed freight and commuter trains, Johansson found that the order of polygonalisation in freight wagon is mostly below 10 [21], and the similar phenomenon is also reported in reference [22]. Therefore, low-order wheel polygonalisation is simulated to establish the mapping relationship between wheel/rail force and wheel polygonalisation in this paper.


### 2.5. Model Validation

The wheel polygonal wear induced wheel/rail contact forces have been widely investigated experimentally and analytically. Most of these studies, however, focus on single order polygonalisation. The validity of the coupled vehicle-track dynamic model in the presence of wheel polygonal wear is thus examined by comparing the simulation results with the reported measured data [5]. The simulation results are obtained for the vehicle-track model considering the reported experimental conditions 0.125 m amplitude of 10th-order polygonalisation, while the wheel polygonal wear is limited to the leading wheelset. Simulation results in terms of peak wheel/rail contact force deviations from the static load are compared with the reported measured data in the 0–100 km/h speed range in Figure 5. The comparisons suggest reasonably good agreements with the experimental results in saturation force and variation tendency. Both the simulation and experimental results show saturation of the peak force in the 50–60 km/h speed range. However, the saturation forces appear at different speeds, which is caused by the structure of the track.Figure 5
Comparisons of the peak wheel/rail contact force deviations from the static load with vehicle speed obtained from coupled vehicle-track model with the reported measured data [5].


## 2.1. Flexible Wheelset Model

The FE model of wheelset is discretized into 22066 Solid 185 elements, as shown in Figure1. The modal characteristics of wheelset are obtained through eigen analysis. The substructure method is introduced to deal with the wheelset FE model to speed up the dynamic calculation, and the model DOF is transformed into the structural model represented by the master DOF. The nodes DOF where the force is applied or output and the boundary conditions are calculated is defined as the master DOF and the other defined as the slave DOF. The substructure method is based on Guyan reduction method condensing a set of elements into a superelement by matrix transformation [17]. The dynamic equation of the FE model can be expressed as(1)Mu¨+Du˙+Ku=F,where M, K, and D are mass matrix, stiffness matrix, and damping matrix, respectively; u is displacement matrix; and F is load matrix. Assuming that there are m master DOF and s slave DOF, the matrix form of (1) can be expressed as(2)MmmMmsMsmMssu¨mmu¨ss+DmmDmsDsmDssu˙mmu˙ss+KmmKmsKsmKssummuss=Fmm0,where Kmm, Kms, Ksm, and Kss are m × m, m × m, s × m, and s × s stiffness matrices, respectively; M and D are the corresponding mass and damping matrices; and Fmm is the load vector acting on the master DOF. Considering that Guyan reduction theory is based on the static transformation of Ku = F [18], uss can be assumed as(3)uss=−Ks−1Ksmumm.Thus,(4)u=ummuss=I−Kss−1Ksmumm.The DOF conversion matrix can be defined as(5)T=I−Kss−1Ksm.Substituting (4) and (5) into equation (1), the motion equation of the finite element model with reduced degrees of freedom is(6)Mmm∗u¨mm+Dmm∗u˙mm+Kmm∗u=F∗,where(7)Μmm∗=TTMT,Dmm∗=TTDT,Kmm∗=TTKT,Fmm∗=TTF.The selected master DOF must retain the structural features of the components, as well as the master nodes such as hinge points and force action points. For this purpose, a total of 132 master nodes located on 13 cross sections of the wheelset along the axle direction are selected. Each of the wheel and axle end has 2 cross-sections. Additional 5 cross-sections are selected on the axle between the wheels. Considering that the wheel/rail impact force induced by wheel polygonal wear can excite some modes of the wheelset, a total of 3 vibration modes with frequency up to 229 Hz are considered to determine the wheelset response and 3 modes of the wheelset are shown in Figure2.Figure 2
Selected vibration modes of the wheelset.


## 2.2. The Ballasted Track Model

Figure3 illustrates the ballasted track is composed of rail, fasteners, sleepers, and ballast. The rail is represented by a finite length continuous Timoshenko beam discretely supported on ballast, which is represented by mass block connected by spring-damping element, via fasteners. The model is formulated for the 100m rail with the sleeper bay of 0.625 m, which is considered to represent characteristics of the infinite length rail with reasonable accuracy and computational efficiency [19].Figure 3
Schematic of ballasted track model: (a) main view and (b) side view.
(a)(b)(c)Both ends of the rail are assumed to be fixed, while each rail is subject to moving loads attributed to the lateral and vertical wheel/rail contact forces,Fwryj(t) and Fwrzj(t) (j = 1∼4). The lateral and vertical forces due to discrete supports are represented as Fsyi(t), Fszi(t) (i = 1∼Ns), where Ns is the number of supports. Figure 3(c) also illustrates the moments caused by the wheel/rail contact forces MGj(t) (j = 1∼4) and the support forces Msi(t) (i = 1∼Ns), which are taken about the center of the rail. xwj (j = 1∼4) and xsi(i = 1∼Ns) define the position of each wheel and the ith discrete support along the rail, respectively.The governing equations describing the verticalz and bending Ψy deflections of the rail subjected to vertical wheel/rail contact forces Fwrzj and support forces Fszi are obtained as [20](8)ρAr∂2zx,t∂t2+κzGAr∂ψyx,y∂x−∂2zx,t∂x2=−∑i=1NsFszitδx−xsi+∑j=1NwFwrzjtδx−xwj,ρIy∂2ψyx,t∂t2+κzGArψyx,t−∂zx,t∂x−EIy∂2ψyx,t∂x2=0,where ρ is the mass density, Ar is the rail cross-sectional area, G is the shear modulus, E is Young’s modulus, and Iy is the second moment of area of the rail cross-section about the y-axis. In the above equations, κz is the vertical shear coefficient, κz = 0.5329. δ(x) is the Dirac delta function, and Nw is the number of wheelset considered in the model.Similarly, the governing equations describing the lateraly and yaw Ψz deflections of the rail subjected to lateral wheel/rail contact forces Fwryj and support forces Fsyi are(9)ρAr∂2yx,t∂t2+κyGAr∂ψzx,tδx−∂2yx,t∂x2=−∑i=1NsFsyitδx−xsi+∑j=1NwFwryjtδx−xwj,ρIz∂2ψzx,t∂t2+κyGArψzx,t−∂yx,t∂x−EIz∂2ψzx,t∂x2=0,where Iz is the second moment of area of the rail cross section about the z-axis and κy is the lateral shear coefficient, κy = 0.4507.The equation of motion describing the torsional deflection Ψx of the rail subjected to the wheel/rail forces and support forces is formulated as(10)ρI0∂2ϕx,t∂t2−GK∂2ϕx,t∂t2=−∑i=1NsMsitδx−xsi+∑j=1NwMGjtδx−xwj,where I0 is the polar moment of inertia of the rail cross section and GK is the torsional stiffness coefficient.The above governing partial differential equations (7)∼(9) describing the deflections of the rail can be converted to a series of ordinary differential equations using the variable separation method in terms of generalized coordinates as(11)zx,t=∑k=1NzZkxqzkt,ψzx,t=∑k=1Nψzψzkxwzkt,yx,t=∑k=1NyYkxqykt,ψyx,t=∑k=1Nψyψykxwykt,ϕx,t=∑k=1NϕϕkxqTkt,where qyk(t), qzk(t), qTk(t), wzkt, and wykt are the vectors of corresponding modal coordinates, N in the above formulation relates to the number of modes considered for each coordinate, and Zk(x), Ψzk(x), Yk(x), Ψyk(x), and Φk(x) are the normalized shape functions of the rail. Using the condition that the rail is simply supported at two ends, the normalized shape functions are given as(12)Zkx=2ρIzlsinkπlx,ψzkx=2ρIylcoskπlx,Ykx=2ρIylsinkπlx,ψykx=2ρIzlcoskπlx,ϕkx=2ρI0lsinkπlx.Upon substituting the above relations in the governing equations of motion, equations (7)∼(9), and applying the normalized mode shape functions of the rails, the partial differential equations of the rail are converted to a set of ordinary differential equations, such that(13)g¨zkt+κzGArmkπl2qzkt−κzGArkπl1mρIywykt=−∑i=1NsFszitZkxsi+∑j=1NwFwrzjtZkxwj,w¨ykt+κzGArρIy+EIyρIykπl2wykt−κzGArkπl1mρIyqzkt=0,(14)g¨ykt+κyGArmkπl2qykt−κyGArkπl1mρIzwzkt=−∑i=1NsFsyitYkxsi+∑j=1NwFwryjtYkxwj,w¨zkt+κyGArρIz+EIzρIzkπl2wzkt−κyGArkπl1mρIzqykt=0,(15)g¨Tkt+GKρI0kπl2qTkt=−∑i=1NsMsitϕkxsi+∑j=1NwMGjtϕkxwj.The solutions of equations (13) ∼ (15) yield deflection responses of the rail, which are applied to the vehicle model to determine the normal contact forces, Fwrzi and Fwryi (i = 1∼4), using the Hertzian contact model available in the SIMPACK platform.The high-magnitude impact forces induced by the wheel defects are transmitted to the ballast through the fasteners, which causes vibration of the ballast. Meanwhile, the vibration of the ballast directly contributes to the rail deflections and thus the wheel/rail contact forces. As Figure3 shows, the fastening is treated as the spring-damper element with vertical stiffness Kfz, vertical damping coefficient Cfz, lateral stiffness Kfy, and lateral damping coefficient Cfz. The sleeper is modeled as a rigid body with mass ms and inertial moment Isx. The motions of the sleeper are described by vertical displacement zs and lateral displacement ys of the mass center of the sleeper and rotation angle Φs. The ballast is modeled as the equivalent mass block with mass mb and only the vertical motion zbL (zbR) is considered. Besides, the equivalent shear spring-damping elements Kr–Cr are set between track bed blocks to consider the shear interaction between left and right, front, and rear adjacent track beds. Parameters of ballasted track model are listed in Table 1.Table 1
Parameters of the ballasted track.
VariableNotationValueEYoung's modulus of rail2.06 × 1011 N/m2GShear modulus of rail7.92 × 1010 N/m2ρDensity of rail7850 kg/m2ACross-section area of rail7.745×10−3m2IySecond moments of area of rail cross section about they-axis5.240×10−6m2IzSecond moments of area of rail cross section about thez-axis3.217×10−5m4I0Solar moment of inertia of rail cross section3.741×10−5m4eDistance from load point to the central line of rail cross section0.023 maDistance from bottle surface to the torsional center of rail cross section0.08147 mbDistance from bottle surface left/right side to the central line of rail cross section0.075 mhRDistance from top surface to the torsional center of rail cross section0.095 mKfzEquivalent vertical stiffness of fastening2.8 × 107 N/mKfyEquivalent lateral stiffness of fastening3.0✕107 N/mCfzEquivalent vertical damping coefficient of fastening3.625 × 104 N/mCfyEquivalent lateral damping coefficient of fastening3.0 × 104 N/mmsMass of sleeper349 kgIsxMoment of inertia of sleeper about thex-axis199 kg m2mbMass of ballast equivalent mass block139.8 kgKfzEquivalent vertical stiffness of fastening2.8 × 107 N/mKfyEquivalent lateral stiffness of fastening3.0 × 107 N/mCfzEquivalent vertical damping coefficient of fastening3.6 × 104N·s/mCfyEquivalent lateral damping coefficient of fastening3.0 × 104N·s/mKsbyLateral equivalent stiffness between sleeper and ballast7.0 × 107 N/mCsbyLateral equivalent damping coefficient between sleeper and ballast6.0 × 104N·s/mKsbzVertical equivalent stiffness between sleeper and ballast7.0 × 107 N/mCsbzVertical equivalent damping coefficient between sleeper and ballast6.0 × 104N·s/mKbzVertical equivalent stiffness between ballast and ground6.7 × 107 N/mKbzVertical equivalent damping coefficient between ballast and ground3.1 × 104N·s/mKrVertical equivalent stiffness between ballasts7.8 × 107 N/mCrVertical equivalent damping coefficient between ballasts8.0 × 104N·s/mdBDistance between mass centers of ballast and sleeper0.6 m


## 2.3. Vehicle Model

The mathematical model coupling the lateral and vertical motions is considered in order to accurately simulate the operation performance of the Chinese open top wagon C80. The model consists of a rigid car body supported on two rigid bolsters via the center plates and side bearings, and four flexible wheelsets coupled to four side frames through the primary rubber suspensions. Meanwhile, two side frames and one bolster are connected by coil springs and friction wedge dampers. Each wheelset is considered as a rotating flexible body, whose deflection responses are obtained from modal superposition, shown in Section2.1.The freight wagon has friction elements such as center plate and side bearing, which will provide rotary friction torques between car body and bogies to ensure the vehicle hunting stability. The rotary friction torque of the center plate can be calculated by the following:(16)Tc=∫0Rc2πr2μcPcAcdr,=23μcPcRc,where Pc is the load on the center plate, Ac and Rc are the area and radius of the center plate, and μc is the friction coefficient of the center plate surface. The side bearing here is a double acting elastic side bearing, which can effectively prevent the excessive increase of the rotary friction torque as the gap reached and improve the curving performance of the freight wagon. The rotary friction torque of the side bearing can be calculated as follows:(17)Ts=Ps+kzΔzμsds,Δz≤Δc,Ps+kzΔcμsds,Δz>Δc,where Ps is side bearing preload, kz is the vertical stiffness of the side bearing, ∆c is side bearing gap with ∆c = 6 mm, μs is the friction coefficient of the side bearing surface, and ds is the distance between the side bearing and the bogie center. In addition, the Chinese type LM wheel profile and CHN60 rail profile are adopted for the simulation. The structural and suspension parameters of freight wagon model are set according to Table 2.Table 2
Freight wagon model parameters.
ParametersValuesCar body mass10829 kgSide frame mass497 kgBolster mass745 kgWheelset mass1171 kgCar body roll moment of inertia aroundx-axis6.932 × 104 kg·m2Car body pitch moment of inertia aroundy-axis1.837 × 105 kg·m2Car body yaw moment of inertia aroundz-axis2.127 × 105 kg·m2Side frame roll moment of inertia aroundx-axis20.735 kg·m2Side frame pitch moment of inertia aroundy-axis188.2 kg·m2Side frame yaw moment of inertia aroundz-axis172.5 kg·m2Bolster roll moment of inertia aroundx-axis259.7 kg·m2Bolster pitch moment of inertia aroundy-axis16.158 kg·m2Bolster yaw moment of inertia aroundz-axis258.08 kg·m2Wheelset roll moment of inertia aroundx-axis700 kg·m2Wheelset pitch moment of inertia aroundy-axis72 kg·m2Wheelset yaw moment of inertia aroundz-axis700 kg·m2Primary suspension vertical stiffness alongz-axis160 MN/mSecondary suspension vertical stiffness alongz-axis2.233 MN/mThe distance between the side bearing and the bogie center0.76 mRadius of friction moment of center plate0.125 mSide bearing preload19.8 kN


## 2.4. Wheel Polygonalisation

Wheel polygonalisation is described by variations in the wheel radius as a function of angular position of the wheelϕh. Figure 4(a) illustrates the OOR deformities measured on a wheel, which suggests nearly periodic variations or wheel polygonalisation. The measured wear profile is idealised by a periodic waveform superimposed on the wheel circumference, as seen in Figure 4(b). The coordinates of a point on the circumference or the contact point can thus be expressed as(18)x=Rw+ApsinNpϕhsinϕh,y=Rw+ApsinNpϕhcosϕh,ϕh∈0,2π,where x and y are the coordinates of a point on the wheel circumference, Ap denotes the amplitude of the polygonal wear, Np is the order of harmonic of the polygonal wear, and Rw is the nominal wheel radius. The radius of the wheel at a given point on the wheel circumference can be given by Rp=Rw+ApsinNpΦh.Figure 4
(a) Measured circumferential polygonal wear and (b) polygonal wear idealised by a harmonic waveform.
(a)(b)Through measurements of circumferential profiles of 99 different wheels employed in high-speed freight and commuter trains, Johansson found that the order of polygonalisation in freight wagon is mostly below 10 [21], and the similar phenomenon is also reported in reference [22]. Therefore, low-order wheel polygonalisation is simulated to establish the mapping relationship between wheel/rail force and wheel polygonalisation in this paper.


## 2.5. Model Validation

The wheel polygonal wear induced wheel/rail contact forces have been widely investigated experimentally and analytically. Most of these studies, however, focus on single order polygonalisation. The validity of the coupled vehicle-track dynamic model in the presence of wheel polygonal wear is thus examined by comparing the simulation results with the reported measured data [5]. The simulation results are obtained for the vehicle-track model considering the reported experimental conditions 0.125 m amplitude of 10th-order polygonalisation, while the wheel polygonal wear is limited to the leading wheelset. Simulation results in terms of peak wheel/rail contact force deviations from the static load are compared with the reported measured data in the 0–100 km/h speed range in Figure 5. The comparisons suggest reasonably good agreements with the experimental results in saturation force and variation tendency. Both the simulation and experimental results show saturation of the peak force in the 50–60 km/h speed range. However, the saturation forces appear at different speeds, which is caused by the structure of the track.Figure 5
Comparisons of the peak wheel/rail contact force deviations from the static load with vehicle speed obtained from coupled vehicle-track model with the reported measured data [5].


## 3. Results and Discussions

In this study, the measured polygonal wear is idealised by a harmonic variation in the wheel radius, as described in section2.4. The vehicle-track model simulations are subsequently performed to evaluate the dynamic responses of the vehicle system under excitations caused by polygonal wear. The simulation results in each case are obtained under excitations due to polygonal wear of both the left and right wheels of the leading wheelset of the front bogie, while assuming negligible phase difference between the two wheels’ deformities.


### 3.1. Influences of Vehicle Speed and Wear Amplitude

Figure6 illustrates the effect of forward speed on the resulting maximum and minimum wheel/rail contact force magnitudes, both empty and loaded vehicles, considering from 0.05 mm to 0.3 mm amplitude 6th-order polygonal wear on both the wheels of the leading wheelset. Take polygonal wear with amplitude of 0.1 mm as an example; the maximum contact forces of empty vehicle increase with vehicle speed in the 20–90 km/h speed range and decrease at speeds exceeding 90 km/h while the maximum contact forces of loaded vehicle reach saturation at speed of 100 km/h. Corresponding to maximum of wheel/rail contact force, the minimum of wheel/rail contact force of loaded vehicles and the maximum wheel-rail force are approximately symmetrical with the static load. Meanwhile, the minimum of wheel/rail contact force of empty vehicle decreases with the speed until reducing to 0 at 90 km/h, which indicates wheel/rail separation.Figure 6
Influence of vehicle speed on the maximum and minimum wheel/rail contact force under the 6th-order polygonal wear: (a) empty vehicle and (b) loaded vehicle.
(a)(b)Apart from the forward speed, the magnitude of the wheel/rail contact force is strongly dependent on the amplitude of the polygonal wear. Figure7 illustrates the effect of wear amplitude on maxima and minima of the wheel/rail contact force of empty and loaded vehicles, respectively, caused by the 6th-order polygonal wear at speed of 90 km/h and 100 km/h. The results of empty clearly show a nearly linear increase and decrease in maximum and minimum contact force, respectively, with increasing wear amplitude at the speed of 90 km/h. Different from the variation at speed of 90 km/h, there is a rapid increase in the maximum contact force while there is a decrease to 0 in the minimum between 0.05 mm amplitude and 0.1 mm amplitude, that the wheel/rail separation contributes to a rapid increase in the maximum contact force. It is necessary to limit the polygonal wear amplitude preventing the separation of wheel and rail. The phenomenon in Figure 7(b) revealed by maximum and minimum contact force is similar to that in Figure 7(a) except for polygonal wear amplitude, which is caused by the different axel load between them.Figure 7
Influence of polygonal wear amplitude on the maxima and minima of the wheel/rail contact forces: (a) empty vehicle and (b) loaded vehicle.
(a)(b)Figures8(a) and 8(b) illustrate the frequency spectra of wheel/rail contact force deviations from the static load of empty vehicle induced by 0.1 mm amplitude and 0.2 mm amplitude, respectively, of 6th-order polygonal wear in the 20–120 km/h speed range. The results suggest that the passing frequency of wheel polygonalisation (foor) increases with the vehicle speed and variation tendency of force peaks along the passing frequency is the same as that of maximum wheel/rail contact force in Figure 6(a), which reveals that additional wheel/rail contact force is induced by polygonal wear. Besides, the spectra also revealed force peaks corresponding to multiples of foor, namely, 2foor and 3foor, appearing the same as that of the minimum of wheel/rail force reduced to 0 in Figure 6(a). These are likely due to wheel/rail separation induced elastic deformations of the wheelset at some speeds. Figures 8(c) and 8(d) illustrate the frequency spectra of wheel/rail contact force of loaded vehicle with 0.1 mm amplitude and 0.2 mm amplitude, respectively, of 6th-order polygonal wear in the 20–120 km/h speed range. Unlike empty vehicles, there are no corresponding to multiples of foor, which is due to no wheel-rail separation shown in Figure 6(b).Figure 8
Frequency spectra of wheel/rail contact force in the 20–120 km/h speed range: (a) empty vehicle (amplitude = 0.1 mm), (b) empty vehicle (amplitude = 0.2 mm), (c) loaded vehicle (amplitude = 0.1 mm), (d) loaded vehicle (amplitude = 0.2 mm).
(a)(b)(c)(d)


### 3.2. Influence of Order of Polygonal Wear

Figure9 illustrates the influence of order of polygonal wear on the maxima of the wheel/rail contact force deviations from the static load for 0.1 mm wear amplitude in the 60–130 km/h speed range. The results reveal that 1st-order and 3rd-order wheel polygonalisation, whether empty or loaded vehicles, have limited influence on wheel/rail contact forces. Furthermore, the maxima contact forces for empty vehicles induced by 5th-order, 7th-order, and 9th-order polygons are saturated at 110 km/h, 80 km/h, and 60 km/h, respectively. By contrast, the saturation forces induced by wheel polygonalisation appear at 115 km/h, 85 km/h, and 65 km/h. This is likely caused by the wheel/rail coupled vertical vibration mode near 60 Hz, as reported in [23, 24]. This relatively low-frequency mode is also referred to as “P2 force” or “P2 resonance” [25] and is related to effective masses of the wheelset and the rail, which cause the different passing frequency between empty and loaded vehicles, and stiffness of the primary suspension and the rail supports. Meanwhile, the passing frequency induced by polygonal wear resonates with P2, which intensifies the wheel/rail contact force resulting in the increase of polygon wear exponentially with time. Additionally, another peak value (113 Hz) induced by 9th-order polygons, whether empty or loaded vehicles, appears near 120 km/h, which is close to the 1st-bending mode (115 Hz) of the wheelset as Figure 2 shows.Figure 9
Effect of order of polygonal wear on maxima of the wheel/rail contact force deviations from the static load in the 60–120 km/h speed range: (a) empty vehicle and (b) loaded vehicle (amplitude = 0.1 mm).


### 3.3. Mapping Relation

As the limited influence is made by 1st-order and 3rd-order wheel polygonalisation, only the mapping relations between induced wheel/rail force and 5th-order, 7th-order, and 9th-order polygonalisation are established in this section. Figure 10 illustrates the mapping relations between wheel/rail forces induced by the polygonalisation with 0.1 mm, 0.2 mm, and 0.3 mm amplitude. The results reveal that amplitude variation of the polygonalisation only affects the amplitude of the saturation force, and the saturation forces of empty and loaded vehicles still appear when the passing frequency is 58 Hz and 113 Hz, 62 Hz, and 113 Hz, respectively, which is primarily affected by P2 force and 1st-bend mode of wheelset. Besides, the difference between saturated wheel/rail forces induced by different polygonal order with the same amplitude is little and the induced wheel/rail forces are almost proportional to the polygonal amplitude, except when the wheel-rail separation occurs.Figure 10
Mapping relation between the wheel/rail contact force deviations from the static load and polygonal wear order with difference amplitude.
(a)(b)The passing frequency induced by polygonal wear resonates with P2 force, which intensifies the wheel/rail contact force resulting in the increase of polygon wear exponentially with time. When the wheel operation time is over a specified maintenance period, polygonal wear develops quickly.Thus, it is very necessary to establish the mapping relationship between wheel/rail force and wheel polygonalisation to detect the state of passing vehicle wheelset by trackside detection, so as to find out and repair in time.


## 3.1. Influences of Vehicle Speed and Wear Amplitude

Figure6 illustrates the effect of forward speed on the resulting maximum and minimum wheel/rail contact force magnitudes, both empty and loaded vehicles, considering from 0.05 mm to 0.3 mm amplitude 6th-order polygonal wear on both the wheels of the leading wheelset. Take polygonal wear with amplitude of 0.1 mm as an example; the maximum contact forces of empty vehicle increase with vehicle speed in the 20–90 km/h speed range and decrease at speeds exceeding 90 km/h while the maximum contact forces of loaded vehicle reach saturation at speed of 100 km/h. Corresponding to maximum of wheel/rail contact force, the minimum of wheel/rail contact force of loaded vehicles and the maximum wheel-rail force are approximately symmetrical with the static load. Meanwhile, the minimum of wheel/rail contact force of empty vehicle decreases with the speed until reducing to 0 at 90 km/h, which indicates wheel/rail separation.Figure 6
Influence of vehicle speed on the maximum and minimum wheel/rail contact force under the 6th-order polygonal wear: (a) empty vehicle and (b) loaded vehicle.
(a)(b)Apart from the forward speed, the magnitude of the wheel/rail contact force is strongly dependent on the amplitude of the polygonal wear. Figure7 illustrates the effect of wear amplitude on maxima and minima of the wheel/rail contact force of empty and loaded vehicles, respectively, caused by the 6th-order polygonal wear at speed of 90 km/h and 100 km/h. The results of empty clearly show a nearly linear increase and decrease in maximum and minimum contact force, respectively, with increasing wear amplitude at the speed of 90 km/h. Different from the variation at speed of 90 km/h, there is a rapid increase in the maximum contact force while there is a decrease to 0 in the minimum between 0.05 mm amplitude and 0.1 mm amplitude, that the wheel/rail separation contributes to a rapid increase in the maximum contact force. It is necessary to limit the polygonal wear amplitude preventing the separation of wheel and rail. The phenomenon in Figure 7(b) revealed by maximum and minimum contact force is similar to that in Figure 7(a) except for polygonal wear amplitude, which is caused by the different axel load between them.Figure 7
Influence of polygonal wear amplitude on the maxima and minima of the wheel/rail contact forces: (a) empty vehicle and (b) loaded vehicle.
(a)(b)Figures8(a) and 8(b) illustrate the frequency spectra of wheel/rail contact force deviations from the static load of empty vehicle induced by 0.1 mm amplitude and 0.2 mm amplitude, respectively, of 6th-order polygonal wear in the 20–120 km/h speed range. The results suggest that the passing frequency of wheel polygonalisation (foor) increases with the vehicle speed and variation tendency of force peaks along the passing frequency is the same as that of maximum wheel/rail contact force in Figure 6(a), which reveals that additional wheel/rail contact force is induced by polygonal wear. Besides, the spectra also revealed force peaks corresponding to multiples of foor, namely, 2foor and 3foor, appearing the same as that of the minimum of wheel/rail force reduced to 0 in Figure 6(a). These are likely due to wheel/rail separation induced elastic deformations of the wheelset at some speeds. Figures 8(c) and 8(d) illustrate the frequency spectra of wheel/rail contact force of loaded vehicle with 0.1 mm amplitude and 0.2 mm amplitude, respectively, of 6th-order polygonal wear in the 20–120 km/h speed range. Unlike empty vehicles, there are no corresponding to multiples of foor, which is due to no wheel-rail separation shown in Figure 6(b).Figure 8
Frequency spectra of wheel/rail contact force in the 20–120 km/h speed range: (a) empty vehicle (amplitude = 0.1 mm), (b) empty vehicle (amplitude = 0.2 mm), (c) loaded vehicle (amplitude = 0.1 mm), (d) loaded vehicle (amplitude = 0.2 mm).
(a)(b)(c)(d)


## 3.2. Influence of Order of Polygonal Wear

Figure9 illustrates the influence of order of polygonal wear on the maxima of the wheel/rail contact force deviations from the static load for 0.1 mm wear amplitude in the 60–130 km/h speed range. The results reveal that 1st-order and 3rd-order wheel polygonalisation, whether empty or loaded vehicles, have limited influence on wheel/rail contact forces. Furthermore, the maxima contact forces for empty vehicles induced by 5th-order, 7th-order, and 9th-order polygons are saturated at 110 km/h, 80 km/h, and 60 km/h, respectively. By contrast, the saturation forces induced by wheel polygonalisation appear at 115 km/h, 85 km/h, and 65 km/h. This is likely caused by the wheel/rail coupled vertical vibration mode near 60 Hz, as reported in [23, 24]. This relatively low-frequency mode is also referred to as “P2 force” or “P2 resonance” [25] and is related to effective masses of the wheelset and the rail, which cause the different passing frequency between empty and loaded vehicles, and stiffness of the primary suspension and the rail supports. Meanwhile, the passing frequency induced by polygonal wear resonates with P2, which intensifies the wheel/rail contact force resulting in the increase of polygon wear exponentially with time. Additionally, another peak value (113 Hz) induced by 9th-order polygons, whether empty or loaded vehicles, appears near 120 km/h, which is close to the 1st-bending mode (115 Hz) of the wheelset as Figure 2 shows.Figure 9
Effect of order of polygonal wear on maxima of the wheel/rail contact force deviations from the static load in the 60–120 km/h speed range: (a) empty vehicle and (b) loaded vehicle (amplitude = 0.1 mm).


## 3.3. Mapping Relation

As the limited influence is made by 1st-order and 3rd-order wheel polygonalisation, only the mapping relations between induced wheel/rail force and 5th-order, 7th-order, and 9th-order polygonalisation are established in this section. Figure 10 illustrates the mapping relations between wheel/rail forces induced by the polygonalisation with 0.1 mm, 0.2 mm, and 0.3 mm amplitude. The results reveal that amplitude variation of the polygonalisation only affects the amplitude of the saturation force, and the saturation forces of empty and loaded vehicles still appear when the passing frequency is 58 Hz and 113 Hz, 62 Hz, and 113 Hz, respectively, which is primarily affected by P2 force and 1st-bend mode of wheelset. Besides, the difference between saturated wheel/rail forces induced by different polygonal order with the same amplitude is little and the induced wheel/rail forces are almost proportional to the polygonal amplitude, except when the wheel-rail separation occurs.Figure 10
Mapping relation between the wheel/rail contact force deviations from the static load and polygonal wear order with difference amplitude.
(a)(b)The passing frequency induced by polygonal wear resonates with P2 force, which intensifies the wheel/rail contact force resulting in the increase of polygon wear exponentially with time. When the wheel operation time is over a specified maintenance period, polygonal wear develops quickly.Thus, it is very necessary to establish the mapping relationship between wheel/rail force and wheel polygonalisation to detect the state of passing vehicle wheelset by trackside detection, so as to find out and repair in time.


## 4. Conclusion

A comprehensive coupled vehicle-track dynamic model of open top wagon C80 is established so as to investigate the effects of low-order wheel polygonalisation on wheel/rail contact force. The model integrates flexible ballasted track and wheelset subsystem models so as to account for elastic deformations that may occur in the presence of repetitive impact loads induced by the wheel polygonalisation. The effects of amplitude and harmonic order of wheel polygonal wear, whether empty and loaded vehicles, are subsequently investigated and the corresponding mapping relations are obtained to detect polygonal wheel by trackside detection. Then, the conclusions can be drawn as follows:(1)
The induced wheel/rail force is almost proportional to the polygon amplitude, except increasing rapidly when the wheel/rail separation occurs(2)
1st-order and 3rd-order polygons, whether empty vehicle and loaded vehicle, have limited influence on wheel/rail contact forces in the 20–130 km/h speed range(3)
The saturation forces of empty and loaded vehicle appear when the passing frequency is 58 Hz and 113 Hz and 62 Hz and 113 Hz, respectively, which is primarily affected by P2 force and 1st-bend mode of wheelset



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*Source: 4677851-2021-10-07.xml*