applied sciences

Article Key Factors of the Initiation and Development of Polygonal Wear in the Wheels of a High-Speed Train

Yue Wu 1, Xuesong Jin 1,*, Wubin Cai 1, Jian Han 2 and Xinbiao Xiao 1

1 State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China; [email protected] (Y.W.); [email protected] (W.C.); [email protected] (X.X.) 2 Mechanical Engineering College, Southwest Jiaotong University, Chengdu 610031, China; [email protected] * Correspondence: [email protected]



Received: 28 July 2020; Accepted: 20 August 2020; Published: 25 August 2020


Abstract: The polygonal wear of train wheels occurs commonly in rail transport and increases the wheel–rail interaction force dramatically and has a bad effect on the safety and comfort of the train. The mechanism of polygonal wear needs to be studied. The characteristics of test data measured from 47,000 sets of polygonal wheels of high-speed trains were analysed statistically. The analysis shows that, in the entire use life cycle of the wheels, the order (wavelength) and development speed of polygonal wear are different; they correspond to different wheel diameters because of wear and re-profiling. A prediction model, which considered the flexibility of the wheelset for the polygonal wear of the wheels of high-speed trains, was developed to explain this phenomenon. This theoretical model analyses the initiation, development, and characteristics of polygonal wear. The analysis includes the effect of the high-frequency flexible deformation of the wheelset, train operation speed, and wheel diameter variation. This study suggests that, if the wheel perimeter is nearly an integral multiple of the wavelength of severe periodic wear along the wheel circumference, the polygonal wear on the wheel can develop quickly. Furthermore, the wavelength of the periodic wear of the wheel relies on the operation speed of the train and wheelset resonant frequency. Therefore, the initiation and development of polygonal wear on wheels depends on the operation speed, wheel diameter, and the resonant frequencies of the wheelset. This conclusion can be applied to research concerning measures associated with the suppression of polygonal wear development.

Keywords: operating speed; polygonal wear; resonance; train–track rigid–flexible coupling dynamics model; wheel diameter

1. Introduction Polygonal wear is uneven wear along the circumference of train wheels and its wavelength varies from a dozen centimetres to the entire circumference of the wheel [1,2]. Polygonal wear with only one wavelength around the wheel circumference is called first-order polygonal wear (i.e., eccentric wear); that with two equal wavelengths is called second-order polygonal wear (i.e., elliptical wear); that with three equal wavelengths is called third-order polygonal wear, and so on. First-, second-, and third-order polygonal wear are considered low-order polygonal wear. In recent years, polygonal wear beyond the eighth order—defined as a high-order polygonal wear in the present research have often been detected on the wheels of high-speed trains, metro trains, and electric locomotives in China [2]. Polygonal wear increases wheel–rail interaction and has a considerable effect on the use life of the components of railway vehicles and tracks. Furthermore, it threatens operation safety [3]. Therefore, this problem has gradually become a matter of urgent public concern in the railway transportation industry in China, and globally, and thus, it has attracted the attention of many scholars over the last few years.

Appl. Sci. 2020, 10, 5880; doi:10.3390/app10175880 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 5880 2 of 23

Kaper identified polygonal wear phenomena on train wheels when he conducted research on the noise of intercity trains and studied the relationship between polygonal wear and noise in the frequency domain [4]. Nielsen and Jin reviewed the literature on wheel polygonal wear in 2000 and 2018, respectively, and in their review, they summarised numerical simulation models of polygonal wear and explanations on its initiation mechanism and development [2,3]. Morys developed a dynamics model of an ICE-1 high-speed Garman train to investigate the polygonal wear mechanism. In the model, the wheelset mass modelling included five parts (two wheels and three brake discs), and they were all regarded as rigid bodies [5–7]. Three-dimensional rotational spring-damper units were used as connections to characterize the bending and torsional properties of the wheelset. The results showed that the bending resonant modes of the wheelset were excited by the fluctuating vertical force of the wheel–rail, and this led to a scenario where the lateral creep force of the wheel–rail varied periodically and the polygonal wear occurred on the rolling circle of the wheel. Based on the Morys’s simulation model, Meinke considered beam elements as connections instead of three-dimensional rotational spring dampers [8]. Then, the effect of rotational inertia, gravity, and the gyroscopic moment of the wheelset on the initiation of polygonal wear was considered. In addition, imbalance calculations and long-term wear simulation were performed. Meywerk established a coupling dynamics model that included an elastic wheelset and two elastic rails to study the formation of polygonal wear [9]. The results showed that flexible deformations of the wheelset and rails, caused by their interaction, resulted in the initiation and development of polygonal wear. The first and second bending modes of the wheelset played an important role in initiating polygonal wear. This literature focused on the characteristics and mechanisms of low-order polygonal wear. Chen proposed a finite element model of a wheelset’s rolling contact with two rails using commercial finite element software to conduct research to identify the root cause of polygonal wear [10]. Chen suggests that the self-excited vibration of the wheel–rail system leads to polygonal wear at a frequency of 150 Hz. A suitable sleeper support stiffness or wheel–rail friction coefficient can be controlled to limit the development of polygonal wear. Tao et al. conducted extensive measurements of polygonal wear on electric locomotive wheels with radii of 1050–1200 mm at the average operation speed of 80 km/h [11,12]. The measurement results showed that the dominant orders of the polygonal wear of locomotive wheels were 18th, 19th, and 24th, and depended on wheel diameter. The initiation mechanism of the polygonal wear of the locomotive wheels is not captured in this paper. Jin et al. studied the mechanism of the ninth-order polygonal wear in wheels of metro trains with a linear induction motor using a detailed site test and observation [13]. They found that the resonant frequency of the first wheelset bending matched the exciting frequency of the ninth-order polygonal wear. When the first wheelset bending mode was excited, the period of lateral creepage and force between wheel and rail occurred. Furthermore, the ninth-order polygonal wear gradually appeared. Wu et al. studied the mechanism of polygonal wear of high-speed train wheels through extensive field tests [14]. The operation speed of the tested train was 250 km/h, and the diameter of the wheels was 860 mm. The results showed that the mechanism of wheel polygonal wear of high-speed trains could be attributed to the excited resonance of the train–track coupling system in operation and changing the operating speed could suppress the development of polygonal wear effectively. In [13,14], the authors conducted many experiments to investigate the mechanism of high-order polygonal wear, but both studies lacked in-depth theoretical research and numerical simulation. Wu et al. studied the root cause of high-order polygonal wear [15]. Their study concluded that there was a ~650 Hz bending mode of the rail segment between the front and rear wheelsets of the same bogie. This bending mode was identified as the primary contributor to polygonal wear. However, in the literature [15], the exciting frequency of the polygonal wear is 550–600 Hz, which does not match the bending mode frequency very well. The conclusions of this paper lack experimental verification. Appl. Sci. 2020, 10, 5880 3 of 23

Cai et al. presented a detailed investigation into the mechanism of metro wheel polygonal wear using on-site experiment numerical simulations [16]. The measurement results showed that more than 70% of the metro wheels had sixth- to eighth-order polygonal wear. In this paper, P2 resonance was suggested as the main contributor to polygonal wear, and the flexibility of the wheelset was also found to influence the formation of polygonal wear. However, this is not enough to explain the higher order polygonal wear, which occurred in high-speed conditions. The literature discussed above provides a certain understanding of wheel polygonal wear problems. However, thus far, consistent understanding on the mechanism of polygonal wear is lacking. To understand the mechanisms of the initiation and development of high-order polygonal wear of railway wheels clearly, this paper conducts a detailed investigation into the high-order polygonal wear of the high-speed trains in China via extensive experimental analysis and numerical simulation. The remainder of this manuscript is organised as follows: Section2 analyses the data measured from more than 1000 sets of polygonal wheels in the field and provides information on the important relationship between polygonal wear order and wheel diameter size. Section3 presents a simulation model to predict the initiation and development of polygonal wear. In Sections4 and5, the model is used to verify the key conditions of initiation and the development of high-order polygonal wear and to calculate the relationship between polygonal wear and high-speed development, train operation speed, and wheel diameter change.

2. Characteristics of Polygonal Wear of High-Speed Wheels Polygonal wear occurs on the wheels of many high-speed trains. The census test of polygonal wear was conducted over a long time in this paper. The tested polygonal wheels comprised about 48,000 sets and the test mileage was >200,000 km after re-profiling. Figure1a shows the Müller–BBM wheel roughness test instrument which was used for measuring polygonal wear, and Figure1b shows a typical polygonal CL60 steel wheelset, which was provided by CSR Tangshan Rolling Stock Co., Ltd. During the test, the measured wheelsets were jacked up using hydraulic equipment so that the wheel could rotate around its axis line freely. The displacement sensor of the roughness test instrument contacts the wheel tread to measure the radial change in the wheel. Figure2 shows the proportion of the polygonal wheel with a diameter change. The proportion of the polygonal wheel represents the quotient, which is the number of polygonal wheels divided by the number of all measured wheels with diameters of 830–840 mm, 840–850 mm, and 910–920 mm. A diameter of 920 mm is the nominal diameterAppl. Sci. 2020 for, 10 new, x FOR high-speed PEER REVIEW wheels. 4 of 22

(a) (b)

Figure 1.1. (a) Test instrument;instrument; ((bb)) PolygonalPolygonal wheel.wheel.

18th order 30 27.30% 25.36% 25 19th order 20 16.10% 15 14.77% 20th order 11.79% 12.80% 10

5 3.74% 1.26% 1.03% 0 40 50 60 70 80 90 00 10 20 8 8 8 9 0-8 0-8 0-9 0-9 830- 840- 85 86 870-8 880- 890- 90 91 Proportion of polygonal wear wheels/% wear polygonal of Proportion Wheel diameter/mm

Figure 2. Statistical results of polygonal wheels.

60 Threshold line of high-speed wheel roughness 50 Ampitude spectrum of polygonal wear Ampitude spectrum of normal wear m

μ 40 30 20 10

Roughness/dB re re 1 Roughness/dB 0 -10 -20 0 4 8 12 16 20 24 28 32 Order Figure 3. Sketch map: Amplitude spectrum of polygonal wear (site measurement) and threshold line of high-speed wheel roughness (referring to the ISO-3095). Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 22

(a) (b) Appl. Sci. 2020, 10, 5880 4 of 23 Appl. Sci. 2020, 10, x FOR PEER REVIEWFigure 1. (a) Test instrument; (b) Polygonal wheel. 4 of 22

18th order 30 27.30% 25.36% 25 19th order 20 16.10% 15 14.77% 20th order 11.79% 12.80% 10

5 3.74% 1.26% 1.03% 0 40 50 60 70 80 90 00 10 20 8 8 8 9 0-8 0-8 0-9 0-9 830- 840- 85 86 870-8 880- 890- 90 91 Proportion of polygonal wear wheels/% wear polygonal of Proportion Wheel diameter/mm Figure(a) 2. Statistical results of polygonal wheels. (b) Figure 2. Statistical results of polygonal wheels. In the analysis, shownFigure in Figure 1. (a2), Test the thresholdinstrument; of (b “high-speed”) Polygonal wheel. rail roughness (ISO-3095 Third edition 1 August 2013)60 was used as the threshold for high-speed wheels because there are no official Threshold line of high-speed wheel roughness partition criteria in China18 railwaysth order so far. Temporary (suggested) partition criteria for polygonal wear 30 50 Ampitude spectrum of polygonal wear and non-polygonal27.30% wheels are designed Ampitude by rolling spectrum stock of companies normal wear and railway transport departments m 25.36% in China25 based on theirμ 40 requirements. The temporary partition criteria consider the rail roughness 19th order threshold20 line specified in ISO-3095 for the maintenance of high-speed wheels with severe polygonal wear. If the depths (peak30 to valley)16.10% of the polygonal wear of some dominant orders exceed the 15 14.77% 20th order high-speed wheel roughness threshold line,11.79% the occurrence12.80% of the polygonal wear phenomenon is confirmed.10 As shown20 by the black line in Figure3, the peak of 18th-order exceeds the roughness threshold5 line; thus, the10 wheel is regarded as a polygonal wheel. For the blue line in Figure3.74%3, there is no peak exceeding the roughness threshold; thus, the wheel is regarded1.26% as1.03% a normal wheel. It is 0 0 noted that ISO-309540 shows re 1 Roughness/dB 50 that the60 threshold70 line of rail8 roughness90 includes00 the wavelengths10 20 of 0.3 cm 8 80 8 9 0-8 0-8 0-9 0-9 to 40 cm only.830- The polygonal840- 85 wear of high-speed86 870-8 wheels880- covers the890- wavelengths90 of91 12 cm to the wholewheels/% wear polygonal of Proportion wheel circumference.-10 However, the wavelengths of the polygonal wear, which are much more Wheel diameter/mm concerned with the point view of controlling dynamic behaviour and noise, are in the range determined -20 by the ISO-3095. 0Figure 4 2. Statistical 8 12 results 16 of polygonal 20 wheels. 24 28 32 Order 60 Figure 3. Sketch map: Amplitude spectrum Threshold of linepolygonal of high-speed wear (sitewheel measurement) roughness and threshold line of high-speed wheel50 roughness (referring Ampitude to the spectrum ISO-3095). of polygonal wear Ampitude spectrum of normal wear m

μ 40 30 20 10

Roughness/dB re re 1 Roughness/dB 0 -10 -20 0 4 8 12 16 20 24 28 32 Order Figure 3. SketchSketch map: map: Amplitude Amplitude spectrum spectrum of of polygonal polygonal wear wear (site (site measurement) measurement) and threshold line of high-speed wheel roughness (referring to the ISO-3095). Appl. Sci. 2020, 10, 5880 5 of 23

Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 22 Through the statistical analysis of the test data, the characteristics of polygonal wear can be concludedThrough as the follows: statistical analysis of the test data, the characteristics of polygonal wear can be concluded as follows: 1. Wheels with different diameters have different proportions of polygonal wheels and different 1. Wheelsdominant with orders different (main diameters wavelength) have of different polygonal proportions wear (Figure of 2polygonal). Furthermore, wheels it seemsand different that the dominantsmaller the orders diameter, (main the wavelength) higher the of proportion polygonal of wear the polygonal(Figure 2). wheels. Furthermore, This depends it seems on that the thedepth smaller of the the decarburised diameter, the layer higher on the the surface proporti ofon the of wheel. the polygonal wheels. This depends on 2. theThe depth proportion of the decarburised of polygonal wheelslayer on exhibits the surface a non-monotonic of the wheel. variation as a function of the wheel 2. Thediameter, proportion including of polygonal three local wheels maxima exhibits around a non-monotonic 830–840 mm, 870–880variation mm, as a and function 910–920 of mm,the wheelcorresponding diameter, toincluding three periods three oflocal high-rate maxima development. around 830–840 mm, 870–880 mm, and 910–920 mm, corresponding to three periods of high-rate development. Figure4 shows the roughness spectra of polygonal wear, measured from di fferent cars for a typicalFigure high-speed 4 shows train. the roughness The subgraphs spectra show of thatpolygo diffnalerent wear, cars havemeasured different from polygonal different wear cars orders.for a typicalThe polygonal high-speed wear train. of the The 20th subgraphs order occurred show that on Cardifferent 1; 19th cars order have occurred different on polygonal Cars 2, 4, andwear 7; orders.18th order The occurredpolygonal on wear Cars of 3, the 6, and20th 8. order occurred on Car 1; 19th order occurred on Cars 2, 4, and 7; 18th order occurred on Cars 3, 6, and 8.

40 th 40 all wheelsets 18 order Car 1 D=920mm 30 th 30 19 order 20 th 20 20 order th 10 10 20 order m) m)

μ 0 0 -10 -10 -20 -20 -30 -30 -40 -40 2 4 6 8 1012141618202224262830 2 4 6 8 1012141618202224262830 40 40 Car 2 D=878mm Car 3 D=837mm th 30 Car 4 D=876mm 30 Car 6 D=841mm 18 order th 20 Car 7 D=877mm 19 order 20 Car 8 D=841mm 10 10 Irregularity amplitude (dB re 1 (dB amplitude Irregularity 0 0 -10 -10 -20 -20 -30 -30 -40 -40 2 4 6 8 1012141618202224262830 2 4 6 8 1012141618202224262830 Order

Figure 4. Results of the measured polygonal wear. Figure 4. Results of the measured polygonal wear. Figure5 denotes the diameter change in the wheels corresponding to the polygonal wear of differentFigure orders 5 denotes that occurred the diameter on this typicalchange train.in the The wh wheelseels corresponding of the tested train to the had polygonal different diameters, wear of differentand this orders was not that done occurred intentionally. on this Thistypical high-speed train. The train wheels was of used the as tested a test train train had for long-termdifferent diameters,follow-up and investigations this was not into done the intentionally. dynamic behaviours This high-speed and wear train states was of used wheels as /arails. test train The diametersfor long- termof the follow-up wheels of investigations the train were into diff erentthe dynamic because ofbehaviours the wear and re-profilingwear states causedof wheels/rails. by an increase The diametersin operational of the mileage. wheels of The the polygonal train were weardifferent and because fatigue statesof the ofwear the and wheels re-profiling determine caused their by metal an increasecutting quantityin operational during mileage. re-profiling. The polygonal Therefore, wear the cutting and fatigue quantities states of of the the wheels wheels in re-profilingdetermine their were metaldifferent. cutting Figure quantity5 shows during that the re-profiling. wheels with Therefore, diameters ofthe ~920 cutting mm generatedquantities a of 20th-order the wheels polygonal in re- profilingwear; those were with different. an 875 Figure mm diameter 5 shows produced that the wheels 19th-order with diameters wear; those of with ~920 830–840mm generated mm diameters a 20th- order polygonal wear; those with an 875 mm diameter produced 19th-order wear; those with 830– 840 mm diameters formed 18th-order polygonal wear. This clearly indicates that a decrease in the diameter leads to a decrease in the polygonal wear order. Furthermore, the measured results of the Appl. Sci. 2020, 10, 5880 6 of 23 formed 18th-order polygonal wear. This clearly indicates that a decrease in the diameter leads to a decreaseAppl. Sci. 2020 in the, 10, polygonalx FOR PEER wearREVIEW order. Furthermore, the measured results of the wheels of car 5 do6 of not 22 indicate polygonal wear in Figures4 and5 because the re-profiling of its wheels was conducted just beforewheels polygonal of car 5 do wear not indicate measurement. polygonal wear in Figures 4 and 5 because the re-profiling of its wheels was conducted just before polygonal wear measurement.

FigureFigure 5.5. PolygonalPolygonal wearwear ofof didifferentfferent ordersorders correspondingcorrespondingto todi differentfferent wheel wheel diameters. diameters. The exciting frequency of the polygonal wear is calculated as: The exciting frequency of the polygonal wear is calculated as: v f = v n (1) f =×πD × n π (1) D where f denotes the exciting frequency of the polygonal wear, v denotes the operating speed of the vehicle,where fD denotesdenotes the the exciting diameter frequency of the wheel, of the and polygonaln represents wear, the v polygonaldenotes the wear operating order. Equationspeed of the (1) indicatesvehicle, D that denotes wheels the with diameter different of the diameters wheel, and have n represents differentpolygonal the polygonal wear wear orders order. atthe Equation same exciting(1) indicates frequency. that wheels For thewith present different test diameters case, f is have ~570–580 different Hz. polygonal This means wear that orders the wheelsat the same are excitedexciting in frequency. the frequency For the range present of 570–580 test case, Hz, f is and ~570–580 could produceHz. This themeans 18th that to 20ththe wheels order polygonalare excited wear,in the corresponding frequency range to di offf erent570–580 wheel Hz, diameters. and could Therefore, produce the it is 18th necessary to 20th to order clearly polygonal find where wear, the 570–580corresponding Hz vibrations to different are coming wheel diameters. from (the vehicle Therefore, or the it trackis necessary or their to coupling clearly find resonance). where the That 570– is the580 keyHz tovibrations explaining are thecoming mechanism from (the of vehicle initiation or andthe track development or their coupling of high-order resonance). polygonal That wear,is the andkey thisto explaining will be further the mechanism discussed later. of initiation and development of high-order polygonal wear, and this Thewill testbe furthe resultsr discussed of the same later. train are shown in Figure6. The first row in Figure6 indicates the wheelThe diameter test results reduction of the from same 920 train to 830 are mm; shown the in second Figure row 6. denotesThe first the row polygonal in Figure wear 6 indicates described the inwheel the polar diameter coordinate reduction system from corresponding 920 to 830 mm; to the di secondfferent wheelrow denotes diameters; the polygonal the third rowwear illustrates described thein the order polar of thecoordinate polygonal system wear, corresponding which is also to called different the roughnesswheel diameters; spectra the of thethird polygonal row illustrates wear. Inthe the order test, of the the train polygonal operation wear, speed which was is 300 also km called/h. The the interval roughness of wheel spectra re-profiling of the polygonal was 200,000 wear. km. In Figurethe test,6 shows the train the operation states of polygonal speed was wear 300 km/h. at six diThffeerent interval wheel of wheel diameters—920, re-profiling 905, was 875, 200,000 860, 840,km. andFigure 830 6 mm. shows The the nominal states of diameter polygonal of new wear wheels at six isdifferent 920 mm, wheel and 830 diameters—920, mm is the diameter 905, 875, close 860, to 840, the endand of830 the mm. wheel The use nominal life. During diameter service, of new the wheels wheel diameteris 920 mm, changes and 830 from mm large is the to diameter small because close ofto wearthe end and of re-profiling. the wheel use Indeed, life. During at wheel service, diameters the wheel of ~920, diameter 875, and changes 830–840 from mm, large a polygonal to small wearbecause of theof wear 20th, and 19th, re-profiling. and 18th orders Indeed, occur, at wheel respectively. diameters According of ~920, to875, Equation and 830–840 (1), the mm, passing a polygonal frequencies wear ofof thethe polygonal 20th, 19th, wear and of three18th diordersfferent occur, orders respectively. or the resultant According exciting frequenciesto Equation between (1), the the passing wheel andfrequencies rail are ~580of the Hz polygonal at 300 km /wearh. Figure of three6 further differ indicatesent orders that or polygonal the resultant wear exciting develops frequencies at a high ratebetween at the the three wheel stages, and at rail which are ~580 the wheel Hz at diameters300 km/h. decreasedFigure 6 further in their indicates whole use that life, polygonal and they wear are, develops at a high rate at the three stages, at which the wheel diameters decreased in their whole use life, and they are, respectively, the initial stage of new wheel use, the wheel diameter near 875 mm, and the stage when the diameter is between 840 and 830 mm, in the present investigation case. Further, this phenomenon can be illustrated using Figure 5. For a wheel diameter around 905 and 860 mm, irregular wear with periodic character or equal-wavelength, or polygonal wear, is not serious. Appl. Sci. 2020, 10, 5880 7 of 23 respectively, the initial stage of new wheel use, the wheel diameter near 875 mm, and the stage when the diameter is between 840 and 830 mm, in the present investigation case. Further, this phenomenon can be illustrated using Figure5. For a wheel diameter around 905 and 860 mm, irregular wear with periodicAppl. Sci. 2020 character, 10, x FOR or PEER equal-wavelength, REVIEW or polygonal wear, is not serious. 7 of 22

Figure 6. Decrease in order with a de decreasecrease in wheel diameter.

From thethe statistical statistical results results of theof trackingthe tracking test intest the in field, the the field, most the prominent most prominent feature of feature polygonal of wearpolygonal is that wear there is nearlythat there a one-to-one is nearly relationship a one-to-one between relationship the wave between number the (or dominantwave number order) (or of thedominant polygonal order) wear of andthe wheelpolygonal diameter. wear Forand di wheelfferent diameter. wheel diameters, For different the wavelengths wheel diameters, (or passing the frequencies)wavelengths of(or polygonal passing frequencies) wear are considerably of polygonal near wear at are a constant considerably operation near speed,at a constant which operation suggests thatspeed, the which vehicle suggests system that in operationthe vehicle generates system in a op vibrationeration generates at nearly a the vibrat sameion resonant at nearly frequency, the same therebyresonant causing frequency, polygonal thereby wear causing on the polygonal wheels. The we resonancear on the wheels. of the vehicle The resonance leads to polygonal of the vehicle wear. Theleads smaller to polygonal the diameter, wear. The the smaller the wavediameter, number. the smaller the wave number.

3. Simulation Model 3.1. Rigid–Flexible Coupling Dynamical Model of Vehicle–Track 3.1. Rigid–Flexible Coupling Dynamical Model of Vehicle–Track Figure7 shows the rigid–flexible coupling dynamical model of vehicle /track [17–23]. The vehicle Figure 7 shows the rigid–flexible coupling dynamical model of vehicle/track [17–23]. The vehicle system model includes a car body, two bogies, and four wheelsets. The car body and frames of the two system model includes a car body, two bogies, and four wheelsets. The car body and frames of the bogies are modelled as rigid bodies, and they each have five degree of freedoms. Wheelset structural two bogies are modelled as rigid bodies, and they each have five degree of freedoms. Wheelset flexibility is modelled using the finite element method and solved using the mode superposition method. structural flexibility is modelled using the finite element method and solved using the mode The primary and secondary suspension systems of the vehicle are modelled with three-dimensional superposition method. The primary and secondary suspension systems of the vehicle are modelled spring-damper elements. The track system is modelled as a monolithic track-bed system that with three-dimensional spring-damper elements. The track system is modelled as a monolithic track- includes rails, fasteners, slabs, and roadbed. The rails are modelled as Timoshenko beams with bed system that includes rails, fasteners, slabs, and roadbed. The rails are modelled as Timoshenko elastic discrete point support. Slab structural flexibility is considered using the finite element beams with elastic discrete point support. Slab structural flexibility is considered using the finite method and solved by using the mode superposition method [19]. The fastener is modelled using element method and solved by using the mode superposition method [19]. The fastener is modelled three-dimensional spring-damper elements. The roadbed support layer is modelled with equally using three-dimensional spring-damper elements. The roadbed support layer is modelled with distributed spring-damper elements. The parameters of the track and vehicle system are listed in equally distributed spring-damper elements. The parameters of the track and vehicle system are Table1. listed in Table 1.

Table 1. Parameter values of the vehicle and track subsystem.

Vehicle System Track System Parameter Value Parameter Value Car body mass (kg) 44,039 Young’s modulus of rail (N/m2) 2.059 × 1011 Bogie mass (kg) 2439 Shear modulus of rail (N/m2) 7.9 × 1010 Wheelset mass (kg) 1881 Density of rail (kg/m3) 7860 Density of wheelset (kg/m3) 7860 Cross-sectional area (m2) 7.745 × 10−3 Primary suspension vertical 9 Bending moment about y-axis (m4) 3.217 × 10−5 stiffness (MN/m) Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 22

Primary suspension vertical 10.5 Bending moment about z-axis (m4) 5.26 × 10−6 damping (kN·s/m) Primary suspension lateral 5.45 Polar moment of inertia (m4) 3.741 × 10−5 stiffness (MN/m) Primary suspension lateral 5.0 Lateral shear coefficient 0.4570 damping (kN·s/m) Secondary suspension vertical 0.24 Vertical shear coefficient 0.5329 stiffness (MN/m) Secondary suspension vertical 25 Density of slab (kg/m3) 2800 damping (kN·s/m) Secondary suspension lateral 0.13 Young’s modulus of slab (N/m2) 3.6 × 1010 stiffness (MN/m) Secondary suspension lateral 15 Rail pad vertical stiffness (MN/m) 40 damping (kN·s/m) Distance between two bogie Rail pad vertical damping 8.9 30 frames (m) (kN·s/m) Appl. Sci. 2020, 10, 5880 8 of 23 Wheelbase (m) 2.5 Sleeper spacing (m) 0.63

Φc Mc Car-body Icx FEM Wheelset

Zc

Ksy

Csy Csz Krx Φt Ksz Bogie frame Timoshenko beam

Cpz Kpz Kpy

Cpy

y Kph ΦrL ΦrR Cph ZrL Rail ZrR FEM Slab z Kpv Cpv

Slab

Ksh Csh Csv Ksv

Figure 7. Vehicle–track rigid–flexible coupling dynamical model. Figure 7. Vehicle–track rigid–flexible coupling dynamical model. Table 1. Parameter values of the vehicle and track subsystem. According to the solid finite element theory, the differential equation of vibration of wheelset is Vehicle System Track System written as Parameter Value Parameter Value Car body massM (kg) u(t )+ K 44,039 u ( t ) = QYoung’s ( t ) modulus ( i = of 1,2,3,4 rail (N/m2) ) 2.059 1011 (2) i i i i i × Bogie mass (kg) 2439 Shear modulus of rail (N/m2) 7.9 1010 where ü i(t) is the acceleration vector of the nodes on the ith wheelset at time t, ui(t) is× the velocity Wheelset mass (kg) 1881 Density of rail (kg/m3) 7860 vector of the nodes on the ith wheelset at time t, Mi is the mass matrix of the ith wheelset, Ki is the 3 2 3 Density of wheelset (kg/m ) 7860( ) Cross-sectional area (m ) 7.745 10− stiffness matrix of the ith wheelset, and Qi t is the force vector of the nodes on the ith× wheelset at Primary suspension vertical stiffness time t, and it contains wheel–rail forces and9 primaryBending suspension moment aboutforces.y-axis (m4) 3.217 10 5 (MN/m) × − The modal superposition method is used to decouple Primary suspension vertical damping 4 6 10.5 Bending moment about z-axis (m ) 5.26 10− (kN s/m) 2 T × · qik( t )+ k q ik ( t ) = ik Q i ( t ) ( i = 1,2,3,4 ) (3) Primary suspension lateral stiffness 5.45 Polar moment of inertia (m4) 3.741 10 5 (MN/m) × − where ik is the vector of the kth order modal displacement on the ith wheelset and 푞ik(t) is the kth Primary suspension lateral damping 5.0 Lateral shear coefficient 0.4570 order modal coordinates(kN s/m) on the ith wheelset at time t. · Secondary suspension vertical 0.24 Vertical shear coefficient 0.5329 stiffness (MN/m) Secondary suspension vertical 25 Density of slab (kg/m3) 2800 damping (kN s/m) · Secondary suspension lateral stiffness 0.13 Young’s modulus of slab (N/m2) 3.6 1010 (MN/m) × Secondary suspension lateral 15 Rail pad vertical stiffness (MN/m) 40 damping (kN s/m) · Distance between two bogie frames 8.9 Rail pad vertical damping (kN s/m) 30 (m) · Wheelbase (m) 2.5 Sleeper spacing (m) 0.63 Appl. Sci. 2020, 10, 5880 9 of 23

According to the solid finite element theory, the differential equation of vibration of wheelset is written as n .. o   M u (t) + K u (t) = Q (t) (i = 1, 2, 3, 4 ) (2) i i i i i ··· .. where ui(t) is the acceleration vector of the nodes on the ith wheelset at time t, ui(t) is the velocity vector of the nodes on the ith wheelset at time t, Mi is the mass matrix of the ith wheelset, Ki is the stiffness matrix of the ith wheelset, and Qi(t) is the force vector of the nodes on the ith wheelset at time t, and it contains wheel–rail forces and primary suspension forces. The modal superposition method is used to decouple

.. n oT q (t) + ω 2q (t) = φ Q (t) (i = 1, 2, 3, 4 ) (3) ik k ik ik i ··· where φik is the vector of the kth order modal displacement on the ith wheelset and qik(t) is the kth order modal coordinates on the ith wheelset at time t. A dynamic explicit integral algorithm is used to solve this equation [17]. Finally, the displacement of the nodes in natural coordinates can be obtained. The finite element method was used to analyse the mode characteristics of the wheelset. The Hypermesh was used to mesh the geometric model of the wheelset, which was provided by the supplier of the wheelset. The element type was Solid45 and the element count was 255,360; the node count was 267,771. Then, the shape testing of all elements was conducted to ensure that a sufficiently fine mesh was used. The summary of the shape testing and the proofs of result independency to mesh size are placed in the AppendixA at the end of the manuscript (Tables A1 and A2; Figure A1). The Block Lanczos method in ANSYS was used for modal analysis. Figure8 shows the bending modes of the wheelset within a range of 800 Hz. The deformation of the wheelset includes mode deformation coupling in the two directions. In the vertical and longitudinal directions, there are same bending modes that have the same frequency and mode shape. Here, the mode at 576 Hz merits attention, because the frequency is very near to the exciting frequency of the polygonal wear, according to Equation (1).

3.2. Method for Online Searching Wheel–Rail Contact Points The bending deformation of the wheelset indicates that the wheel rim has hardly any deformation compared to the wheel tread when the wheelset generates bending deformation within the range of 800 Hz. This feature can be used to help solve the interaction between the geometrical position of the wheel–rail. “The method for online searching wheel-rail contact points” is used to determine the contact geometry relation between a flexible wheelset and a pair of rails [21]. The process is shown in Figure9. The motion of the flexible wheelset includes a rigid motion and flexible deformation (Figure9a,b). Then, a dummy wheelset is used to simulate the flexible wheelset (Figure9c,d). The centre of the dummy wheelset’s coordinates needs to be identified, and the contact geometry relation between the dummy wheelset and the two rails can then be solved by using the trace method. Therefore, solving the problem of the flexible wheelset in rolling contact with the two rails can be transformed effectively into solving the motion of the rigid wheelset. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 22

A dynamic explicit integral algorithm is used to solve this equation [17]. Finally, the displacement of the nodes in natural coordinates can be obtained. The finite element method was used to analyse the mode characteristics of the wheelset. The Hypermesh was used to mesh the geometric model of the wheelset, which was provided by the supplier of the wheelset. The element type was Solid45 and the element count was 255,360; the node count was 267,771. Then, the shape testing of all elements was conducted to ensure that a sufficiently fine mesh was used. The summary of the shape testing and the proofs of result independency to mesh size are placed in the Appendix A at the end of the manuscript (Tables A1 and A2; Figure A1). The Block Lanczos method in ANSYS was used for modal analysis. Figure 8 shows the bending modes of the wheelset within a range of 800 Hz. The deformation of the wheelset includes mode deformation coupling in the two directions. In the vertical and longitudinal directions, there are same bending modes that have the same frequency and mode shape. Here, the mode at 576 Hz merits attention, becauseAppl. Sci.the2020 frequency, 10, 5880 is very near to the exciting frequency of the polygonal wear, according10 of 23 to Equation (1).

FigureFigure 8. Bending 8. Bending modes modes of ofwheelset wheelset (0–800 (0–800 Hz): Hz): ( (a) First bendingbending mode mode at at 87 87 Hz; Hz; (b) ( Secondb) Second bending bending mode at 143 Hz; (c) Third bending mode at 284 Hz; (d) Fourth bending mode at 576 Hz. mode at 143 Hz; (c) Third bending mode at 284 Hz; (d) Fourth bending mode at 576 Hz.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 22

3.2. Method for Online Searching Wheel–Rail Contact Points The bending deformation of the wheelset indicates that the wheel rim has hardly any deformation compared to the wheel tread when the wheelset generates bending deformation within the range of 800 Hz. This feature can be used to help solve the interaction between the geometrical position of the wheel–rail. “The method for online searching wheel-rail contact points” is used to determine the contact geometry relation between a flexible wheelset and a pair of rails [21]. The process is shown in Figure 9. The motion of the flexible wheelset includes a rigid motion and flexible deformation (Figure 9a,b). Then, a dummy wheelset is used to simulate the flexible wheelset (Figure 9c,d)The centre of the dummy wheelset’s coordinates needs to be identified, and the contact geometry relation between the dummy wheelset and the two rails can then be solved by using the trace method. Appl. Sci. 2020, 10, 5880 11 of 23 Therefore, solving the problem of the flexible wheelset in rolling contact with the two rails can be transformed effectively into solving the motion of the rigid wheelset.

FigureFigure 9. ModellingModelling flexible flexible wheelset: wheelset: (a) Static (a) Static wheelset; wheelset; (b) The (b) wheelset The wheelset after rigid after motion; rigid motion; (c) The (wheelsetc) The wheelset after flexibleafter flexibledeformation; deformation; (d) dummy (d) dummy wheelsets; wheelsets; (e) the contact (e) the geometry contact geometry relation between relation betweenthe dummy the wheelset dummy wheelset and the two and rails. the two rails.

TheThe mostmost importantimportant partpart ofof solvingsolving thethe flexibleflexible wheelsetwheelset motionmotion isis determiningdetermining thethe centrecentre ofof thethe dummydummy wheelset’swheelset’s coordinates.coordinates. From Figure9 9e,e, the the displacement displacement vector vector of of thethe centre centre of of the the dummy dummy wheelsetwheelset isis writtenwritten as:as: −−−−−−−→O1ODR1 = −−−−−−−→O 1ADR1 + −−−−−−−−−→ADR1ODR 1 (4) OO=+ OA A O (4) 11DRDRDRDR 11 11 where −−−−−−−→O A denotes the displacement vector of the nominal contact point in the natural system of 1DR1⃗ where O1ADR1 denotes the displacement vector of the nominal contact point in the natural system of coordinates after the wheelset generates rigid motion and flexible deformation. −−−−−−−−−→A Odenotes⃗ the coordinates after the wheelset generates rigid motion and flexible deformation.DR 1ADR1DRO1 DR1 denotes displacementthe displacement vector vector from from the centrethe centre of the of dummythe dummy wheelset wheelset to the to nominalthe nominal contact contact point. point.

 TT    0cossin00  i i   cosψψψr sin ψr 0       rr −−−−−−−→OA =  l    +    1 DR1 =−0   j + −−→uA  − cosφψφr sin ψr cos φφr cos ψψr sin φ φr  (5) OA11DRrrrrr − l 0  ju A  cos− sin cos cos sin  (5) R0 k  sin φr sin ψr sin φr cos ψr cos φr  −−  φψ−− φ ψ φ  R0 k sinrr sin sin r cos r cos r   where l0 is the horizontal distance between the centre of the dummy wheelset and the nominal rolling where l is the horizontal distance between the centre of the dummy wheelset and the nominal rolling circle, R00 is the radius of the nominal rolling circle, −−→uA is the displacement vector of the contact point aftercircle, the R0 wheelset is the radius produces of the flexiblenominal deformation, rolling circle,and uA⃗ it is canthe displacement be solved using vector the of method the contact described point in Section 3.1. φr denotes the angle of roll after the wheelset generates rigid motion and ψr indicates the angle of yaw. The two parameters can be solved using a dynamic explicit integral algorithm. In addition, the transformation matrix between the natural system of coordinates and the wheelset system of coordinates is built using φr and ψr.

−−−−−−−−−→ADR1ODR1 is written as

 T    0   1 0 0  i       −−−−−−−−−→A O =  l     DR1 DR1  0   0 cos φDR1 sin φDR1  j  (6)  −     R 0 sin φ cos φ r k − 0 − DR1 DR1 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 22 after the wheelset produces flexible deformation, and it can be solved using the method described in Section 3.1. φr denotes the angle of roll after the wheelset generates rigid motion and ψr indicates the angle of yaw. The two parameters can be solved using a dynamic explicit integral algorithm. In addition, the transformation matrix between the natural system of coordinates and the wheelset system of coordinates is built using φr and ψr. ⃗ ADR1ODR1 is written as

T Appl. Sci. 2020, 10, 5880 010 0i 12 of 23   AO=− l 0cosφφ sin j (6) DR11 DR  0 DR1DR 1 − − φφ where φDR1 indicates the angle includedR between0 0sincos the axisDR of11 the wheelsetDR r  andk the horizontal line after the wheelset generates rigid motion and flexible deformation. This is written as where φDR1 indicates the angle included between the axis of the wheelset and the horizontal line after the wheelset generates rigid motion and flexible deformation. This is written as φDR1 = φr + φ f (7) φφφ=+ DR1 r f (7) where φf denotes the angle included between the axis of the wheelset and the horizontal line after the wherewheelset φf denotes only produces the angle flexible included deformation. between Figurethe axis 10 of illustrates the wheelset the solving and the method. horizontal First, line two after points the wheelset(B1 and B 2only) at bothproduces sides flexible of the wheel deformation. are selected, Figure and 10 the illustrates connection the linesolving between method. the twoFirst, points two pointsis drawn, (B1 and thisB2) at line both is parallel sides of to the the wheel axis of are the selected, wheelset. and Second, the connection the wheelset line generatesbetween the flexible two pointsdeformation, is drawn, the and two this points line move is parallel to the to two the new axis points of theB wheelset.01 and B02 Second,. Then, thethe positionswheelset ofgeneratesB01 and flexibleB02 are solveddeformation, using the the method two points described move into Sectionthe two 3.1 new. Finally, pointsφ Bf 01can and be B solved.02. Then, the positions of B01 and B02 are solved using f

Figure 10. Solving method of φ f . Figure 10. Solving method of f . In the plane perpendicular to the rails, the displacement vector of the centre of the dummy wheelsetIn the can plane be written perpendicular as: to the rails, the displacement vector of the centre of the dummy wheelset can be written as: −−−−−−−−−→= O1ODROO1 =  11TDR     T    0   i   cos ψ sin ψ 0   0   1 0 0  i     T  r r    T    0c i   ossψψin0  0  100i  (8)     +  rr +       l0  j  −−→uA  cos φr sin ψr cos φr cos ψr sin φr   l0   0 cos φDR1 sin φDR1  j  (8)  − −+−  −  φψ φ ψ φ +−−    φ φ    ll0  juA cosrr sin cos r cos r sin r 011  0 cosDRDR sin j   R0 k  sin φr sin ψr sin φr cos ψr cos φr R0 0 sin φDR1 cos φDR1rk − −−RRk sinφψ sin− −− sin φ cos ψ cos φ− 0− sinφ cos φk  0   rr r r r0  DRDRr11 Then, the motion of the flexible wheelset can be determined by solving the motion of the rigid wheelset.Then, Thethe contactmotion geometryof the flexible relationship wheelset between can be determined the dummy by wheelset solvingand the themotion pair of of the rails rigid can wheelset.be obtained The by contact using suchgeometry a trace relationship method. The between wheel–rail the dummy normal wheelset force can and be calculatedthe pair of using rails can the benonlinear obtained Hertzian by using contact such model,a trace andmethod. the wheel–rail The wheel–rail tangential normal force force is calculated can be calculated using Shen’s using theory. the nonlinearFor details, Hertzian please refer contact to [13 model,]. and the wheel–rail tangential force is calculated using Shen’s theory. For details, please refer to [13]. 3.3. Archard Wear Model 3.3. Archard Wear Model The Archard wear model is used to calculate the wear loss in the contact patch; the wear loss is writtenThe as: Archard wear model is used to calculate the wear loss in the contact patch; the wear loss is Nd written as: V = k (9) wear w H where Vwear is the wear loss in the contact patch, N is the normal contact force, d is the sliding distance, H is the hardness, and kw is the friction coefficient. The contact patch can be divided into many elements of equal size (Figure 11), and the length and width of the element are, respectively, ∆x and ∆y. pz is the normal force on the element, Sx is the lengthwise sliding velocity, Sy is the lateral sliding velocity, and a and b are the semi major axis and the semi minor axis of the contact patch, respectively. Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 22

= Nd Vkwear w (9) H

where Vwear is the wear loss in the contact patch, N is the normal contact force, d is the sliding distance, H is the hardness, and kw is the friction coefficient. The contact patch can be divided into many elements of equal size (Figure 11), and the length

and width of the element are, respectively, Δx and Δy. pz is the normal force on the element, Sx is Appl. Sci. 2020, 10, 5880 13 of 23 the lengthwise sliding velocity, Sy is the lateral sliding velocity, and a and b are the semi major axis and the semi minor axis of the contact patch, respectively.

FigureFigure 11.11. Forces onon elementselements ofof contactcontact patch.patch.

ForFor anan elementelement ofof thethe contactcontact patch,patch, thethe wearwear lossloss is:is:

ppz∆Δd Δ=∆z = kw zd (10) zkw H (10) H The normal force on the element is: The normal force on the element is: r  2  2 3N x 22y pz = pz(x, y) = 3Nxy1   (11) ==2πab − −−a − b (11) ppxyzz(, ) 1   2π ab a  b The sliding displacement is: The sliding displacement is: q 2 2 ∆x ∆d = s ∆t = Sx + Sy (12) | | v Δc 22x where v denotes the velocity of the contactΔ=dst patch. Δ= Combining S + S Equations (10)–(12), the wear loss of(12) any c xyv elements of the contact patch can be written as: c

where vc denotes the velocity of the contactr patch. Combining Equations (10)–(12), the wear loss of  2  y2 q any elements of the contact patch can3 Nkbe wwritten as:x 2 2 ∆x ∆z(x, y) = 1 Sx + Sy (13) 2πabH − a − b vc 22 3Nk xy  Δ where (x, y) is the positionΔ= of the elementwx of the −−contact patch. If the velocity22 + component caused by(13) the zxy(, ) 1   Sxy S 2π abH a b v flexible deformation is ignored, Equation (13) is expressed  as: c r where (x, y) is the position of the element of the2 contact 2 q patch. If the velocity component caused by 3Nkw∆x x y 2 2 vx the flexible deformation∆z(x, y is) =ignored, Equation1 (13) is expressed(ξ φ as:x) + (η + φy) (14) 2πabH − a − b − vc Δ 22 If the rolling contact patch3Nk iswx assumedxy to have  a slight local slip,22 it implies vv xc vx, and thus, Δ=zxy(, ) 1 −−  (ξφ −++ x ) ( ηφ y ) ≈ (14) Equation (14) is expressed as:2π abH a b v   c r q 3Nk ∆ x2  y2 vv≈ If the rolling contact∆z(x, y )patch = is wassumedx 1 to have a slight(ξ localφx) 2slip,+ ( ηit+ impliesφy)2 cx, and thus,(15) Equation (14) is expressed as: 2πabH − a − b − where ξ is the longitudinal creepage, η is the lateral creepage, and φ is the spin creepage. Δ 22 These parameters can be calculated3Nkwx using the vehicle–trackxy  rigid–flexible22 coupling dynamical model. Δ=zxy(, ) 1 −−  (ξφ −++ x ) ( ηφ y ) (15) Equation (15) indicates that the2π wearabH loss of a the contact b patch depends on the normal force and   creepages. When the distribution of these parameters along the wheel circumference fluctuates, thewhere wear ξ lossis the distribution longitudinal along creepage, the wheel η is circumferencethe lateral creepage, also fluctuates, and ϕ is the and spin it can creepage. easily lead These to polygonalparameters wear can underbe calculated certain conditions. using the vehicle–track rigid–flexible coupling dynamical model. Equation (15) indicates that the wear loss of the contact patch depends on the normal force and Selecting the wear coefficient (kw) is important for accurately calculating the wear that occurs on the wheel–rail because of their interaction. In [24], the authors constructed the friction coefficient based on the normal contact compressive stress and sliding velocity between the wheel and the rail obtained through tests. The wear coefficient within the range of 0.0001 to 0.001 means slight wear. In this paper, kw is 0.0004. Using the model discussed above to calculate polygonal wear, for simplicity, the contact points are considered to be distributed along the rolling circle of the wheel between the two adjacent contact points discretely. At these contact points, the normal load and slips of wheel–rail are calculated using the rigid–flexible coupling dynamical model of vehicle–track, discussed above. At each contact point, Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 22 creepages. When the distribution of these parameters along the wheel circumference fluctuates, the wear loss distribution along the wheel circumference also fluctuates, and it can easily lead to polygonal wear under certain conditions.

Selecting the wear coefficient (kw) is important for accurately calculating the wear that occurs on the wheel–rail because of their interaction. In [24], the authors constructed the friction coefficient based on the normal contact compressive stress and sliding velocity between the wheel and the rail obtained through tests. The wear coefficient within the range of 0.0001 to 0.001 means slight wear. In this paper, kw is 0.0004. Using the model discussed above to calculate polygonal wear, for simplicity, the contact points Appl.are considered Sci. 2020, 10, 5880to be distributed along the rolling circle of the wheel between the two adjacent contact14 of 23 points discretely. At these contact points, the normal load and slips of wheel–rail are calculated using the rigid–flexible coupling dynamical model of vehicle–track, discussed above. At each contact point, the wear depthdepth ofof elementselements isis calculated calculated in in the the contact contact patch patch using using the the known known normal normal load load and and slips slips of ofthe the wheel–rail. wheel–rail. Then, Then, the wearthe wear depths depths along along the rolling the rolling circle of circle the wheel of the are wheel summed. are summed. This process This is processrepeated is 50 repeated times. Finally, 50 times. the Finally, maximum the ofmaximum the summated of the wearsummated size along wear the size rolling along circle the rolling of the wheelcircle ofis consideredthe wheel is as considered the polygonal as wearthe polygonal depth after wear the wheeldepth rollsafter 50the times. wheel Because rolls 50 the times. distance Because between the distancethe first andbetween last contact the first points and last is notcontact equal, points the calculatedis not equal, accumulative the calculated wear accumulative curve, which wear indicates curve, polygonalwhich indicates wear discontinues,polygonal wear does discontinues, not fit the actual does scenario. not fit the A fittingactual methodscenario. is appliedA fitting to method solve this is problem;applied to however, solve this due problem; to space however, limitations, due thisto space is not limitations, discussed inthis this is paper.not discussed in this paper. 4. Discussion on the Calculated Results of High-Order Polygonal Wear 4. Discussion on the Calculated Results of High-Order Polygonal Wear Wheels with different diameters have different wavenumbers of high-order polygonal wear; Wheels with different diameters have different wavenumbers of high-order polygonal wear; however, their exciting frequencies are the same and, in general, they are in the range of 570–580 Hz however, their exciting frequencies are the same and, in general, they are in the range of 570–580 Hz (as shown in Section2). The wheelset discussed has a bending mode of 576 Hz (as shown in Section 3.1). (as shown in Section 2). The wheelset discussed has a bending mode of 576 Hz (as shown in Section It is obvious that the modal frequency is close to the exciting frequency of the polygonal wear at 3.1). It is obvious that the modal frequency is close to the exciting frequency of the polygonal wear at 300 km/h. An experimental analysis of the mechanism of high-order polygonal wear of high-speed 300 km/h. An experimental analysis of the mechanism of high-order polygonal wear of high-speed trains was conducted [12]; in that study, the authors suggested that the mechanism of polygonal wear trains was conducted [12]; in that study, the authors suggested that the mechanism of polygonal wear is attributed to the excited resonance of the train–track coupling system in operation through the field is attributed to the excited resonance of the train–track coupling system in operation through the field tests. Therefore, the mechanism of the polygonal wear should be further investigated by developing tests. Therefore, the mechanism of the polygonal wear should be further investigated by developing the simulation model that involves the flexible deformation (bending deformation) of the wheelset in the simulation model that involves the flexible deformation (bending deformation) of the wheelset this paper. in this paper. The numerical simulation uses the vehicle–track rigid–flexible coupling dynamical model described The numerical simulation uses the vehicle–track rigid–flexible coupling dynamical model in Section 3.1, and the track random roughness is used as an exciting input (Figure 12a,b). The speed of described in Section 3.1, and the track random roughness is used as an exciting input (Figure 12a,b). the train is 300 km/h and the diameter of the wheel is 920 mm. The normal contact forces and creepages The speed of the train is 300 km/h and the diameter of the wheel is 920 mm. The normal contact forces are calculated. Figure 12c,d shows the comparisons of the calculated results for the flexible wheelset and creepages are calculated. Figure 12c,d shows the comparisons of the calculated results for the and the rigid wheelset in the frequency domain. flexible wheelset and the rigid wheelset in the frequency domain.

5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 -3 -3 Lateral irregularity/mm Lateral Vertical irregularity/mm -4 -4 -5 -5 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 Distance/m Distance/m (a) (b)

Figure 12. Cont. Appl. Sci. 2020, 10, 5880 15 of 23 Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 22

-6 2 -6 102 10 Flexible Flexible -7 Rigid 10 Rigid 10 132 132144 1 144 1 -8 88 286 10 385 -1 10-1 -9 10-9 576 -2 10-2 -10 -3 10 -3 creepages Lateral 10 creepages Lateral Normal contact force/kN contact Normal Normal contact force/kN contact Normal -4 -11 10-4 10

-5 10 -12 0 100 200 300 400 500 600 700 800 900 1000 10 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Frequency/Hz Frequency/Hz (c) (d)

Figure 12.12. ((aa)) Vertical Vertical irregularity; irregularity; (b) Lateral(b) Lateral irregularity; irregularity; (c) Normal (c) Normal contact forces;contact (d forces;) Lateral (d creepages.) Lateral creepages. As shown in Figure 12c, the normal force of the flexible wheelset is slightly smaller than that of the rigidAs wheelset shown in in the Figure considered 12c, the frequency normal force domain. of the The flexible difference wheelset between is slightly the normal smaller forces than of thethat two of thetypes rigid of wheelset wheelset models in the isconsidered relatively frequency large at 132 doma Hz, whichin. The is thedifference sleeper between passing frequency. the normal Figure forces 12 ofd theshows two the types obvious of wheelset peaks models of the lateral is relatively creepages large at at approximately 132 Hz, which 88, is the 132, sl 144,eeper 286, passing 385, and frequency. 576 Hz, Figureand there 12d are shows large the diff erencesobvious between peaks of the the flexible lateral andcreepages rigid wheelset at approximately at these frequencies. 88, 132, 144, The 286, flexible 385, andwheelset 576 modesHz, and of 88,there 144, are 286, large 385, anddifferences 576 Hz needbetween to have the remarkable flexible and contribution rigid wheelset to the polygonal at these frequencies.wear corresponding The flexible to the wheelset peaks. modes However, of 88, the 144, tested 286, results 385, and in Section 576 Hz1 needshow to that have the remarkable dominant contributionpolygonal wear to the are polygonal in the 18th wear to 20th corresponding order, which to the are peaks. related However, to 576 Hz. the Therefore, tested results the numericalin Section 1simulation show that ofthe the dominant present paperpolygonal focuses wear on are the in influence the 18th to of 20th the wheelsetorder, which vibration are related mode to of 576 576 Hz. Hz Therefore,only and the the other numerical modes simulation (88, 144, 286, of the and presen 385 Hz)t paper are excluded focuses inon thethesimulation influence of program the wheelset when vibrationcalculating mode polygonal of 576 wear.Hz only and the other modes (88, 144, 286, and 385 Hz) are excluded in the simulationThe wear program model when described calculating in Section polygonal3 is used wear. to calculate the wear loss along the circumference of the wheelThe wear with model 50 circles described rolling. in In Section the calculation, 3 is used to the calculate track random the wear roughness loss along is used the circumference as an exciting ofinput. the wheel Figure with13a,b 50 show circles the rolling. calculated In the results calculation, of the wear the losstrack along random the circumferences roughness is used of the as two an excitingwheels with input. 50 Figure rolling 13a,b circles. show One the is a calculated flexible wheelset results andof the the wear other loss is a along rigid wheelset.the circumferences Figure 14a,b of theindicate two wheels their irregularity with 50 rolling spectra circles. corresponding One is a flexible to Figure wheelset 13a,b, and respectively. the otherFrom is a rigid the calculatedwheelset. Figureresult, it14a,b is found indicate that their the flexibleirregula deformationrity spectra corresponding of the wheelset to either Figure greatly 13a,b, contributesrespectively. or From leads the to calculatedpolygonal result, wear generation. it is found that the flexible deformation of the wheelset either greatly contributes or leads to polygonal wear generation. 0 0 -20 0 -20 6.0x10-20 6.0x10 330 30 -20 330 30 -20 330 30 3.0x10-20 3.0x10 0.0 0.0 -20 -20 -3.0x10-20 -3.0x10 -20 -20 300 60 -20 300 60 -6.0x10 300 60 -6.0x10 -20 -20 -9.0x10-20 -9.0x10 -19 -19 -1.2x10-19 -1.2x10 -19 -19 -1.5x10-19 -1.5x10-19 -19 -19 -1.8x10-19 -1.8x10-19

-19 270 90 -19 270 90 -1.8x10-19 -1.8x10-19 -19 -19 -1.5x10-19 -1.5x10-19 -19 -19 Wear loss/mm -19 -19 Wear loss/mm Wear loss/mm -1.2x10 Wear loss/mm -1.2x10 -20 -20 -9.0x10-20 -9.0x10-20 -20 -20 -6.0x10-20 -6.0x10-20 -20 240 120 -20 240 120 -3.0x10-20 -3.0x10-20 0.0 0.0 -20 -20 3.0x10 3.0x10-20 -20 210 150 -20 210 150 6.0x10 6.0x10 210 150 180 180 Angle/° Angle/° (a) (b)

Figure 13. ResultsResults of of wear wear loss loss along along the the circumferences circumferences of of the the two two wheels wheels with with rolling rolling of of 50 50 circles. circles. (a) Flexible wheelset and (b) Rigid wheelset. Appl. Sci. 2020, 10, 5880 16 of 23 Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 22

10-19 10-19 10-20 10-20 10-21 10-21 10-22 10-22 10-23 10-23 10-24 10-24

Wear loss/m Wear -25 10-25 Wear loss/m 10 -26 10-26 10 -27 10-27 10 -28 10-28 10 0 4 8 1216202428323640 0 4 8 12 16 20 24 28 32 36 40 order order (a) (b)

Figure 14. Spectra corresponding to Figure 1313a,b.a,b. (a) Flexible wheelset and (b) Rigid wheelset.

Note thatthat FigureFigure 12 12dd shows shows the the peaks peaks of of the the lateral lateral creepage creepage at 88,at 88, 132, 132, 144, 144, 286, 286, 385, 385, and and 576 576 Hz. Furthermore,Hz. Furthermore, the wheel the wheel circumference circumference is approximately is approximately within the within integral the multiples integral of multiples the wavelengths of the ofwavelengths the vibration of the at 144,vibration 286, andat 144, 576 286, Hz, and at 300 576 kmHz,/h at and 300 atkm/h the and 920 mmat the wheel 920 mm diameter. wheel diameter. This is a basicThis is condition a basic condition for the generation for the generation of polygonal of polygonal wear at wear 144, 286at 144, and 286 576 and Hz. 576 The Hz. polygonal The polygonal wear ofwear the of corresponding the corresponding 5th, 10th, 5th, and10th, 20th and orders 20th orders should should start at start the same at the time same and time develop and develop quickly. However,quickly. However, the calculated the calculated result, as result, shown as inshown Figure in 14Figurea, does 14a, not does show not the show polygonal the polygonal wear of wear the 5thof the and 5th 10th and orders. 10th orders. It is apparently It is apparently contradictory contradictory when when comparing comparing Figure Figure 12d with12d with Figure Figure 14a. In14a. fact, In thefact, polygonal the polygonal wear ofwear 5th andof 5th 10th and orders 10th canorders contribute can contribute prominently prominently if all bending if all modesbending of themodes wheelset of the arewheelset considered are considered for calculating for calculatin polygonalg polygonal wear. However, wear. However, the test results, the test as results, shown inas Figureshown 15in, Figure indicate 15, the indicate dominant the dominant order of only order 20, of and only its 20, corresponding and its corresponding passing frequency passing frequency is close to Appl.576is close HzSci. at2020to a576, speed10 ,Hz x FOR at of PEERa 300 speed km REVIEW/ hofand 300 atkm/h a diameter and at a close diameter to 920 close mm. to 920 mm. 16 of 22 Figure 15 shows the typical site test results. Figure 15 shows the lateral vibration acceleration spectra of the wheelset bearing10 box and bogie frame above the primary suspension when the train is running at 300 km/h after wheel re-profiling Wheelset (no polygonal wear exists). The result of the wheelset 579Hz indicates that, in the actual running condition, Bogie th framee vibration level of the wheelset in the frequency band of 550–600 Hz is more significant than that in other frequency bands. This implies that the mode of the wheelset at 576 Hz is excited1 when the train is running. Furthermore, Figure 15 shows that the vibration of the bogie frame at ~579 Hz (close to a wheelset mode frequency of 576 Hz) is as dominant as that of the wheelset. In such a scenario, the 20th-order polygonal wear was initiated and developed quickly. Then, at frequencies of ~144 Hz and 286 Hz, the uncoupling vibration of the wheelset and bogie probably occurred,Acceleration/g or0.1 other resonances at frequencies near them occurred, as shown in Figure 15. This situation could disturb or suppress the polygonal wear of the 5th and 10th orders. However, this is conjecture that can be verified by improving the present theoretical model. The current model cannot account for the effect of the bogie resonances and the frequency dependent characteristics of the suspension. Thus far,0.01 to emphasise the influence of the wheelset mode of 576 Hz, the effect of 0 100 200 300 400 500 600 700 other modes was intentionally neglected in the calculation using the less advanced model. Frequency/Hz Figure 15. LateralLateral vibration acceleration spectr spectraa of the wheelset and bogie frame.

5. KeyFigure Conditions 15 shows of Initiation the typical an sited Development test results. Figure of Polygonal 15 shows Wear the lateral vibration acceleration spectra of the wheelset bearing box and bogie frame above the primary suspension when the train is 5.1.running Integral at 300Multiple km/h Condition after wheel re-profiling (no polygonal wear exists). The result of the wheelset indicates that, in the actual running condition, the vibration level of the wheelset in the frequency The research results show that periodically fluctuating creepage (or slip) can lead to the initiation band of 550–600 Hz is more significant than that in other frequency bands. This implies that the and development of polygonal wear, and the resonance of the coupling system of train and track can mode of the wheelset at 576 Hz is excited when the train is running. Furthermore, Figure 15 shows cause periodically fluctuating load and creepage of the wheel–rail. If the circumference of the wheel that the vibration of the bogie frame at ~579 Hz (close to a wheelset mode frequency of 576 Hz) is as is nearly an integral multiple of the wavelength of the excited system resonance, it is regarded as a key condition for initiating the high-rate development of polygonal wear. This is called an integral multiple condition. As shown in Figure 16 [12], the basic condition for wheel polygonal wear generation is more rigorous than that for rail corrugation formation because the rail is infinitely long and the corrugation in its development can extend infinitely along with its track. However, the perimeter of the rolling wheel is finite, and uneven wear along the wheel circumference will be superimposed circularly. The referenced start points of the uneven wear on the wheel running surface when the wheel rolls have different phases if the perimeter cannot be divided exactly by the wavelength of the uneven wear caused by the resonance. If so, the uneven wear or wheel polygonal wear develops very slowly. In the other case, wheel polygonal wear can develop very quickly. This case indicates that the changes in wheel perimeter because of the wear and re-profiling is nearly an integral multiple of the wavelength λ of the excited resonance of the wheel–rail system.

Figure 16. Description of basic condition for wheel OOR generation [12].

In the following analysis case, the effect of the wheelset mode of 576 Hz on the development of polygonal wear was analysed under conditions where the operating speed was 300 km/h and the Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 22

10 Wheelset Bogie frame 579Hz

1 Appl. Sci. 2020, 10, 5880 17 of 23 dominant as that of the wheelset. In such a scenario, the 20th-order polygonal wear was initiated and developed quickly.Acceleration/g Then,0.1 at frequencies of ~144 Hz and 286 Hz, the uncoupling vibration of the wheelset and bogie probably occurred, or other resonances at frequencies near them occurred, as shown in Figure 15. This situation could disturb or suppress the polygonal wear of the 5th and 10th orders. However, this is conjecture that can be verified by improving the present theoretical model. The current model cannot0.01 account for the effect of the bogie resonances and the frequency dependent 0 100 200 300 400 500 600 700 characteristics of the suspension. Thus far, to emphasiseFrequency/Hz the influence of the wheelset mode of 576 Hz, the effect of other modes was intentionally neglected in the calculation using the less advanced model. Figure 15. Lateral vibration acceleration spectra of the wheelset and bogie frame. 5. Key Conditions of Initiation and Development of Polygonal Wear 5. Key Conditions of Initiation and Development of Polygonal Wear 5.1. Integral Multiple Condition 5.1. IntegralThe research Multip resultsle Condition show that periodically fluctuating creepage (or slip) can lead to the initiation and developmentThe research results of polygonal show that wear, periodically and the resonance fluctuating of thecreepage coupling (or systemslip) can of lead train to andthe initiation track can andcause development periodically of fluctuating polygonal load wear, and and creepage the resonanc of thee wheel–rail. of the coupling If the system circumference of train ofand the track wheel can is causenearly periodically an integral multiplefluctuating of theload wavelength and creepage of theof t excitedhe wheel–rail. system If resonance, the circumference it is regarded of the as wheel a key iscondition nearly an for integral initiating multiple the high-rate of the developmentwavelength of polygonalthe excited wear. system This resonance, is called an it integralis regarded multiple as a keycondition. condition As shownfor initiating in Figure the 16 high-rate[ 12], the basicdevelopmen conditiont of for polygonal wheel polygonal wear. This wear is generationcalled an integral is more multiplerigorous thancondition. that for As rail shown corrugation in Figure formation 16 [12], because the ba thesic rail condition is infinitely for longwheel and polygonal the corrugation wear generationin its development is more rigorous can extend than infinitely that for alongrail corr withugation its track. formation However, because the the perimeter rail is infinitely of the rolling long andwheel the is corrugation finite, and unevenin its development wear along thecan wheelextend circumference infinitely along will with be superimposedits track. However, circularly. the perimeterThe referenced of the start rolling points wheel of the is unevenfinite, and wear unev onen the wear wheel along running the wheel surface circumference when the wheel will rolls be superimposedhave different phasescircularly. if the The perimeter referenced cannot start be dividedpoints exactlyof the uneven by the wavelength wear on the of thewheel uneven running wear surfacecaused bywhen the the resonance. wheel rolls If so, have the unevendifferent wear phases or wheel if the polygonalperimeter wearcannot develops be divided very exactly slowly. by In the the wavelengthother case, wheel of the polygonaluneven wear wear caused can develop by the reso verynance. quickly. If so, This the caseuneven indicates wear or that wheel the changespolygonal in wearwheel develops perimeter very because slowly. of theIn the wear other and case, re-profiling wheel polygonal is nearly an wear integral can develop multiple very of the quickly. wavelength This caseλ of theindicates excited that resonance the changes of the in wheel–rail wheel perimeter system. because of the wear and re-profiling is nearly an inte

Figure 16. DescriptionDescription of of basic basic condition condition for wheel wheel OOR generation [12]. [12].

InIn the the following following analysis analysis case, case, the the effect effect of of th thee wheelset wheelset mode mode of of576 576 Hz Hz on on the the development development of polygonalof polygonal wear wear was was analysed analysed under under conditions conditions where where the theoperating operating speed speed was was300 300km/h km and/h andthe the wheel diameter was 900 mm and 920 mm. Figure 17 shows the effect of the diameter difference. When the wheel had a 900 mm diameter at 300 km/h, its circumference was not an integral multiple of the wavelength (v = 300 km/h and f = 576 Hz), and when the wheel had a 920 mm diameter, at 300 km/h, the integral multiple condition was nearly satisfied. Thus, the calculated wear loss along Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 22 wheel diameter was 900 mm and 920 mm. Figure 17 shows the effect of the diameter difference. When theAppl. wheel Sci. 2020 had, 10 ,a 5880 900 mm diameter at 300 km/h, its circumference was not an integral multiple 18of ofthe 23 wavelength (v = 300 km/h and f = 576 Hz), and when the wheel had a 920 mm diameter, at 300 km/h, the integral multiple condition was nearly satisfied. Thus, the calculated wear loss along the thethe circumferencesintegral multiple of thecondition two wheels was with nearly 50 rolling satisfied. cycles Thus, showed the acalculated large difference wear atloss the along 20th order the circumferences of the two wheels with 50 rolling cycles showed a large difference at the 20th order circumferencesof polygonal wear, of the as showntwo wheels in Figure with 17 50. rolling cycles showed a large difference at the 20th order of polygonal wear, as shown in Figure 17.

-18 10-18

-19 D=0.92m 10 D=0.90m -20 10-20

-21 10-21

-22 10-22

-23 Wear loss/m Wear loss/m 10

-24 10-24

-25 10-25

-26 10-26 0 4 8 12 16 20 24 28 32 36 40 Order Figure 17. Calculated wear loss along the circumferences of wheels with two different diameters. Figure 17. Calculated wear loss along the circumferences of wheels with two different diameters. As shown in Figure 17, 20th order polygonal wear occurs on wheels with two different diameters. However,As shown it is obvious in Figure that 17, polygonal 20th order wear polygona is more seriousl wearon occurs the wheel on wheels with a diameterwith two of different 920 mm, diameters.that satisfies However, the integral it is multipleobvious conditionthat polygonal nearly. wear is more serious on the wheel with a diameter of 920Figure mm, that18 shows satisfies the the polygonal integral wear multiple results condition obtained nearly. by superposing the uneven wear at different wheelFigure rolling 18 cycles shows (5 cycles,the polygonal 25 cycles, wear and 50results cycles). ob Figuretained 18bya,b superposing indicates the the results uneven of the wear wheels at differentwith diameters wheel ofrolling 920 mm cycles and (5 900 cycles, mm, respectively.25 cycles, an Asd 50 shown cycles). in FigureFigure 18 18a,ba, it isindicates obvious the that results the wear of theloss wheels along thewith circumference diameters of 920 with mm a wear and superposition900 mm, respectively. of different As shown cycles hasin Figure nearly 18a, the it same is obvious phase, thatand the polygonalwear loss along wear increasedthe circumference gradually with and a rapidly. wear superposition As shown in Figureof different 18b, forcycles a wheel has nearly with a thediameter same phase, of 900 mm,and the polygonal wearwear doesincreased not increase gradually considerably, and rapidly. and As it shown appears in similar Figure to 18b, the forpolygonal a wheel wear with pattern a diameter of each ofcycling 900 mm, “rotating” the polygonal or “shifting” wear does around not the increase circumference. considerably, Superposing and it appearsthe uneven similar wear to of the each polygonal cycling with wear di ffpatternerent initial of each phases cycling eliminates “rotating’’ their or peaks “shifting’’ and valleys. around This the is circumference.because the referenced Superposing start pointsthe uneven of the wear uneven of each wear cycling at each rollingwith different cycle have initial diff phaseserent phases eliminates if the theirperimeter peaks is and not valleys. an integral This multiple is because of the wavelengthreferenced start of the points uneven of the wear. unev Ifen so, wear the polygonal at each rolling wear cycledevelops have very different slowly. phases if the perimeter is not an integral multiple of the wavelength of the uneven wear. If so, the polygonal wear develops very slowly. 0 0 -20 0 -20 6.0x10-20 6.0x10 330 30 -20 330 30 -20 330 30 3.0x10-20 3.0x10 0.0 0.0 -20 -20 -3.0x10-20 -3.0x10 -20 300 60 -20 300 60 -6.0x10 300 60 -6.0x10 -20 -20 -9.0x10-20 -9.0x10 -19 -19 -1.2x10-19 -1.2x10 -19 5 cycles -19 -1.5x10 5 cycles -1.5x10 5 cycles -19 -19 5 cycles -1.8x10-19 25 cycles -1.8x10-19

25 cycles

25 cycles -19 270 90 -19 270 90 -1.8x10-19 50 cycles -1.8x10-19 50 cycles Wear loss/mm 50 cycles Wear loss/mm -19 -19 -1.5x10-19 -1.5x10-19 -19 -19 Wear loss/mm -1.2x10 Wear loss/mm -1.2x10 -20 -20 -9.0x10-20 -9.0x10-20 -20 -20 -6.0x10-20 -6.0x10-20 -20 240 120 -20 240 120 -3.0x10-20 -3.0x10-20 0.0 0.0 -20 -20 3.0x10 3.0x10-20 -20 210 150 -20 210 150 6.0x10 210 150 6.0x10 210 150 180 180 Angle/° Angle/° (a) (b)

FigureFigure 18. Polygonal wearwear resultsresults obtained obtained by by superposing superposing the the uneven uneven wear wear of the of wheelsthe wheels rolling rolling with

withcycles, cycles, (a) D (=a) 920D = mm,920 mm, (b) D (b=) 900D = mm.900 mm. Appl. Sci. 2020, 10, 5880 19 of 23

The wheel diameter is an important parameter that can determine whether the wheel satisfies the integral multiple condition. In addition, in the long-time service of a wheel, its diameter changes over a large range because of wear and re-profiling. Considering a high-speed train of China as an example, the diameter for its new wheel is 920 mm. The diameter of the wheel reduces to about 830 mm at the end of its use life under normal use. Therefore, several special values of the diameter during the normal use life can satisfy the integral multiple condition at 300 km/h, and then, the high-order polygonal wear occurs easily and quickly at these diameter values. For the research object in this paper, when the wheel diameters were 920 mm, 875 mm, and 830 mm, the wheels satisfied the integral multiple condition. Thus, wheels with different diameters have different proportions of polygonal wheels and different dominant polygonal orders.

5.2. Effect of Different Diameters and Different Speeds The train is operated at different speeds v on different track lines, and the excited resonance frequency that influences the polygonal wear development does not vary considerably in the analysis during the field tests. Therefore, the operating speed determines the wavelength λ of the polygonal wear, and the integral multiple condition determines the initiation and high-rate development of polygonal wear. Therefore, the high-rate development of the polygonal wear depends on the operation speed and wheel diameter. The analysis of polygonal wear development during the entire normal use life of the wheel was conducted for the following case. The analysis involves different wheel diameters, different operation speeds, and excited resonance frequencies related to the polygonal wear development at a high rate. In this analysis, the diameter was from 820 to 920 mm, the speed was from 150 to 350 km/h, and the excited resonance frequency was 576 Hz, which was assumed to refer to the test results discussed above. For the other excited resonance frequencies, a similar analysis can be conducted. The calculated wear loss along the circumference of the wheel (50 rolling cycles) with different diameters and different speeds is shown in Figure 19. In the figure, the dark areas indicate severe polygonal wear or polygonal wear development at a high rate. Special diameters that would lead to severe polygonal wear are different if the operating speeds are different. At the same speed, there are several special diameters at which the polygonal wear develops fast. Therefore, with the help of numerical results, high-rate polygonal wear development can be predicted theoretically, and the re-profiling rule of polygonal wheels can be built up easily. For example, for a train at an operating speed of 350 km/h, there are two special diameters—860 and 915 mm—at which there are dark areas (Figure 19). Therefore, after the new wheels are placed into operation in the initial stage where the diameter is ~915 mm, polygonal wear occurs; however, it is not considerably serious. The wheel diameter decreases to ~860 mm because of wear and re-profiling, and because another dark area is generated. Thus, it implies that this is a time for high-rate polygonal wear development. The corresponding wheel diameter is the “special diameter.” This time, polygonal wheels need to be re-profiled in time, or the re-profiling interval should be changed, or the train operation speed is changed considerably to decrease the fast development of polygonal wear. For any new type of high-speed train to be placed into commercial operation, the easily excited resonant frequencies related to the initiation of the polygonal wear should be known, and then, figures similar to Figure 19 can be drawn and analysed. Appl. Sci.Appl. 2020 Sci., 102020, x FOR, 10, 5880PEER REVIEW 2019 of 23of 22 heel diameter/mm W

Speed/km·h-1

Figure 19. Polygonal wear development with different operation speeds and different wheel diameters. Figure 19. Polygonal wear development with different operation speeds and different wheel 6. Conclusions diameters. In this study, the characteristics of polygonal wear measured from a large number of wheels of 6. Conclusionshigh-speed trains were statistically analysed, and a prediction model of polygonal wear of high-speed trains was developed. Using the prediction model, the causes and the conditions of the initiation and Indevelopment this study, the of the characteristics polygonal wear of werepolygonal studied. wear The measured main conclusions from a are large given number below. of wheels of high-speed trains were statistically analysed, and a prediction model of polygonal wear of high-speed trains was1. developed.The prominent Using feature the prediction of polygonal model, wear is th thate causes the dominant and the orderconditions and development of the initiation rate ofand developmentpolygonal of the polygonal wear are very wear closely were related stud toied. the The wheel main diameter, conclusions operation are speed, given and below. the resonance frequencies of the system, such as wheelsets. 1. The2. prominentIf the wheel feature circumference of polygonal is nearly wear an integral is that multiple the dominant of the wavelength order and of development the periodical wearrate of polygonalalong wear the rolling are circlevery ofclosely the wheel, related the polygonal to the wearwheel initiates diameter, and develops operation very quickly.speed, Thisand isthe resonancereferred frequencies to as the integral of the system, multiple such condition, as wheelsets. which is a key or basic condition for polygonal wear. 2. If the wheelThis “integral circumference multiple” is isnearly on the an order integral of the multiple dominant of polygonal the wavelength wear. of the periodical wear along3. theWhen rolling the excitedcircle of resonance the wheel, frequency the polygonal of the wear vehicle initiates system and is determined develops very to be quickly. related toThis is referredpolygonal to as wear,the integral the initiation multiple and developmentcondition, which of the is polygonal a key or wear basic depends condition on the for operation polygonal wear. Thisspeed “integral and wheel multiple” diameter. is Their on th relationshipe order of diagramthe dominant is provided, polygonal and it canwear. be used to predict 3. When polygonalthe excited wear resonance development, frequency modify theof re-profilingthe vehicle schedule system of is polygonal determined wheels, to and be eliminaterelated to the polygonal wear on the wheels in time. polygonal wear, the initiation and development of the polygonal wear depends on the operation 4. The effect of flexibility of the wheelset is considered in the prediction model of the polygonal speed and wheel diameter. Their relationship diagram is provided, and it can be used to predict wear of high-speed wheels in this paper. Therefore, the high-frequency characteristics of the polygonal wear development, modify the re-profiling schedule of polygonal wheels, and wheelset bending resonances can be more accurately expressed, and high-order polygonal wear eliminaterelated the topolygonal the wheelset wear bending on the resonances wheels in at time. high frequencies can be numerically reproduced. 4. The effect of flexibility of the wheelset is considered in the prediction model of the polygonal wear ofHowever, high-speed the current wheels simulation in this modelpaper. can Ther onlyefore, reflect the the high-frequency actual scenario of characteristics the train to a certain of the wheelsetextent, and bending some phenomenaresonances cannotcan be bemore accurately accurately reflected. expressed, In addition, and high-order it cannot account polygonal for thewear relatedeffect ofto the the bogie wheelset resonances bending and resonances the frequency at dependent high frequencies characteristics can be of numerically the suspension. reproduced. In short, the current simulation model needs to be improved further. However, the current simulation model can only reflect the actual scenario of the train to a certain extent, and some phenomena cannot be accurately reflected. In addition, it cannot account for the effect of the bogie resonances and the frequency dependent characteristics of the suspension. In short, the current simulation model needs to be improved further.

Author Contributions: Conceptualization, Y.W. and X.J.; methodology, Y.W., J.H., and W.C.; validation, Y.W., X.X., and X.J.; investigation, Y.W.; writing—original draft preparation, Y.W.; writing—review and editing, X.J.; Appl. Sci. 2020, 10, 5880 21 of 23 Appl. Sci. 2020, 10, x FOR PEER REVIEW 20 of 22

Authorsupervision, Contributions: X.J.; projectConceptualization, administration, X.J.; Y.W. funding and X.J.; acquisition, methodology, X.J. All Y.W., authors J.H., have and read W.C.; and validation, agreed to Y.W., the X.X.,published and X.J.; version investigation, of the manuscript. Y.W.; writing—original draft preparation, Y.W.; writing—review and editing, X.J.; supervision, X.J.; project administration, X.J.; funding acquisition, X.J. All authors have read and agreed to the publishedFunding: This version research of the was manuscript. funded by the National Science Foundation of China, grant numbers U1734201 and Funding:51805450. This research was funded by the National Science Foundation of China, grant numbers U1734201 andAcknowledgments: 51805450. The authors wish to thank CSR Tangshan Rolling Stock Co., Ltd. for their assistance in Acknowledgments:conducting the experimentThe authors. We also wish would to thank like to CSR thank Tangshan Editage Rollingfor English Stock language Co., Ltd. editing. for their assistance in conducting the experiment. We also would like to thank Editage for English language editing. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Appendix A The shape testing of all elements is conducted to ensure that a sufficiently fine mesh has been The shape testing of all elements is conducted to ensure that a sufficiently fine mesh has been used. The summary of the shape testing is listed in Table A1. used. The summary of the shape testing is listed in Table A1.

Table A1. The summary of the shape testing. Table A1. The summary of the shape testing. Test Items Number Tested Warning Count Error Count Warn+Err (%) Test Items Number Tested Warning Count Error Count Warn+Err (%) Aspect Ratio 255,360 5232 0 2.05% Aspect Ratio 255,360 5232 0 2.05% Parallel Deviation 255,360 0 0 0 Parallel Deviation 255,360 0 0 0 MaximumMaximum AngleAngle 255,360 255,360 0 0 0 0 0 0 JacobianJacobian RatioRatio 255,360 255,360 0 0 0 0 0 0 WarpingWarping FactorFactor 255,360 255,360 0 0 0 0 0 0

InIn orderorder toto ensureensure thatthat thethe selectedselected mesh size has little eeffectffect on the results of modal analysis, thethe two new new finite finite element element models models with with smaller smaller element element size size were were built, built, as shown as shown in figure in Figure A1. TheA1. Thecomparison comparison results results of modal of modal analysis analysis are shown are shown in Table in Table A2. A2.

(a) (b)

(c)

Figure A1.A1. TheThe finite finite element element model: model: (a) ( Modela) Model 1 (the 1 model(the model in the in manuscript, the manuscript, element element count: 255,360, count: node255,360, count: node 267,771); count: 267,771); (b) Model (b 2) (elementModel 2 (element count: 556,800, count: node556,800, count: node 573,673); count: 573,673); (c) Model (c 3) (elementModel 3 count:(element 834,860, count: node 834,860, count: node 856,285). count: 856,285).

Appl. Sci. 2020, 10, 5880 22 of 23

Table A2. The comparison results of modal analysis.

Model 1 (the Model in the Model 2 Model 3 Manuscript)

Element count 255,360 556,800 The errors refer to 834,860 The errors refer to Node count 267,771 573,673model 1 (%) 856,285 model 1 (%) Order Frequency 1 72.13 72.12 0.01% 72.12 0.01% 2 87.08 87.02 0.07% 86.95 0.15% 3 143.6 143.3 0.21% 142.9 0.49% 4 234.1 233.3 0.34% 232.1 0.85% 5 284.7 284.0 0.25% 283.2 0.53% 6 347.3 346.1 0.35% 344.1 0.92% 7 386.0 385.4 0.16% 384.9 0.28% 8 576.4 575.7 0.12% 574.7 0.29%

References

1. Johansson, A. Out-of-round Railway Wheels—Literature Survey, Field Tests and Numerical Simulations. Licentiate Thesis, Chalmers University of Technology, Göteborg, Sweden, 2003. 2. Jin, X.S.; Wu, Y.; Liang, S.L.; Wen, Z.F. Advance in research on mechanism of out-of-roundness wear of railway vehicle wheels and its solution countermeasure. J. Southwest Jiaotong Univ. 2018, 53, 1–14. 3. Nielsen, J.C.O.; Johansson, A. Out-of-round railway wheels—A literature survey. Proc. Inst. Mech. Eng. F J. Rail Rapid Transit 2000, 214, 79–91. [CrossRef] 4. Kaper, H.P. Wheel corrugation on Netherlands railway (NS): Origin and effects of ‘polygonization’ in particular. J. Sound Vib. 1998, 120, 267–274. [CrossRef] 5. Morys, B.; Kuntze, H.B.; Hirsch, U. Investigation of origin and enlargement of out-of-round phenomena in high speed ICE-wheelsets. In Proceedings of the 10th European ADAMS Users’ Conference, Frankfurt, Germany, 14–15 November 1995; Volume 14, p. 1. 6. Morys, B.; Kuntze, H.B. Simulation analysis and active compensation of the out-of-round phenomena at wheels of high speed trains. In Proceedings of the World Congress on Railway Research, Florence, Italy, 16–19 November 1997. 7. Morys, B. Enlargement of out-of-round wheel profiles on high speed trains. J. Sound Vib. 1999, 227, 965–978. [CrossRef] 8. Meinke, P.; Meinke, S.; Szoc, T. On dynamics of rotating wheel/rail systems in a medium frequency range. In Proceedings of the 4th German-Polish Workshop Dynamical Problems in Mechanical Systems, Warszawa, Poland, 30 July–4 August 1996. 9. Meywerk, M. Polygonalizaiton railway wheels. Arch. Appl. Mech. 1999, 69, 102–120. [CrossRef] 10. Chen, G.X.; Jin, X.S.; Wu, P.B.; Dai, H.Y.; Zhou, Z.R. Finite element study on the generation mechanism of polygonal wear of railway wheels. J. China Railw. Soc. 2011, 33, 14–18. 11. Tao, G.; Wang, L.; Wen, Z.; Guan, Q.; Jin, X. Measurement and assessment of out-of-round electric locomotive wheels. Proc. Inst. Mech. Eng. F 2016, 232, 275–287. [CrossRef] 12. Tao, G.; Wang, L.; Wen, Z.; Guan, Q.; Jin, X. Experimental investigation into the mechanism of the polygonal wear of electric locomotive wheels. Veh. Syst. Dyn. 2018, 56, 1–17. [CrossRef] 13. Jin, X.; Wu, L.; Fang, J.; Zhong, S.; Ling, L. An investigation into the mechanism of the polygonal wear of metro train wheels and its effect on the dynamic behaviour of a wheel/rail system. Veh. Syst. Dyn. 2012, 50, 1817–1834. [CrossRef] 14. Wu, Y.; Du, X.; Zhang, H.J.; Wen, Z.F.; Jin, X.S. Experimental analysis of the mechanism of high-order polygonal wear of wheels of a high-speed train. J. Zhejiang Univ. Sci. A 2017, 18, 579–592. [CrossRef] Appl. Sci. 2020, 10, 5880 23 of 23

15. Wu, X.; Rakheja, S.; Cai, W.; Chi, M.; Ahmed, A.; Qu, S. A study of formation of high order wheel polygonalization. Wear 2019, 424, 1–14. [CrossRef] 16. Cai, W.; Chi, M.; Tao, G.; Wu, X.; Wen, Z. Experimental and numerical investigation into formation of metro wheel polygonalization. Shock Vib. 2019, 2019, 1–18. [CrossRef] 17. Zhai, W.M. Vehicle-Track Coupled Dynamics, 4th ed.; Science Press: Beijing, China, 2007. 18. Zhong, S.; Xiong, J.; Xiao, X.; Wen, Z. Effect of the first two wheelset bending modes on wheel-rail contact behavior. J. Zhejiang Univ. Sci. A 2014, 15, 984–1001. [CrossRef] 19. Xiao, X.B.; Ling, L.; Jin, X.S. A study of the derailment mechanism of a high-speed train due to an earthquake. Veh. Syst. Dyn. 2012, 50, 449–470. [CrossRef] 20. Han, J.; Zhong, S.Q.; Xiao, X.B.; Wen, Z.F.; Zhao, G.T.; Jin, X.S. High-speed wheel/rail contact determining method with rotating flexible wheelset and validation under wheel polygon excitation. Veh. Syst. Dyn. 2018, 56, 1233–1249. [CrossRef] 21. Li, L.; Xiao, X.B.; Jin, X.S. Interaction of subway LIM vehicle with ballasted track in polygonal wheel wear development. Acta Mech. Sin. 2011, 27, 297–307. [CrossRef] 22. Ling, L.; Xiao, X.B.; Xiong, J.Y.; Zhou, L.; Wen, Z.F.; Jin, X.S. Erratum to: A 3D model for coupling dynamics analysis of high-speed train/track system. J. Zhejiang Univ. Sci. A 2015, 15, 964–983. [CrossRef] 23. Han, J.; Zhong, S.Q.; Zhou, X.; Xiao, X.B.; Zhao, G.T.; Jin, X.S. Time-domain model for wheel-rail noise analysis at high operation speed. J. Zhejiang Univ. Sci. A 2017, 18, 593–602. [CrossRef] 24. Jendel, T. Prediction of Wheel Profile Wear-methodology and Verification. Licentiate Thesis, Department of Vehicle Engineering, KTH, Stockholm, Sweden, 2000.

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