Railway Polygonized Wheel Detection Based on Numerical Time-Frequency Analysis of Axle-Box Acceleration

applied sciences

Article Railway Polygonized Wheel Detection Based on Numerical Time-Frequency Analysis of Axle-Box Acceleration

Ying Song 1,2,* , Lei Liang 1, Yanliang Du 2 and Baochen Sun 3

1 Key Laboratory of Traﬃc Safety and Control of Hebei Province, School of Traﬃc and Transportation, Shijiazhuang Tiedao University, Shijiazhuang 050043, Hebei, China; [email protected] 2 School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, Sichuan, China; [email protected] 3 Structure Health Monitoring and Control Institute, Shijiazhuang Tiedao University, Shijiazhuang 050043, Hebei, China; [email protected] * Correspondence: [email protected]; Tel.: +86-0311-879-36613

Received: 28 January 2020; Accepted: 21 February 2020; Published: 28 February 2020

Abstract: The increasing need for repairs of polygonized wheels on high-speed railways in China is becoming problematic. At high speeds, polygonized wheels cause abnormal vibrations at the wheel-rail interface that can be detected via axle-box accelerations. To investigate the quantitative relationship between axle-box acceleration and wheel polygonization in both the time and frequency domains and under high-speed conditions, a dynamics model was developed to simulate the vehicle-track coupling system and that considers both wheel and track ﬂexibility. The calculated axle-box accelerations were analyzed by using the improved ensemble empirical mode decomposition and Wigner-Ville distribution time-frequency method. The numerical results show that the maximum axle-box accelerations and their frequencies are quantitatively related to the harmonic order and out-of-roundness amplitude of polygonized wheels. In addition, measuring the axle-box acceleration enables both the detection of wheel polygonization and the identiﬁcation of the degree of damage.

Keywords: high-speed railway; wheel polygon; axle-box acceleration; time-frequency characterization; detection

1. Introduction The high-speed train system is rapidly developing in China and, as a result of rolling contact and vibration impact, wheel polygonization, which is a type of railway-wheel out of roundness (OOR), is now a growing problem. Wheel-tread polygons have been detected with radial irregularity wavelengths from 0.12 to 3.4 m (which is approximately equal to the circumference of a wheel) and with 1to 25 harmonic orders around the wheel circumference. In the early 1980s, Kreuzen described wheel polygons observed in the Dutch railway system [1]. Wheel polygons can be both dangerous and annoying: the serious Eschede accident of an Inter City Express train in Germany in the summer of 1998, in which 101 people were killed and 194 injured, was attributed to a fatigue crack in a wheel rim caused by a wheel polygon. In recent years, failures in vehicle system components have appeared in the Chinese high-speed railway system, such as gearbox cracks [2], bolt fracture at the axle-box cover and brake disk [3], and other fastener failures that are caused by high-frequency impact due to wheel polygons. In addition, a periodic wheel out-of-roundness leads to noise and vibration levels that annoy passengers, especially at certain train speeds.

Appl. Sci. 2020, 10, 1613; doi:10.3390/app10051613 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 1613 2 of 18

The maximum vertical wheel-rail contact force is increased by wheel OOR, leading to fatigue and reducing the life span of the track and vehicle components, such as wheelset bearings, gearbox, bolts, fastenings, and sleepers [4]. Trained workers are required to analyze wheel polygons by means of a roughness-measuring system such as m|Wheel, which is time-consuming and not particularly precise. Thus, it is essential to develop procedures for real-time, on-line detection of OOR and to find suitable countermeasures. Different approaches for detecting OOR have been reported, including vibration-impact measurement, displacement determination, image and ultrasonic telemetering, laser sensing, and noise detection. For example, Lutzenberge and Wu [5] discussed the Müller-BBM wheel monitoring system allowing for the detection of wheel tread damages. The system, with sensors installed at several track sections to record rail vibration, can identify wheel flats or polygons of all approaching vehicles in an early stage. Much research has been devoted to detect wheel defects based on wheel-rail interaction and track response [6–8]. Lee and Chiu [9] compared three methods of predicting impact force on a railway track-like structure using both finite element and experimental techniques. The results indicated that measuring strain was an effective method of reconstructing dynamic impact force, but that efficacy would be reduced as the impact force increased. Wei et al. [10] described a real-time out-of-roundness wheel-defects monitoring system based on track strain response by using Fiber Bragg Grating sensors. Wayside monitoring systems based on acoustic [11] and ultrasonic [12] sensors have also been employed to detect wheel conditions. However, these sensors can be influenced by electromagnetic interference under a railroad environment. Besides, polygonal-wheel or local spall-wheel defects cannot be detected. Sensors based on Laser-PSD (Position Sensing Detector) technology to measuring out-of-roundness wheel set on-line was reported in [13]. The system can determine the size of wheel out-of-roundness by detecting the displacement of a light spot in PSD based on track dynamics. However, the Laser-PSD sensor requires a precision setup, which is difficult in practice. For most of these measurement systems, the sensors are positioned on rails or sleepers, mainly for OOR detection at speeds below 200 km h 1. · − Vehicle vibrations caused by wheel and track defects affect axle-box accelerations, so some research has been done on using axle-box accelerations to identify short track defects, such as squats, rail spalling, rail corrugation, and track irregularities in several wavelength ranges, and also for more general track surveying. For example, in the late 1990s Coudert [14] described an easy-to-measure method to detect track irregularity of short wavelength by using axle-box acceleration. Tanaka and Furukawa [15] reported a cheap and robust method for estimating wheel load and lateral force caused by short-wavelength track irregularity based on axle-box acceleration measurement, which is difficult to detect with a track inspection car. In particular, they analyzed the relationship between the wheel load, lateral force, and axle-box acceleration of the Shinkansen train. The study revealed that the maximum values of the wheel load and lateral force can be estimated by the use of axle-box acceleration. Molodova et al. [16] discussed the application of axle-box acceleration measurements to assess squats and insulated joints, and determined the quantitative relationship between track defects and the characteristics of excited axle-box acceleration through a finite element model. Salvador [17] detected track defects and track singularities with the aid of axle-box acceleration measurement, and did testing on the Metropolitan Rail Network of Valencia, Spain. The results determined the location of accelerometers on the train and identified track singularities and diverse track defects. However, little research has been reported on the quantitative correlation between wheel polygons and axle-box acceleration. Although Li et al. [18] used axle-box acceleration to detect wheel flats and other wheel damage, they could not assess the damage level of polygonized wheels. One promising, low cost, and relatively simple technique to determine wheel conditions is by detecting axle-box acceleration, so detecting axle-box accelerations for high-speed trains and their time-frequency characteristics merits further study to develop a method to detect wheel polygons. To detect wheel defects based on axle-box acceleration signal analysis, applying appropriate vibration signal processing techniques is essential to determine the state of wheel polygons. Axle-box Appl. Sci. 2020, 10, 1613 3 of 18 acceleration signals caused by polygonized wheels are non-stationary and nonlinear. In the past decades, researchers have proposed several random vibration signal processing approaches, such as short-time Fourier transform (STFT), wavelet transform (WT), Hilbert-Huang transform (HHT), and the Wigner-Ville distribution (WVD). However, the constant time and frequency resolution of STFT make it unsuitable for analyzing a non-stationary signal. In contrast, the time and frequency resolutions of WT are adaptive. For example, Caprioli et al. [19] offered a wavelet survey applied for track diagnosis based on acceleration signal analysis of a running train, discussed the advantages and drawbacks of the discrete wavelet transform, the continuous wavelet transform and the wavelet packets, with respect to the classical Fourier analysis. Jia and Dhanasekar [20] proposed the wavelet local energy averaging approach and average signal wavelet decomposing application for rail wheel flats onboard detection using train acceleration signals. The numerical simulation results demonstrated that the proposed method was effective for monitoring localized rail wheel flats with sizes smaller than predefined threshold. Nevertheless, it is difficult to choose the mother wavelet at present. HHT is an adaptive method for time-frequency analysis, including empirical mode decomposition (EMD) and Hilbert transformation. Li et al. [18] utilized HHT based method to analyze axle-box vibration signal characteristics induced by railway out-of-round wheels. An energy principle algorithm was proposed to suppress mode-mixing phenomenon. The study results show that the revised HHT can be adopted to distinguish out-of-round wheels from normal wheels. However, the mode-mixing problem caused by the intermittent signal in the sifting EMD process still requires attention. Wigner-Ville distribution (WVD) is an ideal approach for a three dimensional representation of non-stationary signals in the time-frequency-amplitude domain, which is effective for monitoring machinery under non-stationary conditions [21]. Liang et al. [22] presented a mixed approach of adaptive noise cancelling (ANC) and time–frequency signal analysis application for detection of wheel flat and rail surface defect. The experimental results show that the combination of ANC technique and WVD can extract weak signals and effectively suppress noises for fault diagnosis. Although the WVD is a valuable tool for time-frequency signal analysis, the cross-term interference problem is the main limitation for its application [23]. Based on the above discussion, a joint time-frequency method based on improved EMD (called the “ensemble EMD”, or EEMD) and the WVD is proposed to extract characteristics from axle-box acceleration signals. This approach minimizes cross-term interferences and permits an adaptive time-frequency-amplitude representation. The aim of this paper is thus to present an analysis of axle-box vibration acceleration using the joint time-frequency method to determine the essential information relating to polygon wheel defects. The first step of this study is to provide quantitative theoretical analysis of the relation between axle-box accelerations and polygonal wheel defects. Then follows the simulation of dynamic responses of axle-box acceleration due to wheel polygons. Based on the theory of vehicle-track coupling dynamics, a dynamic numerical train-track coupling model is developed that considers both elastic wheelsets and elastic tracks implemented in Universal Mechanism (UM) and finite-element analysis (ANSYS) software. The results form a basis for detecting various polygon wheel defects for high-speed railways. Finally, the signal processing of axle-box acceleration signals is performed based on improved EMD and WVD joint time-frequency approaches.

2. Theoretical Analysis Figure1 illustrates a parameter-lumped simple wheel-track model that considers wheel OOR to analyze the axle box vibration caused by wheel polygons. In this model, v is the vehicle velocity, m1 is the unsprung mass, z1 is the vertical displacement of the wheelset, k1 is related to the vertical stiﬀness of the primary spring, m2 is the mass of the track, k2 is the vertical stiﬀness of the track, c2 is the track damping, and z2 is the vertical displacement of the track. The downward direction is positive. Appl. Sci. 20202020, 10, x 1613 FOR PEER REVIEW 44 of of 1819

Train speed v

Carbody

Unsprung mass m1 Wheel out-of-roundness k1 z1-a

Rail m2 k2 c2 z2 Foundation

Figure 1. ParameterParameter-lumped-lumped simple wheel wheel-track-track model after introducing wheel out-of-roundness.out-of-roundness.

Based on conservation of energy, a track may be conceptually modeled as an infiniteinfinite beam with dampers on an elastic foundation. The track track model model allows allows for for deflections deflections and and forces forces in in the the zz direction.direction. The parameters parameters of of the the elastic elastic foundation foundation beam beam are converted are converte into equivalentd into equivalent mass, equivalent mass, equivalent damping, anddamping, an equivalent and an equivalent spring attached spring toattached each wheel. to each Forwheel. a train For witha train a polygonalwith a polygonal wheel wheel of a given of a OORgiven amplitudeOOR amplitudea rolling a alongrolling the along track the at speedtrack atv, thespeed vertical v, the dynamic vertical equationsdynamic equations for the wheel-rail for the system are wheel-rail system are ( .. m1z1 + k1(z1 a z2) = 0  .. m. 1z1  k1 z1 − a − z2   0 (1) m2z2 + c2z2 + k2z2 k1(z1 a z2) = 0 (1) m2 z2  c2 z2  k2 z2− k1 z1 − a −z2   0 The solution to the numerical ordinary differential Equation (1) is The solution to the numerical ordinary differential Equation (1) is  ..   z1z+ z z = a  ω2 1  1z  z2  a  .. 1 2 1− 2  .  (2)  z2 c2 1 k1+k2   2 + k z2 + k z2 z1 = a (2) ωz2 c1  k 1 k  − −  2  2 z   1 2 z  z  a 2 2   2 1 2 k..1  k1  where ω1 = √k1/m1, ω2 = √k1/m2, and z1 is the axle-box acceleration. The initial irregularity of the wheel polygon is defined by a harmonic displacement function where 1  k1 m1 , 2  k1 m2 , and z1 is the axle-box acceleration. The initial irregularity of the wheel apolygon(t) = a sin is (definedωt + ϕ) by a harmonic displacement function (3) at  asint  (3) where a(t) is the time-dependent OOR amplitude, ω is the frequency corresponding to the harmonic orderswherea of(t) theis the OOR, time and-dependentϕ is the phase OOR angleamplitude, between ω is the the two frequency wheels oncorresponding the same axle. to the harmonic ordersThe of Laplacethe OOR, transform and  is of the Equation phase angle (3) is between the two wheels on the same axle. The Laplace transform of Equation (3) is aa(sssinsinθ +ωcoscosθ) a(sa)s=  (4(4)) s22 +ω22 .. To obtain transfer functions of z1and induced by at, we use To obtain transfer functions of z1 and z1 induced by a(t), we use  z s  H sas ( 1 z1 z1(s) = Hz1(s)a(s) (5) z2 s  H z2 sas (5) z2(s) = Hz2(s)a(s) where z1(s) and z2(s) are the Laplace transforms for z1 and z2, respectively. H s1 s and H s2 s are the wheretransferz1 functions(s) and z2 (correspondings) are the Laplace to z transforms1 and z2 under for excitationz1 and z2, respectively.. Hs1(s) and Hs2(s) are the ( ) transferWhen functions the initial corresponding state is zero, to wez1 getand thez2 underLaplace excitation transforma tof. Equation (2): When the initial state is zero, we get the Laplace transform of Equation (2):   s 2    1z1 s z2 s  as  2 2    s +1  ( ) ( ) = ( )   2 1 z1 s z2 s a s (6)  ω1 2 −  s c2 s k2  (6)  z s  s2 c s k 1z s  a s  (1 )   +2 2 + 2 +  2( ) = ()  z1 s 2 k k kk 1 z2 s a s  − ω2 2 1 1 1 1  Using Equation (5) in Equation (6) gives Appl. Sci. 2020, 10, 1613 5 of 18

Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 18 Using Equation (5) in Equation (6) gives 2 s 2 c2 s k2 s + c2s + k2 2 + + ωω2 kk1 kk1 ()= 22 1 1 H z1 s Hz1(s) = 4 3 (7)(7) s 4 cc ss3  1 kk 1  cc ss kk s + 22 + 1 +  22 +11 1 s22 + 22 + 22 22 2 k 2  2 k1 22  k1 k1 ωω1ωωs 1ωω1 ωω2   ωω1 1 s k1 1  2  k1  1  k1 k1 .. When the initial state is zero, the Laplace transform of z1 is When the initial state is zero, the Laplace transform of z1 is

()= 2 ()= 2 ()() z1 s s z1 s s H z1 s a s 4 c s 3 k s 2 s + 2 + 2 ω 2 k k ()θ + ω θ = 2 1 1 ⋅ a ssin cos (8) 2 2 s 4 c s 3  1  k  1  c s k s + ω + 2 + +  2 +1 s 2 + 2 + 2 ω 2ω 2 ω 2 ω 2   ω 2  1 2 k1 1  2 k1 1  k1 k1     (8) 1 c  1  k  1  c k 1 c2 2 1 k2+  2 +1  c2 2 k2 2 Because 2 2 ,, , 2 , + 2 + 1 1 , 2 , and, , andare all positive,are all positive, the Routhcriterion the Routhcriterion is satisfied, is ωω2ωω2 k ω2ω ω2 ω k1  ω2 ωk1 k1 11 2 1k11 1 2  2  k1 1  1  k1 k1 and the characteristic equation becomes satisfied, and the characteristic equation becomes 4 3  !  s 4 c2s 3 1 k2 1  2 c2s k2 s + c2 s +  1 +  k2 + 1 1 s 2 +c2 s +k2 = 0 (9) 2 2 + 2 + 2 + k +1 2s + k + k = 0 (9) ω1ω2ω 2 k1ωω1 2 ωω22  1  ωω21 1 1 1 2 k1 1  2  k1  1  k1 k1 The necessary and sufficient condition for all roots having negative real parts is The necessary and sufficient condition for all roots having negative real parts is X44 A + .. ()= Aii + BsBs +C C z (zs1)s= + (10) 1  + 2 2+ω 2 2 (10) i=1 s +ppii ss + ω i=1 where Ai, pi, B, and C are all constants related to ω1,ω2, c2, k1, k2, and ω. where Ai, pi, B, and C are all constants related to ω1, ω2, c2, k1, k2, and ω. Thus, the inverse Laplace transformation is given by 4 − z =4 Ae pit + Dcos()ωt +ϕ .. 1 X i (11) = pit z1 = i 1Aie− + D cos(ωt + ϕ) (11) i= For Equation (11), the first item is an exponential1 attenuation function independent of frequency, << and, afterFor Equation a long time (11), ( p theit first1), item the second is an exponential term gives attenuationz1 for zero phase function angle independent between the of two frequency, wheels .. onand, the after same a long axle time(φ = (0).pit << 1), the second term gives z1 for zero phase angle between the two wheels on theBased same on axle Equation (ϕ = 0). (11), it can be concluded that the frequency of the axle-box acceleration z .. 1 is theBased same onas the Equation characteristic (11), it canfrequency be concluded of the wheel that the polygon, frequency which ofthe shows axle-box that it acceleration is reasonablez1 tois analyzethe same and as theidentify characteristic wheel polygon frequency defects of thebased wheel on axle-box polygon, vibration. which shows that it is reasonable to analyze and identify wheel polygon defects based on axle-box vibration. 3. Modeling and Signal Analysis Method 3. Modeling and Signal Analysis Method 3.1. Vehicle-Track Rigid-Flexible Coupling Dynamics Model 3.1. Vehicle-Track Rigid-Flexible Coupling Dynamics Model In order to simulate the dynamic responses of axle-box acceleration to polygonal wheel defects, In order to simulate the dynamic responses of axle-box acceleration to polygonal wheel defects, consider as an examplea high-speed electric multiple unit vehicle and a China Railways Track consider as an examplea high-speed electric multiple unit vehicle and a China Railways Track Structure Structure (CRTS) II slab ballastless track of the Chinese high-speed railway lines [24]. Based on the (CRTS) II slab ballastless track of the Chinese high-speed railway lines [24]. Based on the mechanical mechanical model of the vehicle-track coupled system shown in Figure 2a, a numerical model of a model of the vehicle-track coupled system shown in Figure2a, a numerical model of a vehicle-track vehicle-track coupling system was developed, representing a passenger car running on a slab track. coupling system was developed, representing a passenger car running on a slab track. The simulation The simulation is implemented by using both Universal Mechanism (UM) and ANSYS software. is implemented by using both Universal Mechanism (UM) and ANSYS software. The vehicle model consists of one rigid car body, two rigid bogies, and four elastic wheelsets. The car body, bogies, and wheelsets are connected by linear springs and viscous dampers representing the primary and secondary suspensions. The vehicle can have vertical, lateral, and longitudinal motion. The polygonized wheels are suspected of producing vibrationsin the middle- and high-frequency ranges [25]. Therefore, a finite-element model in the ANSYS environment is used to calculate the model wheelset parameters by treating the wheelsets as flexible bodies. The wheel Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 19

we transform the CRTS II slab ballastless track finite-element model into UM. The finite-element model of the CRTS II slab ballastless track is composed of two CHN60 rails, a concrete slab layer, a concrete-asphalt (CA) mortar filling layer, and afoundation, which are all represented by three- dimensional solid elements in ANSYS. The fasteners connecting the rails and the concrete slab layer are modeled as spring elements. The connections between the three layers (i.e., concrete slab layer, CA mortar layer, and the foundation) are modeled as contact elements. Figure 2b shows an enlarged view of the flexible wheelsets and tracks, and Table 1 lists the parameters of the vehicle and track systems [24]. The vertical wheel-rail contact forces are calculated by using the Kik-Poitrowski model, which considers the nonlinear contact mechanics between wheel and rail. Creep and tangential forces in the Appl.contact Sci. 2020 area, 10 are, 1613 calculated by using Kalker’s linearized theory. 6 of 18 The vehicle-track rigid-flexible coupling dynamics model is validated in [26].

Train speed v

Carbody

Bogie

Rail Slab layer CA layer

Foundation

(a)

(b)

FigureFigure 2. 2. Vehicle-trackVehicle-track rigid-ﬂexible rigid-flexible coupling coupling dynamicsdynamics system system for for UM-ANSYS UM-ANSYS simulation:( simulation:a(a)) MechanicalMechanical model model of of train-vehicle train-vehicle coupling coupling system; system (;b ()b Enlarged) Enlarged view view of of ﬂexible flexible wheelsets wheelsets and and tracks. tracks.

The vehicle model consists of one rigid car body, two rigid bogies, and four elastic wheelsets. The car body, bogies, and wheelsets are connected by linear springs and viscous dampers representing the primary and secondary suspensions. The vehicle can have vertical, lateral, and longitudinal motion. The polygonized wheels are suspected of producing vibrationsin the middle- and high-frequency ranges [25]. Therefore, a finite-element model in the ANSYS environment is used to calculate the model wheelset parameters by treating the wheelsets as flexible bodies. The wheel tread is of LMA worn type, and the wheel polygonsaredescribed by sinusoidal functions to determine the harmonic deviation of the wheel radius from a constant value [4]. For the flexible-track model, we transform the CRTS II slab ballastless track finite-element model into UM. The finite-element model of the CRTS II slab ballastless track is composed of two CHN60 rails, a concrete slab layer, a concrete-asphalt (CA) mortar filling layer, and afoundation, which are all represented by three-dimensional solid elements in ANSYS. The fasteners connecting the rails and the concrete slab layer are modeled as spring elements. The connections between the three layers (i.e., concrete slab layer, CA mortar layer, and the foundation) are modeled as contact elements. Figure2b shows an enlarged view of the flexible wheelsets and tracks, and Table1 lists the parameters of the vehicle and track systems [24]. Appl. Sci. 2020, 10, 1613 7 of 18

Table 1. Parameters of vehicle and track systems.

Component Parameter Value Mass (kg) 4.0 104 × Moments of pitch inertia (kg m2) 2.0 106 Carbody · × Moments of roll inertia (kg m2) 1.0 105 · × Moments of yaw inertia (kg m2) 2.0 106 · × Mass (kg) 2.0 103 × Moments of pitch inertia (kg m2) 2.5 103 Bogie · × Moments of roll inertia (kg m2) 1.5 103 · × Moments of yaw inertia (kg m2) 3.5 103 · × Mass (kg) 1.5 103 × Moments of pitch inertia (kg m2) 120 · Wheelset Moments of roll inertia (kg m2) 800 · Moments of yaw inertia (kg m2) 800 · nominal rolling radius (m) 0.46 Mass (kg) 50.0 Moments of pitch inertia (kg m2) 5.0 Axlebox · Moments of roll inertia (kg m2) 1.0 · Moments of yaw inertia (kg m2) 5.0 · Elastic modulus (N m 2) 2.1 1011 · − × CHN60 rail Poisson ratio 0.3 Density(kg m 3) 7800 · − Elastic stiﬀness(MN m 1) 50 · − Rail fastenings Damping coeﬃcient (kN s m 1) 60 · · − Longitudinal spacing(m) 0.65 Elastic modulus(N m 2) 3.9 1010 · − × Poisson ratio 0.2 Concrete slab layer Length width thickness (m) 6.45 2.55 0.20 × × × × Density(kg m 3) 2500 · − Elastic modulus(N m 2) 7.0 109 · − × Poisson ratio 0.167 CA mortar layer Thickness(m) 0.03 Density(kg m 3) 2590 · − Elastic modulus(N m 2) 5.0 109 · − × Poisson ratio 0.2 Foundation Width thickness(m) 3.25 0.3 × × Density(kg m 3) 2500 · −

The vertical wheel-rail contact forces are calculated by using the Kik-Poitrowski model, which considers the nonlinear contact mechanics between wheel and rail. Creep and tangential forces in the contact area are calculated by using Kalker’s linearized theory. The vehicle-track rigid-ﬂexible coupling dynamics model is validated in [26].

3.2. Joint Time-Frequency Method based on EEMD and WVD In order to investigate the simulated axle-box acceleration signals in the time-frequency domain, an efficient algorithm based on EEMD and WVD is proposed: by decomposing a noise-assisted multi-component signal into a number of intrinsic mode functions (IMFs), and then analyzing by using the Wigner-Ville transformation, one cannot only suppress cross-term interference but also exploit the advantages of the WVD. Figure3 shows a flowchart describing the methodology to identify wheel polygons. First, to remove additive noise or unwanted signals from the raw vibration signals, adaptive noise cancellation [22] is Appl. Sci. 2020, 10, 1613 8 of 18 used to pre-process the raw signal. Vibration signals caused by rail irregularities are then filtered and noise is reduced by using an adaptive filter.

Figure 3. Flow chart showing algorithm for joint time-frequency analysis based on Ensemble Empirical Mode Decomposition (EEMD) and Wigner-Ville distribution (WVD). Appl. Sci. 2020, 10, 1613 9 of 18

Next, add Gaussian white noise to the original vibration signal x(t) and decompose the conditioned signal by EEMD into a series of IMFs and a residual:

Xn x(t) = ci(t) + rn(t) (12) i=1 where ci(t) is an IMF, i is the number of modes, and rn(t) is the corresponding residual. The stopping criteria for terminating the sifting process in Figure3 are: (1) that the number of extrema and the number of zero-crossings must differ at most by one, and (2) that the mean between the upper and lowerAppl. Sci. envelopes 2020, 10, x canFOR bePEER considered REVIEW to be zero. The disadvantage of EEMD is that the signatures overlap9 of 19 in both time and frequency because the extremes identify x(t). The correlation coefficient method [25] overlap in both time and frequency because the extremes identify x(t). The correlation coefficient is thus used to eliminate false IMFs by a sifting process. method [25] is thus used to eliminate false IMFs by a sifting process. Third, the remnant IMFs are analyzed by using the WVDas follows: Third, the remnant IMFs are analyzed by using the WVDas follows:  Z *  jf WVDx t, f   ∞ ct  2c x  / 2e dj f τ (13) WVDx(t, f ) = c(t + τ/2)c∗(x τ/2)e− dτ (13) − where f is the frequency,τis the integration −∞variable, and the asterisk (*)indicates complex conjugation. whereFinally,f isthe the WVD frequency,s are summedτ is the integrationtogether to variable,reconstruct and the the WVDs asterisk of (*)indicates the original complex signal: conjugation. Finally, the WVDs are summed together to reconstruct the WVDs of the originaln signal: WVDx t, f   WVDC1 t, f WVDC 2 t, f WVDCn t, f   WVDCi t, f  (14) i1Xn WVDx(t, f ) = WVDC1(t, f ) + WVDC2(t, f ) + ...... + WVDCn(t, f ) = WVDCi(t, f ) (14) 4. Results and Discussions i=1 4. Results and Discussions 4.1. Results of Axle-Box Vibration Acceleration Caused by Wheel Polygons in Time-Domain 4.1. Results of Axle-Box Vibration Acceleration Caused by Wheel Polygons in Time-Domain 4.1.1. Effect of Wheel Polygon 4.1.1. Effect of Wheel Polygon In the section, we present some numerical results showing how the axle-box vibration is related to typesIn the of section,wheel polygons, we present and some examine numerical the time results dependence showing how of the the vertical axle-box acceleration vibration is of related the axle to- typesbox induced of wheel by polygons, wheel polygons, and examine as derived the time from dependence the numerical of the model vertical in accelerationSection 3.1. of the axle-box inducedFigure by wheel 4 shows polygons, the time as derived dependence from of the the numerical vertical model axle-box in Section acceleration 3.1. caused by wheel polygonsFigure of4 showsvarious the harmonic time dependence orders and of the OOR vertical amplitudes axle-box accelerationat a vehicle causedspeed byof wheel350 km·h polygons−1. The ofharmonic various harmonicorders of ordersthe polygonal and OOR wheels amplitudes range atfrom a vehicle 1 and speed25 (i.e., of the 350 polygo km h 1nized. The wheels harmonic have orders 1 to · − of25 thewavelengths polygonal around wheels the range circumference from 1 and 25of (i.e.,the wheel). the polygonized The OOR wheelsamplitudes have range 1 to 25 from wavelengths 0.01 mm aroundto 0.12 mm[ the circumference27]. of the wheel). The OOR amplitudes range from 0.01 mm to 0.12 mm [27].

Figure 4. Vertical acceleration of axle-box as a function of (a) out-of-roundness (OOR) amplitude and (Figureb) harmonic 4. Vertical order acceleration of polygonized of axle wheel-box atas 350a function km h 1 .of (a) out-of-roundness (OOR) amplitude and · − (b) harmonic order of polygonized wheel at 350 km·h−1.

Figure 4a shows that the axle-box acceleration increases approximately linearly with OOR amplitude, with the slope for the highest-order polygonized wheel being several times greater than that for the lowest-order polygonized wheel. Table 2 shows the increase of axle-box acceleration per unit OOR amplitude (m·s−2·mm−1) for the first 25 orders of polygonized wheels. For example, when the velocity is 200 km·h−1, the growth in axle-box acceleration per unit OOR amplitude for the 25th- order polygonized wheel is 1421 m·s−2·mm−1, which is 7.2 times the growth per unit OOR amplitude for a 4th-order polygonized wheel. The growth rate increases gradually as the speed increases, reaching 285 m·s−2·mm−1and 2115 m·s−2·mm−1for the 4th and 25th orders, respectively, at a speed of 350 km·h−1, which is 1.44 and 1.49 times greater than at 200 km·h−1. When the vehicle speed is 350 km·h−1, the growth per unit OOR amplitude of the axle-box acceleration with OOR amplitude for the Appl. Sci. 2020, 10, 1613 10 of 18

Figure4a shows that the axle-box acceleration increases approximately linearly with OOR amplitude, with the slope for the highest-order polygonized wheel being several times greater than that for the lowest-order polygonized wheel. Table2 shows the increase of axle-box acceleration per unit OOR amplitude (m s 2 mm 1) for the ﬁrst 25 orders of polygonized wheels. For example, when the · − · − velocity is 200 km h 1, the growth in axle-box acceleration per unit OOR amplitude for the 25th-order · − polygonized wheel is 1421 m s 2 mm 1, which is 7.2 times the growth per unit OOR amplitude for a · − · − 4th-order polygonized wheel. The growth rate increases gradually as the speed increases, reaching 285 m s 2 mm 1 and 2115 m s 2 mm 1 for the 4th and 25th orders, respectively, at a speed of 350 km h 1, · − · − · − · − · − which is 1.44 and 1.49 times greater than at 200 km h 1. When the vehicle speed is 350 km h 1, the · − · − growth per unit OOR amplitude of the axle-box acceleration with OOR amplitude for the 23rd-order polygonized wheels is 2034 m s 2 mm 1, which is 4.9 times that for a 6th-order polygonized wheel. · − · − Thus, high-order polygonized wheels more strongly aﬀect the axle-box vibration than do low-order polygonized wheels.

Table 2. Growth per unit OOR amplitude of axle-box acceleration due to polygonized wheels, in m s 2 mm 1. · − · − Train Speed(km h 1) · − 200 225 250 275 300 325 350 Harmonic Order Growth Rate (m s 2 mm 1) · − · − 1 108 122 156 170 192 233 242 2 123 155 171 184 224 248 261 3 173 199 214 225 246 255 275 4 198 206 216 229 234 267 285 5 205 221 242 256. 268 294 327 6 213 222 266 301 315 336 415 7 251 258 298 336 355 389 452 8 295 310 316 378 406 446 510 9 324 337 340 434 446 480 542 10 349 398 394 471 465 556 642 11 374 438 456 519 599 675 757 12 426 472 493 651 749 788 936 13 472 519 583 708 872 936 1100 14 491 566 635 804 958 1091 1257 15 520 645 793 904 1009 1215 1447 16 553 672 883 940 1020 1264 1452 17 580 698 898 1063 1090 1289 1475 18 626 755 938 1196 1211 1345 1602 19 744 870 1071 1296 1302 1468 1699 20 829 1012 1240 1450 1483 1619 1832 21 996 1180 1350 1498 1511 1692 1899 22 1023 1354 1467 1611 1685 1721 1975 23 1155 1413 1556 1756 1792 1821 2034 24 1302 1522 1621 1796 1803 1875 2089 25 1421 1611 1743 1882 1901 1968 2115

Figure4b shows that the axle-box acceleration increases with increasing harmonic order and that the growth per harmonic order (m s 2) increases substantially with increasing OOR amplitude. · − Table3 lists the growth per harmonic order of axle-box acceleration for di ﬀerent OOR amplitudes. These results reveal that, for anOOR amplitude in the range 0.01 to 0.12 mm, the growth per harmonic order increases from 0.38 to 6.12 m s 2, at 200 km h 1, and from 1.05 to 10.22 m s 2 at 350 km h 1. It’s · − · − · − · − concluded that the growth per harmonic order of axle-box acceleration due to polygonized wheels increases gradually with increasing velocity at all OOR amplitudes. Appl. Sci. 2020, 10, 1613 11 of 18

Table 3. Growth per harmonic order of axle-box acceleration due to polygonized wheels, in m s 2 order 1. · − · − Appl. Sci. 2020, 10, x FOR PEER REVIEW Speed(km h 1) 11 of 19 · − 0.05 200 2252.94 3.12 2503.52 3.78 275 4.40 4.98 300 5.28 325 350 OOR Amplitude (mm) 0.06 3.22 3.67 4.49Growth 5.28 Rate 5.80 (m s 5.892) 6.24 · − 0.010.07 0.38 0.454.06 4.38 0.624.87 5.58 0.75 6.34 6.92 0.97 7.13 0.99 1.05 0.020.08 0.63 1.154.43 4.58 1.315.25 5.90 1.45 6.60 7.05 1.58 7.78 1.64 1.97 0.030.09 1.51 1.755.01 5.73 1.826.29 7.08 1.97 7.84 8.20 2.17 8.68 2.64 3.73 0.040.10 2.01 2.435.45 6.25 2.556.98 7.55 2.75 8.11 8.86 3.12 9.35 3.20 4.14 0.050.11 2.94 3.125.84 6.77 3.527.11 7.89 3.78 8.76 9.13 4.40 9.87 4.98 5.28 0.060.12 3.22 3.676.12 7.31 4.497.94 8.34 5.28 9.15 9.94 5.80 10.22 5.89 6.24 0.07 4.06 4.38 4.87 5.58 6.34 6.92 7.13 The above0.08 analysis indicates 4.43 that t 4.58he axle-box 5.25 vibrations 5.90 caused by 6.60 polygonized 7.05 wheels 7.78 are almost the0.09 largest in the entire 5.01 vehicle 5.73 system, a 6.29result attributed 7.08 to the 7.84 axle-box 8.20 being directly 8.68 connected0.10 without damping to 5.45 the axle through 6.25 a bearing. 6.98 On 7.55 Chinese high 8.11-speed trains, 8.86 the axle 9.35- 0.11 5.84 6.77 7.11 7.89 8.76 9.13 9.87 box and its related parts commonly suffer from vibration fatigue, which destroys end covers and 0.12 6.12 7.31 7.94 8.34 9.15 9.94 10.22 loosens bolts.

4.1.2.The aboveEffect of analysis Vehicle indicates Speed that the axle-box vibrations caused by polygonized wheels are almost the largestFigure in the 5 shows entire the vehicle maximum system, vertical a result axle-box attributed accelerations to the as a axle-box function of being OOR directly amplitude connected for withouta polygonal damping wheel to thewith axle four through waves and a bearing.with 24 waves On Chinese around its high-speed circumference trains, and thefor train axle-box speeds and its relatedfrom parts 200 to commonly 350 km·h−1 su. Figureffer from 6 shows vibration the calculated fatigue, axle which-box destroysacceleration end as coversa function and of loosens harmonic bolts. order of polygonized wheel and for different vehicle speeds and an OOR amplitude of 0.10 mm. 4.1.2. EffectFigure of Vehicle5 shows Speed that the growth per unit OOR amplitude of the vertical axle-box acceleration increases monotonically with vehicle speed. The calculated results shown in Figure 6 indicate that theFigure maximum5 shows vertical the maximum axle-box acceleration vertical axle-box is linear accelerations in harmonic asorder a function of the polygonal of OOR amplitudewheel and for a polygonalthat this wheel growth with rate four also wavesincreases and with with increasing 24 waves train around speed. its However, circumference the harmonic and for order train has speeds 1 frommuch 200 to less 350 effect km h on− . the Figure growth6 shows rate the than calculated does the axle-box OOR amplitude. acceleration At high as a speeds, function the of OOR harmonic · orderamplitude of polygonized of polygonal wheel wheels and foris what different most affects vehicle the speeds axle-box and acceleration. an OOR amplitude of 0.10 mm.

FigureFigure 5. Vertical 5. Vertical axle-box axle-box acceleration acceleration as a function of of OOR OOR amplitude amplitude at different at diﬀerent speeds speeds for (a) for a (a) a 4th-order4th-order polygonized polygonized wheel wheel and and ( b(b)) a a 24th-order24th-order polygonized polygonized wheel. wheel.

Figure5 shows that the growth per unit OOR amplitude of the vertical axle-box acceleration increases monotonically with vehicle speed. The calculated results shown in Figure6 indicate that the maximum vertical axle-box acceleration is linear in harmonic order of the polygonal wheel and that this growth rate also increases with increasing train speed. However, the harmonic order has much less eﬀect on the growth rate than does the OOR amplitude. At high speeds, the OOR amplitude of polygonal wheels is what most aﬀects the axle-box acceleration. Appl.Appl. Sci.Sci.2020 2020,,10 10,, 1613 x FOR PEER REVIEW 12 ofof 1819

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 19

FigureFigure 6.6. Axle-boxAxle-box accelerationacceleration asas aa functionfunction ofof harmonicharmonic orderorder ofof polygonizedpolygonized wheelwheel (i.e.,(i.e., thethe numbernumber of wavelengthsFigure 6. Axle around-box acceleration the wheel) as for a severalfunction trainof harmonic speeds. order of polygonized wheel (i.e., the number of wavelengthsof wavelengths around around the thewheel) wheel) for for several several train train speeds. speeds. 4.2. Time-Frequency Analysis of Axle-Box Accelerations based on EEMD and WVD 4.2. Time4.2.- TimeFrequency-Frequency Analysis Analysis of Axle of Axle-Box-Box Accelerations Accelerations basedbased on EEMDEEMD and and WVD WVD In this section are presented some results of the calculated axle-box acceleration analyzed by using In thisIn sectionthis section are arepresented presented some some results results of of thethe calculated axle axle-box-box acceleration acceleration analyzed analyzed by by the EEMD-WVD combined method described in Section 3.2, and comparison with the results calculated using usingthe EEMD the EEMD-WVD-WVD combined combined method method described described inin Section 3.2, 3.2, and and comparison comparison with with the theresults results from WVD time-frequency representation. Figure7 shows the time domain acceleration of the axle-box calculatedcalculated from from WVD WVD time time-frequency-frequency representation. representation. Figure 77 s showshows the the time time domain domain acceleration acceleration inducedof the by axle a 24th-order-box induced wheel by a polygon24th-order with wheel OOR polygon amplitude with OOR of amplitude 0.02mmand of 0.02mmand a train speed a train of 300 of the1 axle-box induced by a 24th-order wheel polygon with OOR amplitude of 0.02mmand a train km h−speed. Figure of 3008 shows km·h−1 the. Figure time-frequency 8 shows the time spectrograms-frequency spectrograms of the acceleration. of the acceleration. The result The of the result WVD speed· of 300 km·h−1. Figure 8 shows the time-frequency spectrograms of the acceleration. The result time-frequencyof the WVD analysis time-frequency in Figure 8analysisa reveals in aFigure clear horizontal8a reveals a band clearat horizontal about 690 band Hz, whichat about corresponds 690 Hz, of the WVD time-frequency analysis in Figure 8a reveals a clear horizontal band at about 690 Hz, to the theoreticalwhich corresponds characteristic to the theoretical frequency ch ofaracteristic 691.9 Hz frequency caused by of the 691.9 24th-order Hz caused polygonized by the 24th- wheel,order as which corresponds to the theoretical characteristic frequency of 691.9 Hz caused by the 24th-order per thepolygonized wavelength-ﬁxing wheel, mechanism as f = v/ perλ = nv/ the2πR = 24 wavelength(300/3600-fixing)/(2 π 460 mechanism) = 691.9Hz. polygonized wheel, as per the × wavelength-fixing × mechanism However,f  v unwarranted/   nv 2πR  24 characteristic 300/ 3600/( information2π  460)  691 also.9Hz appears. However, at other unwarranted frequencies, characteristic which is due to f  v /   nv 2πR  24 300/ 3600/(2π  460)  691.9Hz . However, unwarranted characteristic the WVDinformation crossterm also and appears leads at to other an incorrect frequencies, estimation. which is Figuredue to8 theb shows WVD thecrosster time-frequencym and leads to analytic an spectrograminformationincorrect also produced estimation. appears by Figureat the other EEMD-WVD 8b frequencies, shows the combined time which-frequency is method. due analytic to the The WVD spectrogram most crosster prominent producedm and characteristic leads by the to an EEMD-WVD combined method. The most prominent characteristic frequency is again at 690 Hz, frequencyincorrect estimation.is again at 690 Figure Hz, which 8b shows represents the time information-frequency about analytic abnormal spectrogram vibration produced caused by by the the EEMDwhich-WVD represents combined information method. aboutThe most abnormal prominent vibration characteristic caused by the frequency 24th-order is wheel again polygon. at 690 Hz, 24th-orderFurthermore, wheel polygon. no other noticeable Furthermore, characteristic no other frequencies noticeable appear, characteristic unlike in frequenciesFigure 8a. These appear, analytic unlike which represents information about abnormal vibration caused by the 24th-order wheel polygon. in Figureresults8a. show These that analytic the EEMD results-WVD show combined that method the EEMD-WVD suppresses the combined crossterm method of the WVD suppresses but does the Furthermore, no other noticeable characteristic frequencies appear, unlike in Figure 8a. These analytic crosstermnot reduce of the the WVD time but-frequency does not resolution. reduce the time-frequency resolution. results show that the EEMD-WVD combined method suppresses the crossterm of the WVD but does not reduce the time-frequency resolution.

FigureFigure 7. Time-domain 7. Time-domain response response of verticalof vertical axle-box axle-box acceleration acceleration due due to to 24th 24th-order-order wheel wheel polygon polygon with OOR amplitude of 0.02mm at 300 km·h1 −1. with OOR amplitude of 0.02mm at 300 km h− . · Figure 7. Time-domain response of vertical axle-box acceleration due to 24th-order wheel polygon with OOR amplitude of 0.02mm at 300 km·h−1. Appl. Sci. 2020, 10, 1613 13 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 19

(a)

(b)

FigureFigure 8. Spectrogram 8. Spectrogram corresponding corresponding to the to signal the shownsignal inshown Figure in7 :( Figurea) WVD 7: time-frequency (a)WVD time-frequency analytic spectrogram;analytic spectrogram; (b) EEMD-WVD (b) EEMD time-frequency-WVD time- analyticfrequency spectrogram. analytic spectrogram.

FigureFigure9 shows 9 shows partial partial EEMD-WVD EEMD-WVD time-frequency-energy time-frequency-energy spectrograms spectrograms of of calculated calculated axle-box axle-box accelerationacceleration caused caused by by a 24th-ordera 24th-order wheel wheel polygon polygon with with OOR OOR amplitudes amplitudes varying varying from from 0.01 0.01 to to 0.12 0.12 mm.mm. Note Note that that the the largest-power largest-power amplitudes amplitudes in thein the spectrograms spectrograms appear appear around around 690 Hz,690 producedHz, produced by theby 24th-order the 24th-order wheel wheel polygons. polygons. The frequency The frequency band at band zero at amplitude zero amplitude is narrow is (690–700narrow (690 Hz),–700 which Hz), reﬂectswhich a reflects high frequency a high frequency resolution. resolution. In addition, In addition, the peak the amplitude peak amplitude (i.e., peak (i.e., acceleration) peak acceleration) of the of simulatedthe simulated axle-box axle increases-box increases with increasing with increasing OOR amplitudeOOR amplitude (see Figure (see Figure 10). These 10). These results results imply imply that thethat presence the presence of wheel of polygons wheel polygons is clearly is reﬂected clearly inreflected the spectrogram in the spectrogram as a narrow as band a narrow whose band frequency whose dependsfrequency on the depends harmonic on order the harmonic of the polygonized order of the wheels polygonized and on the wheels train speed. and on Furthermore, the train speed. the peakFurthermore, amplitudes the of the peak EEMD-WVD amplitudes time-frequency-energy of the EEMD-WVD time spectrogram-frequency are-energy approximately spectrogram linear are inapproximately the OOR amplitude linear of in the the polygonized OOR amplitude wheels. of Thus, the polygonized given the power wheels. amplitude Thus, given of the the axle-box power acceleration,amplitude theof the train axle speed,-box acceleration, and the harmonic the train order, speed, the OORand the amplitude harmonic of aorder, polygonized the OOR wheel amplitude can beof determined. a polygonized wheel can be determined. Appl. Sci. 2020, 10, 1613 14 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 19

(a) (b)

(e)

FigureFigure 9. EEMD-WVD 9. EEMD-WVD time-frequency-energy time-frequency-energy spectrograms spectrograms corresponding corresponding to the to axle-box the axle acceleration-box causedacceleration by 24th-order caused polygonizedby 24th-order wheels polygonized and for wheels diﬀerent and OORfor different amplitudes: OOR amplitudes (a) OOR amplitude: (a) OOR of0.04 mm; (b) OOR amplitude of0.06 mm; (c) OOR amplitude of0.08 mm; (d) OOR amplitude of0.10 mm; (e) OOR amplitude of0.12 mm. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 19

amplitude of0.04 mm; (b) OOR amplitude of0.06 mm; (c) OOR amplitude of0.08 mm; (d) OOR Appl. Sci. 2020, 10, 1613 15 of 18 amplitude of0.10 mm; (e) OOR amplitude of0.12 mm.

Figure 10. Peak power amplitude of axle-box acceleration due to 24th-order polygonal wheels as a Figure 10. Peak power amplitude of axle-box acceleration due to 24th-order polygonal wheels as a function of OOR amplitude. function of OOR amplitude. 4.3. Comparision with Other Author Results 4.3. Comparision with Other Author Results To verify the validity of the joint time-frequency analysis method, we used the proposed EEMD-WVDTo verify combined the validity method of the tojoint obtain time- thefrequency time-frequency analysis method, characterization we used ofthe vertical proposed axle-box EEMD- accelerationWVD combined caused bymethod areal polygonized to obtain the wheel, time as-frequency reported in characterization [27]. The results of (Figure vertical 11a) axle reveal-box threeacceleration thin horizontal caused by lines areal in thepolygonized spectrogram wheel, at the as characteristicreported in [27 frequencies]. The results of (Figure11a) 31, 710, and reveal 740 Hz.three Furthermore, thin horizontal axle-box lines in vibration the spectrogram caused by at athe random characteri polygonalstic frequencies wheel is of also 31, reﬂected710, and 740 in the Hz. time-frequency-energyFurthermore, axle-box spectrogram vibration caused (Figure by 11 b),a random with the polygonal highest amplitude wheel is occurringalso reflected around in the 740 time Hz, - thefrequency second highest-energy occurringspectrogram around (Figure 710 11b), Hz, andwith the the third highest highest amplitude occurring occurring around around 31 Hz. 740 These Hz, resultsthe second imply highest that the occu randomrring polygonized around 710 wheelHz, and is dominated the third highest by harmonics occurring 1, 23, around and 24 31 around Hz. These the wheelresults circumference, imply that the which random is consistent polygonized with wheel the ﬁeld is dominated measurements by harmonics reported in 1, [27 23,]. and 24 around the Aswheel mentioned circumference, above, Lietwhich al. is [18 consistent] based their with research the field on measurements detecting railway reported out-of-round in[27]. wheel defects on the Hilbert-Huang transform (HHT). However, the HHT cannot assess the severity of wheel defects. The analytic results presented herein illustrate that EEMD-WVD allows us to detect wheel polygons not only by suppressing the WVD crossterms but also by retaining the time-frequency concentration and resolution.

4.4. Limitation and Futherwork Although the numerical simulation has been performed considering the presence of track irregularity PSD of Chinese high-speed railway, there are many other nonlinear factors influencing the calculated axle-box acceleration responses, such as parameters of primary and secondary suspension systems, the stiffness and viscous damping parameters of the ballastless track model, and others. In this regard, experimental tests are under the plan to carry out on a real track, which allows considering the non-linearity of the vehicle-track coupled system and the environmental influences that cannot be easily predicted in the modeling process. Future work will involve measurements in the laboratory and on a running train to prove the effectiveness of the proposed wheel-polygon detection method based on axle-box vibration. This will be followed by the design and development of a real-time system to monitor the development of wheel polygons. Appl. Sci. 2020, 10, 1613 16 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 19

(a)

(b)

Figure 11. Results of EEMD-WVD analysis of axle-box acceleration caused by real polygonized Figure 11. Results of EEMD-WVD analysis of axle-box acceleration caused by real polygonized wheels [27]: (a) time-frequency spectrogram; (b) time-frequency-energy spectrogram. wheels [27]: (a) time-frequency spectrogram;(b) time-frequency-energy spectrogram. 5. Conclusions As mentioned above, Liet al. [18] based their research on detecting railway out-of-round wheel This paper proposes the use of axle-box acceleration to detect railway-wheel polygons of high-speed defects on the Hilbert-Huang transform (HHT). However, the HHT cannot assess the severity of trains. A dynamics model of a vehicle-track rigid-ﬂexible coupled system was designed to capture the wheel defects. The analytic results presented herein illustrate that EEMD-WVD allows us to detect dynamic features of axle-box acceleration associated with polygonized wheels. The time-frequency wheel polygons not only by suppressing the WVD crossterms but also by retaining the time- representation of axle-box accelerations obtained by using the EEMD-WVD combined method clearly frequency concentration and resolution. reveals the presence of polygonized wheels. The main conclusions are as follows: 4.4.(1) LimitationThe quantitative and Futherwork relationship between the characteristics of axle-box acceleration and wheel polygon defects is determined based on a parameter-lumped simple wheel-track model that Although the numerical simulation has been performed considering the presence of track accounts for wheel out-of-roundness. The frequency of the axle-box acceleration coincides with irregularity PSD of Chinese high-speed railway, there are many other nonlinear factors influencing the characteristic frequency of polygonized wheels, which underpins the detection of wheel the calculated axle-box acceleration responses, such as parameters of primary and secondary polygon defects based on axle-box vibration. suspension systems, the stiffness and viscous damping parameters of the ballastless track model, and others.(2) A In dynamic this regard, model experimental of a vehicle-track tests are under coupled the systemplan to withcarry ﬂexibleout on a wheelsetsreal track, which and tracks allows is consideringpresented the tonon determine-linearity the of relationshipthe vehicle-track between coupled wheel system polygon and defects the environmental and axle-box acceleration. influences that cannot be easily predicted in the modeling process. Appl. Sci. 2020, 10, 1613 17 of 18

The results indicate that the maximum amplitude of the vertical axle-box acceleration is linear in both the harmonic order and in OOR amplitude of the polygonal wheel. Furthermore, the growth rate of acceleration increases monotonically with vehicle speed. (3) The EEMD-WVD combined method is used to make a time-frequency analysis of the axle-box acceleration. The frequency and corresponding amplitude of axle-box acceleration are quantitatively related to wheel polygon defects. It’s concluded that EEMD-WVD can be applied for wheel polygons detection and severity assessment. (4) The analysis shows that axle-box acceleration measurements can be used to detect and assess polygon wheels. When the characteristic frequency f 1 ofaxle-box acceleration and the maxima magnitude aaxlebox at a given train speed v1 are determined, the wheel polygonal defects can be detected, provided f 1 does not coincide with the axle-box vibration frequency f 2 excited by round wheels. If f 1, f 2, the harmonic order n of the polygon wheel can be determined by comparing f 1 with the characteristic frequencies corresponding to the ﬁrst 25 harmonics of a polygon wheel at the train speed v1.Based on the peak amplitude and the frequency of axle-box acceleration from the joint time-frequency analysis, the degradation due to wheel polygon defects can be evaluated.

Author Contributions: Conceptualization, Y.S.; Formal analysis, L.L.; Writing—review & editing, Y.D. and B.S.; funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China, grant number11372199, Natural Science Foundation of Hebei Province, grant number E2019210152, and China Postdoctoral Science Foundation Funded Project, grant number 2018M643521. Acknowledgments: We acknowledge TopEdit LLC for the linguistic editing and proofreading during the preparation of this manuscript. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Ahlbeck, D.R. A study of dynamic impact load effects due to railroad wheel profile roughness. Veh. Syst. Dyn. 1988, 17, 13–16. [CrossRef] 2. Liu, J.; Han, J.; Xiao, X.B.; Liu, X.L.; Jin, X.S.; Wang, P. Influence of wheel non-circular wear on axle box cover abnormal vibration in high–speed train. Chin. J. Mech. Eng. 2017, 53, 98–105. [CrossRef] 3. Zhang, F.B.; Wu, P.B.; Wu, X.W. Effects of wheel polygonalization on axle box for high speed train. Zhendong Ceshi Yu Zhenduan 2018, 38, 1063–1068. 4. Johansson, A.; Andersson, C. Out–of–round railway wheels—A study of wheel polygonalization through simulation of three–dimensional wheel–rail interaction and wear. Veh. Syst. Dyn. 2005, 43, 539–559. [CrossRef] 5. Lutzenberge, S.; Wu, Q.L. Condition based maintenance of railway wheel treads and identification of the relevant mechanism with monitoring. In Proceedings of the 18th International Wheelset Congress, Chengdu, China, 7–10 November 2016. 6. Remington, P.; Webb, J. Estimation of wheel/rail interaction forces in the contact area due to roughness. J. Sound Vib. 1996, 1, 83–102. [CrossRef] 7. Attivissimo, F.; Danese, A.; Giaquinto, N.; Sforza, P. A railway measurement system to evaluate the wheel–rail interaction quality. IEEE Trans. Instru. Meas. 2007, 5, 1583–1589. [CrossRef] 8. Stratman, B.; Liu, Y.; Mahadevan, S. Structural health monitoring of railroad wheels using wheel impact load detectors. J. Fail. Anal. Prev. 2007, 3, 218–225. [CrossRef] 9. Lee, M.L.; Chiu, W.K. A comparative study on impact force prediction on a railway track–like structure. Struct. Health. Monit. 2005, 4, 355–376. [CrossRef] 10. Wei, C.; Xin, Q.; Chung, W.H.; Liu, S.-Y.; Tam, H.-Y.; Ho, S.L. Real–time train wheel condition monitoring by Fiber Bragg Grating sensors. Int. J. Distrib. Sens. Netw. 2011, 8, 409048. [CrossRef] 11. Papaelias, M.; Huang, Z.; Amini, A.; Vallely, P.; Day, N.; Sharma, R.; Kerkyras, Y.; Kerkyras, S. Advanced wayside condition monitoring of rolling stock wheelsets. In Proceedings of the 11th ECNDT, Prague, Czech, 6–10 October 2014. Appl. Sci. 2020, 10, 1613 18 of 18

12. Guagliano, M.; Pau, M. An experimental–numerical approach for the analysis of internally cracked railway wheels. Wear 2008, 9–10, 1387–1395. [CrossRef] 13. Yang, K.; Gao, X.R.; Dai, L.X. Research on the principle of railway wheel out-of-roundness on-line dynamic detecting system based on laser measurement. In Proceedings of the 2014 IEEE 11th Far East Forum on Nondestructive Evaluation/Testing: New Technology and Application, Chengdu, China, 20–23 June 2014. 14. Coudert, F.; Sunaga, Y.; Takegami, K. Use of axle box acceleration to detect track and rail irregularities. WCRR 1999, 7, 1–7. 15. Tanaka, H.; Furukawa, A. The estimation method of wheel load and lateral force using the axlebox acceleration. In Proceedings of the World Congress of Rail Research, Seoul, Korea, 22 May 2008. 16. Molodova, M.; Li, Z.; Dollevoet, R. Axle box acceleration: Measurement and simulation for detection of short track defects. Wear 2011, 271, 349–356. [CrossRef] 17. Salvador, P.; Naranjo, V.; Insa, R.; Teixeira, P. Axlebox accelerations: Their acquisition and time–frequency characterisation for railway track monitoring purposes. Meas. J. Int. Meas. Confed. 2016, 82, 301–312. [CrossRef] 18. Li, Y.F.; Liu, J.X.; Li, Z.J. The fault diagnosis method of railway out-of-round wheels using Hilbert–Huang transform. ZhendongCeshi Yu Zhenduan 2016, 36, 734–739. 19. Caprioli, A.; Cigada, A.; Raveglia, D. Rail inspection in track maintenance: A benchmark between the wavelet approach and the more conventional Fourier analysis. Mech. Syst. Signal Process. 2007, 2, 631–652. [CrossRef] 20. Jia, S.; Dhanasekar, M. Detection of rail wheel ﬂats using wavelet approaches. Struct. Health Monit. 2007, 2, 121–131. [CrossRef] 21. Shin, Y.S.; Jeon, J.J. Pseudo Wigner–Ville time-frequency distribution and its application to machinery condition monitoring. Shock Vib. 1993, 1, 65–76. [CrossRef] 22. Liang, B.; Iwnicki, S.; Ball, A.; Young, A.E. Adaptive noise cancelling and time–frequency techniques for rail surface defect detection. Mech. Syst. Signal Process. 2015, 54, 41–51. [CrossRef] 23. Wang, X.L. Detection of Out–Of–Roundness Wheels of Urban Rail Train Based on Improved Wigner–Ville Distribution. Master’s Thesis, Nanjing University of Science & Technology, Nanjing, China, 2017. 24. Xu, P.; Cai, C.B. Dynamic analysis of longitudinally connected ballastless track on earth subgrade. J. SouthwestJiaotong Univ. 2011, 46, 189–194. 25. Ripke, B.; Knothe, K. Simulation of high frequency vehicle—Track interactions. Veh. Syst. Dyn. 1995, 24, 72–85. [CrossRef] 26. Song, Y.; Du, Y.; Zhang, X.; Sun, B. Evaluating the eﬀect of wheel polygons on dynamic track performance in high–speed railway systems using co–simulation analysis. Appl. Sci. 2019, 9, 4165. [CrossRef] 27. Chen, M.; Zhai, W.M.; Ge, X.; Sun, Y. Analysis of wheel-rail dynamic characteristics due to polygonal wheel passing through rail weld zone in high-speedrailways. Chin. Sci. Bull. 2019, 64, 2573–2582. (In Chinese) [CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).