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Quantitative detection of locomotive wheel polygonization under non-stationary conditions by adaptive chirp mode decomposition

Quantitative detection of locomotive wheel polygonization under non-stationary conditions by... Rail. Eng. Science (2022) 30(2):129–147 https://doi.org/10.1007/s40534-022-00272-3 Quantitative detection of locomotive wheel polygonization under non-stationary conditions by adaptive chirp mode decomposition 1 1 1 1 1 1 • • • • • Shiqian Chen Kaiyun Wang Ziwei Zhou Yunfan Yang Zaigang Chen Wanming Zhai Received: 21 December 2021 / Revised: 10 February 2022 / Accepted: 17 February 2022 / Published online: 13 April 2022 The Author(s) 2022 Abstract Wheel polygonal wear is a common and severe After the rotating frequency is obtained, signal resampling defect, which seriously threatens the running safety and and order analysis techniques are applied to an acceleration reliability of a railway vehicle especially a locomotive. signal of an axle box to identify harmonic orders related to Due to non-stationary running conditions (e.g., traction and polygonal wear. Finally, the ACMD is combined with an braking) of the locomotive, the passing frequencies of a inertial algorithm to estimate polygonal wear amplitudes. polygonal wheel will exhibit time-varying behaviors, Not only a dynamics simulation but a field test was carried which makes it too difficult to effectively detect the wheel out to show that the proposed method can effectively detect defect. Moreover, most existing methods only achieve both harmonic orders and their amplitudes of the wheel qualitative fault diagnosis and they cannot accurately polygonization under non-stationary conditions. identify defect levels. To address these issues, this paper reports a novel quantitative method for fault detection of Keywords Wheel polygonal wear  Fault diagnosis  Non- wheel polygonization under non-stationary conditions stationary condition  Adaptive mode decomposition based on a recently proposed adaptive chirp mode Time–frequency analysis decomposition (ACMD) approach. Firstly, a coarse-to-fine method based on the time–frequency ridge detection and ACMD is developed to accurately estimate a time-varying gear meshing frequency and thus obtain a wheel rotating frequency from a vibration acceleration signal of a motor. 1 Introduction Wheel out-of-roundness, especially wheel polygonal wear, & Wanming Zhai is a very common type of wheel defect for a railway wmzhai@swjtu.edu.cn vehicle especially for a locomotive [1–4]. Wheel polygo- Shiqian Chen nization is a phenomenon that the wheel profile periodi- chenshiqian@swjtu.edu.cn cally deviates from an ideal round shape (i.e., periodic Kaiyun Wang radial deviation) along circumference [5]. The defect will kywang@swjtu.edu.cn seriously deteriorate wheel-rail contact behavior, increase Ziwei Zhou wheel-rail vertical force and even cause strong impact load tboweiwer@163.com to locomotive components and infrastructure [6, 7]. As a Yunfan Yang result, wheel polygonization accelerates failure of the yunfanyang525@126.com locomotive and track components and thus seriously Zaigang Chen threatens the locomotive running safety and reliability [6]. zgchen@swjtu.edu.cn In particular, with increasing in the running speed and axle load of modern rail transit equipment, the harms of the Train and Track Research Institute, State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu wheel polygonization behave more obvious. Therefore, 610031, China 123 130 S. Chen et al. researches of the wheel polygonal wear have always been frequency points of strain data as features for training, and hot topics in railway engineering. then applied a sparse Bayesian learning method to achieve In the past decades, most of studies have been carried wheel defect detection. Note that the above methods all out on formation mechanism of the polygonal wear. For rely on track-side sensor data collected in a certain railway instance, Jin et al. [8] investigated the root causes of section, and thus cannot monitor wheel conditions in the polygonal wear of metro train wheels and found that the whole vehicle running lines. Of particular interest to us are formation of the ninth-order polygonal wear is related to fault detection methods using on-board sensor data such as the first bending resonance of the wheelset. Tao et al. [9] vehicle vibration [20, 21]. For instance, Song et al. [22] found the wheelset bending vibration excited by rail cor- identified wheel polygonization by detecting a passing rugation with a specific wavelength can cause higher-order frequency of a polygonal wheel from an axle-box accel- (i.e., the 12th–14th orders) wheel polygonal wear for metro eration, and they also investigated acceleration character- trains. There are also studies on the mechanism of wheel istics induced by wheel polygonal wear with different polygonal wear of electric locomotives [10] and high-speed harmonic orders and amplitudes. This method assumes that trains [11], and most of the researches show that the the vehicle is running under a constant speed condition. polygonal wear is closely related to wheelset vibrations Sun et al. [23] applied an angle domain synchronous (bending or self-excited vibration). On the other hand, averaging technique to enhance fault-related components many studies focus on the influence of polygonal wear on of axle-box acceleration signals, and then constructed a vehicle and infrastructure components. For example, Wang health indicator to reflect roughness levels of wheel et al. [12, 13] investigated the influence of the polygonal polygonal wear. However, this method requires a speed wear on dynamic performance of gearbox housing and coder to measure the vehicle speed, which inevitably axle-box bearing of high-speed trains. The results show increases hardware costs. In addition, it cannot directly that wheel polygonization can increase vibrations and identify amplitudes of the polygonal wear. forces of gearbox and bearing, and thus accelerates A railway vehicle often suffers from non-stationary degeneration of the components. Chen et al. [14] studied running conditions such as traction and braking, resulting the effect of the polygonal wear on subgrade performance in time-varying passing frequencies of a polygonal wheel. by constructing a vehicle–track–subgrade coupled dynam- Traditional frequency spectrum-based fault detection ics model, and showed that vibration and stress of the methods cannot be applied for the time-varying fault fea- subgrade are mainly affected by lower-order polygonal ture extraction. Time–frequency (TF) analysis is a power- wear. It is worth noting that some researchers have inves- ful tool to address this issue. Various TF transforms have tigated vibration characteristics induced by polygonal been developed to generate TF representations (TFRs) of non-stationary signals such as short-time Fourier transform wheels. For instance, Yang et al. [15] compared ampli- tudes, root-mean-square (RMS) and peak-to-valley values (STFT), continuous wavelet transform, TF reassignment of axle box accelerations of heavy-haul locomotives with and synchrosqueezing transform [24–27], some of which and without wheel polygonal wear, and the results indicate have already been widely applied in machine fault diag- that the polygonization significantly increases the loco- nosis [28–30]. However, most of the methods assume that a motive vibration. signal is quasi-stationary and thus cannot accurately ana- Due to the severe harms of polygonal wear, condition lyze strongly modulated non-stationary signals. It is worth monitoring and fault detection of wheel polygonization noting that adaptive signal decomposition methods like have aroused considerable attention. Song et al. [16] empirical mode decomposition, empirical wavelet trans- employed piezoelectric strain sensors installed on the rail form, variational mode decomposition, can be combined to measure wheel-rail forces caused by a passing vehicle, with Hilbert transform to generate improved TFRs [31–33]. and then identified polygonal wheels according to magni- Nevertheless, these methods are mainly designed for nar- tudes of the impact forces. Similarly, Wei et al. [17] row-band signals and thus may suffer from over-decom- defined a wheel condition index based on track strain data position issue for wide-band non-stationary signals. To measured by fiber Bragg grating sensors, and developed a deal with this issue, Chen et al. developed an adaptive chirp real-time wheel condition monitoring system based on the mode decomposition (ACMD) framework and proposed proposed index. Given that the changes of the track strain various methods based on the unified framework [34–38]. caused by wheel minor defects may be unobvious and thus The ACMD is able to accurately estimate instantaneous can be easily affected by wheel load effect, Liu et al. [18] frequencies (IFs) and instantaneous amplitudes (IAs) of employed a Bayesian blind source separation method to non-stationary signals and thus generates high-quality extract defect-sensitive components from raw strain data. TFRs of signals. However, the output results of an ACMD To improve wheel fault detection performance, Ni et al. are sensitive to inputted initial IFs and its applicability in [19] extracted cumulative distribution function values and wheel polygonization detection has not been investigated. Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 131 A ðÞ t , f ðÞ t and u stand for IA, IF and initial phase of the In summary, quantitative detection of harmonic orders p p and amplitudes of railway wheel polygonal wear using p-th signal component, respectively. It is worth noting that vehicle vibration accelerations is a challenging task which the vibration response induced by wheel polygonal wear has not been properly solved by existing methods. In par- often shows a harmonic structure where the IFs of the ticular, non-stationary running conditions of a vehicle signal components are integer multiples of wheel rotating make the issue much more difficult. Utilizing the good frequency. Namely, Eq. (1) can be rewritten as advantages of the ACMD for non-stationary signal pro- P t cessing, a novel method for quantitative fault detection of gtðÞ ¼ A ðÞ t cos 2po f ðÞ t dt þ u ; ð2Þ p p w locomotive wheel polygonization under non-stationary p¼1 conditions is proposed in this paper. Firstly, the ACMD is where f ðÞ t ¼ o f ðÞ t , f ðÞ t is wheel rotating frequency, p p w w combined with a TF ridge detection method to accurately and o is an integer called harmonic order of the polygonal estimate a gear meshing frequency from motor vibration wear. It can be seen that, to effectively detect the wheel acceleration signal and therefore a wheel rotating fre- polygonization, one should accurately estimate f ðÞ t in quency can be obtained according to gear transmission advance. relation. At this stage, the IF initialization issue of ACMD is addressed by TF ridge detection. Then, a signal resam- 2.2 Adaptive chirp mode decomposition pling technique is applied to vibration acceleration of axle box to remove the non-stationary effect of the signal and The ACMD is designed to effectively estimate the IFs and therefore harmonic orders of polygonal wear can be IAs of a multi-component AM-FM signal (also called chirp accurately identified by order analysis. Finally, some signal) and thus achieves reconstruction and separation of polygonal-wear-related harmonics of the axle-box accel- the signal components. The ACMD can accurately analyze eration signal are extracted by ACMD and then an inertial strongly time-varying modulated signals. The basic idea of algorithm is applied to the extracted harmonics to estimate the ACMD is to use a demodulation technique to reduce amplitudes of the polygonal wear. The proposed method modulation degree of an AM-FM signal. Specifically, achieves the wheel rotating frequency estimation and fault Eq. (1) can be rewritten into a frequency demodulated form detection only using locomotive vibration signals (i.e., free as of speed coder). The structure of the paper is organized as follows. In gtðÞ ¼ x ðÞ t cos 2p f ðÞ t dt Sect. 2, the signal model is briefly described at first and p¼1 ð3Þ then the principle of ACMD is introduced. Section 3 details the proposed wheel polygonization detection ~ þy ðÞ t sin 2p f ðÞ t dt method. Effectiveness of the proposed method is demon- 0 strated by dynamics simulation in Sect. 4 and field test in with Sect. 5. Conclusions are drawn in Sect. 6. thi > d x ðÞ t ¼ A ðÞ t cos 2p f ðÞ t  f ðÞ t dt þ u p p p < p Z ; thi 2 Theoretical background y ðÞ t ¼ A ðÞ t sin 2p f ðÞ t  f ðÞ t dt þ u : p p p 2.1 Signal model ð4Þ When a locomotive undergoes non-stationary running where f ðÞ t is a demodulation frequency, and x ðÞ t and conditions, its vibration responses will exhibit time-varying y ðÞ t represent two demodulated signals. Obviously, if amplitude modulated (AM) and frequency modulated (FM) d f ðÞ t ¼ f ðÞ t , the FM effect of the signals in Eq. (4) will be features. In addition, practical vibration responses contain completely removed, resulting in purely AM signals with multiple components induced by different excitation sour- the narrowest bandwidth. Therefore, the ACMD estimates ces. Therefore, a multi-component AM-FM model is often IFs and reconstructs signal components by minimizing employed to characterize the locomotive vibration signal as bandwidths of the demodulated signals [34, 35]as P P X X gtðÞ ¼ g ðÞ t ¼ A ðÞ t cos 2p f ðÞ t dt þ u ; p p p p¼1 p¼1 ð1Þ where P denotes the number of signal components, and Rail. Eng. Science (2022) 30(2):129–147 132 S. Chen et al. no  no the limited space and unpredictable safety concerns, it may min L x ðÞ t ; y ðÞ t ; f ðÞ t s p p x ðÞ t ; y ðÞ t ; f ðÞ t fg p fg p fg not be allowed to install a speed measuring device in a 8 9 < P X 2 2 X railway vehicle. To address this issue, a novel rotating 00 00 ¼ min x ðÞ t þ y ðÞ t þ s gtðÞ  g ðÞ t ; p p ~d : 2 2 frequency estimation method based on vibration signal fg x ðÞ t ;fg y ðÞ t ;fg f ðÞ t p p p p¼1 p¼1 Z  Z t t analysis is proposed in this paper. Note that it is difficult to d d ~ ~ s.t: g ðÞ t ¼ x ðÞ t cos 2p f ðÞ t dt þ y ðÞ t sin 2p f ðÞ t dt ; p p p p p directly detect wheel rotating frequency from vibration 0 0 signal of a vehicle with a healthy wheel shaft. Therefore, it ð5Þ is suggested to extract other dominant frequencies and then where  denotes l norm,  stands for the second kk obtain the rotating frequency according to transmission 2 2 00 00 relation. It is known that gear transmission systems are derivative, x ðÞ t and y ðÞ t are used for assessing p p 2 2 widely used in a locomotive. Due to the continuous signal bandwidths, and s [ 0 is a weighting coefficient. meshing operation, there exists a distinct meshing fre- To efficiently solve the optimization problem in Eq. (5), quency component in vibration response of the gear an iterative algorithm which alternately updates the IFs and transmission system of the locomotive. Therefore, we demodulated signals are available in [34, 35]. Main pro- propose to estimate the gear meshing frequency f ðÞ t at cedures of the algorithm include: (1) input initial IFs; (2) first and then calculate the wheel rotating frequency as calculate demodulated signals with the inputted IFs; (3) f ðÞ t update the IFs using phase information of the demodulated f ðÞ t ¼ ; ð8Þ shaft signals; (4) repeat steps (2) and (3) until the algorithm converges. Assuming that the finally estimated IFs and where Z denotes the number of the teeth of the gear shaft demodulated signals by ACMD are denoted by f ðÞ t and fixedly connected with the wheel shaft. x~ ðÞ t , y~ ðÞ t , for p ¼ 1; 2; ...; P, respectively, then each To accurately estimate the gear meshing frequency, the p p ACMD is employed to analyze vibration acceleration sig- signal component can be reconstructed as nals of motor which is directly connected with gearbox and g~ ðÞ t ¼ x~ ðÞ t cos 2p f ðÞ t dt is less influenced by wheel-rail excitations [39, 40]. Since p p p IF initialization plays an important role in analysis results þ y~ ðÞ t sin 2p f ðÞ t dt : ð6Þ of ACMD, it is necessary to obtain a good estimate of an p p initial IF. To this end, a TF ridge detection technique is Moreover, a high-resolution TFR of the multi- utilized to deal with the IF initialization issue. Firstly, a TFR of the vibration signal is obtained by a proper TF component signal can be constructed as transform. For simplicity, STFT of the signal gtðÞ is cal- ~ ~ culated as ACMDðÞ t; f ¼ A ðÞ t d f  f ðÞ t ; ð7Þ p¼1 þ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  j2pft STFTðÞ t; f ¼ gðÞ t hðÞ t  t e dt; ð9Þ 2 2 where A ðÞ t ¼ x~ ðÞ t þ y~ ðÞ t is estimated IA, and dðÞ p 1 p p denotes Dirac delta function. where j ¼1, htðÞ is a nonnegative, symmetric and real window, and * stands for complex conjugate. Ridge curve of the TFR is often deemed as a good estimate of the IF and 3 Wheel polygonization detection using ACMD can be detected by solving [41, 42] () N1 N1 X X The proposed wheel polygonization detection method 2 2 f ðÞ t ¼ arg max jj STFTðÞ t ; XðÞ t  k jj XðÞ t  XðÞ t ; i i i i1 based on ACMD is presented in this section. The method XðÞ t i¼0 i¼1 mainly includes three steps: (1) wheel rotating frequency ð10Þ estimation; (2) signal resampling and order analysis; (3) where t ¼ t ; t ; .. .; t denote sampling time instants, N polygonal wear amplitude estimation. 0 1 N1 is the number of samples, XðÞ t stands for a set of all the TF paths from t ¼ t to t ¼ t , and k is a penalty factor. 3.1 Wheel rotating frequency estimation 0 N1 Eq. (10) implies that a desired ridge curve should pass As shown in Eq. (2), accurately estimating the wheel through TF points with large magnitudes and there is no significant jump phenomenon between adjacent points (i.e., rotating frequency f ðÞ t is crucial for detection of harmonic orders of the polygonal wear. Most existing methods the curve is smooth enough). The accuracy of IF estimation by TF ridge detection is employ a speed coder to measure the vehicle speed for wheel rotating frequency estimation [23]. However, due to dominated by resolution of the TFR. Due to the limitation Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 133 FM effect and exhibits constant frequencies at o for of traditional TF methods in analyzing strongly modulated signals, it is difficult to get a high-accuracy estimation of p ¼ 1; 2; ; P. Therefore, it is easy to identify harmonic the IF by ridge detection. Therefore, the ACMD is further orders (i.e., o ) by applying Fourier transform to the signal. applied to refine IF estimation. Firstly, with the coarsely In practice, the nonlinear mapping operation (i.e., Eq. (14) estimated gear meshing frequency by ridge detection, i.e., to Eq. (15)) is achieved by signal resampling or data interpolation [43, 44]. Namely, the time-domain signal gtðÞ f ðÞ t , the demodulated signals can be obtained as (see is interpolated from discrete time points fg t to Eq. (5); since ACMD is only employed to estimate the gear n¼0;1;;N1 meshing frequency at this stage, the number of signal discrete angles fg s as l¼0;1;;L1 components is simply set to 1, i.e., P ¼ 1) g ðÞ s ¼ InterpolateðÞ / ðÞ t ;gtðÞ; s ; / l w n n l ~ ð16Þ fg x~ ðÞ t ; y~ ðÞ t ¼ arg min L x ðÞ t ; y ðÞ t ; f ðÞ t ; ð11Þ m s m m m m n ¼ 0; 1; ; N  1; l ¼ 0; 1; ; L  1; x ðÞ t ;y ðÞ t m m where s ¼ l/ðÞ t =ðÞ L  1 for l ¼ 0; 1; ...; L  1 l w N1 where x ðÞ t and y ðÞ t denote the demodulated signals m m denote uniformly discretized samples of angular variable s, corresponding to gear meshing frequency. Then, according and L  N represents the number of angular samples. to Eq. (4), phase information of the demodulated signals can be used for calculating IF deviation as 3.3 Polygonal wear amplitude estimation 1 d y~ ðÞ t Df ðÞ t ¼  arctan 2p dt x~ ðÞ t Apart from harmonic order, harmonic amplitude of the 0 0 1 y~ ðÞ t  x~ ðÞ t  x~ ðÞ t  y~ ðÞ t m m m polygonal wear is also a key factor which directly deter- ¼ ; ð12Þ 2 2 2p x~ ðÞ t þ y~ ðÞ t m m mines whether a wheel should be repaired. In reality, the polygonal wear amplitude is detected through static mea- where  denotes derivative. Finally, estimation of the gear surement of wheel roughness levels using specialized meshing frequency can be improved by error compensation instruments. To the best of our knowledge, on-board as polygonal wear amplitude detection methods based on ~ ~ f ðÞ t ¼f ðÞ t þ‘fg Df ðÞ t ; ð13Þ vehicle vibration signals have hardly been reported yet. m m Note that the wheel polygonal wear along circumferential where ‘fg  represents a low-pass filtering operator [35] for direction can be regarded as a type of irregularity whose reducing noise interference. dynamic effect on a vehicle is similar to that of rail cor- rugation. Therefore, we propose to estimate the polygonal 3.2 Signal resampling and order analysis wear amplitude by combining the ACMD with track irregularity detection methods. In this paper, a well-known With the method proposed in Sect. 3.1, the gear meshing irregularity measurement method called inertial algorithm frequency f ðÞ t can be accurately estimated and a wheel is employed due to its easy implementation and wide rotating frequency f ðÞ t can be calculated according to applicability. As illustrated in Fig. 1, the basic principle of Eq. (8). The obtained f ðÞ t is used as a reference frequency inertial algorithm is that the quadratic integral of acceler- for signal resampling and order analysis. At this stage, a ation gives displacement [45, 46]. Therefore, the track vibration acceleration signal of axle box is analyzed since irregularity can be measured by processing the acceleration the signal is sensitive to the wheel polygonization. Firstly, signal of axle box as an angular variable is defined as s ¼ / ðÞ t ¼ f ðÞ t dt w ZZ and therefore Eq. (2) can be rewritten as dtðÞ ¼ atðÞdtdt; ð17Þ gtðÞ ¼ A ðÞ t cos 2po / ðÞ t þ u : ð14Þ where atðÞ denotes acceleration signal. Note that the inte- p p w p p¼1 gral operation in Eq. (17) may introduce an undesired signal trend which should be removed in a post-processing Then, if the time variable is expressed with the angular step. 1 1 variable as t ¼ / ðÞ s where / denotes inverse function, w w It is worth noting that apart from harmonics induced by Eq. (14) can be mapped into angular domain as polygonal wear, the acceleration signal of axle box con- tains other interference components such as those caused g ðÞ s ¼ A ðÞ s cos 2po s þ u ; ð15Þ / p by random track irregularities. To correctly estimate p¼1 polygonal wear amplitudes, the ACMD is employed to 1 1 extract the harmonics to be interested before the inertial where g ðÞ s ¼g / ðÞ s and A ðÞ s ¼A / ðÞ s . It can be / p w p w seen that the angular-domain signal in Eq. (15) is free of Rail. Eng. Science (2022) 30(2):129–147 134 S. Chen et al. and their amplitudes of wheel polygonal wear without Car body using speed coder. 4 Dynamics simulation Suspension Accelerometer 4.1 Dynamics model Track irregularity In this section, the proposed polygonal wear detection Axle box, method is validated by dynamics simulation. To this end, a wheel locomotive-track coupled dynamics model considering gear transmissions [47–49] is introduced as shown in Fig. 3. The model fully takes into account rail-wheel interaction and thus can well simulate practical locomotive vibration responses. Fig. 1 Illustration of the principle of inertial algorithm The dynamics model contains three sub-models including locomotive, track and gear transmission sub- algorithm is performed. Procedures of the proposed models. In the locomotive sub-model, a car body is linked amplitude estimation method are listed as follows: with two bogie frames via secondary suspension (K –C ) sz sz (1) Construct initial IFs for the harmonics of the accel- while each bogie frame is supported by two wheelsets eration signal as o~ f ðÞ t , p ¼ 1; 2; ; P, according to through primary suspension (K –C ). In each bogie, there pz pz an improved ACMD in [36], where f ðÞ t and o~ w are two gearboxes which are rigidly connected with two denote the estimated rotating frequency and harmonic traction motors, respectively. In the dynamics model, most orders in Sects. 3.1 and 3.2, respectively. of the locomotive components are regarded as rigid bodies (2) Reconstruct harmonics by ACMD according to the represented by mass (M ) and rotational inertia (J ), and q q obtained initial IFs and then summate these harmon- each component has three degrees of freedom, i.e., vertical ics as (Z ), rotational (b ) and longitudinal (X ) motions, where q q q ¼ c; m, w denote the car body, motor and wheelset, gt ~ðÞ ¼ g~ ðÞ t : ð18Þ respectively. The track subsystem consists of rail, rail pads, p¼1 sleepers, ballasts and subgrade. The stiffness and damping of the rail pads, ballasts and subgrade are represented as (3) Calculate quadratic integral of the reconstructed K –C , K –C , and K –C , respectively, where i denotes pi pi bi bi fi fi signal in Eq. (18) to obtain a displacement signal sleeper number. As for the gear transmission subsystem, dtðÞ. Next, least squares spline fitting is applied to the motor with gearbox is suspended through the bearings dtðÞ to obtain a signal trend rt ~ðÞ which is then (K –C ) mounted on the wheel axle and the hanger rod br br subtracted to get the irregularity caused by polygonal (K –C ) linked with bogie frame. The pinion is con- ms ms wear as nected with rotor of the motor and the gear is mounted on ~ ~ ZtðÞ ¼ dtðÞ rt ~ðÞ: ð19Þ the wheel axle. The power of the traction motor is trans- mitted from the pinion to gear via their teeth engagement (4) Perform the order analysis approach in Sect. 3.2 to described by the stiffness and damping elements (K –C ) m m the irregularity ZtðÞ to estimate amplitudes of the along the line of action (LOA). More details about the harmonics of the wheel polygonal wear. model and parameter setting can be found in [47–49]. Note that it is not possible to extract all the harmonics for polygonal wear identification. Therefore, only domi- 4.2 Response analysis nant harmonics (e.g., higher-order harmonics) in order spectrum (in step 1) are reconstructed since they cause The dynamics model in Fig. 3 is employed to calculate stronger vibration to a locomotive. The whole flow chart of locomotive vibration responses caused by wheel polygonal the proposed wheel polygonal wear detection method is wear. Firstly, the irregularity induced by the polygonal presented in Fig. 2. It can be seen that, by synthetically wear with P harmonics is used as an excitation of the analyzing vibration acceleration signals of motor and axle model as box, the proposed method can estimate the wheel rotating frequency and quantificationally detect the harmonic orders Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 135 Acceleration signal of axle box Acceleration signal of motor Resample the signal at a constant angular interval Estimate gear meshing Order frequency by TF ridge analysis Identify harmonic orders of polygonal wear by FFT Wheel Refine the frequency by rotating ACMD frequency estimation Reconstruct harmonics by ACMD Obtain the rotating Amplitude frequency according to Reference estimation transmission relation Estimate polygonal wear frequency amplitudes by inertial algorithm Fig. 2 Flow chart of the proposed wheel polygonization detection method P With the estimated wheel rotating frequency, the signal o utðÞ Z ðÞ t ¼ X cos þ w ; ð20Þ 0 p p resampling technique is applied to acceleration signal of p¼1 axle box and then the order spectrum is obtained and compared with original spectrum as shown in Fig. 7. Due where X , o and w stand for amplitude, harmonic order p p to the time-varying frequency contents, Fourier spectrum and initial phase of the polygonal wear, respectively, utðÞ cannot reveal effective fault information of the polygonal denotes running distance function of locomotive (i.e., the wear. On the contrary, since the non-stationary effect of the integral of locomotive speed), and R is wheel radius (the signal is fully removed by resampling technique, harmonic radius is set to 0.625 m in the simulation). In this section, orders of the polygonal wear can be clearly identified in the polygonal wear with three harmonics is considered and order spectrum (see Fig. 7b). In addition, the order spec- the parameters are provided in Table 1. Figure 4 illustrates trum based on acceleration signal shows much better res- the polygonal wear and its order analysis result. To simu- olution than that in Fig. 4b since the duration of the late the random interference of track irregularity, the acceleration signal is long enough. American fifth-grade track spectrum is employed to gen- Then, polygonal wear amplitude estimation is carried erate random excitation of the dynamics model. It is out. Firstly, based on the order analysis results, the har- assumed that the locomotive is running under a traction monics induced by polygonal wear can be extracted by condition and the speed is gradually increased to 80 km/h. ACMD as shown in Fig. 8. An improved TFR of these The simulated vibration acceleration signals of axle box harmonics by ACMD (see Eq. (7)) is also provided and and motor based on the dynamics model are provided in compared with that by STFT as shown in Fig. 9. It can be Fig. 5. The gear meshing frequency and frequency contents observed that STFT cannot resolve these harmonics due to induced by wheel polygonal wear can be observed in the the poor resolution and interferences from track irregular- TFRs by STFT. However, due to poor resolution of the ity. The TFR obtained by ACMD can clearly uncover time- TFR, details of the wheel polygonization are not available. varying features of the harmonics. Next, inertial algorithm Next, the proposed fault detection method is applied. is applied to the extracted harmonics and the analysis Firstly, as shown in Fig. 6, the gear meshing frequency is results are shown in Fig. 10. It is indicated that the integral estimated and then the wheel rotating frequency can be operation will lead to a complex signal trend with a large calculated according to Eq. (8), where the number of the magnitude and therefore signal detrending is necessary for gear teeth is set to Z ¼120 for the locomotive. The shaft detection of the irregularity caused by polygonal wear. mean-square errors of the estimated gear meshing fre- After detrending, radial deviations (or irregularities) caused quencies by ridge detection and ACMD are 6.43 and by polygonal wear of different wheel rotation cycles are - 5.73 dB, respectively. It can be seen that the IF esti- clearly recovered as illustrated in Fig. 10b. Finally, mation results by ridge detection exhibit staircase effects amplitudes of harmonics of the polygonal wear are esti- due to limited resolution of the TFR (see Fig. 6a, b). After mated by performing order analysis to the recovered radial frequency refinement by ACMD, the IF estimation accu- deviation of a cycle (suggested to use average data of racy can be significantly improved. Rail. Eng. Science (2022) 30(2):129–147 136 S. Chen et al. Fig. 3 Locomotive-track coupled dynamics model considering gear transmissions: a model and b gear transmission systems in one bogie proposed method for quantitative detection of locomotive Table 1 Simulated wheel polygonal wear with three harmonics wheel polygonal wear. p Amplitude X Order Phase (mm) o w 1 0.050 7 0 5 Practical application 2 0.040 8 2p=3 3 0.025 9 4p=3 In this section, the proposed fault detection method is applied to vibration signal analysis of an in-service loco- motive to further demonstrate the effectiveness of the different rotation cycles at lower-speed stage to reduce the method. influence of track irregularities) as shown in Fig. 10c. One can find that the estimated harmonic amplitudes are very close to the true ones showing the effectiveness of the Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 137 Fig. 4 Wheel polygonal wear in the simulation case: a polygonal wear described in the polar coordinate and b order analysis of the polygonal wear (a) (b) 3 1000 -1 Induced by polygonisation -2 -3 0 2 4 6 8 024 6 8 Time (s) Time (s) (c) 1000 (d) -1 Gear meshing frequency -2 0 2 4 6 8 02 4 6 8 Time (s) Time (s) Fig. 5 Simulated vibration signals of the locomotive with wheel polygonization: a and b show waveform and STFT of the acceleration signal of axle box, respectively; c and d illustrate the signal of motor 5.1 Field test Power) carried out a thorough field test for the locomotive [15]. Firstly, wheel roughness levels around circumference The locomotive operated on the Chinese heavy-haul rail- were measured by Mu¨ller-BBM instrument to assess the way from Zhongwei to Yuci. Vibration alarm phenomenon wheel polygonal wear as shown in Fig. 11. The measuring occurred several times for on-board monitoring system of accuracy of the instrument is 0.1 lm with a sampling the locomotive. To investigate root cause of the alarm precision of 1 mm. The roughness measurement was phenomenon, a research group from Train and Track achieved by manually twirling wheels. In addition, vibra- Research Institute (in State Key Laboratory of Traction tion accelerations of key components such as axle box and Rail. Eng. Science (2022) 30(2):129–147 Acceleration (g) Acceleration (g) Frequency (Hz) Frequency (Hz) 138 S. Chen et al. 5.9 6.1 6.3 c d 5.9 6.1 6.3 Time (s) Fig. 6 IF estimation results from the simulated signal in Fig. 5(d): a and b show the estimated gear meshing frequency and wheel rotating frequency by ridge detection, respectively; c and d show the refined results by ACMD (a) (b) 0.08 0.2 0.06 0.15 0.04 0.1 0.02 0.05 0 0 0 20 40 60 0 5 10 Order Frequency (Hz) Fig. 7 Comparison of spectrums of the simulated vibration signals of axle box: a original Fourier spectrum; b order spectrum after signal resampling with the estimated wheel rotating frequency Rail. Eng. Science (2022) 30(2):129–147 Acceleration (g ) Frequency (Hz) Acceleration (g) Quantitative detection of locomotive wheel polygonization... 139 (b) 0.5 1 -0.5 0 .5 0 246 8 0.5 -0.5 246 8 - 0.5 0.5 -1 -0.5 0 2 4 6 0 246 8 Time (s) Time (s) Fig. 8 Harmonic extraction by ACMD for the simulated vibration signals of axle box: a extracted harmonics; b sum of the extracted harmonics (a) (b) Time (s) Fig. 9 Comparison of TFRs of the simulated vibration signals of axle box: a STFT and b ACMD motor were also measured during operation process of the traction condition. Figure 15 illustrates the measured locomotive as shown in Fig. 12. To determine whether the vibration signals and their TF analysis results. It can be root cause of the vibration alarm is the wheel wear, two seen that the dominant frequency contents of the vibration round tests before and after wheel re-profiling were con- signal of axle box are induced by polygonal wear while ducted, respectively. The wheel re-profiling was achieved those of the motor vibration signal are mainly related to by wheel lathe to reduce radial deviation caused by gear meshing operation. Therefore, the proposed method is polygonal wear as shown in Fig. 13. applied to the motor vibration signal to estimate the gear meshing frequency and wheel rotating frequency as shown 5.2 Detection results in Fig. 16, where the number of gear teeth Z is 120. shaft After the rotating frequency is obtained, signal resampling Firstly, the measured data of the locomotive before wheel and order analysis are performed to the vibration signal of re-profiling is analyzed. The measured wheel roughness axle box as illustrated in Fig. 17. Compared with the raw data and its analysis result are provided in Fig. 14.It Fourier spectrum (see Fig. 17a), the developed order clearly shows that the wheel suffers from high-order spectrum (see Fig. 17b) correctly reveals all the harmonic polygonal wear (i.e., the 17th–24th orders). Then, the orders of the polygonal wear and it even shows much better proposed method is applied to locomotive vibration data to resolution than the analysis results based on the measured test whether the polygonal wear information can be wheel roughness in Fig. 14b. TFRs of the polygonal-wear- detected. In this test, the locomotive operates under a related harmonics are shown in Fig. 18. It can be seen that Rail. Eng. Science (2022) 30(2):129–147 Order 9 Order 8 Order 7 Acceleration (g) 140 S. Chen et al. Fig. 10 Polygonal wear amplitude estimation by inertial algorithm in the simulation case: a obtained wheel radial deviation by quadratic integral of the reconstructed harmonics; b radial deviation after detrending; c order analysis of the reconstructed polygonal wear Fig. 11 Field measurement of wheel polygonal wear of the locomotive [15]: a wheel polygonal wear; b polygonal wear measurement by Mu¨ller- BBM instrument Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 141 Wheelset (b) (a) Gear Axle box Motor (c) (d) Bogie Fig. 12 Field measurement of vibration of the locomotive: a the locomotive; b schematic diagram of measurement positions; c Measurement positions of axle box; d measurement positions of motor (a) (b) Polygonised wheel Drive wheel Wheel Lathe tool Fig. 13 Wheel re-profiling with wheel lathe: a schematic diagram and b wheel lathe [15] (a) (b) 0.08 0.06 150 30 0.04 180 0 -0.2 0.02 0.2 10 20 30 40 Order Fig. 14 Measured wheel polygonal wear of the locomotive: a polygonal wear described in the polar coordinate; b order analysis of the polygonal wear Rail. Eng. Science (2022) 30(2):129–147 Polygonal wear (mm) Polygonal wear (mm) 142 S. Chen et al. c d Gear meshing frequency Time (s) Fig. 15 Measured vibration signals of the locomotive with wheel polygonization: a and b show waveform and STFT of the acceleration signal of axle box, respectively; c and d illustrate the signal of motor (b) 5 4.5 3.5 0 2.5 60 60 0 20 40 0 20 40 Time (s) Time (s) Fig. 16 Estimated gear meshing frequency and wheel rotating frequency by ACMD from the signal in Fig. 15d: a gear meshing frequency and b wheel rotating frequency Rail. Eng. Science (2022) 30(2):129–147 e ( Frequency (Hz) Acceleration (g) Acceleration (g) Frequency (Hz) Frequency (Hz) Frequency (Hz) Quantitative detection of locomotive wheel polygonization... 143 (b) 19 21 Fig. 17 Comparison of spectrums of the measured vibration signals of axle box: a original Fourier spectrum; b order spectrum after signal resampling with the estimated wheel rotating frequency (b) Fig. 18 Comparison of TFRs of the measured vibration signals of axle box: a STFT and b ACMD (a) (b) 0.015 0.05 150 30 0.01 -0.05 180 0 0.005 300 40 10 20 30 Order Fig. 19 Measured wheel polygonal wear of the locomotive after wheel re-profiling: a polygonal wear described in the polar coordinate; b order analysis of the polygonal wear Rail. Eng. Science (2022) 30(2):129–147 Polygonal wear (mm) Acceleration (g) Polygonal wear (mm) Acceleration (g) 144 S. Chen et al. -2 -4 -6 50 100 0 50 100 Time (s) -2 Gear meshing frequency -4 0 50 100 50 100 Time (s) Fig. 20 Measured vibration signals of the locomotive after wheel re-profiling: a and b show waveform and STFT of the acceleration signal of axle box, respectively; c and d illustrate the signal of motor ACMD successfully resolves and extracts all the harmonics wheel polygonal wear. Then, the proposed method is and thus generates a high-resolution TFR (see Fig. 18b; applied to demonstrate its usefulness in detecting such a due to the limited space, instead of waveforms, only TFRs slight fault. IF estimation and order analysis results based are provided). These extracted harmonics can be further on the vibration signals are given in Figs. 21 and 22, used for wear amplitude estimation. respectively. It shows that the harmonic orders of the For comparison, wheel roughness and locomotive polygonal wear are clearly identified (see Fig. 22b). vibration data after wheel re-profiling are analyzed. Fig- Comparison of the TFRs obtained by STFT and ACMD is ure 19 shows the measured roughness data. It can be seen illustrated in Fig. 23 indicating higher resolution of ACMD that the wheel lathe does not fully eliminate wheel poly- in separating closely spaced harmonics induced by wheel gonization although the roughness magnitudes and the polygonal wear. number of harmonics are significantly reduced (compared Finally, inertial algorithm is applied to the harmonics with that in Fig. 14). In this case, the dominant harmonic extracted by ACMD for polygonal wear amplitude esti- orders of the polygonal wear are 17 and 24. To show the mation. The amplitude estimation results before and after effectiveness of the method in different working condi- wheel re-profiling are compared in Fig. 24. In both cases, tions, vibration signals of the locomotive during braking the estimated polygonal wear amplitudes based on accel- process are analyzed as illustrated in Fig. 20. It can be eration signals are close to those measured by instrument as found that magnitudes of the vibration acceleration signals shown in Fig. 24. The results indicate that the proposed and energy of the polygonal-wear-induced components are method is effective for detection of wheel polygonal wear decreased evidently. It implies that the vibration alarm of different levels. phenomenon of the locomotive is mainly attributed to the Rail. Eng. Science (2022) 30(2):129–147 Induced by polygonisation Time (s) Time (s) Acceleration (g) Acceleration (g) Frequency (Hz) Frequency (Hz) Quantitative detection of locomotive wheel polygonization... 145 (b) 1000 5 4.5 3.5 2.5 0 50 100 0 50 100 Time (s) Time (s) Fig. 21 Estimated gear meshing frequency and wheel rotating frequency by ACMD from the signal in Fig. 20d: a gear meshing frequency and b wheel rotating frequency (b) 0.05 0.6 24 0.04 0.4 0.03 0.02 0.2 0.01 200 10 20 30 50 100 150 40 Order Frequency (Hz) Fig. 22 Comparison of spectrums of the measured vibration signals of axle box after wheel re-profiling: a original Fourier spectrum; b order spectrum after signal resampling with the estimated wheel rotating frequency (b) 150 150 100 100 50 50 0 50 100 0 50 100 Time (s) Time (s) Fig. 23 Comparison of TFRs of the measured vibration signals of axle box after wheel re-profiling: a STFT and b ACMD Rail. Eng. Science (2022) 30(2):129–147 Acceleration (g) Frequency (Hz) Frequency (Hz) Acceleration (g) Frequency (Hz) Frequency (Hz) 146 S. Chen et al. (b) 0.015 0.08 Measured Measured Estimated Estimated 0.06 0.01 0.04 0.005 0.02 0 0 10 20 30 10 20 30 40 Order Order Fig. 24 Estimated polygonal wear amplitudes by inertial algorithm for the experimental vibration signals: a before wheel re-profiling and b after wheel re-profiling Open Access This article is licensed under a Creative Commons 6 Conclusions Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as In this paper, a novel quantitative detection method for wheel long as you give appropriate credit to the original author(s) and the polygonal wear under non-stationary conditions using loco- source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this motivevibration signals has beendeveloped. Firstly, TF ridge article are included in the article’s Creative Commons licence, unless detection has been combined with ACMD to accurately esti- indicated otherwise in a credit line to the material. If material is not mate the time-varying wheel rotating frequency from motor included in the article’s Creative Commons licence and your intended vibration signal without using speed coder. Next, the order use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright analysis technique based on signal resampling has been holder. To view a copy of this licence, visit http://creativecommons. applied to vibration signal of axle box for harmonic order org/licenses/by/4.0/. detection. Finally, polygonal wear amplitudes have been estimated by integrating the ACMD with inertial algorithm. 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Chen Z, Zhai W, Wang K (2017) Dynamic investigation of a reassignment for planetary gearbox fault diagnosis under non- locomotive with effect of gear transmissions under tractive con- stationary conditions. Mech Syst Signal Process 80:429–444 ditions. J Sound Vib 408:220–233 29. Hua Z, Shi J, Luo Y, Huang W, Wang J, Zhu Z (2021) Iterative 50. ZhaiG,NarazakiY,WangS,ShajihanSAVetal(2022)Syntheticdata matching synchrosqueezing transform and application to rotating augmentation for pixel-wise steel fatigue crack identification using machinery fault diagnosis under nonstationary conditions. Mea- fully convolutional networks. Smart Struct Syst 29(1):237–250 surement 173:108592 Rail. Eng. Science (2022) 30(2):129–147 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Railway Engineering Science Springer Journals

Quantitative detection of locomotive wheel polygonization under non-stationary conditions by adaptive chirp mode decomposition

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Rail. Eng. Science (2022) 30(2):129–147 https://doi.org/10.1007/s40534-022-00272-3 Quantitative detection of locomotive wheel polygonization under non-stationary conditions by adaptive chirp mode decomposition 1 1 1 1 1 1 • • • • • Shiqian Chen Kaiyun Wang Ziwei Zhou Yunfan Yang Zaigang Chen Wanming Zhai Received: 21 December 2021 / Revised: 10 February 2022 / Accepted: 17 February 2022 / Published online: 13 April 2022 The Author(s) 2022 Abstract Wheel polygonal wear is a common and severe After the rotating frequency is obtained, signal resampling defect, which seriously threatens the running safety and and order analysis techniques are applied to an acceleration reliability of a railway vehicle especially a locomotive. signal of an axle box to identify harmonic orders related to Due to non-stationary running conditions (e.g., traction and polygonal wear. Finally, the ACMD is combined with an braking) of the locomotive, the passing frequencies of a inertial algorithm to estimate polygonal wear amplitudes. polygonal wheel will exhibit time-varying behaviors, Not only a dynamics simulation but a field test was carried which makes it too difficult to effectively detect the wheel out to show that the proposed method can effectively detect defect. Moreover, most existing methods only achieve both harmonic orders and their amplitudes of the wheel qualitative fault diagnosis and they cannot accurately polygonization under non-stationary conditions. identify defect levels. To address these issues, this paper reports a novel quantitative method for fault detection of Keywords Wheel polygonal wear  Fault diagnosis  Non- wheel polygonization under non-stationary conditions stationary condition  Adaptive mode decomposition based on a recently proposed adaptive chirp mode Time–frequency analysis decomposition (ACMD) approach. Firstly, a coarse-to-fine method based on the time–frequency ridge detection and ACMD is developed to accurately estimate a time-varying gear meshing frequency and thus obtain a wheel rotating frequency from a vibration acceleration signal of a motor. 1 Introduction Wheel out-of-roundness, especially wheel polygonal wear, & Wanming Zhai is a very common type of wheel defect for a railway wmzhai@swjtu.edu.cn vehicle especially for a locomotive [1–4]. Wheel polygo- Shiqian Chen nization is a phenomenon that the wheel profile periodi- chenshiqian@swjtu.edu.cn cally deviates from an ideal round shape (i.e., periodic Kaiyun Wang radial deviation) along circumference [5]. The defect will kywang@swjtu.edu.cn seriously deteriorate wheel-rail contact behavior, increase Ziwei Zhou wheel-rail vertical force and even cause strong impact load tboweiwer@163.com to locomotive components and infrastructure [6, 7]. As a Yunfan Yang result, wheel polygonization accelerates failure of the yunfanyang525@126.com locomotive and track components and thus seriously Zaigang Chen threatens the locomotive running safety and reliability [6]. zgchen@swjtu.edu.cn In particular, with increasing in the running speed and axle load of modern rail transit equipment, the harms of the Train and Track Research Institute, State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu wheel polygonization behave more obvious. Therefore, 610031, China 123 130 S. Chen et al. researches of the wheel polygonal wear have always been frequency points of strain data as features for training, and hot topics in railway engineering. then applied a sparse Bayesian learning method to achieve In the past decades, most of studies have been carried wheel defect detection. Note that the above methods all out on formation mechanism of the polygonal wear. For rely on track-side sensor data collected in a certain railway instance, Jin et al. [8] investigated the root causes of section, and thus cannot monitor wheel conditions in the polygonal wear of metro train wheels and found that the whole vehicle running lines. Of particular interest to us are formation of the ninth-order polygonal wear is related to fault detection methods using on-board sensor data such as the first bending resonance of the wheelset. Tao et al. [9] vehicle vibration [20, 21]. For instance, Song et al. [22] found the wheelset bending vibration excited by rail cor- identified wheel polygonization by detecting a passing rugation with a specific wavelength can cause higher-order frequency of a polygonal wheel from an axle-box accel- (i.e., the 12th–14th orders) wheel polygonal wear for metro eration, and they also investigated acceleration character- trains. There are also studies on the mechanism of wheel istics induced by wheel polygonal wear with different polygonal wear of electric locomotives [10] and high-speed harmonic orders and amplitudes. This method assumes that trains [11], and most of the researches show that the the vehicle is running under a constant speed condition. polygonal wear is closely related to wheelset vibrations Sun et al. [23] applied an angle domain synchronous (bending or self-excited vibration). On the other hand, averaging technique to enhance fault-related components many studies focus on the influence of polygonal wear on of axle-box acceleration signals, and then constructed a vehicle and infrastructure components. For example, Wang health indicator to reflect roughness levels of wheel et al. [12, 13] investigated the influence of the polygonal polygonal wear. However, this method requires a speed wear on dynamic performance of gearbox housing and coder to measure the vehicle speed, which inevitably axle-box bearing of high-speed trains. The results show increases hardware costs. In addition, it cannot directly that wheel polygonization can increase vibrations and identify amplitudes of the polygonal wear. forces of gearbox and bearing, and thus accelerates A railway vehicle often suffers from non-stationary degeneration of the components. Chen et al. [14] studied running conditions such as traction and braking, resulting the effect of the polygonal wear on subgrade performance in time-varying passing frequencies of a polygonal wheel. by constructing a vehicle–track–subgrade coupled dynam- Traditional frequency spectrum-based fault detection ics model, and showed that vibration and stress of the methods cannot be applied for the time-varying fault fea- subgrade are mainly affected by lower-order polygonal ture extraction. Time–frequency (TF) analysis is a power- wear. It is worth noting that some researchers have inves- ful tool to address this issue. Various TF transforms have tigated vibration characteristics induced by polygonal been developed to generate TF representations (TFRs) of non-stationary signals such as short-time Fourier transform wheels. For instance, Yang et al. [15] compared ampli- tudes, root-mean-square (RMS) and peak-to-valley values (STFT), continuous wavelet transform, TF reassignment of axle box accelerations of heavy-haul locomotives with and synchrosqueezing transform [24–27], some of which and without wheel polygonal wear, and the results indicate have already been widely applied in machine fault diag- that the polygonization significantly increases the loco- nosis [28–30]. However, most of the methods assume that a motive vibration. signal is quasi-stationary and thus cannot accurately ana- Due to the severe harms of polygonal wear, condition lyze strongly modulated non-stationary signals. It is worth monitoring and fault detection of wheel polygonization noting that adaptive signal decomposition methods like have aroused considerable attention. Song et al. [16] empirical mode decomposition, empirical wavelet trans- employed piezoelectric strain sensors installed on the rail form, variational mode decomposition, can be combined to measure wheel-rail forces caused by a passing vehicle, with Hilbert transform to generate improved TFRs [31–33]. and then identified polygonal wheels according to magni- Nevertheless, these methods are mainly designed for nar- tudes of the impact forces. Similarly, Wei et al. [17] row-band signals and thus may suffer from over-decom- defined a wheel condition index based on track strain data position issue for wide-band non-stationary signals. To measured by fiber Bragg grating sensors, and developed a deal with this issue, Chen et al. developed an adaptive chirp real-time wheel condition monitoring system based on the mode decomposition (ACMD) framework and proposed proposed index. Given that the changes of the track strain various methods based on the unified framework [34–38]. caused by wheel minor defects may be unobvious and thus The ACMD is able to accurately estimate instantaneous can be easily affected by wheel load effect, Liu et al. [18] frequencies (IFs) and instantaneous amplitudes (IAs) of employed a Bayesian blind source separation method to non-stationary signals and thus generates high-quality extract defect-sensitive components from raw strain data. TFRs of signals. However, the output results of an ACMD To improve wheel fault detection performance, Ni et al. are sensitive to inputted initial IFs and its applicability in [19] extracted cumulative distribution function values and wheel polygonization detection has not been investigated. Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 131 A ðÞ t , f ðÞ t and u stand for IA, IF and initial phase of the In summary, quantitative detection of harmonic orders p p and amplitudes of railway wheel polygonal wear using p-th signal component, respectively. It is worth noting that vehicle vibration accelerations is a challenging task which the vibration response induced by wheel polygonal wear has not been properly solved by existing methods. In par- often shows a harmonic structure where the IFs of the ticular, non-stationary running conditions of a vehicle signal components are integer multiples of wheel rotating make the issue much more difficult. Utilizing the good frequency. Namely, Eq. (1) can be rewritten as advantages of the ACMD for non-stationary signal pro- P t cessing, a novel method for quantitative fault detection of gtðÞ ¼ A ðÞ t cos 2po f ðÞ t dt þ u ; ð2Þ p p w locomotive wheel polygonization under non-stationary p¼1 conditions is proposed in this paper. Firstly, the ACMD is where f ðÞ t ¼ o f ðÞ t , f ðÞ t is wheel rotating frequency, p p w w combined with a TF ridge detection method to accurately and o is an integer called harmonic order of the polygonal estimate a gear meshing frequency from motor vibration wear. It can be seen that, to effectively detect the wheel acceleration signal and therefore a wheel rotating fre- polygonization, one should accurately estimate f ðÞ t in quency can be obtained according to gear transmission advance. relation. At this stage, the IF initialization issue of ACMD is addressed by TF ridge detection. Then, a signal resam- 2.2 Adaptive chirp mode decomposition pling technique is applied to vibration acceleration of axle box to remove the non-stationary effect of the signal and The ACMD is designed to effectively estimate the IFs and therefore harmonic orders of polygonal wear can be IAs of a multi-component AM-FM signal (also called chirp accurately identified by order analysis. Finally, some signal) and thus achieves reconstruction and separation of polygonal-wear-related harmonics of the axle-box accel- the signal components. The ACMD can accurately analyze eration signal are extracted by ACMD and then an inertial strongly time-varying modulated signals. The basic idea of algorithm is applied to the extracted harmonics to estimate the ACMD is to use a demodulation technique to reduce amplitudes of the polygonal wear. The proposed method modulation degree of an AM-FM signal. Specifically, achieves the wheel rotating frequency estimation and fault Eq. (1) can be rewritten into a frequency demodulated form detection only using locomotive vibration signals (i.e., free as of speed coder). The structure of the paper is organized as follows. In gtðÞ ¼ x ðÞ t cos 2p f ðÞ t dt Sect. 2, the signal model is briefly described at first and p¼1 ð3Þ then the principle of ACMD is introduced. Section 3 details the proposed wheel polygonization detection ~ þy ðÞ t sin 2p f ðÞ t dt method. Effectiveness of the proposed method is demon- 0 strated by dynamics simulation in Sect. 4 and field test in with Sect. 5. Conclusions are drawn in Sect. 6. thi > d x ðÞ t ¼ A ðÞ t cos 2p f ðÞ t  f ðÞ t dt þ u p p p < p Z ; thi 2 Theoretical background y ðÞ t ¼ A ðÞ t sin 2p f ðÞ t  f ðÞ t dt þ u : p p p 2.1 Signal model ð4Þ When a locomotive undergoes non-stationary running where f ðÞ t is a demodulation frequency, and x ðÞ t and conditions, its vibration responses will exhibit time-varying y ðÞ t represent two demodulated signals. Obviously, if amplitude modulated (AM) and frequency modulated (FM) d f ðÞ t ¼ f ðÞ t , the FM effect of the signals in Eq. (4) will be features. In addition, practical vibration responses contain completely removed, resulting in purely AM signals with multiple components induced by different excitation sour- the narrowest bandwidth. Therefore, the ACMD estimates ces. Therefore, a multi-component AM-FM model is often IFs and reconstructs signal components by minimizing employed to characterize the locomotive vibration signal as bandwidths of the demodulated signals [34, 35]as P P X X gtðÞ ¼ g ðÞ t ¼ A ðÞ t cos 2p f ðÞ t dt þ u ; p p p p¼1 p¼1 ð1Þ where P denotes the number of signal components, and Rail. Eng. Science (2022) 30(2):129–147 132 S. Chen et al. no  no the limited space and unpredictable safety concerns, it may min L x ðÞ t ; y ðÞ t ; f ðÞ t s p p x ðÞ t ; y ðÞ t ; f ðÞ t fg p fg p fg not be allowed to install a speed measuring device in a 8 9 < P X 2 2 X railway vehicle. To address this issue, a novel rotating 00 00 ¼ min x ðÞ t þ y ðÞ t þ s gtðÞ  g ðÞ t ; p p ~d : 2 2 frequency estimation method based on vibration signal fg x ðÞ t ;fg y ðÞ t ;fg f ðÞ t p p p p¼1 p¼1 Z  Z t t analysis is proposed in this paper. Note that it is difficult to d d ~ ~ s.t: g ðÞ t ¼ x ðÞ t cos 2p f ðÞ t dt þ y ðÞ t sin 2p f ðÞ t dt ; p p p p p directly detect wheel rotating frequency from vibration 0 0 signal of a vehicle with a healthy wheel shaft. Therefore, it ð5Þ is suggested to extract other dominant frequencies and then where  denotes l norm,  stands for the second kk obtain the rotating frequency according to transmission 2 2 00 00 relation. It is known that gear transmission systems are derivative, x ðÞ t and y ðÞ t are used for assessing p p 2 2 widely used in a locomotive. Due to the continuous signal bandwidths, and s [ 0 is a weighting coefficient. meshing operation, there exists a distinct meshing fre- To efficiently solve the optimization problem in Eq. (5), quency component in vibration response of the gear an iterative algorithm which alternately updates the IFs and transmission system of the locomotive. Therefore, we demodulated signals are available in [34, 35]. Main pro- propose to estimate the gear meshing frequency f ðÞ t at cedures of the algorithm include: (1) input initial IFs; (2) first and then calculate the wheel rotating frequency as calculate demodulated signals with the inputted IFs; (3) f ðÞ t update the IFs using phase information of the demodulated f ðÞ t ¼ ; ð8Þ shaft signals; (4) repeat steps (2) and (3) until the algorithm converges. Assuming that the finally estimated IFs and where Z denotes the number of the teeth of the gear shaft demodulated signals by ACMD are denoted by f ðÞ t and fixedly connected with the wheel shaft. x~ ðÞ t , y~ ðÞ t , for p ¼ 1; 2; ...; P, respectively, then each To accurately estimate the gear meshing frequency, the p p ACMD is employed to analyze vibration acceleration sig- signal component can be reconstructed as nals of motor which is directly connected with gearbox and g~ ðÞ t ¼ x~ ðÞ t cos 2p f ðÞ t dt is less influenced by wheel-rail excitations [39, 40]. Since p p p IF initialization plays an important role in analysis results þ y~ ðÞ t sin 2p f ðÞ t dt : ð6Þ of ACMD, it is necessary to obtain a good estimate of an p p initial IF. To this end, a TF ridge detection technique is Moreover, a high-resolution TFR of the multi- utilized to deal with the IF initialization issue. Firstly, a TFR of the vibration signal is obtained by a proper TF component signal can be constructed as transform. For simplicity, STFT of the signal gtðÞ is cal- ~ ~ culated as ACMDðÞ t; f ¼ A ðÞ t d f  f ðÞ t ; ð7Þ p¼1 þ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  j2pft STFTðÞ t; f ¼ gðÞ t hðÞ t  t e dt; ð9Þ 2 2 where A ðÞ t ¼ x~ ðÞ t þ y~ ðÞ t is estimated IA, and dðÞ p 1 p p denotes Dirac delta function. where j ¼1, htðÞ is a nonnegative, symmetric and real window, and * stands for complex conjugate. Ridge curve of the TFR is often deemed as a good estimate of the IF and 3 Wheel polygonization detection using ACMD can be detected by solving [41, 42] () N1 N1 X X The proposed wheel polygonization detection method 2 2 f ðÞ t ¼ arg max jj STFTðÞ t ; XðÞ t  k jj XðÞ t  XðÞ t ; i i i i1 based on ACMD is presented in this section. The method XðÞ t i¼0 i¼1 mainly includes three steps: (1) wheel rotating frequency ð10Þ estimation; (2) signal resampling and order analysis; (3) where t ¼ t ; t ; .. .; t denote sampling time instants, N polygonal wear amplitude estimation. 0 1 N1 is the number of samples, XðÞ t stands for a set of all the TF paths from t ¼ t to t ¼ t , and k is a penalty factor. 3.1 Wheel rotating frequency estimation 0 N1 Eq. (10) implies that a desired ridge curve should pass As shown in Eq. (2), accurately estimating the wheel through TF points with large magnitudes and there is no significant jump phenomenon between adjacent points (i.e., rotating frequency f ðÞ t is crucial for detection of harmonic orders of the polygonal wear. Most existing methods the curve is smooth enough). The accuracy of IF estimation by TF ridge detection is employ a speed coder to measure the vehicle speed for wheel rotating frequency estimation [23]. However, due to dominated by resolution of the TFR. Due to the limitation Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 133 FM effect and exhibits constant frequencies at o for of traditional TF methods in analyzing strongly modulated signals, it is difficult to get a high-accuracy estimation of p ¼ 1; 2; ; P. Therefore, it is easy to identify harmonic the IF by ridge detection. Therefore, the ACMD is further orders (i.e., o ) by applying Fourier transform to the signal. applied to refine IF estimation. Firstly, with the coarsely In practice, the nonlinear mapping operation (i.e., Eq. (14) estimated gear meshing frequency by ridge detection, i.e., to Eq. (15)) is achieved by signal resampling or data interpolation [43, 44]. Namely, the time-domain signal gtðÞ f ðÞ t , the demodulated signals can be obtained as (see is interpolated from discrete time points fg t to Eq. (5); since ACMD is only employed to estimate the gear n¼0;1;;N1 meshing frequency at this stage, the number of signal discrete angles fg s as l¼0;1;;L1 components is simply set to 1, i.e., P ¼ 1) g ðÞ s ¼ InterpolateðÞ / ðÞ t ;gtðÞ; s ; / l w n n l ~ ð16Þ fg x~ ðÞ t ; y~ ðÞ t ¼ arg min L x ðÞ t ; y ðÞ t ; f ðÞ t ; ð11Þ m s m m m m n ¼ 0; 1; ; N  1; l ¼ 0; 1; ; L  1; x ðÞ t ;y ðÞ t m m where s ¼ l/ðÞ t =ðÞ L  1 for l ¼ 0; 1; ...; L  1 l w N1 where x ðÞ t and y ðÞ t denote the demodulated signals m m denote uniformly discretized samples of angular variable s, corresponding to gear meshing frequency. Then, according and L  N represents the number of angular samples. to Eq. (4), phase information of the demodulated signals can be used for calculating IF deviation as 3.3 Polygonal wear amplitude estimation 1 d y~ ðÞ t Df ðÞ t ¼  arctan 2p dt x~ ðÞ t Apart from harmonic order, harmonic amplitude of the 0 0 1 y~ ðÞ t  x~ ðÞ t  x~ ðÞ t  y~ ðÞ t m m m polygonal wear is also a key factor which directly deter- ¼ ; ð12Þ 2 2 2p x~ ðÞ t þ y~ ðÞ t m m mines whether a wheel should be repaired. In reality, the polygonal wear amplitude is detected through static mea- where  denotes derivative. Finally, estimation of the gear surement of wheel roughness levels using specialized meshing frequency can be improved by error compensation instruments. To the best of our knowledge, on-board as polygonal wear amplitude detection methods based on ~ ~ f ðÞ t ¼f ðÞ t þ‘fg Df ðÞ t ; ð13Þ vehicle vibration signals have hardly been reported yet. m m Note that the wheel polygonal wear along circumferential where ‘fg  represents a low-pass filtering operator [35] for direction can be regarded as a type of irregularity whose reducing noise interference. dynamic effect on a vehicle is similar to that of rail cor- rugation. Therefore, we propose to estimate the polygonal 3.2 Signal resampling and order analysis wear amplitude by combining the ACMD with track irregularity detection methods. In this paper, a well-known With the method proposed in Sect. 3.1, the gear meshing irregularity measurement method called inertial algorithm frequency f ðÞ t can be accurately estimated and a wheel is employed due to its easy implementation and wide rotating frequency f ðÞ t can be calculated according to applicability. As illustrated in Fig. 1, the basic principle of Eq. (8). The obtained f ðÞ t is used as a reference frequency inertial algorithm is that the quadratic integral of acceler- for signal resampling and order analysis. At this stage, a ation gives displacement [45, 46]. Therefore, the track vibration acceleration signal of axle box is analyzed since irregularity can be measured by processing the acceleration the signal is sensitive to the wheel polygonization. Firstly, signal of axle box as an angular variable is defined as s ¼ / ðÞ t ¼ f ðÞ t dt w ZZ and therefore Eq. (2) can be rewritten as dtðÞ ¼ atðÞdtdt; ð17Þ gtðÞ ¼ A ðÞ t cos 2po / ðÞ t þ u : ð14Þ where atðÞ denotes acceleration signal. Note that the inte- p p w p p¼1 gral operation in Eq. (17) may introduce an undesired signal trend which should be removed in a post-processing Then, if the time variable is expressed with the angular step. 1 1 variable as t ¼ / ðÞ s where / denotes inverse function, w w It is worth noting that apart from harmonics induced by Eq. (14) can be mapped into angular domain as polygonal wear, the acceleration signal of axle box con- tains other interference components such as those caused g ðÞ s ¼ A ðÞ s cos 2po s þ u ; ð15Þ / p by random track irregularities. To correctly estimate p¼1 polygonal wear amplitudes, the ACMD is employed to 1 1 extract the harmonics to be interested before the inertial where g ðÞ s ¼g / ðÞ s and A ðÞ s ¼A / ðÞ s . It can be / p w p w seen that the angular-domain signal in Eq. (15) is free of Rail. Eng. Science (2022) 30(2):129–147 134 S. Chen et al. and their amplitudes of wheel polygonal wear without Car body using speed coder. 4 Dynamics simulation Suspension Accelerometer 4.1 Dynamics model Track irregularity In this section, the proposed polygonal wear detection Axle box, method is validated by dynamics simulation. To this end, a wheel locomotive-track coupled dynamics model considering gear transmissions [47–49] is introduced as shown in Fig. 3. The model fully takes into account rail-wheel interaction and thus can well simulate practical locomotive vibration responses. Fig. 1 Illustration of the principle of inertial algorithm The dynamics model contains three sub-models including locomotive, track and gear transmission sub- algorithm is performed. Procedures of the proposed models. In the locomotive sub-model, a car body is linked amplitude estimation method are listed as follows: with two bogie frames via secondary suspension (K –C ) sz sz (1) Construct initial IFs for the harmonics of the accel- while each bogie frame is supported by two wheelsets eration signal as o~ f ðÞ t , p ¼ 1; 2; ; P, according to through primary suspension (K –C ). In each bogie, there pz pz an improved ACMD in [36], where f ðÞ t and o~ w are two gearboxes which are rigidly connected with two denote the estimated rotating frequency and harmonic traction motors, respectively. In the dynamics model, most orders in Sects. 3.1 and 3.2, respectively. of the locomotive components are regarded as rigid bodies (2) Reconstruct harmonics by ACMD according to the represented by mass (M ) and rotational inertia (J ), and q q obtained initial IFs and then summate these harmon- each component has three degrees of freedom, i.e., vertical ics as (Z ), rotational (b ) and longitudinal (X ) motions, where q q q ¼ c; m, w denote the car body, motor and wheelset, gt ~ðÞ ¼ g~ ðÞ t : ð18Þ respectively. The track subsystem consists of rail, rail pads, p¼1 sleepers, ballasts and subgrade. The stiffness and damping of the rail pads, ballasts and subgrade are represented as (3) Calculate quadratic integral of the reconstructed K –C , K –C , and K –C , respectively, where i denotes pi pi bi bi fi fi signal in Eq. (18) to obtain a displacement signal sleeper number. As for the gear transmission subsystem, dtðÞ. Next, least squares spline fitting is applied to the motor with gearbox is suspended through the bearings dtðÞ to obtain a signal trend rt ~ðÞ which is then (K –C ) mounted on the wheel axle and the hanger rod br br subtracted to get the irregularity caused by polygonal (K –C ) linked with bogie frame. The pinion is con- ms ms wear as nected with rotor of the motor and the gear is mounted on ~ ~ ZtðÞ ¼ dtðÞ rt ~ðÞ: ð19Þ the wheel axle. The power of the traction motor is trans- mitted from the pinion to gear via their teeth engagement (4) Perform the order analysis approach in Sect. 3.2 to described by the stiffness and damping elements (K –C ) m m the irregularity ZtðÞ to estimate amplitudes of the along the line of action (LOA). More details about the harmonics of the wheel polygonal wear. model and parameter setting can be found in [47–49]. Note that it is not possible to extract all the harmonics for polygonal wear identification. Therefore, only domi- 4.2 Response analysis nant harmonics (e.g., higher-order harmonics) in order spectrum (in step 1) are reconstructed since they cause The dynamics model in Fig. 3 is employed to calculate stronger vibration to a locomotive. The whole flow chart of locomotive vibration responses caused by wheel polygonal the proposed wheel polygonal wear detection method is wear. Firstly, the irregularity induced by the polygonal presented in Fig. 2. It can be seen that, by synthetically wear with P harmonics is used as an excitation of the analyzing vibration acceleration signals of motor and axle model as box, the proposed method can estimate the wheel rotating frequency and quantificationally detect the harmonic orders Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 135 Acceleration signal of axle box Acceleration signal of motor Resample the signal at a constant angular interval Estimate gear meshing Order frequency by TF ridge analysis Identify harmonic orders of polygonal wear by FFT Wheel Refine the frequency by rotating ACMD frequency estimation Reconstruct harmonics by ACMD Obtain the rotating Amplitude frequency according to Reference estimation transmission relation Estimate polygonal wear frequency amplitudes by inertial algorithm Fig. 2 Flow chart of the proposed wheel polygonization detection method P With the estimated wheel rotating frequency, the signal o utðÞ Z ðÞ t ¼ X cos þ w ; ð20Þ 0 p p resampling technique is applied to acceleration signal of p¼1 axle box and then the order spectrum is obtained and compared with original spectrum as shown in Fig. 7. Due where X , o and w stand for amplitude, harmonic order p p to the time-varying frequency contents, Fourier spectrum and initial phase of the polygonal wear, respectively, utðÞ cannot reveal effective fault information of the polygonal denotes running distance function of locomotive (i.e., the wear. On the contrary, since the non-stationary effect of the integral of locomotive speed), and R is wheel radius (the signal is fully removed by resampling technique, harmonic radius is set to 0.625 m in the simulation). In this section, orders of the polygonal wear can be clearly identified in the polygonal wear with three harmonics is considered and order spectrum (see Fig. 7b). In addition, the order spec- the parameters are provided in Table 1. Figure 4 illustrates trum based on acceleration signal shows much better res- the polygonal wear and its order analysis result. To simu- olution than that in Fig. 4b since the duration of the late the random interference of track irregularity, the acceleration signal is long enough. American fifth-grade track spectrum is employed to gen- Then, polygonal wear amplitude estimation is carried erate random excitation of the dynamics model. It is out. Firstly, based on the order analysis results, the har- assumed that the locomotive is running under a traction monics induced by polygonal wear can be extracted by condition and the speed is gradually increased to 80 km/h. ACMD as shown in Fig. 8. An improved TFR of these The simulated vibration acceleration signals of axle box harmonics by ACMD (see Eq. (7)) is also provided and and motor based on the dynamics model are provided in compared with that by STFT as shown in Fig. 9. It can be Fig. 5. The gear meshing frequency and frequency contents observed that STFT cannot resolve these harmonics due to induced by wheel polygonal wear can be observed in the the poor resolution and interferences from track irregular- TFRs by STFT. However, due to poor resolution of the ity. The TFR obtained by ACMD can clearly uncover time- TFR, details of the wheel polygonization are not available. varying features of the harmonics. Next, inertial algorithm Next, the proposed fault detection method is applied. is applied to the extracted harmonics and the analysis Firstly, as shown in Fig. 6, the gear meshing frequency is results are shown in Fig. 10. It is indicated that the integral estimated and then the wheel rotating frequency can be operation will lead to a complex signal trend with a large calculated according to Eq. (8), where the number of the magnitude and therefore signal detrending is necessary for gear teeth is set to Z ¼120 for the locomotive. The shaft detection of the irregularity caused by polygonal wear. mean-square errors of the estimated gear meshing fre- After detrending, radial deviations (or irregularities) caused quencies by ridge detection and ACMD are 6.43 and by polygonal wear of different wheel rotation cycles are - 5.73 dB, respectively. It can be seen that the IF esti- clearly recovered as illustrated in Fig. 10b. Finally, mation results by ridge detection exhibit staircase effects amplitudes of harmonics of the polygonal wear are esti- due to limited resolution of the TFR (see Fig. 6a, b). After mated by performing order analysis to the recovered radial frequency refinement by ACMD, the IF estimation accu- deviation of a cycle (suggested to use average data of racy can be significantly improved. Rail. Eng. Science (2022) 30(2):129–147 136 S. Chen et al. Fig. 3 Locomotive-track coupled dynamics model considering gear transmissions: a model and b gear transmission systems in one bogie proposed method for quantitative detection of locomotive Table 1 Simulated wheel polygonal wear with three harmonics wheel polygonal wear. p Amplitude X Order Phase (mm) o w 1 0.050 7 0 5 Practical application 2 0.040 8 2p=3 3 0.025 9 4p=3 In this section, the proposed fault detection method is applied to vibration signal analysis of an in-service loco- motive to further demonstrate the effectiveness of the different rotation cycles at lower-speed stage to reduce the method. influence of track irregularities) as shown in Fig. 10c. One can find that the estimated harmonic amplitudes are very close to the true ones showing the effectiveness of the Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 137 Fig. 4 Wheel polygonal wear in the simulation case: a polygonal wear described in the polar coordinate and b order analysis of the polygonal wear (a) (b) 3 1000 -1 Induced by polygonisation -2 -3 0 2 4 6 8 024 6 8 Time (s) Time (s) (c) 1000 (d) -1 Gear meshing frequency -2 0 2 4 6 8 02 4 6 8 Time (s) Time (s) Fig. 5 Simulated vibration signals of the locomotive with wheel polygonization: a and b show waveform and STFT of the acceleration signal of axle box, respectively; c and d illustrate the signal of motor 5.1 Field test Power) carried out a thorough field test for the locomotive [15]. Firstly, wheel roughness levels around circumference The locomotive operated on the Chinese heavy-haul rail- were measured by Mu¨ller-BBM instrument to assess the way from Zhongwei to Yuci. Vibration alarm phenomenon wheel polygonal wear as shown in Fig. 11. The measuring occurred several times for on-board monitoring system of accuracy of the instrument is 0.1 lm with a sampling the locomotive. To investigate root cause of the alarm precision of 1 mm. The roughness measurement was phenomenon, a research group from Train and Track achieved by manually twirling wheels. In addition, vibra- Research Institute (in State Key Laboratory of Traction tion accelerations of key components such as axle box and Rail. Eng. Science (2022) 30(2):129–147 Acceleration (g) Acceleration (g) Frequency (Hz) Frequency (Hz) 138 S. Chen et al. 5.9 6.1 6.3 c d 5.9 6.1 6.3 Time (s) Fig. 6 IF estimation results from the simulated signal in Fig. 5(d): a and b show the estimated gear meshing frequency and wheel rotating frequency by ridge detection, respectively; c and d show the refined results by ACMD (a) (b) 0.08 0.2 0.06 0.15 0.04 0.1 0.02 0.05 0 0 0 20 40 60 0 5 10 Order Frequency (Hz) Fig. 7 Comparison of spectrums of the simulated vibration signals of axle box: a original Fourier spectrum; b order spectrum after signal resampling with the estimated wheel rotating frequency Rail. Eng. Science (2022) 30(2):129–147 Acceleration (g ) Frequency (Hz) Acceleration (g) Quantitative detection of locomotive wheel polygonization... 139 (b) 0.5 1 -0.5 0 .5 0 246 8 0.5 -0.5 246 8 - 0.5 0.5 -1 -0.5 0 2 4 6 0 246 8 Time (s) Time (s) Fig. 8 Harmonic extraction by ACMD for the simulated vibration signals of axle box: a extracted harmonics; b sum of the extracted harmonics (a) (b) Time (s) Fig. 9 Comparison of TFRs of the simulated vibration signals of axle box: a STFT and b ACMD motor were also measured during operation process of the traction condition. Figure 15 illustrates the measured locomotive as shown in Fig. 12. To determine whether the vibration signals and their TF analysis results. It can be root cause of the vibration alarm is the wheel wear, two seen that the dominant frequency contents of the vibration round tests before and after wheel re-profiling were con- signal of axle box are induced by polygonal wear while ducted, respectively. The wheel re-profiling was achieved those of the motor vibration signal are mainly related to by wheel lathe to reduce radial deviation caused by gear meshing operation. Therefore, the proposed method is polygonal wear as shown in Fig. 13. applied to the motor vibration signal to estimate the gear meshing frequency and wheel rotating frequency as shown 5.2 Detection results in Fig. 16, where the number of gear teeth Z is 120. shaft After the rotating frequency is obtained, signal resampling Firstly, the measured data of the locomotive before wheel and order analysis are performed to the vibration signal of re-profiling is analyzed. The measured wheel roughness axle box as illustrated in Fig. 17. Compared with the raw data and its analysis result are provided in Fig. 14.It Fourier spectrum (see Fig. 17a), the developed order clearly shows that the wheel suffers from high-order spectrum (see Fig. 17b) correctly reveals all the harmonic polygonal wear (i.e., the 17th–24th orders). Then, the orders of the polygonal wear and it even shows much better proposed method is applied to locomotive vibration data to resolution than the analysis results based on the measured test whether the polygonal wear information can be wheel roughness in Fig. 14b. TFRs of the polygonal-wear- detected. In this test, the locomotive operates under a related harmonics are shown in Fig. 18. It can be seen that Rail. Eng. Science (2022) 30(2):129–147 Order 9 Order 8 Order 7 Acceleration (g) 140 S. Chen et al. Fig. 10 Polygonal wear amplitude estimation by inertial algorithm in the simulation case: a obtained wheel radial deviation by quadratic integral of the reconstructed harmonics; b radial deviation after detrending; c order analysis of the reconstructed polygonal wear Fig. 11 Field measurement of wheel polygonal wear of the locomotive [15]: a wheel polygonal wear; b polygonal wear measurement by Mu¨ller- BBM instrument Rail. Eng. Science (2022) 30(2):129–147 Quantitative detection of locomotive wheel polygonization... 141 Wheelset (b) (a) Gear Axle box Motor (c) (d) Bogie Fig. 12 Field measurement of vibration of the locomotive: a the locomotive; b schematic diagram of measurement positions; c Measurement positions of axle box; d measurement positions of motor (a) (b) Polygonised wheel Drive wheel Wheel Lathe tool Fig. 13 Wheel re-profiling with wheel lathe: a schematic diagram and b wheel lathe [15] (a) (b) 0.08 0.06 150 30 0.04 180 0 -0.2 0.02 0.2 10 20 30 40 Order Fig. 14 Measured wheel polygonal wear of the locomotive: a polygonal wear described in the polar coordinate; b order analysis of the polygonal wear Rail. Eng. Science (2022) 30(2):129–147 Polygonal wear (mm) Polygonal wear (mm) 142 S. Chen et al. c d Gear meshing frequency Time (s) Fig. 15 Measured vibration signals of the locomotive with wheel polygonization: a and b show waveform and STFT of the acceleration signal of axle box, respectively; c and d illustrate the signal of motor (b) 5 4.5 3.5 0 2.5 60 60 0 20 40 0 20 40 Time (s) Time (s) Fig. 16 Estimated gear meshing frequency and wheel rotating frequency by ACMD from the signal in Fig. 15d: a gear meshing frequency and b wheel rotating frequency Rail. Eng. Science (2022) 30(2):129–147 e ( Frequency (Hz) Acceleration (g) Acceleration (g) Frequency (Hz) Frequency (Hz) Frequency (Hz) Quantitative detection of locomotive wheel polygonization... 143 (b) 19 21 Fig. 17 Comparison of spectrums of the measured vibration signals of axle box: a original Fourier spectrum; b order spectrum after signal resampling with the estimated wheel rotating frequency (b) Fig. 18 Comparison of TFRs of the measured vibration signals of axle box: a STFT and b ACMD (a) (b) 0.015 0.05 150 30 0.01 -0.05 180 0 0.005 300 40 10 20 30 Order Fig. 19 Measured wheel polygonal wear of the locomotive after wheel re-profiling: a polygonal wear described in the polar coordinate; b order analysis of the polygonal wear Rail. Eng. Science (2022) 30(2):129–147 Polygonal wear (mm) Acceleration (g) Polygonal wear (mm) Acceleration (g) 144 S. Chen et al. -2 -4 -6 50 100 0 50 100 Time (s) -2 Gear meshing frequency -4 0 50 100 50 100 Time (s) Fig. 20 Measured vibration signals of the locomotive after wheel re-profiling: a and b show waveform and STFT of the acceleration signal of axle box, respectively; c and d illustrate the signal of motor ACMD successfully resolves and extracts all the harmonics wheel polygonal wear. Then, the proposed method is and thus generates a high-resolution TFR (see Fig. 18b; applied to demonstrate its usefulness in detecting such a due to the limited space, instead of waveforms, only TFRs slight fault. IF estimation and order analysis results based are provided). These extracted harmonics can be further on the vibration signals are given in Figs. 21 and 22, used for wear amplitude estimation. respectively. It shows that the harmonic orders of the For comparison, wheel roughness and locomotive polygonal wear are clearly identified (see Fig. 22b). vibration data after wheel re-profiling are analyzed. Fig- Comparison of the TFRs obtained by STFT and ACMD is ure 19 shows the measured roughness data. It can be seen illustrated in Fig. 23 indicating higher resolution of ACMD that the wheel lathe does not fully eliminate wheel poly- in separating closely spaced harmonics induced by wheel gonization although the roughness magnitudes and the polygonal wear. number of harmonics are significantly reduced (compared Finally, inertial algorithm is applied to the harmonics with that in Fig. 14). In this case, the dominant harmonic extracted by ACMD for polygonal wear amplitude esti- orders of the polygonal wear are 17 and 24. To show the mation. The amplitude estimation results before and after effectiveness of the method in different working condi- wheel re-profiling are compared in Fig. 24. In both cases, tions, vibration signals of the locomotive during braking the estimated polygonal wear amplitudes based on accel- process are analyzed as illustrated in Fig. 20. It can be eration signals are close to those measured by instrument as found that magnitudes of the vibration acceleration signals shown in Fig. 24. The results indicate that the proposed and energy of the polygonal-wear-induced components are method is effective for detection of wheel polygonal wear decreased evidently. It implies that the vibration alarm of different levels. phenomenon of the locomotive is mainly attributed to the Rail. Eng. Science (2022) 30(2):129–147 Induced by polygonisation Time (s) Time (s) Acceleration (g) Acceleration (g) Frequency (Hz) Frequency (Hz) Quantitative detection of locomotive wheel polygonization... 145 (b) 1000 5 4.5 3.5 2.5 0 50 100 0 50 100 Time (s) Time (s) Fig. 21 Estimated gear meshing frequency and wheel rotating frequency by ACMD from the signal in Fig. 20d: a gear meshing frequency and b wheel rotating frequency (b) 0.05 0.6 24 0.04 0.4 0.03 0.02 0.2 0.01 200 10 20 30 50 100 150 40 Order Frequency (Hz) Fig. 22 Comparison of spectrums of the measured vibration signals of axle box after wheel re-profiling: a original Fourier spectrum; b order spectrum after signal resampling with the estimated wheel rotating frequency (b) 150 150 100 100 50 50 0 50 100 0 50 100 Time (s) Time (s) Fig. 23 Comparison of TFRs of the measured vibration signals of axle box after wheel re-profiling: a STFT and b ACMD Rail. Eng. Science (2022) 30(2):129–147 Acceleration (g) Frequency (Hz) Frequency (Hz) Acceleration (g) Frequency (Hz) Frequency (Hz) 146 S. Chen et al. (b) 0.015 0.08 Measured Measured Estimated Estimated 0.06 0.01 0.04 0.005 0.02 0 0 10 20 30 10 20 30 40 Order Order Fig. 24 Estimated polygonal wear amplitudes by inertial algorithm for the experimental vibration signals: a before wheel re-profiling and b after wheel re-profiling Open Access This article is licensed under a Creative Commons 6 Conclusions Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as In this paper, a novel quantitative detection method for wheel long as you give appropriate credit to the original author(s) and the polygonal wear under non-stationary conditions using loco- source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this motivevibration signals has beendeveloped. Firstly, TF ridge article are included in the article’s Creative Commons licence, unless detection has been combined with ACMD to accurately esti- indicated otherwise in a credit line to the material. If material is not mate the time-varying wheel rotating frequency from motor included in the article’s Creative Commons licence and your intended vibration signal without using speed coder. Next, the order use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright analysis technique based on signal resampling has been holder. To view a copy of this licence, visit http://creativecommons. applied to vibration signal of axle box for harmonic order org/licenses/by/4.0/. detection. Finally, polygonal wear amplitudes have been estimated by integrating the ACMD with inertial algorithm. 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Journal

Railway Engineering ScienceSpringer Journals

Published: Jun 1, 2022

Keywords: Wheel polygonal wear; Fault diagnosis; Non-stationary condition; Adaptive mode decomposition; Time–frequency analysis

References