The proposed wheel polygonization detection method based on ACMD is presented in this section. The method mainly includes three steps: (1) wheel rotating frequency estimation; (2) signal resampling and order analysis; (3) polygonal wear amplitude estimation.
Wheel rotating frequency estimation
As shown in Eq. (2), accurately estimating the wheel rotating frequency \(f_{{\text{w}}} \left( t \right)\) is crucial for detection of harmonic orders of the polygonal wear. Most existing methods employ a speed coder to measure the vehicle speed for wheel rotating frequency estimation [23]. However, due to the limited space and unpredictable safety concerns, it may not be allowed to install a speed measuring device in a railway vehicle. To address this issue, a novel rotating frequency estimation method based on vibration signal analysis is proposed in this paper. Note that it is difficult to directly detect wheel rotating frequency from vibration signal of a vehicle with a healthy wheel shaft. Therefore, it is suggested to extract other dominant frequencies and then obtain the rotating frequency according to transmission relation. It is known that gear transmission systems are widely used in a locomotive. Due to the continuous meshing operation, there exists a distinct meshing frequency component in vibration response of the gear transmission system of the locomotive. Therefore, we propose to estimate the gear meshing frequency \(f_{{\text{m}}} \left( t \right)\) at first and then calculate the wheel rotating frequency as
$$f_{{\text{w}}} \left( t \right){ = }\frac{{f_{{\text{m}}} \left( t \right)}}{{Z_{{{\text{shaft}}}} }},$$
(8)
where \(Z_{{{\text{shaft}}}}\) denotes the number of the teeth of the gear fixedly connected with the wheel shaft.
To accurately estimate the gear meshing frequency, the ACMD is employed to analyze vibration acceleration signals of motor which is directly connected with gearbox and is less influenced by wheel-rail excitations [39, 40]. Since IF initialization plays an important role in analysis results of ACMD, it is necessary to obtain a good estimate of an initial IF. To this end, a TF ridge detection technique is utilized to deal with the IF initialization issue. Firstly, a TFR of the vibration signal is obtained by a proper TF transform. For simplicity, STFT of the signal \(g\left( t \right)\) is calculated as
$${\text{STFT}}\left( {t,f} \right) = \int_{ - \infty }^{ + \infty } {g\left( \upsilon \right)h^{ * } \left( {\upsilon - t} \right){\text{e}}^{{ - {\text{j}}2{\uppi }f\upsilon }} {\text{d}}\upsilon },$$
(9)
where \(\text{j}^{2} = - 1\), \(h\left( t \right)\) 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 can be detected by solving [41, 42]
$$\tilde{f}_{{\text{m}}}^{{\text{r}}} \left( t \right) = \mathop {\arg \max }\limits_{\varOmega \left( t \right)} \left\{ {\sum\limits_{i = 0}^{N - 1} {\left| {{\text{STFT}}\left( {t_{i} ,\varOmega \left( {t_{i} } \right)} \right)} \right|^{2} } - \lambda \sum\limits_{i = 1}^{N - 1} {\left| {\varOmega \left( {t_{i} } \right) - \varOmega \left( {t_{i - 1} } \right)} \right|^{2} } } \right\},$$
(10)
where \(t = t_{0} , t_{1},\, \ldots ,\,t_{N - 1}\) denote sampling time instants, N is the number of samples, \(\varOmega \left( t \right)\) stands for a set of all the TF paths from \(t = t_{0}\) to \(t = t_{N - 1}\), and \(\lambda\) is a penalty factor. Eq. (10) implies that a desired ridge curve should pass through TF points with large magnitudes and there is no significant jump phenomenon between adjacent points (i.e., the curve is smooth enough).
The accuracy of IF estimation by TF ridge detection is dominated by resolution of the TFR. Due to the limitation of traditional TF methods in analyzing strongly modulated signals, it is difficult to get a high-accuracy estimation of the IF by ridge detection. Therefore, the ACMD is further applied to refine IF estimation. Firstly, with the coarsely estimated gear meshing frequency by ridge detection, i.e., \(\tilde{f}_{{\text{m}}}^{{\text{r}}} \left( t \right)\), the demodulated signals can be obtained as (see Eq. (5); since ACMD is only employed to estimate the gear meshing frequency at this stage, the number of signal components is simply set to 1, i.e., \(P = 1\))
$$\left\{ {\tilde{x}_{{\text{m}}} \left( t \right),\tilde{y}_{{\text{m}}} \left( t \right)} \right\} = \mathop {\arg \min }\limits_{{x_{{\text{m}}} \left( t \right),y_{{\text{m}}} \left( t \right)}} \left\{ {{\mathcal{L}}_{\tau } \left( {x_{{\text{m}}} \left( t \right),y_{{\text{m}}} \left( t \right),\tilde{f}_{{\text{m}}}^{{\text{r}}} \left( t \right)} \right)} \right\},$$
(11)
where \(x_{{\text{m}}} \left( t \right)\) and \(y_{{\text{m}}} \left( t \right)\) denote the demodulated signals corresponding to gear meshing frequency. Then, according to Eq. (4), phase information of the demodulated signals can be used for calculating IF deviation as
$$\Delta f_{{\text{m}}} \left( t \right) = - \frac{1}{{2{\uppi }}}\frac{{\text{d}}}{{{\text{d}}t}}\left\{ {\arctan \left[ {\frac{{\tilde{y}_{{\text{m}}} \left( t \right)}}{{\tilde{x}_{{\text{m}}} \left( t \right)}}} \right]} \right\} = \frac{1}{{2{\uppi }}}\left\{ {\frac{{\tilde{y}_{{\text{m}}} \left( t \right) \cdot \tilde{x}_{{\text{m}}}^{\prime } \left( t \right) - \tilde{x}_{{\text{m}}} \left( t \right) \cdot \tilde{y}_{{\text{m}}}^{\prime } \left( t \right)}}{{\tilde{x}_{{\text{m}}}^{2} \left( t \right) + \tilde{y}_{{\text{m}}}^{2} \left( t \right)}}} \right\},$$
(12)
where \(\cdot^{\prime}\) denotes derivative. Finally, estimation of the gear meshing frequency can be improved by error compensation as
$$\tilde{f}_{{\text{m}}} \left( t \right){ = }\tilde{f}_{{\text{m}}}^{{\text{r}}} \left( t \right){ + }\ell \left\{ {\Delta f_{{\text{m}}} \left( t \right)} \right\},$$
(13)
where \(\ell \left\{ \cdot \right\}\) represents a low-pass filtering operator [35] for reducing noise interference.
Signal resampling and order analysis
With the method proposed in Sect. 3.1, the gear meshing frequency \(f_{{\text{m}}} \left( t \right)\) can be accurately estimated and a wheel rotating frequency \(f_{{\text{w}}} \left( t \right)\) can be calculated according to Eq. (8). The obtained \(f_{{\text{w}}} \left( t \right)\) is used as a reference frequency for signal resampling and order analysis. At this stage, a vibration acceleration signal of axle box is analyzed since the signal is sensitive to the wheel polygonization. Firstly, an angular variable is defined as \(s = \phi_{{\text{w}}} \left( t \right) = \int_{0}^{t} {f_{{\text{w}}} \left( \upsilon \right){\text{d}}\upsilon }\) and therefore Eq. (2) can be rewritten as
$$g\left( t \right) = \sum\limits_{p = 1}^{P} {A_{p} \left( t \right)\cos \left[ {2{\uppi }o_{p} \phi_{{\text{w}}} \left( t \right) + \varphi_{p} } \right]}.$$
(14)
Then, if the time variable is expressed with the angular variable as \(t = \phi_{{\text{w}}}^{ - 1} \left( s \right)\) where \(\phi_{{\text{w}}}^{ - 1}\) denotes inverse function, Eq. (14) can be mapped into angular domain as
$$g_{\phi } \left( s \right) = \sum\limits_{p = 1}^{P} {A_{p}^{\phi } \left( s \right)\cos \left[ {2{\uppi }o_{p} s + \varphi_{p} } \right]},$$
(15)
where \(g_{\phi } \left( s \right){ = }g\left( {\phi_{{\text{w}}}^{ - 1} \left( s \right)} \right)\) and \(A_{p}^{\phi } \left( s \right){ = }A_{p} \left( {\phi_{{\text{w}}}^{ - 1} \left( s \right)} \right)\). It can be seen that the angular-domain signal in Eq. (15) is free of FM effect and exhibits constant frequencies at \(o_{p}\) for \(p = 1, 2, \cdots ,P\). Therefore, it is easy to identify harmonic orders (i.e., \(o_{p}\)) by applying Fourier transform to the signal. In practice, the nonlinear mapping operation (i.e., Eq. (14) to Eq. (15)) is achieved by signal resampling or data interpolation [43, 44]. Namely, the time-domain signal \(g\left( t \right)\) is interpolated from discrete time points \(\left\{ {t_{n} } \right\}_{n = 0, 1, \cdots ,N - 1}\) to discrete angles \(\left\{ {s_{l} } \right\}_{l = 0, 1, \cdots ,L - 1}\) as
$$\begin{gathered} g_{\phi } \left( {s_{l} } \right) = {\text{Interpolate}}\left( {\phi_{{\text{w}}} \left( {t_{n} } \right),g\left( {t_{n} } \right),s_{l} } \right), \hfill \\ \, n = 0, 1, \cdots , \, N - 1, \, l = 0, 1, \cdots , \, L - 1, \hfill \\ \end{gathered}$$
(16)
where \(s_{l} = {{l\phi_{{\text{w}}} \left( {t_{N - 1} } \right)} \mathord{\left/ {\vphantom {{l\phi_{{\text{w}}} \left( {t_{N - 1} } \right)} {\left( {L - 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {L - 1} \right)}}\) for \(l = 0, 1,\, \ldots ,\,L - 1\) denote uniformly discretized samples of angular variable \(s\), and \(L \ge N\) represents the number of angular samples.
Polygonal wear amplitude estimation
Apart from harmonic order, harmonic amplitude of the polygonal wear is also a key factor which directly determines whether a wheel should be repaired. In reality, the polygonal wear amplitude is detected through static measurement of wheel roughness levels using specialized instruments. To the best of our knowledge, on-board polygonal wear amplitude detection methods based on vehicle vibration signals have hardly been reported yet. Note that the wheel polygonal wear along circumferential direction can be regarded as a type of irregularity whose dynamic effect on a vehicle is similar to that of rail corrugation. Therefore, we propose to estimate the polygonal wear amplitude by combining the ACMD with track irregularity detection methods. In this paper, a well-known irregularity measurement method called inertial algorithm is employed due to its easy implementation and wide applicability. As illustrated in Fig. 1, the basic principle of inertial algorithm is that the quadratic integral of acceleration gives displacement [45, 46]. Therefore, the track irregularity can be measured by processing the acceleration signal of axle box as
$$d\left( t \right) = \iint {a\left( t \right)}{\text{d}}t{\text{d}}t,$$
(17)
where \(a\left( t \right)\) denotes acceleration signal. Note that the integral operation in Eq. (17) may introduce an undesired signal trend which should be removed in a post-processing step.
It is worth noting that apart from harmonics induced by polygonal wear, the acceleration signal of axle box contains other interference components such as those caused by random track irregularities. To correctly estimate polygonal wear amplitudes, the ACMD is employed to extract the harmonics to be interested before the inertial algorithm is performed. Procedures of the proposed amplitude estimation method are listed as follows:
-
(1)
Construct initial IFs for the harmonics of the acceleration signal as \(\tilde{o}_{p} \tilde{f}_{{\text{w}}} \left( t \right)\), \(p = 1, 2, \cdots ,P\), according to an improved ACMD in [36], where \(\tilde{f}_{{\text{w}}} \left( t \right)\) and \(\tilde{o}_{p}\) denote the estimated rotating frequency and harmonic orders in Sects. 3.1 and 3.2, respectively.
-
(2)
Reconstruct harmonics by ACMD according to the obtained initial IFs and then summate these harmonics as
$$\tilde{g}\left( t \right) = \sum\limits_{p = 1}^{P} {\tilde{g}_{p} \left( t \right)}.$$
(18)
-
(3)
Calculate quadratic integral of the reconstructed signal in Eq. (18) to obtain a displacement signal \(\tilde{d}\left( t \right)\). Next, least squares spline fitting is applied to \(\tilde{d}\left( t \right)\) to obtain a signal trend \(\tilde{r}\left( t \right)\) which is then subtracted to get the irregularity caused by polygonal wear as
$$\tilde{Z}\left( t \right) = \tilde{d}\left( t \right) - \tilde{r}\left( t \right).$$
(19)
-
(4)
Perform the order analysis approach in Sect. 3.2 to the irregularity \(\tilde{Z}\left( t \right)\) to estimate amplitudes of the harmonics of the wheel polygonal wear.
Note that it is not possible to extract all the harmonics for polygonal wear identification. Therefore, only dominant harmonics (e.g., higher-order harmonics) in order spectrum (in step 1) are reconstructed since they cause stronger vibration to a locomotive. The whole flow chart of the proposed wheel polygonal wear detection method is presented in Fig. 2. It can be seen that, by synthetically analyzing vibration acceleration signals of motor and axle box, the proposed method can estimate the wheel rotating frequency and quantificationally detect the harmonic orders and their amplitudes of wheel polygonal wear without using speed coder.