Rail corrugation diagnosis from axle box vibration signals using optimized empirical wavelet transform and CNN–LSTM

2.1. Model establishment

In the article, a dynamic model is established based on the parameters of a real urban rail vehicle. In the modeling process, only the primary suspension, secondary suspension, pivot node, primary vertical damper, and secondary air spring are considered. This model focuses only on key structures and ignores other factors with less impact. In the article, the rail is considered as a Timoshenko beam to simulate the continuous interaction between the wheel and the rail.

During the simulation process, the track deflection response is integrated into the vehicle model using the SIMAT co-simulation interface. The specific co-simulation principle is shown in Fig. 1. Firstly, the vehicle speed, wheel-rail contact force, and wheelset position from SIMPACK are inputted into the rail model. After calculation by the rail model, the actual vibration characteristics of the rail are outputted to the SIMPACK model. Due to changes in the rail's vibration characteristics, the wheel-rail contact relationship changes, ultimately leading to changes in the wheel-rail contact force. Then, the new wheel-rail force is inputted into the rail model again.

Fig. 1Co-simulation schematic diagram

2.2. Model validation

A 10-meter section of rail with corrugation was configured in the simulation, exhibiting a waveform illustrated in Fig. 2(a). The corrugation was characterized by a wavelength of 100 mm and a depth of 0.05 mm. In SIMPACK, set up a 100-meter-long straight track and apply the harmonic excitation corresponding to the aforementioned rail corrugation at the range of 55 to 65 meters. The corresponding track irregularity is shown in Fig. 2(b), where the red box indicates the added corrugation excitation, and the vertical irregularity of the remaining track is set to 0.

Fig. 2Corrugation waveform and track irregularity of the simulated rail section

a)

b)

If the axle box vibration acceleration coincides with the wave grinding frequency $\omega$, then the wave grinding pass frequency $f$ satisfies the following relationship with vehicle speed $v$ and wave grinding wavelength $\lambda$:

Set the sampling frequency to 2000 Hz and the vehicle speed to 60 km/h. According to Eq. (1), the pass frequency corresponding to this wave-induced vibration condition is 166.7 Hz.

The vertical axle box vibration signal in the wave-grounding section is shown in Fig. 3(a). The peak values at the start and end points of the wave-grounding section are significantly higher than those in other sections. This indicates that substantial impact vibrations occur when the vehicle enters and exits the wave-grounding area, significantly affecting operational smoothness and safety. The 55-65 m section incorporates wave-induced excitation, where axle box acceleration is markedly higher than in other sections. This confirms axle box acceleration effectively reflects the time-domain characteristics of wave-induced sections.

The frequency domain information of the axle box vertical vibration signal in the wave-grinding section is shown in Fig. 3(b). The first-order vibration frequency of the axle box vertical vibration is 166.7 Hz, consistent with the wave-grinding frequency. This confirms that axle box acceleration can reflect the frequency domain characteristics of the wave-grinding section.

Fig. 3Simulation verification of shaft box vertical vibration acceleration

a) Vertical vibration acceleration signal in spatial domain

b) FFT spectrum

The frequency of axlebox acceleration matches that of wave-induced wear, indicating the high accuracy of the adopted model. This also confirms that vertical vibration acceleration of the axlebox can reflect the time-domain and frequency-domain characteristics of wave-induced wear, making it feasible to diagnose wave-induced wear using vertical vibration acceleration of the axlebox.

3.1. Classification standards for Hamamatsu

The rail wave wear index method, proposed by Liu Zhaojiao et al. [8], enables more effective evaluation of short-wave track irregularities and exhibits excellent repeatability. This paper employs the rail wave wear index as an evaluation metric. The index is computed from the axlebox vertical acceleration after EWT-OSTU reconstruction. First, the moving root mean square (RMS) of the reconstructed signal is calculated with a window length of 0.25 m:

where $N$ corresponds to 0.25 m of track. The track is then divided into 20 m segments, and the maximum RMS value within each segment, $a_{max}$, is extracted. Finally, the corrugation index (in dB) for a segment is defined as:

where $\overline{a}$ is the mean of the maximum RMS values over all segments. The severity levels are categorised as shown in Table 1. This index has been widely adopted in Chinese metro maintenance practice and provides a repeatable proxy for rail surface condition.

3.2. Validation of the EWT-OSTU algorithm based on simulated signals

Simulating severely corrugated track conditions, the vertical acceleration amplitude of axle boxes in corrugated sections significantly exceeds that of normal sections, making them easily distinguishable. To validate the efficacy of the EWT-OSTU algorithm in extracting corrugation features, a dataset representative of mild corrugation (Grade 1) was chosen for analysis. This typical case, characterized by a wavelength of 150 mm and a depth of 0.02 mm, was sourced from the established classification results.

The time-domain plots and energy spectra of the filtered signal and decomposed sub-signals are shown in Fig. 4. After EWT decomposition, two IMF components were obtained: IMF1 contained substantial low-frequency track excitation signals, while the primary wave-grinding components were concentrated in IMF2, exhibiting relatively pure signal composition with a dominant vibration frequency matching the wave-grinding frequency.

Finally, the signal with the highest energy and correlation coefficient was selected for reconstruction using the OSTU algorithm. The reconstructed signal results are shown in Fig. 5, successfully separating the wave-grinding signal from the original signal.

Fig. 4First EWT decomposition of algorithm validation data

Fig. 5Algorithm validation data OSTU reconstruction result map

4.1. CNN–LSTM fault diagnosis model

4.1.3. Overall structure of the fault diagnosis model

We propose a CNN–LSTM model based on empirical wavelet transform, whose structure is shown in Fig. 6.

Fig. 6Structure of CNN-LSTM model based on EWT-OSTU decomposition

The input sequence is first processed by a CNN for spatial feature extraction. The resulting feature sequence is then fed into an LSTM to model the temporal dependencies. This approach effectively reduces the sequence length fed into the LSTM, as the CNN extracts local features from the sequence, allowing the LSTM to focus only on critical information within the sequence and improving training efficiency. Due to the presence of substantial interference noise in the raw axle box signal, training results may be compromised. Following the data preprocessing workflow outlined in the previous section, the raw signals undergo preprocessing. Subsequently, the sub-signals are calculated for their RMS values as described above and used for training the CNN.

The CNN consists of five convolutional layers, each followed by batch normalisation and a ReLU activation. The kernel sizes, strides, and output channels are detailed in Table 2. After each convolution, a max-pooling layer with size 2×1 and stride 2 reduces the spatial dimension. The final CNN output is a 64×30 feature map, which is flattened and fed into a three-layer LSTM with 64 hidden units per layer. A dropout rate of 0.5 is applied after each LSTM layer to prevent overfitting. The model is trained using the Adam optimizer with an initial learning rate of 0.001 for 100 epochs, and the cross-entropy loss is used as the objective function. Early stopping based on validation loss is employed to avoid overtraining.

Table 2Structure and parameters of the CNN–LSTM model

Layer	Number of kernels	Size / stride	Units	Input size	Output size	Activation function	Parameter
Conv1	4	3×2/1×1		1×1000×2	4×1000	relu	24
BN				4×1000	4×1000		8
Pool		2×1/2×1		4×1000	4×500
Conv2	8	3×1/1×1		8×500	8×500	relu	96
BN				8×500	8×500		8
Pool		2×1/2×1		8×500	8×250
Conv3	16	3×1/1×1		16×250	16×250	relu	384
BN				16×250	16×250		32
Pool		2×1/2×1		16×250	16×125
Conv4	32	3×1/1×1		32×125	32×125	relu	1536
BN				32×125	32×125		64
Pool		2×1/2×1		32×125	32×62
Conv5	64	60×1/1×1		32×62	64×61	relu	122880
BN				64×61	64×61		128
Pool		2×1/2×1		64×61	64×30
LSTM			64	64×30	64×1	tanh	89600
Fully connected			3	64×1	1×3		195
Classification			3	1×3	1×3
						Total	214963

The proposed CNN-LSTM intelligent diagnostic model, which utilizes EWT–OSTU for signal decomposition and reconstruction, is architecturally organized into four dedicated functional tiers: the data acquisition layer, the data preprocessing layer, the feature extraction layer, and the fault classification layer. The specific roles of each layer are detailed below:

(1) Raw Data Acquisition Layer:

This layer is responsible for collecting the raw signals of axle box vibration acceleration and vehicle velocity. The collected data is subsequently partitioned into fixed-length segments based on a 20-meter travel distance.

(2) Data Preprocessing Layer:

This layer executes the EWT-OSTU decomposition and reconstruction algorithm on the segmented raw data to generate relevant sub-signals. For each of these sub-signals, it calculates the Root Mean Square (RMS) values for both velocity and axle box vertical vibration acceleration at consistent spatial intervals of 0.02 m, thereby constructing the input samples. Given that the state of rail wave wear is strongly correlated with axlebox vertical vibration acceleration – a parameter that is itself influenced by vehicle speed – a secondary normalization procedure is applied to the acceleration RMS data. This process normalizes the acceleration data using the effective minimum and maximum speeds to derive a standardized speed factor, as defined by the following equation:

5

$v_{rms}^{\text{'}}=\frac{v_{rms}-v_{low}}{v_{high}-v_{low}}.$

(3) CNN Feature Extraction Layer.

This layer is responsible for identifying local spatial patterns within the input samples through a convolutional neural network. The network architecture incorporates five convolutional layers utilizing the ReLU activation function for non-linear transformation. Each convolution is followed by a max-pooling operation with a kernel size of [2,2] to downsample the feature maps. The processing of raw data through this CNN architecture results in a refined feature vector that encapsulates the salient spatial information.

(4) LSTM Fault Classification Layer.

The feature vectors produced by the preceding CNN layer are flattened into a one-dimensional sequence and subsequently introduced to a three-layer LSTM network. This module is tasked with capturing the temporal dynamics and long-range dependencies inherent in the signal. The integration of CNN for spatial analysis and LSTM for temporal modeling creates a synergistic framework that markedly improves the detection of wave abrasion characteristics from the axle box vibration data.

After training through each layer, the training set achieves high accuracy, but the test set accuracy is sometimes unsatisfactory. This may indicate overfitting in the model. To address this, Dropout technology is incorporated into the LSTM layer to mitigate overfitting.

The optimization process involves multiple iterations of parameter adjustment to minimize the objective function, for which the cross-entropy loss is selected.

After EWT-OSTU decomposition, the original data eliminates substantial noise, increasing the proportion of effective information.

4.2. Line test data analysiss

The study selected multi-segment inter-platform data from the full-line dataset as the test model dataset. The dataset was divided into training and test sets. To ensure an unbiased evaluation of generalization capability, the dataset was split based on different operational runs: data from several complete inter-station segments were reserved exclusively for testing, while all remaining segments from other runs were used for training. This separation guarantees that no spatial or temporal overlap exists between the two sets, thus avoiding data leakage and providing a more realistic assessment of model performance on unseen track sections.

Fig. 7 shows the time-domain plots of axlebox vertical acceleration between platforms. In addition to the original axlebox acceleration data, the figure also displays the time-domain plots of processed axlebox acceleration data. This processed data was obtained by stitching together segments processed at 20-meter intervals using the EWT-OSTU method.

Fig. 7Time domain information map of the original data of the test set

Fig. 7 demonstrates the denoising efficacy of the EWT-OSTU algorithm on the original axle box vertical vibration acceleration. The processed signal maintains most of the meaningful information from the raw data but with a substantially improved signal-to-noise ratio. This enhancement facilitates more effective identification and extraction of wave-wear features by the subsequent CNN-LSTM model.

The dataset exhibits a pronounced class imbalance, which is reflected in the composition of the training set. Out of the total 13,000 training samples, the distribution is as follows: 9,000 belong to Class 0, 3,000 to Class 1, and 1,000 to Class 2. To counteract the bias introduced by this imbalance, class weights of 0.0769, 0.2308, and 0.6923 were incorporated into the loss function during model training. For evaluation, the test set consists of 620 data units collected from 8 inter-station segments, distributed as 507 (Class 0), 85 (Class 1), and 28 (Class 2). The test set comprises 620 samples from 8 inter-station segments, with 507, 85, and 28 samples for classes 0, 1, and 2 respectively. Detailed per-class metrics are listed in Table 4. The recall for class 1 is notably lower (0.536), mainly because some impact-rich segments from class 0 were misclassified as class 1. Nevertheless, no misclassifications occurred between class 0 and class 2, indicating that the model reliably distinguishes severe corrugation from healthy track. In practice, false alarms (class  0 → class  1) are far less critical than missed detections of severe corrugation.

The corresponding confusion matrix of the diagnostic results is presented in Fig. 8, where the horizontal and vertical axes denote the true labels and the model’s predictions, respectively. The model demonstrates robust diagnostic performance, achieving recognition rates above 85 % for all categories, which confirms the practical effectiveness of the CNN-LSTM model in identifying wave-induced grinding conditions across most track sections.

Fig. 8CNN-LSTM diagnostic result confusion matrix

A quantitative assessment of the model’s performance on the test set is summarized in Table 4. The results confirm that the CNN-LSTM model achieves accurate identification of wave wear conditions for the majority of track segments. However, a notably lower recall rate is observed for Class 1 labels. This limitation stems mainly from the inherent class imbalance in the dataset, which led to a subset of impact-rich segments from Class 0 being erroneously assigned to Class 1. Despite this, the model attains an overall accuracy of 89.5 %, underscoring its practical utility in diagnosing wave wear under real railway operating conditions.

To more intuitively demonstrate the diagnostic effectiveness of the test set, the diagnostic results were visualized as a spatial domain image, as shown in Fig. 9. The figure reveals that most units were accurately identified, yet some units were misdiagnosed. Among the misdiagnosed data, the majority classified Class 0 samples – units without wave wear – as Class 1 samples, indicating units with minor wave wear. This may occur because localized abnormal impacts between wheelsets and rails are mistakenly identified as wave wear characteristics. Although misclassifications exist, the vast majority of errors involve overestimating the severity of wave wear. Furthermore, there is no overlap between the identified non-wave-worn units and those recommended for grinding due to severe wave wear. Therefore, applying the CNN-LSTM model for wave wear diagnosis and early warning on actual rail lines is feasible.

To provide a preliminary assessment of robustness against speed variations, the test set includes inter-station segments covering a speed range from 40 to 80 km/h. The model maintained high accuracy across this range, with no systematic degradation observed at either low or high speeds. While a more comprehensive evaluation – e.g., training on one speed range and testing on another – would further quantify robustness, the current test set already reflects realistic operational diversity. Future work will include dedicated cross-speed experiments to validate the model’s speed invariance.

Fig. 9Diagnostic results of CNN-LSTM model test set

a) Diagnostic results for all units

b) Error recognition unit

4.3. Model visualization

Generally speaking, deep learning models are black-box models, making it difficult to understand their internal mechanisms. This paper visualizes the hidden layer signals of the CNN-LSTM model to explore the underlying inference process. Fig. 10 displays the activation status of each hidden layer, where yellow indicates activated units and green indicates deactivated units. It is challenging to distinguish fault types from the first convolutional layer, but as the network deepens, the feature maps become increasingly clear. After the feature vectors extracted by the CNN layer undergo further processing in the LSTM layer, the features become more pronounced, facilitating wave grinding recognition.

Fig. 10CNN-LSTM hidden layer signal visualization

Fig. 11CNN extracted data feature map

a) Non bobbing

b) Slight

c) More serious

The fifth-layer convolutional outputs for different data categories were then examined. Inputs with and without wave-grinding signals were converted from 1000×2 input signals into 64×61 matrices according to Table 3. Fig. 11 indicates that different signal types generated distinct feature matrices. The yellow regions in the feature maps extracted by wave-grinding CNNs varied across categories, with corresponding activation neurons also differing. This demonstrates that CNN feature extraction occurs automatically without manual intervention.

Before inputting into LSTM training, the convolutional output vector from the fifth layer undergoes regularization processing, feature extraction via pooling layers, and dimensionality reduction. The resulting feature vector is shown in Fig. 12.

After undergoing the aforementioned processing, the 64×61 matrix transforms into a matrix with more distinct features, facilitating LSTM extraction of signal characteristics and enhancing the accuracy of wave-grinding classification diagnosis.

Fig. 12LSTM input feature vector map

a) Non bobbing

b) Slight

c) More serious