Fault frequency band segmentation and domain adaptation with fault simulated signal for fault diagnosis of rolling element bearings

Abstract

Rolling element bearings are vital components in rotating machinery, and ensuring their reliable operation through robust fault diagnosis is crucial in industrial settings. Deep-learning-based methods have shown promise due to their high accuracy, but they often face challenges in data acquisition and domain shifts between training and inference datasets. Existing approaches have attempted to address these issues through signal generation using simulation models, deep learning techniques, and domain adaptation under partial label scenarios. However, generated signals often lack plausibility or physical fidelity, and partial domain adaptation approaches frequently fail to incorporate fault-related knowledge. This paper proposes a novel method combining fault frequency band segmentation domain adaptation (FBSDA) with fault-added and uncertainty-aware signal simulation. To address the scarcity of fault-labeled signals, the proposed simulation method accounts for uncertainties in the signal acquisition environment by leveraging statistical cyclo-stationary modeling of fault bearings. By adding simulated fault signals to normal signals that contain system characteristic information, the generated signals more accurately reflect real-site environments and physical principles. Additionally, the FBSDA method, a domain adaptation approach focusing on segmenting fault-related information within the fault frequency band, is introduced. To enhance the focus on the fault frequency band, FBSDA employs a fault frequency segmentation module and a loss function inspired by image segmentation techniques. This method effectively reduces the domain gap between source and target domains and simultaneously captures fault information common to both simulated and real signals. The proposed method is validated through two case studies using different testbed datasets under various operating conditions. The results demonstrate the superior performance of our approach in handling domain shifts and different levels of partial labels, outperforming existing signal generation and domain adaptation methods. The proposed method also has a practical value in that the target bearing system can be diagnosed using physical knowledge even in the absence of fault signals that are difficult to obtain.

Graphical Abstract

fault diagnosis, rolling element bearings, deep learning, simulation signal, domain adaptation, fault frequency band segmentation

Highlights

• Simulation method considering environmental situation and uncertainties is introduced.

• The FBSDA focuses on fault-related frequency band using a segmentation label and loss function.

• Good performance across various situations: domain shift causes and partial label scenarios.

List of symbols

 
• s_f (t)

  Encoder function of an autoencoder

 
• h(t)

  Impulse response function

 
• β

  Damping coefficient

 
• f_n

  Resonant frequency

 
• q(t)

  Modulation function by load distribution

 
• i

  Index of each impulse function

 
• T

  Interval between impacts

 
• τ_i

  ith slip of rolling element

 
• A_i

  Amplitude of ith impulse function

 
• σ_τ

  Standard deviation of slip

 
• σ_A

  Standard deviation of amplitude

 
• δij

  Kronecker symbol.

 
• |$\mathcal{L}_{cl}^S$|

  Classification loss on source domain

 
• |$\mathcal{L}_d^{}$|

  Domain classification loss

 
• |${\mathcal{L}_{\it{DANN}}}$|

  DANN total loss

 
• λ

  Trade-off loss of parameter

 
• ϕ

  Loss function for the speed identification

 
• f_r

  Rotation speed

 
• Δf_r

  Variability of rotation speed

 
• f_{unit_FCF}

  Unit fault character frequency

 
• x_f (t)

  Simulated fault signal

 
• N^Real(t)

  Normal data

 
• |$F( {x;{\theta _f}} )$|

  Feature extractor

 
• |$C( {{z_S},{z_T};{\theta _c}} )$|

  State classifier

 
• |$D( {{z_S},{z_T};{\theta _d}} )$|

  Domain claasifier

 
• |$U( {{z_T};{\theta _u}} )$|

  Up-sampler

 
• |${x_S}$|

  Source domain data

 
• |${x_T}$|

  Target domain data

 
• |$N_T^{{\rm{Real}}}$|

  Normal data on target domain

 
• |$F_T^{{\rm{Sim}}}$|

  Fault data

 
• D_S

  Source domain

 
• D_T

  Target domain

 
• |${z_S}$|

  Extracted feature by feature extractor using source domain data

 
• |${z_T}$|

  Extracted feature by feature extractor using target domain data

 
• |$x_i^S$|

  ith data of source data

 
• |$y_i^S$|

  ith label for data of source data

 
• |$x_i^T$|

  ith data of target data

 
• |$y_i^T$|

  ith label for data of target data

 
• C

  The number of state labels

 
• I

  The function that takes 1 when true and 0 otherwise

 
• ω_n

  Natural frequency

 
• ξ

  Damping ratio

 
• |${\widehat y_{FBS}}$|

  Fault frequency band segmentation label

 
• p

  Epoch number

1. Introduction

Robust fault diagnosis for rolling element bearings (REBs) is crucial to mitigate economic loss across various industrial fields (Kim et al., 2022). Among various approaches, vibration-based feature engineering has been considered the most common approach for robust and timely fault diagnosis of REBs. Vibration-based feature engineering can be categorized into several approaches based on the domain from which the fault-related features are derived. Previous approaches have examined the time domain (Jin et al., 2013; Oh et al., 2022), frequency domain (Hasan et al., 2019), and time-frequency domain (Alabsi et al., 2021). In particular, identifying the high-frequency resonance band induced by faulty impulsive components is considered the most common approach; this approach is often implemented through the high-frequency resonance technique (McFadden & Smith, 1984). Furthermore, signal processing methods have been proposed for remaining useful life (RUL) prediction in rotating machinery such as gears, extracting features that respond linearly to the degree of wear using synchronous averaging or cyclo-stationarity properties (Feng et al., 2023; Feng, Ji, Ni et al., 2023). However, feature-based approaches face the challenge of selecting the optimal fault-related features, which are frequently buried by significant external disturbance components, making it difficult to achieve the best diagnostic performance.

In this context, deep-learning-based approaches, which autonomously extract fault-related features while ensuring high performance, have gained significant attention. Initially, various studies on bearing fault diagnosis employed deep-learning architectures, such as long short-term memory (Zhang et al., 2019), deep belief network (Shao et al., 2018), and autoencoder (AE; Jia et al., 2016). Among these, convolutional neural network (CNN)-based structures have demonstrated superior performance in established research fields, such as image data analysis (Krizhevsky et al., 2012; Wen et al., 2018; Wang et al., 2019). In particular, one-dimensional CNNs (1D CNNs) have been the focus of active research due to their ability to handle one-dimensional time-series data, such as the vibration signals commonly used in bearing diagnostics, while requiring fewer parameters than 2D data processing methods (Chen et al., 2018; Kim & Youn, 2019). In addition, it is proposed to utilize different types and levels of noise injection inside the 1D CNN structure for noise-robust diagnosis, which can be easily exposed in real industrial settings (Yang et al., 2023). However, maintaining high diagnostic performance with these models is challenging under certain conditions. First, deep-learning models for classification problems typically have a large number of parameters; this condition necessitates a substantial amount of data for each class to avoid underfitting or overfitting. However, acquiring sufficient training data across various operating conditions is often significantly challenging in actual industrial settings. Additionally, the data used for training and the data used for inference or testing must originate from similar distributions to ensure optimal performance. Consequently, shifts in data distribution that arise due to changes in operating conditions, modifications, or other factors can lead to a decrease in the performance of the model.

To address the challenges associated with insufficient training data, simulation-based and deep-learning-based data augmentation approaches have been actively studied. The simulation-based data augmentation approach typically employs domain knowledge to ensure the cyclo-stationary property of REB faults, where the fault-induced impulse signal is repetitively generated corresponding to the fault characteristic frequency (FCF) with a sliding and slip of the inner components (Urbanek et al., 2013; Kim et al., 2022). Notably, Liu & Gryllias (2022) proposed a simulation-driven method that generates signals that reflect the characteristics of each testbed system, thereby addressing the lack of fault data. Simulation signals that do not reflect specific information, such as fault severity in the source domain, are used to improve fault-diagnosis performance. Additionally, Liu et al. (2023) employed a 5-degree of freedom (DOF) dynamic model to more actively utilize fault-related knowledge, rather than relying solely on cyclo-stationarity for signal simulation. Since all kinds of health-state signals can be generated in this study, the proposed model has been adapted using a signal generated in the source domain. However, when employing the simulation proposed above, a high level of expert and physics-related knowledge is required, along with significant computational costs. Even with these efforts, the discrepancies between simulated signals and actual signals remain substantial. These discrepancies can be attributed to the various uncertainties encountered during real system measurements and the challenges in accurately replicating the experimental environment. Although various efforts have been made to advance the accuracy of simulation models (Wang et al., 2024), there is a clear need to effectively employ a simplified simulation model with the least requirements for expert augmentation of the data. Such an innovation would contribute significantly to the improvement of deep-learning-based REB fault diagnosis. Furthermore, the domain knowledge employed to augment the data using the simulation model can be transferred to the fault-diagnosis model to enhance the model’s performance.

Other recent studies have also focused on deep-learning-based generative models, which do not require domain knowledge when augmenting the training data. Techniques proven effective in image and natural language processing, such as variational AEs and generative adversarial networks (GANs), have been actively employed to generate signals, considering the health state labels of industrial machines (Zheng et al., 2020; Zhang et al., 2022). To validate the effectiveness of these proposed models, previous studies have typically evaluated the accuracy of classification models with and without augmented data or have employed similarity metrics that compare the generated signals to the actual signals. However, it remains challenging to quantitatively evaluate the validity of the generated signals, as compared to the genuine fault-related information, which may be hidden in the time or frequency domain. Furthermore, conventional deep-learning-based data augmentation approaches are typically designed to augment small-sized datasets. These approaches can be impractical in industrial settings where it can often be tremendously difficult to acquire even a single faulty sample. In practice, most target machines operate under normal conditions. Even if a small amount of faulty training samples can be measured from a machine (e.g., from a laboratory), a domain discrepancy with the target machine may still arise (e.g., a machine operated in an industrial setting that is different from the training sample), requiring further treatment of the data. To address this challenge, Rombach et al. (2023) proposed the FaultSignatureGAN, which focuses on generating only fault-induced signatures in the source domain and proposed using this signature to synthesize the faulty data in the target domain. The FaultSignatureGAN approach fully leverages the least amount of faulty data in the source domain to generate sufficient training data in the target domain, enabling training of a target-domain-specific fault-diagnosis model without considering the domain discrepancy issue. However, if the difference in the frequency response is considerable, due to large speed variations or system changes, it is difficult to create an accurate target-domain fault signal by adding normal source-domain signals. This is a common issue with existing deep-learning-based signal generation approaches, which are limited by the difficulty of considering changes in the system itself or rapid changes in operating conditions within the system.

In these approaches, transferring the knowledge acquired from the training dataset to the inference dataset is crucial to resolving the domain-shift problem. This can be achieved through domain adaptation (DA) methods using additional loss functions, such as maximum mean discrepancy (MMD) loss (Tang et al., 2023) and CORrelation ALignment (Sun & Saenko, 2016). Additionally, domain-adversarial neural network (DANN)-based methods, which use adversarial learning to make GANs unable to distinguish between real and generated data, have been proposed to differentiate between domains (Lee et al., 2022; Dai et al., 2023; Yu et al., 2023). Recently, a range of efforts have been pursued to develop DA to cope with the challenging industrial scenario where only normal data are available in the target domain; this is referred to as partial domain adaptation (PDA) (Cao et al., 2018). In this research area, Li & Zhang (2021) proposed a method to enhance conditional alignment by MMD loss and maintain the prediction consistency of the state prediction model by incorporating multiple classification modules. Additionally, Zhu et al. (2023) introduced a technique to calculate and weight the importance of samples using improved convolutional block attention modules to minimize the negative transfer effect caused by outlier class samples in the source domain. Despite the best efforts of researchers to close the gap between the source and target domain by considering distance and marginal distribution in the high-dimensional embedded feature space, conventional DA methods cannot effectively tackle extremely shifted domain problems (e.g., system changes) and imbalanced or partial label situations. For fault diagnosis in imbalanced label situations, researchers have focused on generative models or machine-learning-based oversampling techniques to generate data or embed points in the data’s latent space (Wang et al., 2023; Chen et al., 2024). However, these methods struggle with DA problems, leading to the development of imbalanced domain adaptation (IDA) approaches. IDA approaches use DANN- and MMD-based losses for DA, along with cost-sensitive classification losses to improve classification of minor classes (Wu et al., 2022). These losses assign greater weight to minor classes, reducing misclassification. While these methods address feature shift between domains, they struggle with simultaneous label shift caused by imbalanced distributions in the source and target domains. To address this, Ding et al. (2023) introduced categorical alignment loss to align classes across domains and margin loss regularization to maximize separation between classes, alongside cost-sensitive learning. However, these methods struggle when specific labels are entirely absent in the target domain, making learning challenging.

Other recent studies have shown that it is beneficial to employ DA to address the domain discrepancy between the simulation-based augmented data and actual signals. Liu et al. (2023) employed a 5-DOF dynamic model to augment the insufficient fault-labeled data in the source domain, along with the use of subdomain adaptation using weighted local MMD-based DA. Other work (Zhang et al., 2023) also employed a 5-DOF dynamic model with DANN-based DA to cope with the data scarcity problem. However, the abovementioned studies are limited to the situation of a slight domain-shift problem and do not examine the situation of a considerable change in the operating conditions of the target system. On the other hand, in the RUL estimation of rotating machinery such as gears, a study has also been proposed to perform signal generation using a digital twin based on a dynamic model and DA of MMD loss (Feng, Ji, Zhang et al., 2023). This method relies on actual fault data, which are difficult to obtain, in the target domain during training, making it impractical when such data are unavailable. It was also reported that even a slight domain shift setting caused a performance degradation to the 80% range, due to the lack of similarity between the simulation-based augmented data and the actual signals (Zhang et al., 2023). An alternative method was proposed to address the DA problem using a small amount of real signals (Liu & Gryllias, 2022). This research utilized simulated signals to train a deep-learning-based diagnosis model that considered cyclo-stationarity, rather than the dynamic model. The simulated signals were used as the source domain and adapted to the target domain, where the real signals were obtained. It is common to consider the target domain as the domain with a lack of label information in real-world situations; however, the reverse approach also results in a lack of extendibility of proposed approaches, such as adaptation to other systems. In addition, the lack of physical consistency of features and interpretability of the model, such as focusing on physical features inherent in both real and simulated signals, is another limitation of prior approaches.

Previous studies have shown limitations in the problem of domain shifts or the absence of fault data. Deep-learning-based approaches, including generative models and PDA, become challenging to apply when significant changes occur in the system’s frequency response. Moreover, studies utilizing simulated signals often suffer from low accuracy, as they either fail to address the fundamental differences between generated and real signals or lack a physical interpretation of the diagnostic results. To address the abovementioned limitations of existing methods, this paper proposes fault frequency band segmentation domain adaptation (FBSDA) with the use of simplified simulation-based augmented data. Our proposed method is designed to address the PDA scenario where only normal data are available in the target domain. For this purpose, we augment the fault-labeled data in the target domain using a simplified simulation model while integrating the fault-related physical information used for the simulation into the fault-diagnosis model. The proposed idea enables reliable extraction of only the fault-related features from the augmented fault-labeled data, rather than focusing on the physically inconsistent disturbance components from the augmented data. Furthermore, the proposed method facilitates a visualized validation of the inference results in a physical manner. Specifically, the proposed method consists of two steps: (1) simulation-based faulty signal augmentation in the target domain, considering uncertainty and real-world environments, and (2) segmentation of the fault features where the fault-related information lies. In the first step, we introduce an augmentation method for faulty signals that considers the uncertainty of parameters to simulate the real-world environments encountered during signal acquisition, an aspect that previous methods often overlook. These fault-induced features are then integrated with target-domain normal signals to produce signals that incorporate the system characteristics observed in actual data. In the second step, FBSDA is trained to extract meaningful fault-related features in a physics-informed manner. For this purpose, the fault-diagnosis model is trained to classify the system’s health state based on physically meaningful fault-related information, which was used to augment the fault data in the first step. To ensure the physical consistency of the model, we introduce a fault-segmenter module inspired by image segmentation methods. In addition, DANN-based DA is also employed to further close the gap between the source and target domain. The proposed method is validated by applying it to two bearing datasets—the Seoul National University (SNU) dataset and the Case Western Reserve University (CWRU) dataset—under various domain-shift scenarios and comparing accuracy with previously proposed methods.

The main contributions of this paper are as follows:

1. A simulation-based fault-signal augmentation method is introduced that incorporates environmental uncertainties to generate realistic fault signals by combining them with normal real signals.

2. The FBSDA method is proposed to integrate domain knowledge into the fault-diagnosis model by introducing a frequency band segmentation module that focuses on fault-related information.

3. The proposed approach’s superior performance and robustness against domain shifts and partial label scenarios are demonstrated through validation using testbed datasets acquired under various operating conditions.

The contents of the remainder of this paper are as follows. In Section 2, we present background knowledge related to the proposed method. The proposed FBSDA method is described and experimentally validated in Section 3 and Section 4. After that, the conclusions of this paper are provided in Section 5.

2. Background knowledge

This section provides the background knowledge that helps comprehend the concepts of fault signal simulation and the FBSDA model. The knowledge related to fault signal simulation is described in Section 2.2. The DANN, the backbone of the FBSDA, is explained with its basic structure and training strategy in Section 2.2.

2.1. Simulation for bearing fault signal

The measured vibration signal from a fault bearing contains a lot of inherent information, including operational information, noise, and fault-related information. Among these types of information, operational and noise-related information with relatively high stationarity is common to both the normal and fault signals. In contrast, the fault bearing additionally contains fault components characterized by periodic repetition of impulses caused by the impact between the fault and the rolling element. The repetition of these impulses, which creates second-order cyclo-stationarity, is considered when generating the fault simulation signal (Gardner et al., 2006; Antoni, 2007). To be more realistic and accurate, the signals s_f (t) are generated to reflect statistical features along with periodicity (Liu & Gryllias, 2022).

$$\begin{eqnarray} h(t) = {e^{ - \beta t}}\sin (2\pi {f_n}t) \end{eqnarray}$$

(1)

$$\begin{eqnarray} {s_f}(t) = \sum\limits_{i = - \infty }^\infty {h\left( {t - iT - {\tau _i}} \right) \cdot q\left( {iT} \right){A_i}} + n(t), \end{eqnarray}$$

(2)

where h(t) means the impulse response function, and β and f_n denote the damping coefficient and resonance frequency of h(t) fault impulse signal. q(t) represents modulation generated by load distribution, and n(t) is noise-related information. i is the index of each impulse, and T is the interval between impacts, which can be computed using geometric information of bearings as the inverse of the FCF. The cyclo-stationarity of the fault signal can be modeled by equation (3) using the uncertainty caused by slip and amplitude of ith impulse as variables τ_i and A_i with σ_τ and σ_A the standard deviations, and δij the Kronecker symbol. As a result, the simulated fault signal can be represented in Fig. 1. The random variable used in equation (2) can represent the second-order cyclo-stationarity that the fault signal follows, simulating the theoretical uncertainty of the signal. However, it has the limitation of being unable to simulate the real-world system characteristics or the multiple sources of uncertainty that arise during the measurement process.

$$\begin{eqnarray} E\left\{ {{\tau _i}{\tau _j}} \right\} &=& {\delta _{ij}}{\sigma _\tau }^2\\ E\left\{ {{A_i}^2} \right\} &=& 1 + {\delta _{ij}}{\sigma _A}^2 \end{eqnarray}$$

(3)

Figure 1:

Simulated inner raceway fault signal in (a) the time domain, (b) the frequency domain, and outer raceway fault signal in (c) the time domain, (d) the frequency domain.

2.2 Domain-adversarial learning

Domain differences between the training and inference dataset distribution can be considered one of the main causes of the decline in task performance. Domain-adversarial learning can be a solution to domain discrepancy issues with adversarial learning to extract domain-invariant features (Ganin et al., 2016). The deep learning model trained by domain-adversarial learning is organized in such a way that it fulfills the purpose of extracting features that perform the task of the source domain while being indistinguishable between the source and target domains. The loss function of the model is constructed by subtracting the domain classification loss using both data from the classification loss using the source data to achieve task and domain alignment simultaneously. This method utilizes a minimax optimization for training a GAN model, and the final objective function (⁠|${\mathcal{L}_{\it{DANN}}}$|⁠) is as follows:

$$\begin{eqnarray} {\mathcal{L}_{\it{DANN}}} = \mathcal{L}_{cl}^S - \lambda \mathcal{L}_d^{}, \end{eqnarray}$$

(4)

where |$\mathcal{L}_{cl}^S$| is the classification loss on the source domain, |$\mathcal{L}_d^{}$| is the domain classification loss, and λ is the parameter meaning the trade-off between two losses. However, in the optimization process of the domain classifier, a special layer is required to perform the identity operation that outputs the same as the input in the forward propagation to compute the features but converts the gradient sign in the backpropagation process to update the parameters. This special layer is called a gradient reversal layer (GRL), and the minimax optimization strategy is used with GRL. Domain-invariant features are extracted through adversarial learning, and the features extracted from the two domains are aligned in the high-level feature space. Since domain shifts, such as changes in operating conditions or systems, can occur frequently during bearing fault diagnosis, research on DA has been active.

2.3 Image segmentation

As deep learning techniques for image data have evolved, methods such as image segmentation, which identifies the type and location of information within data, have attracted attention. Unlike object detection, which specifies location through a bounding box, image segmentation assigns each pixel to a specific object, allowing for precise object localization and consideration of image semantics. So, training the model for image segmentation needs the labels for every pixel shown in the upper side of Fig. 2, including the background, and minimizes the loss relative to that label, using methods such as cross-entropy (CE) or dice loss.

Figure 2:

Example of image segmentation (U-Net).

Various representative deep learning models have been developed for image segmentation, including fully convolutional network (Long et al., 2015), U-Net (Ronneberger et al., 2015), SegNet (Badrinarayanan et al., 2016), and DeepLab (Chen et al., 2017). These models share several common features. First, they utilize convolutional layers known for their ability to classify image data. To preserve the localization performance of the extracted features, these models are composed entirely of convolutional layers, avoiding the fully connected (FC) layers used in other CNN models for classification. Second, since the features are compressed by the pooling operator from the input image, the models incorporate an up-sampling component to recover the extracted features, typically achieved through convolutional transpose or deconvolution operations. For example, U-Net, which demonstrates excellent segmentation performance in medical imaging, is designed as an encoder–decoder model, as illustrated in Fig. 2. Finally, the output channels are configured to match the number of classes, enabling the model to classify one object per channel. The model can be trained using binary cross-entropy (BCE) with a single output channel in binary classification. Despite its widespread use in image-based visual inspection and defect detection (Tabernik et al., 2020; Liu et al., 2022), image segmentation has not yet been extensively applied in the field of fault diagnosis using sensor data, such as vibration signals.

3. Proposed method

The proposed FBSDA with fault simulation is described in this section. The fault-added simulation considering uncertainty is explained in Section 3.1. In Section 3.2, the model architecture of FBSDA is provided. Finally, the training procedure and loss function of FBSDA are described in Section 3.3.

3.1 Fault-added simulated signal considering uncertainty

This section presents simulation-based fault-signal augmentation to address the fault data insufficiency problem in the target domain. This method is based on the second-order cyclo-stationary characteristics represented in equations (1) and (2); however, two additional points are considered to make the simulated signal more like the real signal. The first approach is to capture the uncertainty that can occur in real industrial systems. To address this limitation, this paper proposes a simulation method that incorporates parameters with system-related and operation-related uncertainties. Random variables related to system characteristics include the damping coefficient β and natural frequency f_n of the fault-induced impulsive features, expressed in equation (1). These variables determine the resonant and impulsive function h(t) that reflects the inherent characteristics of the system. In addition, random variables related to operation and signal measurement account for the variability that arises due to the uncertain rotation speed and signal condition. Considering the random variables of these operation-related uncertainties, equation (2) can be rewritten as

$$\begin{eqnarray} {s_f}(t) = \sum\limits_{i = 0}^N {h\left( {t - \frac{i}{{{f_{\it{unit}\underline {} FCF}} \times ({f_r} + \Delta {f_r})}} - {\tau _i} - \phi } \right)q(iT){A_i}}, \end{eqnarray}$$

(5)

where ϕ is the signal phase, f_r is rotation speed, Δf_r is the speed variation during control, and f_{unit_FCF} denotes the unit FCF value computed by the bearing’s geometric information. Including the random variables τ_i and A_i used in equation (2), the result can be illustrated as shown in Fig. 3. In general, it is tremendously challenging to determine the optimal values of the design parameters described above. In this paper, we propose to generate a variety of signals by random sampling from the distribution of the parameters, rather than using fixed parameters. Thus, a generalized and robust fault-diagnosis model can be trained while employing physically plausible features. This will be explained further in Section 3.2.

Figure 3:

The details of random variables for fault simulation.

The second additional approach considered here is the integration of real system information. The final generated faulty signal (x_f (t)) can be generated by adding the generated fault-induced features (s_f (t)) to the actual normal signal (N^Real(t)), thus providing a more comprehensive simulation result. The equation governing this enhanced signal simulation method is as follows:

$$\begin{eqnarray} {x_f}(t) = {s_f}(t) + {N^{{\rm{Re}}al}}(t) \end{eqnarray}$$

(6)

The simulated signal derived through the above two processes was compared with the conventionally simulated signal and the real signal in the time and frequency domains; the results are shown in Fig. 4. As shown in Fig. 4a and d, the conventional simulation appears quite different from the real signal (Fig. 4c and f) because the impulse-related information and fault frequency band related to the fault appear too clearly. However, in Fig. 4b and e, the signals generated by the proposed method (x_f (t)) have better similarity to the actual signal by representing the typical vibration response of the system, as well as the fault-induced features.

Figure 4:

Time domain plot of (a) simulated fault signal (s_f(t)), (b) fault-added simulation signal (x_f(t)), (c) real signal, and frequency domain plot of (d) simulated fault signal, (e) fault-added simulation signal, (f) real signal.

3.2 Fault frequency band segmentation domain adaptation

This section introduces a proposed FBSDA architecture. The general concept of the model is described as shown in Fig. 5. The FBSDA approach is composed of four modules: a feature extractor (⁠|$F( {x;{\theta _f}} )$|⁠), a state classifier (⁠|$C( {{z_S},{z_T};{\theta _c}} )$|⁠), a fault-segmenter module (⁠|$S( {{z_T};{\theta _s}} )$|⁠), and a domain classifier (⁠|$D( {{z_S},{z_T};{\theta _d}} )$|⁠). Conventional fault-diagnosis models typically consist of only a feature extractor and a state classifier. However, this simple configuration cannot guarantee physical representation of the features, nor robust diagnosis performance across various domains. To address this issue, this paper introduces a fault-segmenter module to ensure that the fault diagnosis is actually performed based on the physical relationships. Specifically, the fault-segmenter module is designed to utilize the features to predict the frequency range of interest where the fault-related features are included in the frequency domain. This implies that the features used for fault diagnosis are properly defined not only to maximize the statistical classification performance but also to emphasize the frequency range where the actual fault-related information is contained. The fault-segmenter is inspired by the conventional bearing diagnostics of fault frequency band search methods [e.g., kurtogram (Jérôme Antoni, 2007), infogram (Jerome Antoni, 2016)]. Traditional band selection methods rely on fault-related metrics, such as spectral kurtosis or entropy, to identify the optimal frequency band where the metrics are maximized. However, these metrics are susceptible to noise, system characteristics, and other non-fault factors. To address this, we propose a method that directly learns to identify the appropriate frequency band. Drawing from image segmentation techniques—where specific parts of an image are identified as objects within the whole—we train the model to identify the fault band within the entire frequency spectrum This approach ensures the fault frequency band can be identified even when the frequency response changes due to domain shifts. This approach is made feasible by employing a simulation-based augmented signal, where the frequency range of interest can be easily inferred from the simulation design parameters.

Figure 5:

Overview of the proposed FBSDA approach.

At the same time, the feature extractor and the classifier are trained using both real source data and augmented target data. This training strategy enables an emphasis on the fault-related information that remains relatively insensitive to domain variations and is present in both the simulated and real signals. Deep learning models are able to extract a meaningful representation for fault diagnosis despite the differences between the simulated signal and the actual signal (Ha & Fink, 2023). Therefore, we use simulated signals instead of real signals to utilize the physical characteristics of the fault frequency band. Furthermore, we employ a domain classifier that closes the gap between the source and target domains; this approach is inspired by the DANN-based domain adaptation method. The DANN-based DA was applied not only to bridge the gap between the two domains but also to minimize any potential differences between the simulated and actual signals.

The real source data (⁠|${x_S} = \{ {N_S^{\it{Real}},F_S^{\it{Real}}} \}$|⁠) sampled from the source domain (D_S) and target domain data (⁠|${x_T} = \{ {N_T^{\it{Real}},F_T^{Sim}} \}$|⁠) with real normal data (⁠|$N_T^{\it{Real}}$|⁠) sampled from the target domain (D_T), and fault-simulated data (⁠|$F_T^{Sim}$|⁠) are used to train the FBSDA model. The forward process is proceeded using the above source and target dataset. First, the source and target domain data are used as the input of the feature extractor, and each extracted feature is denoted as |${z_S},{z_T}$|⁠.

$$\begin{eqnarray} {z_S} &=& F({x_S};{\theta _f})\\ {z_T} &=& F({x_T};{\theta _f}) \end{eqnarray}$$

(7)

Both extracted features are then fed into the state and domain classifiers to create the output for state and domain classification. However, unlike the state and domain classifiers, the fault-segmenter only uses data from the target domain to create its output. Like the other segmentation models described in Section 2.3, the fault-segmenter restores the features of the target domain data (⁠|${z_T}$|⁠), compressed in the time axis by the feature extractor’s pooling operators, to the same length as the input signal using a transpose convolution. As shown in Fig. 5, the feature extractor (encoder) and the fault-segmenter (decoder) form an encoder–decoder structure, similar to existing image segmentation methods. However, unlike conventional image segmentation, which uses 2D data, it is designed as a 1D CNN to match the dimensionality of the input signal. The restored features found by transpose convolution are converted from the time domain into the frequency domain by a fast Fourier transform (FFT) operation to allow comparison with the segmentation label and calculation of the loss function related to the fault frequency band segmentation. Through this process, the other outputs and loss functions that utilize the input signals of each domain are calculated and trained. This will be further described in Section 3.3.

All layers of the feature extractor consist of five convolutional layers with kernel lengths of 3 and ELU activation functions and pooling layers with a pool size of 2. The fault-segmenter module up-samples the extracted features to 32 times the feature length with the same length as the input data. The extracted features are compressed using a size 1 convolution in the channel dimension. The features are then restored in the time dimension using a convolutional transpose operation with a kernel of size 32. The other two modules (the state classifier and the domain classifier) use three FC layers with the same size parameters and dropouts with a rate of 0.5 after being pooled using global average pooling (GAP). The length of the data size determines the size of these layers. The entire forward propagation process of the proposed model is performed as mentioned above, and information such as the channel kernel size is detailed in Table 1.

Table 1:

Details of the FBSDA model.

Module	Layer type	Channel num	Kernel size	Activation	Stride
Feature extractor	Convolution	8	3	ELU	1
	Max Pool	8	2	−	2
	Convolution	16	3	ELU	1
	Max Pool	16	2	−	2
	Convolution	32	3	ELU	1
	Max Pool	32	2	−	2
	Convolution	64	3	ELU	1
	Max Pool	64	2	−	2
	Convolution	128	3	ELU	1
	Max Pool	128	2	−	2
Fault-segmenter	Convolution	1	1	ELU	1
	Transposed Conv.	1	32	ELU	32
State classifier	GAP	128	Data Size/32		Data Size/32
	FC	−	Data Size/128	ReLU	−
	FC	−	Data Size/512	ReLU	−
	FC	−	# of State	−	−
Domain classifier	GAP	128	Data Size/32		Data Size/32
	FC	−	Data Size/128	ReLU	−
	FC	−	Data Size/512	ReLU	−
	FC	−	2	−	−

3.3 Objective functions and training procedure

The proposed FBSDA model employs three distinct loss functions: classification loss, domain classification loss, and frequency band segmentation (FBS) loss. Each of these loss functions is explained in detail in Sections 3.3.1 to 3.3.3. The formulation of the total loss function and the training procedure are then presented in Sections 3.3.4 and 3.3.5, respectively.

3.3.1 Classification loss

The classification loss is calculated using CE loss. The summation of CE loss using real source data and targeting real normal data and fault-simulated data is expressed as total classification loss. The classification loss function using source domain data (⁠|${x_S} = \{ {x_i^S,y_i^S} \}_{i = 1}^{{n_S}}$|⁠) and target domain data (⁠|${x_T} = \{ {x_i^T,y_i^T} \}_{i = 1}^{{n_T}}$|⁠) is as follows:

$$\begin{eqnarray} {\mathcal{L}_C}\left( {{\theta _f},{\theta _c}} \right) &=& - \frac{1}{{{n_S}}}\sum\limits_{i = 1}^{{n_S}} {\sum\limits_{c = 1}^C {I\left[y_i^S = c\right]} } \log \left( {C\left( {F\left( {x_i^S;{\theta _f}} \right);{\theta _c}} \right)} \right)\\ && -\, \frac{1}{{{n_T}}}\sum\limits_{i = 1}^{{n_T}} {\sum\limits_{c = 1}^C {I\left[y_i^T = c\right]} } \log \left( {C\left( {F\left( {x_i^T;{\theta _f}} \right);{\theta _c}} \right)} \right) \end{eqnarray}$$

(8)

where C is the number of health states, I is the function that takes 1 when true and 0 otherwise, and |${n_S},\,\,{n_T}$| denote the number of source and target domain data, respectively.

3.3.2 Domain classification loss

The domain classification loss for extracting domain-invariant features employs BCE loss. The domain label of the source domain data is 1 and the label of the target data is 0. The domain classification loss function is as follows (Ganin et al., 2016):

$$\begin{eqnarray} {\mathcal{L}_D}\left( {{\theta _f},{\theta _d}} \right) &=& - \frac{1}{{{n_S}}}\sum\limits_{i = 1}^{{n_S}} {\log \left( {D\left( {F\left( {x_i^S;{\theta _f}} \right);{\theta _d}} \right)} \right)} \\ && -\, \frac{1}{{{n_T}}}\sum\limits_{i = 1}^{{n_T}} {\log \left( {1 - D\left( {F\left( {x_i^T;{\theta _f}} \right);{\theta _d}} \right)} \right)} \end{eqnarray}$$

(9)

3.3.3 Fault frequency band segmentation loss

Both real and simulated faulty signals have a resonant fault-related impulse component in a specific frequency band, despite the inevitable inconsistency. Utilizing this fault physics knowledge, the proposed model is guided to focus on the fault frequency band by clarifying the feature-extracting mechanism. For this purpose, segmentation loss, which is commonly used in image segmentation, is used to focus the fault frequency band of the extracted features. To calculate the FBS loss, we first need to define a label for the frequency segmentation. The simulated fault signal is represented by a repetitive form of the impulse signal, determined by the parameters β and f_n, as expressed in Equation (1). However, in order to derive the information band of the impulse signal, the natural frequency and damping ratio must be obtained. The relationship between the parameters can be expressed as follows (Papagiannopoulos & Hatzigeorgiou, 2011):

$$\begin{eqnarray} {\omega _n} = \sqrt {{\beta ^2} + {f_n}^2} \end{eqnarray}$$

(10)

$$\begin{eqnarray} \xi = \frac{\beta }{{{\omega _n}}} \end{eqnarray}$$

(11)

The fault frequency band of the fault simulation data can be determined using the parameters obtained by Equations (10) and (11). Considering the characteristics of the impulse signals, the bandwidth of the resonant frequency band is determined to be 2ξω_n (Papagiannopoulos & Hatzigeorgiou, 2011). Since the center of the fault frequency band is f_n, the frequency band limit is determined as [ω_l, ω_h] = [f_n−ξω_n, f_n + ξω_n]. Therefore, the fault segmentation label ( |${\hat y_{FBS}}$|⁠) is expressed as follows:

$$\begin{eqnarray} {\hat y_{FBS}}\left[ k \right] = \left\{ {\begin{array}{@{}*{1}{l}@{}} {1\quad \left( {(L/{F_S}){\omega _l} \le k \le (L/{F_S}){\omega _h}} \right)}\\ {0\quad \left( {k < (L/{F_S}){\omega _l},\ (L/{F_S}){\omega _h} < k} \right)} \end{array}} \right. \end{eqnarray}$$

(12)

where k is the index label (k = 0, …, L) of L is the data length, and F_S is the sampling rate. The value of the fault frequency band label is determined considering the bandwidth of the fault frequency band and the frequency resolution.

Various loss functions can be employed in a segmentation task, with the CE loss family being particularly prevalent. In this case, the loss is not used to classify the state of a sample but rather to learn to recognize which parts of the data index contain fault-related information. After all, fault segmentation in the frequency domain can be viewed as index-wise binary classification, trained using BCE loss, commonly applied in image segmentation. Specifically, since the fault segmentation label in Fig. 6 is assigned a value of 1 for the fault frequency band of the simulated signal, the restored feature |${z_t}$| in the frequency domain is trained to retain only fault-related information, with other bands reduced to near zero. As a result, non-relevant portions of the restored feature (gray) are reduced to near zero, while the frequency band of interest (black) is preserved. The aforementioned FBS loss function is expressed as follows:

$$\begin{eqnarray*}{\mathcal{L}_{FBS}} = - \frac{1}{{{n_T}}}\sum\limits_{i = 1}^{{n_T}} { - {{\hat y}_{FBS}}\log \left( {S\left( {F\left( {x_T^i} \right)} \right)} \right) - \left( {1 - {{\hat y}_{FBS}}} \right)\log \left( {1 - S\left( {F\left( {x_T^i} \right)} \right)} \right)}\\ \end{eqnarray*}$$

(13)

Figure 6:

Fault frequency band segmentation loss details.

3.3.4 Total loss function

The total loss function is expressed as Equation (14) in a combined form of the loss functions described in Equations (8), (9), and (13). To train the FBSDA model, classification and FBS loss are added, but the domain classification loss is subtracted because of the minimax game optimization problem; this is represented as the GRL form. Also, the parameter λ is used as the weight of the total loss form and adjusted according to epoch number (p), increasing from 0 to 1, as shown in Equation (15):

$$\begin{eqnarray} \mathcal{L}\left( {{\theta _f},{\theta _u},{\theta _c},{\theta _d}} \right) = {\mathcal{L}_{FBS}}\left( {{\theta _f},{\theta _u}} \right) + {\mathcal{L}_C}\left( {{\theta _f},{\theta _c}} \right) - \lambda {\mathcal{L}_D}\left( {{\theta _f},{\theta _d}} \right) \end{eqnarray}$$

(14)

$$\begin{eqnarray} \lambda = \frac{2}{{1 + \exp ( - 10p)}} - 1 \end{eqnarray}$$

(15)

3.3.5 Training procedure

The parameters are updated during optimization using the total loss function, as expressed in Equation (16). Considering the minimax game of the feature extractor and the domain classifier and the task of GRL, the parameters are updated according to the following equations:

$$\begin{eqnarray} \begin{array}{@{}l@{}} {\theta _f} \leftarrow {\theta _f} - \alpha \displaystyle\frac{{\partial \left( {{\mathcal{L}_{FBS}} + {\mathcal{L}_{cl}} - \lambda {\mathcal{L}_d}} \right)}}{{\partial {\theta _f}}}\\ {\theta _u} \leftarrow {\theta _u} - \alpha \displaystyle\frac{{\partial {\mathcal{L}_{FBS}}}}{{\partial {\theta _u}}}\\ {\theta _c} \leftarrow {\theta _c} - \alpha \displaystyle\frac{{\partial {\mathcal{L}_{cl}}}}{{\partial {\theta _c}}}\\ {\theta _d} \leftarrow {\theta _d} - \alpha \left( {\lambda \displaystyle\frac{{\partial {\mathcal{L}_d}}}{{\partial {\theta _d}}}} \right) \end{array} \end{eqnarray}$$

(16)

where α is the learning rate determined as 0.0001 and decaying as half of the previous value at every ten epochs, the total epoch batch size is set as 50 and 128.

4. Experimental validation

In Section 4, we describe the validation of the proposed method using two different REB testbeds; the SNU and CWRU bearing datasets. Section 4.1 provides a detailed description of the SNU and CWRU bearing datasets. Section 4.2 presents the fault-diagnosis results and the domain-adaptation performance for different domain-shift causes, changes in various speed conditions, and system differences. Additionally, the performance of the method is evaluated under varying degrees of label partialities in the target domain.

4.1 Data description and experimental setup

To thoroughly validate the proposed method, datasets were acquired from two different testbeds under various operating conditions. The first dataset, obtained from the SNU bearing testbed, includes data collected under various speed conditions; this dataset is introduced in Section 4.1.1. The second dataset, the CWRU dataset, encompasses data collected under various load conditions, as detailed in Section 4.1.2.

4.1.1 SNU bearing testbed

To evaluate the effectiveness of the proposed method, vibration signals were acquired using the testbed depicted in Fig. 7a. The testbed setup includes a servo motor and two air cylinders capable of generating radial and axial loads. During the entire testing period, the axial and radial loads were maintained at 180kgf and 170kgf, respectively. Signals were acquired using a 3-axis accelerometer (PCB 365A15) and a National Instruments DAQ module (NI 9234) at a sampling rate of 10,240Hz. The experiments utilized SKF 7202a ball bearings with two types of induced faults: inner raceway fault and outer raceway fault, as illustrated in Fig. 7b and c. To assess the performance of the proposed method across various speed conditions, data were collected at three distinct rotational speeds, 1200, 1600 and 2000RPM.

Figure 7:

SNU bearing test bed: (a) configuration, (b) an inner raceway fault specimen, (c) an outer raceway fault specimen (Park et al., 2023).

4.1.2 CWRU bearing testbed

The CWRU bearing dataset, as illustrated in Fig. 8, provided test data of a 6205–2RS JEM SKF deep-groove bearing under three health states: normal, inner raceway fault, and outer raceway fault. Each fault category included three fault sizes with 0.001, 0.002, and 0.003 inch diameters. However, the model was designed to classify among three different health states, considering only the type of fault. Vibration data were collected under four different load conditions (0, 1, 2, 3hp), while maintaining a constant rotational speed of ∼1750RPM; signals were sampled at a rate of 12 000Hz.

Figure 8:

CWRU bearing testbed (“Case Western Reserve University Bearing Data Center Website,”, 2023).

4.2 Case study

This section presents the results of the comparative and ablation studies for the proposed method. For the comparative study, several existing models were selected to benchmark against the proposed method. The comparative methods address the insufficient label data issue with DA or signal generation. First, four DA-based approaches were utilized: PDA (Li & Zhang, 2021), simulation-driven domain adaptation (SDSAN) (Liu et al., 2023), DANN (Ganin et al., 2016), and Domain Adaptation with MMD (Guo et al., 2019). Additionally, FaultSignatureGAN (Rombach et al., 2023), abbreviated as FSG in this paper, was included as a representative comparison method that reflects the real-world environment through the addition of normal data in the source domain with fault signatures generated by the deep-learning model in the frequency domain. PDA suggested the DA model with MMD for conditional data alignment and multiple classification models for unsupervised prediction consistency. SDSAN is composed of signal simulation using the REB dynamic model and subdomain adaptation technique with MMD to tackle the missing health states problem. In addition, since MMD and DANN were introduced to compare DA methods that do not consider the lack of label information among the comparison models, the fault signals (⁠|$F_T^{Sim}$|⁠) generated by the proposed simulation method were used as signals of the lack of labels in the target domain. While the comparison models were trained using different data according to the proposed method, only common real-world datasets were used for validation. The operating conditions utilized for training and validation are described in each case study section, and only real data sets under the same conditions were used for validation. For the ablation study, we studied the effect of each component of the proposed model by excluding the domain adaptation model, uncertainty for the simulation model, and real target normal data used for synthesizing the augmented target fault data. The results from the ablation study were referred to as FBS, Fault addition to normal, or only s_f (t). Since FBS excludes DA-related modules, it is applicable to both target domain alone and combined real data from the source domain, which we denote as FBS(T) and FBS(S + T), respectively.

To train and evaluate the proposed method, we utilized simulation techniques that exploit the uncertainties encountered in a real-world environment, as mentioned earlier. As outlined in Section 3.1, we considered six types of uncertainties, each varying depending on the dataset. The main purpose of the generated signal was to simulate a wide variety of samples for training the generalized deep-learning model, while extracting the fault-related information more accurately. First, the phase variable was set to a uniform distribution within the range of intervals between fault peaks (T = f_{unit_FCF} × f_r), as it is relevant to the cutoff point of the signal to utilize deep learning. Speed variation is an estimated distribution based on the actual measured speed of the data, and for the damping coefficient, slip, and amplitude, the distributions were set based on existing studies (Sobie et al., 2018; Liu & Gryllias, 2022). For the case study with the SNU testbed, the parameters were also set similarly to the parameters from the CWRU dataset. However, the upper limit of amplitude for the SNU dataset was adjusted considering the severity of fault. However, since the two datasets have different system configurations, the natural frequencies may be different. Further, since the higher sampling frequency is likely to find higher natural frequencies in the CWRU dataset, we assumed a higher natural frequency distribution in CWRU to simulate this. We also utilized a uniform distribution to avoid simulating signals around a particular mean. The distributions of the parameters for the two datasets are presented in Table 2. Figure 9 compares the simulated outer raceway fault data with the real CWRU data. Figure 9a and c show that, over the entire segment, the fault impulses have similar periods, despite differences in modulation. Figure 9b and d provide a zoomed-in view of the section with two fault impulses. Although the fault impulses are not identical, they exhibit similar rise and fall patterns and attenuation frequencies. The proposed deep -earning-based diagnosis model is expected to learn fault characteristics by leveraging similarities across simulated signals.

Figure 9:

CWRU bearing outer raceway fault data under load 2hp: (a) simulation data, (b) simulation data in [0.10, 0.12], (c) real data, and (d) real data in [0.10, 0.12].

Table 2:

Uncertainties for the two testbed data simulations.

Types of uncertainty	SNU	CWRU
Phase ϕ	U(0, T)
Speed variation Δfr	N(0, 0.02fr)
Damping coefficient β	U(1000,1500)
Slip of ith impulse τi	N(0, 0.02funit_FCFfr)
Amplitude of ith Impuse Ai	U(0.5, 0.8)	U(0.5, 1.5)
Natural frequency fn	U(3500, 4500)	U(4500, 5500)

4.2.1 Case study 1: domain shift under various operating condition

The performance of the proposed method was evaluated under various speed conditions to assess its robustness to domain shifts. Unlike load, speed can have a large impact on the data, especially in the frequency domain. Thus, we studied the domain changes as the speed changed. In addition, many studies have shown good results when using the CWRU dataset alone (Li & Zhang, 2021; Liu & Gryllias, 2022; Rombach et al., 2023); thus, we utilized the SNU dataset to ensure robust results. Three distinct speeds (1200, 1600, and 2000RPM) were designated as the source and target speed conditions to validate the method’s domain-invariant capabilities across different speed changes. The details and names of each task are provided in Table 3. Normal data are available for both the source and target domains, with the assumption that fault data are not available in the target domain.

Table 3:

Tasks for SNU dataset under various speed conditions.

Task name	Source domain (RPM)	Target domain (RPM)
S12	1200	1600
S13	1200	2000
S21	1600	1200
S23	1600	2000
S31	2000	1200
S32	2000	1600

The validation results across six different tasks are presented in Fig. 10 and Table 4. Figure 10a illustrates the accuracy for all tasks using a radar chart, to facilitate a comprehensive comparison. Meanwhile, Fig. 10b displays the average accuracy for each model as a bar chart. The proposed FBSDA method demonstrates the highest accuracy, above 94%, across all tasks. Notably, the accuracy is generally lower for tasks involving a large difference in transfer speed, particularly when transferring from high to low speed. Consequently, the proposed method achieves an accuracy in the mid-90% range for cases S₁₃ and S₃₁; however, this is higher than the accuracy observed in other cases. Conversely, the proposed method demonstrates a high accuracy of 100% for tasks involving high-speed transfers with a difference of 400RPM. The FBSDA method outperforms others on average, achieving over 95% accuracy. In contrast, DA-based methods achieve an accuracy greater than 80%, while the generative model achieves an average accuracy of less than 80%. It can be seen that the proposed FBSDA method is well applied in the absence of fault data in the target domain, and it secured good performance by utilizing physical knowledge compared to existing DA methods. Besides, the proposed FBSDA method performed well compared to the SDSAN method, a DA method utilizing a dynamic model. It is confirmed that the proposed method, which simulates the fault-related frequency band information and searches for the corresponding band, is more effective than accurately simulating the actual operation of the bearing.

Figure 10:

SNU bearing dataset results: (a) radar chart, and (b) average accuracy results for compared methods.

Table 4:

Accuracy of comparative studies across every task for the SNU testbed dataset.

	Accuracy [%]
Task	FBSDA	PDA	SDSAN	DANN	MMD	FSG
S12	100.00	94.17	91.96	99.62	91.62	82.10
S13	96.76	95.51	95.49	95.43	96.38	68.19
S21	99.05	89.50	88.21	88.19	92.19	59.81
S23	100.00	99.94	99.80	99.81	99.62	98.67
S31	94.10	88.22	86.96	54.10	88.57	57.71
S32	97.14	90.40	87.61	92.38	87.43	75.43
Mean	97.84	92.96	91.67	88.26	92.64	73.65

To further analyze the above-described results, we visualized the features using t-SNE under task S₂₁, as shown in Fig. 11. Figure 11 shows the fault-related features of the source and target domains in the order of the methods presented in Table 4, depicted in Fig. 11a–f. The proposed method demonstrates clustering for each state regardless of the domain differences; this is associated with its high diagnostic accuracy. In contrast, the remaining methods exhibit decreased accuracy due to changes in the features that arise due to the domain shift. Specifically, in Fig. 11b, the normal data are well clustered, while the fault data show clear domain discrepancies. In Figure 11d and e, which utilize data generated by the proposed simulation method, the normal data are well classified; however, the unapparent mechanism of focusing on the fault-related features results in ineffective classification of the different failure modes.

Figure 11:

Feature representation using t-SNE under task S₂₁ for (a) FBSDA, (b) PDA, (c) SDSAN, (d) DANN, (e) MMD, and (f) FSG.

To thoroughly validate the proposed method, we conducted an ablation study examining the impact of each of the components of the model. The results are presented in Table 5. As previously described, four methods were compared against the proposed method. FBS evaluates the results when excluding the DA module, employing both source and target data (i.e., FBS(S + T)) and only target data (i.e., FBS(T)). In addition, “w/o uncertainty” and “only s_f (t)” denote the results when excluding the uncertainty from the simulation model and employing only fault features for simulating the target fault data, respectively. Among the four compared methods, the model without the DA module (i.e., FBS) exhibited lower accuracy, which is explained by the significant contribution of domain adaptation to performance. In addition, “w/o uncertainty” and “only s_f (t)” results showed that extracting a variety of samples considering uncertainty, along with employing the real target normal data, can also contribute to enhancement of the fault-diagnosis performance. The results suggest that the combination of each component has a significant impact on the proposed method.

Table 5:

Result of the ablation study for the SNU dataset.

	Accuracy [%]
Task	Proposed	FBS(S + T)	FBS(T)	w/o uncertainty	Only sf(t)
S12	100.00	94.29	93.52	93.67	95.95
S13	96.76	78.33	76.29	75.95	88.33
S21	99.05	85.00	83.10	92.48	88.86
S23	100.00	99.52	89.76	92.62	97.71
S31	94.10	55.24	56.95	75.00	86.43
S32	97.14	68.81	71.81	84.05	82.38
Mean	97.84	80.2	78.57	85.63	89.94

4.2.2 Case study 2: domain shift for different systems

This section provides the validation results under domain shift conditions that arise due to system variations found in real-world settings. The two testbeds described in Section 4.1 were utilized to assess the proposed method’s effectiveness in addressing these domain shifts. Different tasks were configured based on the degree of partial and missing labels in the target domain. In this case, the SNU and CWRU datasets were set as the source and target domains, respectively. Details of these tasks are summarized in Table 6. The data used for training and testing were obtained from the SNU bearing testbed under three different speed conditions and from the CWRU bearing testbed under four different load conditions, as described in Section 4.1.

Table 6:

Tasks for different systems under various partiality.

Task name	Source domain	Target domain
	CWRU	SNU
CS1	N, I, O	N, I
CS2		N, O
CS3		N
	SNU	CWRU
SC1	N, I, O	N, I
SC2		N, O
SC3		N

The results of the system transfer problem are presented in Fig. 12 and Table 7. The proposed method showed the best diagnostic performance for all tasks. For all domain-shift scenarios, the fault-diagnosis accuracy decreased as the partiality of the obtainable labels in the target domain increased. For tasks CS₃ and SC₃ with the largest partiality, the accuracy of the proposed method drops to about 90%. However, the results show that the proposed method still outperforms the other methods.

Figure 12:

Different system transfer results: (a) radar chart, and (b) average accuracy results.

Table 7:

Accuracy of comparative study for task transferring under different systems.

	Accuracy [%]
Task	FBSDA	PDA	SDSAN	DANN	MMD	FSG
CS1	97.45	91.06	93.00	89.45	66.60	66.60
CS2	99.66	94.75	96.06	84.90	69.66	69.66
CS3	90.00	76.27	82.87	82.86	88.64	88.64
SC1	99.76	90.34	94.65	92.77	77.29	77.29
SC2	96.17	90.43	73.48	73.35	72.98	72.98
SC3	90.85	78.47	72.09	71.78	89.66	89.66
Mean	95.65	86.89	85.36	82.52	77.47	77.47

As shown in Fig. 13, we performed feature representation using t-SNE for task SC₃, in a manner similar to the previous study. Figure 13a–f illustrates the features of each method. The distribution of the normal labels does not appear to differ significantly between the source and target domains. The proposed method displays a well-defined distribution of each label, corresponding to its high accuracy. In contrast, the methods shown in Fig. 13b and c exhibit instances of misdiagnosis for the inner raceway fault. Additionally, the methods with lower accuracy, such as those in Fig. 13e and f, demonstrate poor clustering of the same labels within the dataset.

Figure 13:

Feature representation using t-SNE under task SC₃, for (a) FBSDA, (b) PDA, (c) SDSAN, (d) DANN, (e) MMD, and (f) FSG.

We analyzed whether the FBSDA model effectively extracts features that correspond to the physical meaning in the frequency domain. First, the analysis results using the simulation signal under task CS₃ are shown in Fig. 14. This newly simulated 1600RPM outer raceway fault signal under the target domain was not utilized for training. In Figure 14a, the FFT of the input signal is presented. In Figure 14b, the FFT of all channels of the features extracted from this signal by the feature extractor, along with their average, is represented. This also includes the fault frequency band label utilized during the training procedure. On average, the information is concentrated in the fault frequency band, with certain channels being more sensitive. Figure 14c and d show the signals subsequently restored by the fault-segmenter, along with their FFT and envelope FFT, respectively. These figures also include the fault frequency band and FCF as a red-dotted line. It is evident that the corresponding frequency band is emphasized and the FCF peak is visible.

Figure 14:

Feature representation in frequency domain under task CS₃ for outer raceway fault simulated signal: (a) FFT of signal, (b) FFT of extracted feature (gray) and mean (blue), (c) FFT of up-sampled data with fault frequency area (red-dotted line), and (d) Envelope FFT of up-sampled data (FCF: red-dotted line).

Figure 15 represents the physical analysis results from the real target fault signal utilized in the test. This signal was measured at 1600RPM for the outer raceway fault of the REB. Since the actual fault-related frequency band is unknown from the real fault signal, it is challenging to evaluate the physical meaning of the results intuitively. Thus, we employed a severe fault sample included in the test dataset, which at least vaguely represents the fault-related information in the frequency band. The extracted features and the frequency band segmentation results have energy concentrations around 3500 and 4000 Hz in Fig. 15b and c. Furthermore, it was validated that the up-sampled features clearly represent the expected FCF in Fig. 15d. Even if the fault frequency peak is not as clearly visible as in the analysis using the simulated signal, features are extracted by focusing on specific frequency bands. This suggests that the model effectively focuses on fault-related information, contributing to high fault-diagnosis accuracy.

Figure 15:

Feature representation in frequency domain under task CS₃ for real outer raceway fault signal: (a) FFT of signal, (b) FFT of extracted feature (gray) and mean (red), (c) FFT of up-sampled data with fault frequency area (red-dotted line), and (d) Envelope FFT of up-sampled data (FCF: red-dotted line).

5. Conclusions

This study presents a novel REB fault-diagnosis model to address the critical challenges of data scarcity and domain shift between training and inference datasets. The proposed method introduces uncertainty-included and fault signal simulation, along with a physics-aware FBSDA model. Our proposed simulation method leverages statistical cyclo-stationary modeling to account for uncertainties in the signal acquisition environment. By integrating simulated fault signals with system characteristic signals (i.e., target normal data), the generated signals more accurately reflect real-world environments and the underlying physical principles. To bridge the domain gap between simulated and real-world signals and focus on the fault-related feature, we introduced FBSDA, which incorporates a fault frequency segmentation module and a loss function inspired by image segmentation techniques within a domain adversarial learning framework. This approach effectively mitigates the domain discrepancy between the source and target domains. Validation of our method using two different testbed datasets acquired under various operating conditions demonstrated its superior performance in handling domain shifts and different levels of partial labels. Our approach consistently outperformed existing fault-diagnosis methods under significant domain-shift scenarios, confirming its potential for robustness and reliability in real-world industrial settings. In conclusion, the proposed method significantly advances the field of fault diagnosis for REBs by providing a robust and accurate solution. This method is the first to apply image segmentation to identify fault frequency band in vibration signals for bearing fault diagnosis. Few studies have applied segmentation to the frequency domain of time-series data, so this work contributes to the AI community by embedding expert knowledge into the approach.

This study demonstrates the practical value of the proposed method in generating fault signals that are difficult to obtain on-site and diagnosing using simulation. However, it has limitations in simulating complex faults or faults requiring system-level considerations. Addressing these challenges will require further research into simulations that account for the interrelationships between system components and the physical characteristics of complex faults, enhancing their practical value for real-world applications. Additionally, obtaining accurately labeled fault data in the source domain can be challenging in real environments. Therefore, it is essential to explore methods for handling scenarios where fault signals are unavailable, as well as fully unsupervised settings with no labeled data. Overcoming these challenges could enable effective fault diagnosis even in the absence of actual fault data. This study showed that the differences between generated and actual signals are minimal, and even when the differences occur, robust diagnosis remains achievable due to the generalization capability of the proposed domain adaptation-based deep learning model. However, further research is required to improve the model’s robustness and generalizability. Future work could develop simulations for diagnosis without DA or methods to handle signal differences for cross-domain diagnosis. These advancements would make generated signals applicable for diagnostics across diverse domains.

Conflict of interest statement

The authors declare no conflict of interest.

Author contributions

Jongmin Park (Conceptualization, Investigation, Methodology, Validation, Writing—original draft), Jinoh Yoo (Investigation, Conceptualization), Taehyung Kim (Investigation, Writing—review & editing), Minjung Kim (Investigation), Jonghyuk Park (Software), Jong Moon Ha (Conceptualization, Project administration, Writing—review & editing), and Byeng D. Youn (Supervision, Project administration, Funding acquisition, Writing—review & editing)

Acknowledgments

This research was supported by the International Research & Development Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (No. 2022K1A4A7A04096329).

Data availability

The data underlying this article are available at https://engineering.case.edu/bearingdatacenter/download-data-file