Combining Machine Learning Methods and Data Augmentation for Misaligned Journal Bearings Design

Konstantinos N. Arvanitis1 and Pantelis G. Nikolakopoulos1

1 Machine Design Laboratory, Department of Mechanical Engineering and Aeronautics, University of Patras, 26504 Patras, Greece

Abstract. Shaft misalignment is one of the most common defects observed in rotating systems and has a substantial effect on dynamic behavior, stability, and lifetime. Aim of this study is the binary identification of misalignment using five Machine Learning techniques: Logistic Regression, K-Nearest Neighbors, Support Vector Machines, Decision Tree and Random Forest. Nevertheless, the limited quantity of provided data points coupled with the substantial imbalance between the aligned and misaligned cases necessitated the implementation of oversampling and data augmentation methods. The utilization of SMOTE-LOF for oversampling the minority class, alongside the adoption of a Conditional Tabular GAN for the generation of synthetic data points yielded substantial outcomes. The application of SMOTE with the Local Outlier Factor on the original dataset achieves the oversampling of the minority class by using additional synthetic data, while the LOF factor overcome the noise problem. After the SMOTE-LOF implementation, new synthetic samples of the minority are added to the dataset, eliminating the imbalance, however the ‘cleaned’ dataset’s small number of samples could lead to poor performance of the Machine Learning techniques. Hence, using a Generative Adversarial Network to expand the dataset is requisite, and since the experimental data is tabular, utilizing a Conditional Tabular Generative Adversarial Network is ideal for this case of study. The results of the current study elucidate the dataset that, among the augmented datasets, exhibits the best quality score when contrasted with the original dataset. Furthermore, the dataset that performs the best on the Machine Learning algorithms is determined.

Keywords: data augmentation, resampling, machine learning, journal bearing, misalignment

Introduction

Journal bearings are prevalent engineering components in almost all types of rotating machinery. The identification of defects observed in rotating systems has already attracted several researchers with remarkable results. More specifically, a lot of articles have been examining the diagnosis of misalignment on journal bearings, since it is one of the most severe defects that influences the bearing behavior and consequently the rotor bearing dynamics. Traditional methods such as finite element, finite difference, meshless methods are already being utilized to solve the Reynolds equation for the pressure distribution.

On the other hand, as computer science continues to advance, there has been a notable surge in the utilization of machine learning applications within the field of mechanical engineering. However, within the domain of Tribology, machine learning applications still constitute a new era of research [1], and their growth is steadily extending across various facets of the field.

Gertzos et al. [2] presented a CFD analysis of hydrodynamic lubricated bearings lubricated with a Bingham fluid. Nuruzzaman et al. [3] investigated the pressure distribution and load capacity of a journal bearing using the finite element method (FEM), while Wenjie Zhou et al. [4] proposed a novel numerical method for studying the lubricated characteristics of journal bearings, including the FEM and mixed method. Moreover, J.W. Lund [5] reviewed the dynamic coefficients for fluid film journal bearings, and Zhou et al. [6] proposed a method to find the bearing’s static equilibrium position. In addition, Yiyang Fu et al [7] calculated the pressure development of journal bearings by using inhomogeneous and homogeneous grid in the Finite Difference Method (FDM).

Since misalignment is having a significant impact in the performance of the journal bearing, many researchers have studied misaligned journal bearings, both experimentally and analytically. Jang and Khonsari [8] presented a literature review of the characteristics of the misaligned journal bearings, while Singh and Sinhasan [9] investigated the effects of small misalignment on the dynamic characteristics of an ungrooved big end bearing utilizing the FEM method. Furthermore, Nikolakopoulos and Papadopoulos [10] studied the nonlinear behavior of misaligned journal bearings, while Jun Sun et al [11] presented a hydrodynamic lubrication analysis of misaligned journal bearings due to shaft deformation. Experimental work of axial misalignment effect on the seizure load of journal bearings was executed by Zhang et al [12].

Furthermore, experimental studies of misaligned bearings were conducted by Huber et al. [13] and Bouyer and Fillon [14]. Huber et al. investigated the misaligned plain circular bearing characteristics such as pressure and oil flow under a static load, while Bouyer and Fillon evaluated the performance of a plain journal bearing submitted to a misalignment torque.

The field of artificial intelligence (AI) applied to journal bearing lubrication is relatively new, and although some research has been published there remains ample room for forthcoming contributions. Rosenkranz, Marian et al. [15] presented an overview of the various applications of artificial intelligence (AI) and machine learning (ML) in tribology. Machine learning techniques have been applied for journal bearings’ misalignment monitoring. Saridakis et al. [16] proposed a misalignment diagnosis approach that uses artificial neural networks. The proposed model uses values of journal bearing performance characteristics, such as eccentricity, attitude angle and film thickness and it trains an ANN so that reliable identification is performed. Also, Nikolakopoulos and Kokkinidis [17] made use of ANNs to solve the problem of misaligned hydrodynamic journal bearing. Firstly, the Reynolds equation was solved using finite difference methos (FDM) for different operating and misalignment conditions. The results was used as an input to train four ANNs, one for each design parameter. After the networks were tested for several operational characteristics, the results were compared with the results of the finite difference method. As an outcome, the force and the torque can be predicted with maximum error of 5% with less computational cost than the FDM. Furthermore, the application of ANNs to calculate the hydrodynamic forces and the dynamic of the rotor was investigated by Kornaev et al. [18]. Nevertheless, especially in tribology, collecting extensive datasets can be costly, time-consuming, or even infeasible. Datasets that suffer from imbalance and limited data availability represent common scenarios and frequently result in poor performance of machine learning algorithms. Oversampling technique is widely used in order to handle the imbalance of datasets, however in this case the data distribution is ignored, and the risk of overfitting is increased. Aiming at this problem, Chawla et al. [19] proposed the Synthetic Minority Oversampling Technique (SMOTE). Zhenya Wang et al [20] proposed a method using SMOTE and CNN to deal with imbalanced bearing data, while Xinqian Hu et al. [21] utilized a variation of SMOTE method in order to enhance the variety of fault samples for bearing fault diagnosis. Additionally, data augmentation is an imperative technique in engineering problems that suffer from limited data availability. Data augmentation techniques provide a pragmatic solution by synthetically expanding the dataset through various transformations. Generative Adversarial Networks (GANs) are a popular class of data augmentation models introduced by Ian Goodfellow et al. in 2014 [22] and has few applications on fault diagnosis of rotating systems in case of limited datasets. Wentao Mao et al. used a GAN to improve the generalization ability of the fault diagnosis model of a rolling bearing, and Diwang Ruan et al [23] investigate the bearing fault diagnosis by optimizing a CNN and a GAN collaboratively. Additionally, Cheng Peng et al. [24] substantiate the enhancement of the rolling bearing fault diagnosis method as a result of the GAN utilization.

The aim of this study is the identification of journal bearing misalignment within the context of a limited and imbalanced dataset. The integration of the Synthetic Minority Oversampling Technique (SMOTE) and Conditional Tabular Generative Adversarial Network (CTGAN) is used to effectively address the constraints presented by the limited and imbalanced journal bearing dataset provided by previous work of Saridakis et al. [16]. Then, five machine learning algorithms are utilized: Logistic Regression, K-Nearest Neighbors, Support Vector Machines, Decision Tree and Random Forest. The outcomes of this work elucidate the optimal quantity of generated synthetic samples necessary for the augmented dataset to exhibit the best quality score compared to the original dataset. Additionally, the performance of the machine learning algorithms is taken into consideration.

Theory

Misalignment on journal bearings

Ideally, in a journal bearing, the axes of both shaft and bush are parallel when installed and remain so during the operation under the imposed load and speed. Nevertheless, in practice, this ideal condition rarely exists, and the shaft tends to suffer from some degree of misalignment while rotating inside its bushing. The primary effect is considerable reduction in the minimum thickness of the lubricant film, which is responsible for protecting the surfaces from direct contact.

Substantially, reduced minimum film thickness also alters the entire pressure and temperature fields. In fact, the maximum values of pressure and temperature in misaligned bearings are much higher than those in the aligned ones. To counteract this effect a slight defect in the minimum film thickness area is sometimes added to the bearing geometry to prevent contact. Misalignment can also result from an individual or combination of the following factors: asymmetric bearing loading, elastic deflection of the shaft, under an imposed under load or its own weight., thermal distortion of the shaft, distortions caused by bearing housing support, distortions caused by bearing housing supports., and errors due to installation and assembly defects.

The governing equation is the Reynold’s equation, which can be derived through two distinct approaches: either from Navier-Stokes equations (conservation of mass and momentum) or from equilibrium of an element of the fluid. The simplified equation considering isoviscous lubricant with steady film thickness and unidirectional velocity approximation [25] is,

(1)

The most important effect of bearing misalignment is the drastic alteration of the protective film thickness in both the circumferential and axial directions. Therefore, the Reynolds equation should be modified to allow the variation of film thickness in both directions, meaning that an analytical solution is not available. This needs to be carefully taken into consideration in the analysis. Figure 1.a shows a typical section of the journal and the bearing for the aligned case.

(a)	(b)	(c)

Figure 1: Geometry of aligned and misaligned journal bearing [17]

The sections in the x-z and y-z planes of a misaligned journal bearing are shown in Figure 1.b, while Figure 1.c provides a representation of the geometry in the x-y plane for z=0.

The film thickness prediction plays a significant role in misaligned journal bearings. Nikolakopoulos and Papadopoulos [10] mentioned the following expression describing the film thickness with provision for misalignment:

(2)

Another useful parameter, which indicates the bearing lubrication regime, is the minimum film thickness hmin, which occurs at one of the two ends of the bearing. The minimum film thicknesses for misaligned and worn journal bearings are defined by the following expressions, as mentioned by Saridakis et al. [16]:

(3)
(4)

Synthetic Minority Oversampling Technique (SMOTE)

Imbalanced data typically refers to a condition in which a number of data samples in a certain problem is not equally distributed. This disproportion in distribution occurs when the number of data samples representing a class is significantly lower than those in other classes, thereby leading to the underrepresentation of one or more classes in the dataset. These underrepresented classes are referred to as the minority or positive class, while others are called the majority or negative class. Generally, there are two traditional techniques for handling class imbalance: undersampling and oversampling. Undersampling involves removing instances of the majority class while in oversampling, instances of the minority class are duplicated.

Another canonical technique is Synthetic Minority Oversampling Technique (SMOTE). The SMOTE [19] is based on the idea of the nearest neighbor algorithm (kNN) and assumes that a synthetic data sample can be interpolated between members of the minority class in “feature space” (as opposed to “data space”). Concretely, the SMOTE algorithm chooses k nearest neighbors for each data point based upon the similarities in feature vectors and creates a synthetic sample at a random point on the line between each data point and chosen neighbor. As a result, the original data obtained using SMOTE are used to synthesize new minority data that are different from the original ones, thereby alleviating the impact of overfitting on the minority class, which is a common issue caused by random oversampling. This technique boosts class membership whilst introducing enough variety to encourage classifier generalization and grow minority class decision regions.

The minority class is over-sampled by taking each minority class sample and introducing synthetic examples along the line segments joining any/all of the k minority class nearest neighbors (usually five nearest neighbors are used). Depending upon the amount of over-sampling required, neighbors from the k nearest neighbors are randomly chosen. Synthetic samples are generated in the following way: Take the difference between the feature vector (sample) under consideration and its nearest neighbor. Multiply this difference by a random number between 0 and 1 and add it to the feature vector under consideration. This causes the selection of a random point along the line segment between two specific features. This approach effectively forces the decision region of the minority class to become more general. The synthetic examples cause the classifier to create larger and less specific decision regions, rather than smaller and more specific ones that may lead to overfitting.

Figure 2: Example of generating synthetic data with SMOTE [19].

However, SMOTE can produce synthetic minority data samples considered as noise, which is also part of the majority classes. Therefore, an improvement of the technique by adding the Local Outlier Factor (LOF) has been introduced. [26] The proposed method is called SMOTE-LOF and produces better accuracy, AUC and f-measure than the SMOTE. An outlier is the odd point in a dataset. It is also known as abnormalities, discordances, deviations or anomalies, while noise can be defined as the wrong label (class noise) or errors in the attribute value (attribute noise). Typically, most of outliers are noise, however, sometimes they are correct data. Therefore, out of a number of outliers detected, the ones identified as noise are derived from synthetic minority class samples generated by the SMOTE. The LOF value is the mean ratio of mean reachability distance of each minority sample divided by the mean reachability distance of each k-neighbor.

(5)
(6)

Where:

lof(x): Local Outlier Factor of object x in dataset o

lrd(o): Local Reachability Density of objects in dataset o

lrd(x): Local Reachability Density of object x

Figure 3: Noise samples in a dataset [26]

Conditional Tabular Generative Adversarial Network (CTGAN)

Generative Adversarial Networks [22], or GANs for short, are an approach to generative modeling using deep learning methods, such as convolutional neural networks. Generative modeling is an unsupervised learning task in machine learning that involves automatically discovering and learning the regularities or patterns in input data in such a way that the model can be used to generate or output new examples that plausibly could have been drawn from the original dataset.

The basic principle of GANs is inspired by two-player zero-sum game, in which the total gains of two players are zero, and each player’s gain or loss of utility is exactly balanced by the loss or gain of the utility of another player. GANs often comprise a generator and a discriminator that learn simultaneously. The generator tries to capture the potential distribution of real samples and generates new data samples. The discriminator is often a binary classifier, discriminating real samples from the generated samples as accurately as possible. Both the generator and the discriminator can adopt the structure of currently popular deep neural networks. The optimization process of GANs is a minimax game process, and the optimization goal is to reach Nash equilibrium, where the generator is considered to have captured the distribution of real samples.

Figure 4: The structure of a Generative Adversarial Network (GAN) [27].

The discriminator evaluates the quality of the data created by the generator [27]. It attempts to predict the source of the sample, using samples that come from two sources:

Real data instances. The discriminator uses these instances as positive examples during training.

Fake data instances created by the generator. The discriminator uses these instances as negative examples during training.

The discriminator connects to two loss functions, the discriminator one and the generator one. During its training, it ignores the generator loss and just uses the discriminator loss. Generator loss is used during generator training.

During discriminator training:

The discriminator classifies both real data and fake data from the generator.

The discriminator loss penalizes the discriminator for misclassifying a real instance as fake or a fake instance as real.

The discriminator updates its weights through backpropagation from the discriminator loss through the discriminator network.

Figure 5: Backpropagation in discriminator training [27].

The generator learns to map a latent space to the distribution of the data it aims to reproduce, so that when fed with a noise vector from the latent space, it predicts a sample from the estimated distribution [27]. The generator is evaluated by the discriminator, meaning that its goal is to create data samples that are similar to those in the original data set. The discriminator penalizes the generator for producing implausible results. The generator output is connected directly to the discriminator input. Through backpropagation, the discriminator's classification provides a signal that the generator uses to update its weights.

Generator training requires tighter integration between the generator and the discriminator than discriminator training requires. The portion of the GAN that trains the generator includes: random input (noise), generator network, which transforms the random input into a data instance, discriminator network, which classifies the generated data, discriminator output., and generator loss, which penalizes the generator for failing to fool the discriminator.

Figure 6: Backpropagation in generator training [27].

To train a neural net, we alter the net's weights to reduce the error or loss of its output. In GAN, however, the generator is not directly connected to the loss that we're trying to affect. The generator feeds into the discriminator net, and the discriminator produces the output we're trying to affect. The generator loss penalizes the generator for producing a sample that the discriminator network classifies as fake.

This extra chunk of the network must be included in backpropagation. Backpropagation adjust each weight in the right direction by calculating the weight’s impact on the output – how the output would change if the weight is changed. However, the impact of the generator weight depends on the impact of the discriminator weights it feeds into. So backpropagation starts at the output and flows back through the discriminator into the generator.

Meanwhile, the discriminator must not change during generator training. So, the generator is trained with the following procedure [27]: sample random noise, , produce generator output from sampled random noise, get discriminator ‘Real’ or ‘Fake’ classification for generator output, calculate loss from discriminator classification, backpropagate through both the discriminator and generator to obtain gradients, and use these gradients to change only the generator weights.

Similarly, discriminator is constant during generator training phase. Otherwise the generator would be trying to hit a moving target and might never converge. It’s this back and forth that allows GANs to solve generative problems that would otherwise be insoluble. The discriminator will try to get better at distinguishing fake and real data, while the generator will progressively get better at generating data that is closer to the original.

On the other hand, for a GAN, convergence is often a fleeting, rather than stable, state. As the generator improves with training, discriminator performance declines because it can’t easily tell the difference between real and fake data. In case that the generator succeeds perfectly, then the discriminator has a 50% accuracy, meaning that it flips a coin to make its prediction. As a result, discriminator feedback gets less meaningful over time. To better understand the minimax optimization in GANs, one needs to first answer the following question: What is the proper notion of equilibrium in GAN zero-sum games? In other words, what should be the optimality criteria in the GAN’s minimax optimization problem? A classical notion of equilibrium in the game theory literature is the Nash equilibrium, a state in which no player can improve its individual gain by choosing a different strategy. As a well-established result, for a generator G expressive enough to reproduce the distribution of observed samples, Nash equilibrium exists for that generator function producing the data distribution [28].

Most commonly, GANs are used for image or even audio/speech generation. However, there are plenty of Real-world domains that are often described by “tabular data”, i.e., data that can be structured and organized in a table-like format. As a standard, features (sometimes called “variables” or “attributes”) are represented in columns, whereas observations (or “records”) correspond to the rows. Additionally, real-world data usually comprises both numeric and categorical features. Numeric features (also called “continuous”) are those that encode quantitative values, whereas categorical (also called “discrete”) represent qualitative measurements. CTGAN (Conditional Tabular Generative Adversarial Network) was conceptualized to partially “capture” this heterogeneity of real-world data and compared to other architectures such as WGAN and WGAN-GP, has proven to be more robust and generalizable for a variety of datasets [29]. The "conditional" part of CTGAN refers to the fact that it can also generate synthetic data that is conditioned on certain input variables, such as gender or age.

Contrary to image data, where pixel values normally follow a Gaussian-like distribution, continuous features in tabular data are often non-Gaussian. Moreover, they tend to follow multimodal distributions, where probability distributions have more than one mode, i.e., they present distinct local maxima (or “peaks”).

Figure 7: Gaussian versus Multimodal Distribution [29].

To capture these behaviors, CTGAN uses mode-specific normalization. Using a VGM (Variational Gaussian Mixture) model, each value in a continuous feature is represented by a one-hot vector indicating its sampled mode and a scalar that represents the value normalized according to that mode. An example of mode-specific normalization is presented in Figure 8. The VGM finds three modes, namely n1, n2 and n3. Cij represents a value in a feature j (e.g., j=”Age”), for which ρ3 is picked. Cij is therefore represented by a one-hot vector indicating the third node, and a scalar , where φ3 represents the standard deviation of the third mode (ρ3).

The representation of a row become the concatenation of continuous and discrete columns.

(7)

Where di,j is one-hot representation of a discrete value.

Figure 8: Example of mode-specific normalization [29].

On the other hand, CTGAN aims to address essentially two main challenges introduced by categorical features. The first challenge is the sparsity of one-hot-encoded vectors in real-world data. While the generator outputs the probability distributions over all possible categorical values, the original “real” categorical values are directly encoded in a one-hot-vector. These are easily distinguishable by the discriminator by comparing the distribution sparseness between real and synthetic data. The second challenge is the imbalance associated to some categorical features. If some categories of a feature are underrepresented, they cannot be learned adequately by the generator. If we were concerned with predictive modeling or classification tasks, data oversampling could be a solution to alleviate this issue. However, since the goal of synthetic data generation is to mimic the properties of the original data, this is not an option.

CTGAN introduces a Conditional Generator to deal with the challenges imposed by imbalanced categories which usually lead to GAN’s infamous mode collapse. Specifically, the goal is to resample efficiently in a way that all the categories from discrete attributes are sampled evenly (but not necessary uniformly) during the training process, and to recover the (not-resampled) real data distribution during test. Let k∗ be the value from the i∗th discrete column Di∗ that has to be matched by the generated samples , then the generator can be interpreted as the conditional distribution of rows given that particular value at that particular column, i.e. . For this reason, it is named Conditional Generator and the GAN is referred as Conditional GAN.

Figure 9: Overview of the CTGAN model. With training-by-sampling, examples are conditioned on the possible values of categorical features, sampled according to their log-frequency [29].

The key to training a GAN and evaluate its performance over time are the loss functions, a set of formulas that indicate each network how to improve after each training iteration (or epoch). The discriminator and the generator each have their own loss values. Epoch after epoch, these networks learn by trying to minimize their loss function.

Loss functions are an area of active research, and many approaches have been proposed for different uses. In CTGAN, the custom loss functions formulated by SDV [30] for the purposes of creating synthetic data are used.

(8)
(9)

Where:

x: real data

x’: synthetic data

D(x): Discriminator’s output given the real data

D(x’): Discriminator’s output given the synthetic data

The discriminator is learning to produce low values if the data is synthetic and high values if it is real. The range of these values is dependent on linear transformations and dimensions of the input data. When CTGAN performs well, both the generator and discriminator loss will eventually stabilize when they have run through enough epochs [30].

Results and Discussion

Using the results of the previous work, the primary objective of this current study is the identification of misalignment of a journal bearing. The 13×1331 dataset from [16] was used for implementation of a variety of Machine Learning algorithms., while a binary column named ‘Misaligned’ was added in order to determine if the bearing is aligned or not. The journal bearing was judged to be aligned if dimensionless misalignment angles were less than 0.05. It is mentioned that wear depth is used as a training input, but it is not a target variable, nevertheless.

Figure 10: Dataset used in this study (first 5 rows).

Two primary issues were identified through the application of a preliminary analysis to the data:

Limited dataset. The Machine Learning algorithms would not perform well given the 1331 experimental data samples.

Imbalanced dataset. Aligned cases outnumber the misaligned cases by a wide margin, making the Machine Learning models prone to overfitting.

Figure 11: Flowchart for this study	This work introduces a new model for the identification of misalignment, particularly when faced with limited and imbalanced dataset. Initially, the imbalance within the minority class is addressed through the application of SMOTE, coupled with the LOF mechanism to mitigate noise interference. Subsequently, the resampled, ‘cleaned’ dataset is further augmented with synthetic samples using a CTGAN, thereby ensuring a sufficient volume of data samples. Finally, the expanded dataset is subjected to various Machine Learning algorithms for predictive modeling. A visual representation of this workflow is depicted in Figure 12. The Aligned – Misaligned cases on the resampled and the cleaned dataset are presented, where resampled is the dataset after the SMOTE implementation and the cleaned is the one after the SMOTE-LOF implementation. It is clear that the SMOTE method oversampled the minority class (aligned case), and as an outcome the number of aligned cases is equal with the misaligned ones. Then, the 258 generated outliers are erased using the LOF, and the misaligned cases subsequently outnumber the aligned cases by a small margin.

(a)	(b)	(c)

Figure 12: Aligned-Misaligned cases on the original, resampled and cleaned dataset

Additionally, some pairwise relationships of the cleaned dataset variables, with the noisy samples highlighted, are presented as well.

Figure 13: Cleaned data with noisy samples highlighted

Then, using the SDV library in Python, a CTGAN model is created and fit to the cleaned dataset and new synthetic data is generated. It is noted that all the features of the input data is numeric; the “Misaligned” column is erased and then added to the generated dataset, depending on the values of the nondimensional misalignment angle. The CTGAN model is based on the GAN-based Deep Learning data synthesizer which was proposed in [29].

In CTGAN, the mode-specific normalization is invented to overcome the non-Gaussian and multimodal distribution. A conditional generator and training-by-sampling are designed to deal with the imbalanced discrete columns. Finally, fully-connected networks and several recent techniques are utilized to train a high-quality model.

Define the following notations:

x1 ⊕ x2 ⊕ . . .: concatenate vectors x1, x2, . . .

gumbelτ(x): apply Gumbel softmax with parameter τ on a vector x

leakyγ(x): apply a leaky ReLU activation on x with leaky ratio γ

FCu→υ(x): apply a linear transformation on a u-dim input to get a υ-dim output

Tanh, ReLU, softmax, BN are used for batch normalization, as well as drop for dropout.

Since columns in a row do not have local structure, fully-connected networks are used in generator and discriminator to capture all possible correlations between columns. Specifically, two fully-connected hidden layers are utilized in both generator and discriminator. In generator, batch-normalization and Relu activation function are used. After two hidden layers, the synthetic row representation is generated using a mix activation functions. The scalar values αi is generated by tanh, while the mode indicator βi and discrete values di is generated by gumbel softmax. In discriminator, leaky relu function and dropout on each hidden layer are used.

Finally, the conditional generator G(z, cond) can be formally described as:

(10)

For the discriminator, the PacGAN framework is used with 10 samples in each pac to prevent mode collapse. The architecture of the critic (with pac size 10) C(r1, . . . , r10, cond1, . . . cond10) can be formally described as:

(11)

The model is trained using WGAN loss with gradient penalty. Adam optimizer is used with learning rate 2*10-4.

Using the SDV library in Python, new samples of data are generated. In this study we train the CTGAN for 1000 epochs, while both the generator and the discriminator have two fully connected hidden layers with 256 neurons each. Also, the batch size (the number of samples used in each step) is set to 500.

Consequently, a number of generated synthetic samples are added to the original dataset, resulting to the new, expanded dataset. The performance of the Machine Learning algorithms is evaluated for 3000, 4000, 5000 and 8000 synthetic samples.

SDV library in Python provides a variety of metrics to evaluate the quality of the generated samples compared to the original dataset. For this study, quality report as well as visualizing the synthetic data are used. Additionally, the distribution of generator and discriminator loss per epoch is presented.

The quality report provides an overall quality score, column shapes and column pair trends. Column shapes is a property that looks at the overall shape the columns to see if the synthetic data matches the real data, while the column pair trends between two columns describe how they vary in relation to each other, for example the correlation. The higher the score, the more the trends are alike.

The results of this study for 4000 generated samples are presented in the table below:

Table 1: CTGAN metrics for each expanded dataset.

Metric	Overall Quality Score	Column Shapes	Columns Pair Trends
3000 Generated Samples	89.86%	91.64%	88.08%
4000 Generated Samples	87.34%	89.8%	84.89%
5000 Generated Samples	93.47%	91.2%	95.73%
8000 Generated Samples	88.07%	89.93%	86.21%

It is noticed that the expanded dataset of 5000 generated samples has the best overall quality score and column pair trends, while the column shapes value is above 90%. Then, we plot the column shape for each feature of this specific dataset.

Figure 14: CTGAN Column shape for each feature of the expanded dataset of 5000 synthetic samples.

It is observed that the data distribution of the synthetic data samples is satisfying similar to the original ones, with some noticeable inconsistencies especially at the distribution of column “f”.

Finally, the generator and the discriminator loss are plotted for each epoch.

Figure 15: Generator and Discriminator loss

As mentioned beforehand, generator and discriminator loss stabilization over time indicates a good performance of the CTGAN. From the plot it is observed that after approximately 400 epochs both generator and discriminator loss eventually stabilize and get less noisier, suggesting that CTGAN performs well. It is noteworthy that the selection of 1000 epochs for runtime proved judicious, as both the discriminator and generator loss reached a state of stabilization at an extended number of epochs.

After the data augmentation, the machine learning techniques are applied to each new extended dataset, utilizing the sklearn library in Python.

Table 2: Accuracy and AUC of each ML algorithm on the augmented datasets.

ML algorithm	Logistic Regression	ΚΝΝ	SVM	Decision Tree	Random Forest
Accuracy	AUC	Accuracy	AUC	Accuracy	AUC	Accuracy	AUC	Accuracy	AUC
3000 Generated Samples	86.52%	0.88	88.78%	0.89	90.16%	0.9	85.58%	0.85	89.09%	0.88
4000 Generated Samples	89.39%	0.9	91.50%	0.92	90.61%	0.91	87.28%	0.87	91.03%	0.91
5000 Generated Samples	89.98%	0.9	90.66%	0.91	89.48%	0.9	86.79%	0.86	90.16%	0.9
8000 Generated Samples	85.75%	0.87	87.01%	0.87	87.53%	0.88	82.10%	0.81	87.04%	0.87

Conclusions

This work resulted in some important conclusions regarding the application of data augmentation and Machine Learning methods to the tribological design of misaligned journal bearings. The rarity of comparable studies in scientific literature not only amplifies the significance of the findings, but also initiates the exploration of this new field of research. To sum up, the application of the SMOTE-LOF eliminated the imbalance of the original dataset. Then, four (4) augmented datasets were generated by utilizing a Convolutional Tabular GAN (CTGAN), each one consisted of 3000,4000,5000 and 8000 synthetic data samples accordingly. Finally, the Machine Learning algorithms were utilized on each expanded dataset. For this study, a variety of ML techniques were chosen to utilize on the augmented datasets, in order to determine which one is the ideal for this kind of problem. Furthermore, the utilization of multiple ML algorithms on the augmented datasets affords a more comprehensive assessment on the datasets’ overall performance, thereby facilitating the selection of the most suitable dataset. More specifically, the following are observed:

The SMOTE technique eliminates the original dataset’s imbalance effectively, while the LOF erases a considerable amount of noisy samples.

The augmented dataset consisted of 5000 synthetic data samples has the best performance and quality compared to the original dataset.

A good performance of the CTGAN is indicated by the stabilization of both the discriminator and generator loss after an extended number of epochs.

The augmented dataset consisted of 4000 synthetic samples exhibits the superior performance on the Machine Learning algorithms.

KNN and SVM are the algorithms with the best performance on the augmented datasets.

Therefore, in the context of monitoring applications, it is judicious to opt for the generation of 4000 synthetic samples, since the augmented dataset exhibits superior performance when subjected to Machine Learning techniques. As a generic conclusion, the proposed methodology can effectively employed for the identification of the misalignment of journal bearings, especially when dealing with limited and imbalanced data. This is a very helpful tool when collecting a sufficient amount of experimental data is not feasible.

References

M. Marian and S. Tremmel, “Current Trends and Applications of Machine Learning in Tribology—A Review,” Lubricants, vol. 9, no. 9, 2021, doi: https://doi.org/10.3390/lubricants9090086.

K. P. Gertzos, P. G. Nikolakopoulos, and C. A. Papadopoulos, “CFD analysis of journal bearing hydrodynamic lubrication by Bingham lubricant,” Tribology International, vol. 41, no. 12, 2008, doi: https://doi.org/10.1016/j.triboint.2008.03.002.

D. M. Nuruzzaman, M. K. Khalil, M. A. Chowdhury, and M. L. Rahaman, “Study on Pressure Distribution and Load Capacity of a Journal Bearing Using Finite Element Method and Analytical Method,” International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS, vol. 10, no. 5, 2010.

W. Zhou, Y. Wang, G. Wu, B. Gao, and W. Zhang, “Research on the lubricated characteristics of journal bearing based on finite element method and mixed method,” Ain Shams Engineering Journal, vol. 13, no. 4, 2022, doi: https://doi.org/10.1016/j.asej.2021.11.007.

J. W. Lund, “Review of the Concept of Dynamic Coefficients for Fluid Film Journal Bearings,” Journal of Tribology, vol. 109, no. 41, 1987, doi: https://doi.org/10.1115/1.3261324.

W. Zhou, X. Wei, L. Wang, and G. Wu, “A superlinear iteration method for calculation of finite length journal bearing’s static equilibrium position,” Royal Society Open Science, vol. 4, no. 5, 2017, doi: https://doi.org/10.1098/rsos.161059.

Y. Fu, J. Jin, and X. Chen, “Numerical analysis of Journal Bearing based on Finite Difference Method with Inhomogeneous Grid,” Journal of physics, vol. 1064, 2018, doi: https://doi.org/10.1088/1742-6596/1064/1/012059.

J. Jang and M. Khonsari, “On the Characteristics of Misaligned Journal Bearings,” Lubricants, vol. 3, no. 1, 2015, doi: https://doi.org/10.3390/lubricants3010027.

D. Singh, R. Sinhasan, and R. Pal, “Performance Characteristics of an Ungrooved Big-End Bearing with Misalignment,” Tribology Transactions, vol. 32, no. 2, 1989, doi: https://doi.org/10.1080/10402008908981883.

P. G. Nikolakopoulos and C. A. Papadopoulos, “Non-linearities in misaligned journal bearings,” Tribology International, vol. 27, no. 4, 1994, doi: https://doi.org/10.1016/0301-679x(94)90004-3.

J. Sun and G. Changlin, “Hydrodynamic lubrication analysis of journal bearing considering misalignment caused by shaft deformation,” Tribology International, vol. 37, no. 10, 2004, doi: https://doi.org/10.1016/j.triboint.2004.05.007.

X. Liang, Z. Yin, and Q. Dong, “An experimental study of axial misalignment effect on seizure load of journal bearings,” Tribology International, vol. 131, pp. 476–487, 2019, doi: https://doi.org/10.1016/j.triboint.2018.11.014.

M. Huber, S. Strzelecki, and W. Steinhilper, “Theoretical and Experimental Determination of the Performances of Misaligned and Statically Loaded Cylindrical Journal Bearings,” Tribology at Work: Proceedings of the 5th International Tribology Conference in Australia, 1998.

J. Bouyer and M. Fillon, “An Experimental Analysis of Misalignment Effects on Hydrodynamic Plain Journal Bearing Performances,” Journal of Tribology, vol. 124, no. 2, 2002, doi: https://doi.org/10.1115/1.1402180.

A. Rosenkranz, M. Marian, F. J. Profito, N. Aragon, and R. Shah, “The Use of Artificial Intelligence in Tribology—A Perspective,” Lubricants, vol. 9, no. 1, 2020, doi: https://doi.org/10.3390/lubricants9010002.

K. M. Saridakis, P. G. Nikolakopoulos, C. Papadopoulos, and A.J. Dentsoras, “Identification of wear and misalignment on journal bearings using artificial neural networks,” Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, vol. 226, 2012, doi: https://doi.org/10.1177/1350650111424237.

K.A. Kokkinidis and P. G. Nikolakopoulos, “Simulation of Misaligned Journal Bearings Using Neural Networks,” Tribology in industry, vol. 44, no. 1, 2022, doi: https://doi.org/10.24874/ti.1078.03.21.10.

Alexey Kornaev, Nickolay Kornaev, E. Kornaeva, and L. Savin, “Application of Artificial Neural Networks to Calculation of Oil Film Reaction Forces and Dynamics of Rotors on Journal Bearings,” International Journal of Rotating Machinery, vol. 2017, 2017, doi: https://doi.org/10.1155/2017/9196701.

N. V. Chawla, K. W. Bowyer, L. O. Hall, and W. P. Kegelmeyer, “SMOTE: Synthetic Minority Over-sampling Technique,” Journal of Artificial Intelligence Research, vol. 16, Jun. 2002, doi: https://doi.org/10.1613/jair.953.

Z. Wang, T. Liu, X. Wu, and C. Liu, “A Diagnosis Method for Imbalanced Bearing Data Based on Improved SMOTE Model Combined with CNN-AM,” Journal of Computational Design and Engineering, vol. 10, no. 5, 2023, doi: https://doi.org/10.1093/jcde/qwad081.

X. Hu, J. Man, H. Yang, J. Deng, and Y. Liu, “An Improved Metalearning Framework to Optimize Bearing Fault Diagnosis under Data Imbalance,” Journal of Sensors, vol. 2022, 2022, doi: https://doi.org/10.1155/2022/1809482.

I. Goodfellow et al., “Generative Adversarial Nets,” 2014.

D. Ruan, X. Song, Clemens Gühmann, and J. Yan, “Collaborative Optimization of CNN and GAN for Bearing Fault Diagnosis under Unbalanced Datasets,” Lubricants, vol. 9, no. 10, 2021, doi: https://doi.org/10.3390/lubricants9100105.

P. Cheng, S. Zhang, and C. Li, “A Rolling Bearing Fault Diagnosis Based on Conditional Depth Convolution Countermeasure Generation Networks under Small Samples,” Sensors, vol. 22, no. 15, 2022, doi: https://doi.org/10.3390/s22155658.

G. W. Stachowiak and A. W. Batchelor, Engineering tribology. Amsterdam: Butterworth-Heinemann, 2016.

Asniar, N. U. Maulidevi, and K. Surendro, “SMOTE-LOF for noise identification in imbalanced data classification,” Journal of King Saud University - Computer and Information Sciences, vol. 34, 2021, doi: https://doi.org/10.1016/j.jksuci.2021.01.014.

Google Developers, “Overview of GAN Structure | Generative Adversarial Networks,” 2019. https://developers.google.com/machine-learning/gan/gan_structure

Farzan Farnia and Asuman Ozdaglar, “Do GANs always have Nash equilibria,” International Conference on Machine Learning, 2020.

L. Xu, M. Skoularidou, A. Cuesta-Infante, and K. Veeramachaneni, “Modeling Tabular Data using Conditional GAN,” 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada, 2019.

SDV, “Demystifying the CTGAN Loss Function | Synthetic Data Modeling · sdv-dev/SDV · Discussion #980,” GitHub, 2022. https://github.com/sdv-dev/SDV/discussions/980