# A Wavelet Transform-Based Transfer Learning Approach for Enhanced Shaft Misalignment Diagnosis in Rotating Machinery

Houssem Habbouche, Tarak Benkedjouh, Yassine Amirat, Mohamed Benbouzid

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Houssem Habbouche, Tarak Benkedjouh, Yassine Amirat, Mohamed Benbouzid. A Wavelet Transform-Based Transfer Learning Approach for Enhanced Shaft Misalignment Diagnosis in Rotating Machinery. Electronics, 2025, 14 (2), pp.341. 10.3390/electronics14020341. hal-05352324

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Article

# A Wavelet Transform-Based Transfer Learning Approach for Enhanced Shaft Misalignment Diagnosis in Rotating Machinery

Houssem Habbouche <fnref n="1" />, Tarak Benkedjouh <fnref n="1" />, Yassine Amirat <fnref n="2" /> and Mohamed Benbouzid <fnref n="3" />,<fnref n="4" />*

**Abstract:** Rotating machines are vital for ensuring reliability, safety, and operational availability across various industrial sectors. Among the faults that can affect these machines, shaft misalignment is particularly critical due to its impact on other components connected to the shaft, making it a key focus for diagnostic systems. Misalignment can lead to significant energy losses, and therefore, early detection is crucial. Vibration analysis is an effective method for identifying misalignment at an early stage, enabling corrective actions before it negatively impacts equipment efficiency and energy consumption. To improve monitoring efficiency, it is essential that the diagnostic system is not only intelligent but also capable of operating in real-time. This study proposes a methodology for diagnosing shaft misalignment faults by combining wavelet transform for feature extraction and transfer learning for fault classification. The accuracy of the proposed soft real-time solution is validated through a comparison with other time-frequency transformation techniques and transfer learning networks. The methodology also includes an experimental procedure for simulating misalignment faults using a laser measurement tool. Additionally, the study evaluates the thermal impacts and vibration signature of each type of misalignment fault through multi-sensor monitoring, highlighting the effectiveness and robustness of the approach. First, wavelet transform is used to obtain a good representation of the signal in the time-frequency domain. This step allows for the extraction of key features from multi-sensor vibration signals. Then, the transfer learning network processes these features through its different layers to identify the faults and their severity. This combination provides an intelligent decision-support tool for diagnosing misalignment faults, enabling early detection and real-time monitoring. The proposed methodology is tested on two datasets: the first is a public dataset, while the second was created in the laboratory to simulate shaft misalignment using a laser alignment tool and to demonstrate the effect of this defect on other components through thermal imaging. The evaluation is carried out using various criteria to demonstrate the effectiveness of the methodology. The results highlight the potential of implementing the proposed soft real-time solution for diagnosing shaft misalignment faults.

**Keywords:** detection; diagnosis; shafts misalignment; wavelet transform; multi sensor; transfer learning

## 1. Introduction

The digital transformation of industrial systems has promised considerable economic benefits. In particular, the implementation of the Internet of Things, Cloud Computing,

<figure><img src="image_2.png" /></figure>

check for
updates

Academic Editor: Davide Astolfi

Received: 19 December 2024

Revised: 9 January 2025

Accepted: 15 January 2025

Published: 17 January 2025

**Citation:** Habbouche, H.; Benkedjouh, T.; Amirat, Y.; Benbouzid, M. A Wavelet Transform-Based Transfer Learning Approach for Enhanced Shaft Misalignment Diagnosis in Rotating Machinery. *Electronics* **2025**, *14*, 341. https://doi.org/10.3390/electronics14020341

**Copyright:** © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

<footnote n="1">¹ Mechanical Structures Laboratory, Ecole Militaire Polytechnique, Algiers 16046, Algeria; houssem.habbouche@emp.mdn.dz (H.H.); tarak.benkedjouh@emp.mdn.dz (T.B.)</footnote>

<footnote n="2">² L@bIsen, ISEN Yncrea Ouest, 29200 Brest, France; yassine.amirat@isen-ouest.yncrea.fr</footnote>

<footnote n="3">³ Institut de Recherche Dupuy de Lôme (UMR CNRS 6027), University of Brest, 29238 Brest, France</footnote>

<footnote n="4">⁴ Logistics Engineering College, Shanghai Maritime University, Shanghai 201306, China</footnote>

<footnote>* Correspondence: mohamed.benbouzid@univ-brest.fr; Tel.: +86-332-9801-8007</footnote>
and Big Data in factories allows for the better monitoring of a production system’s current state [1]. The maintenance sector is one of the pillars of industry, which ensures the availability of equipment and the improvement of production [2]. This activity has evolved over time to meet industry needs. Corrective maintenance is a repair action performed after the machine has stopped due to a fault [3]. The appearance of preventive maintenance helps overcome the drawback of corrective maintenance, but on the other hand, it is considered costly because of the replacement of parts before their end of life. At this level, the new maintenance strategy appears, called “predictive maintenance”. According to the literature, a predictive maintenance extension called “Condition Based Maintenance (CBM)” is interested in diagnosing rotating machines based on their health state [4].

A misalignment fault is considered as one of the critical faults that can affect rotating machines. Such a fault leads to the rapid wear of the various mechanical components attached to misaligned shafts such as bearings and gears [5]. On the other hand, a machine that operates with a misalignment fault causes excessive energy consumption and decreases energy efficiency [6]. Researchers in the field of CBM have defined two diagnostic approaches: model-based and data-driven. The model-based approach requires prior knowledge of the system (geometry, materials, etc.), which is not very practical. The data-driven approach extracts information from monitoring data (acoustics, acceleration, etc.) to diagnose the machine [7]. Unfortunately, the collected data are often noisy and require processing to extract useful information. This processing is ensured by the different techniques of signal processing: temporal, frequency, and time-frequency analysis [8]. Temporal analysis is based on the calculation of statistical indicators (RMS, skewness, kurtosis, etc.). Frequency analysis is based on the transformation of the signal from the time domain to the frequency domain, consisting of tracing the spectrum based on techniques such as Fourier transformation. These two types of analysis are often influenced by the temporal evolution of the frequency, such as the appearance of a new fault and the change in rotation frequency, in addition to the nature of the non-stationary monitoring signals. To remedy these disadvantages, the time-frequency analysis makes it possible to follow the fault in time as well as the frequency and allows for the detection of the appearance of new faults [7]. The 1D time-domain vibration signal is transformed into a 2D time-frequency representation, which is subsequently used as input for a robust 2D CNN model. Various methods have been employed for this transformation, including the Short-Time Fourier Transform (STFT), Kurtogram, Hilbert–Huang Transform (HHT), and Wavelet Transform (WT). Freire et al. [9] utilized the STFT to extract the time-frequency components of torsional vibration signals from diesel engines. These representations serve as input to a convolutional neural network (CNN), enabling accurate fault classification even in the presence of noise. Luczak et al. [10] applied STFT to visualize the time-frequency characteristics of vibration signals, providing a basis for comparison with the CWTx6-CNN method. While the STFTx6-CNN method demonstrates good performance, CWTx6-CNN proves more effective in concentrating frequency components and enhancing defect representation. Peng et al. [11] combined the Kurtogram with band decomposition via wavelet packet transform and a robust indicator to extract transient fault signatures, even under external impulsive noise. Chen et al. [12] integrated the Fast Kurtogram into a synchrosqueezed transform-based method to filter and demodulate signals, improving time-frequency resolution and effectively handling long signals. This enables the precise detection of fault characteristic frequencies under variable-speed conditions. Indragandhi et al. [13] employed the HHT to decompose vibration signals of an induction machine into intrinsic mode functions, revealing frequency characteristics under power quality disturbances. These features, combined with time-domain data, are used as input for machine learning algorithms, notably support vector machines, to achieve accurate classification. Kwon et al. [14] applied the HHT after
empirical mode decomposition, to diagnose eccentricity faults by analyzing frequency distortions and instantaneous energy variations in the Hilbert spectrum. The method effectively distinguishes eccentricity faults from demagnetization faults across various loads and speeds. Among the most used time-frequency techniques is the Wavelet Transform (WT). This latter presents several advantages, such as the multi-resolution representation of a signal and insensitivity to noise; it allows one to decompose the signal at different scales, which makes it possible to detect fine details in high frequencies and more global characteristics in low frequencies [9,15]. Another advantage is that it can detect anomalies in signals by identifying sudden variations in wavelet coefficients at different scales. This capability can be used in many areas, such as health monitoring. Several frameworks in the field of diagnosis have been based on WT for processing. Bang et al. [16] proposes a diagnostic methodology of pumps; WT is used for 1D-signals conversion to 2D-images, to be used for training the GoogleNet. Du et al. [17] combined integrated gradient with the WT technique to keep only the frequency range containing the necessary information for bearing diagnosis. With the same aim of improving the quality of the bearing faults diagnosis, Mo et al. [18] used WT with adaptive windowing, allowing data extraction from noise-affected signals. In order to improve the multi-resolution representation of the WT technique, Tang et al. [19] used an improved version of the already proposed synchrosqueezed wavelet transform based on the compression and rearrangement of WT coefficients. These images are used for training an improved ResNet for the classification of bearing faults. Rezazadeh et al. [20] addressed the challenges of overfitting and data distribution shifts caused by operational variability. By combining WT with advanced machine learning methods, including a multi-layer perceptron and a transformer encoder, their framework enables accurate diagnostics with minimal target data.

Following the advent of artificial intelligence, scholars pursued automation processes across various domains, including the diagnosis of rotating machinery [21]. This discipline has evolved with the development of computational powers to discover machine learning and then deep learning. Among the latter's networks, convolution neural networks have been very successful in the area of facial recognition and object detection [22]. These networks have been developed in the context of transfer learning, which consists of completing the learning of a machine learning model, which was previously trained to solve a given task, in order to enable it to solve a similar task, generally more precisely [23]. Among these transfer models, AlexNet is a deep convolution neural network developed by Krizhevsky et al. [24] in 2012. This deep architecture allows the model to capture increasingly complex features as you progress through the layers, resulting in better classification accuracy. Recent frameworks are interested in this network. Lu et al. [25] treated the diagnosis of bearing faults, transforming vibratory signals into spectrograms (2D-images), which were used for training AlexNet. For the same purpose of diagnosing the rolling fault, Shi et al. [26] combined variational mode decomposition and hierarchical fuzzy entropy for signal decomposition and then transformed them into images using WT to train AlexNet for classification. Bai et al. [27] used compressive sensing theory to reproduce the original signal to minimize noise. WT later transforms these reproduced signals into images and, at the end, for AlexNet training.

Several studies focus on diagnosing rotating machines using data-driven approaches; however, few address misalignment faults as presented in previous works. Moreover, even studies that focus on misalignment faults often fail to provide details on the procedure for measuring these faults, the necessary tools, and the impact of such faults. Additionally, some works in the literature present diagnostic results for each sensor separately, whereas the results could be enhanced through data fusion.
To go back to the disadvantages mentioned above, this framework presents the following contributions:

* The integration of information from multiple sensors (accelerometers, infrared camera, etc.) to improve the accuracy and reliability of misalignment fault diagnosis.

* The use of wavelet transform to capture the characteristic signature of misalignment in vibration signals, highlighting its temporal and frequency components.

* The automation of soft real-time monitoring through transfer learning, enabling the rapid adaptation of models to new data.

* The validation of the proposed methodology through two experimental setups, confirming its effectiveness under various conditions.

* The development of a test bench using a laser alignment tool to accurately replicate misalignment scenarios and evaluate their effects.

* The analysis of the effects of misalignment on adjacent components, by visualizing the thermal distribution caused by this fault.

The paper is organized as follows: Section 2 introduces different formulations to take into account the shaft misalignment, identifying rotation axes of transmission components, and presents the basic theory of WT-based time-frequency transformed generation and transfer learning for fault classification. Section 3 focuses on validation on a published dataset. Section 4 is devoted to experiments and the comparative validation of the proposed fault diagnosis methods. Section 5 presents the conclusions.

## 2. Proposed Methodology for Misalignment Fault Diagnosis

The proposed methodology proceeds as shown in Figure 1, starting with the emulation of misalignment faults and the collection of vibration data using an acquisition chain. The collected data are often noisy because of the imperfection of the mechanical parts and harmonics of the studied fault; for this purpose, the wavelet transform technique is used to extract useful information concerning the misalignment fault. The obtained scalograms are then used to train a learning transfer network called AlexNet for the automation of the monitoring and diagnosis processes. At the end, there is an evaluation of the learning process based on the calculation of accuracy and confusion matrices. This methodology is a soft real-time solution that can be implemented on a microprocessor to detect faults in real time.

### 2.1. Wavelet Transform

Wavelet transform is a time-frequency analysis technique. It decomposes the signal in terms of a family of wavelets. The basic idea of WT is to represent the signal to be analyzed as a superposition of wavelets. It allows for the use of long time intervals where more precise low-frequency information is desired and also permits the use of shorter time intervals where accurate high-frequency information is selected [28].

Generally, the wavelet transform can be represented in continuous (i.e., continuous wavelet transform) as well as in discrete forms (i.e., discrete wavelet transform). The formula for the continuous wavelet transform is [29]:

$$WT(a; b) = \frac{1}{\sqrt{a}} \int_{-\infty}^{+\infty} x(t) \psi(\frac{t-b}{a}) dt \quad (1)$$

where $\psi$ is the wavelet analyzing function, and $a$ and $b$ are scale and position parameters, respectively. By varying the position of wavelet $b$ and its frequency $a$, it gives a time-frequency image depending on $a$ (ordinate) and $b$ at the output (abscissa). Several wavelets can be used for filtering; Morse wavelet is particularly well-suited for analyzing signals with varying frequency content over time. It is widely used due to its flexibility in capturing both high- and low-frequency features.
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Flowchart of the proposed shaft misalignment fault diagnosis.</figcaption></figure>

## 2.2. Transfer Learning

Convolution neural networks (CNNs) are a type of artificial neural network operating in two essential stages. The first stage is to extract the relevant characteristics from the training data, using convolution layers with different filters and pooling layers (max, min, or mean). These characteristics can be discriminating to differentiate between classes or to recognize data from the same class. The second stage of the CNN is the learning of the characteristics extracted before. This part can be a simple neural network or a network of another type [30].

The turning point was the contribution of Krizhevsky at the ImageNet Challenge in December 2012. The proposed CNN, called AlexNet (Figure 2), won this competition with a wide margin. In computer vision, deep convolution networks have become the technique of choice [24]. This model is very effective in the image analysis because it is based on the inspection of images pixel by pixel with the application of small convolutions on the image. There are several architectures of CNNs; the most common are AlexNet, GoogLeNet, ResNet, etc.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** AlexNet architecture.</figcaption></figure>

AlexNet consists of eight learning layers, five convolution layers, and three fully connected layers. The output of the final fully connected layers is a Softmax function that converts the weights to the probability distributions of the given 1000 classes. The input images are of 227 × 227 × 3 resolution; they are fed to the first convolution layer, which filters the image using 96 kernels of size 11 × 11 × 3 at a stride of 4 pixels. Following this, the third, fourth, and final convolution layers are connected to one another without and to intermediate layers like pooling and normalization.

### 3. First Experimental Validation

#### 3.1. *Experimental Dataset Presentation*

For the first validation, a published experimental dataset is used [31]. This dataset was collected using a test bench consisting of a motor connected to a receiver via an elastic coupling, with a shaft rotation frequency of 24.68 Hz. Various misalignment faults (horizontal, vertical, and normal) are included, each with different degrees of severity (0.5 and 1.15 mm for horizontal misalignment, and 0.34, 0.8, and 1.09 mm for vertical misalignment). Monitoring signals were captured using accelerometers mounted on the radial (horizontal and vertical) and axial directions of the bearings. The recordings lasted 8.43 s with a sampling frequency of 25,597 Hz.

For this first validation, multi-sensor data fusion is adopted. It consists of merging signals before learning, named early fusion [32]. Six different classes are presented in the following Table 1.

<table><caption>**Table 1.** Misalignment types.</caption><thead><tr><th>Misalignment Type</th><th>Accelerometers Axis</th></tr></thead><tbody><tr><td>No misalignment</td><td>Horizontal</td></tr><tr><td>0.5 mm horizontal</td><td></td></tr><tr><td>1.15 mm horizontal misalignment</td><td>Radial horizontal</td></tr><tr><td>0.34 mm vertical misalignment</td><td>Radial vertical</td></tr><tr><td>0.8 mm vertical misalignment</td><td>Axial directions</td></tr><tr><td>1.09 mm vertical misalignment</td><td></td></tr></tbody></table>
## 3.2. Wavelet Transform-Based Signal Processing

Starting with pre-processing signals by subdividing them into frames, each frame is transformed into a time-frequency image using WT. This transformation, based on the Morse wavelet, is known for its excellent multi-resolution representation in both the temporal and frequency domains. It enables the decomposition of signals at different scales, allowing for the detection of fine details at high frequencies and more global characteristics at low frequencies.

The calculation of the absolute values of the WT coefficients to obtain scalograms images is presented in Figure 3.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** WT of different signals classes: (a) no misalignment, (b) horizontal misalignment with 0.5 mm, (c) horizontal misalignment with 1.15 mm, (d) vertical misalignment with 0.34 mm, (e) vertical misalignment with 0.8 mm, and (f) vertical misalignment with 1.09 mm.</figcaption></figure>

The misalignment fault is a low-frequency fault [33], which can be detected on the lower part of the obtained images in the Figure 3. On the other hand, the upper part of the images is affected by other faults such as the bearing fault and the harmonics of the misalignment fault. By comparing the obtained images, it is easy to distinguish between the healthy (no misalignment) in Figure 3a and the others (Figure 3b–f). Regarding the second comparison of the severity of faults, between (b) and (c) for horizontal misalignment with 0.5 mm and 1.15 mm, respectively, the latter has more contrast than the former, indicating that the signature of the fault is more severe. Similarly, in figures (d, e, and f) for vertical misalignment with 0.34 mm, 0.8 mm, and 1.09 mm, respectively, the contrast of the lower part is low on (d), medium on (e), and intense on (f).

These scalograms illustrate the non-stationary nature of the monitoring signals through the emergence of frequency components over time. This behavior is linked to external noise caused by mechanical imperfections and acquisition noise [34]. To further justify this characteristic, a time-domain analysis was performed by calculating time features (RMS in Figure 4a, kurtosis in Figure 4b, skewness in Figure 4c, and crest factor in Figure 4d) for 100 samples from the normal class.
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Statistical features evolution for normal state: (a) RMS, (b) kurtosis, (c) skewness, and (d) crest factor.</figcaption></figure>

Figure 4 highlights the differences between each sample within the same class, demonstrating the non-stationary nature of the signal.

### 3.3. *Learning and Test*

The obtained WT scalograms undergo a resizing step to adapt them to different transfer networks, which can operate with varying input layer sizes. This step is also crucial for comparing transformation techniques such as STFT and spectral kurtosis, which produce images with different resolutions and require standardization without losing information. It is important to note that this step has no impact on the defect representation in the image. The images obtained after this step form a dataset that is randomly subdivided into two parts, 80% (240 images) and 20% (60 images), for training and test, respectively. Regarding network training, the adopted options are a batch size equal to 10 and a learning rate equal to $10^{-4}$. After training, the accuracy reaches 92%, with a matrix of confusion displayed in Figure 5.

When analyzing the resulting confusion matrix, the first remark is about the fault detection (the class has no misalignment against the other fault classes), which is 100% (60/60). The second remark concerns the class with maximum missed images, which is the vertical misalignment of 0.34 mm, with 10 missed images with the same fault with a different severity of 0.8 mm, which represents 2.8% of the accuracy, which is considered acceptable.

The result shows interest in the proposed methodology (WT-AlexNet) for the diagnosis of misalignment faults at different degrees of severity.
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Confusion matrix, the blue color represents correctly classified samples, while other colors indicate misclassified ones.</figcaption></figure>

A study of the computational and memory costs was conducted to evaluate the deployment of the soft real-time solution. The analysis was performed on a computer equipped with an Intel Core i7-6700K processor (manufactured by Intel Corporation, headquartered in Santa Clara, CA, USA) and 32 GB of RAM using MATLAB (from Matworks, licenced to Ecole Militaire Polytechnique, Algiers, Algeria). During the offline phase, which primarily involves training the AlexNet network, the training process required 84 min and utilized 1.2 GB of GPU memory. The resulting trained model has a size of 201 MB. In the online phase, the model demonstrates efficient performance, with estimated response times for key processes as follows: 0.0178 ms for the wavelet transform, 0.0145 ms for resizing, and 0.0223 ms for classification. This ensures the feasibility of the proposed solution for real-time applications.

To demonstrate the effectiveness of the proposed combination, this section presents two comparisons: first, of the signal processing techniques, and second, of the transfer network performance. The two signal processing techniques selected for comparison with the wavelet transform (WT) are the Short-Time Fourier Transform (STFT) and the Kurtogram. The STFT calculates the discrete Fourier transform over a sliding window along the signal, while the Kurtogram computes the spectral kurtosis using a similar sliding window approach. These techniques produce an output image that is subsequently resized to fit the input layer of the AlexNet network. The final results showed an accuracy of 83.89% for STFT and 30.83% for the Kurtogram, compared to 92% for the WT. This performance difference is primarily due to the fixed window size used in STFT and the Kurtogram, which limits their ability to adapt to local frequency changes in the signal. In contrast, the WT, with its multi-scale nature, enables a more detailed and adaptive analysis of frequency variations, leading to improved fault detection and classification. On the other hand, the Wavelet Transform (WT) exhibits better resistance to noise, enabling the more effective detection of defects with periodic characteristics [9]. This result highlights the advantage of using the WT as the signal processing technique within this methodology [10]. This comparison showed that the WT transform can be used even for detecting low-frequency faults, such as shaft misalignment, which in our case occur at 49.36 Hz and 74.04 Hz (2 and 3 times the rotational speed of 24.68 Hz). Additionally, WT has been incorporated into other frameworks for detecting faults at even lower frequencies. In [35], WT is used to detect imbalance faults on a turbine at a frequency lower than 10 Hz. In [36], WT enables the detection of rotor imbalance in wind turbines at 0.24 Hz.
The second comparison focuses on transfer networks. In our case, the comparison includes SqueezeNet, ResNet50, and EfficientNet-b0 versus AlexNet, all of which are pre-trained networks differing in their architectures and complexities. These networks were tested with images obtained from the WT. The accuracies achieved were 88.89% for SqueezeNet, 89.44% for ResNet50, and 86.11% for EfficientNet-b0, compared to 92% for AlexNet. These differences can be attributed to the fact that AlexNet, with its simpler architecture, appears better suited to this specific type of data generated by the WT. Conversely, the similar performances of SqueezeNet, ResNet50, and EfficientNet-b0 demonstrate their robustness, but their additional complexity does not provide significant improvements in this specific context.

# 4. Second Experimental Validation

This part is divided into three essential stages, which are:

1. Step 1: Emulate a known misalignment faults using the Easy-laser E540 system (manufactured by Easy-Laser AB, headquartered in Mölndal, Sweden);

2. Step 2: Recording vibration signals using acquisition chain;

3. Step 3: Application of the proposed methodology for data processing and network training.

## 4.1. Step 1

After the design and completion of the misalignment fault test bench (Figure 6), it should be noted that the base of the motor is rotating to allow yaw axis adjustment Figure 7b, in addition to the possibility of adding motor adjustment wedges for parallel misalignment adjustment Figure 7a. An elastic coupling between the motor and the receiving shaft is used to simulate faults Figure 7c.

<figure><img src="image_8.png" /><figcaption>Figure 6. Misalignment test bench: (a) model; (b) real.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7. Misalignment control parts, (a) Soft foot, (b) Flexible coupling, and (c) Motor support.</figcaption></figure>
An asynchronous motor of type AT150 GMBH (Hamburg, Germany) with maximum speed and power of 6000 rpm and 1.5 kW, respectively, is pictured above. It is attached to a control unit of type GUNT AT150 (sourced from the manufacturer GUNT, Barsbüttel, Germany), allowing for control speed and torque. This electrical system is used for powering the bench.

To create misalignment faults of different types (parallel and angular) with different degrees of severity, a laser alignment system named Easy-Laser E540 (manufactured by Easy-Laser AB, headquartered in Mölndal, Sweden) is used (Figure 8). The steps for creating the misalignment faults are as follows: start with the assembly of the two units (one on the motor shaft and the other on the driven shaft) and connect them to the measurement unit for data acquisition. The next step involves calibrating the measurement system by rotating the driven shaft to specific positions (3 o'clock, 12 o'clock, and 9 o'clock). Finally, the results provide the offset misalignment (parallel misalignment) measured in millimeters and the angular misalignment measured in mm/100 mm for both the horizontal and vertical directions.

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Experimental setup and data acquisition system for shaft misalignment fault diagnosis using shaft misalignment system Easy-Laser E540.</figcaption></figure>

The experimental tests carried out on the test bench with different configurations are shown in Table 2. The recording time must be short, to avoid system deterioration. Even simulated faults must not exceed the tolerance limit, to avoid shaft bending and bearing failure caused by periodic fatigue [5], as can be seen in Figure 9, concerning the two bearings and that of the motor. The thermal image is captured using infrared camera Fluke Ti32 (manufactured by Fluke Corporation, headquartered in Everett, WA, USA).

<table><caption>**Table 2.** Misalignment fault of experimental tests.</caption><thead><tr><th>Fault</th><th>Type</th><th>Vertical</th><th>Horizontal</th></tr></thead><tbody><tr><td rowspan="2">Fault N°1</td><td>Offset (mm)</td><td>-0.05</td><td>+0.01</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.06</td><td>-0.02</td></tr><tr><td rowspan="2">Fault N°2</td><td>Offset (mm)</td><td>-0.07</td><td>+0.08</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.06</td><td>-0.12</td></tr><tr><td rowspan="2">Fault N°3</td><td>Offset (mm)</td><td>-0.07</td><td>-0.12</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.07</td><td>+0.27</td></tr><tr><td rowspan="2">Fault N°4</td><td>Offset (mm)</td><td>-0.06</td><td>+0.32</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.08</td><td>-0.18</td></tr></tbody></table>
<table><caption>**Table 2. Cont.**</caption><thead><tr><th>Fault</th><th>Type</th><th>Vertical</th><th>Horizontal</th></tr></thead><tbody><tr><td rowspan="2">Fault N°5</td><td>Offset (mm)</td><td>-0.04</td><td>+0.38</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.10</td><td>-0.21</td></tr><tr><td rowspan="2">Fault N°6</td><td>Offset (mm)</td><td>-0.10</td><td>+0.51</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.09</td><td>-0.30</td></tr><tr><td rowspan="2">Fault N°7</td><td>Offset (mm)</td><td>-0.03</td><td>+0.38</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.08</td><td>-0.41</td></tr><tr><td rowspan="2">Fault N°8</td><td>Offset (mm)</td><td>-0.04</td><td>+0.00</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.05</td><td>-0.12</td></tr><tr><td rowspan="2">Fault N°9</td><td>Offset (mm)</td><td>+0.04</td><td>-0.13</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.09</td><td>+0.19</td></tr><tr><td rowspan="2">Fault N°10</td><td>Offset (mm)</td><td>+0.14</td><td>-0.03</td></tr><tr><td>Angular (mm/100 mm)</td><td>-0.01</td><td>+0.21</td></tr><tr><td rowspan="2">Fault N°11</td><td>Offset (mm)</td><td>+0.26</td><td>+0.18</td></tr><tr><td>Angular (mm/100 mm)</td><td>-0.02</td><td>-0.06</td></tr><tr><td rowspan="2">Fault N°12</td><td>Offset (mm)</td><td>+0.38</td><td>+0.28</td></tr><tr><td>Angular (mm/100 mm)</td><td>+0.32</td><td>-2.88</td></tr><tr><td rowspan="2">Fault N°13</td><td>Offset (mm)</td><td>-0.16</td><td>+0.48</td></tr><tr><td>Angular (mm/100 mm)</td><td>-0.03</td><td>-0.89</td></tr><tr><td rowspan="2">Fault N°14</td><td>Offset (mm)</td><td>+0.17</td><td>-0.01</td></tr><tr><td>Angular (mm/100 mm)</td><td>-0.01</td><td>-0.02</td></tr><tr><td rowspan="2">Fault N°15</td><td>Offset (mm)</td><td>-0.26</td><td>+0.08</td></tr><tr><td>Angular (mm/100 mm)</td><td>-0.05</td><td>-0.16</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Thermal image in the case of Defaut N°1: speed of rotation 3000 RPM using infrared camera Fluke Ti32.</figcaption></figure>

To ensure the detection of the misalignment fault, two tests were conducted at 3000 RPM with an offset configuration of -0.01 mm and -0.48 mm for the vertical and horizontal directions, respectively, and an angular misalignment of 0.10 mm/100 mm and
−0.30 mm/100 mm. The vibration signal is displayed in Figure 10a, and its corresponding spectrum is shown in Figure 10b.

<figure><img src="image_12.png" /><figcaption>Figure 10. (a) Vibration signal of the misalignment fault; (b) spectrum of the vibration signal.</figcaption></figure>

As observed in the spectrum, the presence of the rotational frequency (50 Hz) is evident, along with the misalignment fault frequency, which is twice the rotational speed, and its harmonics.

## 4.2. Step 2

To build the dataset of simulated cases presented in Table 2, signals were acquired through three accelerometers 4507-B-001 from Bruël & Kjær (manufactured by Bruël & Kjær, headquartered in Naerum, Denmark). The electrical signal of the accelerometers was acquired using a Bruël & Kjær 3050-A-060 (manufactured by Bruël & Kjær, headquartered in Naerum, Denmark) acquisition board, which has six BNC channels and a maximal sampling frequency of 50 kHz. Accelerometers are mounted on the first and second bearing at the positions shown in Figure 11.

<figure><img src="image_13.png" /><figcaption>Figure 11. Sketch of the test bench and acquisition chain.</figcaption></figure>

Through this acquisition chain, two-second signals are recorded, and each signal is subdivided into 90 frames. The rotation speed is set at 1000 rpm, which is just right for picking up misalignment issues at double the rotational speed.

## 4.3. Step 3

At this level, there is a dataset of 36 frames for 15 classes, from 1st, 2nd, and 3rd sensors named Data N°1, Data N°2, and Data N°3, respectively. Each frame is transformed into a 2D-image using the WT technique (Figure 12), followed by the resizing step to the format 227 × 227 × 3.
<figure><img src="image_14.png" /><figcaption>**Figure 12.** WT of signal from sensor 3: (a) Fault N°1; (b) Fault N°6.</figcaption></figure>

The signal dataset is transformed into an image dataset, which will then be used for training and testing the AlexNet network. The dataset is subdivided into 70% for training, 20% for validation, and 10% for testing.

The training options are as follows:

* The number of the epoch is 2000;

* The batch size is 10;

* The image dimension is 227;

* The learning rate is equal to $10^{-6}$.

The test results after training are displayed in Table 3.

<table><caption>**Table 3.** Accuracy results.</caption><thead><tr><th></th><th>Data N°1</th><th>Data N°2</th><th>Data N°3</th></tr></thead><tbody><tr><td>Accuracy</td><td>94.07%</td><td>99.26%</td><td>95.56%</td></tr></tbody></table>

The precision results show the performance of the proposed methodology, despite combining WT for the signal processing and AlexNet for feature extraction and classification. To see the classification in more detail, the test confusion matrices of the three datasets (Data N°1, N°2, and N°3) are presented in Figures 13–15.

<figure><img src="image_15.png" /><figcaption>**Figure 13.** Confusion matrix of Data N°1, the blue color represents correctly classified samples, while other colors indicate misclassified ones.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Confusion matrix of Data N°2, the blue color represents correctly classified samples, while other colors indicate misclassified ones.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 15.** Confusion matrix of Data N°3, the blue color represents correctly classified samples, while other colors indicate misclassified ones.</figcaption></figure>

As shown in the three confusion matrices, the detection of misalignment defect severity is of very high quality, with accuracies of 94.07%, 99.26%, and 95.56%. The first accelerometer (vertical accelerometer) showed lower accuracy compared to the second accelerometer (horizontal accelerometer) because the bearing is stiffer in the axial direction than in the vertical direction, making vibration transmission more pronounced in the horizontal direction. The third accelerometer (axial) also exhibited lower accuracy compared to the
second accelerometer, as the misalignment defect manifests more prominently in the radial direction than in the axial direction.

## 5. Conclusions

This framework is applied to the diagnosis of rotating machines, specifically for shaft misalignment faults. The proposed methodology consists of an intelligent decision-support tool for real-time machine monitoring. It processes multi-sensor data from the machine's monitoring system. The obtained signals are transformed into the time-frequency domain using the Wavelet Transform technique, which allows for early fault detection through the appearance of its characteristic frequency. The result of this transformation is scalograms, which are then used to train a transfer learning network called AlexNet. This architecture is employed for feature extraction and fault classification.

The experimental validation is conducted in two stages: the first using a publicly available dataset, and the second through laboratory testing, where a laser alignment system is employed to create the faults and thermal imaging is used to assess the impact of these faults on the machine. The results obtained in both cases are satisfactory, showcasing the potential of this approach for deployment in new industries.

In the future, the dynamic modeling of forces generated by shaft misalignment faults will be explored. This will ultimately enable the development of a hybrid approach for diagnosing such faults. Another research direction involves the design of digital twins to predict and anticipate the onset of shaft misalignment.

**Author Contributions:** Conceptualization, H.H.; Methodology, H.H., T.B., Y.A. and M.B.; Software, H.H.; Validation, T.B., Y.A. and M.B.; Formal analysis, H.H., T.B., Y.A. and M.B.; Writing—original draft, H.H.; Writing—review & editing, T.B., Y.A. and M.B.; Visualization, T.B., Y.A. and M.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

**Conflicts of Interest:** The authors declare no conflicts of interest.

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