Predictive machine health monitoring using deep convolution neural network for noisy vibration signal of rotating machine using empirical mode decomposition

Abstract

In a noisy industry environment, to predict machine faults using vibration signals, a specially designed Deep Convolution Neural Network (DCNN) with an additional noisy layer has been recently demonstrated. On the contrary, this paper presents a noise-susceptible fault classification using frequency spectrums in standard DCNN. The study involves two types of spectrum representation (a) Short-time Fourier Transform (STFT)of raw original signal and (b) Hilbert Huang Transform (HHT) of Empirical mode decomposition-intrinsic mode function (EMD-IMFs) and two different datasets (i) CWRU Bearing dataset and (ii) Nasa Milling Dataset which is non-stationary. Three binary DCNN classification problems are performed. For bearing data both representations maintained 100% classification accuracies for noise range up to 10 dB. HHT-EMD-IMF performs better for the non-stationary milling dataset. HHT-EMD-IMF is extended to bearing data set multiclass fault classification problem. The performance is compared with state-of-the-art work. The study compares all IMFs of clean and noisy signals to quantify the impact of noise on EMD for 8 different specific faults of the CWRU bearing dataset. The analysis of average normalized noise shows that EMD-IMF₀ has minimum deviation due to noise for all noise ranges. The DCNN model maintains 100% classification accuracy at high noise levels up to 10 dB and is better than the state-of-the-art noisy CNN approach. An ablation study shows that the proposed method is highly susceptible to impulse noise as well. It is also shown that the proposed method does not need additional computation time for training as in noisy layer CNN.

Article Highlights

A novel way of using frequency spectrum of noisy vibration signal in a deep convolution neural network is proposed for machine fault classification of bearing data set and shown that it is faster to train and performs better than an existing noisy layered CNN method.

Two datasets are utilised a standard bearing data set with multiple faults and a milling dataset which is non stationary and two approaches binary class classification and multi-class classification are considered and experimented for two spectrum representations in which Hilbert transform of empirical mode decomposed intrinsic mode functions performed better for binary classification of noisy vibration signals even for additive white Gaussian noise range of 10 dB.

For multiclass classification that comprises of 8 faults, HHT of EMD-IMF₀ performs better with 100% classification accuracy up to 10 dB additive Gaussian noise which is better than the existing noisy layered DCNN and in the propsed method the noisy training process consumes same time as clean signal training. The method is good as well for impulse noise for all noise ranges.

Introduction

Predictive maintenance requires regular monitoring of the state of bearings, which are crucial parts of spinning machinery [1]. Bearing vibration signals are frequently employed for failure detection, although noise greatly affects these signals [2]. Research indicates that the integration of noise-resistant techniques, like the use of a Gaussian noise layer, enhances the efficacy of machine learning models in the diagnosis of bearing defects [3]. Specifically, in real-world contexts, ensemble approaches and deep learning models trained with noisy data have shown better accuracy and robustness. Predictive maintenance models have a major problem with noise in sensor data [4]. Although machine learning techniques and conventional signal processing techniques [5] have been used to reduce the impact of noise, these approaches have drawbacks. Developing models that are resistant to noise can be facilitated by the utilization of Gaussian noise layers, which is one of the latest developments in deep learning. The other way of improving the durability of predictive models is to use ensemble approaches [6]. To further increase the dependability of predictive maintenance systems, future research should concentrate on tackling the problems of noise variability and model interpretability. The goal of predictive maintenance is to anticipate and stop equipment problems by utilizing data from machinery [7]. The difficulty in this problem stems from the noise in the sensor data, which can have a big impact on how well machine learning models work. Numerous things, including inaccurate sensors, external disturbances, and interference from electrical components, might introduce noise. Thus, creating models that are resistant to noise is essential for accurate predictive maintenance [8]. Due to the inherent noise in sensor data, false alarms or the failure to notice possible breakdowns may occur from faulty predictions. Noise decreases the signal quality and interferes with the feature extraction process, making the input to machine learning models unreliable. This issue is especially common in industries where sensor data is gathered in challenging and variable environments. In these kinds of situations, vibration, temperature changes, electromagnetic interference, and mechanical wear and tear on sensors can all produce noise [9]. Many methods have been used to lower noise in sensor data. Conventional methods involve signal processing techniques including Fourier transforms, wavelet transforms, and filtering [10]. To eliminate high-frequency noise components, filtering techniques such as low-pass filters are employed [11]. To remove noise in particular frequency bands, wavelet transformations have been used to break down the signal into its component frequency parts. These techniques, however, might not work in every situation and need careful tuning and domain expertise. This is especially true when the noise characteristics are unknown or change over time.

In sensor data, machine learning models are being employed more and more for robust prediction and noise reduction. To extract features that are resistant to noise and less susceptible to variations in the data, some research has concentrated on feature engineering [12]. Robust learning algorithms that can manage noisy labeling and outliers are used in alternative methods. Support vector machines (SVM) [13] and ensemble techniques like Random Forests, for example, have been used because of their noise resistance [14]. Deep learning models have demonstrated promise in processing noisy sensor data because of their capacity to extract complex patterns from data [15]. Time-series data from machinery has been fed into convolutional and recurrent neural networks (RNNs) for predictive maintenance and defect diagnostics. According to [16], these models can learn hierarchical representations of the input data, which aids in preserving the underlying structure even in the presence of noise. Deep learning models, however, are susceptible to noisy inputs, therefore regularization strategies like weight decay and dropout are frequently employed to increase their robustness. [17] Adding a Gaussian noise layer during training is a current method for enhancing deep learning model resilience to noise. This technique involves adding Gaussian noise to the neural network input data. Usually, during the training phase, the noise intensity is lowered to help the model progressively learn to adjust to noise. This method, which simulates different noise circumstances during training, improves the model ability to generalize to previously unknown noisy data. It is inspired by denoising autoencoders [18].

Though the ensemble-based techniques are more accurate they are complex and need a strategical selection of kernel functions and hyperparameter tuning in the training phase. There are a lot of merits in developing a noise susceptible signal representation by using advanced signal processing concepts and using the standard machine learning (ML) models. In our previously published research [19], we utilized the short-time Fourier transform (STFT) to obtain spectrum images of empirical mode decomposition-intrinsic mode function (EMD-IMF) signals. These images were then trained using a deep convolutional neural network (DCNN). The results showed that this method achieved a higher classification accuracy for fault classification compared to using spectrum images of the original signals even when the vibration signal is non-stationary in nature. The advantage of using DCNN is that it eliminates the need for a separate feature extraction stage, as it automatically learns features from the training images. To validate the effectiveness of the EMD method, we conducted a numerical experiment [19] using a bearing dataset for fault classification using spectrum images of original vibration signals and spectrum images of EMD-IMFs. Interestingly, both methods achieved a validation accuracy of 100%. The experiment was extended to a milling dataset which is non-stationary in nature. The results demonstrated that the EMD-IMF method outperformed the method that utilized the spectrum of the original signal. Specifically, for the milling dataset which consists of highly non-stationary signals, [20] the validation accuracy of the DCNN model using the spectrum of original signals was 40%, while the EMD-IMF method achieved an 81% validation accuracy. This highlights the performance improvement of the EMD method in non-stationary scenarios.

The experiments bring out the fact that in general vibration signal frequency-spectrum learning using DCNN helps better fault classification. Recent literature shows that much work has not been done to analyze the frequency spectrum using DCNN for fault classification in a noisy environment. In a novel way, in this paper, an attempt is made to study the results of using the frequency spectrum of noisy vibration signals for fault classification and to compare the results with the existing work [3] in which the classification is susceptible to noise up to 15 dB. The novelty is validated by applying it to 2 different spectrum representation.

• Short-time Fourier Transform (STFT) of raw original signal

• Hilbert Huang Transform (HHT) of Empirical mode decomposition-intrinsic mode function (EMD-IMF)

for 2 different datasets.

• Bearing dataset from Case Western Reserve University (CWRU) [21]

• NASA Milling Dataset [22] (non-stationary in nature [20]

for 2 different random noise.

• Additive white Gaussian noise 30 to − 3 dB which models the thermal noise involved in digital electronic systems

• Impulse noise 30 to − 3 dB which models the sudden impact from the external mechanical source

The study is done in two approaches. In the first approach spectrum images of the original noisy vibration signals and the spectrum images of EMD-IMF of the noisy signals are used for 3 different binary classification problems. The binary classification is to classify the signal as either good or faulty. The bearing dataset and milling dataset are used in this approach and more details about the dataset will be given later. The results are promising for both spectrum representations for the bearing data set even at a noise level of 10 dB. However, for the milling dataset problem which is one of the three binary classification problems and non-stationary in nature, though both the methods are equally robust to noise to some extent, the classification accuracy of EMD-IMF spectrum images is good and acceptable.

With this background, as the method of analyzing EMD-IMFs suits even well for non-stationary signals, an attempt is made in this proposed paper as an approach 2 to study the performance of EMD-based noisy vibration analysis using standard convolution neural network (CNN) for multiclass classification (8 classes) of bearing dataset. For this study, the work [3] which was discussed previously is taken as a benchmark and this paper investigates how noisy training affects deep convolutional neural networks (DCNN) performance on real-world data for multi-class problems. In this [3], two different procedures are studied for 5 fault classification. In the first procedure the clean vibration signals are used for training and noisy vibration is used for test. In the second procedure the noisy vibration signals are used in training and noisy vibration signals are used for test. The study shows that the noisy training procedure performs better than the other procedure. The ensemble approach [3] performed well in every scenario and was trained to identify higher rotational speeds. For accelerometer and microphone data, noisy training increases the test accuracy in DCNNs; the largest improvement was shown in DCNNs featuring a Gaussian noise layer. It was discovered that models using accelerometer data retain over 94% accuracy for no noise, when noise is added, the accuracy drops to less than 90% at 24 dB. Similarly, for microphone data, for no noise, 98% accuracy is maintained. When noise is added, the accuracy is dropped to lower than 90% at 12 dB. Meanwhile, the disadvantage of a noisy CNN layer is that it leads to computational complexity and also increases exponentially in the length of the code.

Approach 2 of the proposed work of this paper investigates the effectiveness of learning the HHT spectrum of EMD-IMF using DCNN for multiclass (8 classes) fault classification in a simulated noisy environment that is done by adding additive white Gaussian noise in various dB from 30 to − 3 dB. This study proposes a method that.

not only enhances the model accuracy but also has significantly less parameters and highlights the model lightweight design, the workflow is shown in Fig. 1. The performance is evaluated under various noise ranges during testing and training, comparing noisy and clean data. The study also examines the model resistance to noise, highlighting the importance of noise in model development. For the case of multiclass (8 classes) fault classification of the bearing dataset, the model shows that 100% classification accuracy remains almost perfect at high noise levels 30–10 dB, for noisy training and noisy testing. At moderate noise levels from 10 to 1 dB, the accuracy drops significantly. Meanwhile, the values show a decline, indicating more misclassifications as noise increases. For clean training and noisy testing, 100% classification accuracy is maintained for 20 dB noise for EMD-IMF_0.

Fig. 1 [Images not available. See PDF.]

Block diagram for proposed DCNN mode

Similarly, the experiment is done for IMF_1, IMF_2, and IMF₃. As the decomposed signals IMF_1-3 hold less information in the signal the prediction accuracy is decreased. 80% classification accuracy is maintained for 25 dB, for IMF₁ for noise training, and 64% for clean training. 80% classification accuracy is maintained for 20 dB, for IMF₂ for noise training, and 24% for clean training. 59% classification accuracy is maintained for 30 dB, for IMF₃ for noise training, and 58% for clean training. Thus, it is inferred that the best accuracy is given by DCNN models trained on HHT-EMD-IMF₀ spectrum images for additive white Gaussian noisy vibration signals. An ablation study is made to check the performance of the DCNN model by using impulse noise in place of additive white Gaussian noise, for which the model performed well for a noise range from 30 to − 3 dB with very good accuracies close to 100 for all noise range. The computation time is analyzed for approach 2 for the DCNN training and HHT spectrum generation. It is observed that both clean and noisy training took same time. This feature is better than noisy CNN layer architecture that is used in an existing work [3] as adding noisy layer in CNN increase the converging time and need careful hyperparameter tuning.

To summarize, the contribution of the paper following points are made.

• As a novel way, frequency spectrum learning of DCNN is studied for accurate fault classification using vibrational signals in a noisy environment in two approaches in which overall two different datasets, two different spectrum representations, and two different noises are used.

• Approach 1 brings out the fact that both representations, STFT of original vibration signals and HHT of EMD-IMF are robust for additive white Gaussian noise for binary classification problems by giving an accuracy of 100% at 10 dB, but still the HHT-EMD-IMF gave higher performance even when the dataset is the non-stationary for the case of milling dataset.

• Approach 2 brings out the fact that learning HHT of EMD-IMF spectrums is robust for additive white Gaussian noise for multi-class classification (8 classes) problems for bearing datasets and the model performance is better than the existing method [3] even at a 10 dB noise level.

• Apart from additive white Gaussian noise, the efficacy of noise robustness of the HHT-EMD-IMF method is studied and justified by using impulsive noise.

• The computation time is analyzed for approach2.

The points that support the novelty of the work for noisy vibration signal DCNN are listed here,

• This is the first work in our knowledge to use frequency domain analysis of vibration signals for noise robust fault classification using DCNN.

• Using frequency domain analysis of EMD-IMF that accommodates non-stationary signal.

• Validation using Additive white Gaussian and Impulse noise models.

• Noise analysis of EMD IMF signals quantitatively comparing the noisy and clean signals to determine its noise susceptibility,

Related works

The goal of this work is to investigate effective techniques for converting machine vibration signals into images that can be utilized for exact fault classification with DCNN. The technique will produce extremely accurate machine fault classification because DCNN is an excellent classifier for visual pictures. It is suggested in this work to use the [21] CWRU bearing vibration signal database from Case Western Reserve University, which is a widely used benchmark dataset by many researchers [23, 24, 25, 26, 27, 28, 29, 30, 31–32]. In order to detect sorted bearing faults in brushless DC motors, the research introduces a novel hierarchical image-based time–frequency convolutional neural network (HTFICNN). [33] The HTFICNN converts vibration and current data into time–frequency information (TFI) by combining three distinct time–frequency visualization techniques. It surpasses the highest precision of other models with precision, sensitivity, specificity, and F measures of 99.8594%, 99.8594%, 99.9531%, and 99.8593%, respectively. This study suggests an SSDAE-based bearing vibration signal diagnostic technique. MPE is utilized for feature extraction, and EEMD is used for adaptive multi-scale decomposition. High classification accuracy of 99.55% and 97.98%, respectively, were attained by the approach [26]. In this [34] study, a method for detecting rolling element-bearing flaws in brushless DC motors running in non-stationary settings is presented. Measurements of stator current and lateral vibration are employed as fault indicators, and features are extracted using a discrete wavelet transform. To see if the trained RNN could identify and categorize the various defects, the OFNDA features were then given into its input exactly. The current approach has an overall classification rate of almost 97%. This article [35] suggests wavelet-LSTM, Hilbert-LSTM, and modified LSTM techniques. The suggested models capture both short-term fluctuations and long-term trends by combining the multiresolution analysis capabilities of transforms with the temporal dependence identification capability of LSTM. When compared to current models, the suggested Hilbert-LSTM and wavelet-LSTM approaches exhibit better performance parameters, leading to better RMSE and performance metrics. This study is to use a convoluted neural network (CNN) in MATLAB to create a deep learning model for the precise diagnosis. [36] An accuracy of 85% was attained by the model, which was trained using the SGDM optimization approach. The study demonstrates the promise of machine learning algorithms and imaging tools for classification. [37] This study employs a deep learning method based on convolutional neural networks to distinguish between healthy and unhealthy. The approach employs an activation function, a pooling layer, and a dense layer to three different matrices with varying pixel values. Training is done with the Adam optimizer. The algorithm validation accuracy was 93.28%, and the mean average precision and recall scores were 0.85 and 0.88, respectively. [38] The study suggests two predictive approaches including convolutional neural networks for identification, logistic regression, K-nearest neighbors (KNN)machine learning techniques for classification, and GLCM and 2-D discrete wavelet transform and support vector machine. The model based on convolutional neural networks performs the best accuracy at 97%, SVM at 93%, logistic regression at 91%, and KNN at 89%. [39] The conjugate gradient approach of supervised learning utilizing artificial neural networks (ANN) is used to extract and classify a total of 21 time–frequency domain features using the HHT method. At the IMF-1 level, the average categorization accuracy was 86.2%. The outcomes of applying the ANN classification based on the HHT are highly favorable. For a more accurate classification of machine problems, it is suggested that the STFT of raw original noisy-based spectrums and the HHT of EMD-IMF of vibration signals be tested. This research also uses another NASA Milling Dataset [22] to demonstrate the robustness of the suggested HHT of the EMD-IMF method and its superiority over the raw signal STFT of raw original noisy method. In order to investigate the wear and tear of the tool used in the milling process, this dataset includes vibration signals and a variety of other data. It has been used in numerous studies [40, 41, 42, 43, 44, 45, 46, 47, 48–49] to develop machine learning algorithms that use all of the database data columns to estimate the tool remaining useful life (RUL). This work is limited to using the VB spindle alone.

Theoretical background of EMD algorithm

EMD was initially proposed by Huang et al. [50] as a method for adaptively processing signals. It decomposes nonlinear and nonstationary time series into distinct spectrum modes known as intrinsic mode functions (IMFs). These IMFs are amplitude-modulated frequency modulated (AM-FM) signals that represent specific frequency bands of the original time series, ranging from high-frequency first IMF to low-frequency bands last IMF [51].

Each IMF adheres to the following properties: (1) the difference between the number of zero-crossings and local extrema is at most one, and (2) the mean value of the upper and lower envelopes of an IMF, identified by local maxima and minima, is always zero at any given time.

A simple outline of how the EMD algorithm works:

1. Start with the original signal $x(t)$.

2. Extract the first IMF:

• Identify local maxima and minima.

• Interpolate to form upper and lower envelopes.

• Calculate the mean of the envelopes.

• Subtract the mean from the signal to get the first IMF.

• Iterate until the first IMF meets the IMF criteria.

1. Subtract the IMF from the original signal to obtain the residual.

2. Repeat the process on the residual to extract subsequent IMFs.

By applying these steps iteratively, EMD decomposes the signal into its constituent IMFs and a residual, facilitating detailed analysis of complex signals. First IMF typically represents the highest frequency oscillations within the signal [52]. The equation for adding additive white Gaussian noise to a signal and then applying EMD:

Apply EMD to the noisy Signal,

Decompose the noisy signal $x_{\mathit{noisy}}\left(t\right)$ using EMD:

1

$$x_{\mathit{noisy}}\left(t\right)=\sum _{\text{j}=1}^{\text{n}}{c}_j\left(t\right)+r_n\left(t\right)$$where:

$x_{\mathit{noisy}}\left(t\right)$ noisy signal.

$c_j\left(t\right)$$j$-th intrinsic mode function from the EMD of the noisy signal.

$r_n\left(t\right)$ Residual after extracting $n$ IMFs.

By adding noise and then applying EMD, the decomposition may become more robust to mode mixing for comprehensive results, ensemble EMD (EEMD) is generally preferred. In this paper, IMF_0,1,2,3 are used. The residual signal mostly will have noise and it is discarded for the analysis remaining 4 IMFs are used for analysis and in previous work it is shown that analyzing IMF₀ is more accurate [19, 20]. Figure 2 shows the vibration signal decomposition of the HHT of EMD-IMFs that is used in this proposed work.

Fig. 2 [Images not available. See PDF.]

HHT-EMD-IMFs decomposition

Merits of HHT analysis of EMD-IMF

The EMD method for studying nonlinear and non-stationary data. This technique breaks down complicated data into IMF that can be converted into Hilbert transformations. [50] These functions are often short and finite immediate frequencies as functions of time are created using this effective and adaptive approach, which may be used for nonlinear and non-stationary processes. For complex data sets, the technique makes use of instantaneous frequencies and localized signal features.

Hilbert Huang transform-based [53, 54–55] fault feature ratio is a common demodulation approach for vibration-based defect diagnostics. It is commonly known that the HHT highlights the local characteristics of $x\left(t\right)$ in the following ways: It is defined as the convolution of the signal $x\left(t\right)$ with $\frac{1}{\pi t}$.

2

$$m(t)=\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{x(\tau )}{t-\tau }d\tau$$

Both time, as well as motion variables, are denoted by $t$ and $\tau$, respectively. It is commonly known that the HHT is a frequency-independent ${90}^0$ phase shifter and a time-domain convolution that converts one real-valued time into another. Therefore, it does not affect the modulating signals non-stationary features. In practical applications, machine malfunctions are typically the cause of modulation. Therefore, demodulation is required to identify fault-related signals. Fortunately, this condition may be satisfied by providing an analytical signal, coupling the $x\left(t\right)$ and $m\left(t\right)$ we can have an analytic signal $g\left(t\right)$ as

3

$$g\left(t\right)=x\left(t\right)=im\left(t\right)=c\left(t\right)e^{i\phi (t)}$$

4

$$c\left(t\right)=\sqrt{x^2(t)+y^2(t)}$$

5

$$\phi \left(t\right)=\text{arctan}\frac{y(t)}{x(t)}$$$\phi \left(t\right)$ is the instantaneous phase of $x\left(t\right)$, and $c\left(t\right)$ is the instantaneous amplitude of $x\left(t\right)$, which might represent how the energy of the $x\left(t\right)$, changes with time. The envelope spectrum can be determined by doing spectrum analysis on the envelope signal $c\left(t\right).$ The information about the impulse in each period and its severity is contained in the signal. The spectrum will clearly show some of the typical frequencies when a bearing has local problems. The study investigates the HHT potential for detecting bearing faults. It illustrates how a signal spectrum analysis increases the amplitude of bearing fault signals for visual inspection in noisy condition signals.

Proposed work

The study examines fault classification using frequency spectrum image representation of noisy vibration signals in DCNN. Two different forms of signal representations are used (i) STFT of noisy raw original signal and (ii) HHT of EMD-IMFs. The study is done on two different data sets, a Bearing dataset from Case Western Reserve University (CWRU), [21] and a NASA Milling Dataset [22]. The NASA milling dataset signals are highly non-stationary which is shown in [20]. In the previous study on clean vibration signal without noise, it is inferred that for the case of the bearing dataset, both the representations performed equally well with validation accuracy close to 100%. For the case of the non-stationary milling dataset clean signals, it is shown that the EMD-IMF spectrum image representation gives a validation accuracy of 81.25% and outperforms the other representation which gives a validation accuracy of 40.62%. With this background of the performance of the frequency-based analysis of vibration signals for fault classification of clean signals, in this paper, two different approaches are proposed to study the efficacy of the spectrum analysis for noisy vibration signals.

The approach 1 is a generalized approach to compare the noise robustness of the spectrum-based method using both the representations (i) and (ii) for three DCNN-based binary vibration signals classification problems as given below.

1. Noisy Bearing dataset normal versus ball fault

2. Noisy Bearing dataset normal versus orthogonal fault.

3. Noisy Milling dataset good versus bad.

The approach 2 is a specifically focused approach to compare the HHT of EMD -IMF representation with an existing method [20] for multiclass bearing data set faults. Both approaches are performed by covering a noise range of 30 to − 3 dB. Approach 1 brings out the merits of HHT of EMD-IMF than that of STFT of raw signals for noisy environment. Approach 2 brings out the merits of the HHT of the EMD-IMF method over existing work [20] in a noisy environment.

Generation of STFT spectrum and HHT-EMD-IMF spectrum images

Vibration signals for various unhealthy and healthy conditions are taken from the dataset for generating spectrums. To enhance the complexity of the signal and to study the noise robustness performance of DCNN, additive white Gaussian noise is added in a controlled manner. Subsequently, the signal is divided into $Q$, overlapping signals for $T$ time units. Each of these $Q$ signals is considered as a representative of good and fault class. The $Q$ depends upon the dataset we have considered in this study. For each signal, the STFT and EMD-IMF-HHT techniques are applied. In STFT, the original time-domain signal viewed through the window is transformed into the frequency-domain using STFT, and peaks that correspond to the spectrum components are generated. The spectrum display makes use of the magnitude value and the complex-valued STFT. The spectrum illustrates how the frequency changes over time. For the HHT-EMD-IMF, the S intrinsic mode functions (IMF) are then subjected to the HHT using a window length of $N$=256. The total number of frames is 50% overlapping. The spectrum images obtained using EMD with HHT encompass comprehensive information about the time-domain signal. The IMF, derived from EMD, serves as a representation of the signal in an approximate parallel structure.

Dataset selection in bearing data

Our methodology effectiveness was demonstrated through experiments conducted on the CWRU [21] bearing vibration signal database. The experimental setup is shown in Fig. 3 which consists of a dynamometer, a torque transducer, and a 2 HP motor. The test bearings supported the motor shaft, and electro-discharge machining was employed to create single-point defects on the test bearings, with dimensions of 7, 14, 21, and 28 mils (one mil is equal to 0.001 inch). Digital data was recorded at a rate of 12,000 pulses per second for drive-end bearing problems. The torque transducer was used to manually record and measure the speed and horsepower. Accelerometers will be utilized to collect vibration data. The experimental parameters of bearing conditions are shown in Table 1. The bearing vibration signal database encompasses a wide range of signal elements, including various fault types that may occur in experimental equipment such as inner race (IR), ball (BA), center (C), opposite (OPP), and orthogonal (OR) defects. This experiment considered 8 classes of fault signals. The proposed experiment utilizes drive end (DE) accelerometer data to analyze the vibration signals and corresponding fault signals.

Fig. 3 [Images not available. See PDF.]

CWRU bearing setup

Table 1. Experimental parameters & Bearing conditions

Aspects	Specification
Motor speed(rpm)	1730
Motor load (HP)	3
Sampling rate	Digital data was collected at 12,000 samples per second
Types of bearing	Healthy bearing
	Faulty bearing (inner race, ball & outer race like centered, orthogonal, opposite)
Faulty diameter	7 mils, 21 mils, (1 mil = 0.001 inches)

Following the procedures outlined in Sect. 4.1, the original DE vibration signals from the bearing dataset were subjected to the addition of additive white Gaussian noise ranging from 30 to − 3 dB. These signals were extracted using STFT and HHT of EMD-IMFs. For binary class classification normal vibration signal and ball, orthogonal faulty vibration signal were considered which is transformed to image data with $N$=256, to get 4737 normal images and 1096 images corresponding to the ball and orthogonal fault as shown in Tables 2 and 3.

Table 2. Bearing data description of normal and ball fault binary classification for STFT of noisy raw original signal and HHT of EMD-IMFs

Data set images	Normal	Fault Ball (B021 3)
Total images	4737	1096
Selected training images	876	876
Selected testing images	220	220

Table 3. Bearing data description of normal and orthogonal fault binary classification for STFT of noisy raw original signal and HHT of EMD-IMFs

Data set images	Normal	Fault Orthogonal (OR021@33)
Total images	4737	1108
Selected training images	876	876
Selected testing images	220	220

For multi-class classification, the first four IMF signals were selected for further analysis. The HHT was applied to each signal before converting them into image data with N = 256, resulting in 1100 images for each fault class. Table 4 shows the experimental parameters of the bearing condition. The total number of spectrum images for one IMF for multiple faults is 8,807, and the total number of spectrums for all IMFs across the entire range of noise is 7,04,560. Equation 6 shows the total number of spectrum images and 7 & 8 show the training and testing image computation. In case (1) the clean train and noisy test with DCNN and in case (2) noisy train and noisy test with DCNN is taken. Further elaboration on this will be provided in the results and discussion Sect. 6.

6

$$\sum _{i=1}^n{x}_i=\sum _{i=1}^n{y}_i+\sum _{i=1}^n{z}_i$$were $$x_i=Totalimages(i=\text{1,2},3,\cdots \cdots )$$$$y_i=Trainingimages(i=\text{1,2},3,\cdots \cdots )$$$$z_i=Testingimages(i=\text{1,2},3,\cdots \cdots )$$

Table 4. Bearing data description for multi-class classification for HHT of EMD-IMF

EMD-IMFs	IMF0	IMF0	IMF0	IMF0	IMF0	IMF0	IMF0	IMF0
Fault name	IR0073	B007 3	OR007@6 3	OR007@33	OR007@123	IR021 3	OR021@6 3	OR021@12 3
Total images	1100	1096	1100	1099	1103	1096	1105	1108
Training	880	876	880	879	882	876	884	886
Testing	220	220	220	220	220	220	220	220

7

$$\sum _{i=1}^n{y}_i=\frac{4}{5}\sum _{i=1}^n{x}_i(i=\text{1,2},3,\cdots \cdots )$$

8

$$\sum _{i=1}^n{z}_i=\frac{1}{5}\sum _{i=1}^n{x}_i(i=\text{1,2},3,\cdots \cdots )$$

Dataset selection in milling data

Milling data [22] is considered to demonstrate the usefulness of the suggested approach. The data in this set represents experiments from runs on a milling machine under various operating conditions. In particular, tool wear was investigated (Goebel 1996) in a regular cut as well as an entry cut and exit cut. Data sampled by three different types of sensors (acoustic emission sensor, vibration sensor, current sensor) were acquired at several positions. Milling data comprises of 1 × 167 struct array with fields: Case, run, VB, time, DOC, feed, material, smcAC, smcDC, vib_table, vib_spindle, AE_table, AE_spindle. The basic setup encompasses the spindle and the table of the Matsuura machining center MC-510 V. An acoustic emission sensor and a vibration sensor are each mounted to the table and the spindle of the machining center. The signals from all sensors are amplified and filtered, then fed through two RMSs before they enter the computer for data acquisition. This struct array corresponds to 167 recordings of the experiment done and in this, we use the vibration data vib-spindle for analysis. For every recording, the VB data represents the flank wear and tear involved in the tool at the end of the experiment. In our analysis, we considered two cases Normal and Fault. The data with less wear and tear corresponding to VB of range (0–2) are considered normal and data with high wear and tear corresponding to VB > 3.8 are considered Fault.

Following the procedures outlined in Sect. 4.1, the original Vib-spindle vibration signals from the milling dataset were subjected to the addition of additive white Gaussian noise ranging from 30 to − 3 dB. These signals were extracted using STFT and HHT of EMD-IMFs. Which is transformed into image data with $N$=256, to get 54 normal images and 54 images with corresponding faults as shown in Table 5.

Table 5. Milling dataset Vib-spindle normal and fault for STFT of noisy raw original signal and HHT of EMD-IMFs

Data set images	Normal	Fault
Total images	54	54
Training images	34	34
Testing images	20	20

DCNN

Deep learning convolution neural network experiment is an exploration into the field of artificial intelligence that focuses on the ability of machines to imitate intelligent human behavior. Figure 4 shows the DCNN architecture. This subfield utilizes multiple layers of processing and incorporates both structured and unstructured data for training purposes. Deep learning has practical applications in various industries such as virtual assistants, entertainment, image coloring, and robotics. In this particular study, the CNN model demonstrated excellent performance in spectrum image classification, highlighting its effectiveness in image classification tasks. Additionally, recent advancements in machine learning-based models have shown their utility in accurately and efficiently recognizing multiple faults.

Fig. 4 [Images not available. See PDF.]

DCNN architecture

CNN learn feature automatically

The proposed work showcases the implementation of CNN demonstrating the automatic learning of features. CNN has been proven to eliminate the necessity for handcrafted features and parameter selection in various studies. When representing a vibration signal as time–frequency spectrum images, the y-axis shows frequency variations for a specific time frame, while the x-axis displays variations in different time frames. This distribution captures the vibration signal entirely, reflecting its characteristics in the image features. CNN can effectively learn from these vibration images. The convolution layer of CNN learns filters automatically based on output feedback that distinguishes classes, and the flattened output of the convolution layers can be viewed as extracted image features. These features can then be utilized in subsequent discriminative layers, eliminating the need for a separate parameter computation stage.

DCNN parameter setting for proposed work

In the DCNN, spectrum images are classified through a series of sequential steps. The model type in this study is sequential, utilizing the Keras library. Initially, 64 filters are applied with the ReLu activation function. Subsequently, the number of filters is increased to 128 with maximum pooling and ReLu activation. Image preprocessing involves rescaling, shear range, zoom range, and horizontal flipping. Both shear and zoom ranges are set at 0.2, horizontal flipping is enabled, and the class mode is categorical. Following the pooling stage, the system progresses to the flattening stage, which are shown in Table 6. During the training process, the proposed model successfully identified and differentiated between the prominent and less significant characteristics. It then utilized a fully connected layer to accurately classify texture images. For multi-class classification, the softmax activation function is employed. The model parameter output shape of DCNN is shown in Table 7.

Table 6. Model parameters of DCNN

Parameters	Values
Filter size	32 × 32, 64 × 64,
Batch size	32
Kernel size	3 × 3
Shear range	0.2
Zoom range	0.2
Layers	Conv2D, Max Pooling2D, Global AveragePooling2D, Dropout, Flatten, Dense
Activation layer	Relu, Softmax
Model type	Sequential
Optimizer	Adam
Flip type	Horizontal
Metrics	Accuracies
Loss function	Categorical-cross entropy
Epoch	25–100
Dense	256
Dropout	0.5

Table 7. Model parameter and output shape of DCNN

Layer type	Output shape	Parameter
conv2d (Conv2D)	(62, 62, 64)	1792
max_pooling2d (MaxPooling2D)	(31, 31, 64)	0
conv2d_1 (Conv2D	(29, 29, 128)	73,856
max_pooling2d_1 (MaxPooling2D)	(14, 14, 128)	0
flatten (Flatten)	(25,088)	0
dense (Dense)	(128)	3,211,392
dense (Dense)	(1)	129
Total params: 3,287,169 (12.54 MB)
Trainable params: 3,287,169 (12.54 MB)
Non-trainable params: 0 (0.00 Byte)

Detailed analysis of health monitoring using vibration signal numerical experimental validation and results

The objective of the numerical experiment is to construct various ranges of additive white Gaussian noise vibration signal spectrum of STFT and HHT of EMD-IMFs with a DCNN for the classification of binary classes, and the conclusions drawn from the findings are explained here. The performance of the suggested approach is contrasted with methods, and the multiclass extension of the problem is also addressed for bearing datasets. The experiment is performed in two different approaches as explained in Sect. 4. Two different accuracies are used as metrics one is validation accuracy which is done on 20% of test data randomly picked and the other one is referred to as classification accuracy which is done on 20% of test vibration signals handpicked and used to determine the confusion matrix. Classification accuracy is important for comparison as the same vibration signals are used for various spectrum representation methods.

Approach 1–generalized binary classification

As we already discussed, in Approach 1, noisy spectrum-based images for the three binary classification problems are applied for both representations (i) STFT of noisy raw original signal and (ii) HHT of EMD-IMFs. The three different problems related to binary classification are the noisy bearing dataset normal versus ball fault, noisy bearing dataset normal versus orthogonal fault, and a non-stationary nature noisy Milling dataset good versus bad.

The numerical experiment is conducted for all three problems and the conclusions are derived. For problem 1, binary classification of noisy bearing dataset normal versus ball fault, the spectrum image training and testing dataset listed in Table 2 is used in DCNN classification whose parameter is tabulated in Table 6. The results are shown in Table 8 and it can be concluded that the classification accuracy which corresponds to the bearing dataset, is highly promising for both spectrum creation cases. For both methods, STFT of noisy raw original signal and HHT of EMD-IMFs 100% validation accuracy and classification accuracy is maintained till 10 dB, and 80% of accuracy is maintained till 1 dB noise. The confusion matrix for normal and ball faults for both cases is shown in Fig. 5.

Table 8. Binary classification for normal and ball fault of bearing with various ranges of additive white Gaussian noise on CWRU bearing dataset for STFT of noisy raw original signal and HHT of EMD-IMFs

Normal (100)-Ball (0.21) STFT of noisy raw original signal			Normal (100)-Ball (0.21) HHT of EMD- IMF0
Noise range in dB	Validation accuracy (%) Tested on random 20% test data	Classification accuracy (%) Tested on handpicked 20% test data	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data
30	100	100	100	100
25	100	100	100	100
20	100	100	100	100
15	100	100	100	100
10	100	99.09	100	100
5	72.16	59.54	92.05	94.5
2	51.14	48.18	90.72	78
1	52.35	52.72	85.47	70.5
− 3	46.47	53.40	63.24	55.5
− 2	51.76	49.31	59.36	50

Fig. 5 [Images not available. See PDF.]

Confusion matrix for normal and ball fault binary class for additive white Gaussian. a STFT of noisy raw original signal, b HHT-EMD-IMF

For problem 2, binary classification of noisy bearing dataset normal versus orthogonal fault, the spectrum image training and testing dataset listed in Table 3 is used in DCNN classification whose parameter is tabulated in Table 6. The results are shown in Table 9 and it can be concluded that the classification accuracy which corresponds to the bearing dataset, is highly promising for both spectrum creation cases. For both methods, STFT of noisy raw original signal and HHT of EMD-IMFs 100% validation accuracy and classification accuracy is maintained till 5 dB, and 80% of accuracy is maintained till -2 dB noise. The confusion matrix for normal and ball faults for both cases is shown in Fig. 6.

Table 9. Binary classification for normal and orthogonal fault of bearing with various ranges of additive white Gaussian noise on CWRU bearing dataset for STFT and EMD- IMF₀

Normal (100)-Orthogonal (0.21) STFT of noisy raw original signal			Normal (100)-Orthogonal (0.21) HHT of EMD- IMF0
Noise range in dB	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data
30	100	100	100	100
25	100	100	100	100
20	100	100	100	100
15	100	100	100	100
10	100	100	100	100
5	100	98	100	100
2	77.84	86.59	99.91	95
1	85.23	80.90	98.23	80.5
− 3	56.25	54.31	98.86	77.5
− 2	50	53.18	89.01	70.5

Fig. 6 [Images not available. See PDF.]

Confusion matrix for normal and orthogonal fault binary class for additive white Gaussian. a STFT of noisy raw original signal, b HHT-EMD-IMF

For problem 3, the noisy milling dataset good versus bad spectrum image training and testing dataset has been listed in Table 5 for the DCNN parameter as tabulated in Table 6 the result can be concluded that the classification accuracy shown in Table 10, corresponds to the milling dataset. As this problem is non-stationary, it can be observed that the accuracy falls when the noise level increases. It is been clearly visible that HHT-EMD- IMF₀ is better for noisy vibration signal of bearing dataset compared with STFT of noisy raw original signal. For STFT of noisy raw original signal 82% classification accuracy is been dropped to 67% for the 10 dB noise range similarly 71% of validation accuracy drops to 57%. In the case of HHT of EMD-IMFs, 92% of classification accuracy drops to 75% for the 10 dB noise range similarly 85% accuracy drops to 62%. It can be inferred that both methods are robust to noise in the same degree but in general, the classification and validation accuracy of HHT-EMD-IMF is better and acceptable.

Table 10. Binary classification of milling dataset for STFT of noisy raw original signal and HHT of EMD-IMFs

Milling STFT of noisy raw original signal			Milling and HHT of EMD-IMFs
Noise range in dB	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data
30	71.43	82.5	85.71	92.5
25	42.86	85	85.71	90.5
20	42.86	77.5	77.14	85.5
15	57.14	70	71.43	83.5
10	57.14	67.5	62.86	75
5	57.14	60	62.86	72.49
2	42.86	57.49	58.86	69.5
1	42.86	57.49	56.86	65.5
− 3	42.86	51	52.86	60
− 2	42.86	45.5	51.43	57.5

Approach 2 HHT-EMD-IMF multi-class classification

This approach has a specific goal to compare the proposed method with an existing method to classify noisy vibration signals. The results of approach 1 suggest that frequency spectrum representations are susceptible to noise in DCNN learning and the general HHT of EMD-IMF method is better for non-stationary cases. Thus this approach is limited to only HHT-EMD-IMF representation.

Using the spectrum images generated by HHT of EMD-IMFs in the DCNN provides a robust result for a binary classification task, as shown in the result and explanation in 5.1. The confusion matrix is studied and this approach is expanded for a multiclass classification problem. Table 4 tabulates the HHT of EMD-IMFs of bearing data for eight distinct faults, as shown in the dataset description [21], Following training and testing, the DCNN model accuracy was maintained for a certain range of additive white Gaussian. For all the bearing fault classes RMS of the difference between original and noisy IMFs is determined, and the averaged normalized noise is determined for vibration signals of specific faults. This is then compared with other IMFs and this comparison helps to determine that IMF₀ is the most robust.

Computation of normalized noise

This methodology quantifies the impact of noise on EMD by comparing IMFs of clean and noisy signals. The accumulated IMF and signal errors highlight how noise distorts decomposed modes, and the normalized error provides an interpretable measure of noise sensitivity in EMD. These error metrics offer valuable insights for applications in fault diagnosis, signal processing, and any domain where noise-robust decomposition is critical. The EMD is applied for every frame and its noisy version by adding noise of specific dB, for every frame the Root means square (RMS) of difference between the original IMF and noisy IMF is determined which is developed as ${\mathit{IMFnoise}}_{\mathit{RMS}}$ and given in Eq. 9. And ${\mathit{IMFnoise}}_{\mathit{RMS}}$ is normalized by the RMS value of ${\mathit{IMF}}_{\mathit{original}}$ which is called normalized noise and denoted has ${\mathit{NN}}_{\mathit{RMS}}$ is given in Eq. 10. This RMS value quantifies how much the addition of noise affects the decomposition result across multiple segments. For the vibration signal of certain fault ${\mathit{NN}}_{\mathit{RMS}}$ is obtained to every frame of length $N$ for various noise added averaged normalized noise is determined ${\mathit{AvgNN}}_{\mathit{RMS}}$ is shown in Fig. 7.

Fig. 7 [Images not available. See PDF.]

Normalized noise for all the fault classes of CWRU bearing vibration signal. a Ball (B007), b Center (C007), c Center (C21), d Inner race (IR007), e Inner race (IR21), f Opposite (OPP007), g Opposite (OP21), h Orthogonal (OR007)

For each segment $M$:

9

$${\mathit{IMFnoise}}_{\mathit{RMS}}=RMS({\mathit{IMF}}_{\mathit{original}}-{\mathit{IMF}}_{\mathit{noisy}})$$

${\mathit{IMF}}_{\mathit{original}}$ is obtained by decomposing the segment $M$

${\mathit{IMF}}_{\mathit{noisy}}$ is obtained by decomposing the segment $M$ 1(noisy version)

10

$${\mathit{NN}}_{\mathit{RMS}}=\frac{{\mathit{IMFnoise}}_{\mathit{RMS}}}{RMS({\mathit{IMF}}_{\mathit{original}})}$$

From Fig. 7 it can be inferred that for all dB s ${\mathit{NN}}_{\mathit{RMS}}$ is much less for IMF₀ which implies IMF₀ is less susceptible to noise.

Inference of multiclass classification

In this proposed work the additive white Gaussian is added to the bearing vibration signal for 8 different fault classes. The frequency spectrum is generated using the HHT of EMD-IMFs. The comparative study is made for the decomposed signal obtained from the EMD-IMFs which are IMF₀, IMF_1, IMF_2, and IMF₃ the numerical experiment performance has been detailed below. The accuracy is drawn for the noisy training versus noisy testing similarly for clean training versus noisy testing.

For approach 2, multiclass classification of the noisy bearing dataset, the spectrum image training and testing dataset listed in Table 4 is used in DCNN classification whose parameter is tabulated in Table 6. The results are shown in Table 11 and it can be concluded that the classification accuracy which corresponds to the HHT of EMD-IMF₀ is the best for all dBs which complies with the discussion made in Sect. 6.2.1. About less ${\mathit{NN}}_{\mathit{RMS}}$ of EMD IMF₀ and its high susceptibility to noise as in Fig. 6.

Table 11. DCNN based noisy training and clean training accuracy for HHT of EMD-IMF₀

Model with noisy signal for both training and testing				Model trained with clean signal and tested with noisy signal
Noise range in dB	Validation accuracy (%) Tested on random 20%test data		Classification accuracy (%) Tested on handpicked 20% test data	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data
30DB	99.65	100		99.83	100
25DB	100	100		99.83	100
20	99.83	100		100	100
15	100	100		99.65	75.625
10	96.88	100		100	63.125
5	77.95	94.375		99.83	12.5
2	64.24	74.375		99.65	18.75
1	58.51	78.125		99.83	12.5
− 2	45.66	60.624		99.65	12.5
− 3	41.67	40.625		99.83	12.5

Noise levels of 30–15 dB for noisy training and testing, validation accuracy remains almost perfect close to 100% where the classification accuracy is maintained till 10 dB. Significantly the classification accuracy drops after 2 dB. For clean training and testing, accuracy is maintained till 20 dB then it gradually decreases as the noise range increases. Both the validation accuracy and classification accuracy values show a significant decline, indicating more misclassifications as noise increases. The confusion matrix is shown in Fig. 8 for 8 class classifications for 10 dB noise which represents 100% classification accuracy.

Fig. 8 [Images not available. See PDF.]

Confusion matrix for multiple fault classes

The spectrum image training and testing dataset listed in Table 4 is used in DCNN classification whose parameter is tabulated in Table 6. The results are shown in Table 12 and it can be concluded that the classification accuracy which corresponds to the HHT of EMD-IMF₁ Noise levels of 25 dB for noisy training and testing, validation accuracy remains almost close to 80% where the classification accuracy is maintained till 20 dB. Significantly the classification accuracy drops after 15 dB. For clean training and testing, accuracy is maintained till 20 dB is 57% then it decreases as the noise range increases.

Table 12. DCNN based noisy training and clean training accuracy for HHT of EMD-IMF₁

Model with noisy signal for both training and testing			Model trained with clean signal and tested with noisy signal
Noise range in dB	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data
30DB	60.42	75.625	72.92	91.25
25DB	60.59	80.625	69.27	64.37
20	54.17	71.25	71.01	57.49
15	41.32	61.25	71.7	12.5
10	27.6	32.5	70.49	12.5
5	14.93	12.5	75.35	12.5
2	13.19	10	69.44	12.5
1	21.35	12.5	70.14	12.5
− 2	20.14	13.75	70.66	12.5
− 3	20.14	14.37	68.06	12.5

The spectrum image training and testing dataset listed in Table 4 is used in DCNN classification whose parameter is tabulated in Table 6. The results are shown in Table 13 and it can be concluded that the classification accuracy corresponds to the HHT of EMD-IMF_2. Noise levels of 20 dB for noisy training and testing, validation accuracy remains almost close to 80% where the classification accuracy is maintained till 20 dB. Significantly the classification accuracy drops after 15 dB. For clean training and testing, accuracy is maintained till 30 dB then it gradually decreases as the noise range increases.

Table 13. DCNN based noisy training and clean training accuracy for HHT of EMD-IMF₂

Model with noisy signal for both training and testing			Model trained with clean signal and tested with noisy signal
Noise range in dB	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data
30DB	70.14	77.5	57.36	70
25DB	66.15	83.75	67.08	43
20	68.75	80	63.75	24
15	45.31	53.75	62.22	17
10	32.81	32.5	63.33	10
5	15.45	12.5	62.08	10
2	15.45	12.5	61.67	10
1	13.37	12.5	58.61	10
− 2	14.76	13.75	62.78	10
− 3	12.15	14.37	61.67	10

The spectrum image training and testing dataset listed in Table 4 is used in DCNN classification whose parameter is tabulated in Table 6. The results are shown in Table 14 and it can be concluded that the classification accuracy corresponds to the HHT of EMD-IMF₃. Noise levels of 30 dB for noisy training and testing, validation accuracy 62.5% where the classification accuracy is maintained till 25 dB. Significantly the classification accuracy drops after 20 dB. For clean training and testing, validation accuracy obtained is 58.75% for 30 dB then it gradually decreases as the noise range increases.

Table 14. DCNN based noisy training and clean training accuracy for HHT of EMD-IMF₃

Model with noisy signal for both training and testing			Model trained with clean signal and tested with noisy signal
Noise range in dB	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data
30DB	62.5	59.375	55.6	58.75
25DB	67.19	58.12	57.81	55
20	54.69	50	54.17	25
15	43.75	38.75	56.77	19.375
10	34.38	31.25	55.56	10.625
5	29.69	13.75	53.82	13.7
2	17.19	11.875	55.73	13.125
1	18.75	11.25	54.17	13.75
− 2	21.88	25	54.86	13.125
− 3	28.12	4.375	56.6	14.3

The above numerical experiment concludes that the decomposition of the first IMF of EMD holds more clean information less susceptible to white Gaussian noise than the other IMFs which helps DCCN to learn features from the frequency spectrum information. So, the validation accuracy and the classification accuracy obtained by HHT of EMD-IMF₀ are high when compared with other IMFs. As the non-stationary signal is decomposed into $S$ modes apart from the first IMF all other IMFs hold specific limitations characteristic of the signal hence other IMFs unable to predict the fault pattern present in the signal. From this, it is concluded that the HHT of EMD-IMF is robust to noisy non-stationary vibration signals which can adaptively decompose signals making it a valuable product for effectively handling noise in various signal processing applications.

HHT of EMD-IMFs for impulse noise bearing dataset

An ablation study is made by replacing the additive white Gaussian with impulse noise to verify the performance of HHT of EMD-IMFs in DCNN learning for IMF₀. The impulse noise can rise rapidly in signal amplitude and also can appear as short peaks that interrupt the regular flow of signals. Figure 9 shows the impulse noise bearing dataset samples for the − 3 dB range for HHT of EMD-IMF₀. As in Approach 2, the experiment is performed with impulse noise for multi-class classification of bearing dataset noisy spectrum image training, and the testing dataset listed in Table 4 is used in DCNN classification whose parameter is tabulated in Table 6. The results are shown in Table 15 and it can be concluded that the classification accuracy which corresponds to the bearing dataset, is highly promising for spectrum creation cases of HHT of EMD-IMFs. The impulse noise range is 30 dB − 3 dB 100% classification accuracy is maintained.

Fig. 9 [Images not available. See PDF.]

HHT of EMD-IMFs for impulse noise bearing dataset for 10 fault classes and normal a normal, b ball-0.007, c ball-0.21, d centered-0.007, e centered-0.21, f inner race-0.007, g inner race-0.21, h opposite-0.007, i opposite-0.21, j orthogonal-0.007, k orthogonal-0.21

Table 15. DCNN based impulse noise training accuracy for HHT of EMD-IMF₀

Model with noisy signal for both training and testing
Noise range in Db	Validation accuracy (%) Tested on random 20%test data	Classification accuracy (%) Tested on handpicked 20% test data
30	100	100
25	100	100
20	100	100
15	100	100
10	100	100
5	100	100
2	100	100
1	100	100
− 3	100	100
− 2	100	100

Inference

In this paper an attempt is made to study the noisy robustness of DCNN vibration signal fault classification using frequency spectrum representation for the Bearing vibration dataset from Case Western Reserve University (CWRU), [21] and the NASA Milling vibration Dataset [22]. Two different frequency representations are used and they are STFT of noisy raw original signal and HHT of EMD-IMFs. In the numerical experiment, the noise is added in a controlled manner and maintained in a range from − 3 to 30 dB. The DCNN classification is done on these noisy signals and two different accuracies are used as metrics. Validation accuracy is obtained by training the model using randomly selected 80% of data and testing the model using randomly selected 20% of data. The classification accuracy is obtained by using hand-picked training data (80%) and testing data (20%) to ensure the comparison is bias-free. The experiment is done in two approaches.

In approach 1, three different binary classification problems are used. In problem 1, noisy bearing dataset normal versus ball fault binary classification, for both the spectrum methods, 100% validation accuracy and classification accuracy are maintained till 10 dB, and 80% of accuracy is maintained till 1 dB noise. In problem 2, noisy bearing dataset normal versus orthogonal fault binary classification, for both the spectrum methods, 100% validation accuracy and classification accuracy are maintained till 5 dB, and 80% of accuracy is maintained till − 2 dB noise. In problem 3, non-stationary noisy milling datasets are good versus bad for both the spectrum methods, better classification and validation accuracy are maintained till 25 dB. And 80% of classification accuracy is maintained till 25 dB for STFT of noisy raw original signal and 80% classification accuracy is maintained till 15 dB for HHT of EMD- IMF₀.

As the binary classification problem obtained good results for a noisy vibration signal, for frequency spectrum representation HHT of EMD-IMFs, the problem is extended to multiclass (8 classes) classification of bearing data set faults in approach 2. In this approach 2 the performance of the model trained on two different conditions noisy training and clean training data and tested on noisy data across various noise levels, measured in decibels (dB). It highlights the interpretability of noise-resistant models, especially deep-learning models. Understanding how these models handle noise and make predictions. So the RMS of the difference between original and noisy IMFs is determined, and the averaged normalized noise is determined for vibration signals of specific faults. This is then compared with other IMFs and this comparison helps to determine that IMF₀ is the most robust.

Table 16 shows the comparison of the model with the existing method [3] and it is visible that for the 15 dB noise range 75% of accuracy is obtained for noisy training and 66% is obtained for clean training. Our proposed experiment shows that for EMD-IMF₀, the classification accuracy is 100% for noisy training and 75.625% for clean training. For the 10 dB noise range classification accuracy of 48% is obtained for noisy training and 46% is obtained for clean training. Our proposed experiment shows that for EMD-IMF₀ the classification accuracy is 100% for noisy training and 63.125% for clean training.

Table 16. Comparison table for proposed DCNN based noisy training and clean training accuracy for HHT of EMD-IMFs

Author	Method	Noise in dB	Noisy training accuracy (%)	Clean training accuracy (%)
Priscile Fogou Suawa, et al. (2023)	A noisy convolutional deep learning method based on the insertion of a noisy Gaussian layer	15 dB	75	66
		10 dB	48	46
Proposed work	Added an additive white Gaussian in vibration signal and given to EMD decomposed into (N) IMFs	15 dB
	HHT of EMD-IMF0		100	75.625
	HHT of EMD-IMF1		61.25	12.5
	HHT of EMD-IMF2		53.75	17
	HHT of EMD-IMF3		38.75	19.375
	HHT of EMD-IMF0	10 dB	100	63.125
	HHT of EMD-IMF1		32.5	12.5
	HHT of EMD-IMF2		32.5	10
	HHT of EMD-IMF3		31.25	10.625

The values in bold implies the best performance

All numerical computations are performed in a desktop system. The system parameters in which the experiment is done on 11th Gen Intel(R) Core (TM) i5-1135G7 processor at a clock of 2.40 GHz with RAM of 8.00 GB, system type 64-bit operating system, × 64-based processor, and Windows 11.Table 17 shows the time taken for HHT frequency spectrum generation for CWRU-bearing dataset for a selected fault which is around 5 s and acceptably low for 1100 sample signals. This time includes the time taken for EMD and HHT for each sample. Table 18 shows the time taken for the execution of the proposed method with the DCNN model for multiclass classification of Bearing datasets both for clean and noisy versions. The time taken for both noisy training and clean training is same which implies there is no additional computation time needed to handle the noise.

Table 17. Computation time taken for HHT-EMD-IMF spectrum image generation

Dataset	Sample number	CPU time (minute)
Bearing CWRU orthogonal fault	1100	5 min 94 s

Table 18. Computation time taken for the DCNN model for a random trial

Dataset	Number of faults	Epoch	Sample number	CPU time (minute)
Bearing CWRU Clean	8	25	Training = 7043 Testing = 1760	27 min 03 s
Bearing CWRU noisy	8	25	Training = 7043 Testing = 1760	27 min 43 s

Conclusion

This paper investigates the robustness of DCNN vibration signal fault classification using frequency spectrum representation for Bearing and NASA milling dataset vibration signals. It is a novel idea to use the time–frequency image of a noisy vibration signal to classify the machine fault using DCNN. Two different frequency representations STFT of noisy raw original signals and HHT of EMD IMF signals are used, with additive Gaussian noise added in a controlled manner in a range of 30 to − 3 dB. The model classification accuracy is evaluated using hand-picked training and testing data, ensuring bias-free comparisons for both spectrum methods. Approach 1 uses three binary classification problems: noisy bearing dataset normal versus ball fault, noisy bearing dataset normal versus orthogonal fault, and non-stationary noisy milling datasets good versus bad. The results of approach 1 bring out that frequency spectrum learning using DCNN gives a classification accuracy close to 100% for both the methods in case of the Bearing dataset. It is also shown that for the nonstationary milling dataset, the HHT-EMD-IMF performs better than STFT of raw original signal, The binary classification problem for noisy vibration signals in HHT of EMD-IMFs is extended to the multiclass classification of bearing data set faults in the approach 2 that comprises of 8 different faults. This approach tests the model interpretability and predictability, comparing noisy and clean training data. The RMS of the difference between original and noisy IMFs is determined, and the averaged normalized noise is determined for vibration signals of specific faults. This is then compared with other IMFs and this comparison helps to determine that IMF₀ is the most robust. The work demonstrates that DCNN learning of Hilbert transforms spectrum images of EMD-IMF₀ of noisy vibration signal performs better with a classification accuracy of 100% even when a noise level of 10 dB is added to the vibration signal which is been compared with the existing state of art method which needs a special noisy CNN layer. The performance of the model of the existing method is good for a certain level of noise range which is about 15 dB and gradually decreases as the noise level increases which shows the proposed HHT of EMD-IMF-DCNN model is better for multiple fault classification. As an ablation study, the proposed HHT of EMD-IMF-DCNN for multi-class fault classification is extended to another noise case, impulse noise in place of additive Gaussian noise and the result obtained is very promising as the classification accuracy is 100% unchanged for all noise levels from 30 to − 3 dB. The computation time for the complete process of training and testing for the approach is reported and the time taken by the training process for both noisy and clean signal are same and thus indicates no extra computation cost for dealing the noise.

Acknowledgements

None

Author contributions

Pavithra R programming, numerical experiments, manuscript writing, testing. Prakash R—idea, architecture, validation and manuscript review.

Funding

Open access funding provided by Vellore Institute of Technology.

Data availability

All the experimental dataset of this study are available from public data repository at the website of “Download a Data File|Case School of Engineering |Case Western Reserve University.” https://engineering.case.edu/bearingdatacenter/download-data-file (accessed Jul. 05, 2022).

Materials availability

All the experiments were conducted using Matlab R2023a for this study with 64-bit operating system with 2.60 GHz intel core i9 processor having 16 GB RAM and 4 GB dedicated Graphic card.

Code availability

Yes.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.

Abbreviations

EMD

Empirical mode decomposition

IMFs

Intrinsic mode functions

EMD-IMFs

Empirical mode decomposition-intrinsic mode functions

HHT

Hilbert Huang transform

RMS

Root means square

dB

Decibel

DCNN

Deep convolution neural network

CNN

Convolution neural network

SVM

Support vector machines

RNNs

Recurrent neural networks

ML

Machine learning

STFT

Short-time Fourier transform

AM-FM

Amplitude-modulated frequency modulated

EMD-IMF₀

Empirical mode decomposition-first intrinsic mode functions

EEMD

Ensemble empirical mode decomposition

CWRU

Case Western Reserve University

IR

Inner race

BA

Ball

C

Center

OPP

Opposite

OR

Orthogonal

DE

Drive end

${\mathit{NN}}_{\mathit{RMS}}$

Normalized noise

${\mathit{AvgNN}}_{\mathit{RMS}}$

Averaged normalized noise

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Pavithra R, Ramachandran P. An overview of predictive maintenance for industrial machine using vibration analysis. In: 3rd IEEE Int. Virtual Conf. Innov. Power Adv. Comput. Technol. i-PACT 2021. 2021. no. 1997, pp. 1–7. https://doi.org/10.1109/i-PACT52855.2021.9696751.

2. Liang, P; Deng, C; Wu, J; Yang, Z. Intelligent fault diagnosis of rotating machinery via wavelet transform, generative adversarial nets and convolutional neural network. Meas J Int Meas Confed; 2020; 159, [DOI: https://dx.doi.org/10.1016/j.measurement.2020.107768] 107768.

3. Suawa, PF; Halbinger, A; Jongmanns, M; Reichenbach, M. Noise-Robust Machine Learning Models for Predictive Maintenance Applications. IEEE Sens J; 2023; 23, 13 pp. 15081-15092. [DOI: https://dx.doi.org/10.1109/JSEN.2023.3273458] 1481.35092

4. Civera, M; Surace, C. A comparative analysis of signal decomposition techniques for structural health monitoring on an experimental benchmark. Sensors; 2021; 21, 5 pp. 1-35. [DOI: https://dx.doi.org/10.3390/s21051825] 0805.58039

5. Hui, KH; Hee, LM; Leong, MS; Abdelrhman, AM. Vibration condition monitoring: latest trend and review. Appl. Mech. Mater.; 2015; 773–774, pp. 139-143. [DOI: https://dx.doi.org/10.4028/www.scientific.net/amm.773-774.139] 1346.20045

6. Wu, Z; Huang, NE. Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv Adapt Data Anal; 2009; 1, 1 pp. 1-41. [DOI: https://dx.doi.org/10.1142/S1793536909000047] 1463.65224

7. Examples, S; Vibration, OF; Motor, ANE. The role of vibration monitoring in predictive maintenance Part 2: some illustrative examples. Maint Asset Manag; 2010; 25, 2 pp. 38-44.

8. Yan, X; Liu, Y; Jia, M. Research on an enhanced scale morphological-hat product filtering in incipient fault detection of rolling element bearings. Meas J Int Meas Confed; 2019; 147, [DOI: https://dx.doi.org/10.1016/j.measurement.2019.106856] 1501.92207106856.

9. Asilturk I, Aslanci H, Ozmen U. Machinery monitoring using vibration signal analysis.In: 40th IASTEM Int. Conf. Kuala Lumpur, Malaysia. 2016. no. 2, pp. 978–93.

10. Wang, H; Li, R; Tang, G; Yuan, H; Zhao, Q; Cao, X. A Compound fault diagnosis for rolling bearings method based on blind source separation and ensemble empirical mode decomposition. PLoS One; 2014; [DOI: https://dx.doi.org/10.1371/journal.pone.0109166]

11. Dekys, V. Condition monitoring and fault diagnosis. Procedia Eng.; 2017; 177, pp. 502-509. [DOI: https://dx.doi.org/10.1016/j.proeng.2017.02.252] 1378.00049

12. Jia, F; Lei, Y; Lin, J; Zhou, X; Lu, N. Deep neural networks: a promising tool for fault characteristic mining and intelligent diagnosis of rotating machinery with massive data. Mech Syst Signal Process; 2016; 72–73, pp. 303-315. [DOI: https://dx.doi.org/10.1016/j.ymssp.2015.10.025] 1414.93014

13. Liu, X; Bo, L; Luo, H. Bearing faults diagnostics based on hybrid LS-SVM and EMD method. Meas J Int Meas Confed; 2015; 59, pp. 145-166. [DOI: https://dx.doi.org/10.1016/j.measurement.2014.09.037] 0925.70012

14. Jin Z, Shang J, Zhu Q, Ling C, Xie W, Qiang B. RFRSF: employee turnover prediction based on random forests and survival analysis. In: Lect. Notes Comput. Sci. (including Subser. Lect. Notes Artif. Intell. Lect. Notes Bioinformatics), vol. 12343 LNCS. 2020. pp. 503–515, https://doi.org/10.1007/978-3-030-62008-0_35.

15. Zhang, W; Li, C; Peng, G; Chen, Y; Zhang, Z. A deep convolutional neural network with new training methods for bearing fault diagnosis under noisy environment and different working load. Mech Syst Signal Process; 2018; 100, pp. 439-453. [DOI: https://dx.doi.org/10.1016/j.ymssp.2017.06.022] 1363.90113

16. Wen, L; Li, X; Gao, L. A new two-level hierarchical diagnosis network based on convolutional neural network. IEEE Trans Instrum Meas; 2020; 69, 2 pp. 330-338. [DOI: https://dx.doi.org/10.1109/TIM.2019.2896370] 1497.92133

17. Zhang Z, Lee WS. Deep graphical feature learning for the feature matching problem. In: Proc. IEEE Int. Conf. Comput. Vis. vol. 2019-Octob, pp. 5086–5095, 2019 https://doi.org/10.1109/ICCV.2019.00519.

18. Vincent P, Larochelle H, Bengio Y, Manzagol PA. Extracting and composing robust features with denoising autoencoders. In: Proc. 25th Int. Conf. Mach. Learn. 2008. pp. 1096–1103, https://doi.org/10.1145/1390156.1390294.

19. Pavithra, R; Ramachandran, P. Deep convolution neural network for machine health monitoring using spectrograms of vibration signal and its EMD-intrinsic mode functions. J Intell Fuzzy Syst; 2023; 44, 6 pp. 8827-8840. [DOI: https://dx.doi.org/10.3233/JIFS-223012] 07767156

20. Pavithra, R; Ramachandran, P. Non-stationary signal analysis utilizing VMD and EMD algorithm for multiple fault classification for industrial applications. J. Intell. Fuzzy Syst.; 2024; [DOI: https://dx.doi.org/10.3233/JIFS-237546] 1498.68169

21. Download a Data File | Case School of Engineering | Case Western Reserve University. https://engineering.case.edu/bearingdatacenter/download-data-file. Accessed 05 Jul 2022.

22. NASA Milling Dataset | Kaggle. https://www.kaggle.com/datasets/vinayak123tyagi/milling-data-set-prognostic-data. Accessed 05 Jul 2022.

23. Xie, Y; Zhang, T. Fault diagnosis for rotating machinery based on convolutional neural network and empirical mode decomposition. Shock Vib.; 2017; [DOI: https://dx.doi.org/10.1155/2017/3084197] 1399.93099

24. Li, DC et al. Fault diagnosis of rotating machinery based on dual convolutional-capsule network (DC-CN). Meas J Int Meas Confed; 2021; 187, 2022. [DOI: https://dx.doi.org/10.1016/j.measurement.2021.110258] 1498.62304

25. Huo, Z; Zhang, Y; Jombo, G; Shu, L. Adaptive multiscale weighted permutation entropy for rolling bearing fault diagnosis. IEEE Access; 2020; 8, pp. 87529-87540. [DOI: https://dx.doi.org/10.1109/ACCESS.2020.2992935]

26. Dai, J; Tang, J; Shao, F; Huang, S; Wang, Y. Fault diagnosis of rolling bearing based on multiscale intrinsic mode function permutation entropy and a stacked sparse denoising autoencoder. Appl Sci.; 2019; [DOI: https://dx.doi.org/10.3390/APP9132743] 1510.68093

27. Chen, Y; Zhang, T; Luo, Z; Sun, K. A novel rolling bearing fault diagnosis and severity analysis method. Appl Sci.; 2019; [DOI: https://dx.doi.org/10.3390/app9112356] 1488.76045

28. Lu, T; Yu, F; Han, B; Wang, J. A generic intelligent bearing fault diagnosis system using convolutional neural networks with transfer learning. IEEE Access; 2020; 8, pp. 164807-164814. [DOI: https://dx.doi.org/10.1109/ACCESS.2020.3022840]

29. Xu, Y; Li, Z; Wang, S; Li, W; Sarkodie-Gyan, T; Feng, S. A hybrid deep-learning model for fault diagnosis of rolling bearings. Meas J Int Meas Confed; 2021; 169, 108502. [DOI: https://dx.doi.org/10.1016/j.measurement.2020.108502] 1525.65037

30. Yang, Y; Fu, P; He, Y. Bearing fault automatic classification based on deep learning. IEEE Access; 2018; 6, pp. 71540-71554. [DOI: https://dx.doi.org/10.1109/ACCESS.2018.2880990] 1349.93300

31. Eren, L; Ince, T; Kiranyaz, S. A generic intelligent bearing fault diagnosis system using compact adaptive 1D CNN classifier. J Signal Process Syst; 2019; 91, 2 pp. 179-189. [DOI: https://dx.doi.org/10.1007/s11265-018-1378-3]

32. Hendriks, J; Dumond, P; Knox, DA. Towards better benchmarking using the CWRU bearing fault dataset. Mech Syst Signal Process; 2022; 169, 108732. [DOI: https://dx.doi.org/10.1016/j.ymssp.2021.108732] 1486.92220

33. Ali, AK; Rafa Abed, W. hierarchical deep learning for bearing fault detection in BLDC motors using time-frequency analysis. J Electr Comput Eng; 2024; 2024, pp. 15-17. [DOI: https://dx.doi.org/10.1155/2024/3376733]

34. Abed, W; Sharma, S; Sutton, R; Motwani, A. A robust bearing fault detection and diagnosis technique for brushless DC motors under non-stationary operating conditions. J Control Autom Electr Syst; 2015; 26, 3 pp. 241-254. [DOI: https://dx.doi.org/10.1007/s40313-015-0173-7] 1338.93376

35. Shejul, K; Harikrishnan, R; Kukker, A. Short-term electricity price forecasting using the empirical mode decomposed Hilbert-LSTM and Wavelet-LSTM models. J Electr Comput Eng; 2024; [DOI: https://dx.doi.org/10.1155/jece/4575735]

36. Kukker A, Dhar S, Singh V, Amitabh V, Likhite S. Deep learning for pulmonary disease identification. In: 2023 14th Int. Conf. Comput. Commun. Netw. Technol. ICCCNT 2023. 2023. pp. 1–6. https://doi.org/10.1109/ICCCNT56998.2023.10307502

37. Pandey G, Sharma R, Kukker A. Multilayer convolutional neural network for maize leaf disease classification. In: 2023 7th Int. Conf. Comput. Appl. Electr. Eng. Adv. Sustain. Transp. Syst. CERA 2023. 2023. pp. 1–5. https://doi.org/10.1109/CERA59325.2023.10455141.

38. Pandey G, Sharma R, Kukker A. Solanaceae family plants disease classification using machine learning techniques. In: 2023 IEEE Pune Sect. Int. Conf. PuneCon 2023, no. Ccv. 2023. pp. 1–6. https://doi.org/10.1109/PuneCon58714.2023.10450081.

39. Kukker A, Sharma R, Malik H. Forearm movements classification of EMG signals using Hilbert Huang transform and artificial neural networks. In: 2016 IEEE 7th Power India Int. Conf. PIICON 2016. 2017. pp. 0–5. https://doi.org/10.1109/POWERI.2016.8077417.

40. Zhou, Y; Sun, W. Tool wear condition monitoring in milling process based on current sensors. IEEE Access; 2020; 8, pp. 95491-95502. [DOI: https://dx.doi.org/10.1109/ACCESS.2020.2995586]

41. Aghazadeh, F; Tahan, A; Thomas, M. Tool condition monitoring using spectral subtraction algorithm and artificial intelligence methods in milling process. Int J Mech Eng Robot Res; 2018; 7, 1 pp. 30-34. [DOI: https://dx.doi.org/10.18178/ijmerr.7.1.30-34] 1412.20022

42. Assafo, M; Langendorfer, P. A TOPSIS-assisted feature selection scheme and SOM-based anomaly detection for milling tools under different operating conditions. IEEE Access; 2021; 9, pp. 90011-90028. [DOI: https://dx.doi.org/10.1109/ACCESS.2021.3091476] 0972.68516

43. Shao, Y; Deng, X; Yuan, Y; Mechefske, CK; Chen, Z. Characteristic recognition of chatter mark vibration in a rolling mill based on the non-dimensional parameters of the vibration signal. J Mech Sci Technol; 2014; 28, 6 pp. 2075-2080. [DOI: https://dx.doi.org/10.1007/s12206-014-0106-6] 1207.92011

44. P. Wright, D. Dornfeld, and N. Ota, “Green Manufacturing and Sustainable Manufacturing WIRELESS SENSOR NETWORKS ( WSNs ),” 2008.

45. Gittler, T; Glasder, M; Öztürk, E; Lüthi, M; Weiss, L; Wegener, K. International conference on advanced and competitive manufacturing technologies milling tool wear prediction using unsupervised machine learning. Int J Adv Manuf Technol; 2021; 117, 7–8 pp. 2213-2226. [DOI: https://dx.doi.org/10.1007/s00170-021-07281-2]

46. Yu, W; Kim, IY; Mechefske, C. Remaining useful life estimation using a bidirectional recurrent neural network based autoencoder scheme. Mech Syst Signal Process; 2019; 129, pp. 764-780. [DOI: https://dx.doi.org/10.1016/j.ymssp.2019.05.005] 1242.74073

47. Mohanraj, T; Yerchuru, J; Krishnan, H; Nithin Aravind, RS; Yameni, R. Development of tool condition monitoring system in end milling process using wavelet features and Hoelder’s exponent with machine learning algorithms. Meas J Int Meas Confed; 2021; 173, 108671. [DOI: https://dx.doi.org/10.1016/j.measurement.2020.108671]

48. Xie, R; Wu, D. Optimal transport-based transfer learning for smart manufacturing: tool wear prediction using out-of-domain data. Manuf Lett; 2021; 29, pp. 104-107. [DOI: https://dx.doi.org/10.1016/j.mfglet.2021.08.004] 1244.62019

49. Bazi, R; Benkedjouh, T; Habbouche, H; Rechak, S; Zerhouni, N. A hybrid CNN-BiLSTM approach-based variational mode decomposition for tool wear monitoring. Int J Adv Manuf Technol; 2022; 119, 5–6 pp. 3803-3817. [DOI: https://dx.doi.org/10.1007/s00170-021-08448-7]

50. Huang, NE et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc A; 1996; 454, 1971 pp. 903-995.1631591[DOI: https://dx.doi.org/10.1098/rspa.1998.0193] 0945.62093

51. Flandrin, P; Rilling, G; Gonçalvés, P. Empirical mode decomposition as a filter bank. IEEE Signal Process Lett.; 2004; 11, pp. 112-114. [DOI: https://dx.doi.org/10.1109/LSP.2003.821662] 1390.94382

52. Mohanty S, Gupta KK, Raju KS. Comparative study between VMD and EMD in bearing fault diagnosis. In: 9th Int. Conf. Ind. Inf. Syst. ICIIS 2014. 2015. https://doi.org/10.1109/ICIINFS.2014.7036515.

53. Kerschen, G; Vakakis, AF; Lee, YS; McFarland, DM; Bergman, LA. Toward a fundamental understanding of the Hilbert-Huang transform in nonlinear structural dynamics. JVC/J Vib Control; 2008; 14, 1–2 pp. 77-105.2385237[DOI: https://dx.doi.org/10.1177/1077546307079381] 1229.70069

54. Wang, D; Miao, Q; Fan, X; Huang, HZ. Rolling element bearing fault detection using an improved combination of Hilbert and wavelet transforms. J Mech Sci Technol; 2009; 23, 12 pp. 3292-3301. [DOI: https://dx.doi.org/10.1007/s12206-009-0807-4] 1360.90162

55. Carcaterra, A; Sestieri, A. Complex envelope displacement analysis: a quasi-static approach to vibrations. J Sound Vib; 1997; 201, 2 pp. 205-233. [DOI: https://dx.doi.org/10.1006/jsvi.1996.0748] 0858.73051