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Abstract

This paper proposes a new criterion for selecting the optimum number of Intrinsic Mode Functions (IMFs) from vibration signals measured on rotating machines. The criterion is based on calculating the variation of normalized Shannon entropy ( $Δ_{NSE}$ ) between two successive IMFs, enabling the optimization of the number of processed IMFs. The method combines Variational Mode Decomposition (VMD) with Wavelet Multi-Resolution Analysis (WMRA) using the newly introduced criterion. The key steps involve determining the optimal number of IMFs, decomposing the vibration signals using VMD, and applying WMRA to the selected IMFs. Numerical simulations and experimental results demonstrate its effectiveness in identifying various faults, including gear defects, shaft misalignment, insufficient backlash, and belt defects. These latter flaws are the primary cause of the elevated noise levels in the measured signals. Finally, to validate our approach in an industrial setting, we diagnosed a turbofan, which revealed the presence of several typical faults for this type of installation. The proposed approach addresses a critical industry need by increasing fault diagnosis accuracy in noisy environments. Its practical impact stems from its ability to improve early fault detection, which is critical for predictive maintenance strategies aimed at reducing equipment downtime and extending its lifespan.

Keywords

Selection criterion for IMFs variational mode decomposition wavelet multi-resolution analysis normalized Shannon entropy highly noisy signals

Introduction

In industrial settings, fault detection in rotating machinery has always been critical for predictive maintenance procedures to ensure equipment reliability and peak performance. Vibration analysis is one of the primary diagnostic tools for detecting faults, as it can identify potential issues throughout the equipment through measurement and signal processing. Effective vibration diagnostics can help prevent catastrophic failures that result in unplanned downtime, production losses, and safety hazards. It also helps to plan maintenance operations like adjustment, troubleshooting, and repair. This is why fault detection based on vibration monitoring has remained a popular research topic.

The literature is rich with numerous studies focusing on diagnostics using classical methods, which include time-domain, frequency-domain, and time-frequency domain analyses.¹ Early works concentrated on time-domain analysis of raw vibration signals,^2,3 using scalar indicators sensitive to signal shape to indicate fault presence.^4–7 Subsequently, frequency-domain analysis via FFT has been extensively employed, yielding satisfactory results,^8,9 especially with the development of FFT algorithms such as power spectra¹⁰ and bi-coherence.¹¹ Cepstral analysis has also been introduced and proven advantageous for rapid fault detection.^12,13 High-frequency resonance techniques like envelope analysis have been utilized in the time-frequency domain.^14,15 More recent studies have incorporated advanced algorithms, such as enhanced Kurtogram¹⁶ for efficient extraction of fault-induced impulses cyclostationary¹⁷ analysis, which has shown its superiority over other methods. Kebabsa et al.^18,19 used frequency modulation techniques to detect bearing and gear faults in an industrial environment, employing the cyclostationarity method. Other decomposition techniques, such as empirical mode decomposition (EMD)²⁰ and its variants, have been used to improve fault detection, including Ensemble Empirical Mode Decomposition (EEMD),²¹ Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN),²² and Improved CEEMDAN (ICEEMDAN).²³

Recent advances in fault diagnosis research have centered on improving accuracy and reliability through novel methods. Kebabsa et al.,²⁴ for example, developed an innovative spectral analysis control approach that optimizes predictive maintenance in critical industrial machinery by using the Overall Level (OL) indicator. Hou and Pan²⁵ used the Teager-Kaiser Energy Operator (TKEO) and envelope spectral analysis to detect faults in reciprocating compressors, which improved signal clarity by removing noise. Damou et al.²⁶ developed a hybrid method for automated bearing fault classification that combines ensemble empirical mode decomposition (EEMD) and wavelet packet transform (WPT), resulting in high diagnostic accuracy using machine learning algorithms. Furthermore, Imane et al.²⁷ addressed the issue of variable operating speeds in machinery by proposing a multistage classifier that efficiently diagnoses bearing faults using empirical wavelet transforms. These recent contributions represent significant advances in fault diagnosis, combining advanced signal processing techniques with machine learning algorithms.

Wavelet analysis, widely adopted in recent research, both in its continuous^28,29 and discrete forms,³⁰ has yielded promising results. The initial use of wavelets for vibration signal analysis was pioneered by Wang and McFadden,³¹ who applied it to gearbox vibrations of a helicopter, successfully localizing gear faults. Nikolaou and Antoniadis³² introduced wavelet transform as an alternative method to traditional time-frequency analysis for bearing fault detection, using data collected on a test platform, showing promising results for monitoring rotating machinery conditions. Other studies have utilized continuous wavelet transform (CWT) as diagnostic tools for gear fault detection, as seen in the works of Sung et al.,³³ Akira et al.,³⁴ and Meltzer and Dien.³⁵ In Djebala et al.’s study,³⁶ the optimization of Wavelet Multi-Resolution Analysis (WMRA) using kurtosis as an optimization criterion was conducted to determine the number of decomposition levels, relevant detail, and other parameters. This technique has been successfully applied in Djebala et al.³⁷ for bearing fault detection, demonstrating excellent performance in fault detection. Additionally, various studies have combined wavelet analysis with other analytical methods, resulting in significant advancements in the field.^38–41

In the literature, research on Variational Mode Decomposition (VMD) focuses on two main axes: its application for fault diagnosis and the optimization of its parameters.⁴² In the first axis, notable studies include Mohanty et al.,⁴³ who employed VMD to decompose measured signals from a ball bearing. FFT analysis of the resulting Intrinsic Mode Functions (IMFs) demonstrated VMD’s superiority in health monitoring of bearings compared to EMD, even in the presence of noise. An et al.⁴⁴ performed envelope demodulation on IMFs obtained via VMD from a gear fault signal, showing the method’s effectiveness in both simulated and real signal processing for gear fault detection. Mahgoun et al.⁴⁵ investigated VMD’s efficacy in diagnosing gear faults under variable speed conditions. Wang et al.⁴⁶ applied VMD combined with envelope spectrum for gear fault detection. Chen et al.⁴⁷ proposed a scheme for gearbox fault detection using VMD to extract acoustic emission features, successfully diagnosing a failure. Lin et al.⁴⁸ further advanced by combining VMD with probabilistic neural networks for gear fault detection, achieving excellent results that affirm the advantages of VMD. In the second axis, focusing on VMD optimization, Li et al.⁴⁹ introduced an independent-oriented VMD using locally weighted scatterplot smoothing (LOWESS) for a more reliable and stable selection of the number of intrinsic modes. Feng et al.⁵⁰ suggested choosing an appropriate number of IMFs (K) based on spectral characteristics. Zhang et al.⁵¹ used correlation and energy ratio iteratively to determine K. Jiang et al.⁵² utilized initial central frequencies as an optimization strategy to obtain effective intrinsic modes. Jiang et al.⁵³ proposed a coarse-to-fine decomposition technique where VMD is applied to evaluate different signal types for early fault detection in rotating machinery. Numerous other studies have explored various techniques,^54–59 significantly advancing vibrational analysis in this field.

Despite extensive research into fault detection using VMD, current methods for optimizing the number of intrinsic modes continue to face challenges in terms of accuracy, computational complexity, and robustness, particularly when dealing with highly noisy industrial signals. Existing techniques frequently lack simplicity and effectiveness, particularly in real-time applications for rotating machinery. To improve fault detection in the presence of highly noisy signals, a simpler and more effective method for determining the optimal number of intrinsic modes is needed. This article aims to close these gaps by proposing a new optimization criterion for IMFs, which will improve the VMD process and be combined with WMRA to improve fault detection performance.

This manuscript is structured as follows: Section “Mathematical formulation” describes the mathematical formulations of the methods used in this article. Next, the proposed approach is explained in Section “Proposed approach.” Section “Numerical simulation” contains a numerical simulation. We move on to Section “Experimental study,” where we present the experimental study, followed by the results and discussion in Section “Results and discussions.” Section “Industrial application of the combined and optimized VMD-WMRA approach” gives an industrial application of the proposed approach. Finally, a brief conclusion is presented in Section “Conclusion.”

Mathematical formulation

This section gives an overview of the signal decomposition process using Variational Mode Decomposition (VMD) and Shannon Entropy (SE) theory, as well as the fundamentals of Wavelet Multi-Resolution Analysis (WMRA). The aim is to help readers acquire a more comprehensive understanding of the background and functioning of these techniques.

Variational mode decomposition (VMD)

Variational mode decomposition (VMD) is an adaptive and innovative data-driven signal processing method proposed by Dragomiretskiy and Zosso⁶⁰ that decomposes signal S(t) into their band-limited Intrinsic Modes Functions (IMFs) named $u_{k}$ , each cracterized by a specific center frequency $ω_{k}$ and bandwidth. This method entails searching for the best solution iteratively⁶⁰ by constantly updating each IMF and center frequency (equation (1)):

min_{{u_{k}}, {ω_{k}}} {\sum_{k} {‖ \partial}_{t} [(δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j ω_{k} t} ‖_{2}^{2}}

(1)

where, $\sum_{k} u_{k} = s (t)$ , * represents the convolution, $‖ . ‖_{2}^{2}$ is the squared $L^{2} - norm$ , and $δ$ is the Dirac function. ${u_{k}}, {ω_{k}}$ represents the IMFs of the signal and their central frequencies, respectively.

The IMFs provide several components related to detecting faults that cause resonance in the measured signal in the context of diagnosing faults in rotating machinery. The augmented Lagrangian L⁶⁰ was introduced to address the minimization problem proposed in equation (1):

\begin{matrix} L ({u_{k}}, {ω_{k}}, λ) : = α \sum_{k} ‖ \partial_{t} [(δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j ω_{k} t} ‖_{2}^{2} \\ {+ ‖ S (t) - \sum_{k} u_{k} (t) ‖}_{2}^{2} + 〈 λ (t), S (t) - \sum_{k} u_{k} (t) 〉 \end{matrix}

(2)

where, $α$ is the moderate bandwidth constraint.

The solution is then obtained through an iterative sub-optimization sequence known as the alternate direction method of multipliers (ADMM),⁶⁰ where we can find an estimation of the components $u_{k}$ and $ω_{k}$ (equations (3) and (4), respectively):

{\hat{u}}_{k}^{n + 1} (ω) = \frac{\hat{S} (ω) - \sum_{i \neq k} {\hat{u}}_{i} (ω) + \frac{\hat{λ} (ω)}{2}}{1 + 2 α {(ω - ω_{k})}^{2}}

(3)

ω_{k}^{n + 1} = \frac{\int_{0}^{\infty} ω {| {\hat{u}}_{k} (ω) |}^{2} d ω}{\int_{0}^{\infty} {| {\hat{u}}_{k} (ω) |}^{2} d ω}

(4)

In equations (3) and 4), $\hat{.}$ denotes the Fourier transform.

Shannon entropy (SE)

Shannon entropy (SE) is a measure of the uncertainty or randomness of a probability distribution.⁶¹ It is calculated as follows (equation (5)):

SE (n) = - \sum_{i = 1}^{N} p (n_{i}) \log_{2} (p (n_{i}))

(5)

where, $p (n_{i})$ is the probability of obtaining the value $n_{i}$ .

SE is a measure of the uncertainty and information in the vibration signal produced by a gearbox in gear fault detection. It is used to quantify the degree of randomness obtained from gears, which aids in the detection of gear failures early.⁶² Thus, SE can offer a useful criterion for analyzing and measuring the similarity of vibratory signals.⁶³ The similarity between the IMFs of a signal in this work case is based on changes in the frequency spectrum of the vibration signal.

Wavelet multi-resolution analysis (WMRA)

Wavelet multi-resolution analysis (WMRA) is a highly effective signal-processing method that has grown in popularity in recent years. It employs the wavelet transform, a mathematical technique for representing a signal S(t) using functions derived from the expansion and translation of a singular function known as the wavelet mother $ψ (t)$ .⁶⁴

The wavelet family (equation (6)) takes the following form⁶⁵:

ψ_{a, b} (t) = \frac{1}{\sqrt{a}} ψ (\frac{t - b}{a})

(6)

$a, b$ define the expansion and translation parameters, respectively.

Giving that $ψ^{*}$ is the conjugate of $ψ$ , we can define the continuous wavelet transform (CWT) (equation (7)) of the function $S (t)$ by⁶⁵:

CWT (a, b) = \frac{1}{\sqrt{a}} \int_{- \infty}^{+ \infty} S (t) * ψ^{*} (\frac{t - b}{a}) dt

(7)

If we replace $a$ by $2^{m}$ and $b$ by $n 2^{m}$ , where $n$ and $m$ are integers, we obtain the expression (equation (8)) of the Discrete Wavelet Transform (DWT)⁶⁵:

DWT (m, n) = 2^{- \frac{m}{n}} \int_{- \infty}^{+ \infty} S (t) * ψ^{*} (2^{- m} t - n) dt

(8)

DWT then involves a waterfall decomposition, as proposed by Mallat,⁶⁶ which separates the signal using low and high pass filters, resulting in two vectors containing:

- approximation coefficients $c A_{j}$ for low-frequency components,

- detail coefficients $c D_{j}$ for the highest frequency components.

During the decomposition process, these vectors are down-sampled and then passed through two additional reconstruction filters to obtain approximations $A_{j}$ and details $D_{j}$ . Following the modal (equation (9)) proposed by Mallat,⁶⁶ these sub-signals can be reconstructed as follows:

\begin{matrix} A_{j - 1} = A_{j} + D_{j} \\ S = A_{j} + \sum_{i \leq j}^{n} D_{j} \end{matrix}

(9)

Djebala et al.³⁶ proposed an optimization of WMRA (OWMRA) that is specifically designed to analyze shock signals. Using kurtosis as an optimization criterion, this optimized version selects various parameters to determine the number of levels required for analyzing gear defects. The number of levels³⁶ is expressed as:

n \leq 1.44 \log (\frac{F_{max} (S)}{F_{c}})

(10)

where, $F_{\max}$ is the maximum frequency of a signal $S (t)$ , and $F_{c}$ is the shock frequency.

Proposed approach

The approach aims to detect gear faults in highly noisy signals. To improve diagnostic results, we propose a new criterion that uses the normalized variation of Shannon entropy between two successive IMFs to determine the optimal number of IMFs required for decomposing vibration signals using Variational Mode Decomposition (VMD). WMRA is then used to demodulate the obtained IMFs, which improves fault detection. The detailed methodology of the proposed procedure, as shown in the flowchart of Figure 1, is outlined below:

Measure the vibrational signals and calculate their Shannon Entropies (SE).

Introduce the signal decomposition parameters for VMD, such as the initial number of IMFs, which should be set relatively high (k₀ = 20 to 30) for fault detection. Other parameters, such as the bandwidth constraint (α) and tolerance, are chosen based on the works of Dragomiretskiy and Zosso⁶⁰ and Zhang et al.⁵¹

Start the iterations at the VMD level, where j varies from 2 to k₀ with a step of 1, calculating the Shannon Entropy for each IMF, denoted as $S E_{IMF}$ . This indicator is effective for pattern recognition and is subsequently normalized with respect to $SE$ obtaining its normalized value $NS E_{IMF}$ using equation (11):

NS E_{IMF} = \frac{S E_{IMF}}{SE}

(11)

Verify the proposed stopping criterion $Δ_{NSE}$ after each step by calculating the difference between the two consecutive NSE_IMF values $Δ_{NSE} = NS E_{IM F_{(j)}} - NS E_{IM F_{(j - 1)}}$ . If $Δ_{NSE}$ is below a 1% threshold, it indicates that the similarity between the two IMF components in the same decomposition is greater than 99%, and thus, we terminate the iteration.

By stopping the iterations, we are able to determine the appropriate number of IMFs by taking $K = j - 1$ iterations. This is done to avoid the problem of over-segmentation, which occurs when two IMFs share the same information.⁶⁰ Following that, we redo the signal decomposition with the new value of K to obtain the optimal IMFs for detecting faults.

Finally, the selected IMFs will be processed using WMRA to identify the IMFs’ demodulation frequencies (for example, bearing resonance frequencies, gear meshing frequencies, and belt resonance frequencies), allowing us to understand the nature of the fault.

Figure 1.

The flowchart of the proposed approach for fault detection.

Numerical simulation

Determining the number of IMFs in a sinusoidal signal

Tests on a simulated signal $X (t)$ ⁶⁷ were carried out to verify the effectiveness of the proposed criterion for determining the number of IMFs in the decomposition using the VMD method. This signal is created by summing three distinct sinusoidal signals $X 1 (t), X 2 (t)$ and $X 3 (t)$ , as described in equation (12). Figures 2 and 3 show the three simulated signals as well as the overall signal obtained by summing them.

\begin{matrix} X (t) = X 1 (t) + X 2 (t) + X 3 (t) \\ X 1 (t) = 3 \sin (2 π 5 t) \\ X 2 (t) = 0.4 \sin (2 π 200 t) \\ X 3 (t) = 1.2 \sin (2 π 50 t) \end{matrix}

(12)

Figure 2.

Three simulated sinusoidal signals.

Figure 3.

The signal obtained from the sum of the three simulated signals.

In this example application, we already know the exact number of IMFs in the signal $X (t)$ , which is three. The goal is to validate the efficacy of the proposed criterion for selecting the optimal number of IMFs during decomposition.

Each of the IMFs obtained should represent a different component of the signal $X (t)$ constructed. We can then use this criterion in the decomposition of simulated and measured gear signals after validating its ability to accurately extract the various components of the original signal.

The decomposition of the signal $X (t)$ given by equation (12) using the approach proposed in Figure 1 yields the value of the proposed criterion $Δ_{NSE}$ after each iteration. Table 1 and Figure 4 show the values of $Δ_{NSE}$ . It can be noted that for $Δ_{NSE}$ values below a certain threshold of $0.01$ , the number of IMFs is $j = 4$ . Therefore, the appropriate number of IMFs to choose is $K = j - 1 = 3$ , which corresponds to the three simulated signals.

Table 1.

Criterion values $Δ_{NSE}$ and the corresponding number of IMFs.

$Δ_{NSE}$ values	0.7159	0.0136	0.0099	0.0043	0.0056	0	0.0025	0.0051	0.003
No. of IMFs j	2	3	4	5	6	7	8	9	10

Figure 4.

Variation of the proposed criterion $Δ_{NSE}$ for t he signal $X (t)$ .

Figure 5 depicts the decomposition of the signal $X (t)$ by VMD with the optimal number of IMFs determined by the proposed criterion. The IMFs obtained from this decomposition highlight the three components of the original signal, confirming the efficacy of the proposed criterion in determining the optimal number of IMFs.

Figure 5.

The resulting IMFs from VMD for K = 3.

To further validate our criterion, Figure 6(a) and (b) show the representations of the IMFs obtained from the VMD decomposition for $K = 2$ and $K = 4$ , respectively. These figures allow for a close examination of the properties of the IMFs obtained with various values of K. In Figure 6(a) ( $K = 2$ ), we can clearly observe mode mixing and its influence on IMF1. This is an example of under-segmentation, where two combined IMFs are insufficient to capture all of the original signal’s features.

Figure 6.

Resulted IMFs from VMD: (a) for K = 2 and (b) for K = 4.

In Figure 6(b), ( $K = 4$ ), we can notice shared information between IMF2 and IMF4. This is an example of over-segmentation, in which an excessive number of IMFs are generated, some of which may contain redundant or similar data. These findings correspond to the scenarios of under- and over-segmentation discussed in Dragomiretskiy and Zosso.⁶⁰ Thus, we can confirm in detail the performance of our criterion in identifying the optimal number of IMFs for an accurate and comprehensive signal decomposition by examining them.

Numerical simulation of gear faults

Gear faults appear on the spectra and envelope spectra at the rotation frequencies of the shafts on which the gear wheels are mounted. For each pair of gears, the amplitudes of these rotation frequencies will be amplified at the gear mesh frequencies and their harmonics. As a result, it is prudent to choose these resonance frequencies and conduct in-depth analyses around them. The combined approach proposed in this article, “VMD-WMRA,” based on the proposed criterion, can be used to diagnose this type of fault very effectively.

A gear fault signal S_en(t) is simulated in this section using the mathematical model provided by Capdessus and Sidahmed.⁶⁸ Figure 7 depicts a gearbox with three shafts rotating at frequencies of 15, 12, and 17 Hz, respectively. The gear wheels mounted on the first and third shafts have $Z_{1} = 42$ and $Z_{4} = 45$ teeth, respectively. On the second shaft, two gear wheels are mounted with $Z_{2} = 50$ and $Z_{3} = 65$ teeth. We can generate the signal S_en(t) for studying gear faults using the expression in equation (13). By analyzing this signal with the proposed approach, it is possible to identify gear faults in this gearbox.

\begin{matrix} S_{en} (t) = (\sum_{n = - \infty}^{+ \infty} S_{e} (t - n \times τ_{e})) * \\ (\begin{matrix} 1 + \sum_{m = - \infty}^{+ \infty} S_{r 1} (t - m \times τ_{r 1}) + \\ \sum_{p = - \infty}^{+ \infty} S_{r 2} (t - p \times τ_{r 2}) \end{matrix}) + n (t) \end{matrix}

(13)

Figure 7.

Gearbox configuration: schematic representation.

where, $S_{e}$ is the response signal of gears meshing; $τ_{e}$ represents the meshing periods: $τ_{e} = 1 / F_{m}$ , $; τ_{r 2}$ are the rotation periods of the defected gears:

τ_{r 1} = 1 / F_{r 1}, τ_{r 2} = 1 / F_{r 2};

$n (t)$ is the white Gaussian noise for a realistic effect;

${F_{m}}_{1} = {F_{r}}_{1} \times Z_{1} = {F_{r}}_{2} \times Z_{2} = 588 Hz$ is the first meshing frequency,

${F_{m}}_{2} = {F_{r}}_{2} \times Z_{3} = {F_{r}}_{3} \times Z_{4} = 760 Hz$ is the second meshing frequency.

The simulated gearbox has two gear mesh frequencies, F_m₁ and F_m₂, which correspond to the meshing of the first and second shafts, as well as the second and third shafts. Faults were simulated on the intermediate shaft gearwheel Z₂ and output shaft gearwheel Z₄.

The simulated signal and its spectrum of the two gear faults are shown in Figure 8. The appearance of the two gear mesh frequencies (F_m₁ and F_m₂) and their harmonics is clearly visible in the spectrum. A quick zoom around the gear mesh frequencies reveals that they are modulated by the rotation frequencies F_r₂ and F_r₃, which correspond to faults on gearwheels Z₂ and Z₄, respectively. Therefore, the simulated signal accurately represents the frequency characteristics of the gearbox gears.

Figure 8.

Simulated signal $S_{en} (t)$ of the combined gear fault in the time and frequency domains.

Figure 9 shows the decomposition of the signal-simulating gear faults using the VMD method based on the proposed criterion, which yields four IMFs. The proposed criterion based on Shannon Entropy has the primary advantage of reducing the number of IMFs that may contain information about the presence of potential faults. Thus, it simplifies and improves the combination of the two signal processing methods, VMD-WMRA. VMD can provide an accurate diagnosis for simulated signals or signals measured with minimal noise. However, as we will see in the following section, for superior diagnostic effectiveness in the case of highly noisy signals, VMD must be combined with WMRA.

Figure 9.

Variation of the $Δ_{NSE}$ for $S_{en} (t)$ .

The signals and spectra of the four IMFs are shown in Figure 10. It is possible to detect extremely effective isolation of the gear mesh frequencies and some of their harmonics, allowing us to perform a more precise diagnosis of potential gearbox failures.

Figure 10.

IMFs obtained from VMD of simulated signal $S_{en} (t)$ in the time (left) and frequency (right) domains.

The application of WMRA to each of the four IMFs from Figure 10 yields envelope spectra for each, as shown in Figure 11. As shown in Figure 11(a), IMF 1, which covers the first gear mesh frequency ${F_{m}}_{1} = 588 Hz$ , is modulated by the rotation frequency of the intermediate shaft ${F_{r}}_{2} = 12 Hz$ on which we simulated the fault. Meanwhile, IMF 2, covering the second gear mesh frequency ${F_{m}}_{2} = 760 Hz$ , is modulated by the rotation frequency of the output shaft ${F_{r}}_{3} = 17 Hz$ , as seen in Figure 11(b). IMFs 3 and 4, covering the combination of $3 {F_{m}}_{2}$ plus $4 {F_{m}}_{1}$ and $4 {F_{m}}_{2}$ plus $5 {F_{m}}_{1}$ , respectively, exhibit envelope spectra that represent the combination of the two fault frequencies, F_r₂ and F_r₃.

Figure 11.

Envelope spectra obtained from WMRA for each IMF: (a) IMF 01, (b) IMF 02, (c) IMF 03, and (d) IMF 04.

Experimental study

The vibrational signals processed in this section were measured on the test bench designed and manufactured at the Laboratory of Mechanics and Structures at the University of Guelma, Algeria (Figure 12(a)). It comprises the following components:

a 2.2 kW electric motor with a rotation speed of 1500 rpm,

an elastic coupling for transmitting motion between the motor and the gearbox,

a gearbox with mechanical characteristics provided in Table 2,

an electromagnetic brake for applying various loads to the gearbox, and

a V-ribbed belt for transmitting motion between the output shaft and the electromagnetic brake.

Figure 12.

The experimental setup: (a) test rig and (b) defected gear.

Table 2.

Used gears and characteristic frequencies.

Gear type	Straight teeth
Gear characteristics	Transmission ratios	$U 1 = 42 / 50 = 0.84$ $U 2 = 65 / 45 = 1.444$
	Rotation frequencies	First case:	$F_{r 1} = 14$ Hz $F_{r 2} = 12$ Hz $F_{r 3} = 17$ Hz
		Second case:	$F_{r 1} = 23$ Hz $F_{r 2} = 20$ Hz $F_{r 3} = 28$ Hz
Meshing frequency	First case:	$F_{m 1} = 588 Hz$ $F_{m 2} = 761 Hz$
	Second case:	$F_{m 1} = 966 Hz$ $F_{m 2} = 1300 Hz$

An experimental setup was designed with the same mechanical components as the previously simulated gearbox to validate the results of the numerical simulation presented in Section “Numerical simulation.”

Table 2 lists all the mechanical characteristics of the gears, including the transmission ratios for each mesh stage (U1 and U2), the rotation frequencies of the three shafts (F_r₁, F_r₂ and F_r₃) and gear mesh frequencies (F_m₁ and F_m₂), for two different rotation speeds of the electric motor.

The vibrational signals were measured experimentally for both rotation frequencies of 14 and 23 Hz, with and without gear faults, using the Bruel & Kjær PULSE 16.1 analyzer and the Pulse LabShop acquisition software, as shown in Figure 12(a) Three Bruel & Kjær type 4533-B accelerometers were used, one on each meshing stage of the gearbox housing in the horizontal direction.

Four vibrational signals in the frequency range of 6400 Hz were measured and will be processed using the proposed method: two without gear faults (S1, S2) and two with gear faults (S3, S4) for both rotation frequencies, as shown in Figure 13. These signals will be processed using the proposed method.

Figure 13.

Vibratory signals measured for different cases: (a) S1 without gear defect and $F_{r 1} = 14 Hz$ , (b) S2 without gear defect and $F_{r 1} = 23 Hz$ , (c) S3 with gear defect and $F_{r 1} = 14 Hz$ , and (d) S4 with gear defect and $F_{r 1} = 23 Hz$ .

In order to achieve an accurate diagnosis of the measured signals, an analysis of scalar indicators is essential. In this study, a set of six scalar indicators were calculated and are presented in Table 3. Some of these indicators are sensitive to the signal’s energy, such as the peak value (PV), root mean square (RMS), energy (E), and power (P). Other indicators are sensitive to the signal’s shape, such as kurtosis (K) and crest factor (CF). By examining the magnitudes of the calculated indicators, it is clear that the signals contain defects other than the experimentally simulated defects on the gear wheels and are also highly noisy. For signal S1, without gear fault, the kurtosis value reaches 5.08, whereas, in the absence of shock defects, its value should not normally exceed 3. The presence of a high noise level in the various signals will make fault detection difficult and will require a high-performance diagnostic method. These observations highlight the importance of proposing a robust diagnostic method to identify faults despite the significant noise in the signals. An in-depth analysis based on the proposed approach will be necessary for accurate fault detection in the studied gearbox.

Table 3.

Scalar indicators’ value.

Signals		RMS	CF	PV	E × 10³	K	P
Without gear defect	S1	5.56	5.96	33.10	506.22	5.08	30.90
	S2	3.77	4.38	16.90	116.29	3.76	14.19
With gear defect	S3	4.80	6.55	33.60	378.84	6.12	23.12
	S4	10.04	6.20	82.10	165.07	6.05	100.75

Results and discussions

This section is devoted to the experimental validation of the proposed method for detecting gearbox faults. As previously stated, the selected signals from the experimental setup (S1, S2, S3, and S4) are measured for two different system health states, the first without gear faults (S1 and S2) and the second with a gear fault (S3 and S4), at two different speeds, $F_{r 1} = 14 Hz$ and $F_{r 1} = 23 Hz$ .

The work begins with WMRA processing the four signals. The obtained results will be compared to those obtained by applying the proposed approach, which consists of determining the optimal number of IMFs based on the proposed criterion, decomposing the same signals using VMD, and finally processing each IMF obtained using WMRA.

Signal processing without gear fault

The signals S1 and S2, along with their corresponding spectra, are presented in Figure 14. These signals, without gear faults in the gearbox, show abnormal values of the scalar indicators, indicating the presence of another type of defect.

Figure 14.

Signals without gear defects and their corresponding spectra: (a) S1 with F_r₁ = 14 Hz and (b) S2 with F_r₁ = 23 Hz.

In the case of S1, shown in Figure 14(a), spectral analysis shows the dominance of the meshing frequency F_m₁, which has overwhelmed the rest of the frequencies, making the diagnosis difficult and thus requiring more advanced signal processing methods. On the other hand, for S2, presented in Figure 14(b), we can clearly see the noise present in the signal. The corresponding spectrum shows the appearance of the meshing frequency F_m₁ and its harmonic 2F_m₁, where the amplitude of 2F_m₁ is about 1.5 times higher than that of F_m₁, which corresponds to an insufficient inter-axial clearance defect (insufficient backlash at the tooth root).

Results obtained by WMRA

The processing of the two signals, S1 and S2, by WMRA for the two rotational speeds gives the envelope spectra shown in Figure 15. It can be observed that the amplitudes of the harmonics of the input shaft’s rotational frequencies 2F_r₁, 3F_r₁, and 4F_r₁ are higher than that of the fundamental frequency F_r₁, indicating the presence of misalignment of the input shaft. The misalignment is much more pronounced, especially at the rotational frequency of 14 Hz, as shown in Figure 15(a).

Figure 15.

Envelope spectra obtained from WMRA: (a) S1 with F_r₁ = 14 Hz and (b) S2 with F_r₁ = 23 Hz.

It is noted that the vibrational amplitude at this frequency is twice that observed at the 23 Hz frequency, as shown in Figure 15(b). This result is confirmed by the thesis work of J. Bouyer,⁶⁹ which shows that the amplitude of misalignment increases with the intensity of the misalignment torque, especially when the load or rotational speed is low.

Results obtained by VMD-WMRA combination

The decomposition of the same signals using VMD based on the proposed criterion provides the signals and spectra of the different IMFs, as shown in Figure 16. For S1 (see Figure 16(a)), it can be observed that the spectra of the obtained IMFs perfectly isolate the meshing frequency F_m₁ (IMF1) and some of its harmonics (IMFs: 2, 3, and 4). Similarly, for S2 (see Figure 16(b)), the isolation of the meshing frequency F_m₁ (IMF2) and its fifth harmonic (IMF3) is noted.

Figure 16.

Resulting IMFs and the corresponding spectrum obtained from VMD: (a) S1 with F_r₁ = 14 Hz and (b) S2 with F_r₁ = 23 Hz.

Additionally, IMF1 isolates a resonance at the frequency of 422 Hz, with an amplitude lower than that of F_m₁. After investigating this resonance and following the calculations below, we conclude that this resonance represents the natural frequency of the belt, F_Br. The values of the frequency F_Br for the two rotational speeds are presented in Table 4. The absence of the frequency F_Br in S1 is due to its close value to F_m₁, which has a large amplitude, thereby overshadowing F_Br, unlike in S2.

Table 4.

Calculation results for the two rotation frequencies

Rotation frequency F_r₁	Input torque C_e	Output torque C_s	Belt tension T	The resonant frequency $F_{Br}$
14 Hz	25.01 N.m	20.68 N.m	827 N	535 Hz
23 Hz	15.22 N.m	12.55 N.m	502 N	417 Hz

The following expressions can obtain the values of the belt resonance frequency:

$C_{e} = 9.555 \times \frac{P}{N_{e}}$ , where C_e is the input torque, P_e and N_e are the power and the rotational speed of the input shaft, respectively.

$C_{s} = \frac{C_{e}}{u}$ , where C_s is the output torque, and u is the total transmission ratio.

$T = \frac{P}{V_{s}}$ , where T is the belt tension and V_s is the linear speed of the belt.

The natural frequency of the belt $F_{Br}$ is calculated by the following formula:

$F_{Br} = \frac{1}{2 L_{b}} \sqrt{\frac{T}{μ}}$ , where L_b is the length of the free span, and μ is the linear mass of the belt.

To diagnose faults, we apply the WMRA to each IMF for the two rotational speeds, as shown in Figures 17 and 18. For S1, the presence of misalignment of the input shaft is confirmed by the modulation of the harmonics of the rotational frequency (2F_r₁, 3F_r₁), which exceed the amplitude of the rotational frequency itself, as shown in IMFs 1, 2, and 4

Figure 17.

Envelope spectra obtained by the WMRA for each IMF of S1.

Figure 18.

Envelope spectra obtained by the WMRA for each IMF of S2.

For S2, shown in Figure 18, the misalignment of the input shaft is confirmed by the presence of the 2F_r₁ peak, which exceeds the rotational frequency F_r₁, as seen in IMFs 2 and 3. The presence of misalignment on the input shaft, along with the insufficient inter-axial distance between the two gears, identified in the spectrum in Figure 14(b), caused a misalignment on the second shaft, which is detected in IMFs 1 and 2 by the presence of a peak at the frequency 2F_r₂ in the absence of the rotational frequency F_r₂. Additionally, in IMF3, a peak at a frequency of 4 Hz is observed.

After investigating the condition of the belt, we found two defects on the grooves of the V-ribbed belt, as shown in Figure 19. These defects justify the presence of the peak at the frequency 4 Hz, which corresponds to the belt passing frequency F_B and a large number of its harmonics.

Figure 19.

Defects identified on the V-ribbed belt.

Signal processing with gear fault

The signals showing gear defects, S3 and S4, as illustrated in Figure 20, were also analyzed in this section. Due to the presence of a defect on a tooth of the driven gear rotating at the frequency F_r₂, we observe an increase in the vibration amplitudes compared to the case without gear defects in the signals and spectra.

Figure 20.

Signals with gear faults and their corresponding spectra: (a) S3 with F_r₁ = 14 Hz and (b) S4 with F_r₁ = 23 Hz.

For signal S3, shown in Figure 20(a), there are impacts with higher amplitudes compared to S1, and the corresponding spectrum still shows the dominance of the meshing frequency F_m₁. The spectrum of S4, illustrated in Figure 20(b), indicates that the presence of a defect on the gear rotating at 23 Hz has exacerbated the inter-axial insufficiency defect, given that the amplitude of 2F_m₁ exceeds that of F_m₁ by approximately 3.5 times. We also observe a sevenfold increase in the amplitude of the belt resonance frequency $F_{Br}$ compared to S2, with a frequency shift up to 502 Hz. This frequency shift is due to the increased belt tension T caused by the exacerbation of the inter-axial insufficiency defect as well as the misalignment on the various shafts.

Results obtained by WMRA

In the envelope spectra obtained by processing the two signals using WMRA, shown in Figure 21, we observe the absence of F_r₂ harmonics for both rotational speeds of 14 and 23 Hz. Therefore, it can be said that WMRA does not allow the localization of the gear fault on the driven wheel.

Figure 21.

Envelope spectra obtained by WMRA: (a) S3 with F_r₁ = 14 Hz and (b) S4 with F_r₁ = 23 Hz.

It was found that during the disassembly and reassembly of wheel Z₂ to create the defect on the flank of a tooth, a misalignment was induced on the intermediate shaft. This latter defect is represented by the presence of a peak at 2F_r₂, as well as the confirmation of the misalignment of the input shaft by a peak at 3F_r₁, as shown in Figure 21(a).

For case S4, shown in Figure 21(b), a very significant increase in vibration amplitudes of approximately 20 times is observed, due in part to the increase in rotational speed in the presence of defects and, on the other hand, to the belt resonance, which allows the appearance of the third harmonic of the belt defect at 3F_B and misalignment defects on both shafts.

Results obtained by VMD-WMRA combination

In Figure 22, the IMFs and their spectra obtained by VMD using the proposed criterion for signals S3 and S4 are presented. It is observed that the IMFs perfectly isolate the different frequencies (belt resonance frequency, meshing frequency, and its harmonics).

Figure 22.

The IMFs obtained by VMD and the corresponding spectra: (a) S3 with F_r₁ = 14 Hz and (b) S4 with F_r₁ = 23 Hz.

Figure 22(a) demonstrates the isolation of F_m₁ and some of its harmonics (3F_m₁, 4F_m₁, 5F_m₁, 10F_m₁), while Figure 22(b) shows the appearance of the belt resonance frequency $F_{Br}$ . and some harmonics of the meshing frequency (2F_m₁, 3F_m₁, 5F_m₁). The envelope spectra of each IMF for the rotation frequency of 14 Hz, obtained by applying WMRA, are presented in Figure 23. On the envelope spectra of IMF1, IMF2, and IMF3, the modulation of F_m₁ by the rotation frequency of the intermediate shaft F_r₂, on which the wheel with the tooth defect is mounted, is clearly observed. This result was not achievable when using WMRA alone.

Figure 23.

Envelope spectra obtained by the WMRA for each IMF of S3.

Figure 24 displays the envelope spectra obtained by the combination of VMD and WMRA. On the envelope spectrum of IMF1, a significant number of harmonics of half the belt defect frequency F_B are observed, lower in amplitude compared to 2F_B, indicating the presence of a dual belt defect as previously illustrated in Figure 19. On the envelope spectrum of IMF2, the modulation of F_m₁ by the rotation frequency of the driven shaft F_r₂ is observed. In contrast to case S3, diagnosing the gear fault is less straightforward due to the presence of significant misalignment on the input shaft, indicated by a peak at 2F_r₁ dominating the harmonics of F_r₂. The envelope spectrum of IMF3 shows the presence of F_r₁ harmonics, attributed to insufficient backlash. Furthermore, IMF4 confirms the presence of belt defects by the appearance of numerous harmonics of F_B.

Figure 24.

Envelope spectra obtained by the WMRA for each IMF of S4.

Industrial application of the combined and optimized VMD-WMRA approach

To demonstrate the proposed approach’s effectiveness, the proposed method was applied to signals measured in an industrial setting, specifically on the cooling turbofan (101BJT), a key machine in the ammonia manufacturing process at Algeria’s largest fertilizer company (FERTIAL).⁷⁰ The 101BJT is made up primarily of an electric motor, a fan, a speed reducer, and a turbine, as shown in Figure 25.

Figure 25.

Kinematic scheme of the 101BJT with measuring points.

The 101BJT turbofan comprises two drives: a main drive powered by a steam turbine operating at F_r₁ = 83.5 Hz (equivalent to 5010 rpm) and a backup drive powered by an electric motor operating at F_r₂ = 15.5 Hz (equivalent to 930 rpm).

The turbine and speed reducer are connected to the fan via a clutch coupling, which disengages automatically when the turbine speed falls below that of the fan drive shaft. Vibration measurements were taken in three directions at turbofan bearing points numbered P1 through P10. Figure 26(a) illustrates the use of two accelerometers: a Bruel & Kjaer triaxial accelerometer 4524B-00 and an industrial accelerometer Bruel & Kjær type 4511-001. Signals were collected using the Bruel & Kjær Pulse 16.1 analyzer and Pulse LabShop acquisition software, as shown in Figure 26(b). Post-processing was done in MATLAB.

Figure 26.

Taking measurements: (a) sensor positions and (b) pulse analyzer 16.1.

Based on the overall RMS levels of the measured signals, which were notably high at bearings P2 and P5, these signals were processed in both low-frequency (0–800 Hz) and high-frequency (0–12,800 Hz) bands, as shown in Figure 27.

Figure 27.

Signals measured in the P2 and P5 bearings at low and high frequencies.

Table 5 presents the characteristic frequencies of various types of faults that may occur in bearings P2 and P5, such as F_ow (oil whirl frequency), F_if (impact friction frequency), F_Bp (blade passing frequency), F_PEFB (passing frequency of each set of fixed blades), N_fb (number of groups of fixed blades), and F_m (meshing frequency), with N_b representing the number of blades in both turbine wheels, Z₁ the number of teeth on the driving wheel, and Z₂ the number of teeth on the driven wheel. These characteristic frequencies play an important role in vibration analysis and fault detection. Identifying these characteristic frequencies in the measured signals at bearings P2 and P5 allows for an accurate diagnosis of the type of fault present.

Table 5.

Characteristic frequencies of possible defects in P2 and P5.

Defect type	Characteristic frequency
Oil whirl	F _ow = (0.3–0.5) × F_r₁F_ow = 25.05–41.75 Hz
Impact friction	F _if = (1/4–1/3) × F_r₁F_if = 20.87–27.83 Hz
Blade passing (N_b = 162 blades, N_fb = 6 blades)	F _Bp = F_r1 × N_P = 8517 HzF_PEFB = F₁ × N_fb = 501 Hz
Gears (Z₁ = 47 teeth, Z₂ = 252 teeth)	F _m = F_r₁ × Z₁ = F_r₂ × Z₂F_m = 3924.5 Hz

Analysis of signals in the low-frequency band allows investigation of faults manifesting within this frequency range. Applying the proposed criterion for VMD decomposition yields two IMFs for bearing P2 and three IMFs for bearing P5, subsequently processed by WMRA to obtain envelope spectra, as shown in Figure 28. In IMF1 spectra of signals from bearings P2 and P5, the frequencies of oil whirl defect F_ow and impact friction defect F_if are clearly visible. Both defects are more pronounced on the turbine’s P2 bearing compared to the reducer’s P5 bearing, as indicated by the amplitude differences between the two spectra.

Figure 28.

Envelope spectra obtained by WMRA for IMFs of low-frequency signals: (a) bearing P2 and (b) bearing P5.

Similarly, applying the proposed approach to high-frequency signals results in four IMFs for bearing P2 and five IMFs for bearing P5. The envelope spectra obtained by WMRA for these IMFs are illustrated in Figure 29. Investigation of the IMFs from high-frequency signals reveals the presence of a gear defect through the appearance of the input shaft rotation frequency F_r₁ and its harmonics in IMFs 1 and 4 for bearing P2, as well as IMFs 1 and 3 for bearing P5.

Figure 29.

Envelope spectra obtained by WMRA for IMFs of high-frequency signals: (a) bearing P2 and (b) bearing P5.

Conclusion

This article presented a combined approach to fault detection in industrial installations that uses Variational Mode Decomposition and Wavelet Multi-Resolution Analysis. The primary goal was to create a reliable and effective method for diagnosing faults in extremely noisy vibration signals. To accomplish this, we proposed a new criterion based on the variation of normalized Shannon entropy that maximizes the number of Intrinsic Mode Functions by effectively isolating the various carrier frequencies present in the spectrum. The use of this criterion and the VMD-WMRA combination on both numerically simulated and experimentally measured signals demonstrated the effectiveness of the proposed approach. This method provides a promising solution for detecting faults in rotating machinery while overcoming challenges posed by vibrational signal disturbances such as shaft misalignment and insufficient backlash.

The results from simulation and experimentation show that this approach has the potential to improve the reliability and accuracy of fault diagnosis. This could have a significant impact on preventive maintenance, helping to improve operational efficiency and extend equipment lifespan in industrial settings. Furthermore, the method was applied to signals collected from an industrial cooling turbofan, allowing for the diagnosis of gear, oil whirl, and impact friction faults in the turbine and reducer bearings. While the proposed method has demonstrated efficacy in early fault diagnosis, even within industrial settings, several prospective avenues for future research are identified: the incorporation of the optimized VMD-WMRA approach into machine learning algorithms to facilitate automated fault classification and prediction, as well as the application of cyclostationary methods to analyze signals recorded under fluctuating operating conditions, thereby enhancing diagnostic precision in progressively noisy environments.

Footnotes

Handling Editor: Aarthy Esakkiappan

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The current research was conducted by the “Structural Dynamics & Industrial Maintenance” research group at the Mechanics & Structures Laboratory (LMS) of the 08 May 1945 University, Guelma, Algeria, under the funding of the General Directorate of Scientific Research and Technological Development (DGRSDT) through the PRFU research project: A11N01UN240120220004.

ORCID iDs

Ammar Mrabti

Zakarya Ouelaa

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Diagnosis of faults using a new VMD-WMRA approach,optimized by a new criterion: Industrial application

Abstract

Keywords

Introduction

Mathematical formulation

Variational mode decomposition (VMD)

Shannon entropy (SE)

Wavelet multi-resolution analysis (WMRA)

Proposed approach

Numerical simulation

Determining the number of IMFs in a sinusoidal signal

Numerical simulation of gear faults

Experimental study

Results and discussions

Signal processing without gear fault

Results obtained by WMRA

Results obtained by VMD-WMRA combination

Signal processing with gear fault

Results obtained by WMRA

Results obtained by VMD-WMRA combination

Industrial application of the combined and optimized VMD-WMRA approach

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References