A remaining useful life prediction approach for ball bearing by acoustic emission signal and physics-informed neural network

Abstract

Ensuring the safety and reliability of key components, such as bearings, is crucial in machinery systems. Therefore, academia and industry have long tried to develop prognostic and health management technology. In particular, it is necessary to carry out methods to predict the remaining useful life (RUL) of the bearing, which can reduce economic losses and significantly prevent safety accidents. To overcome the deficiency of current investigations on bearing RUL prediction, a new method based on acoustic emission signals and a physics-informed neural network (PINN) is proposed in this paper. A new health index is constructed by developing an improved diffusion entropy, which outperforms current time-domain features. Four different labeling functions are compared to demonstrate that the exponential function has the best capacity to represent the bearing degradation process. The physical knowledge in this paper is derived from reliability engineering, that is, the failure procedure described via the Weibull distribution, and this knowledge is incorporated into a long short-term memory neural network to construct the PINN model by developing a Weibull-loss function. Moreover, this knowledge integration can improve prediction performance and help circumvent some obstacles, including poor quality or limited historical data and the obscurity of the underlying processes. Finally, the effectiveness and superiority of the proposed method are validated through experimental results and comparison with other existing methods.

Keywords

Prognostics and health management remaining useful life ball bearing acoustic emission physics-informed neural network

Introduction

Prognostic and health management (PHM),¹ which aims to optimize and timely maintenance of machinery and equipment, has attracted more and more attention in both academia and industry. The reason is that multiple machinery and equipment, such as helicopters, wind turbines, induction motors, aero engines, and high-speed trains, are becoming increasingly complex with the rapid development of technology and suffer harsh working environments chronically, thus leading to sudden failure and accompanying maintenance costs and downtime losses. Compared to traditional maintenance strategies that depend on operators to observe and identify the health status of machinery and equipment, PHM can greatly reduce manual labor and improve maintenance efficiency by incorporating four procedures, that is, monitoring, diagnosis, prognosis, and health management, as depicted in Figure 1. Monitoring refers to the health condition surveillance of machinery and equipment, which depends on multiple sensors (e.g., vibration sensors or temperature sensors) to trace the corresponding health state through data processing.^2,3 Diagnosis means the identification, location, and detection of the type, position, and degree of the faults, respectively, and advanced signal processing techniques (e.g., feature extraction and selection) and emerging artificial intelligence (AI) approaches (particularly machine learning and deep learning) have facilitated these decisions.^4,5 The prognosis aims to estimate the progress of degradation and further predicts the remaining useful life (RUL) of mechanical equipment, which can be defined as “the length from the current time to the end of useful life.”⁶ Health management integrates the former three procedures into an optimal maintenance strategy that can balance maintainability and economic cost. In addition, prediction of RUL^7,8 is one of the most important tasks in PHM, as it is the ultimate goal of prognostics that can guide maintenance plans.⁹ In recent decades, more and more attention^10–18 has been paid to the study of RUL prediction of machinery, going to great lengths to solve two questions:

(i) How can we achieve the RUL prediction through the monitoring data of machinery?

(ii) How to estimate the RUL prediction performance?

Figure 1.

A typical PHM strategy for machinery and equipment.

In multiple modern rotating machinery and equipment, rolling bearings, as the most critical mechanical components, are widely used to facilitate linear motion or rotary motion by reducing friction between moving parts. However, they are vulnerable due to several issues, including extensive load, high speed, and poor lubrication. Particularly, bearings may suffer overheating and increased clearance over time under such harsh conditions and finally be prone to collapse. For instance, about 80% of the gearbox problems can be attributed to bearings in the wind energy industry,¹⁹ and several accidents of aircraft caused by bearing failure have been reported.²⁰ Therefore, it is essential to conduct investigations on bearing RUL prediction, which can effectively improve the reliability of rotating machinery and reduce maintenance and downtime costs correspondingly. A typical bearing RUL prediction procedure is illustrated in Figure 2, and it employs a mathematical model to fit existing data at $t_{FPT}$ time to project the $RU L_{optimal}$ for the bearing lifetime via a predefined failure threshold. Moreover, uncertainties associated with the degradation process and unknown future working conditions should be considered, that is, the confidence interval estimation, including upper confidence bound ( $RU L_{\max}$ ) and lower confidence bound ( $RU L_{\min}$ ). Numerous studies²¹ have been accomplished to contribute to the development of multiple mathematical models, which can be divided into three categories: physics-driven models, data-driven models, and hybrid models.²² Physics-driven models are developed via failure mechanisms or the first principle of damage to describe the degradation process, and the parameters in physics-driven models are generally identified via finite element analysis or specific experiments. Xu et al.²³ predicted bearing RUL by revealing the relationship between feature values of vibration signals and the variable in the Paris equation, which can characterize the failure mechanism of materials and predict fatigue life or residual fatigue life. An improved unscented particle filter (PF) was developed by Teng et al.²⁴ to identify parameters in an exponential model, which was derived from the Lundberg-Palmgren rule, to implement the RUL prediction of bearings in wind turbines. The effectiveness of these two physics-driven models was demonstrated via experimental verifications. When a thorough understanding of the failure mechanisms and accurate estimation of parameters can be satisfied, the physics-driven models can provide an accurate prediction of bearings RUL, while this situation is always unrealistic for complex mechanical systems. Subsequently, the data-driven models, which transform acquired and monitored data into relevant models to describe degradation behavior without any physics or principles, have become the most popular category. Generally, the relevant models here can be conducted via statistics (such as autoregressive (AR), Wiener process, Gamma process, Gaussian process, Markov model, Kalman filters, and PFs) or AI. Kundu et al.²⁵ conducted a Weibull accelerated failure time regression (WAFTR) model that considered both operating condition parameters and condition monitoring signals to implement bearing RUL prediction. An extended Kalman filter was employed by Singleton et al.²⁶ to forecast bearing RUL under different operating conditions, and its parameters were learned via an affine function. On the other hand, AI-enabled data-driven models have attracted more and more attention recently due to the rapid development of AI techniques that are capable of dealing with RUL prediction issues of complex mechanical systems. Previous investigations on AI-enabled data-driven models focus on machine learning (ML) methods consisting of two steps, that is, handcraft feature extraction from acquired and monitored data (such as vibration signals) and training a regression model based on extracted features to implement RUL prediction. For instance, Rai and Upadhyay²⁷ extracted time-domain and frequency-domain features from vibration signals to train a self-organizing map classifier to construct the health index (HI) to describe the degradation process, and then a support vector regression prediction model was trained via HI to predict RUL of bearings. Then, recent research has contributed more to deep learning (DL) methods for bearing RUL prediction, such as convolutional neural networks²⁸ and recurrent neural networks (RNNs), particularly long short-time memory (LSTM)²⁹ since DL methods can learn the representation features automatically from data such as vibration signals to avoid drawbacks of handcraft features, including high time-cost and severe experience-orientation. Besides those supervised DL methods, unsupervised DL methods, which require no label information to extract excellent data representation for RUL prediction, have also been investigated recently, and their performance approaches or even surpasses that of supervised DL methods. Similarly, Deutsch and He³⁰ proposed a deep belief network-feedforward neural network (DBN-FNN) to implement bearings’ RUL prediction. Moreover, considering the difficulty of obtaining enough run-to-failure data in real-world scenarios, some scholars have conducted investigations on transfer learning (TL).⁸ As one paradigm of DL methods, TL can be used to achieve bearing RUL prediction, since it can extract domain-invariant features and transfer the degradation knowledge acquired under different operating conditions through domain adaptation. For instance, inspired by the TL strategy, Miao et al.³¹ proposed a sparse domain adaptation network (SDAN) to predict bearing RUL under different working conditions, and SDAN can reduce the feature distribution shift across domains effectively. Finally, some researchers have noticed the capacity of hybrid-based models, which integrate the advantages of both physics-driven models and data-driven models. Qian et al.³² proposed an approach to track defects and predict RUL of rolling bearings, which employed both the physics-driven model (Paris crack growth model) and the data-driven model (enhanced phase space warping) to describe and characterize the defect propagation. A hybrid approach for bearing RUL prediction was proposed by Rezamand et al.,³³ and it first employed Kernel Fuzzy C-Means, Hidden Markov Model (HMM), and damage progression model (DPM) to discriminate and determine the acquired data in different states and then applied the adaptive Bayesian algorithm to forecast RUL.

Figure 2.

A typical bearing RUL prediction procedure.

Though the above-mentioned investigations have proved that the data-driven models, particularly based on vibration signals, can be used to predict bearing RUL, some limitations should be noticed, including difficulty in detecting impending failure before damage occurs and vulnerability of operating speed. Therefore, several works^34–36 have ascertained the feasibility of acoustic emission (AE) to implement PHM of bearings, since AE is capable of detecting incipient failures by tracking stress waves that originate from released strain energy in a material when deformation or fracture occurs. Particularly, the root mean square (RMS) and Kurtosis extracted from raw AE signals show a good correlation to the degradation process of bearings. Similar to bearing RUL prediction via vibration signals, AE-based bearing RUL prediction depends on feature extractions to characterize the degradation process (i.e., HI construction) and develops physics-driven or data-driven models to implement RUL prediction. Elforjani³⁷ employed AE signals to predict bearing RUL by proposing a new HI called signal intensity estimator (SIE) and training an artificial neural network (ANN) to correlate SIE with corresponding bearing degradation. Furthermore, in terms of bearing RUL prediction via AE signals, Elforjani and Shanbr³⁸ compared the performance of multiple HIs and data-driven models, including ANN, support vector machine regression (SVMR), and Gaussian process regression (GPR), and they noted that proper model structure and sufficient data were incredibly essential. Aye and Heyns³⁹ proposed an optimal GPR model based on a novel HI, which was derived from the integration or combination of existing simple mean and covariance functions of AE signals, to forecast bearing RUL. Moreover, DL-based models have been utilized in bearing RUL prediction via AE signals, for instance, Motahari-Nezhad and Jafari^40,41 extracted multiple time-domain and frequency-domain features to construct HI (considered prognosability, monotonicity, and trendability) and trained different multilayer perceptron (MLP) models and LSTM models for the bearing RUL prediction.

According to the above introduction, we can see that DL-based data-driven models have been implemented and applied to bearing RUL prediction via AE signals. However, little investigation contributes to developing DL models incorporated with physical knowledge (i.e., feeding physical knowledge of bearing degradation into the DL model training process to improve interpretability and generalization for different application scenarios⁴²) for bearing RUL prediction, particularly via AE signals. Here, the interpretability, to some extent, means the explanation of model prediction results and the working mechanisms behind, helping us to understand how the model makes the prediction. In other words, it is essential to build connections between the mapping relationship of the model and our prior knowledge, further enhancing the degree of comprehension of predictions. Therefore, incorporating physics-based knowledge into the model (e.g., via the loss function) can restrain and supervise the training process of the model, making the prediction more consistent with prior knowledge, that is, higher interpretability of predictions. A physics-informed DL model aims to extract better features that can characterize bearing degradation information than purely data-driven models by leveraging domain knowledge to alleviate the prediction error that is contrary to real-world physics.^43,44 Using an HI constructed via vibration signals and knowledge of monotonic degradation described by temperature signals, Chen et al.⁴⁵ developed a physics-informed neural network (PINN) named degradation consistency recurrent neural network to overcome the drawbacks of purely data-driven methods and improve bearing RUL prediction performance. Yang et al.⁴⁶ proposed a physics-informed bearing RUL prediction approach that employed a dynamic adaptive inverse discrete Fourier transform (IDFT) frequency-domain block to incorporate physical knowledge, and a residual self-attention multistate gated control unit (RSA-MSGCU) was used to make bearing RUL prediction. According to the literature review, to the best of our knowledge, no complete investigation has been conducted on bearing RUL prediction via AE signals and PINN until now. Therefore, in this paper, we study the feasibility of AE signals and PINN to forecast bearing RUL for the first time, and the main contributions are summarized as follows.

(i) A novel PINN model is developed to forecast bearing RUL, and its performance is better than current DL-based models since the physical knowledge of bearing degradation is leveraged via a customized data-physics-driven loss function. In addition, the developed PINN provides a higher level of compliance with physics, which enhances the interpretability and generalization capacity.

(ii) A new HI called improved diffusion entropy (IDEn), which can estimate signal complexity, is proposed to process AE signals and characterize the bearing degradation process in this paper. Compared to current HIs constructed via time-domain and frequency-domain features (RMS, Kurtosis, etc.), the proposed IDEn is more promising, reliable, robust, and sensitive to bearing degradation.

(iii) The effectiveness of the proposed method is verified via a set of laboratory tests on run-to-failure bearings. The priority of the proposed method is confirmed by comparison with other existing methods.

The rest of this paper is organized as follows: the details of the proposed method are given in “Methodology,” including a brief introduction of the AE signal and a presentation of the proposed IDE and PINN; the experimental setup, including the test rig and measurement procedure, and results are illustrated and discussed in the third and fourth sections, respectively; and the fifth section concludes the entire article and provides future research directions.

Methodology

In this paper, a novel bearing RUL prediction method is proposed, and its flowchart is depicted in Figure 3. First, AE signals from the ball bearing under run-to-failure tests are acquired. Then, we employ a new entropy indicator, that is, IDEn to implement feature extraction and HI construction from AE signals. Finally, a PINN model that incorporates LSTM and physical knowledge based on Weibull distribution is developed and trained via IDE-enabled HI to forecast bearing RUL with promising performance.

Figure 3.

Flowchart of the proposed method.

Acoustic emission

AE refers to the generation of transient elastic waves, which are caused by the release of energy (i.e., a rapid redistribution of stress) in a material when external stimuli such as changes in pressure, load, and temperature are applied. Multiple events from earthquakes and rock bursts down to initiation and growth of cracks can be sources of AE signals, and detection or analysis of AE signals can provide valuable information (e.g., origin and process) of discontinuity in the material, which is an attractive versatility for PHM in modern industry.^47,48 Generally, there are two kinds of AE signals (burst and continuous signals), and the AE signals caused by bearing degradation belong to the burst signal (the so-called “hit”). A typical AE burst signal is illustrated in Figure 4, and we can see that several parameters are employed to characterize AE signals. $Amplitude$ refers to the maximum peak of the AE waveform, and it can be expressed in dB as

\begin{matrix} Amplitude = 20 \cdot \log (V_{\max} / 1 μ v) - PG \end{matrix}

(1)

where $V_{\max}$ is the maximum voltage value of the AE waveform, $PG$ is the preamplifier gain in dB. $Duration$ is the time interval as

\begin{matrix} Duration = t_{1} - t_{0} \end{matrix}

(2)

where $t_{0}$ is the time that the AE signal crosses the threshold for the first time, $t_{1}$ is the time that the AE signal falls below the threshold finally. $Energy$ is the integral of the AE waveform (in terms of rectified voltage) over the duration as

\begin{matrix} Energy = \int_{t_{0}}^{t_{1}} V (t) d t \end{matrix}

(3)

where $V (t)$ is the rectified voltage signal of the AE hit. $Counts$ means the number of AE signals that traverse over the threshold. Moreover, another three timing parameters, that is, peak definition time (PDT), hit definition time (HDT), and hit lockout time (HLT), are also important to ensure AE acquisition performance. PDT is utilized to determine the AE signal peak correctly for amplitude and rise time, and proper HDT can ensure uniqueness (only one hit) of the AE signals in a duration. Spurious measurements as well as acquisition redundancy can be avoided by selecting HLT prudently.

Figure 4.

A typical AE signal and corresponding parameters.

Entropy

Entropy that works as a statistical measure metric can be used to estimate the complexity of information of data series, and several entropy metrics have been widely used in PHM, including Shannon entropy and Gnome entropy⁴⁹. Entropy refers to the uncertainty of a data series; specifically, its value increases with more complex data series. In this paper, a new entropy indicator named IDEn is proposed by upgrading the current diffusion entropy analysis (DEA)⁵⁰ via Deng entropy,⁵¹ and its detailed procedure depicted in Figure 5 is given as follows.

(i) In terms of a time series $x (T)$ , we first apply stripes (the number of stripes $N_{stripes}$ is:

\begin{matrix} w_{x} = | \max x (T) - \min x (T) | \end{matrix}

(4)

w_{s} = \frac{w_{x}}{N_{stripes}}

(5)

(ii) The time series $x (T)$ is then uniformly scaled to $X (T)$ via the width of stripes, which means stripes have a width of 1.

\begin{matrix} X (T) = \frac{x (T)}{w_{s}} \end{matrix}

(6)

(iii) Events can be recorded when the time series crosses or intersects a stripe at time T, which can be determined via two conditions:

X_{T} (T) < floor (X_{T - 1} (T) + 1)

(7)

X_{T} (T) > ceil (X_{T + 1} (T) - 1)

(8)

It is worth noting that a crossing or intersection must happen at time T when either of these two conditions is false. Subsequently, 1 (a crossing happens at time T) or 0 (on the contrary) can be appended to the event array $E_{T}$ .

(iv) Based on the event array, a diffusion trajectory $Y_{T}$ can be constructed by taking the cumulative sum of $E_{T}$ as:

Y_{T} = \sum_{t = 0}^{n_{T}} E_{T}

(9)

where $n_{T}$ is the length of $E_{T}$ .

(v) A set of moving windows is employed to characterize the diffusion trajectory, that is, each moving window with a defined length l steps along the diffusion trajectory to generate many slices. Each slice has a distance from the origin along the y-axis, and these displacements are binned in a histogram, which can be used to develop a probability distribution $P_{l}$ via normalization.

(vi) For the moving window with length l, we can calculate the Deng entropy $D_{m} (Y_{l})$ as:

D_{m} (Y_{l}) = - \sum m (y_{i}) \ln \frac{m (y_{i})}{e^{| y_{i} |} - 1}

(10)

where $Y_{l}$ is the diffusion trajectory’s value at time $T = l$ , $m (\cdot)$ is the mass function (probability distribution), and $y_{i}$ is the possible values of $Y_{l}$ , $| y_{i} |$ is the cardinality of $y_{i}$ .

(vii) After calculating corresponding Deng entropy values under different lengths of moving windows, we can obtain a principal relation curve, that is, Deng entropy versus natural log of moving window lengths, as:

\begin{matrix} D_{m} (Y_{l}) = δ \ln (l) + C \end{matrix}

(11)

where $δ$ is the scaling of the time series $x (T)$ , that is, the proposed IDEn, and it can be calculated via the Siegel and Theil-Sen methods since the relation is linear in log-scale, C is a constant. It is worth noting that the range of $δ$ is between 0.5 (non-complex process) and 1 (process at criticality), which means: the closer it is to 1, the more complex the process is.

Figure 5.

Flowchart of the proposed IDEn.

Finally, IDEn of AE signals from the ball bearing under run-to-failure tests is employed to construct HI for further RUL forecasting via the PINN, which will be introduced in detail in the next subsection.

Physics-informed neural network

According to the earlier introduction, it is well known that AI-enabled data-driven models, including ML and DL-based methods, have attracted a lot of attention in the field of bearing RUL prediction. However, it is worth noting that a sufficient amount of training data is required for these ML and DL-based methods, which is challenging and time-consuming in real-world industrial applications. In addition, the black-box nature of these methods cannot provide a satisfactory understanding of the underlying processes and even struggles to adhere to physics constraints. Thus, a rising investigation area aiming to enhance current data-driven models is developed via the integration of explicit prior knowledge in recent years, that is, PINN. In this paper, as depicted in Figure 6, the external knowledge is derived from reliability engineering, which employs the probabilistic relationship of the Weibull distribution (in particular, the Weibull cumulative distribution function (CDF)) to describe the failure procedure of the machinery (i.e., “bathtub” curve of a bearing), and it is integrated into an LSTM network through a Weibull-based loss function. According to reliability engineering, a bearing always fails following a “bathtub” curve illustrated in Figure 7:

(i) a higher probability of failure during the initial period (i.e., run-in stage), which can be attributed to manufacturing defects and installation errors.

(ii) a relatively low and constant probability of failure during the design life.

(iii) an obvious increasing probability of failure during the stage of wearout. Subsequently, the Weibull distribution can be implemented to characterize the above bearing’s failing probability, since it is intuitive and capable of modeling multiple failure types. Particularly, the Weibull CDF $F (t)$ at time t is deployed in this paper, which can be given as:

\begin{matrix} F (t) = 1 - e^{- {(\frac{t}{η})}^{β}} \end{matrix}

(12)

where $β$ is the shape parameter that reveals when a bearing always fails at the end of its useful life: during run-in stage $β < 1$ , during design life $β \approx 1$ , during the stage of wearout $β > 1$ ; $η$ is the characteristic life of a bearing, which means 63.2% of the units have failed. Generally, $β$ and $η$ should be determined via a lot of historical data and prudent analyses of failures, while the Weibayes method⁵² can be used to estimate $η$ via a reasonably defined $β$ in such a situation of insufficient failure history as:

η = {[\sum_{i = 1}^{N} \frac{T_{i}^{β}}{r}]}^{1 / β}

(13)

where $T_{i}$ is the test time for each sample, r is the number of failures, N is the total size of samples. In this paper, $β$ is set to be 2, which is a recommended value for ball bearing in the previous investigation.⁵³

Figure 6.

Flowchart of the proposed PINN.

Figure 7.

The “bathtub” curve of a bearing and corresponding Weibull CDF function.

After determining $β$ and $η$ , we develop a Weibull-based loss function, which consists of traditional mean squared error (mse) loss and the Weibull-loss function, to integrate knowledge into an LSTM network to construct the PINN model for bearing RUL forecasting. LSTM is an advanced RNN that can be utilized to predict time series data with diminished gradient exploding or vanishing problems. A typical LSTM cell architecture is shown in Figure 8, and its forward propagation is implemented through the following equations:

\begin{matrix} i_{t} = σ (x_{t} W_{xi} + h_{t - 1} W_{hi} + b_{i}) \end{matrix}

(14)

\begin{matrix} f_{t} = σ (x_{t} W_{xf} + h_{t - 1} W_{hf} + b_{f}) \end{matrix}

(15)

\begin{matrix} o_{t} = σ (x_{t} W_{xo} + h_{t - 1} W_{ho} + b_{o}) \end{matrix}

(16)

\begin{matrix} c_{t} = f_{t} \otimes c_{t - 1} + i_{t} \otimes \tanh (x_{t} W_{xc} + h_{t - 1} W_{hc} + b_{c}) \end{matrix}

(17)

\begin{matrix} h_{t} = o_{t} \otimes \tanh (c_{t}) \end{matrix}

(18)

where $x_{t}$ is the input at time t, $i_{t}$ is the input gate, $o_{t}$ is the output gate, $f_{t}$ is the forget gate, $h_{t}$ is the hidden state, $c_{t}$ is the cell status at time t, $σ (\cdot)$ is the active function, namely sigmoid, W terms denote weight matrices of the LSTM model (e.g., $W_{ih}$ is the weight matrix between input and hidden state), b terms are bias vectors (e.g., $b_{i}$ is the bias vector of the input gate), ⊗ denotes the element-by-element multiplication. The architecture of the LSTM model used in this paper is given in Table 1.

Figure 8.

A typical LSTM cell architecture.

Table 1.

The architecture of LSTM model.

No.	Layer	Parameters
1	Input	25 $\times$ 1
2	LSTM	unit = 128
3	Dropout	0.2
4	LSTM	unit = 128
5	Dense	unit = 100
6	Dropout	0.2
7	Dense	unit = 1
8	Output	1

LSTM: long short-term memory.

The training of the LSTM network is achieved via the minimization of a loss function, that is, the back-propagation process, and the Weibull-based loss function $L (t, \hat{t})$ used in this paper can be described as:

L (t, \hat{t}) = \underset{mse - loss}{\underset{︸}{\frac{1}{n} \sum_{i = 1}^{n} {(t_{i} - \hat{t_{i}})}^{2}}} + λ \underset{Weibull - loss}{\underset{︸}{\frac{1}{n} \sum_{i = 1}^{n} {(F (t_{i}) - F (\hat{t_{i}}))}^{2}}}

(19)

where n is the number of samples in the data set; $F (t_{i})$ is the true fraction failing time at time t, and it can be calculated via Equation (12), that is, CDF function; $F (\hat{t_{i}})$ is the estimated fraction failing at time t, and it is obtained through the output of the LSTM model after each training step.

Finally, the PINN model (i.e., LSTM model enhanced by the Weibull-based loss function) proposed in this paper can be trained via extracted IDEn-enabled HI and employed to forecast bearing RUL, which should outperform counterparts with traditional loss functions. Generally, several issues, including poor quality (e.g., mess with noise, or lower informational quality) and insufficient data, can affect the stability and convergence of the loss function in a DL model. The Weibull-loss function can apply a corrective force on the prediction model during the training process (because features extracted from poor-quality data that do not conform to the Weibull distribution will be dropped) to alleviate negative effects from poor-quality data. Moreover, this custom loss function’s interpretability can provide a possibility to counteract the nongeneralizable performance of the model that only depends on insufficient historical data.

Experimental setup

Test rig

In this paper, as depicted in Figure 9, a self-developed test rig is built to implement the run-to-failure testing on bearings, that is, no defects are initiated on the bearing to ensure the degraded bearing whose operating conditions are close to real cases. An asynchronous motor within the speed range (between 0 and 3000 rpm) is employed to drive the whole test rig. A compliant and rigid shaft coupling is utilized to connect the asynchronous motor and the testing part, which consists of two accompaniment pillow blocks with bearings (the left accompaniment pillow block has a pair of tapered rolling bearings, right accompaniment pillow block has a ball bearing) and the tested bearing (type 62052Z deep groove ball bearing, SKF, Sweden, and more detailed parameters are given in Table 2). The axial and radial forces (ranges are between 0 and 15,000 N) are applied to the tested bearing by two Electrical Hydrostatic Actuators (EHA) with force transducers, respectively, which can reduce the tested bearing’s life duration in the case of overload (i.e., more than the bearing’s maximum dynamic load). The rotating speed of the asynchronous motor and the loading force of the EHA are controlled by an electric control cabinet that equips corresponding programmable logic controllers (PLC). Via the developed test rig, we can obtain real experimental data (e.g., AE signals in this paper) to characterize the whole degradation procedure of ball bearings, and the detail is discussed in the next subsection.

Figure 9.

Photo of the test rig.

Table 2.

Parameters of SKF 62052Z deep groove ball bearing.

Parameter	Value
Inner diameter	25 mm
Outer diameter	52 mm
Roller number	9
Width	15 mm
Sealing	Shield on both sides
Lubricant	Grease

Measurement procedure

On the one hand, the temperature monitoring is implemented via an resistance temperature detector (RTD) platinum PT100 probe, which is placed inside the pillow block and close to the bearing’s external ring. On the other hand, the AE measurement on the test rig was conducted via a USB AE node (type 1283, MISTRAS, Princeton, NJ, USA) as well as a NANO-30 AE sensor, and the bandwidth of the AE node and the AE sensor is 1 kHz–1 MHz and 125 kHz–750 kHz, respectively. The installation location of the AE sensor on the test rig was illustrated in Figure 10, and the coupling between the sensor head and bearing housing was accomplished via a couplant (type high vacuum grease, Molykote, Midland, MI, USA). Moreover, according to the DIN EN 13477-2:2010, the Hsu-Nielson approach was utilized to confirm this coupling, that is, the AE responses caused by pencil-lead breaks on the bearing housing were observed. During the whole testing, a 40 dB threshold value was set to reduce the effect of noise, that is, signals with low power were neglected. Moreover, a 1000 Hz high-pass filter is employed to ensure that AE signals can be devoid of noise and interference from external sources. The AE acquisition was triggered when signals exceeded the preset threshold value, and it was pre-amplified and sampled with a rate of 1 MHz and a resolution of 18 bits. Moreover, the PDT, HDT, and HLT were set to 1, 2, and 20 ms, respectively. After triggering, an illustration of one hit is given in Figure 11. The above operation can ensure a high sample rate of AE signals with a feasible acquisition rate for the measuring system. In this paper, four run-to-failure tests under a certain working condition (rotating speed: 2000 rpm, radial force: 12,000 N) are conducted, and more details are discussed in the next section.

Figure 10.

Setup of AE measurement.

Figure 11.

Illustration of one hit (AE signal).

Results and discussion

Experimental results

Four run-to-failure tests (working conditions: the rotating speed is 2000 rpm, and the radial force is 12,000 N) are conducted on the proposed test rig, and the temperature monitoring is used to determine the time when the test is terminated (i.e., a bearing failure occurs when the temperature value exceeds a certain value, which is in this paper). In terms of AE measurement, the signal collection is implemented every 10 s, and each recorded AE signal has 7168 data points, as shown in Figure 11. Then, using the proposed IDEn, the lifetime procedures of four testing bearings are depicted in Figure 12, and the detailed lifetime information is summarized in Table 3. It can be seen that the degradation process is different for distinct bearings (e.g., different lifetimes), while all testing bearings represent a slowly monotonic tendency, which is an ideal case.

Figure 12.

IDEn computed on AE signals of four testing bearings.

Table 3.

Summary of experimental results.

No.	Train/Test	Name	Lifetime (×10 s)
1	Train	Bearing1	1658
2	Train	Bearing2	969
3	Test1	Bearing3	1216
4	Test2	Bearing4	741

Evaluation of features and HI construction

As discussed earlier, the extraction of features and HI construction are essential procedures for bearing RUL prediction. Therefore, in this subsection, three evaluation indices, including monotonicity indicator ( $Mon$ ), robustness indicator ( $Rob$ ), and trendability indicator ( $Tre$ ), are employed to estimate the performance of the extracted feature, that is, the IDEn. Since the degradation process of bearing is irreversible, a promising HI should have a monotonic (increasing or decreasing) trend, and the absolute difference between feature’s positive and negative derivatives determines the $Mon$ (its range is from 0 to 1, and a higher value represents better monotonicity), which can be expressed as:

\begin{matrix} \begin{matrix} Mon {(f_{k})}_{k = 1 : K} = \\ \frac{1}{K - 1} | No . of d / d f > 0 - No . of d / d f < 0 | \end{matrix} \end{matrix}

(20)

where ${(f_{k})}_{k = 1 : K}$ is the HI value at time $t_{k}$ , K is the total number of HI values in the sequence, $d / d f = f_{k + 1} - f_{k}$ means the difference of the HI, $No . of d / d f > 0$ and $No . of d / d f < 0$ denote the number of positive and negative differences, respectively.

Due to noises under varying working conditions and stochasticity of the degradation processes, the HI may show random fluctuations, which reduce the prediction stability. Therefore, a promising HI should be robust to interferences and present a smooth degradation process. $Rob$ (range is from 0 to 1) that denotes more fluctuations (i.e., prediction uncertainty) in a smaller score can be computed as:

\begin{matrix} Rob {(f_{k})}_{k = 1 : K} = \frac{1}{K} \sum_{k = 1}^{K} \exp (- | \frac{f_{k} - f_{k}^{T}}{f_{k}} |) \end{matrix}

(21)

where ${(f_{k})}^{T}$ is the average trend value of the HI at time $t_{k}$ , and it is always obtained by implementing the smoothing method on the original HI sequence.

Besides the monotonicity and robustness, we always expect the HI’s trend to correlate with the bearing’s lifetime, that is, trendability, which changes between −1 and 1. Specifically, $Tre = 1$ means that HI has a strong positive correlation with time, and HI has a strong negative correlation with time when $Tre = - 1$ . The expression of $Tre$ is given as:

\begin{matrix} \begin{matrix} Tre {(f_{k})}_{k = 1 : K} = \\ \frac{K (\sum_{k = 1}^{K} f_{k} t_{k}) - (\sum_{k = 1}^{K} f_{k}) (\sum_{k = 1}^{K} t_{k})}{\sqrt{AB}} \\ A = [K \sum_{k = 1}^{K} f_{k}^{2} - {(\sum_{k = 1}^{K} f_{k})}^{2}] \\ B = [K \sum_{k = 1}^{K} t_{k}^{2} - {(\sum_{k = 1}^{K} t_{k})}^{2}] \end{matrix} \end{matrix}

(22)

In this paper, to verify the efficacy of the proposed HI in this paper, we implement a comparison between the proposed HI and several typical time-domain features, including RMS, skewness (Skw), kurtosis (Kur), crest factor (Cf), and several typical entropy, including approximate entropy (ApEn), sample entropy (SpEn), and permutation entropy (PeEn). The comparison results are summarized in Table 4, and we can see that the proposed IDEn almost has the best performance, which demonstrates the efficacy of the proposed IDEn for HI construction.

Table 4.

Summary of comparison results between the proposed IDEn and other typical time-domain features.

Features	$Mon$	$Rob$	$Tre$	Computational complexity
IDEn	0.34	0.92	0.46	$O (n^{2})$
RMS	0.29	0.89	0.45	$O (n)$
Skw	0.02	0.42	0.03	$O (n)$
Kur	0.09	0.55	0.06	$O (n)$
Cf	0.24	0.83	0.42	$O (n)$
ApEn	0.19	0.67	0.25	$O (n^{2})$
SpEn	0.30	0.85	0.44	$O (n^{2})$
PeEn	0.30	0.88	0.46	$O (n^{2})$

IDEn: improved diffusion entropy; Skw: skewness; Kur: kurtosis; Cf: crest factor; ApEn: approximate entropy; SpEn: sample entropy; PeEn: permutation entropy. The bold values indicate that IDEn is the proposed feature, which has better performance than other features.

Finally, we construct the HI based on the proposed IDEn, which can be expressed as:

\begin{matrix} HI = 1 - Normalization (0, 1) [IDEn] \end{matrix}

(23)

The above equation can construct the HI via the proposed IDEn in a decreasing trend, which will represent better consistency of the labeling function discussed in the next subsection.

RUL prediction performance

After constructing HI, we train the PINN model via the training dataset (Bearing1 and Bearing2) to forecast bearing RUL, and the prediction performance is verified through two testing datasets (i.e., Bearing3 and Bearing4). First, the sliding time window method is used to construct the sample data of the training and testing dataset, which should satisfy the input requirements of the PINN (specifically, LSTM) model. In terms of the input array for the LSTM model, the dimension size is $N_{sample} \times N_{feature}$ , where $N_{sample}$ and $N_{feature}$ are the numbers of sampling points and features, respectively. Based on the trial-and-error, the length of the sliding time window $N_{length}$ in this paper is set to be 25 to construct the input sample, which corresponds to the LSTM’s architecture given in Table 1. Moreover, four different labeling functions, including the linear function, Piecewise function, Quadratic function, and exponential function, are employed to determine the best sample label (the output of labeling functions at the time of the $N_{length}$ sampling point is set as the label). Subsequently, the sliding window moves backward along the sampling time axis with one unit to construct a series of input samples, and it is worth noting that this overlap between adjacent samples can make the LSTM model fit the training data more fully. The illustration of four labeling functions is depicted in Figure 13 (take Bearing1 as an example), and corresponding expressions are given as:

\begin{matrix} Linear function : f_{L} (t) = 1 - (\frac{t_{k}}{t_{K}}) \end{matrix}

(24)

\begin{matrix} \begin{matrix} Piecewise function : \\ f_{P} (t) = {\begin{matrix} 1, t_{k} \leq t_{init} \\ (\frac{t_{k}}{t_{init} - t_{K}}) + (\frac{t_{K}}{t_{K} - t_{init}}), t_{k} > t_{init} \end{matrix} \end{matrix} \end{matrix}

(25)

\begin{matrix} Quadratic function : f_{Q} (t) = 1 - (\frac{t_{k}^{2}}{t_{K}^{2}}) \end{matrix}

(26)

\begin{matrix} Exponential function : f_{E} (t) = d - \exp (t_{k} τ + α) \end{matrix}

(27)

Figure 13.

Illustration of labeling functions.

where $t_{K}$ is the whole lifetime of the bearing, $t_{init}$ is the initial degradation time of the bearing, $α$ is the convergence rate, which is set to be −20 based on the experience, d and $τ$ are two hyperparameters, which can be obtained via the following equation:

\begin{matrix} \begin{matrix} {\begin{matrix} f_{E} (t_{\min}) = 1 \\ f_{E} (t_{\max}) = 0 \end{matrix} \end{matrix} \end{matrix}

(28)

The architecture of the LSTM model is given in Table 1, and the model is trained under a Python (version 3.6.3) environment with the library of PyTorch (version 2.1.6). The hardware condition is a workstation that contains an Intel I9-13900K processor, a memory of 128 GB, and one Nvidia GeForce RTX 4090 Graphics processing unit (GPU). To reveal the effectiveness of key hyperparameters (which are learning rate and batch size in this paper) on the model’s performance, we conduct a sensitivity analysis via grid search, and details are given in Table 5. Using the loss as an estimation index, we depict the sensitivity analysis results in Figure 14 to determine the optimized hyperparameters (by selecting the minimum loss value), which would also guide further tuning for different scenarios in future research.

Table 5.

Summary of hyperparameters in the sensitive analysis.

Hyperparameter	Value 1	Value 2	Value 3
Learning rate	0.01	0.001	0.0001
Batch size	16	32	64

Figure 14.

Sensitivity analysis results among different hyperparameters.

According to the sensitive analysis, the batch size is set to 64, and the initial learning date of $1 \times 10^{- 3}$ (Adam optimizer) is employed, and the training epoch is set to 200. Subsequently, we illustrate the loss values over epochs of the training procedure for Test 1 (i.e., Bearing3) in Figure 15, and it can be seen that loss values are converged to a small value after 100 epochs, which demonstrates that the physical knowledge (i.e., the reliability engineering and failure procedure described via the Weibull distribution) is well integrated during the training process. The RUL prediction results of Test 1 with labels of linear function and exponential function are compared in Figure 16, and we can see that the prediction performance with the label of exponential function is better than the counterpart with the label of linear function, especially at the initial stage. Moreover, using the label of linear function, the predicted values at the initial stage are lower than real values, and the trend fluctuates and tends to be horizontal (only a slightly downward tendency). The reason can be attributed to linearly declining labels, which cannot make the trained model predict RUL accurately, since the bearing is still in a healthy condition (i.e., actual performance does not change) at the initial stage.

Figure 15.

Learning curves of Test 1.

Figure 16.

RUL prediction results of Test 1 with labels of linear function and exponential function.

To quantitatively evaluate the prediction performance of the proposed method, the root mean squared error (RMSE), which is a commonly used metric for prediction, is used in this paper:

\begin{matrix} RMSE = \sqrt{\frac{1}{n_{sample}} \sum_{i = 1}^{n_{sample}} {(p_{i} - \hat{p_{i}})}^{2}} \end{matrix}

(29)

where $n_{sample}$ is the number of samples, $p_{i}$ and $\hat{p_{i}}$ are predicted and actual values, respectively.

The prediction results of Test 1 and Test 2 under four labeling functions are depicted in Figure 17, and we can see that the predicted values using the exponential labeling function are the smallest, compared to other labeling functions. This phenomenon can be attributed to the reason that the degradation process behaves with a slow monotonic tendency (i.e., a gentle curve in the early stage, and an abrupt change at a later period), which aligns better with the exponential labeling function than the other three functions. Moreover, to demonstrate the superiority of the PINN (i.e., the embedded physical knowledge of reliability engineering and failure procedure described via the Weibull distribution), a traditional LSTM model is also trained to forecast bearing RUL. The constructed HI, training and testing datasets, architecture of LSTM, and training hyperparameters are all set the same as the proposed PINN model, while the only difference is the loss function, that is, no Weibull-loss function is employed. The comparison results are given in Table 6, and we can see that the proposed PINN model outperforms the traditional LSTM model, which verifies the benefits of incorporating physical knowledge.

Figure 17.

Comparison results among different labeling functions.

Table 6.

Comparison of prediction performance between PINN and LSTM.

Model	Test 1 RMSE	Test 2 RMSE
PINN	0.08	0.06
LSTM	0.22	0.19

RMSE: root mean squared error; PINN: physics-informed neural network; LSTM: long short-term memory. PINN is the proposed method in this paper, which has better performance than its counterpart, i.e., LSTM that has no integrated knowledge.

Advantages analysis

Finally, to further improve the advantages of the proposed method in this paper, we implement a comparative study between the proposed method (IDEn-enabled HI and PINN) and other existing methods in the literature. Elforjani^37,38 extracted RMS and SIE from AE signals to construct HI and implement bearing RUL prediction by training an ANN model, which has three hidden layers and one output layer. Moreover, the sigmoid function and resilient back-propagation algorithm are used to improve the prediction results. Motahari-Nezhad and Jafari⁴¹ computed a series of time-domain metrics of AE signals and selected prominent features via the correlation-based feature selection approach. They found that the Jordan feedback neural network with the Bayesian Regularization training algorithm is capable of forecasting bearing RUL with the least error. In addition, Motahari-Nezhad and Jafari⁴⁰ also determined significant frequency-domain features from AE signals and trained a radial basis function (RBF) neural network to realize bearing RUL prediction with promising performance. The results between the proposed method and the existing methods mentioned above are compared to confirm the superiority of the proposed method. In addition, to better position the proposed method within the existing body of research, two traditional prognostic techniques belonging to the statistical approach, that is, the AR model and the PF model, are employed as counterparts. IDEn works as a feature to construct the HI curve, and the prediction models for bearing forecasting are developed via the AR and the PF model, respectively. The final results of the comparison are illustrated in Figure 18, and we can observe that the proposed method has the lowest value of the RMSE, that is, the highest prediction accuracy between the predicted and actual RUL. In other words, the proposed method in this article outperforms current methods, which can provide good guidance for future work.

Figure 18.

Comparison results between the proposed method and other existing methods.

Conclusion

Recently, data-driven methods, particularly DL-based models, have been employed for RUL prediction of bearings. However, these methods cannot incorporate physical knowledge that can improve interpretability and generalization for multiple application scenarios. Therefore, in this paper, we propose a novel bearing RUL prediction method based on AE signals and a PINN model. Experimental tests and corresponding results verify the effectiveness and superiority of the proposed method. Some conclusions are given as follows.

(i) Experimental results indicate that the proposed method can forecast bearing RUL accurately, and this excellent prediction performance has good potential in practical applications.

(ii) Compared to current time-domain features, the proposed IDEn in this paper can characterize AE signals and construct HI for RUL prediction with better performance, including monotonicity, robustness, and trendability. Thus, we may expect further application of IDEn for signal processing in the future.

(iii) Four different labeling functions are compared in this paper, and the results suggest that the exponential function has the best capacity to represent the degradation process of bearing, which provides an important reference for future investigations.

(iv) The physical knowledge, that is, reliability engineering and failure procedure described via the Weibull distribution, is incorporated into the LSTM model to construct a PINN by developing a Weibull-loss function. According to experimental results, this PINN model outperforms the LSTM model that utilizes traditional loss function only, that is, the integrated physical knowledge provides significant benefits. The integration method of knowledge in this paper is easy to implement, and more work should be encouraged to improve bearing RUL prediction performance in the future.

Some limitations and the resulting future work in this paper should also be taken seriously, for example, (1) the computation cost of feature extraction should be reduced by proposing more convenient methods or using other high-performance languages such as Rust to speed up the calculation of IDEn, (2) the instability caused by selecting improper shape parameters in the Weibull function, which may be rectified by changing model architectures and hyperparameters. In addition, only stable working conditions are considered in this paper, further investigations should focus on variations of operating conditions, which are more common and practical in real industry. It is worth noting that TL has demonstrated immense potential in solving RUL prediction problems under varying working conditions. Thus, we can expect that the proposed method might be adapted to handle such variations via the TL strategies. For example, after constructing HI curves via IDEn, we can train a new LSTM prediction model that has better generalization ability using the adversarial domain adaptation technique to ensure RUL forecasting performance under varying working conditions. Finally, in this paper, only complete and clear AE signals are considered; further investigations should be conducted to handle cases where AE signals are missing or corrupted, for example, using generative adversarial nets (GAN) to fill in deficient data.

Footnotes

Acknowledgements

Some of the PINN codes in this paper were based on those by Tim von Hahn and Chris K Mechefske from Queen’s University ().

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: This research was partially supported by the Natural Science Foundation of China (Grant Number: 52205061), the Hong Kong Scholar Program (Grant Number: XJ2022017), and the Chinese University of Hong Kong (Project ID: 3110174). We greatly appreciate their financial support.

ORCID iD

Furui Wang

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