Prognostics health management (PHM) of rotating machinery has become an important process for increasing reliability and reducing machine malfunctions in industry. Bearings are one of the most important equipment parts and are also one of the most common failure points. To assess the degradation of a machine, this paper presents a bearing remaining useful life (RUL) prediction method. The method relies on a novel health indicator and a linear degradation model to predict bearing RUL. The health indicator is extracted by using Hilbert–Huang entropy to process horizontal vibration signals obtained from bearings. We present a linear degradation model to estimate RUL using this health indicator. In the training phase, the degradation detection threshold and the failure threshold of this model are estimated by the distribution of 600 bootstrapped samples. These bootstrapped samples are taken from the six training sets. In the test phase, the health indicator and the model are used to estimate the bearing’s current health state and predict its RUL. This method is suitable for the degradation of bearings. The experimental results show that this method can effectively monitor bearing degradation and predict its RUL.
Ball bearings are one of the most widely used universal parts in various pieces of mechanical equipment. The operational failure of ball bearings is a major reason for the failure of mechanical equipment. Compared with that of other mechanical parts, the service lives of bearings originating from the same batch can vary greatly in terms of the fault occurrence time, fault type, and severity. In this way, some bearings operate normally over the entire design life, and some do not reach the end of the design life and experience early failure. Therefore, the normal operation of a ball bearing cannot be ensured by timely maintenance based on only the design life. When a bearing fails, it causes noise and reduces the accuracy of the work, causing severe vibration and even damage to the equipment, which stops production and results in workplace hazards. Therefore, condition monitoring and fault diagnosis must be carried out during bearing operations in order to change the traditional timing for predictive maintenance. In this way, we can identify bearing faults and replace bearings prior to the occurrence of failure. Using realtime monitoring, the replacement time can be forecasted so that the bearing can be utilized to its maximum extent and that unexpected equipment issues resulting from degraded bearings are avoided, no mechanical equipment is destroyed, and personal safety is protected. Therefore, research on bearing fault diagnosis and residual life predictions is of great practical significance [
Technology addressing the remaining useful life prediction of a bearing is known as fault prediction and health management (PHM). PHM consists of two aspects: fault prediction and health management. Fault prediction mainly includes monitoring of the health status of equipment, predicting the remaining life, or predicting the future state. Health management relates to making full use of the forecasted information to make decisions related to the safety and reliability of equipment and to prolong equipment life. In recent years, PHM technology has gradually matured to practical field applications. Usually, for composite objects, it is difficult to establish an accurate failure physical model to predict the RUL. Therefore, the method driving the data is based on a large number of historical failure points that are used to predict the RUL. Datadriven prediction methods mainly use pattern recognition and machine learning techniques combined with changes in the equipment operating conditions to achieve forecasting. Traditional datadriven methods for nonlinear systems mainly include regressionbased models [
When a ball bearing fails, the vibration signal has a tendency to exhibit strong nonstationary and nonlinear characteristics. Therefore, obtaining fault characteristic information from these nonstationary and nonlinear signals is essential for ball bearing fault diagnosis. Using the Hilbert–Huang transform (HHT), vibration signals can be adaptively decomposed to provide local and timedomain signal information [
In this study, horizontal vibration acceleration signals from ball bearings are utilized to extract the health indicator by Hilbert–Huang entropy. This indicator is the input to the linear degradation model. If the indicator reaches the degradation detection threshold, its RUL is predicted using this model.
The ball bearingaccelerated life test data used throughout this paper are derived from the IEEE PHM 2012 Data Challenge. Under normal circumstances, the RUL of a bearing is within the range of tens of thousands of hours, but such a long running time is impossible to achieve in a laboratory environment. Therefore, an accelerated failure is achieved by imposing an additional load on the bearing or increasing the rotational speed on the PRONOSTIA platform. The purpose of the PRONOSTIA testbed is to test and verify the bearing fault detection and fault prediction models. Using this platform, a bearing can be rapidly invalidated within a few hours via an accelerated life test so that the bearing failure data can be acquired. PRONOSTIA provides bearing degradation data under various operating conditions. The experimental data in this paper are horizontal acceleration data from bearings from the IEEE PHM 2012 Data Challenge dataset [
The RUL of a machine is the expected life or usage time remaining before the machine requires repair or replacement. Predicting RUL from system data is a central goal of predictivemaintenance algorithms. To estimate the RUL of a system, a model that can perform an estimation based on the time evolution or statistical properties of condition indicator values needs to be developed.
Predictions from such models are statistical estimates with the associated uncertainty. They provide a probability distribution of the RUL of the test machine. Developing a model for RUL prediction is the next step in the algorithmdesign process after identifying promising condition indicators. Because the developed model uses the time evolution of condition indicator values to predict RUL, this step is often iterative with the step of identifying condition indicators.
There are three families of RUL estimation models: similarity model, degenerate model, and survival model.
A similarity model is based on a historical database that combines the RUL prediction of a test machine with the behavior of a known machine. Such a model compares the trend of test data or conditional indicator values with the same information extracted from other similar systems.
Many scholars have proposed various similarity models for RUL. A similaritybased prognostic approach is developed for estimating the RUL of a valve asset using fullscope highfidelity simulators to generate the runtofailure data [
Using a similar model requires running fault data similar to systems (components). “Running to the fault” data are the data that begin during normal operation and end when the machine is in a nearfailure or maintenance state. “Run to the failure” data show similar degradation behavior. In other words, as the system degrades, the data change in a particular way. IEEE PHM 2012 data support the entire sixtrainingrun failed dataset, which does not exhibit very similar degradation behavior. It is not easy to choose an appropriate model based on similarity.
This model infers the behavior of the past to predict future conditions. This type of RUL calculates the condition indicator for a degenerate indicator that fits the linear or exponential model, giving a degenerate contour in your ensemble. It then uses the degraded contour of the test component to count the remaining time until the indicator reaches a specified threshold. These models are most useful when there is a status indicator of a known value indicating failure. The two available degradation model types are the linear degenerate model and the exponential degenerate model.
The linear degenerate model describes the degenerate behavior as a linear stochastic process with an offset. Linear degenerate models are useful when the system does not experience cumulative degradation. A nonparametric estimator is proposed for percentiles of the timetofailure distribution obtained from a linear degradation model using the kernel density method in [
The exponential degenerate model describes the degenerate behavior as an exponential stochastic process with an offset. The exponential degradation model is useful when the test component undergoes cumulative degradation. This model has been presented in many literatures with specific applications. The variation of the ONstate resistance is identified as the failure precursor, and an exponential degradation model that fits successfully with the experimental data is developed [
The linear and exponential degradation models are intuitive and clear, and the method of parameter calculation is easy. These models are therefore preferred.
The survival model is a statistical method for statistical time event data. This is useful when the user does not have a complete history of running failures. The survival model is mainly used in the food and finance industry and is not applied much in the machinery industry. A survival model is introduced to contemplate two simultaneous accelerating factors affecting a food product’s shelf life: temperature and illumination [
The modelbased method presented in this paper is shown in Figure
Flow chart of the proposed method.
The construction of health indicators is the key to predicting the remaining service life. Bearings are the most common mechanical part in rotating machinery, and its health indicators have attracted much attention in practice. Given that, in many cases, it is difficult to measure and quantify the health indicators of a bearing, many vibrationbased methods have been proposed to construct a bearing health indicator. Although many bearing health indicators have been proposed for a constant operating environment in the past few years, most bearing health indicators do not have a theoretical lower limit, so theoretical baselines do not exist. More theoretical studies should be conducted. In addition, performance indicators based on monotonicity, variance, trend, predictability, early fault detection, and calculation time should be proposed [
Ball bearing fault diagnosis signals include vibration, temperature, chemical analysis of the lubricant, and sound intensity. Vibration detection is the most effective method for bearing fault diagnosis. There are three analysis methods used for vibration detection.
Timedomain analysis is the first detection method in the vibration detection method, and it determines if the mechanical equipment has failed by calculating timedomain statistical characteristic parameters (mean value, variance, kurtosis, etc.) of the vibration signal. For the mechanical equipment vibration signal calculated and analyzed only in the time domain, it can only determine if the overall operational status of the equipment is normal and cannot determine the severity of a fault and predict any RUL [
Frequencydomain analysis is based on Fourier analysis to observe the operation state of the device by examining the frequency of faultbased characteristics over the spectrum diagram and the corresponding amplitude of the related frequencies. Commonly used frequencydomain vibration signal detection methods include fast Fourier transforms, power spectra, and filtering. However, each of these methods has flaws and loses some characteristics of nonlinear vibrations, meaning that the monitoring frequency band has a decisive impact on the analysis results [
When a ball bearing fails, its vibration signal often has both nonstationary and nonlinear characteristics. The time and frequencydomain analyses can only describe the signal as a whole and not the local characteristics of a signal for a certain time or a specified frequency range. Timefrequency analyses can be used to analyze the time and frequency of signals simultaneously. The timefrequency analysis method is the most effective method for the analysis of nonstationary and nonlinear vibration signals associated with ball bearings. The timefrequency analysis method can accurately describe the time and frequency characteristics of fault signals and has more advantages. Conventional timefrequency analysis methods include shorttime Fourier transform (STFT), Wigner–Ville distribution (WVD), wavelet transform (WT), and EMD [
In this study, a novel Hilbert–Huang entropy as a new bearing health indicator is addressed. In a ball bearing vibration test, the timedomain signal describes the ball bearing vibration amplitude with time. The vibration acceleration signal is collected at a time scale of 0.1 s using PRONOSTIA. Therefore, the bearing health indicator is also conducted in time slices on signal slice
Hilbert–Huang entropy.
Hilbert–Huang transform is a timefrequency method widely used in speech recognition and seismic signal analysis. The Hilbert–Huang transform consists of empirical mode decomposition (EMD) and Hilbert transform. EMD can decompose the signal into a small amount of intrinsic mode function (IMF) components, which is very adaptive and efficient. The decomposition method is suitable for nonlinear and nonstationary signal analysis and timefrequency energy representation because the base of the expansion is adaptive. The method has a powerful ability to reveal the real physical significance of data checking because of its complete empirical characteristics [
The marginal Hilbert spectrum was calculated as follows:
To assess the health status of the bearings, a feature representing the degraded condition of the bearing needs to be identified. This section employs a new feature, known as Hilbert–Huang entropy (HHE), which is measured by the Shannon entropy based on the horizontal and vertical acceleration spectra [
The HHE of the horizontal acceleration over six training sets is presented in Figure
HHE in training sets. (a) Training set 1. (b) Training set 2. (c) Training set 3. (d) Training set 4. (e) Training set 5. (f) Training set 6.
For comparing the health indicator of the standard model with the HHE of the training set 1, we need to transform the HHE into a standard format (Figure
Comparison of HHE and the standard health indicator. (a) The health indicator of the standard model. (b) HHE in the training set 1.
We now consider signalamplitude transformations. Amplitude transformations follow the rules as follows:
The new form of Hilbert–Huang entropy in all training sets can be seen in Figure
New form of Hilbert–Huang entropy. (a) Training set 1. (b) Training set 2. (c) Training set 3. (d) Training set 4. (e) Training set 5. (f) Training set 6.
Envelope analysis is one of the most successful and widely used methods in early fault diagnosis of rolling bearings. These technologies have been comprehensively described and analyzed in the field of rotating machinery, especially in the diagnosis of bearings. In this study, we use the peak envelopes and use spline interpolation with notaknot conditions over local maxima separated by at least
Cubic spline interpolation smooths curves without adding irregularities to the signal, and therefore, it was selected as an appropriate technique to apply to the data. Assuming that there are
Let
The fault of the machine is the life expectancy or time remaining before the machine needs to be repaired or replaced. Predicting residual service life from system data is the core goal of the predictive maintenance algorithm. A model that fits the time evolution of a health indicator and predicts how long it will last before a health indicator crosses the threshold that indicates a failure threshold, and one that compares the time evolution of a health indicator from a system that runs to fault, can calculate the most likely failure time for the current system. In this study, we consider the following simple linear degradation model. The linear degradation model object implements the following continuoustime linear degradation model:
Based on the degradation paths described in Figure
Health indicator. (a) Training set 1. (b) Training set 2. (c) Training set 3. (d) Training set 4. (e) Training set 5. (f) Training set 6.
It is difficult to determine degradation detected threshold (DDT) and failure threshold (FT) because at failure, values of different machines generally have a large variation range. For datadriven methods, RUL is obtained when a health indicator exceeds a predefined DDT, and the machine should stop when a health indicator exceeds a predefined FT. However, the DDT and FT are usually determined experimentally because the health indicator values of different bearings at a failure time are generally different [
DDT and FT labeled. (a) Training set 1. (b) Training set 2. (c) Training set 3. (d) Training set 4. (e) Training set 5. (f) Training set 6.
Distribution and parameter estimation.
Symbol  Before bootstrap sampling  After bootstrap sampling 

DDT 


CI: [2.28619, 3.6382]  CI: [2.9561, 2.99284]  

 
CI: [0.402092, 1.57988]  CI: [0.216847, 0.242873]  


FT 


CI: [4.88223, 6.86253]  CI: [5.85236, 5.90777]  

 
CI: [0.588944, 2.31406]  CI: [0.327067, 0.366321] 
CI: confidence interval.
To verify the model prediction results in this paper, we used eleven test sets and determined the RUL of these test sets using the health indicator and linear degradation model. According to the 2012 IEEE PHM Data Challenge criterion, we computed the scores and mean errors for these eleven RULs using the model and its parameters. All the results are listed in Table
The accuracy of the RUL in different algorithms.
Error (%)  This paper (DDT = 2.97; FT = 5.88)  Tianyi Wang  Edwin Sutrisno 

B1_3  92.44  91.4  37.0 
B1_4  100.00  97.1  80.0 
B1_5  20.43  69.6  9.0 
B1_6  7.76  66.4  −5.0 
B1_7  82.29  93.5  −2.0 
B2_3  82.93  94.6  64.0 
B2_4  3.22  70.5  10.0 
B2_5  58.77  86.7  −440.0 
B2_6  5.63  68.2  49.0 
B2_7  −121.94  29.3  −317.0 
B3_3  −54.38  56.1  90.0 
Mean error  57.25  74.9  100.3 
Score  0.2992  0.0981  0.3066 
In Section
Scores of different FT values and DDT values in linear degradation model.
DDT  

2.95  2.96  2.97  2.98  2.99  3  
FT  5.85  0.2942  0.2943  0.2944  0.2945  0.2944  0.2945 
5.86  0.2958  0.2959  0.2959  0.2961  0.2960  0.2961  
5.87  0.2974  0.2975  0.2976  0.2977  0.2976  0.2977  
5.88  0.2990  0.2991  0.2992  0.2993  0.2992  0.2993  
5.89  0.3007  0.3008  0.3008  0.3009  0.3008  0.3009  
5.9  0.3023  0.3024  0.3024  0.3025  0.3025  0.3026  
5.91  0.3039  0.3040  0.3041  0.3042  0.3041  0.3042  
5.92  0.3056  0.3057  0.3057  0.3058  0.3058  0.3059  
5.93  0.3072  0.3074  0.3074  0.3075  0.3074  0.3075  
5.94  0.3089  0.3090  0.3091  0.3092  0.3091  0.3092  
5.95  0.3106  0.3107  0.3107  0.3109  0.3108  0.3109  
5.96  0.3123  0.3124  0.3124  0.3125  0.3125  0.3126  
5.97  0.3140  0.3141  0.3141  0.3142  0.3142  0.3143  
5.98  0.3157  0.3158  0.3158  0.3160  0.3159  0.3160 
The estimation of FT is based on the 6 training sets, and the optimal value and confidence interval of FT is obtained by bootstrap sampling. The experiment results show that when the FT value is greater than the upper limit of the confidence interval, the score of this algorithm can be improved. When FT ≥ 5.93, the scores of our method are higher than those of the two winners. In future research, we will analyze how many training sets are needed to get the optimal FT value.
In Table
Theoretically, the RUL can be predicted by the exponential degradation model. In our research, a comparison is conducted using the same values of DDT and FT in the linear degradation model and the exponential degradation model. We used the method in Section
Scores of exponential degradation model.
DDT  

2.95  2.96  2.97  2.98  2.99  3  
FT  5.85  0.2018  0.2011  0.1996  0.2026  0.2079  0.2087 
5.86  0.2009  0.2003  0.1988  0.2018  0.2083  0.2092  
5.87  0.2001  0.1995  0.198  0.201  0.2088  0.2097  
5.88  0.1994  0.1987  0.1973  0.2002  0.2093  0.2101  
5.89  0.1986  0.1979  0.1965  0.1994  0.2097  0.2106  
5.9  0.1978  0.1972  0.1958  0.1986  0.2102  0.2111 
In this study, a novel health indicator algorithm and a method to compute the degradation detected threshold and failure threshold of the linear degradation model for a bearing RUL prediction are developed. Horizontal vibration signals collected from accelerated degradation bearings tests are utilized to demonstrate the effectiveness of the method. This health indicator is processed by Hilbert–Huang entropy and amplitude transformations. The DDT and FT values are estimated by bootstrapped sampling and parameter estimation of these samples distribution. The RUL prediction model uses a linear degradation model. The results demonstrate that the proposed modelbased method’s performance is close to the best winners’ methods in the 2012 IEEE PHM Data Challenge. In the future, we will use this proposed method to test more open experimental datasets and to establish more robust wearout period detection and RUL prediction models, which will be able to describe anomalies in the degradation trends.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was partially supported by the Appropriative Researching Fund for Guangdong Provincial Key Laboratory of Precision Equipment and Manufacturing Technology under Grant PEM201604 and Bureau of Education of Guangzhou Municipality under Grant 2017192201.