A Nonlinear Degradation Model Based Method for Remaining Useful Life Prediction of Rolling Element Bearings Yaguo Lei*, Naipeng Li, Feng Jia, Jing Lin, Saibo Xing
State Key Laboratory for Manufacturing Systems Engineering Xi'an Jiaotong University No. 28 Xianning West Road 710049, Xi'an, China *Corresponding author:
[email protected] (Yaguo Lei)
Abstract-This paper proposes a nonlinear degradation model based method for remaining useful life
(RUL)
prediction of
rolling element bearings. First, a new nonlinear degradation model is constructed which considers four variable sources of stochastic degradation processes of bearings simultaneously, i.e., the
temporal
variability,
the
unit-to-unit
variability,
the
measurement variability and the nonlinear variability. Then a Kalman particle filtering (KPF) algorithm is applied to estimate the state and predict the
RUL
of bearings. The effectiveness of
the nonlinear degradation model based method is demonstrated using
simulated
degradation
processes
and
accelerated
degradation tests of rolling element bearings. The results show that the nonlinear model performs better than the linear model in describing the degradation processes of bearings, and KPF is more effective in the state estimation and
RUL
prediction of
bearings than the Kalman filtering and the particle filtering algorithms.
Keywords-remaining degradation
useful
life
prediction;
model; Kalman particle filtering;
nonlinear
rolling
element
bearings
I.
INTRODUCTION
Prognostics and health management (PHM) of rolling element bearings has attracted substantial attention recently due to its importance for reducing unscheduled risks and increasing the reliability of machinery. PHM generally includes two key processes, i.e., diagnostics and prognostics [1]. The task of diagnostics is to detect fault presence in bearings as early as possible and identify types, locations and degrees of faults. Nowadays, with the development of professional instruments for conducting fault detection of bearings, more and more sophisticated diagnostic methods have been proposed. However, diagnostics can only be conducted after fault occurrence, which is unable to prevent the happening of catastrophes especially for significant machinery. In order to reduce unscheduled risks and maintenance costs, prognostics should be conducted during the health management process. Prognostics are to forecast the future performance of bearings using prediction methods and obtain the available time before bearings lose their operation ability. Accurate RUL prediction of bearings allows for preventative maintenance and
replacement, thus reducing costly unscheduled maintenance. Consequently, RUL prediction of rolling element bearings has attracted more and more attention in recent years [2]. The commonly used RUL prediction methods for mechanical systems are model-based methods, which attempt to set up empirical or physical models to describe degradation processes of machinery, and update model parameters according to real-time information from measurements. Rolling element bearings generally present stochastic degradation processes. Therefore, it is natural to model the degradation process of a bearing as a stochastic process. So far, different kinds of stochastic process models have been developed and applied to predict the RUL of mechanical systems, such as Gamma processes [3], inverse Gaussian processes [4], random coefficient regression models [5] and Wiener processes [6]. The uncertainty of stochastic degradation processes of rolling element bearings are mainly caused by the following four variable sources: 1) the temporal variability, 2) the unit-to unit variability, 3) the measurement variability and 4) the nonlinear variability. The temporal variability is referred to as the inherent uncertainty of the degradation process over time, which is usually represented by random noises of the stochastic process [7]. The unit-to-unit variability is caused by the diversity of the health states and operation conditions among different units. In a stochastic process, the unit-to-unit variability should be described by introducing random variances in specified model parameters [8]. The measurement variability means the difference between the measurements and the actual health states of bearings, which is usually described by measurement noises in the stochastic process [9]. The nonlinear variability should be explained as follows. With the development of the fault severity, the degradation speed of bearings changes because of the variation of the health states, which is always presented as a gradually accelerated degradation process. Therefore, a nonlinear degradation model should be constructed to describe the accelerated degradation process. An appropriate degradation model should consider the above four variable sources simultaneously in describing the stochastic degradation processes of rolling element bearings.
This research is supported by National Natural Science Foundation of China (51222503,51475355),Provincial NaturalScience Foundation RP0035 research project ofShaanxi (2013JQ7011) and Fundamental Research Funds (or the Central Universities (2012jdgzO1).
2015 Prognostics and System Health Management Conference-Beijing (2015 PHM-Beijing)
However, most degradation models published in literature only considered some of the four variable sources. Gebraeel et al. established an exponential degradation model incorporating the temporal variability and the unit-to-unit variability [10]. Peng et a1. proposed a linear degradation model considering the temporal variability, unit-to-unit variability and measurement variability simultaneously [8]. Ye et a1. also considered the same three variable sources and constructed a degradation model based on Wiener processes [9]. In contrast, little research has been done on RUL prediction using degradation models which consider the four variable sources. Based on the work of the former researchers, this paper proposes a new nonlinear degradation model which considers the four variable sources simultaneously. Constructing an appropriate degradation model is not enough for predicting the RUL of rolling element bearings accurately. Another major task is to estimate model parameters and health states adaptively according to real-time information from measurements. Kalman filtering (KF) is one of the commonly used state estimation algorithms and has been applied to the parameter and state estimation in RUL prediction. Wang et al. used KF to estimate a drift parameter of a linear degradation model [6]. Si et al. utilized KF to estimate model parameters and health states of a linear degradation model simultaneously [11]. KF is applicable for a linear Gaussian system. However, most mechanical systems are nonlinear and non-Gaussian. Therefore, particle filtering (PF) is introduced into the RUL prediction of mechanical systems because of its superiority in the state estimation for nonlinear and/or non Gaussian systems [12]. Orchard et a1. proposed a PF-based prognostic method and applied this method to the RUL prediction of planetary gearboxes [13]. Chen et a1. constructed a degradation model based on neuro-fuzzy systems and estimated the system state using high-order PF [14]. Zio et al. proposed a PF-based prognostic method for RUL prediction of mechanical components subject to fatigue crack growth [15]. A common problem in PF is the particle degeneracy, i.e., after a few iterations, all but one particle have negligible weights [12]. The particle degeneracy is able to be restricted by sampling particles from an optimal importance density function. However, the optimal importance density function is impossible to be acquired in real application. In order to restrict the particle degeneracy, a Kalman particle filtering (KPF) algorithm is applied in this paper which uses KF to construct an approximated optimal importance density function for the particle sampling of PF. Based on the above analysis, this paper proposes a nonlinear model based method for RUL prediction of rolling element bearings. First, a novel nonlinear degradation model is constructed to describe degradation processes of rolling element bearings, which considers four variable sources of stochastic degradation processes simultaneously, i.e., the temporal variability, the unit-to-unit variability, the measurement variability and the nonlinear variability. Second, a KPF algorithm is applied to the RUL prediction of bearings for restricting the particle degeneracy of PF. The rest of the paper is organized as follows. Section II describes the nonlinear degradation model based method for RUL prediction of rolling element bearings. In Section III, simulations of degradation
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processes are utilized to evaluate the proposed method. In Section IV, the demonstrated using data of accelerated rolling element bearings. Conclusions are II. A. tk
performance of the proposed method is degradation tests on drawn in Section V.
NONLINEAR DEGRADATION MODEL BASED METHOD
Development of the Nonlinear Degradation Model In the nonlinear degradation model, the state of a system at is described as follows: (1)
where Xo is the initial state of the system and is set to be zero for simplification. rocess
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Figure 2. State estimation and RUL predicton results ofthe linear degradation process: (a) state estimation results and (b) RUL prediction results.
To quantitatively compare the performance of the four methods, a commonly used performance measure Cumulative Relative Accuracy (CRA) [20] is calculated and displayed in
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The simulated nonlinear process is also analyzed using the same methods applied in the linear one. The state estimation and RUL prediction results using Ml, M2, M3 and M4 are shown in Fig. 4. From the state estimation results, it is hard to compare the performance of the four methods. While from the RUL prediction results, it is seen that the linear model based method Ml is unable to predict the RUL effectively. Other three methods have much better performance than Ml. To further compare the RUL prediction accuracy of the four methods, the CRA values of their results are calculated and displayed in Table III. It is seen that the CRA value of Ml is much smaller than other methods. Among the three nonlinear model based methods, M4 performs a little better than others, which means that KPF is more effective in state estimation and RUL prediction than KF and PF.
RP0035
2015 Prognostics andSystem Health Management Conference-Beijing (2015 PHM-Beijing)
-
-
-
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degradation trends of the bearings. What's more, this stage is more meaningful for the RUL prediction than the normal operation stage, because we want to know more about the health states of the bearings when they are closer to the fmal failure. Considering the above reasons, the RUL of the bearings is just predicted during the degradation stage, and the first predicting time (FPT) of the bearings is show in Fig. 6.
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Fig. 5 shows the vibration signals of two tested bearings during the whole lifetime. It is seen that the amplitudes of the vibration signals show rapid increases at the end of the lifetime, which implies that the bearings have abrupt degradation processes. The root mean square (RMS) of the vibration signals is extracted as the health indicator of the tested bearings as is shown in Fig. 6. It is shown that RMS contains two distinct different stages, i.e., the normal operation stage and the failure stage. During the normal operation stage, the RMS values present little variation. It is difficult to predict the degradation trends of the bearings based on the information of this stage. On the contrary, the degradation stage is informative for
978-1-4673-8554-1/15/$31.00 ©2015 IEEE
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Introduction to the Tests and Degradation Data
The tests are conducted on an experimental system named PRONOSTIA [21] which is designed to test methods for bearing fault detection, diagnosis and RUL prediction. In order to conduct accelerated degradation tests of bearings during a few hours, a radial force which is equal to the bearings' maximum dynamic load 4 kN, is applied on the tested bearings. The rotating speed keeps 1,800 rpm. Accelerometers are fixed on the outer race of the bearings and vibration signals are captured with a sampling frequency of 25.6 kHz. Each sample contains 2,560 data points, and the sampling is repeated every 10 s. The tested bearings are naturally degraded during the tests without seeding a fault in advance.
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(5)
Vibration signals of the tested bearings: (a) bearing 1 and (b) bearing 2.
1.5
In order to demonstrate the effectiveness of the proposed method in RUL prediction of rolling element bearings, vibration signals of the whole lifetime collected from the accelerated degradation tests of rolling element bearings are applied to verify the benefits of the proposed method. A.
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Figure 4. State estimation and RUL predicton results of the nonlinear degradation process: (a) state estimation results and (b) RUL prediction results.
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B. RUL Prediction of the Tested Beariangs The RMS values of the tested bearings during the degradation stage are input into the nonlinear model proposed in this paper, and the model parameters are initialized using the MLE algorithm. Then, the RUL is predicted during the degradation stage. To compare the four different prediction methods, i.e., Ml, M2, M3 and M4 named in Section III, all of them are applied to the RUL prediction of the tested bearings. The expectations of the RUL predicted using the four methods for the two tested bearings are displayed in Fig. 7and Fig. 8, respectively. It is seen from the RUL prediction results that Ml, which is based on the linear model, presents the worst results among
RP0035
2015 Prognostics and System Health Management Conference-Beijing (2015 PHM-Beijing)
these methods. In order to further compare these methods quantitatively, the CRA values of them are calculated and displayed in Table IV. The results show that the nonlinear model based methods M2, M3 and M4 perfonn better than the linear model based method Ml especially for bearing 2. What's more, among the three nonlinear model based methods, M4 has the largest CRA value. The above results demonstrate that the nonlinear model perfonns better than the linear model in the RUL prediction of the bearings. In addition, KPF is able to further improve the prediction accuracy of the nonlinear model compared with KF and PF. 500,,��--�------------------, 400 300 200
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