- Email: [email protected]
Adaptive prognostics for rolling element bearing condition
Mechanical Systems and Signal Processing (1999) 13(1), 103–113 Article No. mssp.1998.0183, available online at http://www.idealibrary.com on
ADAPTIVE PROGNOSTICS FOR ROLLING ELEMENT BEARING CONDITION Y. L, S. B, C. Z, T. K, S. D S. L George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 -0405 , U.S.A.
(Received February 1998 , accepted after revisions June 1998) Rolling element bearing failure is one of the foremost causes of breakdown in rotating machinery. This paper proposes a remaining life adaptation methodology based on mechanistic modeling and parameter tuning. Vibration measurement is used to estimate defect severity by monitoring the signals generated from rotating bearings. Through a defect propagation model and defect diagnostic model, an adaptive algorithm is developed to fine tune the parameters involved in the propagation model by comparing predicted and measured defect sizes. In this manner, the instantaneous rate of defect propagation can be captured despite defect growth behavior variation. Therefore, a precise estimation of the remaining life can be determined. Simulations and experimental results are presented to illustrate the implementation principles and to verify the applicability of the adaptive prognostic methodology.
7 1999 Academic Press
1. INTRODUCTION AND MOTIVATION
Rolling element bearing failure is one of the foremost causes of breakdown in rotating machinery. Such bearing failure can be catastrophic in certain situations, such as in helicopters and in automatic processing machines. Reliable estimation of a bearing’s remaining utility presents the most challenging aspect in maintenance optimisation and catastrophic failure avoidance. Currently there are two basic methods for bearing maintenance: the statistical method of bearing life estimation, and bearing condition monitoring and diagnostics. Rolling contact fatigue is by and large probabilistic in nature. Lundberg and Palmgren [1] modeled bearing survival probability in terms of dynamic load rating and equivalent load. The method provided a statistical approach to predict a bearing’s fatigue life. However, the life of a bearing operating under a specific environment can differ significantly from another apparently identical unit. Therefore, bearing maintenance based on statistical techniques has many limitations. Indeed, it is not uncommon to replace a bearing that has longer remaining life by a new one with a shorter life. Unusual operating conditions can dramatically shorten a bearing’s life and lead to unexpected breakdown. Therefore, the alternative bearing maintenance method, bearing condition monitoring that employs techniques such as vibration analysis and acoustic emission analysis, has received considerable attention. Currently available methods of bearing health monitoring focus on the identification of faults that have taken place on a bearing. Generally speaking, they can be classified into either time domain or frequency domain methods. 0888–3270/99/010103 + 11 $30.00/0
7 1999 Academic Press
104
. .
Time domain methods are based on the statistically distinctive behaviors between good and defective bearings in the time domain such as showk pulse counting [2], root mean square (rms), peak value, crest factor and kurtosis [3–5]. Frequency domain methods are based on the fact that a localised defect generates periodic signals with unique characteristic defect frequency. The identification of the periodic signals indicates a certain type of defect. They include the signal averaging method [6] bicoherence analysis [7], cepstrum analysis [8], and the high-frequency response technique (HFRT) [9, 10]. A signal-averaging method is used to identify the periodic defective bearing signals through time domain averaging. Bicoherence analysis reveals the statistics dependence of the family of the harmonics of characteristic defect frequencies [7]. Cepstrum analysis involves taking the spectrum of a spectrum. It is sensitive to side bands. The high-frequency resonance technique (HFRT) takes advantage of the fact that most of the signal’s energy generated by a defect is concentrated in the high-frequency resonance range. HFRT can provide envelope signals with high signal-to-noise ratios which are associated with the periodicity of a defective bearing signal. An adaptive noise-cancellation method has also been developed to enhance the envelope spectrum obtained by HFRT [11]. Currently detectable bearing defects [11, 12] are much smaller than 6.25 mm2 (0.01 in2), which is commonly considered to be a fatal failure size by industry standards [13]. However, to determine just the existence of a bearing defect is insufficient for the purposes of condition-based maintenance and catastrophic failure avoidance. When a fatal defect is diagnosed, the machinery is often forced to shut down at inconvenient times with a tremendous loss of time, productivity and capital. Therefore, it is important to predict the growth rate of defects and the remaining life of bearings in a prognostic mode in addition to a diagnostic mode. With reliable prognostic capability, bearing maintenance and replacement can be scheduled at optimal times. The primary difficulty for effective implementation of bearing prognostics is the highly stochastic nature of defect growth. For example, the propagation of a rolling contact fatigue spall beyond its initial appearance is a highly variable process [13]. The variation to reach the final failure size from the point where a defect can be detected may be greater than its L10 life (the number of revolutions, or hours, at a specified constant speed that 90% of the bearing population will endure [14]). Deterministic models based on fracture or damage mechanics [15, 16] do not accurately describe the variable process of bearing defect propagation. Condition symptoms such as vibration and acoustic emission are closely related to wear processes in machinery [17]. Reliability analysis and prognostics can be achieved by using Weibull and Frechet symptom models [18]. Therefore it is possible to predict the breakdown time of machinery at the running stage based on vibration condition monitoring techniques. However, the techniques have not shown the capability to deal with the time-varying nature of bearing defect propagation. As a result, reliable bearing prognostic techniques have been poorly developed to date. This paper presents the formulation of a bearing prognostic methodology based on the in-process adaptation of defect propagation rate with vibration signal analysis. It utilises a deterministic defect propagation model and an adaptive algorithm to fine-tune the predicted rate of defect propagation in a real-time manner. The variable nature of defect propagation is addressed by a mechanistic model with time-varying parameters. The adaptive alteration of the model parameters offers the best prediction, in the least square error sense, of the bearing future state for any given diagnostic system. The following section discusses the theoretical basis of the prognostic system. It is followed by numerical simulations to illustrate the implementation principles and an experimental study to verify its effectiveness.
105
2. THEORETICAL BACKGROUND
2.1. Results of laboratory experiments and observations of in-service structures clearly indicate that fatigue crack growth is affected by a variety of factors (e.g. stress states, material properties, temperature, lubrication, and other environmental effects). The following empirical and deterministic fatigue crack propagation model based on Paris’s formula [18] has been widely accepted for many years: da = C0 (DK)n dN
(1)
where a is the instantaneous length of a dominant crack and N represents running cycles. The parameters C0 and n are regarded as material-dependent constants, and are related to factors such as material properties, environment, etc. The term DK represents the range of stress intensity factor over one loading cycle. Equation (1) states that crack growth rate in terms of length per running cycle is exponential function of stress intensity factor range DK. Based upon experimental observations, other factors such as stress ratio and threshold stress intensity factors have been proposed and added to equation (1) [18]. In the area of bearing defect analysis in industry, the defect severity of a bearing is represented by the surface area size, rather than the length, of the defect [13]. Thus, a deterministic bearing defect growth model can be given in a manner similar to Paris’s formula as follows: D =
dD = C0 (D)n dt
(2)
which states that the rate of defect growth is related to the instantaneous defect area D under a constant operating condition. The parameters C0 and n are material constants that need to be determined experimentally, and often vary with factors other than the instantaneous defect size. 2.2. An adaptive prognostic system is developed to estimate the size of defect and the rate of defect growth, as shown in Fig. 1. The system includes the deterministic defect propagation model, as specified by equation (2), that calculates the future bearing defect size at time t + D based on given bearing running condition and the defect size at the current time t. Note that the defect area size at the current time is often unavailable from direct measurements without interrupting the machinery operation; therefore, indirect and non-intrusive measurements (such as the monitoring of vibration, temperature, or acoustic emission) are commonly exercised to infer the defect area through signal processing and diagnostic models. The future bearing defect size as forecasted by the prognostic model is compared to the measurement-inferred bearing condition at time t + D. The comparison shows a certain amount of prediction error due to the fact that the stochastic nature of defect growth cannot be accurately described by constant parameters C0 and n. Therefore, an adaptive algorithm is employed herein to take advantage of the prediction error for the purpose of fine-tuning the model parameters. In this manner the propagation model is expected to improve its accuracy continuously in following the time-varying defect growth behavior. An adaptive algorithm based on the recursive least square principle is discussed in the following section.
106
. .
Figure 1. Adaptive prognostic methodology.
2.3. - () Through time domain integration, equation (2) can be written in the form of ln (D) = a + b ln (t + t0 )
(3)
where t0 = (C0 /(1 − n))D0n + 1 is the time when the smallest defect area D0 occurs, D0 is the smallest defect area that can be detected by a given diagnostic system, a = (1/(n − 1)) ln (C0 /(1 − n)), b = (1/(1 − n)), and t is bearing running time relative to t0 . There are three parameters, namely a, b, and t0 , that need to be estimated in the model. Since these parameters are time-varying in a defect propagation process, a recursive least square (RLS) algorithm with a forgetting factor is used to adaptively update the values of a, b, and t0 [19]. The RLS algorithm is given as the following: e(t) = Y(t) − Y (t, u (t − 1))
(4)
Figure 2. Simulated true (——) and estimated (with standard deviation of 0.4; . . . . ) defect areas at the continuous growth of defect area.
107
T 1 Initial estimation of defect propagation model parameters in the simulation of continuous and discontinuous growth of defect area Simulation of continuous growth of defect area
Simulation of discontinuous growth of defect area
Prediction no.
a
b
t0
a
b
t0
1 2 3 4
−4.2 −4.2 −4.2 −5.0
2.5 3.4 2.5 2.5
2.0 2.4 2.4 2.0
−3.5 −3.5 −3.5 −2.0
2.2 2.2 2.5 2.2
1.8 1.5 1.8 1.8
c(t) =
0
dY (t, u) du
P(t) = l−1 P(t − 1) −
b
(5)
u = u (t − 1)
P(t − 1)c(t)c T(t)P(t − 1) l + c T(t)P(t − 1)c(t)
u (t) = u (t − 1) + P(t)c(t)e(t)
1
(6) (7)
where the vector u(t) of unknown parameters is u(t) = [a b t0 ]T, e(t) is the prediction error, Y(t) = ln (D), Y (t) is the estimated value of Y(t) and P(t) is covariance matrix. Initial covariance matrix is chosen as a unit matrix scaled by a positive scalar that is typically in the region of 1–1000. The scalar reflects uncertainty of a system. Without prior knowledge of a system, large scalar should be selected. The forgetting factor l falls within the range of 0 Q l E 1. 3. NUMERICAL SIMULATION ANALYSIS
Numerical simulations are performed to examine the feasibility of the adaptive prognostic system. In the first set of simulations, a defect is assumed to grow continuously according to the deterministic propagation model of equation (2) with a = −4.5, b = 3.2
Figure 3. Prediction with adaptation with different initial parameter values at the continuous growth of defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
108
. .
Figure 4. True and estimated a at the continuous growth defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
and t0 = 1.6. The growth of defect area is compared to the result of a generic diagnostic model. Diagnostic methods in practical applications always rely upon the signal processing of in-direct measurements, therefore they are vulnerable to disturbances such as noise and physics model mismatch. In accounting for the effect of disturbances, the generic diagnostic model is patterned after equation (2) but with added corruption from a normally distributed noise component of zero mean and 0.4 standard deviation. Figure 2 shows both the true and diagnosed defect areas in this case. To perform adaptive prognostics, initial values of the model parameters a, b, and t0 are assigned (Table 1). The forgetting factor l used in the simulations is 0.999. Figure 3 shows the predictions based on equations (3)–(6) in coupling with the noise-corrupted diagnostic model. It is seen that the difference of initial parameter values does not lead to significantly different prediction results. Figures 4–6 show the tuning of parameter values by the adaptation algorithm. The parameters are seen to converge to values different than the true ones. Since the adaptation algorithm, equations (4)–(7), only forces the minimisation of error in the prediction of defect size, D, it does not necessarily guarantee that the parameter values will converge to their true values. Defect area predictions based on constant prognostic model parameters without adaptation are shown in Fig. 7. At the lack of parameter fine-tuning routine, small amount
Figure 5. True and estimated b at the continuous growth defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
109
Figure 6. True and estimated t0 at the continuous growth defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
of parameter difference results in large prediction errors with the increase of bearing running cycles. Typically, a bearing spall defect grows when subsurface cracks propagate to the surface and material is broken away. Energy collections are needed for such a process. Therefore, a spall defect grows discontinuously and its spall area increases abruptly whenever materials are broken away from a running surface. The second set of simulations is to test the response of the adaptive diagnostics to this discontinuous growing process. Figure 8 presents the defect area growth in comparison to diagnostic model predictions corrupted by normally distributed noise of zero mean and 0.5 standard deviation. Predictions with adaptation and without adaptation at different initial parameter values listed in Table 1 are shown in Figs 9 and 10, respectively. It is evident that the adaptive prognostic model quickly tracks the defect propagation trend and makes accurate predictions. 4. EXPERIMENTAL INVESTIGATIONS
An experimental set-up is used to perform bearing life test in evaluating the effectiveness of the adaptive prognostic methodology. The set-up consists of a test housing, a hydraulic
Figure 7. Prediction without adaptation with different initial parameter values at the continuous growth of defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
110
. .
Figure 8. Simulated true (——) and estimated (with standard deviation of 0.5; . . . . ) defect areas at the discontinuous growth of defect area.
load applicator, a shaft drive, accelerometers and a data acquisition system (Fig. 11). The test housing has a 127-mm (5-in) bore and a built-in radial load cylinder. The house accommodates four bearings with the leftmost one being the test bearing. Timken LM50130 cup and LM501349 cone bearings were used in the study. The rated radial load for this type of bearings is 1491 kg (3314 lb) for a 90 million cycle life rating. The radial load is provided by a Power Team P59 hydraulic hand pump that provides pressure to the load cylinder on the housing. The shaft is driven by a d.c. servomotor with a speed controller. A Kistler tri-axial accelerometer is attached to the housing to acquire vibration signals. Measured signals are low-pass filtered at 10 kHz for anti-aliasing before sampling. The sampling rate in the experiments is 30 000 points/s. In order to accelerate a defect propagation process, an initial defect is artificially generated on the cup raceway by an electrical discharge machine. The defect is a crack oriented along the bearing axial direction to simulate a real-time fatigue crack with width of 300 mm. In order to simulate the propagation of a natural defect, the prognostic scheme is not exercised until the bearing has run for 20 million cycles. At that point the maximum width of defect increases to 1000 mm and a natural spall defect shape is generated. Experiments were performed at a shaft speed of 1600 rpm, a preload of 585 kg (1300 lb), and a radial load of 2485 kg (5522 lb) which is about 167% of the rated radial load. The
Figure 9. Predictions with adaptation at the discontinuous growth of defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
111
Figure 10. Predictions without adaptation at the discontinuous growth of defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
defect is positioned in the loading zone. The system is lubricated by thin spindle oil with viscosity of 54–60 SSU at 37.7°C (100°F). The experimental precedure began by recording the accelerometer measurements during the running of a defected bearing. The running was interrupted about every 10 h, the defective bearing removed from the test set-up, and the defect size measured with a Hommelwerke T8000 profilometer. Then the bearing was re-assembled into the set-up to repeat the experimental procedure. The growth of bearing defect as measured by the profilometer is shown by the cross-marks in Fig. 12. In the study, the diagnostic model estimated the defect size based on the RMS level of the accelerometer signal in the radial direction over the frequency band of 3000–5000 Hz where vibrations generated by other mechanical systems are absent. This frequency range is where most of vibratory energy due to bearing defect is concentrated. It is found that rms of band-passed signals around the band is highly correlated to defect area size. The defect areas are modeled to be related linearly to the rms of acceleration signal by D = 3.28 + 4.56R (mm2) where R is the rms value and of which the standard deviation is 0.52.
Figure 11. Life test housing schematic.
(8)
112
. .
Figure 12. Predicted defect areas by the adaptive algorithm and measured defect areas by the profilometer with respect to running cycle numbers. +, Measurement; w, prediction 1; *, prediction 2; r, prediction 3; e, prediction 4, ×, prediction 5.
With various initial values of the model parameters, the bearing defect areas as forecasted by the adaptive prognostic model are shown in Fig. 12. Note that the prognostic model utilised only the diagnostic model, equation (8), for adaptation purpose while remaining ignorant of the profilometer measurements. This situation emulates practical applications in which bearing operations cannot be interrupted for physical defect inspection. The results in Fig. 12 imply that the adaptive prognostic system can effectively predict the bearing defect propagation process. The fact that the prediction accuracy is not strongly affected by the choice of initial parameter values further suggests that the prognostic system can perform well without a priori knowledge of the model parameters. This aspect is particularly important for real life applications since a priori and precise knowledge of the fracture mechanics model is usually unavailable. The major prediction error sources are attributed to the uncertainty in the defect propagation process as well as in the diagnostic model based on accelerometer signals. Since the prognostic system cannot rely upon accurate measurement of the defect size in practical application cases, the reliability of the diagnostic model can be a critical factor to the overall performance of the prognostic system. 5. CONCLUSIONS
This paper presents an adaptive prognostic methodology for the prediction of bearing defect growth and remaining life. The defect size as predicted by a fatigue crack propagation model is compared to the estimation from a diagnostic model in the future to fine-tune the propagation model parameters. This methodology accounts for the time-variant nature of defect growth while providing the best prediction possible, in the least square error sense, for any given diagnostic system. Numerical simulations revealed that in both continuous and discontinuous defect growth cases, the adaptive prognostic methodology outperforms the deterministic models without adaptation. Experimental study with vibration measurement in bearing life testing was also performed to evaluate the capability of the proposed methodology. It was seen that the adaptive prognostics effectively predicted the bearing defect propagation process without a priori knowledge of the prognostic model parameters. In addition, the reliability of the diagnostic model is suggested to be a critical factor to the overall performance of the adaptive prognostic system.
113
ACKNOWLEDGEMENTS
This work was funded by the Office of Naval Research under research grant number N00014-95-10539, entitled ‘Integrated Diagnostics’. The Timken Company and The Torrington company have also provided support. Any opinions, findings, conclusions or recommendations are those of the authors and do not necessarily reflect the views of the Office of Naval Research, The Timken Company, or The Torrington Company. REFERENCES 1. G. L and A. P 1952 Acta Polytechnica 96 Mechanical Engineering Series 2. Dynamic capacity of rolling bearings. 2. G. O. G and T. T 1962 ASLE Transactions 5, 197–209. Detection of damage of assembled rolling element bearings. 3. D. D and R. M. S 1978 Transactions of the ASME Journal of Mechanical Design 100, 229–235. Detection of rolling element bearing damage by statistical vibration analysis. 4. R. J. A and J. M 1985 The Institution of Engineers, Australia, Mechanical Engineering Transactions 10, 102–107. Time domain methods for monitoring the condition of rolling element bearings. 5. H. R. M and F. H 1995 Applied Acoustics 44, 67–77. Application of statistical moments to bearing failure detection. 6. S. B and B. D 1979 Transactions of the ASME Journal of Mechanical Design 101, 118–125. Analysis of roller/ball bearing vibrations. 7. J. C. L, J. M and B. H 1995 Transactions of the ASME Journal of Engineering for Industry 117, 625–629. Bearing localized defect detection by bicoherence analysis of vibrations. 8. N. T 1994 Measurement 12, 285–289. A comparison of some vibration parameters for the condition monitoring of rolling element bearings. 9. P. D. MF and J. D. S 1984 International Journal of Tribology 17, 1–18. Vibration monitoring of rolling element bearings by the high frequency resonance technique—a review. 10. Y.-T. S and S.-J. L 1992 Journal of Sound and Vibration 155, 75–84. On initial fault detection of a tapered rolling bearing: frequency domain analysis. 11. Y. L, J. S, S. D, T. K and S. Y. L 1997 21st Biennial Conference on Reliability, Stress Analysis and Failure Prevention (RSAFP) , 763–772. Bearing fault detection via high frequency resonance technique and adaptive line enhancer. 12. J. S, Y. L, S. L, T. K and S. D 1997 Mechanical Systems and Signal Processing 11, 693–705. Bearing condition diagnostics via multiple sensors. 13. M. R. H 1992 Transactions of the ASME Journal of Tribology 114, 328–333. Rolling element bearing fatigue damage propagation. 14. T. A. H 1991 Rolling Bearing Analysis, 3rd edn. New York: John Wiley. 15. Y. M, M. K and H. Y 1985 ASLE Transactions 28, 60–68. Analysis of surface crack propagation in lubricated rolling contact. 16. L. M. K and M. D. B 1983 Transactions of the ASME Journal of Tribology 105, 198–205. A pitting model for rolling contact fatigue. 17. C. C 1985 Wear 105, 297–305. Teh tribovibroacoustic model of machines. 18. C. C, H. G. N and M. T 1997 Mechanical Systems and Signal Processing 11, 107–117. A passive diagnostic experiment with ergodic properties. 18. V. Z. P and E. M. M 1985 Mechanics of Elastic-plastic Fracture. Hemisphere Publishing Corporation. 19. G. C. G and K. S. S 1984 Adaptive Filtering Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall.
ADAPTIVE PROGNOSTICS FOR ROLLING ELEMENT BEARING CONDITION Y. L, S. B, C. Z, T. K, S. D S. L George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 -0405 , U.S.A.
(Received February 1998 , accepted after revisions June 1998) Rolling element bearing failure is one of the foremost causes of breakdown in rotating machinery. This paper proposes a remaining life adaptation methodology based on mechanistic modeling and parameter tuning. Vibration measurement is used to estimate defect severity by monitoring the signals generated from rotating bearings. Through a defect propagation model and defect diagnostic model, an adaptive algorithm is developed to fine tune the parameters involved in the propagation model by comparing predicted and measured defect sizes. In this manner, the instantaneous rate of defect propagation can be captured despite defect growth behavior variation. Therefore, a precise estimation of the remaining life can be determined. Simulations and experimental results are presented to illustrate the implementation principles and to verify the applicability of the adaptive prognostic methodology.
7 1999 Academic Press
1. INTRODUCTION AND MOTIVATION
Rolling element bearing failure is one of the foremost causes of breakdown in rotating machinery. Such bearing failure can be catastrophic in certain situations, such as in helicopters and in automatic processing machines. Reliable estimation of a bearing’s remaining utility presents the most challenging aspect in maintenance optimisation and catastrophic failure avoidance. Currently there are two basic methods for bearing maintenance: the statistical method of bearing life estimation, and bearing condition monitoring and diagnostics. Rolling contact fatigue is by and large probabilistic in nature. Lundberg and Palmgren [1] modeled bearing survival probability in terms of dynamic load rating and equivalent load. The method provided a statistical approach to predict a bearing’s fatigue life. However, the life of a bearing operating under a specific environment can differ significantly from another apparently identical unit. Therefore, bearing maintenance based on statistical techniques has many limitations. Indeed, it is not uncommon to replace a bearing that has longer remaining life by a new one with a shorter life. Unusual operating conditions can dramatically shorten a bearing’s life and lead to unexpected breakdown. Therefore, the alternative bearing maintenance method, bearing condition monitoring that employs techniques such as vibration analysis and acoustic emission analysis, has received considerable attention. Currently available methods of bearing health monitoring focus on the identification of faults that have taken place on a bearing. Generally speaking, they can be classified into either time domain or frequency domain methods. 0888–3270/99/010103 + 11 $30.00/0
7 1999 Academic Press
104
. .
Time domain methods are based on the statistically distinctive behaviors between good and defective bearings in the time domain such as showk pulse counting [2], root mean square (rms), peak value, crest factor and kurtosis [3–5]. Frequency domain methods are based on the fact that a localised defect generates periodic signals with unique characteristic defect frequency. The identification of the periodic signals indicates a certain type of defect. They include the signal averaging method [6] bicoherence analysis [7], cepstrum analysis [8], and the high-frequency response technique (HFRT) [9, 10]. A signal-averaging method is used to identify the periodic defective bearing signals through time domain averaging. Bicoherence analysis reveals the statistics dependence of the family of the harmonics of characteristic defect frequencies [7]. Cepstrum analysis involves taking the spectrum of a spectrum. It is sensitive to side bands. The high-frequency resonance technique (HFRT) takes advantage of the fact that most of the signal’s energy generated by a defect is concentrated in the high-frequency resonance range. HFRT can provide envelope signals with high signal-to-noise ratios which are associated with the periodicity of a defective bearing signal. An adaptive noise-cancellation method has also been developed to enhance the envelope spectrum obtained by HFRT [11]. Currently detectable bearing defects [11, 12] are much smaller than 6.25 mm2 (0.01 in2), which is commonly considered to be a fatal failure size by industry standards [13]. However, to determine just the existence of a bearing defect is insufficient for the purposes of condition-based maintenance and catastrophic failure avoidance. When a fatal defect is diagnosed, the machinery is often forced to shut down at inconvenient times with a tremendous loss of time, productivity and capital. Therefore, it is important to predict the growth rate of defects and the remaining life of bearings in a prognostic mode in addition to a diagnostic mode. With reliable prognostic capability, bearing maintenance and replacement can be scheduled at optimal times. The primary difficulty for effective implementation of bearing prognostics is the highly stochastic nature of defect growth. For example, the propagation of a rolling contact fatigue spall beyond its initial appearance is a highly variable process [13]. The variation to reach the final failure size from the point where a defect can be detected may be greater than its L10 life (the number of revolutions, or hours, at a specified constant speed that 90% of the bearing population will endure [14]). Deterministic models based on fracture or damage mechanics [15, 16] do not accurately describe the variable process of bearing defect propagation. Condition symptoms such as vibration and acoustic emission are closely related to wear processes in machinery [17]. Reliability analysis and prognostics can be achieved by using Weibull and Frechet symptom models [18]. Therefore it is possible to predict the breakdown time of machinery at the running stage based on vibration condition monitoring techniques. However, the techniques have not shown the capability to deal with the time-varying nature of bearing defect propagation. As a result, reliable bearing prognostic techniques have been poorly developed to date. This paper presents the formulation of a bearing prognostic methodology based on the in-process adaptation of defect propagation rate with vibration signal analysis. It utilises a deterministic defect propagation model and an adaptive algorithm to fine-tune the predicted rate of defect propagation in a real-time manner. The variable nature of defect propagation is addressed by a mechanistic model with time-varying parameters. The adaptive alteration of the model parameters offers the best prediction, in the least square error sense, of the bearing future state for any given diagnostic system. The following section discusses the theoretical basis of the prognostic system. It is followed by numerical simulations to illustrate the implementation principles and an experimental study to verify its effectiveness.
105
2. THEORETICAL BACKGROUND
2.1. Results of laboratory experiments and observations of in-service structures clearly indicate that fatigue crack growth is affected by a variety of factors (e.g. stress states, material properties, temperature, lubrication, and other environmental effects). The following empirical and deterministic fatigue crack propagation model based on Paris’s formula [18] has been widely accepted for many years: da = C0 (DK)n dN
(1)
where a is the instantaneous length of a dominant crack and N represents running cycles. The parameters C0 and n are regarded as material-dependent constants, and are related to factors such as material properties, environment, etc. The term DK represents the range of stress intensity factor over one loading cycle. Equation (1) states that crack growth rate in terms of length per running cycle is exponential function of stress intensity factor range DK. Based upon experimental observations, other factors such as stress ratio and threshold stress intensity factors have been proposed and added to equation (1) [18]. In the area of bearing defect analysis in industry, the defect severity of a bearing is represented by the surface area size, rather than the length, of the defect [13]. Thus, a deterministic bearing defect growth model can be given in a manner similar to Paris’s formula as follows: D =
dD = C0 (D)n dt
(2)
which states that the rate of defect growth is related to the instantaneous defect area D under a constant operating condition. The parameters C0 and n are material constants that need to be determined experimentally, and often vary with factors other than the instantaneous defect size. 2.2. An adaptive prognostic system is developed to estimate the size of defect and the rate of defect growth, as shown in Fig. 1. The system includes the deterministic defect propagation model, as specified by equation (2), that calculates the future bearing defect size at time t + D based on given bearing running condition and the defect size at the current time t. Note that the defect area size at the current time is often unavailable from direct measurements without interrupting the machinery operation; therefore, indirect and non-intrusive measurements (such as the monitoring of vibration, temperature, or acoustic emission) are commonly exercised to infer the defect area through signal processing and diagnostic models. The future bearing defect size as forecasted by the prognostic model is compared to the measurement-inferred bearing condition at time t + D. The comparison shows a certain amount of prediction error due to the fact that the stochastic nature of defect growth cannot be accurately described by constant parameters C0 and n. Therefore, an adaptive algorithm is employed herein to take advantage of the prediction error for the purpose of fine-tuning the model parameters. In this manner the propagation model is expected to improve its accuracy continuously in following the time-varying defect growth behavior. An adaptive algorithm based on the recursive least square principle is discussed in the following section.
106
. .
Figure 1. Adaptive prognostic methodology.
2.3. - () Through time domain integration, equation (2) can be written in the form of ln (D) = a + b ln (t + t0 )
(3)
where t0 = (C0 /(1 − n))D0n + 1 is the time when the smallest defect area D0 occurs, D0 is the smallest defect area that can be detected by a given diagnostic system, a = (1/(n − 1)) ln (C0 /(1 − n)), b = (1/(1 − n)), and t is bearing running time relative to t0 . There are three parameters, namely a, b, and t0 , that need to be estimated in the model. Since these parameters are time-varying in a defect propagation process, a recursive least square (RLS) algorithm with a forgetting factor is used to adaptively update the values of a, b, and t0 [19]. The RLS algorithm is given as the following: e(t) = Y(t) − Y (t, u (t − 1))
(4)
Figure 2. Simulated true (——) and estimated (with standard deviation of 0.4; . . . . ) defect areas at the continuous growth of defect area.
107
T 1 Initial estimation of defect propagation model parameters in the simulation of continuous and discontinuous growth of defect area Simulation of continuous growth of defect area
Simulation of discontinuous growth of defect area
Prediction no.
a
b
t0
a
b
t0
1 2 3 4
−4.2 −4.2 −4.2 −5.0
2.5 3.4 2.5 2.5
2.0 2.4 2.4 2.0
−3.5 −3.5 −3.5 −2.0
2.2 2.2 2.5 2.2
1.8 1.5 1.8 1.8
c(t) =
0
dY (t, u) du
P(t) = l−1 P(t − 1) −
b
(5)
u = u (t − 1)
P(t − 1)c(t)c T(t)P(t − 1) l + c T(t)P(t − 1)c(t)
u (t) = u (t − 1) + P(t)c(t)e(t)
1
(6) (7)
where the vector u(t) of unknown parameters is u(t) = [a b t0 ]T, e(t) is the prediction error, Y(t) = ln (D), Y (t) is the estimated value of Y(t) and P(t) is covariance matrix. Initial covariance matrix is chosen as a unit matrix scaled by a positive scalar that is typically in the region of 1–1000. The scalar reflects uncertainty of a system. Without prior knowledge of a system, large scalar should be selected. The forgetting factor l falls within the range of 0 Q l E 1. 3. NUMERICAL SIMULATION ANALYSIS
Numerical simulations are performed to examine the feasibility of the adaptive prognostic system. In the first set of simulations, a defect is assumed to grow continuously according to the deterministic propagation model of equation (2) with a = −4.5, b = 3.2
Figure 3. Prediction with adaptation with different initial parameter values at the continuous growth of defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
108
. .
Figure 4. True and estimated a at the continuous growth defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
and t0 = 1.6. The growth of defect area is compared to the result of a generic diagnostic model. Diagnostic methods in practical applications always rely upon the signal processing of in-direct measurements, therefore they are vulnerable to disturbances such as noise and physics model mismatch. In accounting for the effect of disturbances, the generic diagnostic model is patterned after equation (2) but with added corruption from a normally distributed noise component of zero mean and 0.4 standard deviation. Figure 2 shows both the true and diagnosed defect areas in this case. To perform adaptive prognostics, initial values of the model parameters a, b, and t0 are assigned (Table 1). The forgetting factor l used in the simulations is 0.999. Figure 3 shows the predictions based on equations (3)–(6) in coupling with the noise-corrupted diagnostic model. It is seen that the difference of initial parameter values does not lead to significantly different prediction results. Figures 4–6 show the tuning of parameter values by the adaptation algorithm. The parameters are seen to converge to values different than the true ones. Since the adaptation algorithm, equations (4)–(7), only forces the minimisation of error in the prediction of defect size, D, it does not necessarily guarantee that the parameter values will converge to their true values. Defect area predictions based on constant prognostic model parameters without adaptation are shown in Fig. 7. At the lack of parameter fine-tuning routine, small amount
Figure 5. True and estimated b at the continuous growth defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
109
Figure 6. True and estimated t0 at the continuous growth defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
of parameter difference results in large prediction errors with the increase of bearing running cycles. Typically, a bearing spall defect grows when subsurface cracks propagate to the surface and material is broken away. Energy collections are needed for such a process. Therefore, a spall defect grows discontinuously and its spall area increases abruptly whenever materials are broken away from a running surface. The second set of simulations is to test the response of the adaptive diagnostics to this discontinuous growing process. Figure 8 presents the defect area growth in comparison to diagnostic model predictions corrupted by normally distributed noise of zero mean and 0.5 standard deviation. Predictions with adaptation and without adaptation at different initial parameter values listed in Table 1 are shown in Figs 9 and 10, respectively. It is evident that the adaptive prognostic model quickly tracks the defect propagation trend and makes accurate predictions. 4. EXPERIMENTAL INVESTIGATIONS
An experimental set-up is used to perform bearing life test in evaluating the effectiveness of the adaptive prognostic methodology. The set-up consists of a test housing, a hydraulic
Figure 7. Prediction without adaptation with different initial parameter values at the continuous growth of defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
110
. .
Figure 8. Simulated true (——) and estimated (with standard deviation of 0.5; . . . . ) defect areas at the discontinuous growth of defect area.
load applicator, a shaft drive, accelerometers and a data acquisition system (Fig. 11). The test housing has a 127-mm (5-in) bore and a built-in radial load cylinder. The house accommodates four bearings with the leftmost one being the test bearing. Timken LM50130 cup and LM501349 cone bearings were used in the study. The rated radial load for this type of bearings is 1491 kg (3314 lb) for a 90 million cycle life rating. The radial load is provided by a Power Team P59 hydraulic hand pump that provides pressure to the load cylinder on the housing. The shaft is driven by a d.c. servomotor with a speed controller. A Kistler tri-axial accelerometer is attached to the housing to acquire vibration signals. Measured signals are low-pass filtered at 10 kHz for anti-aliasing before sampling. The sampling rate in the experiments is 30 000 points/s. In order to accelerate a defect propagation process, an initial defect is artificially generated on the cup raceway by an electrical discharge machine. The defect is a crack oriented along the bearing axial direction to simulate a real-time fatigue crack with width of 300 mm. In order to simulate the propagation of a natural defect, the prognostic scheme is not exercised until the bearing has run for 20 million cycles. At that point the maximum width of defect increases to 1000 mm and a natural spall defect shape is generated. Experiments were performed at a shaft speed of 1600 rpm, a preload of 585 kg (1300 lb), and a radial load of 2485 kg (5522 lb) which is about 167% of the rated radial load. The
Figure 9. Predictions with adaptation at the discontinuous growth of defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
111
Figure 10. Predictions without adaptation at the discontinuous growth of defect area (simulated test). ——, Real value; . . . . , prediction 1; –––, prediction 2; –·–·, prediction 3; –-–-, prediction 4.
defect is positioned in the loading zone. The system is lubricated by thin spindle oil with viscosity of 54–60 SSU at 37.7°C (100°F). The experimental precedure began by recording the accelerometer measurements during the running of a defected bearing. The running was interrupted about every 10 h, the defective bearing removed from the test set-up, and the defect size measured with a Hommelwerke T8000 profilometer. Then the bearing was re-assembled into the set-up to repeat the experimental procedure. The growth of bearing defect as measured by the profilometer is shown by the cross-marks in Fig. 12. In the study, the diagnostic model estimated the defect size based on the RMS level of the accelerometer signal in the radial direction over the frequency band of 3000–5000 Hz where vibrations generated by other mechanical systems are absent. This frequency range is where most of vibratory energy due to bearing defect is concentrated. It is found that rms of band-passed signals around the band is highly correlated to defect area size. The defect areas are modeled to be related linearly to the rms of acceleration signal by D = 3.28 + 4.56R (mm2) where R is the rms value and of which the standard deviation is 0.52.
Figure 11. Life test housing schematic.
(8)
112
. .
Figure 12. Predicted defect areas by the adaptive algorithm and measured defect areas by the profilometer with respect to running cycle numbers. +, Measurement; w, prediction 1; *, prediction 2; r, prediction 3; e, prediction 4, ×, prediction 5.
With various initial values of the model parameters, the bearing defect areas as forecasted by the adaptive prognostic model are shown in Fig. 12. Note that the prognostic model utilised only the diagnostic model, equation (8), for adaptation purpose while remaining ignorant of the profilometer measurements. This situation emulates practical applications in which bearing operations cannot be interrupted for physical defect inspection. The results in Fig. 12 imply that the adaptive prognostic system can effectively predict the bearing defect propagation process. The fact that the prediction accuracy is not strongly affected by the choice of initial parameter values further suggests that the prognostic system can perform well without a priori knowledge of the model parameters. This aspect is particularly important for real life applications since a priori and precise knowledge of the fracture mechanics model is usually unavailable. The major prediction error sources are attributed to the uncertainty in the defect propagation process as well as in the diagnostic model based on accelerometer signals. Since the prognostic system cannot rely upon accurate measurement of the defect size in practical application cases, the reliability of the diagnostic model can be a critical factor to the overall performance of the prognostic system. 5. CONCLUSIONS
This paper presents an adaptive prognostic methodology for the prediction of bearing defect growth and remaining life. The defect size as predicted by a fatigue crack propagation model is compared to the estimation from a diagnostic model in the future to fine-tune the propagation model parameters. This methodology accounts for the time-variant nature of defect growth while providing the best prediction possible, in the least square error sense, for any given diagnostic system. Numerical simulations revealed that in both continuous and discontinuous defect growth cases, the adaptive prognostic methodology outperforms the deterministic models without adaptation. Experimental study with vibration measurement in bearing life testing was also performed to evaluate the capability of the proposed methodology. It was seen that the adaptive prognostics effectively predicted the bearing defect propagation process without a priori knowledge of the prognostic model parameters. In addition, the reliability of the diagnostic model is suggested to be a critical factor to the overall performance of the adaptive prognostic system.
113
ACKNOWLEDGEMENTS
This work was funded by the Office of Naval Research under research grant number N00014-95-10539, entitled ‘Integrated Diagnostics’. The Timken Company and The Torrington company have also provided support. Any opinions, findings, conclusions or recommendations are those of the authors and do not necessarily reflect the views of the Office of Naval Research, The Timken Company, or The Torrington Company. REFERENCES 1. G. L and A. P 1952 Acta Polytechnica 96 Mechanical Engineering Series 2. Dynamic capacity of rolling bearings. 2. G. O. G and T. T 1962 ASLE Transactions 5, 197–209. Detection of damage of assembled rolling element bearings. 3. D. D and R. M. S 1978 Transactions of the ASME Journal of Mechanical Design 100, 229–235. Detection of rolling element bearing damage by statistical vibration analysis. 4. R. J. A and J. M 1985 The Institution of Engineers, Australia, Mechanical Engineering Transactions 10, 102–107. Time domain methods for monitoring the condition of rolling element bearings. 5. H. R. M and F. H 1995 Applied Acoustics 44, 67–77. Application of statistical moments to bearing failure detection. 6. S. B and B. D 1979 Transactions of the ASME Journal of Mechanical Design 101, 118–125. Analysis of roller/ball bearing vibrations. 7. J. C. L, J. M and B. H 1995 Transactions of the ASME Journal of Engineering for Industry 117, 625–629. Bearing localized defect detection by bicoherence analysis of vibrations. 8. N. T 1994 Measurement 12, 285–289. A comparison of some vibration parameters for the condition monitoring of rolling element bearings. 9. P. D. MF and J. D. S 1984 International Journal of Tribology 17, 1–18. Vibration monitoring of rolling element bearings by the high frequency resonance technique—a review. 10. Y.-T. S and S.-J. L 1992 Journal of Sound and Vibration 155, 75–84. On initial fault detection of a tapered rolling bearing: frequency domain analysis. 11. Y. L, J. S, S. D, T. K and S. Y. L 1997 21st Biennial Conference on Reliability, Stress Analysis and Failure Prevention (RSAFP) , 763–772. Bearing fault detection via high frequency resonance technique and adaptive line enhancer. 12. J. S, Y. L, S. L, T. K and S. D 1997 Mechanical Systems and Signal Processing 11, 693–705. Bearing condition diagnostics via multiple sensors. 13. M. R. H 1992 Transactions of the ASME Journal of Tribology 114, 328–333. Rolling element bearing fatigue damage propagation. 14. T. A. H 1991 Rolling Bearing Analysis, 3rd edn. New York: John Wiley. 15. Y. M, M. K and H. Y 1985 ASLE Transactions 28, 60–68. Analysis of surface crack propagation in lubricated rolling contact. 16. L. M. K and M. D. B 1983 Transactions of the ASME Journal of Tribology 105, 198–205. A pitting model for rolling contact fatigue. 17. C. C 1985 Wear 105, 297–305. Teh tribovibroacoustic model of machines. 18. C. C, H. G. N and M. T 1997 Mechanical Systems and Signal Processing 11, 107–117. A passive diagnostic experiment with ergodic properties. 18. V. Z. P and E. M. M 1985 Mechanics of Elastic-plastic Fracture. Hemisphere Publishing Corporation. 19. G. C. G and K. S. S 1984 Adaptive Filtering Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall.