International Journal of Performability Engineering, Vol. 6, No. 5, September 2010, pp. 425-434.
© RAMS Consultants
Printed in India
Roller Bearing Defect Prognosis using Likelihood
Parameters and Proportional Hazards Model
A. K. VERMA1, B. SREEJITH2 and A. SRIVIDYA3
1
Department of Electrical Engineering,
Interdisciplinary Programme in Reliability Engineering,
3
Department of Civil Engineering,
Indian Institute of Technology Bombay, Mumbai - 400076, India.
2
(Received on August 31, 2009, revised on March 17, 2010)
Abstract: Bearings are critical components employed virtually in all rotating
machines and automobiles to alleviate friction between surfaces during relative
motion. In traditional approaches, rolling element bearing failures are predicted
based on either historical time-to-failure data (event data) or condition
monitoring (CM) data. Prediction methods using event data are of little value to
maintenance decision making since they render general forecasts for the total
population of identical units instead of forecast for a particular unit presently
operating in the machine. Prognosis based on CM data provides short term
predictions which may not be useful in maintenance scheduling. Proportional
hazards model (PHM) can be used to predict hazard rates and reliability of
machines and its components using both event data and CM data.
This paper presents a method for defect prognosis of roller bearings using
Weibull proportional hazards model (WPHM) based on parameters obtained
from vibration analysis and historical event data. Morlet wavelet filter (MWF) is
used for denoising of vibration signals. Time domain parameters extracted from
the denoised vibration signals are used as covariates in the WPHM. Use of loglikelihood parameters as covariates in WPHM is explored and their performance
is compared with that of other parameters. The proposed approach helps in early
estimation of hazard and reliability with more accuracy, eventually increasing the
effectiveness of condition based maintenance and reducing maintenance costs.
Keywords: Roller bearing, event data, condition monitoring, log-likelihood parameters,
proportional hazards model.
1. Introduction
Bearings are critical components employed virtually in all rotating machines and
automobiles. Roller bearings are the common type of bearings employed in many of the
applications. Earlier, the maintenance techniques employed for roller bearings included
unplanned breakdown maintenance and time based preventive maintenance. Later,
condition based maintenance (CBM) strategies were applied to bearings. In a CBM
program, maintenance actions are scheduled depending on the health of the equipment.
CBM helps to ward off unwarranted maintenance activities, and hence significantly
reduces the maintenance cost and increases the availability.
__________________________________________
*Corresponding author’s email: [email protected]
425
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A. K.Verma, B. Sreejith and A. Srividya
Methods available to assess the condition of rotating machines are visual inspection,
performance monitoring, vibration monitoring, wear debris monitoring and temperature
monitoring [1]. Vibration monitoring is cost effective and it can be applied without
interfering with machinery operation. This method is widely used to assess the condition
of roller bearings.
Various stages in CBM are data acquisition, data processing, feature extraction, defect
diagnosis, prognosis and maintenance decision making. Two types of data involved in
CBM are event data and condition monitoring (CM) data. Data pertaining to the
equipment regarding starting of operation, maintenance performed, replacement of
components, breakdowns, stoppages etc. are all event data. Life of the
equipment/component before failure/replacement can be obtained from the event data.
Vibration data can be acquired using an accelerometer located near the component to be
monitored. Common features extracted from the vibration signal for condition monitoring
of rolling element bearing are peak value (Pv), RMS value (RMS), standard deviation
(SD), kurtosis value (Kv), crest factor (Crf), clearance factor (Clf ), impulse factor (Imf)
and shape factor (Shf) [2]. Log-likelihood values [3] are also used for feature extraction
from vibration signals [4]. Acquired time-domain vibration data can be processed further
using frequency domain techniques or time-frequency/scale analysis methods. Defect
diagnosis involves detection and identification of the location of defects.
Impulses are introduced in the vibration signal due to localized bearing defects. The
signal may become non-stationary due to time variations in properties such as amplitude
and frequency composition. Some of the causes of time variations in properties are sliding
effects, instantaneous variations in the contact angle, variations in operating conditions
and internal natural deterioration of the components. Morlet wavelet filter (MWF) can be
used for improving the signal-to-noise ratio of vibration signals for detecting impulses due
to localized bearing defects [5, 6]. The admissibility conditions of the wavelet and the
consequences due to violations of these conditions are discussed by Vaas and Cristalli [7].
Prognosis involves prediction of failures before it actually occur [8]. Prognosis helps in
making decisions regarding condition monitoring interval and maintenance scheduling.
Traditional methods for predicting bearing failure use historical time-to-failure data and
parametric failure model based on Weibull distribution. Prediction methods using event
data are of little value to maintenance decision making since they render general forecasts
for the total population of identical units instead of forecast for a particular unit presently
operating in the machine [9]. Prognosis can also be performed using CM data. Various
approaches for prognosis of individual units based on acquired CM data are discussed in
[8 - 10]. These methods provide short term condition predictions and the time interval
may not be sufficient for optimal maintenance scheduling and preparing spares and human
resources. Sometimes, features extracted from CM data contradict one another and trends
may not be consistent due to randomness in the data, leading to missed alarms and false
alarms.
Proportional hazards model (PHM) [11, 12] predicts hazard rates and reliability of
machines and its components using both historical event data and CM data. PHM using
hazard rate of two parameter Weibull distribution as baseline hazard known as Weibull
proportional hazards model (WPHM) was applied for aircraft engine, marine gas turbine
[13] and rail diesel engine [14] to improve the accuracy of the hazard estimates from
spectrometric analysis of lubricating oil. Application of WPHM to centrifugal pump [15]
and gearbox [16] with features obtained from the processed vibration signal as covariates
have been reported. Time domain parameters computed from band pass filtered vibration
signals were also used as covariates in WPHM for predicting bearing deterioration [17].
Roller Bearing Defect Prognosis using Likelihood Parameters and Proportional Hazards Model
427
On comparison of the performances of WPHM and logistic regression model, it was
observed that WPHM provided better reliability prediction for bearings [18]. A study on
the suitability of various time domain features for prognosis of roller bearings shows that
Pv, RMS and log-likelihood parameters will provide accurate prognosis compared to other
parameters [4].
This paper provides an approach for hazard estimation and reliability modeling of
roller bearings using acquired vibration signals. Suitability of various traditional timedomain features and log-likelihood features for bearing prognosis is examined. Vibration
signals are preprocessed using a Morlet wavelet filter (MWF) algorithm and features are
extracted from the processed signal. WPHM is used to synthesize time-to-failure data and
CM data for defect prognosis. Run to failure experiment data was processed and analyzed
using the proposed model. Relevance of the covariates obtained from CM data in the
model is examined using likelihood ratio test [19]. Residual analysis was used to test the
goodness of fit of the model.
2.
Signal processing
The vibration signals are processed using optimized MWF. The Morlet wavelet function
is defined as
Ψ(t) = exp(-β2t2/2)Cos(ωct)
(1)
where ωc is the center frequency and β is the shape parameter. The admissibility condition
requires zero mean value for the wavelet function ψ(t). The value of center frequency is
selected as 7π/4 and the shape factor is varied in the range <0.52, 1.83> to ensure the
fulfillment of the zero mean condition. Since the vibration signal measured for a damaged
bearing contain impulses, a wavelet generating fewest wavelet transformation coefficients
of the signal is desirable. In the proposed method, Shannon entropy is used to identify the
shape factor of the wavelet producing sparsest wavelet coefficients. The coefficients {ci};
i=1,2, …….n are normalized as
pi =
ci
n
∑cj
j =1
where
n
∑p
i
=1
(2)
i =1
and the Shannon entropy is computed as
n
E ( p ) = − ∑ pi log pi
(3)
i =1
The Shannon entropy is calculated for wavelet coefficients corresponding to various shape
factors in the selected range. A wavelet with shape factor corresponding to minimum
Shannon entropy generates sparsest set of coefficients. Hence, the shape factor is selected
using minimum Shannon entropy criterion. After fixing the shape factor, kurtosis values
of the wavelet coefficients for different scales are computed. Since the signals with higher
impulse content possess larger kurtosis value, the scale corresponding to maximum
kurtosis value is selected. The filtered signal is the wavelet transform of the signal
obtained using Morlet wavelet with optimized values of shape factor and scale.
3.
Feature Extraction
In addition to Pv and RMS, Weibull negative log-likelihood value (Wnl) and normal
negative log-likelihood value (Nnl) of the time domain vibration signals are used for
defect prognosis of roller bearings in this study. The performances of log-likelihood
parameters are compared with that of traditional time-domain parameters. For a vibration
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A. K.Verma, B. Sreejith and A. Srividya
signal with n sampling points with amplitudes {x1, x2, ………., xn}, the negative loglikelihood function is defined as,
n
−Λ = − ∑ log  f ( xi ;θ1 , θ 2 ) 
(4)
i =1
where f(xi;θ1,θ2) denotes the probability density function (pdf). The pdfs of Weibull
negative log-likelihood function and normal negative log-likelihood function are
computed as follows,
Weibull pdf, f (xi; β, η) = βη-β|xi|β-1exp [-(|xi|/η) β]
(5)
where the shape and scale parameters are denoted by β and η, respectively.
Normal pdf, f(x i ;µ , σ ) =
1
exp
σ 2π
{
 − ( xi − µ )2 


2σ 2
}
(6)
where mean and the standard deviation are denoted by µ and σ, respectively.
4.
Reliability Modeling
PHM is a semi-parametric model used to predict hazard of system from event data and
CM data. In this model, it is assumed that the hazard rate at time t, h(t) is the product of
the baseline hazard function, h0(t) and a covariate function exp(b. z(t)) as follows,
h(t, z(t)) = h0(t) exp(b.z(t))
(7)
where b is the vector of coefficients and z(t) is the vector of covariates. In WPHM, the
baseline hazard function is assumed to be the hazard function of Weibull distribution and
the hazard rate is modeled as,
βt 
h(t, z(t)) =  
η η
β-1
exp(b.z(t))
(8)
where β and η are shape parameter and scale parameter respectively. The cumulative
hazard rate is given by,
t
t
0
0
H(t,z(t)) = ∫ h(t, z(t))dt= ∫
βt 
 
η η
β -1
exp(b.z(t))dt
The reliability, R(t) can be computed as,
R(t) = exp [-H(t, z(t))]
If f(t) is the failure probability density function, the likelihood function is given by,
n
m
L = ∏ f (ti ).∏ R (t j )
i =1
(9)
(10)
(11)
j =1
where i denotes failures and j denotes suspensions. Since the failure probability density
function can be represented as,
f(t) = h(t).R(t)
(12)
the likelihood function in Eq. (11) can be represented as,
n
k
L = ∏ h(ti ).∏ R (tl )
i =1
(13)
l =1
where k=m+n and l denotes failures and suspensions. Substituting for h(t) and R(t) from
Eqs. (8), (9) and (10) in Equation (13),
Roller Bearing Defect Prognosis using Likelihood Parameters and Proportional Hazards Model
429
k
 tl β  t  β-1

(14)
exp(b.zi (ti )).∏ exp - ∫  l  exp(b.zl (tl ))dt 
i=1
l=1
 0 η  η 

The log-likelihood function can be obtained as,
β-1
β-1
k tl
β n t   n
βt 
(15)
Λ= lnL= nln + ∑ln   i   + ∑b.zi (ti )- ∑ ∫  l  exp(b.zl (tl ))dt
η i=1   η   i=1
l=1 0 η  η 
The parameters β, η and b can be estimated by maximizing the log-likelihood function.
Hazard rate can be computed by substituting the values of parameters in Equation (8).
Then, reliability can be computed using Equations (9) and (10). The method of prognosis
is illustrated in Fig. 1.
n
L=∏
β  ti 
 
ηη
β-1
Figure 1: Flowchart of prognosis using MWF and WPHM.
5.
Vibration data
The original data provided by the Center for Intelligent Maintenance Systems (IMS),
University of Cincinnati through the NASA Prognostic Data Repository [20] contains
bearing vibration data collected during accelerated bearing deterioration experiment
conducted on a specially designed test rig under elevated loads and rotational speed. The
test rig contains four Rexnord ZA-2115 self aligned double row spherical roller bearings
installed on a shaft driven by an AC motor and coupled by rub belts. A radial load of
430
A. K.Verma, B. Sreejith and A. Srividya
26.689 kN was added to the shaft and bearing by a spring mechanism and the rotation
speed was kept constant at 2000 rpm. The data collected using DAQCarde-6062E data
acquisition card and LabVIEW software contains vibration signal measured at a sampling
rate of 20 kHz for one second every 20 minutes until failure using accelerometers installed
on each bearing housing.
Four bearing failure events were reported in the three tests. During the first test, the
third bearing failed due to inner race defect and the fourth bearing failed due to rolling
element defect. Failures were reported due to defects in the outer race for first and third
bearings during second and third tests respectively. Since the tests were terminated when a
bearing failure was observed, the remaining bearings which are not damaged at the end of
the test are considered as suspensions.
6.
Prognosis
The vibration signals are filtered using MWF, and the time-domain parameters of the
filtered signals are calculated. The time domain parameters considered in this study are
Pv, RMS, Wnl and Nnl. WPHMs are fitted and model parameters are calculated using R
language and software environment. The resulting hazard functions are shown in Table 1.
A simple Weibull model developed without incorporating the covariates is also shown in
the table as model 5.
Significance of covariates in the WPHM is verified by comparing with simple Weibull
model using likelihood ratio test. Twice the difference between log-likelihood of the data
for WPHM and simple Weibull model is the test statistic. The test statistic, LR has an
approximate chi-square distribution with degrees of freedom equal to the number of
covariates in the WPHM. The null hypothesis states that the covariate is significant in the
model. The likelihood ratio test rejects the null hypothesis if the value of LR is too small.
Results of the likelihood ratio tests performed for the WPHMs are shown in Table 2.
Since p values lie between 0.025 and 0.05, the covariates are statistically significant in the
models. One can also observe that Pv and Nnl are the most significant covariates. Since
Nnl in comparison with Pv, is more robust with changes in operating conditions, model 4
is considered for further analysis.
Table 1: WPHMs built for the bearing data with vibration signals filtered using MWF
Model
No.
Covariate
1
Pv
h(t) =
2.08 
t



4082.906  4082.906 
2
RMS
h(t) =
1.961 
t



4726.302  4726.302 
Hazard function
1.080
3
Wnl
4
Nnl
5
-
exp (1.344 Pv(t) )
0.961
exp ( 8.067 RMS(t) )
0.941
h(t) =
1.941 
t



620.296  620.296 
h(t) =
1.961 
t



1720.223  1720.223 
exp (1.286 ×10 −4 Wnl(t) )
0.961
1.028
h(t) =
2.028 
t



1138.195  1138.195 
exp (1.427 ×10−4 Nnl(t) )
Roller Bearing Defect Prognosis using Likelihood Parameters and Proportional Hazards Model
431
Table 2: Likelihood ratio tests of WPHMs built for the bearing data
Model No.
1
2
3
4
5
Covariate
Pv
RMS
Wnl
Nnl
-
Log-likelihood
-22.165
-22.539
-22.454
-22.188
-24.643
Test statistic, LR
4.956
4.208
4.378
4.91
-
p
0.025 < p < 0.05
0.025 < p < 0.05
0.025 < p < 0.05
0.025 < p < 0.05
The plots of hazard rate and reliability for bearings failed due to outer race, inner race and
rolling element defects obtained using model 4 are shown in Figs. 2-4. It can be observed
that hazard rates increase significantly before failures occur. In the reliability plots, RA
and RW are the actual reliability and the reliability computed using simple Weibull model
without including the covariates. RU and RL are the reliabilities computed with covariate
values corresponding to normal and damaged conditions, respectively. The actual failure
rate RA is close to RU initially and reduces significantly before failure occurs. Hence,
maintenance planning can be performed depending on actual reliability. If simple Weibull
model is used with estimated life corresponding to 90% reliability as replacement period,
bearing will be replaced after an operating life of 400 hours. In such a situation, a
breakdown occurs for first bearing in the second test (Fig. 2). In the other two cases
considered (Figs. 3 and 4), bearing will be replaced at 400 hours even though the useful
lives of the bearings are more than 800 hours. Hence, it can be observed that the use of
WPHM helps to reduce breakdowns and unnecessary replacement of bearings resulting in
optimized maintenance and reduction of maintenance costs and downtime of equipments.
1
x 10
-4
1
0.8
RL
RU
Reliahility
H azard rate, h
0.8
0.6
0.4
RW
0.4
0.2
0.2
0
0
RA
0.6
50
100
Time (hours)
150
0
0
100
200
300
Time (hours)
Figure 2: Hazard rate and reliability of a bearing failed due to outer race defect
400
500
432
A. K.Verma, B. Sreejith and A. Srividya
5
x 10
-4
1
4.5
0.8
RL
RU
3.5
R e lia h ility
H a z a rd ra te , h
4
3
2.5
2
0.6
RA
RW
0.4
1.5
0.2
1
0.5
0
0
0
100
200
300
400
500
600
700
800
0
500
1000
1500
2000
2500
3000
Time (hours)
Time (hours)
Figure 3: Hazard rate and reliability of a bearing failed due to inner race defect
7.
Conclusions
Since the impulses due to rolling bearing localized defects are small in magnitude and
variations are present in the interval between impulses, MWF can be used to increase the
signal-to-noise ratio of vibration signals. WPHM model helps to estimate the hazard and
reliability of bearings with more accuracy. Time domain parameters computed from
filtered vibration signals can be used as covariates in the WPHM. Nnl is a time domain
parameter which can be used for prognosis of the rolling bearing using WPHM.
Application of WPHM helps in maintenance decision making to avoid breakdowns and
unwanted maintenance, eventually increasing the effectiveness of condition based
maintenance and reducing maintenance costs. Effectiveness of the proposed procedure is
demonstrated using actual bearing vibration data.
4
x 10
-4
1
RL
0.8
RU
RA
R e lia h ility
Hazard rate, h
3
2
1
0
0
RW
0.6
0.4
0.2
100
200
300
400
500
Time (hours)
600
700
800
0
0
500
1000
1500
2000
2500
Time (hours)
Figure 4: Hazard rate and reliability of a bearing failed due to rolling element defect
3000
Roller Bearing Defect Prognosis using Likelihood Parameters and Proportional Hazards Model
433
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A. K. Verma is a Professor at the Indian Institute of Technology Bombay. He
received his Ph. D. degree in Reliability Engineering from Indian Institute of
Technology Kharagpur, where he also earned a B. Tech. (Hons) degree. His areas of
research in Reliability Engineering are interdisciplinary applications in Software
Engineering, Computing, Maintenance and Power Systems. He is an author/editor of
several books and an editor in several journals.
B. Sreejith received his M. Tech. from Indian Institute of Technology Madras and is
currently working towards his Ph. D. in Reliability Engineering at the Indian Institute
of Technology Bombay. The topic of his research is the application of likelihood
parameters, wavelet transforms, artificial neural networks and Weibull proportional
hazard models for defect diagnosis and prognosis of roller bearings.
A. Srividya received her M. Tech. and Ph. D. from Indian Institute of Technology
Bombay and is now a Professor in the same institute. She has published several
journal papers and is an author/editor of several books and journals. Her current
research interests include Structural Reliability; Reliability based Optimal Designs,
Simulation studies for Reliability Estimation and Quality Benchmarking studies for
Service Industries.