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Reliability estimation for cutting tools based on logistic regression model using vibration signals
Mechanical Systems and Signal Processing 25 (2011) 2526–2537
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Reliability estimation for cutting tools based on logistic regression model using vibration signals Baojia Chen, Xuefeng Chen n, Bing Li, Zhengjia He, Hongrui Cao, Gaigai Cai State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China
a r t i c l e in f o
abstract
Article history: Received 1 June 2010 Received in revised form 28 September 2010 Accepted 2 March 2011 Available online 10 March 2011
As an important part of CNC machine, the reliability of cutting tools influences the whole manufacturing effectiveness and stability of equipment. The present study proposes a novel reliability estimation approach to the cutting tools based on logistic regression model by using vibration signals. The operation condition information of the CNC machine is incorporated into reliability analysis to reflect the product time-varying characteristics. The proposed approach is superior to other degradation estimation methods in that it does not necessitate any assumption about degradation paths and probability density functions of condition parameters. The three steps of new reliability estimation approach for cutting tools are as follows. First, on-line vibration signals of cutting tools are measured during the manufacturing process. Second, wavelet packet (WP) transform is employed to decompose the original signals and correlation analysis is employed to find out the feature frequency bands which indicate tool wear. Third, correlation analysis is also used to select the salient feature parameters which are composed of feature band energy, energy entropy and time-domain features. Finally, reliability estimation is carried out based on logistic regression model. The approach has been validated on a NC lathe. Under different failure threshold, the reliability and failure time of the cutting tools are all estimated accurately. The positive results show the plausibility and effectiveness of the proposed approach, which can facilitate machine performance and reliability estimation. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Reliability estimation Cutting tool Logistic regression model Wavelet packet decomposition Correlation analysis
1. Introduction In modern manufacturing systems, machine tools are the major equipment and play a very important role. The malfunction of machine tools may result in the halt of the whole production and bring about tremendous financial losses. For example, in the case of complex installations such as automobile assembly lines, it can be as high as $20,000 per minute [1]. Therefore, manufacturers have been paying great attention to machine reliability improvement to reduce unexpected downtime and raise product quality [2,3]. A cutting tool is an important part of machine tools and its reliability influences the total manufacturing effectiveness and stability of machine tools. With an accurate estimate of tool lifetime, worn tools can be changed in time to reduce waste product and tools costs noticeably. It is even possible to guarantee a certain surface quality. Tool failure and lifetime is judged by its wear measurement described in several standards (ISO3685, ISO8688 and ANSI/ASME B94.55M) [4]. These
n
Corresponding author. Tel./fax: þ 86 29 82663689. E-mail address: [email protected] (X. Chen).
0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.03.001
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standards provide wear threshold values at several points of the worn region which is acquired by its direct measurement (e.g., CCD cameras). Due to the constraints of high cost, discontinuity and susceptibility to operational environment, the direct methods are restricted to a very narrow range of applications and need to be improved. In indirect methods, the wear state can be estimated by the cutting force, torque, temperature, acoustic emission and vibration [5–7]. As one of the indirect sensorbased methods, vibration measurement is widely used due to its low price, easy implementation and on-line continuous testing characteristic. Some investigations have been carried out to correlate vibration signals to tool wear [8,9]. Their results show that vibration signature features extracted from time and frequency domains are sensitive to the tool wear. Now, the key issue to estimate cutting tools reliability indices is how to combine these vibration features using an appropriate reliability model. Traditional approaches to reliability analysis of physical and other electro-mechanical products are predominantly based on lifetime of large sample size [10]. Such reliability analyses have been extensively studied in the past [11,12]. With few exceptions, all these analyses are geared towards estimating population characteristics of a system, subsystem or component. However, for a single and small sample device(s), such statistic data are of little meaning. People are particularly interested in the life margin and current reliability of items used in their systems. Wherefore degradation measures often provide more information about device performance and precision during operation [13,14]. It is helpful to determine the product time-varying characteristics and reflect the relationship between failure and performance degradation. In general, degradation is the reduction in performance, reliability and life span of assets. There are a number of researches and applications on reliability assessment by using degradation data. Lu and Meeker [15] introduced a general nonlinear mixed-effects model and developed a Monte Carlo simulation procedure to calculate an estimate of the distribution function of the time-to-failure. Chen and Zheng [16] proposed an alternative approach which makes inference directly on the lifetime distribution. Chinnam and Rai [17] collected and analyzed thrust-force and torque signals generated by drilling operation to predict on-line reliability of the drill-bit. Lin et al. [18] demonstrated a procedure to extract useful condition indicators from vibration signals and use the proportional hazards model (PHM) to develop optimal maintenance policies for the gearboxes. Lin and Tseng [19] combined the Weibull PHM and vibration-based machine condition monitoring techniques to estimate several machine reliability statistics. However, the above methods require a specific mechanical knowledge and make many assumptions about condition parameters degradation paths and their distribution probability density functions. Logistic regression is a robust tool that can easily represent multiple variables as a dichotomous problem [20]. It has been widely used in bionomics, biology, epidemiology, etc. [21–23]. The main advantages are that logistic regression does not rely on assumptions of normality for the predictor variables or the errors and it allows the selection effect to vary nonlinearly [24]. The independent variable can be continuous, discrete or dummy variable. Based on its likelihood function, it needs less computation effort to estimate parameters rather than PHM. [3] Assuming a facility performance state is a process from normal to failure, and may be reflected and interpreted by multiple feature parameters. The logistic regression model could be used to analyze and describe the failure probability of facilities under certain conditions, which would benefit decision making (using, repairing or replacing). Some exploratory work has been done on mechanical equipment performance evaluation and remaining useful life (RUL) prediction. Yan and Lee [25] explored a method to assess real-time performance of elevator door system by inputting features of online data to the logistic model. Liao et al. [26] presented a combination of the proportional hazards model and logistic regression model to predict the unit RUL. In the present study, vibration signals and tool wear were measured together during machining process. The salient features indicative of tool wear were extracted through signal processing techniques and integrated in a logistic regression model to evaluate reliability of lathe cutting tools. The rest of the paper is organized as follows: the reliability assessment method based on logistic regression model is described in Section 2, the experimental design and setup are introduced in Section 3, the vibration signals analysis, reliability modeling and application results are discussed in Section 4. Finally, some concluding remarks are given in Section 5.
2. Reliability estimation method based on logistic regression model 2.1. Wavelet packet decomposition Given a finite energy signal whose scaling space is assumed as U00 , wavelet packet (WP) transform can decompose U00 into small subspaces Ujn in dichotomous way [27]. The dichotomous way is realized by the following recursive scheme: Ujnþ 1 ¼ Uj2n Uj2n þ 1
j 2 Z, n 2 Z þ
ð1Þ
where j (jr0) is the resolution level and denotes orthogonal decomposition. Z denotes the domain of integers. Z þ denotes the domain of positive integers. Ujnþ 1 , Uj2n and Uj2n þ 1 are the three close spaces corresponding to cn(t), c2n(t) and c2n þ 1(t). cn(t) satisfies the following equation: pffiffiffiX pffiffiffiX hðkÞcn ð2tkÞ c2n þ 1 ðtÞ ¼ 2 gðkÞcn ð2tkÞ
c2n ðtÞ ¼ 2
k2Z
k2Z
ð2Þ
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Note that the first wavelet is the so-called mother wavelet function
c0 ðtÞ ¼ jðtÞ, c1 ðtÞ ¼ cðtÞ
ð3Þ
The discrete filters h(k) and g(k) are quadrature mirror filter (QMF) associated with the scaling function j(t) and the mother wavelet function c(t). The sub-signal at Ujn1 , the nth subspace on the jth level can be reconstructed by a linear j,n combination of WP function ck ðtÞ as follows: X j,n j,n Dk ck ðtÞ k 2 Z ð4Þ snj ðtÞ ¼ k2Z
can be obtained from The WP coefficients Dj,n k Z þ1 j,n ¼ f ðtÞck ðtÞdt Dj,n k
ð5Þ
Since the wavelet cj,k(t) is an orthogonal basis at L2(R), the energy of the sub-signal snj ðtÞ is calculated by X j,n 2 En ¼ Dk
ð6Þ
The normalized energy of snj ðtÞ is ! X En Pn ¼ En =
ð7Þ
1
k
n
Energy entropies is a frequently used feature to detect the change of signal energy in different frequency bands, which can reveal the amount of information stored in observed signal. It is defined as X I ¼ En log En ð8Þ n
2.2. Feature definition Both amplitude and distribution of the vibration signals change along with tools’ state from sharp to worn. Previous studies have shown that some time-domain features can indicate tool wear [28,29]. In this work, 11 time-domain feature parameters are extracted: mean (xm), peak (xp), root amplitude (xra), root mean square (xrms), standard deviation (xstd), skewness (xske), kurtosis (xk), crest (xc) margin (xma), shape (xsha) and impulse factor (xi). The first four parameters reflect the vibration amplitude and energy in time domain. The remaining parameters represent the time series distribution of the signal in time domain [30]. A brief mathematical description of these features is shown in Table 1. 2.3. Feature selection based on correlation analysis Although the above mentioned features are indicative of tool wear from different aspects, they have different important degrees. Some features are salient and closely related to the wear, but others are not. Thus, before a feature vector is input into logistic regression model, it is necessary to select salient features of tool wear state from the feature set and discard the irrelevant or redundant features to improve the estimating accuracy and avoid the problem of dimensionality. Here, correlation analysis is employed to select the salient features. The correlation coefficient (CC) r(U, V) between two
Table 1 The feature parameters. xm ¼
PN n ¼ 1
xðnÞ
rP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N
xstd ¼ xma ¼
xp ¼ max9xðnÞ9
N
ðxðnÞxm Þ2 N1
n ¼ 1
xp xra
xske ¼ xsha ¼
PN n ¼ 1
ðxðnÞxm Þ3
ðN1Þx3std
xrms xm
where x(n) is a signal series for n¼1,2, y, N, N is the number of data points.
xra ¼ xk ¼ xi ¼
PN n ¼ 1
pffiffiffiffiffiffiffiffiffi2 jxðnÞj
N
PN n ¼ 1
ðxðnÞxm Þ4
ðN1Þx4std
xp xm
rP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N
xrms ¼ xc ¼
xp xrms
n ¼ 1
N
ðxðnÞÞ2
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one-dimensional vectors or signals U and V of the same length is calculated by equation ðUUÞðVVÞ
rðU,VÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð9Þ
ðUUÞðUUÞ0 ðVVÞðVVÞ0 Þ
where U and V are the mean values of U and V, respectively. The value of r ranges between 1 and þ1. r 4 0, r o0 and r ¼0 indicate a positive correlation, negative correlation and irrelevance, respectively. The larger the absolute value of r, the higher the degree of correlation. 2.4. Logistic regression model Wear state of cutting tools can be indicated by a set of characteristic parameters. If observed samples which consist of characteristic parameters and cutting tool state are acquired, the logistic regression can be used to set up the relationship between normality and failure [31]. Suppose at time ti, the cutting tool condition feature is a kþ1 dimensional vector Xi ¼(1, x1i, x2i, y, xki)0 and tool state is yi (under normal state, yi ¼1, otherwise yi ¼0), the reliability function of the cutting tool can be described as Rðti 9Xi Þ ¼ Pðyi ¼ 19Xi Þ ¼
expðBXi Þ 1 þ expðBXi Þ
ð10Þ
where B ¼ ðb0 , b1 , . . ., bk Þ is the model parameter vector and b0 40. The logistic or logit regression model is LogitðyÞ ¼ ln
Rðti 9Xi Þ ¼ BXi 1Rðti 9Xi Þ
ð11Þ
The log-likelihood function (LLF) of the observed degradation features can be expressed as X ½yi BXi lnð1 þexpðBXi ÞÞ ln½LðBÞ ¼
ð12Þ
i
Since logistic regression is nonlinear, the model parameters can be obtained by maximizing the log-likelihood function through the Nelder–Mead’s algorithm. Once the model parameters are identified, the reliability and 95% confidence interval (CI) indicated by feature vector Xj can be calculated as Rðtj Þ ¼ Pðyj ¼ 19Xj Þ ¼ 2 6 6 4
^ jÞ expðlog^ itðyj ÞÞ expðBX ¼ ^ jÞ 1 þ expðlog^ itðyj ÞÞ 1 þexpðBX
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ j 1:96 VarðBX ^ jÞ exp BX qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ^ j 1:96 VarðBX ^ jÞ 1þ exp BX
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ^ j þ1:96 VarðBX ^ jÞ exp BX 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7 5 ^ j þ 1:96 VarðBX ^ jÞ 1 þexp BX
ð13Þ
ð14Þ
^ j Þ is the variance of model parameters and can be calculated as where VarðBX ^ j Þ ¼ X0 COV ðBÞXj VarðBX j
ð15Þ
where COV(B) is model parameters covariance matrix. 2.5. The proposed reliability estimating method In the field of rotating machinery, the running state of the machine can usually be identified by finding certain characteristic frequencies and changes of their amplitude. However, there are not such particular frequencies to indicate the cutting tools conditions of lathe. As reported in [32], the characteristics associated with tool wear mainly focus on a specific frequency band. Therefore, WP transform is employed to decompose vibration signals into different frequency bands and correlation analysis is used to select salient features indicative to tool wear. The proposed method is described in Fig. 1. It includes the following procedures. First step is data acquisition. The vibration signals and tool wear are measured by an accelerometer and a micro-optical system. Second step is feature extraction and selection. When the original vibration signals are decomposed to the fourth resolution level (j ¼ 4) with WP transform, the whole scaling space is divided into 16 frequencies bands. The WP energy and energy entropy are calculated by Eqs. (6) and (8). By correlation analysis, the feature frequency bands sensitive to wear change can be selected out. Similarly, the 11 timedomain features of each feature band are extracted and salient features are selected out by applying correlation analysis. The feature vector X can be obtained by combining the time-domain features with energy entropy and feature band energy. If flank wear value VB is smaller than the threshold Vt, the tool state index Y¼1, otherwise Y ¼0. Vt is selected as
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Accelerometer
micro-optical system
Vibration Signals
Tool Wear
Data Acquisition
NC lathe
WPD
WP Energy
No
VB≤Vt
Feature Band Time - domain feature Energy
Correlation analysis
Salient feature
X
Yes
Y=0
Y=1
Y
Logistic regression model
Reliability indices
Reliability Estimation
WP entropy
Features Extraction and Selection
Correlation analysis
Fig. 1. Flow chart of reliability estimation procedure.
0.6 mm in the study with reference to ISO3685. Finally, by taking X and Y as input, logistic regression model is established and applied to estimate the reliability of other tools being used. 3. Experimental design and setup The experimental setup for the in-process detection of tool wear is illustrated in Fig. 2. An CNC lathe is employed for the interrupted cutting of steel bars fixed to the work holder. The cutting tool used in this test is a carbide tool and the experiment conditions and specifications are shown in Table 2. Tool wear VB is measured by a micro-optical system with a CCD camera, an adjustable circular-ring-mode LED light and a built-micrometer. The maximum resolution is 0.01 mm. The tool vibration signals are monitored by an accelerometer and then sent to a data acquisition system (LMS SCADAS305) and a portable computer. At the same time, the acoustic emission (AE) signals, the workpiece surface roughness data and the current signals of spindle motor and Z-servo motor are also measured, respectively, for further research. The type and model of all used sensors are listed in Table 3. The installation of micro-optical system, accelerometer and AE sensor are shown in Fig. 3. 4. Result and discussion 4.1. Vibration signal analysis In the manufacturing process, tool vibration is caused by a variety of factors, such as cutting force, built up edge, material inhomogeneity, ambient conditions and tool wear. Hence the main purpose of vibration analysis is to find the features indicative of tool wear. In the present experiment, a total of 12 cutting tools’ vibration signals and wear data were measured. It is assumed that all investigated tools have the same wear mechanism and vibration characteristic with increasing tool wear. Among them, tool no. 4 is taken as an example to show the whole analysis process. The on-line vibration signals were measured and the sampling frequency was 32,768 Hz, which was determined by the frequency
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Roughometer Workpiece
Current Sensor
Spindle motor
Cutting tool Current Sensor Micro-optical system
AE sensor
Accelerometer Z-servo motor
LMS Signal Acquisition and Analysis Instrument
Portable computer
Digital phosphor oscilloscope
Fig. 2. The schematic diagram of the experimental setup.
Table 2 Experimental conditions. Lathe
Cutting Tool
Workpiece Material Cutting conditions
Type: horizontal Model: FTC-20 Spindle speed: 45–4500 rpm Rated power: 11 kW Type: diamond carbide tool Model: CNMG120408-HM Material: 42CrMo4 45# Steel bars Feed-rate f: 0.15 mm/rev Cutting speed vc: 200 m/min Depth of cut ap: 2 mm
Table 3 Type and model of all sensors. Signal
Sensor type
Sensor model
Tool wear Vibration AE
Micro-optical system Accelerometer AE Sensor Coupler Roughometer Hall current sensor
MZDH0670 PCB ICP352C34 Kistler 8152B Kistler 5125 SJ-201 GAA-KY1
Roughness Current
characteristics of tool vibration. The vibration signal and its spectrum of tool no. 4 after running for 78 min are shown in Fig. 4. It can be observed from the figure that the signal contains abundant frequency components and the energy is mainly focused on two band ranges of 2000–4000 and 7000–10,000 Hz. As shown in Fig. 4, the vibration signals are decomposed by WP transform. Because Daubechies family of wavelet packets seems to resemble the cutting tools vibration signal most and Daubechies 10 is adopted as the mother wavelet. In order to find out the feature frequency band indicative of tool wear, the WP energy of different sampling time is calculated and normalized. The results are shown in Fig. 5(a)–(d). It can be clearly observed that the signal energy mainly concentrates on band interval 7–10 and the maximum energy concentrates on band 9. With the processing time increasing, the signals’ energy gradually transfers from high frequency to low frequency. As expressed in Fig. 5, the energy
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CCD camera
Bracket
Micro-optical system
AE Sensor
Cutting tool
Accelerometer
Fig. 3. The installation of micro-optical system, accelerometer and AE sensor.
Fig. 4. Vibration signal and its spectrum of tool no. 4 after running for 78 min.
transfers from band interval 8–10 to band interval 3–7. But the energy above band 11 changes little. For a detailed observation, the energy changing trends over time of bands 7 and 9 are depicted in Fig. 6(a)–(b). From time t ¼69–93 min, the energy ratio of band 7 P7 increases from 0.02 to 0.13. In contrast, the energy ratio of band 9 P9 reduced from 0.48 to 0.30. This phenomenon may be caused by the increase of tool wear, which results in increasing contact area between the tool and workpiece. Fig. 6(c)–(d) are energy entropy and tool wear changing trends. The tool wear is monotonically increasing. But the band energy and the entropy are non-monotonically increasing or decreasing. Their trends are conspicuous and closely related to the wear. The above procedure for analysis is also applied to other 11 tools. Results are similar to those of tool no. 4, which bears out our earlier assumption. The CCs between WP band energy, entropy and corresponding tool wear are also calculated. In order to eliminate the randomness, the averaged coefficients are calculated and displayed in Fig. 7. It can be observed that the CCs of bands 4, 7, 9 energy and entropy are larger, reaching more than 0.6. Significantly, the CCs of bands 7 and 9 are more than 0.7. In the next step, sub-signals of bands 7 and 9 are selected to calculate the 11 time-domain features listed in Table 1 to acquire more salient features. According to Fig. 1, the analyzing procedure is implemented and the averaged CCs between the time-domain features and tool wear are obtained and listed in Fig. 7. Compared with band 9, the correlation degrees of band 7 features are higher, especially the CCs of the 1st, 3rd, 4th and 5th time-domain feature corresponding to xm7, xra7, xrms7 and xstd7, respectively, are also larger than 0.6. The total number of selected features by correlation analysis is 4 time, 3 WP energy and 1 entropy, whose CCs are larger than 0.6, which yields a total of 8. They compose the input vector X¼(xm7 xra7 xrms7 xstd7 P4 P7 P9 I)0 of logistic regression model.
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Fig. 5. The WP energy spectrums at different sampling time: (a) t¼ 69 min, (b) t ¼73 min, (c) t ¼81 min and (d) t ¼89 min.
Fig. 6. The changing trends of different feature parameters (a) energy change of band 7, (b) energy change of band 9, (c) energy entropy change of tool no. 4 and (d) tool wear of tool 4.
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Fig. 7. The CCs between WP energy, entropy, time-domain features and tool wear.
Table 4 Information measures of different models. No. Feature set
Model expression (log^ itðyÞ ¼ )
AIC
SC
BIC
1 2
7.39 17.53P7 þ32.19P9 þ3.11I 2.26 164.2xm7 þ 141.3xra7 7.92xrms7 þ 64.47xstd þ0.32P4 21.25P7 þ 29.6P9 þ 2.86I 5.29 þ92.37xm7 104.9xra7 7.49xrms7 17.61P7 þ 20.4P9
68.14 79.64 559.01 69.61 95.49 543.16
3
Optimal subset: P7,P9,I All features: xm7, xra7, xrms7, xstd7, P4, P7, P9, I CC40.65 subset: xm7, xra7, xrms7, P7, P9
4
CC40.7 subset: P7, P9
2.02 19.33P7 þ19.33P9
72.25 80.87 557.78
69.07 86.32 552.33
4.2. Reliability modeling and estimation After data preparation, by taking X as independent variable and tool state y as dependent variable, 131 samples were obtained from the selected salient features and corresponding state data of tool nos. 1–11. Logistics regression model was set up by using SAS software [33]. To find out the optimal subset, forward stepwise, backward stepwise and combined stepwise methods were applied. Score statistics and Wald statistics were used to add and delete variable. The three methods have the same model as p^ i ð16Þ ¼ BXi ¼ 7:3917:53 P7 þ 32:19 P9 þ 3:11I log^ itðyÞ ¼ ln 1p^ i It can be observed that the calculated model has no time-domain features although their CCs also above 0.6. This is because the model variable selection is an automatic process according to specific statistic criterion. Only those parameters that make the greatest contribution to the model LLF defined as Eq. (12) could be selected based on a certain significance level. Maybe the contributions of time-domain features are all great when they act as a signal model variable. But if they are correlated with other variables (e.g., energy features), multicollinearity will weaken their significance level and make them not appear in the model. At the same time, three other models were also set up and their information measures are listed in Table 4. It is obvious that the Akaike information criterion (AIC) and Schwarts criterion (SC) of model 1 are smaller than those of the three other models, and the absolute value of Bayesian information criterion (BIC) is larger than those of the other three models. According to [33], it implies that the fit goodness of model 1 is higher than other models. It means that the model 1 better describes the relationship between indicators and tool state. So model 1 (i.e. Eq. (16)) has been used as the model for cutting tool reliability evaluation. To identify the validity of the proposed model, the 11 condition feature vectors of tool no. 12 were input into Eqs. (16) and (5) to calculate the reliability and 95% CI. As depicted with thin solid and dashed lines in Fig. 8, the estimated reliability indices show an expected decreasing trend with the growth of cutting time. Due to the discreteness and randomness of the sampling data, the reliability curves are discontinuous and non-monotonically declining. In order to represent the actual continuous degradation process of cutting tool, single variable logistics regression model is employed
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1 0.9 0.8
Reliability R
0.7 0.6 0.5 0.4 0.3 estimated reliability 95% CI of estimated reliability fitted reliability fitted 95% CI
0.2 0.1 0 75
80
85
90 Time t/min
95
100
105
Fig. 8. The estimated reliability indices (Vt ¼ 0.6 mm).
1 0.9 0.8
Reliability R
0.7 0.6 0.5 0.4 0.3 0.2 0.1
estimated reliability 95% CI of estimated reliability fitted reliability fitted 95% CI
0 80
85
90 Time t/min
95
100
Fig. 9. The estimated reliability indices (Vt ¼ 0.5 mm).
to fit the reliability curve and its 95% CI. The smoothed reliability curves are shown with thick solid and dashed lines in Fig. 8. From time t ¼92–98 min, the fitted model curve has some deviation from the estimated. It is caused by fitting error. Compared with the cutting tool life, the region is so narrow that it has little influence on the estimation result from the point of engineering application. As the logistics model is set up on the basis of tools’ two state (normality or failure), we decided that the maximum reliability allowed is 50%. If the reliability value decreases below 0.5, the tool is considered to be invalid, i.e. the tool wear is getting larger than 0.6 mm. The real measured failure time of tool no. 12 is 98.6 min (VB ¼0.607 mm) and the estimated failure time of the smoothed model is 100 min. The estimated error is 1.42%. In practical production, the threshold of flank wear may be set flexibly to satisfy the requirement of surface quality. Supposing the threshold Vt ¼0.5 mm, the re-estimated reliability model is shown as p^ i ð17Þ log^ itðyÞ ¼ ln ¼ BXi ¼ 1:38727:0757xrms7 þ 1:8269xstd7 6:2087 P7 þ 12:5146 P9 þ 1:4842I 1p^ i Similarly, the estimated reliability curves and smoothed reliability curves are also estimated and shown in Fig. 9. Compared with Fig. 8, the decreasing trend is relatively smaller and the 95% CI is larger, which may be due to the more failure samples with the lowered limit. The measured real failure time of tool 12 is 93.8 min (VB ¼0.512 mm) and estimated failure time is 96.76 min. The estimated error is 3.6%. It can be seen that the two established models both accurately estimate the reliability and failure time of tool 12.
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5. Conclusion The present study proposes a reliability estimation approach to cutting tools, which is based on a logistic regression model by using vibration signals. WP decomposition and correlation analysis are applied to extract and select the salient features. The selected band energy, energy entropy and time-domain features show a high degree of correlation to tool wear. Without requiring a specific mechanical knowledge and making many assumptions about probability density functions of variables, the logistic regression model for reliability estimation can be established just by acquiring the tools’ feature vector and their corresponding state. A case study shows that the reliability indices and failure time of the tool can be accurately assessed by using the obtained model and the approach is also proved effective when the threshold is changed. The positive results of this research clearly demonstrate the potential of this approach in evaluating performance and reliability. Future work can be centered on model validity in case of small samples and combining with other methods to predict machine tool performance. In addition, we look forward to do more research on the robustness of the proposed method and the influence of tool material, workpiece material, cutting parameters to the model.
Acknowledgments This work is jointly supported by the key project of National Nature Science Foundation of China (No. 51035007), the National S&T Major Projects of China (Grant no. 2009ZX04014-015) and the National Basic Research Program of China (Grant no. 2009CB724405). References [1] S.A. Spiewak, R. Duggirala, K. Barnett, Predictive monitoring and control of the cold extrusion process, Annals of CIRP 49 (1) (2000) 383–386. [2] L. Liao, J. Lee, Design of a reconfigurable prognostics platform for machine tools, Expert Systems with Applications 37 (1) (2010) 240–252. [3] N. Gorjian, L. Ma, M. Mittinty, P. Yarlagadda, Y. Sun, A review on reliability models with covariates, in: Proceedings of the Fourth World Congress on Engineering Asset Management, Athens, Greece, 2009. [4] B. Sick, On-line and indirect tool wear monitoring in turning with artificial neural networks: a review of more than a decade of research, Mechanical Systems and Signal Processing 16 (4) (2002) 487–546. [5] D.E. Dimla Sr., P.M. Lister, On-line metal cutting tool condition monitoring: I: force and vibration analyses, International Journal of Machine Tools and Manufacture 40 (5) (2000) 739–768. [6] X. Li, A brief review: acoustic emission method for tool wear monitoring during turning, International Journal of Machine Tools and Manufacture 42 (2) (2002) 157–165. [7] C. Scheffer, P.S. Heyns, Wear monitoring in turning operations using vibration and strain measurements, Mechanical Systems and Signal Processing 15 (6) (2001) 1185–1202. [8] H. Cao, X. Chen, Y. Zi, F. Ding, H. Chen, J. Tan, Z. He, End milling tool breakage detection using lifting scheme and Mahalanobis distance, International Journal of Machine Tools and Manufacture 48 (2) (2008) 141–151. [9] D.E. Dimla Snr., The correlation of vibration signal features to cutting tool wear in a metal turning operation, The International Journal of Advanced Manufacturing 19 (10) (2002) 705–713. [10] P.D.T. O’Connor, Commentary: reliability-past, present, and future, IEEE Transactions on Reliability 54 (1) (2005) 3–10. [11] K. Kwon, D.M. Frangopol, Bridge fatigue reliability assessment using probability density functions of equivalent stress range based on field monitoring data, International Journal of Fatigue 32 (8) (2010) 1221–1232. [12] M. Haiba, D.C. Barton, P.C. Brooks, M.C. Levesley, Review of life assessment techniques applied to dynamically loaded automotive components, Computers & Structures 80 (5) (2002) 481–494. [13] N. Gorjian, L. Ma, M. Mittinty, P. Yarlagadda, Y. Sun, A review on degradation models in reliability analysis, in: Proceedings of the Fourth World Congress on Engineering Asset Management, Athens, Greece, 2009. [14] D. Xu, W. Zhao, Reliability prediction using multivariate degradation data, in: Proceedings of the 51st Annual Reliability and Maintainability Symposium, 2005, pp. 337–341. [15] C.J. Lu, W.Q. Meeker, Using degradation measures to estimate a time-to-failure distribution, Technometrics 35 (2) (1993) 161–174. [16] Z. Chen, S. Zheng, Lifetime distribution based degradation analysis, IEEE Transactions on Reliability 54 (1) (2005) 3–10. [17] R. Chinnam, B. Rai, Computation intelligence in online reliability monitoring, Intelligence in Reliability Engineering 40 (2007) 223–260. [18] D. Lin, M. Wiseman, D Banjevic, A.K.S. Jardine, An approach to signal processing and condition-based maintenance for gearboxes subject to tooth failure, Mechanical Systems and Signal Processing 18 (2004) 993–1007. [19] C.C. Lin, H.Y. Tseng., A neural network application for reliability modelling and condition-based predictive maintenance, International Journal of Advanced Manufacturing Technology 25 (2005) 174–179. [20] S.W. Menard, Applied logistic regression analysis, Sage publication Inc., London, United Kingdom, 2002. [21] J. Cai, R. Liu, Logistic regression model for isoconversional kinetic analysis of cellulose pyrolysis, Energy and Fuels 22 (2) (2008) 867–870. [22] P.M. Ridker, R.J. Glynn, C.H. Hennekens, Logistic regression model for determining risks factor for hypertensive disorders in pregnancy, WSEAS Transactions on Systems 5 (9) (2006) 2135–2139. [23] A. Jalkanen, U. Mattila, Logistic regression models for wind and snow damage in northern Finland based on the National Forest Inventory data, Forest Ecology and Management 135 (1) (2000) 315–330. [24] F.J. Janzen, H.S. Stern, Logistic regression for empirical studies of multivariate selection, Society for the Study of Evolution 52 (6) (1998) 1564–1571. [25] J. Yan, J. Lee, Degradation assessment and fault modes classification using logistic regression, Transactions of the ASME, Journal of Manufacturing Science and Engineering 127 (4) (2005) 912–914. [26] H. Liao, W. Zhao, H. Guo, Predicting remaining useful life of an individual unit using proportional hazards model and logistic regression model, in: Proceedings of Annual Reliability and Maintainability Symposium, 2006, pp. 127–132. [27] X. Hu, Z. Wang, X. Ren, Classification of surface EMG signal using relative wavelet packet energy, Computer Methods and Programs in Biomedicine 79 (3) (2005) 189–195. [28] S. Li, M.A. Elbestawi, Fuzzy clustering for automated tool condition monitoring in machining, Mechanical Systems and Signal Processing 10 (5) (1996) 533–550. [29] R.G. Silva, R.L. Reuben, K.J. Baker, S.J. Wilcox, Tool wear monitoring of turning operations by neural network and expert system classification of a feature set generated form multiple sensors, Mechanical Systems and Signal Processing 12 (2) (1998) 319–332.
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Reliability estimation for cutting tools based on logistic regression model using vibration signals Baojia Chen, Xuefeng Chen n, Bing Li, Zhengjia He, Hongrui Cao, Gaigai Cai State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China
a r t i c l e in f o
abstract
Article history: Received 1 June 2010 Received in revised form 28 September 2010 Accepted 2 March 2011 Available online 10 March 2011
As an important part of CNC machine, the reliability of cutting tools influences the whole manufacturing effectiveness and stability of equipment. The present study proposes a novel reliability estimation approach to the cutting tools based on logistic regression model by using vibration signals. The operation condition information of the CNC machine is incorporated into reliability analysis to reflect the product time-varying characteristics. The proposed approach is superior to other degradation estimation methods in that it does not necessitate any assumption about degradation paths and probability density functions of condition parameters. The three steps of new reliability estimation approach for cutting tools are as follows. First, on-line vibration signals of cutting tools are measured during the manufacturing process. Second, wavelet packet (WP) transform is employed to decompose the original signals and correlation analysis is employed to find out the feature frequency bands which indicate tool wear. Third, correlation analysis is also used to select the salient feature parameters which are composed of feature band energy, energy entropy and time-domain features. Finally, reliability estimation is carried out based on logistic regression model. The approach has been validated on a NC lathe. Under different failure threshold, the reliability and failure time of the cutting tools are all estimated accurately. The positive results show the plausibility and effectiveness of the proposed approach, which can facilitate machine performance and reliability estimation. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Reliability estimation Cutting tool Logistic regression model Wavelet packet decomposition Correlation analysis
1. Introduction In modern manufacturing systems, machine tools are the major equipment and play a very important role. The malfunction of machine tools may result in the halt of the whole production and bring about tremendous financial losses. For example, in the case of complex installations such as automobile assembly lines, it can be as high as $20,000 per minute [1]. Therefore, manufacturers have been paying great attention to machine reliability improvement to reduce unexpected downtime and raise product quality [2,3]. A cutting tool is an important part of machine tools and its reliability influences the total manufacturing effectiveness and stability of machine tools. With an accurate estimate of tool lifetime, worn tools can be changed in time to reduce waste product and tools costs noticeably. It is even possible to guarantee a certain surface quality. Tool failure and lifetime is judged by its wear measurement described in several standards (ISO3685, ISO8688 and ANSI/ASME B94.55M) [4]. These
n
Corresponding author. Tel./fax: þ 86 29 82663689. E-mail address: [email protected] (X. Chen).
0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.03.001
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standards provide wear threshold values at several points of the worn region which is acquired by its direct measurement (e.g., CCD cameras). Due to the constraints of high cost, discontinuity and susceptibility to operational environment, the direct methods are restricted to a very narrow range of applications and need to be improved. In indirect methods, the wear state can be estimated by the cutting force, torque, temperature, acoustic emission and vibration [5–7]. As one of the indirect sensorbased methods, vibration measurement is widely used due to its low price, easy implementation and on-line continuous testing characteristic. Some investigations have been carried out to correlate vibration signals to tool wear [8,9]. Their results show that vibration signature features extracted from time and frequency domains are sensitive to the tool wear. Now, the key issue to estimate cutting tools reliability indices is how to combine these vibration features using an appropriate reliability model. Traditional approaches to reliability analysis of physical and other electro-mechanical products are predominantly based on lifetime of large sample size [10]. Such reliability analyses have been extensively studied in the past [11,12]. With few exceptions, all these analyses are geared towards estimating population characteristics of a system, subsystem or component. However, for a single and small sample device(s), such statistic data are of little meaning. People are particularly interested in the life margin and current reliability of items used in their systems. Wherefore degradation measures often provide more information about device performance and precision during operation [13,14]. It is helpful to determine the product time-varying characteristics and reflect the relationship between failure and performance degradation. In general, degradation is the reduction in performance, reliability and life span of assets. There are a number of researches and applications on reliability assessment by using degradation data. Lu and Meeker [15] introduced a general nonlinear mixed-effects model and developed a Monte Carlo simulation procedure to calculate an estimate of the distribution function of the time-to-failure. Chen and Zheng [16] proposed an alternative approach which makes inference directly on the lifetime distribution. Chinnam and Rai [17] collected and analyzed thrust-force and torque signals generated by drilling operation to predict on-line reliability of the drill-bit. Lin et al. [18] demonstrated a procedure to extract useful condition indicators from vibration signals and use the proportional hazards model (PHM) to develop optimal maintenance policies for the gearboxes. Lin and Tseng [19] combined the Weibull PHM and vibration-based machine condition monitoring techniques to estimate several machine reliability statistics. However, the above methods require a specific mechanical knowledge and make many assumptions about condition parameters degradation paths and their distribution probability density functions. Logistic regression is a robust tool that can easily represent multiple variables as a dichotomous problem [20]. It has been widely used in bionomics, biology, epidemiology, etc. [21–23]. The main advantages are that logistic regression does not rely on assumptions of normality for the predictor variables or the errors and it allows the selection effect to vary nonlinearly [24]. The independent variable can be continuous, discrete or dummy variable. Based on its likelihood function, it needs less computation effort to estimate parameters rather than PHM. [3] Assuming a facility performance state is a process from normal to failure, and may be reflected and interpreted by multiple feature parameters. The logistic regression model could be used to analyze and describe the failure probability of facilities under certain conditions, which would benefit decision making (using, repairing or replacing). Some exploratory work has been done on mechanical equipment performance evaluation and remaining useful life (RUL) prediction. Yan and Lee [25] explored a method to assess real-time performance of elevator door system by inputting features of online data to the logistic model. Liao et al. [26] presented a combination of the proportional hazards model and logistic regression model to predict the unit RUL. In the present study, vibration signals and tool wear were measured together during machining process. The salient features indicative of tool wear were extracted through signal processing techniques and integrated in a logistic regression model to evaluate reliability of lathe cutting tools. The rest of the paper is organized as follows: the reliability assessment method based on logistic regression model is described in Section 2, the experimental design and setup are introduced in Section 3, the vibration signals analysis, reliability modeling and application results are discussed in Section 4. Finally, some concluding remarks are given in Section 5.
2. Reliability estimation method based on logistic regression model 2.1. Wavelet packet decomposition Given a finite energy signal whose scaling space is assumed as U00 , wavelet packet (WP) transform can decompose U00 into small subspaces Ujn in dichotomous way [27]. The dichotomous way is realized by the following recursive scheme: Ujnþ 1 ¼ Uj2n Uj2n þ 1
j 2 Z, n 2 Z þ
ð1Þ
where j (jr0) is the resolution level and denotes orthogonal decomposition. Z denotes the domain of integers. Z þ denotes the domain of positive integers. Ujnþ 1 , Uj2n and Uj2n þ 1 are the three close spaces corresponding to cn(t), c2n(t) and c2n þ 1(t). cn(t) satisfies the following equation: pffiffiffiX pffiffiffiX hðkÞcn ð2tkÞ c2n þ 1 ðtÞ ¼ 2 gðkÞcn ð2tkÞ
c2n ðtÞ ¼ 2
k2Z
k2Z
ð2Þ
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Note that the first wavelet is the so-called mother wavelet function
c0 ðtÞ ¼ jðtÞ, c1 ðtÞ ¼ cðtÞ
ð3Þ
The discrete filters h(k) and g(k) are quadrature mirror filter (QMF) associated with the scaling function j(t) and the mother wavelet function c(t). The sub-signal at Ujn1 , the nth subspace on the jth level can be reconstructed by a linear j,n combination of WP function ck ðtÞ as follows: X j,n j,n Dk ck ðtÞ k 2 Z ð4Þ snj ðtÞ ¼ k2Z
can be obtained from The WP coefficients Dj,n k Z þ1 j,n ¼ f ðtÞck ðtÞdt Dj,n k
ð5Þ
Since the wavelet cj,k(t) is an orthogonal basis at L2(R), the energy of the sub-signal snj ðtÞ is calculated by X j,n 2 En ¼ Dk
ð6Þ
The normalized energy of snj ðtÞ is ! X En Pn ¼ En =
ð7Þ
1
k
n
Energy entropies is a frequently used feature to detect the change of signal energy in different frequency bands, which can reveal the amount of information stored in observed signal. It is defined as X I ¼ En log En ð8Þ n
2.2. Feature definition Both amplitude and distribution of the vibration signals change along with tools’ state from sharp to worn. Previous studies have shown that some time-domain features can indicate tool wear [28,29]. In this work, 11 time-domain feature parameters are extracted: mean (xm), peak (xp), root amplitude (xra), root mean square (xrms), standard deviation (xstd), skewness (xske), kurtosis (xk), crest (xc) margin (xma), shape (xsha) and impulse factor (xi). The first four parameters reflect the vibration amplitude and energy in time domain. The remaining parameters represent the time series distribution of the signal in time domain [30]. A brief mathematical description of these features is shown in Table 1. 2.3. Feature selection based on correlation analysis Although the above mentioned features are indicative of tool wear from different aspects, they have different important degrees. Some features are salient and closely related to the wear, but others are not. Thus, before a feature vector is input into logistic regression model, it is necessary to select salient features of tool wear state from the feature set and discard the irrelevant or redundant features to improve the estimating accuracy and avoid the problem of dimensionality. Here, correlation analysis is employed to select the salient features. The correlation coefficient (CC) r(U, V) between two
Table 1 The feature parameters. xm ¼
PN n ¼ 1
xðnÞ
rP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N
xstd ¼ xma ¼
xp ¼ max9xðnÞ9
N
ðxðnÞxm Þ2 N1
n ¼ 1
xp xra
xske ¼ xsha ¼
PN n ¼ 1
ðxðnÞxm Þ3
ðN1Þx3std
xrms xm
where x(n) is a signal series for n¼1,2, y, N, N is the number of data points.
xra ¼ xk ¼ xi ¼
PN n ¼ 1
pffiffiffiffiffiffiffiffiffi2 jxðnÞj
N
PN n ¼ 1
ðxðnÞxm Þ4
ðN1Þx4std
xp xm
rP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N
xrms ¼ xc ¼
xp xrms
n ¼ 1
N
ðxðnÞÞ2
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one-dimensional vectors or signals U and V of the same length is calculated by equation ðUUÞðVVÞ
rðU,VÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð9Þ
ðUUÞðUUÞ0 ðVVÞðVVÞ0 Þ
where U and V are the mean values of U and V, respectively. The value of r ranges between 1 and þ1. r 4 0, r o0 and r ¼0 indicate a positive correlation, negative correlation and irrelevance, respectively. The larger the absolute value of r, the higher the degree of correlation. 2.4. Logistic regression model Wear state of cutting tools can be indicated by a set of characteristic parameters. If observed samples which consist of characteristic parameters and cutting tool state are acquired, the logistic regression can be used to set up the relationship between normality and failure [31]. Suppose at time ti, the cutting tool condition feature is a kþ1 dimensional vector Xi ¼(1, x1i, x2i, y, xki)0 and tool state is yi (under normal state, yi ¼1, otherwise yi ¼0), the reliability function of the cutting tool can be described as Rðti 9Xi Þ ¼ Pðyi ¼ 19Xi Þ ¼
expðBXi Þ 1 þ expðBXi Þ
ð10Þ
where B ¼ ðb0 , b1 , . . ., bk Þ is the model parameter vector and b0 40. The logistic or logit regression model is LogitðyÞ ¼ ln
Rðti 9Xi Þ ¼ BXi 1Rðti 9Xi Þ
ð11Þ
The log-likelihood function (LLF) of the observed degradation features can be expressed as X ½yi BXi lnð1 þexpðBXi ÞÞ ln½LðBÞ ¼
ð12Þ
i
Since logistic regression is nonlinear, the model parameters can be obtained by maximizing the log-likelihood function through the Nelder–Mead’s algorithm. Once the model parameters are identified, the reliability and 95% confidence interval (CI) indicated by feature vector Xj can be calculated as Rðtj Þ ¼ Pðyj ¼ 19Xj Þ ¼ 2 6 6 4
^ jÞ expðlog^ itðyj ÞÞ expðBX ¼ ^ jÞ 1 þ expðlog^ itðyj ÞÞ 1 þexpðBX
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ j 1:96 VarðBX ^ jÞ exp BX qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ^ j 1:96 VarðBX ^ jÞ 1þ exp BX
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ^ j þ1:96 VarðBX ^ jÞ exp BX 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7 5 ^ j þ 1:96 VarðBX ^ jÞ 1 þexp BX
ð13Þ
ð14Þ
^ j Þ is the variance of model parameters and can be calculated as where VarðBX ^ j Þ ¼ X0 COV ðBÞXj VarðBX j
ð15Þ
where COV(B) is model parameters covariance matrix. 2.5. The proposed reliability estimating method In the field of rotating machinery, the running state of the machine can usually be identified by finding certain characteristic frequencies and changes of their amplitude. However, there are not such particular frequencies to indicate the cutting tools conditions of lathe. As reported in [32], the characteristics associated with tool wear mainly focus on a specific frequency band. Therefore, WP transform is employed to decompose vibration signals into different frequency bands and correlation analysis is used to select salient features indicative to tool wear. The proposed method is described in Fig. 1. It includes the following procedures. First step is data acquisition. The vibration signals and tool wear are measured by an accelerometer and a micro-optical system. Second step is feature extraction and selection. When the original vibration signals are decomposed to the fourth resolution level (j ¼ 4) with WP transform, the whole scaling space is divided into 16 frequencies bands. The WP energy and energy entropy are calculated by Eqs. (6) and (8). By correlation analysis, the feature frequency bands sensitive to wear change can be selected out. Similarly, the 11 timedomain features of each feature band are extracted and salient features are selected out by applying correlation analysis. The feature vector X can be obtained by combining the time-domain features with energy entropy and feature band energy. If flank wear value VB is smaller than the threshold Vt, the tool state index Y¼1, otherwise Y ¼0. Vt is selected as
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Accelerometer
micro-optical system
Vibration Signals
Tool Wear
Data Acquisition
NC lathe
WPD
WP Energy
No
VB≤Vt
Feature Band Time - domain feature Energy
Correlation analysis
Salient feature
X
Yes
Y=0
Y=1
Y
Logistic regression model
Reliability indices
Reliability Estimation
WP entropy
Features Extraction and Selection
Correlation analysis
Fig. 1. Flow chart of reliability estimation procedure.
0.6 mm in the study with reference to ISO3685. Finally, by taking X and Y as input, logistic regression model is established and applied to estimate the reliability of other tools being used. 3. Experimental design and setup The experimental setup for the in-process detection of tool wear is illustrated in Fig. 2. An CNC lathe is employed for the interrupted cutting of steel bars fixed to the work holder. The cutting tool used in this test is a carbide tool and the experiment conditions and specifications are shown in Table 2. Tool wear VB is measured by a micro-optical system with a CCD camera, an adjustable circular-ring-mode LED light and a built-micrometer. The maximum resolution is 0.01 mm. The tool vibration signals are monitored by an accelerometer and then sent to a data acquisition system (LMS SCADAS305) and a portable computer. At the same time, the acoustic emission (AE) signals, the workpiece surface roughness data and the current signals of spindle motor and Z-servo motor are also measured, respectively, for further research. The type and model of all used sensors are listed in Table 3. The installation of micro-optical system, accelerometer and AE sensor are shown in Fig. 3. 4. Result and discussion 4.1. Vibration signal analysis In the manufacturing process, tool vibration is caused by a variety of factors, such as cutting force, built up edge, material inhomogeneity, ambient conditions and tool wear. Hence the main purpose of vibration analysis is to find the features indicative of tool wear. In the present experiment, a total of 12 cutting tools’ vibration signals and wear data were measured. It is assumed that all investigated tools have the same wear mechanism and vibration characteristic with increasing tool wear. Among them, tool no. 4 is taken as an example to show the whole analysis process. The on-line vibration signals were measured and the sampling frequency was 32,768 Hz, which was determined by the frequency
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Roughometer Workpiece
Current Sensor
Spindle motor
Cutting tool Current Sensor Micro-optical system
AE sensor
Accelerometer Z-servo motor
LMS Signal Acquisition and Analysis Instrument
Portable computer
Digital phosphor oscilloscope
Fig. 2. The schematic diagram of the experimental setup.
Table 2 Experimental conditions. Lathe
Cutting Tool
Workpiece Material Cutting conditions
Type: horizontal Model: FTC-20 Spindle speed: 45–4500 rpm Rated power: 11 kW Type: diamond carbide tool Model: CNMG120408-HM Material: 42CrMo4 45# Steel bars Feed-rate f: 0.15 mm/rev Cutting speed vc: 200 m/min Depth of cut ap: 2 mm
Table 3 Type and model of all sensors. Signal
Sensor type
Sensor model
Tool wear Vibration AE
Micro-optical system Accelerometer AE Sensor Coupler Roughometer Hall current sensor
MZDH0670 PCB ICP352C34 Kistler 8152B Kistler 5125 SJ-201 GAA-KY1
Roughness Current
characteristics of tool vibration. The vibration signal and its spectrum of tool no. 4 after running for 78 min are shown in Fig. 4. It can be observed from the figure that the signal contains abundant frequency components and the energy is mainly focused on two band ranges of 2000–4000 and 7000–10,000 Hz. As shown in Fig. 4, the vibration signals are decomposed by WP transform. Because Daubechies family of wavelet packets seems to resemble the cutting tools vibration signal most and Daubechies 10 is adopted as the mother wavelet. In order to find out the feature frequency band indicative of tool wear, the WP energy of different sampling time is calculated and normalized. The results are shown in Fig. 5(a)–(d). It can be clearly observed that the signal energy mainly concentrates on band interval 7–10 and the maximum energy concentrates on band 9. With the processing time increasing, the signals’ energy gradually transfers from high frequency to low frequency. As expressed in Fig. 5, the energy
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CCD camera
Bracket
Micro-optical system
AE Sensor
Cutting tool
Accelerometer
Fig. 3. The installation of micro-optical system, accelerometer and AE sensor.
Fig. 4. Vibration signal and its spectrum of tool no. 4 after running for 78 min.
transfers from band interval 8–10 to band interval 3–7. But the energy above band 11 changes little. For a detailed observation, the energy changing trends over time of bands 7 and 9 are depicted in Fig. 6(a)–(b). From time t ¼69–93 min, the energy ratio of band 7 P7 increases from 0.02 to 0.13. In contrast, the energy ratio of band 9 P9 reduced from 0.48 to 0.30. This phenomenon may be caused by the increase of tool wear, which results in increasing contact area between the tool and workpiece. Fig. 6(c)–(d) are energy entropy and tool wear changing trends. The tool wear is monotonically increasing. But the band energy and the entropy are non-monotonically increasing or decreasing. Their trends are conspicuous and closely related to the wear. The above procedure for analysis is also applied to other 11 tools. Results are similar to those of tool no. 4, which bears out our earlier assumption. The CCs between WP band energy, entropy and corresponding tool wear are also calculated. In order to eliminate the randomness, the averaged coefficients are calculated and displayed in Fig. 7. It can be observed that the CCs of bands 4, 7, 9 energy and entropy are larger, reaching more than 0.6. Significantly, the CCs of bands 7 and 9 are more than 0.7. In the next step, sub-signals of bands 7 and 9 are selected to calculate the 11 time-domain features listed in Table 1 to acquire more salient features. According to Fig. 1, the analyzing procedure is implemented and the averaged CCs between the time-domain features and tool wear are obtained and listed in Fig. 7. Compared with band 9, the correlation degrees of band 7 features are higher, especially the CCs of the 1st, 3rd, 4th and 5th time-domain feature corresponding to xm7, xra7, xrms7 and xstd7, respectively, are also larger than 0.6. The total number of selected features by correlation analysis is 4 time, 3 WP energy and 1 entropy, whose CCs are larger than 0.6, which yields a total of 8. They compose the input vector X¼(xm7 xra7 xrms7 xstd7 P4 P7 P9 I)0 of logistic regression model.
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Fig. 5. The WP energy spectrums at different sampling time: (a) t¼ 69 min, (b) t ¼73 min, (c) t ¼81 min and (d) t ¼89 min.
Fig. 6. The changing trends of different feature parameters (a) energy change of band 7, (b) energy change of band 9, (c) energy entropy change of tool no. 4 and (d) tool wear of tool 4.
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Fig. 7. The CCs between WP energy, entropy, time-domain features and tool wear.
Table 4 Information measures of different models. No. Feature set
Model expression (log^ itðyÞ ¼ )
AIC
SC
BIC
1 2
7.39 17.53P7 þ32.19P9 þ3.11I 2.26 164.2xm7 þ 141.3xra7 7.92xrms7 þ 64.47xstd þ0.32P4 21.25P7 þ 29.6P9 þ 2.86I 5.29 þ92.37xm7 104.9xra7 7.49xrms7 17.61P7 þ 20.4P9
68.14 79.64 559.01 69.61 95.49 543.16
3
Optimal subset: P7,P9,I All features: xm7, xra7, xrms7, xstd7, P4, P7, P9, I CC40.65 subset: xm7, xra7, xrms7, P7, P9
4
CC40.7 subset: P7, P9
2.02 19.33P7 þ19.33P9
72.25 80.87 557.78
69.07 86.32 552.33
4.2. Reliability modeling and estimation After data preparation, by taking X as independent variable and tool state y as dependent variable, 131 samples were obtained from the selected salient features and corresponding state data of tool nos. 1–11. Logistics regression model was set up by using SAS software [33]. To find out the optimal subset, forward stepwise, backward stepwise and combined stepwise methods were applied. Score statistics and Wald statistics were used to add and delete variable. The three methods have the same model as p^ i ð16Þ ¼ BXi ¼ 7:3917:53 P7 þ 32:19 P9 þ 3:11I log^ itðyÞ ¼ ln 1p^ i It can be observed that the calculated model has no time-domain features although their CCs also above 0.6. This is because the model variable selection is an automatic process according to specific statistic criterion. Only those parameters that make the greatest contribution to the model LLF defined as Eq. (12) could be selected based on a certain significance level. Maybe the contributions of time-domain features are all great when they act as a signal model variable. But if they are correlated with other variables (e.g., energy features), multicollinearity will weaken their significance level and make them not appear in the model. At the same time, three other models were also set up and their information measures are listed in Table 4. It is obvious that the Akaike information criterion (AIC) and Schwarts criterion (SC) of model 1 are smaller than those of the three other models, and the absolute value of Bayesian information criterion (BIC) is larger than those of the other three models. According to [33], it implies that the fit goodness of model 1 is higher than other models. It means that the model 1 better describes the relationship between indicators and tool state. So model 1 (i.e. Eq. (16)) has been used as the model for cutting tool reliability evaluation. To identify the validity of the proposed model, the 11 condition feature vectors of tool no. 12 were input into Eqs. (16) and (5) to calculate the reliability and 95% CI. As depicted with thin solid and dashed lines in Fig. 8, the estimated reliability indices show an expected decreasing trend with the growth of cutting time. Due to the discreteness and randomness of the sampling data, the reliability curves are discontinuous and non-monotonically declining. In order to represent the actual continuous degradation process of cutting tool, single variable logistics regression model is employed
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1 0.9 0.8
Reliability R
0.7 0.6 0.5 0.4 0.3 estimated reliability 95% CI of estimated reliability fitted reliability fitted 95% CI
0.2 0.1 0 75
80
85
90 Time t/min
95
100
105
Fig. 8. The estimated reliability indices (Vt ¼ 0.6 mm).
1 0.9 0.8
Reliability R
0.7 0.6 0.5 0.4 0.3 0.2 0.1
estimated reliability 95% CI of estimated reliability fitted reliability fitted 95% CI
0 80
85
90 Time t/min
95
100
Fig. 9. The estimated reliability indices (Vt ¼ 0.5 mm).
to fit the reliability curve and its 95% CI. The smoothed reliability curves are shown with thick solid and dashed lines in Fig. 8. From time t ¼92–98 min, the fitted model curve has some deviation from the estimated. It is caused by fitting error. Compared with the cutting tool life, the region is so narrow that it has little influence on the estimation result from the point of engineering application. As the logistics model is set up on the basis of tools’ two state (normality or failure), we decided that the maximum reliability allowed is 50%. If the reliability value decreases below 0.5, the tool is considered to be invalid, i.e. the tool wear is getting larger than 0.6 mm. The real measured failure time of tool no. 12 is 98.6 min (VB ¼0.607 mm) and the estimated failure time of the smoothed model is 100 min. The estimated error is 1.42%. In practical production, the threshold of flank wear may be set flexibly to satisfy the requirement of surface quality. Supposing the threshold Vt ¼0.5 mm, the re-estimated reliability model is shown as p^ i ð17Þ log^ itðyÞ ¼ ln ¼ BXi ¼ 1:38727:0757xrms7 þ 1:8269xstd7 6:2087 P7 þ 12:5146 P9 þ 1:4842I 1p^ i Similarly, the estimated reliability curves and smoothed reliability curves are also estimated and shown in Fig. 9. Compared with Fig. 8, the decreasing trend is relatively smaller and the 95% CI is larger, which may be due to the more failure samples with the lowered limit. The measured real failure time of tool 12 is 93.8 min (VB ¼0.512 mm) and estimated failure time is 96.76 min. The estimated error is 3.6%. It can be seen that the two established models both accurately estimate the reliability and failure time of tool 12.
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5. Conclusion The present study proposes a reliability estimation approach to cutting tools, which is based on a logistic regression model by using vibration signals. WP decomposition and correlation analysis are applied to extract and select the salient features. The selected band energy, energy entropy and time-domain features show a high degree of correlation to tool wear. Without requiring a specific mechanical knowledge and making many assumptions about probability density functions of variables, the logistic regression model for reliability estimation can be established just by acquiring the tools’ feature vector and their corresponding state. A case study shows that the reliability indices and failure time of the tool can be accurately assessed by using the obtained model and the approach is also proved effective when the threshold is changed. The positive results of this research clearly demonstrate the potential of this approach in evaluating performance and reliability. Future work can be centered on model validity in case of small samples and combining with other methods to predict machine tool performance. In addition, we look forward to do more research on the robustness of the proposed method and the influence of tool material, workpiece material, cutting parameters to the model.
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