condition-based maintenance optimization for motorized...

Research ArticleCondition-Based Maintenance Optimization for MotorizedSpindles Integrating Proportional HazardModel with SPC Charts

Xuejiao Du 1 Jingbo Gai 1 and Cen Chen 2

1College of Aerospace and Civil Engineering Harbin Engineering University Harbin 150001 China2Department of Electrical Engineering Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Xuejiao Du duxuejiaohrbeueducn

Received 22 May 2020 Accepted 29 June 2020 Published 21 July 2020

Academic Editor Francesco Pellicano

Copyright copy 2020 Xuejiao Du et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Reliability of motorized spindles has a great effect on the performance and productivity of computer numerical control (CNC)machine tools for intelligent manufacturing Condition-based maintenance (CBM) is an efficient method to prevent seriousfailures to improve system reliability and to reduce management costs for motorized spindles However owing to variousdegradation features acquired during condition monitoring the challenge is to propose an appropriate feature to evaluate thereliability level of motorized spindles and to set up optimal CBM policies Based on the motivation a three-stage approach isproposed in this paper In the first stage proportional hazard model (PHM) is developed to describe the reliability consideringfailure events together with multiple degradation features Next statistical process control (SPC) charts are constructed forcondition monitoring and anomaly detection in order to achieve early detection of potential failures At last a CBM schedule ismodeled in consideration of maintenance cost minimization the maintenance plan is optimized by determining the optimalcontrol limits of SPC charts

1 Introduction

As an emerging technology in the field of intelligentmanufacturing the motorized spindle of the computernumerical control (CNC) machine tool is regarded as a corecomponent of all manufacturing facilities its performancecould determine the machining accuracy and productionefficiency of CNC machine tools even the wholemanufacturing supply chain to a large extent [1] (us itsreliability improvement and health management causewidespread concern For the purpose of improving theavailability and reducing production expenses of the mo-torized spindle condition-based maintenance (CBM) hasdrawn much attention [2] It focuses on utilizing the con-dition monitoring information collected in real time toassess health status and to develop optimal maintenancestrategies [3]

As a condition monitoring method control charts of thestatistical process control (SPC) theory can be incorporatedwith maintenance analysis to realize CBM optimization

Shrivastava et al [4] integrated the cumulative sum(CUSUM) control chart with preventive maintenance (PM)planning for joint optimization of maintenance activitiesand chart parameters Integrated models combining the X

control chart with PM actions were proposed in [5 6] inorder to protect the equipment from quality declineLampreia et al [7] modified CUSUM and exponentiallyweighted moving average (EWMA) control charts for onlinevibration monitoring thus reducing false alarms andplanning early interventions (e similar modelling idea wasalso adopted for the economic design of maintenance planswith the Shewhart control chart chi-square control chartetc as discussed in [8ndash12] Mehrafrooz and Noorossana [13]developed a model under the assumption of perfect main-tenance the model could express six alternative scenarios ofthe monitored process to achieve the most economicmaintenance schedule (e integrated model was furtherdeveloped by Yin et al [14] considering ten scenarios of thedelayed monitoring process From the perspective of reli-ability improvement some SPC charts were also established

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7618376 11 pageshttpsdoiorg10115520207618376

to monitor the feature of time-between-events (TBE) as-suming that TBE followed the gamma distribution [15] thenormal distribution [16] the exponential distribution [17]the Weibull distribution [18] etc

However these aforementioned control charts are allunivariate which are commonly used to monitor indi-vidual observations of a certain type of characteristicvariable For the case of the motorized spindle conditionmonitoring usually contains multiple sensor signals suchas vibration bearing preload acoustic emission soundand temperature [19] and the signals are often auto-correlated [20] (erefore the conventional SPC charts willnot be effective to extract useful and reliable informationfrom the monitoring process leading to larger errors inCBM optimization

To overcome this problem the paper develops a three-stage approach for the purpose of extracting the fusionfeature monitoring the health degradation and optimizingthe CBM strategy for motorized spindles (e main con-tributions of the proposed approach are described as follows

(1) Proportional hazard model (PHM) is introduced toextract the health indicator (e benefits can bedemonstrated in two aspects on the one hand PHMcan integrate multivariate information frommultiplemonitoring signals together with failure data whichresults in the more accurate degradation detectionfor CBM optimization On the other hand thehazard rate from PHM is selected as the health in-dicator of SPC charts It can reduce random inter-ference for SPC charts when monitoring theautocorrelation process the detection error such asthe false alarm or undetected failure can be effec-tively avoided

(2) CBM optimization is combined with SPC chartdesign based upon economic considerations thehealth condition of motorized spindles can beconstantly updated and two types of the failuremechanism including the random failure and thefatigue failure and four possible scenarios are takeninto consideration (e integrated model can avoidunnecessary or untimely maintenance and achieveconsiderable economic benefit by reducing main-tenance costs and raising production efficiencywhich provide effective service and guidance forpractitioners

(e remainder of the paper is organized as followsSection 2 describes the CBM optimization problem andassumptions in detail in which the failure mechanism SPCcharts and maintenance policy are discussed In Section 3the integrated model for combined SPC chart design andCBM scheduling is developed (e hazard rate extractedfrom the PHM is used to characterize the health conditionand the SPC scheme considering four possible scenarios isestablished to monitor the degradation trend in real time(en Section 4 introduces the proposed three-stage ap-proach for optimizing the CBM schedule upon economicconsiderations In association with the PHM and the SPC

charts the average long-run cost during a renewal cycle isminimized by optimizing the upper control limit (UCL)Next a real case of the motorized spindle is studied to verifythe feasibility and practicability of the proposed approach inSection 5 Finally conclusions are presented in the lastsection

2 Problem Statement and Assumptions

For this research the degradation of the motorized spindleduring the long life cycle is due to both external factors likecumulative shocks environmental stress etc and internalfactors like continuous aging etc (erefore the system willsuffer from random failures and fatigue failures Multiplecondition features together with failure event data can beused as the health indicator to characterize the effects ofdegradation

(en the SPC chart is used as a failure diagnosis tool formonitoring the health indicator and detecting the two typesof failures the random failure also known as the hardfailure which is self-announcing and could be detectedimmediately and the fatigue failure also known as the softfailure which is defined to occur only when the degree ofdegradation reaches the preset control limits (e SPC chartscheme is shown in Figure 1 and its control limits can beusually computed following the formula [21] in Table 1according to different SPC charts

When combined withmaintenance planning the controllimits of SPC charts can be regarded as the trigger of amaintenance action the optimization of the CBM strategy istransformed into determining the optimal control limits inorder to minimize the maintenance cost from the per-spective of economic efficiency

(e integrated CBM model is developed under the as-sumptions as follows

(1) (e changing trend of the health indicator may notbe obvious over time and increase rapidly after thesystem enters the wear-out stage In order to graspsystem degradation level and implement mainte-nance actions in time inspections are scheduledevery Δ time unit to detect the health conditionperiodically (e inspection time is short so that theinspection activities are considered instantaneousand nondestructive (e cost per inspection Ci ispredetermined as a constant according to the actualcondition

(2) (e two types of failures correspond to two types ofmaintenance activities which are the preventivereplacement and the corrective replacement re-spectively To be more specific if a hard failureoccurs unexpectedly the corrective replacement isperformed instantly and the average cost per cor-rective replacement Cc covers all costs with respect tothe failure and its associated problems On thecontrary if the monitoring health indicator is de-tected to exceed the control limits at the inspectiontime which is interpreted as a soft failure thepreventive replacement is carried out (e average

2 Mathematical Problems in Engineering

cost per preventive replacement Cp covers the costsfor performance maintenance improvement andother related operations In generalCc gtCp since thecorrective replacement is relatively more complexthan the preventive replacement

(3) After each replacement the condition of the mon-itoredmotorized spindle will be restored to the initialstate representing that the system is under the ldquoasgood as newrdquo maintenance strategy In addition it isassumed that the time for both preventive andcorrective replacements is not considered in theCBM model

3 Condition-Based Maintenance (CBM)Model Development

(e integrated model for SPC chart design and CBMscheduling is developed from an economic point of viewwhose objective is to minimize the average long-run costconsidering the costs of all inspections and maintenanceactivities

31 Proportional Hazard Model (PHM) PHM proposed byCox [22] is a popular model which is used to describe therelationship between the degradation influence and thehazard rate (e degradation level is characterized bymultiple features during lifetime operation which areconsidered as covariates(e advantage of using the PHM in

CBM is that it could deduce the hazard rate considering allcovariates in the current state such as the failure event andcondition monitoring factors According to the PHM thehazard rate at time t is defined by

h(t Z(t)) h0(t)exp(cZ(t)) (1)

where h0(t) is the baseline hazard rate at time t and it will becontinually modified by Z(t) let Z(t) [Z1(t) Z2(t) middot middot middot

Zn(t)] be a vector made up of values of n conditionmonitoring features at time t which is a time-dependentstochastic process representing the degradation evolutionAnd c is a vector made up of regression coefficients whichdetermines the influence weight of the monitoring featureson the failure rate

(e time to failure of the electromechanical systemcommonly follows the Weibull distribution [23] As formotorized spindles their baseline hazard rate of the PHM isassumed to be derived from a two-parameter Weibull dis-tribution which has the expression as

h0(t) βη

t

η1113888 1113889

βminus1

(2)

where βgt 0 represents the shape parameter of the Weibulldistribution and ηgt 0 represents the scale parameter of theWeibull distribution

Substituting the Weibull baseline hazard rate functionZ(t) and c vectors into the PHM Weibull proportionalhazard model (WPHM) is derived as

Table 1 Characteristics of four typical SPC charts

Feature Monitored statistic Control limitsShewhart zt yt UCL μ0 + nσ0 LCL μ0 minus nσ0MA zt (y1 + y2 + middot middot middot + ytminusw+1w) UCL μ0 + n(σ0

w

radic) LCL μ0 minus n(σ0

w

radic)

CUSUM zt max(0 ztminus1 + yt minus μ0 minus k) UCL dkEWMA zt yt + (1 minus λ)ztminus1 UCL μ0 + nσ0

(λ2minusλ)

1113968LCL μ0 minus nσ0

(λ2minusλ)

1113968

yt Monitored variable at t μ0 (e mean of in-control dataσ0 (e standard deviation of in-control data n Control limit widthW Moving average width k (e slope of the lower armD Distance typically set to 10 λ (e weighted parameter 0lt λle 1UCL Upper control limit LCL Lower control limit

Mon

itore

d he

alth

indi

cato

r UCL

Fatigue failure

Random failureInspection

point

0 t1 t2 t3 t t1 t2 tk tk+1 tk+2

Time

Figure 1 (e SPC chart scheme of motorized spindles

Mathematical Problems in Engineering 3

h(t Z(t)) βη

t

η1113888 1113889

βminus1

exp c1Z1(t) + middot middot middot + cnZn(t)( 1113857 (3)

Given theWPHM then the conditional reliability can beobtained as

R(t Z(t)) P(Tgt t | Z(s) 0le sle t)

exp minus 1113946t

0h(s Z(s))ds1113888 1113889

(4)

where the data of failure time are denoted by T and Z(s) isthe actual value of Z(t) at time s

(e corresponding probability density function (PDF) isderived as

f(t Z(t)) h(t Z(t))R(t Z(t)) h0(t)exp(cZ(t))

exp minus 1113938t

0 h(s Z(s))ds1113872 1113873

(5)

Based on the condition monitoring system the degra-dation features are acquired in real time and the covariatescould be predicted and updated expressed asZ(v) 0lt vltinfin (erefore the remaining useful life ofmotorized spindles can be predicted as derived in [24] by

L(t) E[T minus t | Tgt t] 1113946infin

0exp minus 1113946

t+τ

th(υ Z(υ))dυ1113888 1113889dτ

(6)

According to the monitoring data of multiple featuresand the failure event data the parameters of the WPHM canbe estimated by the maximum likelihood method Since themotorized spindle may run to failure or be maintainedbefore failures in the actual CBM process the failure eventdata collected involve both time to failure and censored timeTaking these data types into consideration the likelihoodfunction is given by

L(middot) 1113945n

i1f ti Z ti( 1113857( 1113857 1113945

m

s1R ts Z ts( 1113857( 1113857 (7)

where ti represents the value of time to failure ts representsthe value of censored time n is the number of failure timesand m is the number of censored time (e parameter es-timate of β η and c can be calculated by maximizingequation (7)

As a result the hazard rate in real-time operation can beextracted by the PHM it is adopted as the health indicatorinterpreting the current performance state In the next stepSPC techniques will be applied to monitor the changingtrend of the hazard rate and provide CBM decision support

32 Integrated Model and Optimization In the case ofcombining SPC charts with CBM scheduling for mainte-nance strategy optimization of motorized spindles fourpossible scenarios of the integrated CBM model can occurduring a renewal cycle as shown in Figure 2 A renewal cycleis identified as the period between start of process

monitoring and a maintenance activity [25] and the fourdifferent scenarios are further illustrated as follows

Scenario 1 (S1) the performance of the motorizedspindle remains in control with no false out-of-controlalarm which is interpreted as h(t Z(t))leUCLWhen ahard failure occurs in the interval(kΔ (k + 1)Δ) k 0 1 middot middot middot n a corrective replace-ment is immediately implemented and restores thespindle to its initial conditionScenario 2 (S2) the performance of the motorizedspindle remains in control until a false out-of-controlalarm is given at (k + 1)Δ k 0 1 middot middot middot n A preventivereplacement is performed and restores the spindle to itsinitial conditionScenario 3 (S3) the performance of the motorizedspindle shifts to the out-of-control condition in theinterval (k1Δ (k1 + 1)Δ) k1 0 1 middot middot middot n which isinterpreted as h(t Z(t))leUCL However the SPCchart fails to capture an alert signal until the systemresults in a hard failure in the interval(k2Δ (k2 + 1)Δ) k2 0 1 middot middot middot n A corrective re-placement is then executed and restores the spindle toits initial conditionScenario 4 (S4) the performance of the motorizedspindle degrades to the out-of-control condition in theinterval (k1Δ (k1 + 1)Δ) k1 0 1 middot middot middot n minus 1 and theSPC chart succeeds to detect the out-of-control signalat k2Δ k2 k1 + 1 middot middot middot n A preventive replacement iscarried out and restores the spindle to its initialcondition

Considering the four possible scenarios the averagelong-run CBM cost c can be transformed to the expectedcost over the expected length of a renewal cycle [26] which isgiven by

c E(C(T))

E(T) (8)

where E(C(T)) 11139364i1 p(Si) 1113936

mi

j1 E(C | Sij)p(Sij) andE(T) 1113936

4i1 p(Si) 1113936

mi

j1 E(T | Sij)p(Sij) p(Si) represents theprobability of the ith scenario p(Sij) represents the proba-bility of the jth state in the ith scenario and mi represents thenumber of states of the ith scenario which are expressed as

p Si( 1113857 p Si1( 1113857 + p Si2( 1113857 + middot middot middot + p Simi1113872 1113873

m1 n + 1 m2 n

m3 ((n + 1)(n + 2)2)

m4 ((n + 1)n2)

(9)

(e optimization objective is to obtain the minimumvalue of the average long-run CBM cost by setting up anoptimal control limit UCLlowast as follows

UCLlowast argmin(c) (10)

Taking the situation when n 1for illustration the CBMelements of each scenario are presented in Tables 2ndash4


Startmonitoring

System state

In control

Out of control

Control chartmonitoring

No signal

False alarm

No signal

Alert signal

Random failure

Random failure

Fatigue failure

Fatigue failure

Scenarios

Correctivereplacement


Preventivereplacement


Scenario 1

Scenario 2

Scenario 3

Scenario 4

Figure 2 Four possible scenarios of the integrated CBM model

Table 2 Occurrence probability of each scenario in the integrated model n 1

Scenario Symbol Probability of occurrence

(S1)p(S11) (1 minus exp(minus 1113938

Δ0 h0(t) 1113938

infin0 φ(z(t))fZ(t)dzdt))P(h(t)leUCL)

p(S12) exp(minus 1113938Δ0 h0(t) 1113938

D

0 φ(z(t))fZ(t)dzdt)(1 minus exp(minus 11139382Δ0 h0(t) 1113938

D

0 φ(z(t))fZ(t)dzdt))P(h(t)leUCL)Pn

(S2) p(S2) exp(minus 1113938Δ0 h0(t) 1113938

D

0 φ(z(t))fZ(t)dzdt)Pf

(S3)

p(S31) (1 minus exp(minus 1113938Δ0 h0(t) 1113938

infin0 φ(z(t))fZ(t)dzdt))P(h(t)gtUCL)

p(S32) exp(minus 1113938Δ0 h0(t) 1113938

infin0 φ(z(t))fZ(t)dzdt)P(h(Δ)gtUCL)Pu(1 minus exp(minus 1113938

2Δ0 h0(t) 1113938


p(S33) exp(minus 1113938Δ0 h0(t) 1113938

D


infin0 φ(z(t))fZ(t)dzdt))P(h(t)gtUCL)Pn

(S4) p(S4) exp(minus 1113938Δ0 h0(t) 1113938

infin0 φ(z(t))fZ(t)dzdt)P(h(Δ)gtUCL)Pd

D (e threshold of the degradation feature which has a relationship with UCL as D 1cIn(UCLh0(t))

Pd (e probability of the signal when the process is out of controlPu (e probability of no signal when the process is out of controlPf (e probability of the signal when the process is in controlPn (e probability of no signal when the process is in control

Table 3 Expected cycle time of each scenario in the integrated model n 1

Scenario Symbol Expected time per cycle

S1 E(T | S12)Δ minus 1113938

Δ0 (Δ minus t)f(t)dt

2Δ minus 11139382ΔΔ (2Δ minus t)f(t)dt

S2 E(T | S2) Δx

S3

E(T | S31) Δ minus 1113938Δ0 (Δ minus t)f(t)dt

E(T | S32) 2Δ minus 11139382ΔΔ (2Δ minus t)f(t)dt


S4 E(T | S4) Δx

Table 4 Expected cycle cost of each scenario in the integrated model n 1

Scenario Symbol Expected cost per cycleS1 E(C | S1) (Ccp(S11) + (Cc + Ci)p(S12))p(S1)

S2 E(C | S2) Cp + Ci

S3 E(C | S3) (Ccp(S31) + (Cc + Ci)p(S32) + (Cc + Ci)p(S33))p(S3)



4 The Three-Stage Approach forCBM Optimization

In this section the proposed approach developed to monitorthe health degradation and optimize the CBM schedule formotorized spindles is described in Figure 3 (e approach iscomposed of three stages involving health indicator ex-traction degradation condition monitoring and CBM op-timization In order to implement this approach the PHM isconstructed and the hazard rate is estimated which is thenapplied in SPC charts as the monitoring statistic At lastcombining SPC with CBM management an integratedoptimization model is given to develop the maintenanceplan andminimize the average long-run cost of maintenanceby searching for the optimal control limits

(e three stages of the proposed approach are outlined asfollows

Stage 1 (construction of the PHM) the model isadopted to extract the estimation of the Stage 2 (es-tablishment of the SPC charts) as long as the pa-rameters of the PHM are estimated with the trainingdata the test data can be collected and plugged into themodel to obtain the hazard rate as the degradationfeature to be monitored (e samples of the hazard rateare then plotted on the control chart with the corre-sponding central line (CL) and UCL which can becomputed as Table 1 according to the selected charttypeStage 2 (establishment of the SPC charts) as longas the parameters of the PHM are estimated with thetraining data the test data can be collected and pluggedinto the model to obtain the hazard rate as the deg-radation feature to be monitored (e samples of thehazard rate are then plotted on the control chart withthe corresponding central line (CL) and UCL whichcan be computed as Table 1 according to the selectedchart typehazard rate by the fusion of event data andcondition monitoring data (is stage is illustrated indetail as follows

Step 1 (data acquisition) (e performance of mo-torized spindles mainly depends on their key func-tional components such as the bearings the toolclamping system and the rotary unions [26](erefore these components would be focused byinstalling additional sensors and different signals arecollected comprising vibration current forceacoustic emission temperature leakage flow etc [19]in order to describe the degradation characteristics ofthe system more accuratelyStep 2 (data preprocessing and feature extraction)Using the collected life cycle signals of the monitoredspindle to select the typical features in time fre-quency and time-frequency domains approachessuch as principal component analysis can be adoptedfor the purpose of reducing the dimension of featureswhile keeping the information with the largest con-tribution to describe system degradation processStep 3 (health indicator estimation) Combining theextracted features from the condition monitoring

system with failure event data in the lifetime ofmotorized spindles the PHM is constructed and itsparameters are estimated with the training samples(erefore the hazard rate observation can be obtainedas the health indicator for the next stage of degra-dation condition monitoring

Stage 2 (establishment of the SPC charts) as long as theparameters of the PHM are estimated with the trainingdata the test data can be collected and plugged into themodel to obtain the hazard rate as the degradationfeature to be monitored (e samples of the hazard rateare then plotted on the control chart with the corre-sponding central line (CL) and UCL which can becomputed as Table 1 according to the selected charttype(e chart is then adopted as a tool for anomaly de-tection and a trigger for maintenance activities Inparticular if a sample point plotted on the SPC chart iswithin the UCL the system is considered as in-controland the periodic inspection and daily health manage-ment are maintained Otherwise the system is definedas out of control which implies that the degradationlevel exceeds an acceptable threshold and a preventivereplacement is performed Meanwhile a failure of thesystem could occur during both in-control and out-of-control phases which will trigger a correctivereplacementStage 3 (optimization of the CBM) based on theaforementioned maintenance strategy the monitoringsystem integrating SPC charts and PHM is furtherdeveloped for CBM optimization Four differentscenarios that can describe the possibilities in mon-itoring process are taken into account in order toestablish the CBM model Each scenario is ended by amaintenance activity and the monitored system isrestarted as a new cycle with an ldquoas good as newrdquocondition Finally the SPC chart is updated and theoptimal CBM schedule is obtained through mini-mizing the average long-run cost during a renewalcycle by optimizing the UCL

5 Case Study

(eproposed three-stage approach is applied to a real case ofthe motorized spindle in order to illustrate its strongpracticability (e developing process of degradation andfailure of motorized spindles are mainly blamed on bearingfailures clamping device rubbing and rotary union loos-ening which can be characterized by vibration signals [27](erefore the case reported here is based on vibration signalanalysis in particular

51 Experimental Setup (e failure events and conditionmonitoring data sets are generated from the motorizedspindle going through run-to-failure experiments underconstant rotary speed at 4000 RPM (e experiment rig isshown in Figure 4


(e experimental setup is designed considering thestructure of the motorized spindle which is shown inFigure 4(a) the spindle can be subjected to the cuttingtorque provided by the dynamometer joined to the toolholder with the couplings and the biaxial cutting forcesprovided by the axial and radial loading systems(e spindleis kept running 12 hours per day except for irregularequipment adjustment and unexpected failures

Based on experimental requirements two PCB 356A15triaxial high sensitivity ceramic shear ICP accelerometersare installed on the front face and the back face of themotorized spindle respectively (e specific placement is

circled in Figures 4(b) and 4(c) and the vibration signal iscollected every 4 hours by ADLINKrsquos USB-24054-CH 24-bit 128 kSs dynamic signal acquisition (e test and thecondition monitoring system are controlled by LabVIEWprogram

52 Feature Extraction and PHMConstruction (e trainingdata collected include time of inspections time to failurecensored time and vibration signal Each inspection lasts atleast 30 seconds which contains 150000 vibration signalswith the sampling rate of 5 kHz A total of 246 inspectionsspanning over 1000-hour operation test on the tested mo-torized spindle are conducted During the whole experi-mental period 7 hard failures and 5 suspensions areobserved Among them the 7 failures are considered asunexpected functional failures and the 5 suspensions aredue to adjustments of experimental equipment

Only the x-axial vibration measurements were used foranalysis and three commonly used time-domain features areextracted from the lifetime vibration data of the testedspindle(e technicians believe that the selected features cancontain essential information to interpret the specific failurewell [28] (e features as derived in [29] are summarized inTable 5

PHMs with different features are fitted and K-S good-ness-of-fit tests are performed to obtain a better model (eparameters and the test statistics of hazard rate functions areestimated and listed in Table 6 (e traditional Weibulldistribution without incorporating any feature is also fittedand its result is shown in Table 6 as Model 1

It implies in Table 6 that the significance of RMS in thePHM is greater than kurtosis and crest factor In additionthe fitting effect of the PHM integrating with condition

Stage 1Construction of PHM

Condition monitoringData acquisition

Signal denoising andfeature extraction

PHM development toobtain health indicator

Compute the controllimits (UCL and CL)based on health data

Stage 2Establishment of SPC charts

Preventive maintenancetriggered

Yes Abnormaldetected or not

No

Yes

Health indicatorexceeds the control limits

(UCL) or not

CBM optimized to obtainoptimal control limits

CBM model constructionconsidering four scenarios

Testing data monitoredperiodically

Stage 3Optimization of CBM

No NoHard failureoccurs or not

Corrective maintenancetriggered

Yes

Figure 3 Flowchart of the proposed three-stage approach

Frontface

HSK gripper Bearings

Clampingspring

Rotaryfeedthrough

Back face

Spindle

StatorRotor

Bearings (a)

a

c

Vibration sensors

Axial and radial loading

b

Montorized spindle

Tools holder

Dynamometer

Lodaing unit

(b)

(c)

Figure 4 (e experiment rig for motorized spindle monitoring


monitoring features is better than the traditional Weibulldistribution Since the PHM with RMS has the lowest K-Sstatistic of 02190 and the highest p value of 08255 com-paring with the other models in the table the PHM withRMS is selected for further degradation monitoring andmaintenance decision of the motorized spindle and itsparameters are estimated as β 20046 η 1513602 andc1 10768

53 SPC Establishment and Degradation Monitoring For amonitored renewal cycle of themotorized spindle a life cycleof 372 hours with 93 inspections in total is adopted to obtainthe 93 testing values of RMS as the lifetime features whichare shown in Figure 5(a) (en the set of RMS is pluggedinto the PHM obtained from the training data set in Section52 and the hazard rate is calculated and plotted inFigure 5(b)

(e hazard rate set of the testing cycle is then used asmonitored health indicators (e SPC charts introduced inSection 2 are constructed as supports for anomaly detectionand CBM optimization (e Shewhart I-chart and theEWMA chart are taken as examples which are shown inFigures 6(a) and 6(b) In the case of monitoring the hazardrate only UCL and CL are essential (e red solid linesrepresent the UCLs of the charts the green-dotted linesrepresent the CLs of the charts the black dots are corre-sponding statistics derived from the testing data of thehazard rate and the marked purple circles are degradationstates determined to be out of control

To be more specific the two charts are establishedfollowing the rules in Table 1 As for the Shewhart I-chartin Figure 6(a) the estimated process mean μ0 00219the estimated process standard deviation σ0 00013UCL 00258 CL 00219 and the control limit widthn 3 and the process is determined to be out of control atthe 62nd inspection Figure 6(b) depicts the results of theEWMA chart whose weight parameter is set to beλ 025 the UCL is stable at UCL 00234 and theprocess is determined to be out of control at the 60thinspection

Since the Shewhart I-chart monitors the performance ofthe motorized spindle only based on observed data at in-dividual inspections it is less sensitive in detecting degra-dation with small and moderate change trends incomparison with the EWMA chart [30] However in thecase of monitoring the hazard rate of the tested spindle theresults of the EWMA chart show little significant deviationcomparing to the Shewhart I-chart which is mostly theconsequence of using the hazard rate from the proposedPHM as the monitoring statistic (e model could com-promise multiple information based on time series in orderto eliminate autocorrelation in the process and reduce theinfluence of the random error As a result the two types ofchartsrsquo abilities for gradual degradation detection aresatisfactory

54 CBM Schedule Optimization To build the final CBMmodel the established SPC chart is keen to be updated andoptimized by finding optimal UCLlowast which is used to triggercorresponding maintenance actions so that the inspectionand maintenance cost can be minimized

In the current case the tested spindle is inspected every 4 hours the cost of each inspection Ciis ￥30 (epreventive replacement cost Cpis ￥500 and the correctivereplacement cost Ccis ￥6000 To simplify the analysisPdand Pfare set to be 0 and Puand Pnare set to be 1 whichreduce the four scenarios to two for the control chart Fi-nally the CBM model can be interpreted as follows

Scenario 1 (S1) the motorized spindle remains incontrol until a hard failure occurs in the interval(4k 4(k + 1)) A corrective replacement is imple-mented and restores the spindle to its initial conditionScenario 2 (S2) the motorized spindle degrades to theout-of-control condition in the interval (4k 4(k + 1))and the SPC chart succeeds to detect it at 4(k + 1) Apreventive replacement is carried out and restores thespindle to its initial condition

As a result the long-run cost rate is obtained through thefollowing formula

Table 5 Summary of selected time-domain features

Feature Definition Formula

RMS Root mean square the square root of the mean square

(1N) 1113936Ni1 (xi minus x)2

1113969

Kurtosis A measure of the vibratory magnitudes of the probability distribution in the time domain (1N)1113936Ni1(xi minus xRMS)4

Crest factor A measure of a waveform showing the ratio of peak values to the RMS value (max(xi)RMS)

Table 6 Estimation results for the selected PHMs

Modelh0(t) ci K-S p

β η c1 (RMS) c2 (kurtosis) c3 (crest factor)

1 17634 1622420 mdash mdash mdash 03028 045512 20046 1513602 10768 mdash mdash 02190 082553 22251 1247858 mdash 00822 mdash 02595 064424 20661 1688726 mdash mdash 01218 02846 05302


c 1113936

930 Cc + kCi( 1113857pk S1( 1113857 + 1113936

930 Cp +(k + 1)Ci1113872 1113873pk S2( 1113857

1113936930 4(k + 1) minus 1113938

4(k+1)

4k(4(k + 1) minus t)f(t)dt1113874 1113875pk S1( 1113857 + 1113936

930 4(k + 1)pk S2( 1113857

(11)

Based on the vibration signals at each inspection of thedegradation process RMS is found to follow the normaldistribution with parameters μ(t)and σ(t) (e changes ofthe two parameters with time are studied with respect to the93 inspections which are expressed as

μ(t) 011 exp 114 times 10minus 3t1113872 1113873 + 896 times 10minus16 exp(009t)

σ(t) 004 exp 265 times 10minus 4t1113872 1113873 + 187 times 10minus8 exp(004t)

(12)

(erefore the probability that the hazard rate exceedsUCL at time t is derived as

P(h(t)gtUCL) 1 minusΦD minus μ(t)

σ(t)1113888 1113889 (13)

(rough optimization of equation (11) the optimalCBM schedule is finally obtained with UCLlowast 00524 andthe minimal long-run cost rate is c 151942 which isshown as the red-marked circle in Figure 7(a) According toequation (3) the RMS threshold versus the testing time ofthe motorized spindle can be obtained and depicted inFigure 7(b) (e result also has certain directive significancein practical application because the preventive replacementcan be simply triggered when the RMS observation at theinspection exceeds the RMS threshold

0

02

04

06

08

1RM

S

Time (hours)0 100 200 300 400

(a)

Time (hours)

0

002

004

006

008

01

Haz

ard

rate

0 100 200 300 400

(b)

Figure 5 Lifetime features for the tested motorized spindle (a) RMS (b) (e hazard rate estimates

0

002

004

006

008

01

Shew

hart

I-ch

art s

tatis

tic

In-control sampleOut-of-control sample

UCLCL

Time (hours)0 100 200 300 400

(a)

0

002

004

006EW

MA

stat

istic


UCLCL

Time (hours)0 100 200 300 400

(b)

Figure 6 Two types of SPC charts for the tested motorized spindle (a) (e Shewhart I-chart (b) (e EWMA chart


6 Conclusions

(e proposed three-stage approach in this paper is com-mitted to fusing various degradation features monitoringthe degradation process and developing the optimal CBMplans for motorized spindles (e combined effect of con-dition features degrading over time together with failureevents is interpreted by the PHM and SPC charts are thenintroduced to monitor the changing trend of the hazard rateobtained from the PHM Application of the SPC charts withperiodic hazard rate inspections helps in CBM modellingand four possible scenarios are defined to constitute themaintenance model At last the CBM plan with minimumlong-run cost is obtained by seeking for the optimal UCL

(e feasibility and practicability of the approach areproved by a case study using the actual spindle vibrationdata (e analysis results indicate that the PHM contributesto describing the degradation states of the motorized spindlemore accurately and three different time-domain featuresare extracted from the raw vibration signals and are used ascovariates in the PHM RMS is verified to have the bestfitting effect of the experimental data

(e proposed approach allows not only maintenanceoptimization but also accurate characterization of healthdegradation It can help maintenance technicians andmanagers reduce breakdown time delete unnecessarymaintenance actions and minimize maintenance costs in anintuitive and efficient way (e approach in this paper canalso be applied for the motorized spindle operating in CNCmachine tools for intelligent manufacturing in order toachieve better performance and a longer life of the wholemanufacturing supply chain

Data Availability

No data were used to support this study

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is research was supported by the Fundamental ResearchFunds for the Central Universities (Grant no3072020CFJ0202)

References

[1] T Liu W Gao D Zhang et al ldquoAnalytical modeling forthermal errors of motorized spindle unitrdquo InternationalJournal of Machine Tools and Manufacture vol 112 pp 53ndash70 2017

[2] S Alaswad and Y Xiang ldquoA review on condition-basedmaintenance optimization models for stochastically deterio-rating systemrdquo Reliability Engineering amp System Safetyvol 157 pp 54ndash63 2017

[3] Y Lei N Li L Guo N Li T Yan and J Lin ldquoMachineryhealth prognostics a systematic review from data acquisitionto RUL predictionrdquo Mechanical Systems and Signal Process-ing vol 104 pp 799ndash834 2018

[4] D Shrivastava M S Kulkarni and P Vrat ldquoIntegrated designof preventive maintenance and quality control policy pa-rameters with CUSUM chartrdquo gte International Journal ofAdvanced Manufacturing Technology vol 82 no 9-12pp 2101ndash2112 2016

[5] Y Li E Pan and Z Chen ldquoConsidering machine healthcondition in jointly optimizing predictive maintenance policyand X-bar control chartrdquo in Proceedings of the InternationalConference On Grey Systems And Intelligent Services (GSIS2017 pp 328ndash337 Stockholm Sweden August 2017

[6] A Farahani H Tohidi and A Shoja ldquoAn integrated opti-mization of quality control chart parameters and preventivemaintenance using Markov chainrdquo Advances in ProductionEngineering amp Management vol 14 no 1 pp 5ndash14 2019

[7] S P G F d S Lampreia J F G Requeijo J A M DiasV M Vairinhos and P I S Barbosa ldquoCondition monitoringbased on modified CUSUM and EWMA control chartsrdquoJournal of Quality in Maintenance Engineering vol 24 no 1pp 119ndash132 2018

[8] P Charongrattanasakul and A Pongpullponsak ldquoMinimizingthe cost of integrated systems approach to process control andmaintenance model by EWMA control chart using geneticalgorithmrdquo Expert Systems with Applications vol 38 no 5pp 5178ndash5186 2011

Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)

Figure 7 (e CBM optimization results (a) (e average long-run cost rate versus UCL (b) (e RMS threshold versus time


[9] M A Ardakan A Z Hamadani M Sima et al ldquoA hybridmodel for economic design of MEWMA control chart undermaintenance policiesrdquogte International Journal of AdvancedManufacturing Technology vol 83 no 9-12 pp 2101ndash21102016

[10] J Zhong and Y Ma ldquoAn integrated model based on statisticalprocess control and maintenance for two-stage dependentprocessesrdquo Communications in Statistics Simulation ampComputation vol 46 no 1 pp 106ndash126 2017

[11] H Rasay M S Fallahnezhad and Y Zare Mehrjerdi ldquoAnintegrated model for economic design of chi-square controlchart and maintenance planningrdquo Communications in Sta-tistics - gteory and Methods vol 47 no 12 pp 2892ndash29072018

[12] W H Zhou and G L Zhu ldquoEconomic design of integratedmodel of control chart and maintenance managementrdquoMathematical amp Computer Modelling vol 47 no 11-12pp 1389ndash1395 2008

[13] Z Mehrafrooz and R Noorossana ldquoAn integrated modelbased on statistical process control and maintenancerdquoComputers amp Industrial Engineering vol 61 no 4pp 1245ndash1255 2011

[14] H Yin G Zhang H Zhu et al ldquoAn integrated model ofstatistical process control and maintenance based on thedelayed monitoringrdquo Reliability Engineering System Safetyvol 133 pp 323ndash333 2015

[15] CW Zhang M Xie J Y Liu and T N Goh ldquoA control chartfor the Gamma distribution as a model of time betweeneventsrdquo International Journal of Production Research vol 45no 23 pp 5649ndash5666 2007

[16] Y He F Liu J Cui et al ldquoReliability-oriented design ofintegrated model of preventive maintenance and qualitycontrol policy with time-between-events control chartrdquoComputers amp Industrial Engineering vol 129 pp 228ndash2382019

[17] R Li and X Zhang ldquoPreventive maintenance interval opti-mization for continuous multistate systemsrdquo MathematicalProblems in Engineering vol 2020 Article ID 294294010 pages 2020

[18] Y Li Z Chen and E Pan ldquoJoint economic design of CUSUMcontrol chart and age-based imperfect preventive mainte-nance policyrdquo Mathematical Problems in Engineeringvol 2018 pp 9246372ndash11 2018

[19] H Cao X Zhang and X Chen ldquo(e concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2017

[20] F Kadri F Harrou S Chaabane Y Sun and C TahonldquoSeasonal ARMA-based SPC charts for anomaly detectionapplication to emergency department systemsrdquo Neuro-computing vol 173 pp 2102ndash2114 2016

[21] D CMontgomery Introduction to Statistical Quality ControlJohn Wiley amp Sons New York NY USA 2012

[22] D R Cox ldquoRegression models and life-tablesrdquo in Break-throughs in Statistics S Kotz and N L Johnson Edspp 527ndash541 Springer Berlin Germany 1992

[23] J I McCool Using the Weibull Distribution ReliabilityModelling and Inference John Wiley amp Sons New York NYUSA 2012

[24] H Liao W Zhao and H Guo ldquoPredicting remaining usefullife of an individual unit using proportional hazards modeland logistic regression modelrdquo in Proceedings Of the AnnualReliability And Maintainability Symposium (RAMS rsquo06)pp 127ndash132 Newport Beach CA USA January 2006

[25] B Liu Z Liang A K Parlikad M Xie and W KuoldquoCondition-based maintenance for systems with aging andcumulative damage based on proportional hazards modelrdquoReliability Engineering amp System Safety vol 168 pp 200ndash2092017

[26] R Neugebauer J Fischer and M Praedicow ldquoCondition-based preventive maintenance of main spindlesrdquo ProductionEngineering vol 5 no 1 pp 95ndash102 2011

[27] D Goyal and B S Pabla ldquoCondition based maintenance ofmachine tools-A reviewrdquo CIRP Journal of ManufacturingScience and Technology vol 10 pp 24ndash35 2015

[28] C De Castelbajac M Ritou S Laporte and B FuretldquoMonitoring of distributed defects on HSM spindle bearingsrdquoApplied Acoustics vol 77 pp 159ndash168 2014

[29] S Gunes M Dursun K Polat et al ldquoSleep spindles recog-nition system based on time and frequency domain featuresrdquoExpert Systems with Applications vol 38 no 3 pp 2455ndash2461 2011

[30] W A Jensen L A Jones-Farmer C W Champ andW H Woodall ldquoEffects of parameter estimation on controlchart properties a literature reviewrdquo Journal of QualityTechnology vol 38 no 4 pp 349ndash364 2006


to monitor the feature of time-between-events (TBE) as-suming that TBE followed the gamma distribution [15] thenormal distribution [16] the exponential distribution [17]the Weibull distribution [18] etc

However these aforementioned control charts are allunivariate which are commonly used to monitor indi-vidual observations of a certain type of characteristicvariable For the case of the motorized spindle conditionmonitoring usually contains multiple sensor signals suchas vibration bearing preload acoustic emission soundand temperature [19] and the signals are often auto-correlated [20] (erefore the conventional SPC charts willnot be effective to extract useful and reliable informationfrom the monitoring process leading to larger errors inCBM optimization

To overcome this problem the paper develops a three-stage approach for the purpose of extracting the fusionfeature monitoring the health degradation and optimizingthe CBM strategy for motorized spindles (e main con-tributions of the proposed approach are described as follows

(1) Proportional hazard model (PHM) is introduced toextract the health indicator (e benefits can bedemonstrated in two aspects on the one hand PHMcan integrate multivariate information frommultiplemonitoring signals together with failure data whichresults in the more accurate degradation detectionfor CBM optimization On the other hand thehazard rate from PHM is selected as the health in-dicator of SPC charts It can reduce random inter-ference for SPC charts when monitoring theautocorrelation process the detection error such asthe false alarm or undetected failure can be effec-tively avoided

(2) CBM optimization is combined with SPC chartdesign based upon economic considerations thehealth condition of motorized spindles can beconstantly updated and two types of the failuremechanism including the random failure and thefatigue failure and four possible scenarios are takeninto consideration (e integrated model can avoidunnecessary or untimely maintenance and achieveconsiderable economic benefit by reducing main-tenance costs and raising production efficiencywhich provide effective service and guidance forpractitioners

(e remainder of the paper is organized as followsSection 2 describes the CBM optimization problem andassumptions in detail in which the failure mechanism SPCcharts and maintenance policy are discussed In Section 3the integrated model for combined SPC chart design andCBM scheduling is developed (e hazard rate extractedfrom the PHM is used to characterize the health conditionand the SPC scheme considering four possible scenarios isestablished to monitor the degradation trend in real time(en Section 4 introduces the proposed three-stage ap-proach for optimizing the CBM schedule upon economicconsiderations In association with the PHM and the SPC

charts the average long-run cost during a renewal cycle isminimized by optimizing the upper control limit (UCL)Next a real case of the motorized spindle is studied to verifythe feasibility and practicability of the proposed approach inSection 5 Finally conclusions are presented in the lastsection

2 Problem Statement and Assumptions

For this research the degradation of the motorized spindleduring the long life cycle is due to both external factors likecumulative shocks environmental stress etc and internalfactors like continuous aging etc (erefore the system willsuffer from random failures and fatigue failures Multiplecondition features together with failure event data can beused as the health indicator to characterize the effects ofdegradation

(en the SPC chart is used as a failure diagnosis tool formonitoring the health indicator and detecting the two typesof failures the random failure also known as the hardfailure which is self-announcing and could be detectedimmediately and the fatigue failure also known as the softfailure which is defined to occur only when the degree ofdegradation reaches the preset control limits (e SPC chartscheme is shown in Figure 1 and its control limits can beusually computed following the formula [21] in Table 1according to different SPC charts

When combined withmaintenance planning the controllimits of SPC charts can be regarded as the trigger of amaintenance action the optimization of the CBM strategy istransformed into determining the optimal control limits inorder to minimize the maintenance cost from the per-spective of economic efficiency

(e integrated CBM model is developed under the as-sumptions as follows

(1) (e changing trend of the health indicator may notbe obvious over time and increase rapidly after thesystem enters the wear-out stage In order to graspsystem degradation level and implement mainte-nance actions in time inspections are scheduledevery Δ time unit to detect the health conditionperiodically (e inspection time is short so that theinspection activities are considered instantaneousand nondestructive (e cost per inspection Ci ispredetermined as a constant according to the actualcondition

(2) (e two types of failures correspond to two types ofmaintenance activities which are the preventivereplacement and the corrective replacement re-spectively To be more specific if a hard failureoccurs unexpectedly the corrective replacement isperformed instantly and the average cost per cor-rective replacement Cc covers all costs with respect tothe failure and its associated problems On thecontrary if the monitoring health indicator is de-tected to exceed the control limits at the inspectiontime which is interpreted as a soft failure thepreventive replacement is carried out (e average












h0(t) βη

t

η1113888 1113889

βminus1

(2)





w


w

radic)


(λ2minusλ)


(λ2minusλ)

1113968


Mon

itore

d he

alth

indi

cato

r UCL

Fatigue failure


point

0 t1 t2 t3 t t1 t2 tk tk+1 tk+2

Time



h(t Z(t)) βη

t

η1113888 1113889

βminus1




exp minus 1113946t

0h(s Z(s))ds1113888 1113889

(4)




exp minus 1113938t

0 h(s Z(s))ds1113872 1113873

(5)



0exp minus 1113946

t+τ

th(υ Z(υ))dυ1113888 1113889dτ

(6)


L(middot) 1113945n

i1f ti Z ti( 1113857( 1113857 1113945

m

s1R ts Z ts( 1113857( 1113857 (7)







c E(C(T))

E(T) (8)

where E(C(T)) 11139364i1 p(Si) 1113936

mi


4i1 p(Si) 1113936

mi



m1 n + 1 m2 n

m3 ((n + 1)(n + 2)2)

m4 ((n + 1)n2)

(9)





Startmonitoring

System state

In control

Out of control


No signal

False alarm

No signal

Alert signal

Random failure

Random failure

Fatigue failure

Fatigue failure

Scenarios





Scenario 1

Scenario 2

Scenario 3

Scenario 4





Δ0 h0(t) 1113938


p(S12) exp(minus 1113938Δ0 h0(t) 1113938

D


D


(S2) p(S2) exp(minus 1113938Δ0 h0(t) 1113938

D


(S3)



p(S32) exp(minus 1113938Δ0 h0(t) 1113938


2Δ0 h0(t) 1113938


p(S33) exp(minus 1113938Δ0 h0(t) 1113938

D



(S4) p(S4) exp(minus 1113938Δ0 h0(t) 1113938






S1 E(T | S12)Δ minus 1113938



S2 E(T | S2) Δx

S3




S4 E(T | S4) Δx














5 Case Study



















No

Yes


(UCL) or not







Yes


Frontface


Clampingspring

Rotaryfeedthrough

Back face

Spindle

StatorRotor

Bearings (a)

a

c

Vibration sensors


b

Montorized spindle

Tools holder

Dynamometer

Lodaing unit

(b)

(c)















(1N) 1113936Ni1 (xi minus x)2

1113969




Modelh0(t) ci K-S p




c 1113936

930 Cc + kCi( 1113857pk S1( 1113857 + 1113936

930 Cp +(k + 1)Ci1113872 1113873pk S2( 1113857

1113936930 4(k + 1) minus 1113938

4(k+1)


930 4(k + 1)pk S2( 1113857

(11)




(12)



σ(t)1113888 1113889 (13)


0

02

04

06

08

1RM

S

Time (hours)0 100 200 300 400

(a)

Time (hours)

0

002

004

006

008

01

Haz

ard

rate

0 100 200 300 400

(b)


0

002

004

006

008

01

Shew

hart

I-ch

art s

tatis

tic


UCLCL

Time (hours)0 100 200 300 400

(a)

0

002

004

006EW

MA

stat

istic


UCLCL

Time (hours)0 100 200 300 400

(b)



6 Conclusions




Data Availability




Acknowledgments


References









Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)




































h0(t) βη

t

η1113888 1113889

βminus1

(2)





w


w

radic)


(λ2minusλ)


(λ2minusλ)

1113968


Mon

itore

d he

alth

indi

cato

r UCL

Fatigue failure


point

0 t1 t2 t3 t t1 t2 tk tk+1 tk+2

Time



h(t Z(t)) βη

t

η1113888 1113889

βminus1




exp minus 1113946t

0h(s Z(s))ds1113888 1113889

(4)




exp minus 1113938t

0 h(s Z(s))ds1113872 1113873

(5)



0exp minus 1113946

t+τ

th(υ Z(υ))dυ1113888 1113889dτ

(6)


L(middot) 1113945n

i1f ti Z ti( 1113857( 1113857 1113945

m

s1R ts Z ts( 1113857( 1113857 (7)







c E(C(T))

E(T) (8)

where E(C(T)) 11139364i1 p(Si) 1113936

mi


4i1 p(Si) 1113936

mi



m1 n + 1 m2 n

m3 ((n + 1)(n + 2)2)

m4 ((n + 1)n2)

(9)





Startmonitoring

System state

In control

Out of control


No signal

False alarm

No signal

Alert signal

Random failure

Random failure

Fatigue failure

Fatigue failure

Scenarios





Scenario 1

Scenario 2

Scenario 3

Scenario 4





Δ0 h0(t) 1113938


p(S12) exp(minus 1113938Δ0 h0(t) 1113938

D


D


(S2) p(S2) exp(minus 1113938Δ0 h0(t) 1113938

D


(S3)



p(S32) exp(minus 1113938Δ0 h0(t) 1113938


2Δ0 h0(t) 1113938


p(S33) exp(minus 1113938Δ0 h0(t) 1113938

D



(S4) p(S4) exp(minus 1113938Δ0 h0(t) 1113938






S1 E(T | S12)Δ minus 1113938



S2 E(T | S2) Δx

S3




S4 E(T | S4) Δx














5 Case Study



















No

Yes


(UCL) or not







Yes


Frontface


Clampingspring

Rotaryfeedthrough

Back face

Spindle

StatorRotor

Bearings (a)

a

c

Vibration sensors


b

Montorized spindle

Tools holder

Dynamometer

Lodaing unit

(b)

(c)















(1N) 1113936Ni1 (xi minus x)2

1113969




Modelh0(t) ci K-S p




c 1113936

930 Cc + kCi( 1113857pk S1( 1113857 + 1113936

930 Cp +(k + 1)Ci1113872 1113873pk S2( 1113857

1113936930 4(k + 1) minus 1113938

4(k+1)


930 4(k + 1)pk S2( 1113857

(11)




(12)



σ(t)1113888 1113889 (13)


0

02

04

06

08

1RM

S

Time (hours)0 100 200 300 400

(a)

Time (hours)

0

002

004

006

008

01

Haz

ard

rate

0 100 200 300 400

(b)


0

002

004

006

008

01

Shew

hart

I-ch

art s

tatis

tic


UCLCL

Time (hours)0 100 200 300 400

(a)

0

002

004

006EW

MA

stat

istic


UCLCL

Time (hours)0 100 200 300 400

(b)



6 Conclusions




Data Availability




Acknowledgments


References









Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)


























h(t Z(t)) βη

t

η1113888 1113889

βminus1




exp minus 1113946t

0h(s Z(s))ds1113888 1113889

(4)




exp minus 1113938t

0 h(s Z(s))ds1113872 1113873

(5)



0exp minus 1113946

t+τ

th(υ Z(υ))dυ1113888 1113889dτ

(6)


L(middot) 1113945n

i1f ti Z ti( 1113857( 1113857 1113945

m

s1R ts Z ts( 1113857( 1113857 (7)







c E(C(T))

E(T) (8)

where E(C(T)) 11139364i1 p(Si) 1113936

mi


4i1 p(Si) 1113936

mi



m1 n + 1 m2 n

m3 ((n + 1)(n + 2)2)

m4 ((n + 1)n2)

(9)





Startmonitoring

System state

In control

Out of control


No signal

False alarm

No signal

Alert signal

Random failure

Random failure

Fatigue failure

Fatigue failure

Scenarios





Scenario 1

Scenario 2

Scenario 3

Scenario 4





Δ0 h0(t) 1113938


p(S12) exp(minus 1113938Δ0 h0(t) 1113938

D


D


(S2) p(S2) exp(minus 1113938Δ0 h0(t) 1113938

D


(S3)



p(S32) exp(minus 1113938Δ0 h0(t) 1113938


2Δ0 h0(t) 1113938


p(S33) exp(minus 1113938Δ0 h0(t) 1113938

D



(S4) p(S4) exp(minus 1113938Δ0 h0(t) 1113938






S1 E(T | S12)Δ minus 1113938



S2 E(T | S2) Δx

S3




S4 E(T | S4) Δx














5 Case Study



















No

Yes


(UCL) or not







Yes


Frontface


Clampingspring

Rotaryfeedthrough

Back face

Spindle

StatorRotor

Bearings (a)

a

c

Vibration sensors


b

Montorized spindle

Tools holder

Dynamometer

Lodaing unit

(b)

(c)















(1N) 1113936Ni1 (xi minus x)2

1113969




Modelh0(t) ci K-S p




c 1113936

930 Cc + kCi( 1113857pk S1( 1113857 + 1113936

930 Cp +(k + 1)Ci1113872 1113873pk S2( 1113857

1113936930 4(k + 1) minus 1113938

4(k+1)


930 4(k + 1)pk S2( 1113857

(11)




(12)



σ(t)1113888 1113889 (13)


0

02

04

06

08

1RM

S

Time (hours)0 100 200 300 400

(a)

Time (hours)

0

002

004

006

008

01

Haz

ard

rate

0 100 200 300 400

(b)


0

002

004

006

008

01

Shew

hart

I-ch

art s

tatis

tic


UCLCL

Time (hours)0 100 200 300 400

(a)

0

002

004

006EW

MA

stat

istic


UCLCL

Time (hours)0 100 200 300 400

(b)



6 Conclusions




Data Availability




Acknowledgments


References









Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)


























Startmonitoring

System state

In control

Out of control


No signal

False alarm

No signal

Alert signal

Random failure

Random failure

Fatigue failure

Fatigue failure

Scenarios





Scenario 1

Scenario 2

Scenario 3

Scenario 4





Δ0 h0(t) 1113938


p(S12) exp(minus 1113938Δ0 h0(t) 1113938

D


D


(S2) p(S2) exp(minus 1113938Δ0 h0(t) 1113938

D


(S3)



p(S32) exp(minus 1113938Δ0 h0(t) 1113938


2Δ0 h0(t) 1113938


p(S33) exp(minus 1113938Δ0 h0(t) 1113938

D



(S4) p(S4) exp(minus 1113938Δ0 h0(t) 1113938






S1 E(T | S12)Δ minus 1113938



S2 E(T | S2) Δx

S3




S4 E(T | S4) Δx














5 Case Study



















No

Yes


(UCL) or not







Yes


Frontface


Clampingspring

Rotaryfeedthrough

Back face

Spindle

StatorRotor

Bearings (a)

a

c

Vibration sensors


b

Montorized spindle

Tools holder

Dynamometer

Lodaing unit

(b)

(c)















(1N) 1113936Ni1 (xi minus x)2

1113969




Modelh0(t) ci K-S p




c 1113936

930 Cc + kCi( 1113857pk S1( 1113857 + 1113936

930 Cp +(k + 1)Ci1113872 1113873pk S2( 1113857

1113936930 4(k + 1) minus 1113938

4(k+1)


930 4(k + 1)pk S2( 1113857

(11)




(12)



σ(t)1113888 1113889 (13)


0

02

04

06

08

1RM

S

Time (hours)0 100 200 300 400

(a)

Time (hours)

0

002

004

006

008

01

Haz

ard

rate

0 100 200 300 400

(b)


0

002

004

006

008

01

Shew

hart

I-ch

art s

tatis

tic


UCLCL

Time (hours)0 100 200 300 400

(a)

0

002

004

006EW

MA

stat

istic


UCLCL

Time (hours)0 100 200 300 400

(b)



6 Conclusions




Data Availability




Acknowledgments


References









Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)

































5 Case Study



















No

Yes


(UCL) or not







Yes


Frontface


Clampingspring

Rotaryfeedthrough

Back face

Spindle

StatorRotor

Bearings (a)

a

c

Vibration sensors


b

Montorized spindle

Tools holder

Dynamometer

Lodaing unit

(b)

(c)















(1N) 1113936Ni1 (xi minus x)2

1113969




Modelh0(t) ci K-S p




c 1113936

930 Cc + kCi( 1113857pk S1( 1113857 + 1113936

930 Cp +(k + 1)Ci1113872 1113873pk S2( 1113857

1113936930 4(k + 1) minus 1113938

4(k+1)


930 4(k + 1)pk S2( 1113857

(11)




(12)



σ(t)1113888 1113889 (13)


0

02

04

06

08

1RM

S

Time (hours)0 100 200 300 400

(a)

Time (hours)

0

002

004

006

008

01

Haz

ard

rate

0 100 200 300 400

(b)


0

002

004

006

008

01

Shew

hart

I-ch

art s

tatis

tic


UCLCL

Time (hours)0 100 200 300 400

(a)

0

002

004

006EW

MA

stat

istic


UCLCL

Time (hours)0 100 200 300 400

(b)



6 Conclusions




Data Availability




Acknowledgments


References









Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)









































No

Yes


(UCL) or not







Yes


Frontface


Clampingspring

Rotaryfeedthrough

Back face

Spindle

StatorRotor

Bearings (a)

a

c

Vibration sensors


b

Montorized spindle

Tools holder

Dynamometer

Lodaing unit

(b)

(c)















(1N) 1113936Ni1 (xi minus x)2

1113969




Modelh0(t) ci K-S p




c 1113936

930 Cc + kCi( 1113857pk S1( 1113857 + 1113936

930 Cp +(k + 1)Ci1113872 1113873pk S2( 1113857

1113936930 4(k + 1) minus 1113938

4(k+1)


930 4(k + 1)pk S2( 1113857

(11)




(12)



σ(t)1113888 1113889 (13)


0

02

04

06

08

1RM

S

Time (hours)0 100 200 300 400

(a)

Time (hours)

0

002

004

006

008

01

Haz

ard

rate

0 100 200 300 400

(b)


0

002

004

006

008

01

Shew

hart

I-ch

art s

tatis

tic


UCLCL

Time (hours)0 100 200 300 400

(a)

0

002

004

006EW

MA

stat

istic


UCLCL

Time (hours)0 100 200 300 400

(b)



6 Conclusions




Data Availability




Acknowledgments


References









Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)






































(1N) 1113936Ni1 (xi minus x)2

1113969




Modelh0(t) ci K-S p




c 1113936

930 Cc + kCi( 1113857pk S1( 1113857 + 1113936

930 Cp +(k + 1)Ci1113872 1113873pk S2( 1113857

1113936930 4(k + 1) minus 1113938

4(k+1)


930 4(k + 1)pk S2( 1113857

(11)




(12)



σ(t)1113888 1113889 (13)


0

02

04

06

08

1RM

S

Time (hours)0 100 200 300 400

(a)

Time (hours)

0

002

004

006

008

01

Haz

ard

rate

0 100 200 300 400

(b)


0

002

004

006

008

01

Shew

hart

I-ch

art s

tatis

tic


UCLCL

Time (hours)0 100 200 300 400

(a)

0

002

004

006EW

MA

stat

istic


UCLCL

Time (hours)0 100 200 300 400

(b)



6 Conclusions




Data Availability




Acknowledgments


References









Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)


























c 1113936

930 Cc + kCi( 1113857pk S1( 1113857 + 1113936

930 Cp +(k + 1)Ci1113872 1113873pk S2( 1113857

1113936930 4(k + 1) minus 1113938

4(k+1)


930 4(k + 1)pk S2( 1113857

(11)




(12)



σ(t)1113888 1113889 (13)


0

02

04

06

08

1RM

S

Time (hours)0 100 200 300 400

(a)

Time (hours)

0

002

004

006

008

01

Haz

ard

rate

0 100 200 300 400

(b)


0

002

004

006

008

01

Shew

hart

I-ch

art s

tatis

tic


UCLCL

Time (hours)0 100 200 300 400

(a)

0

002

004

006EW

MA

stat

istic


UCLCL

Time (hours)0 100 200 300 400

(b)



6 Conclusions




Data Availability




Acknowledgments


References









Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)


























6 Conclusions




Data Availability




Acknowledgments


References









Upper control limit

15

16

17

18

19

20

21Lo

ng-r

un co

st ra

te

003 004 005 006 007 008 009

(a)

Time (hours)

0

2

4

6

8

10

12

14

RMS

thre

shol

d

0 100 200 300

(b)


























condition-based maintenance optimization for motorized...

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