# Roller Bearing Monitoring by New Subspace-Based Damage Indicator

Authors: G. Gautier; R. Serra; J.-M. Mencik
Journal: Shock and Vibration
 (2015)

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## Abstract

A frequency-band subspace-based damage identification method for fault diagnosis in roller bearings is presented. Subspace-based damage indicators are obtained by filtering the vibration data in the frequency range where damage is likely to occur, that is, around the bearing characteristic frequencies. The proposed method is validated by considering simulated data of a damaged bearing. Also, an experimental case is considered which focuses on collecting the vibration data issued from a run-to-failure test. It is shown that the proposed method can detect bearing defects and, as such, it appears to be an efficient tool for diagnosis purpose.


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## Body

## 1. Introduction

Successfully implementing a condition monitoring procedure allows a mechanical system to operate at full capacity without the need of shutting down the process for periodic inspections. In this context, vibration-based structural health monitoring (SHM) techniques [1] can be considered which involve measuring the vibration signals of a structure in the time domain, as well as proposing damage indicators and efficient statistical analysis for determining its current state of structural health [2].Vibration-based damage detection methods have gained a large popularity over the last two decades, especially for rotating machines [3]. Rolling element bearings are essential components of rotating machines, which constitute the primary cause of breakdowns [4]. When a rolling element moves through a faulty surface, an impact is induced which in turn excites the resonance frequency of the bearing system. As the bearing rotates, these impulses occur periodically at the characteristic frequency of the defect. This explains why the detection of faults in bearings is commonly achieved by identifying those bearing characteristic frequencies (BCFs) [5]. Notice however that these impulses are very weak at the early stage of fault generation and are thus usually overwhelmed by measurement noises and other vibration sources such as rotor unbalance [6] or gear mesh, which induce difficulties for detecting these emerging defects.For the purpose of weak signature enhancement, a variety of signal processing techniques have been proposed. They mainly include scalar indicators [7], high frequency resonance techniques [8], spectral kurtosis analysis [9, 10], wavelet analysis [11, 12], empirical mode decomposition [13, 14], or PeakVue analysis [15].The present work is concerned with the use of subspace identification methods for damage diagnosis [16, 17]. The originality of the present work is to assess the effectiveness and applicability of such methods to identify damage in roller bearing when the output signals are only considered, that is, when the input data are not known a priori. To that aim, vibration response data are periodically collected and are used to obtain a so-called observability matrix which is well suited for damage detection [18]. Here, a subspace-based damage detection procedure is proposed which is combined with a pass-band filtering approach [19]. Clearly speaking, the output signals are filtered around the bearing characteristic frequencies so as to provide efficient subspace-based damage indicators.The rest of the paper is organized as follows. The basics of the subspace-based methods, for damage detection, are recalled. Also a new subspace damage indicator is proposed. A presentation of roller bearings is proposed in Section3 along with a damage identification strategy combined with the aforementioned pass-band filtering procedure. A theoretical study of a damaged bearing and an experimental validation based on a run-to-failure test are proposed in Sections 4 and 5, respectively. This paper is an extended version with several improvements of the scientific and experimental contents of an earlier conference paper version [20].


## 2. Subspace-Based Damage Detection Procedure

### 2.1. Subspace-Based Methods

Consider the following2nth order discrete-time linear state space system for k=0,1,…,N-1 [21]:(1)xk+1=Axk+wk,yk=Cxk+vk,where xk is a 2n×1 state vector defined at the discrete time tk (k=1,2,…,N); yk is a l×1 vector of output data; also, A, B, C, and D are system matrices of respective sizes 2n×2n, 2n×m, l×2n, and l×m. Besides, vk and wk are 2n×1 and l×1 vectors of process and measurement noises, respectively, which are supposed to be white Gaussian with zero-mean distributions and joint covariance matrix [22](2)EwpvpwqTvqT=QSSTRδpq,where E· is the expectation operator, while δpq is the Kronecker delta.Some notations used for the subspace algorithms are introduced here. A block Hankel matrix of output data is first considered, which can be partitioned into past (p) and future (f) outputs as follows:(3)
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### 2.2. Subspace-Based Damage Indicator

The key idea behind the proposed subspace-based damage detection procedure consists in comparing the extended observability matrix of the safe structure (say,Γi=Γs) with that of the structure in an unknown health condition (say, Γi=Γu). This yields the consideration of an error norm Γu-ΓsF=ΓeuTu-ΓesTsF, where ·F is the Frobenius norm, while Γes=U1s and Γeu=U1u are experimental extended observability matrices; see (13).Here, an alternative error norm is considered which involves right-multiplying the residualΓeuTu-ΓesTs by (Tu)-1; that is,(14)Γeu-ΓesTFwhereT=TsTu-1.Besides, an approximate expression of the matrix T is considered which involves solving the following least squares problem Γu=ΓsT, which leads to T=(Γs)+Γu. The relevance of this approximate expression can be justified in the sense that the error norm Γu-ΓsTF is supposed to be small. Hence, (14) can be rewritten as(15)Γeu-ΓesΓes+ΓeuF.Assume that the QR decomposition of Γes and Γeu is expressed as(16)Γes=Q1Q2R10=Q1R1,Γeu=Q1Q2R1R2=Q1R1+Q2R2, which appears to be consistent with the fact that the error norm (15) is small (indeed, both QR decomposition types are based on the same subspaces ran(Q1), while R2F is supposed to be small compared to R1F); this yields Γeu-Γes(Γes)+ΓeuF=Q2(Q2)TΓeuF. The proof of this result lies in the fact that the pseudoinverse of Q1R1 is given by (R1)-1(Q1)T, where (Q1)T(Q1)=I. By considering the fact that the Frobenius norm is unitarily invariant, this yields Q2(Q2)TΓeuF=(Q2)TΓeuF. Hence, a subspace damage indicator can be expressed as follows:(17)Γeu-ΓesTF≈Q2TΓeuF.It is expected that structural damage is detected by an increase of the subspace damage indicator in (17). The important point is to identify the relevant changes of the damage indicator, that is, those which are not due to noise. To address this issue, it is proposed to normalize the damage indicator through the consideration of nref reference data sets for the safe structure. Within this framework, a residual covariance matrix Σ can be defined as follows:(18)Σ=1nref-1∑k=1nrefrkrkT,where rk refers to the residual vector for a given data set k (k=1,…,nref):(19)rk=vec⁡Q2TΓeuk,where vec· denotes the column stacking operator. As a result, a normalized damage indicator can be defined as follows:(20)ζ=rTΣ-1r,where r=vec(Q2)TΓeu is the residual vector that concerns any arbitrary data set k of the structure in an unknown health condition.Defineζ¯ and σ as the mean value and standard deviation of the error indicator ζ over the first nref reference data sets for the safe structure. Then, an X-bar control chart [24] can be considered which consists in a centerline (CL) with upper and lower control limits (UCL and LCL) as follows:(21)CL=ζ¯;UCL=ζ¯+3σ;LCL=ζ¯-3σ.By considering such a control chart, this yields an efficient means to assess the variation of a process and clearly identify damage when the damage indicator ζ exceeds the UCL [25].


## 2.1. Subspace-Based Methods

Consider the following2nth order discrete-time linear state space system for k=0,1,…,N-1 [21]:(1)xk+1=Axk+wk,yk=Cxk+vk,where xk is a 2n×1 state vector defined at the discrete time tk (k=1,2,…,N); yk is a l×1 vector of output data; also, A, B, C, and D are system matrices of respective sizes 2n×2n, 2n×m, l×2n, and l×m. Besides, vk and wk are 2n×1 and l×1 vectors of process and measurement noises, respectively, which are supposed to be white Gaussian with zero-mean distributions and joint covariance matrix [22](2)EwpvpwqTvqT=QSSTRδpq,where E· is the expectation operator, while δpq is the Kronecker delta.Some notations used for the subspace algorithms are introduced here. A block Hankel matrix of output data is first considered, which can be partitioned into past (p) and future (f) outputs as follows:(3)
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e
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.Here, the numberi of block rows of Yp and Yf should be greater than the order of the system—that is, 2n—in order to identify the system; also, the number j of columns is usually chosen so that it is equal to N-2i+1, where N is the number of time samples. In the same way as (3), block Hankel matrices of measurement noises can be defined as follows:(4)V0∣2i-1=defVpVf,W0∣2i-1=defWpWf.Also,2n×j matrices of state sequences Xp and Xf can be defined as follows:(5)Xp=defx0x1⋯xj-1,Xf=defxixi+1⋯xi+j-1.Consider now the following(li)×2n so-called extended observability matrix Γi and (li)×(li) block Toeplitz matrix Hiw:(6)Γi=defCCACA2⋮CAi-1,Hiw=def00C0⋮⋱⋱CAi-2⋯C0.By considering (1) and (3)–(6), the following matrix equations can be derived for the past and future parts, respectively [16]:(7)Yp=ΓiXp+HiwWp+Vp,Yf=ΓiXf+HiwWf+Vf.The key idea behind subspace methods is to identify the extended observability matrixΓi, and further the system matrix A. The strategy consists in projecting the row space of the future outputs Yf on the row space of the past outputs Yp, as follows:(8)YfYp=YfYpTYpYpT+Yp,where (·)+ denotes the Moore-Penrose pseudoinverse. By considering (7), this yields(9)YfYp=ΓiXfYp+HiwWfYp+VfYp.The procedure enables one to remove the noise terms; it is understood that the row spaces ofWf and Vf are perpendicular to the row space of Yp when the number of samples grows to infinity. A proof of this statement lies in the consideration of (2) [22]. Hence, the vector of state sequence Xf belongs to the joint row space of Yp; that is,(10)limN→∞⁡1NYfYp=ΓiXfYp.As a result, one has(11)YfYp≈ΓiXfYp.In other words, there exists some direct connection between the extended observability matrixΓi and the matrix Yf which is supposed to be known from measurement of the output signals [23]. For practical purpose, a singular value decomposition (SVD) of Yf/Yp is usually considered as follows:(12)YfYp=U1U2S1000V1TV2T≃U1S1V1T,where S1 is a 2n×2n diagonal matrix whose components are the 2n largest singular values of Yf/Yp, while U1 and V1 are (li)×2n and j×2n matrices of orthogonal vectors. In other words, it is assumed here that rank(Yf/Yp)=2n. Assume that the matrix Γi is full column rank; that is, rank(Γi)=2n, which is certainly true because N≫2n. Thus it turns out that the column space of Γi is almost the same as the space spanned by the first 2n left singular vectors of the matrix Yf/Yp, that is, those associated with its 2n largest singular values.From (11) and (12), one has ΓiXf/Yp≈U1S1V1T. This particularly means that the column space of Γi matches that of U1; that is, ran(Γi)≈ran(U1). Hence an estimate of the extended observability matrix can be defined as Γe=U1. The determination of the true observability matrix Γi follows as(13)Γi=ΓeTi,where Ti is a 2n×2n full rank matrix, while Γe is the so-called experimental extended observability matrix that is built from output measurements.


## 2.2. Subspace-Based Damage Indicator

The key idea behind the proposed subspace-based damage detection procedure consists in comparing the extended observability matrix of the safe structure (say,Γi=Γs) with that of the structure in an unknown health condition (say, Γi=Γu). This yields the consideration of an error norm Γu-ΓsF=ΓeuTu-ΓesTsF, where ·F is the Frobenius norm, while Γes=U1s and Γeu=U1u are experimental extended observability matrices; see (13).Here, an alternative error norm is considered which involves right-multiplying the residualΓeuTu-ΓesTs by (Tu)-1; that is,(14)Γeu-ΓesTFwhereT=TsTu-1.Besides, an approximate expression of the matrix T is considered which involves solving the following least squares problem Γu=ΓsT, which leads to T=(Γs)+Γu. The relevance of this approximate expression can be justified in the sense that the error norm Γu-ΓsTF is supposed to be small. Hence, (14) can be rewritten as(15)Γeu-ΓesΓes+ΓeuF.Assume that the QR decomposition of Γes and Γeu is expressed as(16)Γes=Q1Q2R10=Q1R1,Γeu=Q1Q2R1R2=Q1R1+Q2R2, which appears to be consistent with the fact that the error norm (15) is small (indeed, both QR decomposition types are based on the same subspaces ran(Q1), while R2F is supposed to be small compared to R1F); this yields Γeu-Γes(Γes)+ΓeuF=Q2(Q2)TΓeuF. The proof of this result lies in the fact that the pseudoinverse of Q1R1 is given by (R1)-1(Q1)T, where (Q1)T(Q1)=I. By considering the fact that the Frobenius norm is unitarily invariant, this yields Q2(Q2)TΓeuF=(Q2)TΓeuF. Hence, a subspace damage indicator can be expressed as follows:(17)Γeu-ΓesTF≈Q2TΓeuF.It is expected that structural damage is detected by an increase of the subspace damage indicator in (17). The important point is to identify the relevant changes of the damage indicator, that is, those which are not due to noise. To address this issue, it is proposed to normalize the damage indicator through the consideration of nref reference data sets for the safe structure. Within this framework, a residual covariance matrix Σ can be defined as follows:(18)Σ=1nref-1∑k=1nrefrkrkT,where rk refers to the residual vector for a given data set k (k=1,…,nref):(19)rk=vec⁡Q2TΓeuk,where vec· denotes the column stacking operator. As a result, a normalized damage indicator can be defined as follows:(20)ζ=rTΣ-1r,where r=vec(Q2)TΓeu is the residual vector that concerns any arbitrary data set k of the structure in an unknown health condition.Defineζ¯ and σ as the mean value and standard deviation of the error indicator ζ over the first nref reference data sets for the safe structure. Then, an X-bar control chart [24] can be considered which consists in a centerline (CL) with upper and lower control limits (UCL and LCL) as follows:(21)CL=ζ¯;UCL=ζ¯+3σ;LCL=ζ¯-3σ.By considering such a control chart, this yields an efficient means to assess the variation of a process and clearly identify damage when the damage indicator ζ exceeds the UCL [25].


## 3. Roller Bearing Damage Identification

### 3.1. Introduction

Rolling bearings are mechanical systems whose components—that is, rolling elements, inner raceway, outer raceway, and cage—usually induce a complex vibration behavior [26]. The vibration signature of a defective bearing is characterized by harmonics at particular bearing frequencies [27]. These harmonics can be identified from subspace identification methods but can be blurred by higher energy vibrations which are generated by other components of the same machine. Therefore a signal processing technique is required so as to clearly identify defects in rolling element bearings. This consists in applying a pass-band filter, which focuses on selecting a frequency region of interest.


### 3.2. Characteristic Frequencies

Generally, rolling bearings consist of two concentric rings, called the inner and outer races, with a set of rolling elements running on their tracks. Standard shapes of rolling elements include the ball, cylindrical roller, tapered roller, needle roller, and symmetrical and unsymmetrical barrel roller (see Figure1). Typically, the rolling elements in a bearing are guided in a cage that ensures uniform spacing and prevents mutual contact.Figure 1
Schematic representation of a roller bearing.There are four basic motions that are used to describe the bearing dynamics whose corresponding frequencies are called the bearing characteristic frequencies (BCFs). These frequencies relate the fundamental train frequency (FTF), the ball passing frequency inner race (BPFI), the ball passing frequency outer race (BPFO), and the ball spin frequency (BSF) and depend on the rotation speed. These are defined as follows [28]:(i)
The fundamental train frequency is related to the motion of the cage:(22)FTF=fr21-dBdPcos⁡θ,where NB, dP, dB, θ, and fr are the number of balls, the pitch diameter, the ball diameter, the contact angle, and the rotation frequency of the bearing.(ii)
The ball passing frequency inner race indicates the rate at which the balls pass a point on the track of the inner race:(23)BPFI=NB2fr1+dBdPcos⁡θ.(iii)
The ball passing frequency outer race is defined as the rate at which the balls pass a point on the track of the outer race:(24)BPFO=NB2fr1-dBdPcos⁡θ.(iv)
The ball spin frequency is the rate of rotation of a ball about its own axis in a bearing:(25)BSF=dP2dBfr1-dBdPcos⁡θ2.Those four bearing characteristic frequencies can be determined provided that the following assumptions are satisfied [29]: (1)
The balls/rollers have the same diameter.(2)
The interactions between the balls, inner race, and outer race are only due to rolling contacts.(3)
There is no slipping between the shaft and the bearing.(4)
The outer race is fixed, while the inner race is in rotation.


### 3.3. Damage Identification Strategy

The subspace-based damage detection method can be achieved by filtering the vibration signals of a bearing around theBCFs. Here, a pass-band filtering technique is considered [27], which focuses on the consideration of narrow bands around theBCFs. The method is detailed as follows:(i)
It includes determination of the center frequency of each pass-band filter, defined as a specificBCF.(ii)
It includes determination of the bandwidth, around eachBCF. Notice that the bandwidths should not overlap each other, that is, among the set of filters which are considered.(iii)
It includes generation of the filtered data, for eachBCF.(iv)
It also includes calculation of the subspace-based damage indicator (Section2.2) for eachBCF.The flowchart depicted in Figure 2 summaries the proposed methodology.Figure 2
Flowchart of the subspace-based damage identification strategy.


## 3.1. Introduction

Rolling bearings are mechanical systems whose components—that is, rolling elements, inner raceway, outer raceway, and cage—usually induce a complex vibration behavior [26]. The vibration signature of a defective bearing is characterized by harmonics at particular bearing frequencies [27]. These harmonics can be identified from subspace identification methods but can be blurred by higher energy vibrations which are generated by other components of the same machine. Therefore a signal processing technique is required so as to clearly identify defects in rolling element bearings. This consists in applying a pass-band filter, which focuses on selecting a frequency region of interest.


## 3.2. Characteristic Frequencies

Generally, rolling bearings consist of two concentric rings, called the inner and outer races, with a set of rolling elements running on their tracks. Standard shapes of rolling elements include the ball, cylindrical roller, tapered roller, needle roller, and symmetrical and unsymmetrical barrel roller (see Figure1). Typically, the rolling elements in a bearing are guided in a cage that ensures uniform spacing and prevents mutual contact.Figure 1
Schematic representation of a roller bearing.There are four basic motions that are used to describe the bearing dynamics whose corresponding frequencies are called the bearing characteristic frequencies (BCFs). These frequencies relate the fundamental train frequency (FTF), the ball passing frequency inner race (BPFI), the ball passing frequency outer race (BPFO), and the ball spin frequency (BSF) and depend on the rotation speed. These are defined as follows [28]:(i)
The fundamental train frequency is related to the motion of the cage:(22)FTF=fr21-dBdPcos⁡θ,where NB, dP, dB, θ, and fr are the number of balls, the pitch diameter, the ball diameter, the contact angle, and the rotation frequency of the bearing.(ii)
The ball passing frequency inner race indicates the rate at which the balls pass a point on the track of the inner race:(23)BPFI=NB2fr1+dBdPcos⁡θ.(iii)
The ball passing frequency outer race is defined as the rate at which the balls pass a point on the track of the outer race:(24)BPFO=NB2fr1-dBdPcos⁡θ.(iv)
The ball spin frequency is the rate of rotation of a ball about its own axis in a bearing:(25)BSF=dP2dBfr1-dBdPcos⁡θ2.Those four bearing characteristic frequencies can be determined provided that the following assumptions are satisfied [29]: (1)
The balls/rollers have the same diameter.(2)
The interactions between the balls, inner race, and outer race are only due to rolling contacts.(3)
There is no slipping between the shaft and the bearing.(4)
The outer race is fixed, while the inner race is in rotation.


## 3.3. Damage Identification Strategy

The subspace-based damage detection method can be achieved by filtering the vibration signals of a bearing around theBCFs. Here, a pass-band filtering technique is considered [27], which focuses on the consideration of narrow bands around theBCFs. The method is detailed as follows:(i)
It includes determination of the center frequency of each pass-band filter, defined as a specificBCF.(ii)
It includes determination of the bandwidth, around eachBCF. Notice that the bandwidths should not overlap each other, that is, among the set of filters which are considered.(iii)
It includes generation of the filtered data, for eachBCF.(iv)
It also includes calculation of the subspace-based damage indicator (Section2.2) for eachBCF.The flowchart depicted in Figure 2 summaries the proposed methodology.Figure 2
Flowchart of the subspace-based damage identification strategy.


## 4. Numerical Simulation

### 4.1. Roller Bearing Damaged Model

A number of well-established models which describe the vibration signals produced by faulty bearings have already been proposed by using finite element model updating [30] or analyzing different physical effects [31–33]. In general, all of these models simulate bearing signals as a series of exponentially decaying high frequency oscillations, which appear repeatedly due to the contact between a fault and the mating surface and low-frequency phenomena which act as amplitude modulators. These signals can be generally decomposed into three components:(26)yt=y1t+y2t+nt,where (i)
y1(t) represents the periodic component in the signal, given by(27)y1t=∑i=1NA1isin⁡2πf1it+θ1i,where A1(i), θ1(i), and f1(i) are the amplitude, initial phase, and frequency of the ith sinusoidal element, respectively. Here, two frequencies f1(1)=40 Hz and f1(2)=250 Hz are chosen to construct the periodic signal y1(t). The values of the parameters used are displayed in Table 1.(ii)
y2(t) represents the transient component of the signal. Physically, it reflects the evolution of the structural defects within the bearing, such as spalling on the surface of the bearing raceways or rolling elements. The transient component is modeled as a series of M exponentially attenuated vibrations, given by(28)y2t=∑i=1MA2iΘt-tie-Ct-tisin⁡2πf2t-ti+θ2,where A2(i), C, ti, θ2, and f2 are the amplitude, attenuation factor, time-delay, initial phase, and frequency of the ith impact, respectively. A2(i) is modeled as a random variable that varies between 0 and A2, while ti is defined as ti=i×1/BCF, whereBCF is the monitored frequency. Also, Θ(t-ti) in (28) is the conventional Heaviside function:(29)Θt-ti=1,t-ti≥00,t-ti<0.The values of the parameters used are displayed in Table 2.(iii)
n(t) is a white Gaussian noise, with signal-to-noise ratio (SNRdB) defined as(30)SNRdB=10×log10⁡AsignalAnoise2,where A is the root mean square amplitude of the signal and the noise.The signal depicted in Figure 3 is obtained from the chosen parameters reported in Tables 1 and 2, with SNRdB=100 dB and a signal length and sampling frequencies of 0.5 s and 10000 Hz, respectively.Table 1
Parameters related to the simulated test signaly1(t).
Parameter
Sinusoidal element (N)
(
i
)
1
2
A
1
(
i
) [m]
0.033
0.012
f
1
(
i
) [Hz]
250
40
θ
1
(
i
) [rad]
0
0.32πTable 2
Parameters related to the simulated test signaly2(t).
A
2
f
2
C
B
CF
θ
2
(m)
(Hz)
(s−1)
(Hz)
(rad)
1.10−6
2000
500
150
0Figure 3
Vibration data of the damaged bearing.


### 4.2. Damage Identification

To validate the proposed detection procedure,500 samples of bearing fault data are simulated. Every 50 samples, the amplitude of the fault A2 in (28) is increased, starting from 10-7 m (reference state) until 10-6 m (maximum damage).TheBCF subspace damage indicator is obtained from the whole set of simulated data. Here, the number of reference data sets is fixed to nref=50 and the filter bandwidth is chosen as [BCF±0.1BCF]. The order of the observability matrices is chosen as i=10.The results are reported in Figure4 regarding the damage indicator ζBSF; see (17). It is shown that the damage is identified as soon as the amplitude of the fault increases from 10-7 m to 2×10-7 m (i.e., from the undamaged state to the first level of damage). The value of the damage indicator ζBSF increases in good agreement with the evolution of the amplitude of the fault, which fully gives credit to the proposed methodology.Figure 4
Evolution of the subspace-based damage indicator.


## 4.1. Roller Bearing Damaged Model

A number of well-established models which describe the vibration signals produced by faulty bearings have already been proposed by using finite element model updating [30] or analyzing different physical effects [31–33]. In general, all of these models simulate bearing signals as a series of exponentially decaying high frequency oscillations, which appear repeatedly due to the contact between a fault and the mating surface and low-frequency phenomena which act as amplitude modulators. These signals can be generally decomposed into three components:(26)yt=y1t+y2t+nt,where (i)
y1(t) represents the periodic component in the signal, given by(27)y1t=∑i=1NA1isin⁡2πf1it+θ1i,where A1(i), θ1(i), and f1(i) are the amplitude, initial phase, and frequency of the ith sinusoidal element, respectively. Here, two frequencies f1(1)=40 Hz and f1(2)=250 Hz are chosen to construct the periodic signal y1(t). The values of the parameters used are displayed in Table 1.(ii)
y2(t) represents the transient component of the signal. Physically, it reflects the evolution of the structural defects within the bearing, such as spalling on the surface of the bearing raceways or rolling elements. The transient component is modeled as a series of M exponentially attenuated vibrations, given by(28)y2t=∑i=1MA2iΘt-tie-Ct-tisin⁡2πf2t-ti+θ2,where A2(i), C, ti, θ2, and f2 are the amplitude, attenuation factor, time-delay, initial phase, and frequency of the ith impact, respectively. A2(i) is modeled as a random variable that varies between 0 and A2, while ti is defined as ti=i×1/BCF, whereBCF is the monitored frequency. Also, Θ(t-ti) in (28) is the conventional Heaviside function:(29)Θt-ti=1,t-ti≥00,t-ti<0.The values of the parameters used are displayed in Table 2.(iii)
n(t) is a white Gaussian noise, with signal-to-noise ratio (SNRdB) defined as(30)SNRdB=10×log10⁡AsignalAnoise2,where A is the root mean square amplitude of the signal and the noise.The signal depicted in Figure 3 is obtained from the chosen parameters reported in Tables 1 and 2, with SNRdB=100 dB and a signal length and sampling frequencies of 0.5 s and 10000 Hz, respectively.Table 1
Parameters related to the simulated test signaly1(t).
Parameter
Sinusoidal element (N)
(
i
)
1
2
A
1
(
i
) [m]
0.033
0.012
f
1
(
i
) [Hz]
250
40
θ
1
(
i
) [rad]
0
0.32πTable 2
Parameters related to the simulated test signaly2(t).
A
2
f
2
C
B
CF
θ
2
(m)
(Hz)
(s−1)
(Hz)
(rad)
1.10−6
2000
500
150
0Figure 3
Vibration data of the damaged bearing.


## 4.2. Damage Identification

To validate the proposed detection procedure,500 samples of bearing fault data are simulated. Every 50 samples, the amplitude of the fault A2 in (28) is increased, starting from 10-7 m (reference state) until 10-6 m (maximum damage).TheBCF subspace damage indicator is obtained from the whole set of simulated data. Here, the number of reference data sets is fixed to nref=50 and the filter bandwidth is chosen as [BCF±0.1BCF]. The order of the observability matrices is chosen as i=10.The results are reported in Figure4 regarding the damage indicator ζBSF; see (17). It is shown that the damage is identified as soon as the amplitude of the fault increases from 10-7 m to 2×10-7 m (i.e., from the undamaged state to the first level of damage). The value of the damage indicator ζBSF increases in good agreement with the evolution of the amplitude of the fault, which fully gives credit to the proposed methodology.Figure 4
Evolution of the subspace-based damage indicator.


## 5. Experimental Validation

### 5.1. Preliminary Comments

The proposed damage identification procedure is applied to an experimental test rig which hosts four bearings on one shaft as shown in Figure5. The shaft is driven by an AC motor and is connected to rubber belts. A radial load of 6000 lbs is applied to the shaft and the bearing by means of a spring mechanism. A magnetic plug is installed in the oil feedback pipe to collect debris, from the oil, as an evidence of bearing degradation.Figure 5
Experimental bearing test rig.High sensitive ICP piezoaccelerometers sensors are placed on the bearing housing, in radial horizontal direction as shown in Figure5. The vibration data are collected every 10 minutes with a sampling rate of 20 kHz. Each data sample contains 20480 points. Data collection is conducted by a NI Labview program and is generated by IMS Center with support from Rexnord Corp. in Milwaukee, WI [34].The984 data pieces are recorded during 7 days, by considering sample of about 140 data pieces each day, until a significant amount of metal debris is found on the magnetic plug of the test bearing. The test stops until the accumulated debris adherent to the magnetic plug exceeds a certain threshold. At this time, a visual inspection is made. All failures occurred once the designed lifetime of the bearing is reached which is 100 million revolutions.Four Rexnord ZA-2115 double row bearings are mounted on the shaft. According to the geometric parameters of the bearing listed in Table 3, the nominalBCFs are calculated for a constant rotation speed of 2000 RPM and listed in Table 4.Table 3
Geometric parameters of the roller bearing.
Pitch diameter
Roller diameter
Contact angle
Number of rollers
d
P (mm)
d
B (mm)
(degree)
N
B
71.5
8.4
15.17
16Table 4
Rexnord ZA-2115  BCFs for running speed 2000 RPM.
Bearing characteristic frequencies
Theoretical values (Hz)
FTF
14.78
BSF
280.08
BPFO
236.43
BPFI
296.90


### 5.2. Run-to-Failure Test

From visual inspection, an outer race defect is discovered in bearing1. A sample of temporal and frequency representations of the data set is shown in Figure 6.Figure 6
Normalized amplitude of the acceleration over time and frequency domains for the 900th data sample on accelerometer1.Subspace damage indicators are obtained for each accelerometer and eachBCF from the whole set of accelerometers data. Here, the number of reference data sets is nref=140 and the filter bandwidth is chosen as [BCF±0.1BCF]. The order of the observability matrices (3) is chosen as i=12 by analyzing the singular values of Γi (as shown in Figure 7).Figure 7
Model order selection (here2n=6).Results of subspace damage indicatorsζFTF, ζBSF, ζBPFI, and ζBPFO are reported in Figures 8, 9, 10, and 11, respectively. A comparison of results for eachBCF and each accelerometer position shows that the first bearing is identified as damaged approximately between two and three days before the experiment is stopped. It is seen that the subspace damage indicator ζBPFO on accelerometer 1 has highest level and earliest sensitivity. According to this result, it is thus expected that the outer ring of the first bearing is the most damaged component, which is successfully demonstrated by visual inspection.Figure 8
Damage indicators associated with theFTF frequency from each accelerometer.
(a)
(b)
(c)
(d)Figure 9
Damage indicators associated with theBSF frequency from each accelerometer.
(a)
(b)
(c)
(d)Figure 10
Damage indicators associated with theBPFI frequency from each accelerometer.
(a)
(b)
(c)
(d)Figure 11
Damage indicators associated with theBPFO frequency from each accelerometer.
(a)
(b)
(c)
(d)It can be shown thatζBPFO also increases on accelerometers 3 and 4 probably because the dynamic behavior of the damaged bearing in position 1 is also recorded on the whole set of accelerometers or because bearings 3 and 4 start undergoing some deterioration, which seems to be the most probable situation although it is not validated by visual inspection at the end of the test.


## 5.1. Preliminary Comments

The proposed damage identification procedure is applied to an experimental test rig which hosts four bearings on one shaft as shown in Figure5. The shaft is driven by an AC motor and is connected to rubber belts. A radial load of 6000 lbs is applied to the shaft and the bearing by means of a spring mechanism. A magnetic plug is installed in the oil feedback pipe to collect debris, from the oil, as an evidence of bearing degradation.Figure 5
Experimental bearing test rig.High sensitive ICP piezoaccelerometers sensors are placed on the bearing housing, in radial horizontal direction as shown in Figure5. The vibration data are collected every 10 minutes with a sampling rate of 20 kHz. Each data sample contains 20480 points. Data collection is conducted by a NI Labview program and is generated by IMS Center with support from Rexnord Corp. in Milwaukee, WI [34].The984 data pieces are recorded during 7 days, by considering sample of about 140 data pieces each day, until a significant amount of metal debris is found on the magnetic plug of the test bearing. The test stops until the accumulated debris adherent to the magnetic plug exceeds a certain threshold. At this time, a visual inspection is made. All failures occurred once the designed lifetime of the bearing is reached which is 100 million revolutions.Four Rexnord ZA-2115 double row bearings are mounted on the shaft. According to the geometric parameters of the bearing listed in Table 3, the nominalBCFs are calculated for a constant rotation speed of 2000 RPM and listed in Table 4.Table 3
Geometric parameters of the roller bearing.
Pitch diameter
Roller diameter
Contact angle
Number of rollers
d
P (mm)
d
B (mm)
(degree)
N
B
71.5
8.4
15.17
16Table 4
Rexnord ZA-2115  BCFs for running speed 2000 RPM.
Bearing characteristic frequencies
Theoretical values (Hz)
FTF
14.78
BSF
280.08
BPFO
236.43
BPFI
296.90


## 5.2. Run-to-Failure Test

From visual inspection, an outer race defect is discovered in bearing1. A sample of temporal and frequency representations of the data set is shown in Figure 6.Figure 6
Normalized amplitude of the acceleration over time and frequency domains for the 900th data sample on accelerometer1.Subspace damage indicators are obtained for each accelerometer and eachBCF from the whole set of accelerometers data. Here, the number of reference data sets is nref=140 and the filter bandwidth is chosen as [BCF±0.1BCF]. The order of the observability matrices (3) is chosen as i=12 by analyzing the singular values of Γi (as shown in Figure 7).Figure 7
Model order selection (here2n=6).Results of subspace damage indicatorsζFTF, ζBSF, ζBPFI, and ζBPFO are reported in Figures 8, 9, 10, and 11, respectively. A comparison of results for eachBCF and each accelerometer position shows that the first bearing is identified as damaged approximately between two and three days before the experiment is stopped. It is seen that the subspace damage indicator ζBPFO on accelerometer 1 has highest level and earliest sensitivity. According to this result, it is thus expected that the outer ring of the first bearing is the most damaged component, which is successfully demonstrated by visual inspection.Figure 8
Damage indicators associated with theFTF frequency from each accelerometer.
(a)
(b)
(c)
(d)Figure 9
Damage indicators associated with theBSF frequency from each accelerometer.
(a)
(b)
(c)
(d)Figure 10
Damage indicators associated with theBPFI frequency from each accelerometer.
(a)
(b)
(c)
(d)Figure 11
Damage indicators associated with theBPFO frequency from each accelerometer.
(a)
(b)
(c)
(d)It can be shown thatζBPFO also increases on accelerometers 3 and 4 probably because the dynamic behavior of the damaged bearing in position 1 is also recorded on the whole set of accelerometers or because bearings 3 and 4 start undergoing some deterioration, which seems to be the most probable situation although it is not validated by visual inspection at the end of the test.


## 6. Conclusion

A damage identification procedure has been proposed which makes use of a subspace method combined with a pass-band data filtering technique. Within this framework, several damage indicators have been considered in the diagnosis of faults in rotating machines. The present methodology has been successfully applied to identify damage in a roller bearing, by considering simulated data. Also, the efficiency of the method has been highlighted regarding an experimental test that consists in monitoring real bearings. It has been shown that a roller bearing defect can be detected at an early stage with accurate precision. Future works may concern the comparison of this new indicator with other methods and extend the diagnosis to other rotating machinery components such as gearbox.



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*Source: 828093-2015-12-06.xml*