# Dynamic Analysis of a Rotor-Bearing-SFD System with the Bearing Inner Race Defect

Authors: Junhong Zhang; Xin Lu; Jiewei Lin; Liang Ma; Jun Wang
Journal: Shock and Vibration
 (2017)

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## Abstract

In this paper, the dynamic behavior of a rotor-bearing-SFD system with the inner race defect of bearing is investigated. The contact force between the rolling element and the race is calculated in Hertzian contact and elastohydrodynamic lubrication condition. The supporting force of the SFD is simulated by integrating the pressure distribution derived from Reynolds’s equation. The equations of motion of the rotor-bearing-SFD system are derived and solved using the fourth-order Runge-Kutta method. The dynamic behavior and the fault characteristics are analyzed with two configurations of the SFD:(1) mounted on the unfaulted bearing and (2) mounted on the faulty bearing. According to the analysis of time-frequency diagram, waterfall plot, and spectral diagram, the results show that the characteristics of inner race defects on bearing frequencies are related to the characteristic multiple frequency of the inner race defect and the fundamental frequency. The speed and defect width have different influence on the distribution and amplitude of frequency. The SFD can enhance the system stability under the bearing fault but the enhancement decreases with the increasing speed. Meanwhile, the beneficial effect of the SFD varies according to the mounted position in the rotor system.


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## Body

## 1. Introduction

Bearings are most widely used in rotating machinery as one of the most critical components. However, for the complex running condition of rotating machinery, bearings easily fail. According to research, over 40% failures are due to bearing defects among failures of motors [1]. Bearing faults may cause unstable vibration of rotor system which leads to serious economic and increase the downtime [2, 3]. Therefore, it is essential to understand the dynamic and intrinsic transient characteristics of vibration caused by bearing faults.Bearing faults can be divided into distributed defects and localized defects [4, 5]. The localized defects, also called single-points defects, include the cracks, pits on the surface of the inner race, outer race, and ball race. The distributed defects include surface roughness, waviness, misalignment races, and off-size rolling elements. In order to study the dynamic characteristics of ball bearing, Kankar et al. [6] developed a model of localized defects on a ball bearing and employed the Fast Fourier Transformation (FFT) and the Poincare maps to analyze the defects of bearing components. Arslan and Aktürk [7] developed a shaft-bearing model and analyzed the characteristics of bearing defects in frequency and time domains; the frequency characteristics of fault were studied. Patil et al. [8] used the time domain and the power spectrum to study the frequency and acceleration of vibration components of ball bearings and discussed the effects of the defects size and locations.According to the recent research, the vibration signals are nonstationary and time-varying with the effect of speed changing and the nonlinear characteristics of the rotor system caused by faults. To well extract the transient characteristics from signal data, many signal denoising methods have been developed [9], such as time-domain method, band-pass filtering, frequency-domain threshold, empirical mode decomposition (EMD), wavelet transform (WT), time-frequency analysis (TFA), manifold learning, and matching pursuit. However, there are still some issues that remained to be studied among these denoising methods. In these methods, time-frequency analysis can reflect the change of signal characteristics with time and is widely used in faults diagnosis [10]. Ding and He [9] proposed a novel transient signal reconstruction method for bearing fault features extraction and find this method is effective to verify faults characteristics of bearing fault. Liu et al. [11] used a novel fault detection method to study the impact time-frequency dictionary and faulty features of roller bearing defects.In order to reduce the nonlinear vibration, rotors are often supported by ball bearings with squeeze film dampers (SFD) in many applications [12]. SFD can be designed through introducing a centralized spring and filling the clearance with lubrication oil. In a SFD mounted rotor-bearing system, the bearing can be held in its position statically by the supporting fluid pressure. Meanwhile, the critical speed of the rotor system as well as the stability is able to be controlled by adjusting the SFD parameters. Many scholars focused on the dynamic characteristics of the rotor system affected by SFD [13–15]. Previous work on SFD on a two-disk rotor system with misalignment and the effects of SFD was discussed [16].However, although various available models represent the dynamics characteristics of rolling element bearings, there is no universal and detailed platform for the whole system [17]. There also need the intensive study of the bearing fault laws about the faults parameters to understand the nature of the fault and the occurrence and development of the fault, so as to timely and accurate to all kinds of anomalies or failure of the emergence and development of the trend to make judgments.On the basis of previous work [16], this paper developed a model of bearing which considered both Hertz contact and elastohydrodynamic lubrication. This model was focused on studying the fault characteristics of the rotor system caused by inner race defects. The waterfall map, time-frequency map, and spectral diagram were introduced to analyze the dynamic and signal characteristics of inner race defects, studying the law of the frequency characteristics changing with defect parameters. In the end, the effects of SFD to the dynamic characteristics of faults rotor system were discussed.


## 2. Model of the Rotor-Bearing-SFD System

### 2.1. SFD Model

The schematic of the SFD used in this paper is shown in Figure1. The instantaneous pressure distribution ps can be calculated using the Reynolds equations [18]:(1)1Rs2∂∂θhs3∂ps∂θ+hs3∂2ps∂x2=-12μsΩ∂hs∂θ+12μs∂hs∂t.Figure 1
Model of the SFD.In order to solve the Reynolds equation, the Sommerfeld transformation was introduced [19]; after integrating twice ps can be written as(2)ps=6μsεsΩsin⁡θ+ε˙scos⁡θx2-ls2/4cs21+εscos⁡θ3+ps,2-ps,1xls+12ps,2+ps,1,where hs=cs1+εscos⁡θ, μs is the oil viscosity, Ω is the precession angular velocity of the journal, cs is the radial clearance of the SFD, and εs is the eccentric ratio of the bearing and given as εs=x2+y2/cs, ps,1 and ps,2 are the SFD boundary pressures, and ls is the bearing width.According to the Reynolds boundary condition,ps,1 and ps,2 are assumed equal to 0; the supporting force of the oil film can be obtained as [19](3)Fs,y=-μsRsls3cs2y2+x2yε˙sI1+εsΩI2-xε˙sI2+εsΩI3,Fs,x=-μsRsls3cs2y2+x2xε˙sI1+εsΩI2+yε˙sI2+εsΩI3,where Rs is radius of SFD, I1, I2, and I3 are classic Sommerfeld integral equations; they can be written as(4)I1=∫θ1θ1+πcos2⁡θ1+εscos⁡θ3dθ,I2=∫θ1θ1+πsin⁡θcos⁡θ1+εscos⁡θ3dθ,I3=∫θ1θ1+πsin2⁡θ1+εscos⁡θ3dθ.


### 2.2. Ball Bearing Model with Faults

#### 2.2.1. Ball Bearing Model

The sketch of the ball bearing is shown in Figure2. For the bearing model ignoring the effect of the lubrication oil, the load distribution of the bearing can be determined based on the Hertzian contact theory [20, 21]. Due to the working condition of the rotor system, the centrifugal forces of the rolling elements should be considered. The model of ball bearing can be developed by following assumptions:(1)
The outer rings are connected rigidly to the support and inner rings are fixed rigidly to the shaft.(2)
The inner race, ball, and outer race move in the plane of bearing only.(3)
The temperature of bearing is stable when the bearing is in operation.Figure 2
Schematic diagram of the ball bearing.According to Harris’ work [22], the contact stiffness of the ball bearing can be calculated as follows:(5)kb,jH=∂Qb,ji∂δb,j=32Qb,ji1/3kc,ji-2/3+kc,jo-2/31+Fb,cQb,ji2/3-kc,jo-2/31+Fb,cQb,ji-1/3Qb,ji-1-1,where Qb is the load of ball, kc is the contact deformation coefficient of the ball, and Fb,c is the centrifugal force of the ball. The subscript “j” refers to the ball number, and the superscripts “i” and “o” refer to the inner and outer races, respectively.Regarding the point contact, the oil film thickness equation proposed by Sarangi et al. [21] is appropriate for the ball bearing:(6)Hb=2.69U0.67G0.53W-0.0671-0.61exp⁡-0.73eb,where U=μbu/2E′Rb is a nondimensional velocity, G=αE′ is a nondimensional material parameter, W=Qb/E′Rb2 is a nondimensional load parameter, eb is the ellipticity of the rolling element, μb is the dynamic viscosity, u is the linear velocity, E′=E/1-ν2 is the equivalent Young’s modulus, υ is the Poisson’s ratio, and α is the pressure-viscosity coefficient.Then the oil film stiffness between the rolling element and the inner race can be calculated as(7)kEHLi=∂Qj∂Hbi=∂Hbi∂Qj-1=-10.18023U--0.67G--0.53Qj1.067E′-0.067Rbi-1.1341-0.61e-0.73Keli-1.The oil film stiffness between the rolling element and the outer race,kEHLo, can be calculated in the same way. Thus, the total oil film stiffness of the bearing can be obtained:(8)kb,jE=1kEHLi+1kEHLo-1.The global stiffness of the rolling element can be obtained:(9)kb,j=1kb,jH+1kb,jE-1.Accordingly, the radial contact load of a rolling element can be calculated as(10)Qb,ji=kb,j·δb,j.Then the supporting forces of the ball bearing in the y- and x-directions are given as(11)Fb,y=-Qbyδb,Fb,x=-Qbxδb.


#### 2.2.2. Bearing Model with Inner Race Defect

When defect occurred on inner race of bearing, Figure2 was transferred into Figure 3 in detail. The displacement of the ith rolling element can be calculated as [6](12)rj=xcos⁡θj+ysin⁡θj-μb-μinner,where μinner is the additional displacement of the rolling element due to the inner race defect which is given as(13)μinner=Rinner-Rinner2-Li22,where Db is the radius of the rolling element and Li is the width of the defect on inner race.Figure 3
The Inner race defect of the ball.The contact angle,θi, of the ith rolling element and the defect angle, Φi, are given as(14)θj=2πNt-1+ωcaget,(15)Φj=arcsin⁡Li2Rinner,where N is the number of the rolling element, ωcage is the angular velocity of the cage, and Rinner is the radius of the defect.(16)ωcage=ωrotorrR+r,where R and r are the radiuses of the outer and the inner races, respectively.Substituting Eq. (12)–(15), the additional displacement of the rolling element, μinner, per cycle can be obtained:(17)μinner=Rinner-Rinner2-Li220<mod⁡θj2π-θinner<Φj0otherwise.


### 2.3. Rotor System Model

The rotor-bearing-SFD system is developed based on the Jeffcott rotor (shown in Figure4) in which the ball bearing B (on the right end) has the inner race defect and the SFD is mounted on one bearing at a time (e.g., the SFD was mounted on bearing B in Figure 4, and in this paper, the SFD mounted both A and B were discussed). The model is established based on the following assumptions:(1)
The movements of the rotor in axial and torsional directions are negligible.(2)
Each lumped mass point has four degrees of freedom including two translations and two rotations.(3)
The mass of spindle is lumped at the location of two bearings and the rotor mass is lumped at a point at the middle of the rotor.Figure 4
The model of the rotor-bearing-SFD system.The equations of motion of the dynamic system are derived as(18)mrx¨r+cx˙r+kxr-xA+kxr-xB=mreω2cos⁡θ,mry¨r+cy˙r+kyr-yA+kyr-yB=mreω2sin⁡θ-mg,mAx¨A+cox˙A+kxA-xr=FAb,x,mAy¨A+coy¨A+kyA-yr=FAb,y-mAg,mBx¨B+cox˙B+kxB-xr=FBb,x,mBy¨B+coy¨B+kyB-yr=FBb,y-mBg,msx¨s+kaxs=Fs,x-FBb,x,msy¨s+kays=Fs,y-FBb,y-msg,where mr, mA, and mB are the lumped masses of the rotor at the disk and the bearing positions respectively, c and co are viscous damping factors of the disk and the bearing, respectively, k is the stiffness of the shaft, FA(b,x), FA(b,y), FB(b,x), and FB(b,y) are the supporting forces of the bearings in the x- and y-directions, ms is the lumped mass of the bearing outer race, ka is the supporting stiffness of the SFD, and Fs,x and Fs,y are the supporting forces of the SFD in the x- and y-directions.


### 2.4. Numerical Method and Simulation Parameters

In the paper, the differential equations of motion are solved by the Runge-Kutta method. The time step of the iterative procedure isΔt=1×10-5 s. The time-varying data corresponding to the first 500 periods generated by the numerical integration is deliberately excluded in order to discard the transient solutions. The parameters of the system are shown as follows:Parameters of the System d m: 47e-3 m D b: 8.731e-3 m m r: 15 kg m A: 1 kg m B: 1 kg c: 1500 N∗s/m c o: 2100 N∗s/m k a: 2e6 N/m k: 2.5e7 N/m μ s: 0.02 Pa∗S c s: 8e-5 m l s: 11e-3 m R s: 0.03 m


## 2.1. SFD Model

The schematic of the SFD used in this paper is shown in Figure1. The instantaneous pressure distribution ps can be calculated using the Reynolds equations [18]:(1)1Rs2∂∂θhs3∂ps∂θ+hs3∂2ps∂x2=-12μsΩ∂hs∂θ+12μs∂hs∂t.Figure 1
Model of the SFD.In order to solve the Reynolds equation, the Sommerfeld transformation was introduced [19]; after integrating twice ps can be written as(2)ps=6μsεsΩsin⁡θ+ε˙scos⁡θx2-ls2/4cs21+εscos⁡θ3+ps,2-ps,1xls+12ps,2+ps,1,where hs=cs1+εscos⁡θ, μs is the oil viscosity, Ω is the precession angular velocity of the journal, cs is the radial clearance of the SFD, and εs is the eccentric ratio of the bearing and given as εs=x2+y2/cs, ps,1 and ps,2 are the SFD boundary pressures, and ls is the bearing width.According to the Reynolds boundary condition,ps,1 and ps,2 are assumed equal to 0; the supporting force of the oil film can be obtained as [19](3)Fs,y=-μsRsls3cs2y2+x2yε˙sI1+εsΩI2-xε˙sI2+εsΩI3,Fs,x=-μsRsls3cs2y2+x2xε˙sI1+εsΩI2+yε˙sI2+εsΩI3,where Rs is radius of SFD, I1, I2, and I3 are classic Sommerfeld integral equations; they can be written as(4)I1=∫θ1θ1+πcos2⁡θ1+εscos⁡θ3dθ,I2=∫θ1θ1+πsin⁡θcos⁡θ1+εscos⁡θ3dθ,I3=∫θ1θ1+πsin2⁡θ1+εscos⁡θ3dθ.


## 2.2. Ball Bearing Model with Faults

### 2.2.1. Ball Bearing Model

The sketch of the ball bearing is shown in Figure2. For the bearing model ignoring the effect of the lubrication oil, the load distribution of the bearing can be determined based on the Hertzian contact theory [20, 21]. Due to the working condition of the rotor system, the centrifugal forces of the rolling elements should be considered. The model of ball bearing can be developed by following assumptions:(1)
The outer rings are connected rigidly to the support and inner rings are fixed rigidly to the shaft.(2)
The inner race, ball, and outer race move in the plane of bearing only.(3)
The temperature of bearing is stable when the bearing is in operation.Figure 2
Schematic diagram of the ball bearing.According to Harris’ work [22], the contact stiffness of the ball bearing can be calculated as follows:(5)kb,jH=∂Qb,ji∂δb,j=32Qb,ji1/3kc,ji-2/3+kc,jo-2/31+Fb,cQb,ji2/3-kc,jo-2/31+Fb,cQb,ji-1/3Qb,ji-1-1,where Qb is the load of ball, kc is the contact deformation coefficient of the ball, and Fb,c is the centrifugal force of the ball. The subscript “j” refers to the ball number, and the superscripts “i” and “o” refer to the inner and outer races, respectively.Regarding the point contact, the oil film thickness equation proposed by Sarangi et al. [21] is appropriate for the ball bearing:(6)Hb=2.69U0.67G0.53W-0.0671-0.61exp⁡-0.73eb,where U=μbu/2E′Rb is a nondimensional velocity, G=αE′ is a nondimensional material parameter, W=Qb/E′Rb2 is a nondimensional load parameter, eb is the ellipticity of the rolling element, μb is the dynamic viscosity, u is the linear velocity, E′=E/1-ν2 is the equivalent Young’s modulus, υ is the Poisson’s ratio, and α is the pressure-viscosity coefficient.Then the oil film stiffness between the rolling element and the inner race can be calculated as(7)kEHLi=∂Qj∂Hbi=∂Hbi∂Qj-1=-10.18023U--0.67G--0.53Qj1.067E′-0.067Rbi-1.1341-0.61e-0.73Keli-1.The oil film stiffness between the rolling element and the outer race,kEHLo, can be calculated in the same way. Thus, the total oil film stiffness of the bearing can be obtained:(8)kb,jE=1kEHLi+1kEHLo-1.The global stiffness of the rolling element can be obtained:(9)kb,j=1kb,jH+1kb,jE-1.Accordingly, the radial contact load of a rolling element can be calculated as(10)Qb,ji=kb,j·δb,j.Then the supporting forces of the ball bearing in the y- and x-directions are given as(11)Fb,y=-Qbyδb,Fb,x=-Qbxδb.


### 2.2.2. Bearing Model with Inner Race Defect

When defect occurred on inner race of bearing, Figure2 was transferred into Figure 3 in detail. The displacement of the ith rolling element can be calculated as [6](12)rj=xcos⁡θj+ysin⁡θj-μb-μinner,where μinner is the additional displacement of the rolling element due to the inner race defect which is given as(13)μinner=Rinner-Rinner2-Li22,where Db is the radius of the rolling element and Li is the width of the defect on inner race.Figure 3
The Inner race defect of the ball.The contact angle,θi, of the ith rolling element and the defect angle, Φi, are given as(14)θj=2πNt-1+ωcaget,(15)Φj=arcsin⁡Li2Rinner,where N is the number of the rolling element, ωcage is the angular velocity of the cage, and Rinner is the radius of the defect.(16)ωcage=ωrotorrR+r,where R and r are the radiuses of the outer and the inner races, respectively.Substituting Eq. (12)–(15), the additional displacement of the rolling element, μinner, per cycle can be obtained:(17)μinner=Rinner-Rinner2-Li220<mod⁡θj2π-θinner<Φj0otherwise.


## 2.2.1. Ball Bearing Model

The sketch of the ball bearing is shown in Figure2. For the bearing model ignoring the effect of the lubrication oil, the load distribution of the bearing can be determined based on the Hertzian contact theory [20, 21]. Due to the working condition of the rotor system, the centrifugal forces of the rolling elements should be considered. The model of ball bearing can be developed by following assumptions:(1)
The outer rings are connected rigidly to the support and inner rings are fixed rigidly to the shaft.(2)
The inner race, ball, and outer race move in the plane of bearing only.(3)
The temperature of bearing is stable when the bearing is in operation.Figure 2
Schematic diagram of the ball bearing.According to Harris’ work [22], the contact stiffness of the ball bearing can be calculated as follows:(5)kb,jH=∂Qb,ji∂δb,j=32Qb,ji1/3kc,ji-2/3+kc,jo-2/31+Fb,cQb,ji2/3-kc,jo-2/31+Fb,cQb,ji-1/3Qb,ji-1-1,where Qb is the load of ball, kc is the contact deformation coefficient of the ball, and Fb,c is the centrifugal force of the ball. The subscript “j” refers to the ball number, and the superscripts “i” and “o” refer to the inner and outer races, respectively.Regarding the point contact, the oil film thickness equation proposed by Sarangi et al. [21] is appropriate for the ball bearing:(6)Hb=2.69U0.67G0.53W-0.0671-0.61exp⁡-0.73eb,where U=μbu/2E′Rb is a nondimensional velocity, G=αE′ is a nondimensional material parameter, W=Qb/E′Rb2 is a nondimensional load parameter, eb is the ellipticity of the rolling element, μb is the dynamic viscosity, u is the linear velocity, E′=E/1-ν2 is the equivalent Young’s modulus, υ is the Poisson’s ratio, and α is the pressure-viscosity coefficient.Then the oil film stiffness between the rolling element and the inner race can be calculated as(7)kEHLi=∂Qj∂Hbi=∂Hbi∂Qj-1=-10.18023U--0.67G--0.53Qj1.067E′-0.067Rbi-1.1341-0.61e-0.73Keli-1.The oil film stiffness between the rolling element and the outer race,kEHLo, can be calculated in the same way. Thus, the total oil film stiffness of the bearing can be obtained:(8)kb,jE=1kEHLi+1kEHLo-1.The global stiffness of the rolling element can be obtained:(9)kb,j=1kb,jH+1kb,jE-1.Accordingly, the radial contact load of a rolling element can be calculated as(10)Qb,ji=kb,j·δb,j.Then the supporting forces of the ball bearing in the y- and x-directions are given as(11)Fb,y=-Qbyδb,Fb,x=-Qbxδb.


## 2.2.2. Bearing Model with Inner Race Defect

When defect occurred on inner race of bearing, Figure2 was transferred into Figure 3 in detail. The displacement of the ith rolling element can be calculated as [6](12)rj=xcos⁡θj+ysin⁡θj-μb-μinner,where μinner is the additional displacement of the rolling element due to the inner race defect which is given as(13)μinner=Rinner-Rinner2-Li22,where Db is the radius of the rolling element and Li is the width of the defect on inner race.Figure 3
The Inner race defect of the ball.The contact angle,θi, of the ith rolling element and the defect angle, Φi, are given as(14)θj=2πNt-1+ωcaget,(15)Φj=arcsin⁡Li2Rinner,where N is the number of the rolling element, ωcage is the angular velocity of the cage, and Rinner is the radius of the defect.(16)ωcage=ωrotorrR+r,where R and r are the radiuses of the outer and the inner races, respectively.Substituting Eq. (12)–(15), the additional displacement of the rolling element, μinner, per cycle can be obtained:(17)μinner=Rinner-Rinner2-Li220<mod⁡θj2π-θinner<Φj0otherwise.


## 2.3. Rotor System Model

The rotor-bearing-SFD system is developed based on the Jeffcott rotor (shown in Figure4) in which the ball bearing B (on the right end) has the inner race defect and the SFD is mounted on one bearing at a time (e.g., the SFD was mounted on bearing B in Figure 4, and in this paper, the SFD mounted both A and B were discussed). The model is established based on the following assumptions:(1)
The movements of the rotor in axial and torsional directions are negligible.(2)
Each lumped mass point has four degrees of freedom including two translations and two rotations.(3)
The mass of spindle is lumped at the location of two bearings and the rotor mass is lumped at a point at the middle of the rotor.Figure 4
The model of the rotor-bearing-SFD system.The equations of motion of the dynamic system are derived as(18)mrx¨r+cx˙r+kxr-xA+kxr-xB=mreω2cos⁡θ,mry¨r+cy˙r+kyr-yA+kyr-yB=mreω2sin⁡θ-mg,mAx¨A+cox˙A+kxA-xr=FAb,x,mAy¨A+coy¨A+kyA-yr=FAb,y-mAg,mBx¨B+cox˙B+kxB-xr=FBb,x,mBy¨B+coy¨B+kyB-yr=FBb,y-mBg,msx¨s+kaxs=Fs,x-FBb,x,msy¨s+kays=Fs,y-FBb,y-msg,where mr, mA, and mB are the lumped masses of the rotor at the disk and the bearing positions respectively, c and co are viscous damping factors of the disk and the bearing, respectively, k is the stiffness of the shaft, FA(b,x), FA(b,y), FB(b,x), and FB(b,y) are the supporting forces of the bearings in the x- and y-directions, ms is the lumped mass of the bearing outer race, ka is the supporting stiffness of the SFD, and Fs,x and Fs,y are the supporting forces of the SFD in the x- and y-directions.


## 2.4. Numerical Method and Simulation Parameters

In the paper, the differential equations of motion are solved by the Runge-Kutta method. The time step of the iterative procedure isΔt=1×10-5 s. The time-varying data corresponding to the first 500 periods generated by the numerical integration is deliberately excluded in order to discard the transient solutions. The parameters of the system are shown as follows:Parameters of the System d m: 47e-3 m D b: 8.731e-3 m m r: 15 kg m A: 1 kg m B: 1 kg c: 1500 N∗s/m c o: 2100 N∗s/m k a: 2e6 N/m k: 2.5e7 N/m μ s: 0.02 Pa∗S c s: 8e-5 m l s: 11e-3 m R s: 0.03 m


## 3. Results and Discussion

### 3.1. Model Validation

#### 3.1.1. Bearing Model

In [12], a ball bearing dynamic model was established by Zhou et al. In order to validate the ball bearing dynamic model the simulation results of the model should be compared with the results of [12] on the same computation conditions. In this paper, the response of the rotor is shown in Figure 5; it is easy to find the VC frequency and 2, 3 VC frequency which represents the ball passage frequency. The same conclusion was found in [12]; therefore, the model established in this paper is further validated.Figure 5
Rotor response of frequency spectrum (a) in this paper (b) in [11].
(a) (b)


#### 3.1.2. SFD Model

In [23], the dynamic model of an unbalanced rotor supported on SFD without ball bearing was established by Zhu et al.; analytical method and numerical integration method were employed to obtain the responses of the rotor system. In order to validate the ball bearing dynamic model, the simulation results of the model should be compared with the results of [23] on the same computation conditions. It is shown in Figure 6 that the rotor system has multiple solutions in the region of (1480, 1550) rad/s. The same phenomenon of multiple solution was found in [23]. By using this method, the model established in this paper is validated.Figure 6
Steady state responses of rotor in this paper (a) in this paper (b) in [22].
(a) (b)


### 3.2. Effects of Inner Race Defect

#### 3.2.1. Effects of Rotating Speed

For a ball bearing, the characteristic frequency of the inner race defectfi can be calculated as(19)fi=N21+DbDm·f,where Db is the diameter of the rolling element, Dm is the pitch diameter of the bearing, N is the number of the rolling elements, and f is the rotating frequency of the rotor.In order to study the fault characteristics of the bearing inner race defect, the waterfall plot of the faulted rotor system in the range of (50, 250) rad/s was shown in Figure7. When the speed is at 50 rad/s, the maximum frequency is 4fi+f (4.5e-7 m), the maximum frequency became fi (1.01e-6 m) at 250 rad/s, and it can be found that, with the increasing of speed, the amplitude of high frequencies higher than 10x fundamental frequency decreased obviously, the maximum frequency moved to low frequency, and the amplitude of the maximum frequency had great increasing.Figure 7
Waterfall plot of inner race defect bearing system with speed at width1e-3 m.From Figure8, when the speed is at 100 rad/s, many multiple frequency components appear due to the inner race defect, in which the dominant peaks at 2fi, 2fi+f and 2fi+2f are much higher than that at the fundamental frequency. In addition, the amplitudes at 3fi-f and 2fi-f are also greater than that at the fundamental frequency. Among the frequency components, the maximum amplitude at 2fi+f (8.1×10-7 m) is nearly four times that at the fundamental frequency. Besides, the sideband frequencies centering on fi and its multiple frequencies with an interval f (e.g., 2fi-f, 2fi+f) can be found in the faulted rotor system.Figure 8
The spectral diagrams of the fault system with bearing inner race defect atω=100 rad/s and width 1e-3 m.At 150 rad/s case, in Figure9 for the faulted system, three dominant frequency components can be found which are 2fi-2f, fi+2f, and 2fi-f. The maximum amplitude is nearly 7.9e-7 m at 2fi-2f which is far higher than the fundamental frequency. Similar to the 100 rad/s case, sideband frequencies from fi to 2fi can be found. Comparing to the faulted system at 100 rad/s, the amplitudes at the sideband frequencies increased while that centering at 2fi decreased.Figure 9
The spectral diagrams of the fault system with bearing inner race defect atω=150 rad/s and width 1e-3 m.By analyzing these pictures, it can be concluded that the fault characteristics of the bearing inner race defect are significant at a lower rotating speed of the rotor system. With increasing speed, the significance of the bearing fault characteristic becomes much lower due to the amplitude of the fundamental frequency increases greatly, while that of the fault characteristic frequencies only has a very slight increment.


#### 3.2.2. Effects of Defects Width

From Figure10, it is found that the maximum frequency is 2fi+f no matter how width changed, with the increasing of width in range of (0.5e-3 m, 2e-3 m), the amplitude of 2fi+f is increased from 3.87e-7 m to 1.381e-6 m, with a little decrease when width at 2.5e-3 m. The sideband besides fi has great increasing while it, besides 3fi, has no obvious changing. From Figure 11, the same phenomenon is found, only the maximum frequency is 2fi-2f, and it increased from 3.67e-7 m (at 0.5e-3 m) to 1.569e-6 m (at 2.5e-3 m). From Figures 10 and 11, it can be found that the width of defect has great influence to the amplitude of frequency. With the increasing of width, the amplitude has great change. But the width has no obvious effect to the distribution of frequency.Figure 10
Waterfall plot of inner race defect bearing system with width of defect at 100 rad/s.Figure 11
Waterfall plot of inner race defect bearing system with width of defect at 150 rad/s.


### 3.3. Effects of SFD

#### 3.3.1. SFD Mounted on Bearing A

To study the effect of the SFD on the dynamic behavior of the rotor system with bearing fault, the SFD is mounted on one of the two bearings at a time. The spectral diagrams of the faulted system with the SFD mounted on the normal bearing at the speed of 100 rad/s, 150 rad/s are shown in Figures12 and 13.Figure 12
The spectral diagrams of the faulted system with SFD mounted on the normal bearing atω=100 rad/s and width = 1e-3 m.Figure 13
The spectral diagrams of the faulted system with SFD mounted on the normal bearing atω=150 rad/s and width = 1e-3 m.Comparing with the system without SFD (Figure8), it can be found that the amplitude of the fundamental frequency does not change, while the amplitudes at the fault characteristic frequencies and the sideband frequencies are decreased. For instance, the amplitude at 2fi+f decreased from 8.1 × 10−7 m to 6.7 × 10−7 m, that at 2fi decreased from 5.3 × 10−7 m to 4.8 × 10−7 m, and that at 2fi+2f decreased from 6.6 × 10−7 m to 5.3 × 10−7 m, while the maximum frequency still locates at 2fi+f the same as the system without SFD. It can be seen that the introduction of the SFD on the normal bearing does not have much effect on the frequency characteristics of the rotor system at low speed.


#### 3.3.2. SFD Mounted on Bearing B

Figures13–15 are the spectral diagrams of the rotor system with the SFD mounted on the fault bearing at 100 rad/s and 150 rad/s, respectively.Figure 14
The spectral diagram of the faulted system with SFD mounted on the faulted bearing atω=100 rad/s.Figure 15
The spectral diagram of the faulted system with SFD mounted on the faulted bearing atω=150 rad/s.It can be seen that the amplitude at the fundamental frequency declines from 2.2 × 10−7 m (shown in Figure 7(b)) to 1.5 × 10−7 m (shown in Figure 13) when the rotating speed is 100 rad/s. The amplitudes at the fault characteristic frequencies also decrease a lot, such as from 8.1 × 10−7 m to 1.52 × 10−7 m at 2fi+f, from 5.3 × 10−7 m to 1.9 × 10−7 m at 2fi, and from 6.6 × 10−7 m to 6.5 × 10−8 m at 2fi+2f. In addition, dominant frequency shifts to 2fi.At 150 rad/s case, the amplitude of the fundamental frequency of the system with the SFD mounted on the faulted bearing does not change much comparing with that without the SFD. However, the amplitudes at fault characteristic frequencies decrease significantly due to the adoption of the SFD (from 1.04 × 10−6 m to 1.02 × 10−7 m at fi+f and from 5.3 × 10−7 m to 2.01 × 10−7 m at fi). Apart from that, the amplitudes at fault characteristic frequencies are even smaller than that of the fundamental frequency which may be a considerable influence on system response caused by the SFD mounted on the faulted bearing. It is found that the stability of the rotor system is enhanced due to the fact that fault characteristics are suppressed a lot by the SFD on the faults side compared to the normal side.


### 3.4. Time-Frequency Analysis

#### 3.4.1. For Effect of Faults Characteristics

Time-frequency diagrams of the faulted rotor system at different rotating speeds are shown in Figure16. It is found that the energy of the signal concentrates around the 10x fundamental frequency, which is close to the peak at the 2fi+f frequency in the spectral diagram. The signal of the bearing fault performs as impacts in the high frequency range without any patterns, while the signal of the fundamental frequency is continuous in the same frequency range. Figure 16(b) indicates that the system energy centralizes at the fundamental frequency and between 5 and 10x fundamental frequency, and in Figure 16(c) the energy centralizes at 5x fundamental frequency. The amplitude of the max energy has a little change with the speed increasing.Figure 16
Time-frequency diagrams of the faulted system at (a) 100 rad/s, (b) 150 rad/s, and (c) 200 rad/s.
(a) (b) (c)The time-frequency diagrams of the faulted rotor system at different defect width were shown in Figures16 and 17. With the increasing of defects width, the distribution of signal energy changed a little, and the characteristic of the signal is changeless, but the amplitude of energy increased a lot with width increasing.Figure 17
Time-frequency diagrams of the faulted system at 100 rad/s with different defect width: (a)0.5e-3 m; (b) 1e-3 m; (c) 1.5e-3 m; (d) 2e-3 m; (e) 2.5e-3 m.
(a) (b) (c) (d) (e)


#### 3.4.2. For Effect of SFD

Time-frequency diagrams of the faulted rotor system with the SFD mounted on bearing A (normal bearing) are shown in Figure18. It can be seen that the rotor signal in high frequency range is still remittent, while the distribution of the signal energy is much looser with some components moving to the higher frequency at 100 rad/s. With the speed increased to 200 rad/s, the signal energy moves to 5x fundamental frequency (fi+f in the spectral diagram), and the energy in high frequency is reduced. Thus the system signal energy does not show apparent changes when the SFD is mounted on the normal bearing side. Comparing to Figure 16, the energy of the signal has no obvious decreasing.Figure 18
The time-frequency diagrams of the system with the SFD mounted on the normal bearing at (a) 100 rad/s and (b) 150 rad/s and (c) 200 rad/s.
(a) (b) (c)Time-frequency diagrams of the rotor system with the SFD mounted on the bearing B (faulted bearing) are shown in Figure19. At low rotating speed, the energy of the signal decreases sharply especially at the frequency higher than 10x fundamental frequency (2fi+f in the spectral diagram). As the speed increased to 200 rad/s, the energy of signal shifts to the fundamental frequency and that of the fault signal at high frequency reduces a lot.Figure 19
The time-frequency diagram of the system with the SFD mounted on the faulted bearing at (a) 100 rad/s and (c) 150 rad/s and (c) 200 rad/s.
(a) (b) (c)


## 3.1. Model Validation

### 3.1.1. Bearing Model

In [12], a ball bearing dynamic model was established by Zhou et al. In order to validate the ball bearing dynamic model the simulation results of the model should be compared with the results of [12] on the same computation conditions. In this paper, the response of the rotor is shown in Figure 5; it is easy to find the VC frequency and 2, 3 VC frequency which represents the ball passage frequency. The same conclusion was found in [12]; therefore, the model established in this paper is further validated.Figure 5
Rotor response of frequency spectrum (a) in this paper (b) in [11].
(a) (b)


### 3.1.2. SFD Model

In [23], the dynamic model of an unbalanced rotor supported on SFD without ball bearing was established by Zhu et al.; analytical method and numerical integration method were employed to obtain the responses of the rotor system. In order to validate the ball bearing dynamic model, the simulation results of the model should be compared with the results of [23] on the same computation conditions. It is shown in Figure 6 that the rotor system has multiple solutions in the region of (1480, 1550) rad/s. The same phenomenon of multiple solution was found in [23]. By using this method, the model established in this paper is validated.Figure 6
Steady state responses of rotor in this paper (a) in this paper (b) in [22].
(a) (b)


## 3.1.1. Bearing Model

In [12], a ball bearing dynamic model was established by Zhou et al. In order to validate the ball bearing dynamic model the simulation results of the model should be compared with the results of [12] on the same computation conditions. In this paper, the response of the rotor is shown in Figure 5; it is easy to find the VC frequency and 2, 3 VC frequency which represents the ball passage frequency. The same conclusion was found in [12]; therefore, the model established in this paper is further validated.Figure 5
Rotor response of frequency spectrum (a) in this paper (b) in [11].
(a) (b)


## 3.1.2. SFD Model

In [23], the dynamic model of an unbalanced rotor supported on SFD without ball bearing was established by Zhu et al.; analytical method and numerical integration method were employed to obtain the responses of the rotor system. In order to validate the ball bearing dynamic model, the simulation results of the model should be compared with the results of [23] on the same computation conditions. It is shown in Figure 6 that the rotor system has multiple solutions in the region of (1480, 1550) rad/s. The same phenomenon of multiple solution was found in [23]. By using this method, the model established in this paper is validated.Figure 6
Steady state responses of rotor in this paper (a) in this paper (b) in [22].
(a) (b)


## 3.2. Effects of Inner Race Defect

### 3.2.1. Effects of Rotating Speed

For a ball bearing, the characteristic frequency of the inner race defectfi can be calculated as(19)fi=N21+DbDm·f,where Db is the diameter of the rolling element, Dm is the pitch diameter of the bearing, N is the number of the rolling elements, and f is the rotating frequency of the rotor.In order to study the fault characteristics of the bearing inner race defect, the waterfall plot of the faulted rotor system in the range of (50, 250) rad/s was shown in Figure7. When the speed is at 50 rad/s, the maximum frequency is 4fi+f (4.5e-7 m), the maximum frequency became fi (1.01e-6 m) at 250 rad/s, and it can be found that, with the increasing of speed, the amplitude of high frequencies higher than 10x fundamental frequency decreased obviously, the maximum frequency moved to low frequency, and the amplitude of the maximum frequency had great increasing.Figure 7
Waterfall plot of inner race defect bearing system with speed at width1e-3 m.From Figure8, when the speed is at 100 rad/s, many multiple frequency components appear due to the inner race defect, in which the dominant peaks at 2fi, 2fi+f and 2fi+2f are much higher than that at the fundamental frequency. In addition, the amplitudes at 3fi-f and 2fi-f are also greater than that at the fundamental frequency. Among the frequency components, the maximum amplitude at 2fi+f (8.1×10-7 m) is nearly four times that at the fundamental frequency. Besides, the sideband frequencies centering on fi and its multiple frequencies with an interval f (e.g., 2fi-f, 2fi+f) can be found in the faulted rotor system.Figure 8
The spectral diagrams of the fault system with bearing inner race defect atω=100 rad/s and width 1e-3 m.At 150 rad/s case, in Figure9 for the faulted system, three dominant frequency components can be found which are 2fi-2f, fi+2f, and 2fi-f. The maximum amplitude is nearly 7.9e-7 m at 2fi-2f which is far higher than the fundamental frequency. Similar to the 100 rad/s case, sideband frequencies from fi to 2fi can be found. Comparing to the faulted system at 100 rad/s, the amplitudes at the sideband frequencies increased while that centering at 2fi decreased.Figure 9
The spectral diagrams of the fault system with bearing inner race defect atω=150 rad/s and width 1e-3 m.By analyzing these pictures, it can be concluded that the fault characteristics of the bearing inner race defect are significant at a lower rotating speed of the rotor system. With increasing speed, the significance of the bearing fault characteristic becomes much lower due to the amplitude of the fundamental frequency increases greatly, while that of the fault characteristic frequencies only has a very slight increment.


### 3.2.2. Effects of Defects Width

From Figure10, it is found that the maximum frequency is 2fi+f no matter how width changed, with the increasing of width in range of (0.5e-3 m, 2e-3 m), the amplitude of 2fi+f is increased from 3.87e-7 m to 1.381e-6 m, with a little decrease when width at 2.5e-3 m. The sideband besides fi has great increasing while it, besides 3fi, has no obvious changing. From Figure 11, the same phenomenon is found, only the maximum frequency is 2fi-2f, and it increased from 3.67e-7 m (at 0.5e-3 m) to 1.569e-6 m (at 2.5e-3 m). From Figures 10 and 11, it can be found that the width of defect has great influence to the amplitude of frequency. With the increasing of width, the amplitude has great change. But the width has no obvious effect to the distribution of frequency.Figure 10
Waterfall plot of inner race defect bearing system with width of defect at 100 rad/s.Figure 11
Waterfall plot of inner race defect bearing system with width of defect at 150 rad/s.


## 3.2.1. Effects of Rotating Speed

For a ball bearing, the characteristic frequency of the inner race defectfi can be calculated as(19)fi=N21+DbDm·f,where Db is the diameter of the rolling element, Dm is the pitch diameter of the bearing, N is the number of the rolling elements, and f is the rotating frequency of the rotor.In order to study the fault characteristics of the bearing inner race defect, the waterfall plot of the faulted rotor system in the range of (50, 250) rad/s was shown in Figure7. When the speed is at 50 rad/s, the maximum frequency is 4fi+f (4.5e-7 m), the maximum frequency became fi (1.01e-6 m) at 250 rad/s, and it can be found that, with the increasing of speed, the amplitude of high frequencies higher than 10x fundamental frequency decreased obviously, the maximum frequency moved to low frequency, and the amplitude of the maximum frequency had great increasing.Figure 7
Waterfall plot of inner race defect bearing system with speed at width1e-3 m.From Figure8, when the speed is at 100 rad/s, many multiple frequency components appear due to the inner race defect, in which the dominant peaks at 2fi, 2fi+f and 2fi+2f are much higher than that at the fundamental frequency. In addition, the amplitudes at 3fi-f and 2fi-f are also greater than that at the fundamental frequency. Among the frequency components, the maximum amplitude at 2fi+f (8.1×10-7 m) is nearly four times that at the fundamental frequency. Besides, the sideband frequencies centering on fi and its multiple frequencies with an interval f (e.g., 2fi-f, 2fi+f) can be found in the faulted rotor system.Figure 8
The spectral diagrams of the fault system with bearing inner race defect atω=100 rad/s and width 1e-3 m.At 150 rad/s case, in Figure9 for the faulted system, three dominant frequency components can be found which are 2fi-2f, fi+2f, and 2fi-f. The maximum amplitude is nearly 7.9e-7 m at 2fi-2f which is far higher than the fundamental frequency. Similar to the 100 rad/s case, sideband frequencies from fi to 2fi can be found. Comparing to the faulted system at 100 rad/s, the amplitudes at the sideband frequencies increased while that centering at 2fi decreased.Figure 9
The spectral diagrams of the fault system with bearing inner race defect atω=150 rad/s and width 1e-3 m.By analyzing these pictures, it can be concluded that the fault characteristics of the bearing inner race defect are significant at a lower rotating speed of the rotor system. With increasing speed, the significance of the bearing fault characteristic becomes much lower due to the amplitude of the fundamental frequency increases greatly, while that of the fault characteristic frequencies only has a very slight increment.


## 3.2.2. Effects of Defects Width

From Figure10, it is found that the maximum frequency is 2fi+f no matter how width changed, with the increasing of width in range of (0.5e-3 m, 2e-3 m), the amplitude of 2fi+f is increased from 3.87e-7 m to 1.381e-6 m, with a little decrease when width at 2.5e-3 m. The sideband besides fi has great increasing while it, besides 3fi, has no obvious changing. From Figure 11, the same phenomenon is found, only the maximum frequency is 2fi-2f, and it increased from 3.67e-7 m (at 0.5e-3 m) to 1.569e-6 m (at 2.5e-3 m). From Figures 10 and 11, it can be found that the width of defect has great influence to the amplitude of frequency. With the increasing of width, the amplitude has great change. But the width has no obvious effect to the distribution of frequency.Figure 10
Waterfall plot of inner race defect bearing system with width of defect at 100 rad/s.Figure 11
Waterfall plot of inner race defect bearing system with width of defect at 150 rad/s.


## 3.3. Effects of SFD

### 3.3.1. SFD Mounted on Bearing A

To study the effect of the SFD on the dynamic behavior of the rotor system with bearing fault, the SFD is mounted on one of the two bearings at a time. The spectral diagrams of the faulted system with the SFD mounted on the normal bearing at the speed of 100 rad/s, 150 rad/s are shown in Figures12 and 13.Figure 12
The spectral diagrams of the faulted system with SFD mounted on the normal bearing atω=100 rad/s and width = 1e-3 m.Figure 13
The spectral diagrams of the faulted system with SFD mounted on the normal bearing atω=150 rad/s and width = 1e-3 m.Comparing with the system without SFD (Figure8), it can be found that the amplitude of the fundamental frequency does not change, while the amplitudes at the fault characteristic frequencies and the sideband frequencies are decreased. For instance, the amplitude at 2fi+f decreased from 8.1 × 10−7 m to 6.7 × 10−7 m, that at 2fi decreased from 5.3 × 10−7 m to 4.8 × 10−7 m, and that at 2fi+2f decreased from 6.6 × 10−7 m to 5.3 × 10−7 m, while the maximum frequency still locates at 2fi+f the same as the system without SFD. It can be seen that the introduction of the SFD on the normal bearing does not have much effect on the frequency characteristics of the rotor system at low speed.


### 3.3.2. SFD Mounted on Bearing B

Figures13–15 are the spectral diagrams of the rotor system with the SFD mounted on the fault bearing at 100 rad/s and 150 rad/s, respectively.Figure 14
The spectral diagram of the faulted system with SFD mounted on the faulted bearing atω=100 rad/s.Figure 15
The spectral diagram of the faulted system with SFD mounted on the faulted bearing atω=150 rad/s.It can be seen that the amplitude at the fundamental frequency declines from 2.2 × 10−7 m (shown in Figure 7(b)) to 1.5 × 10−7 m (shown in Figure 13) when the rotating speed is 100 rad/s. The amplitudes at the fault characteristic frequencies also decrease a lot, such as from 8.1 × 10−7 m to 1.52 × 10−7 m at 2fi+f, from 5.3 × 10−7 m to 1.9 × 10−7 m at 2fi, and from 6.6 × 10−7 m to 6.5 × 10−8 m at 2fi+2f. In addition, dominant frequency shifts to 2fi.At 150 rad/s case, the amplitude of the fundamental frequency of the system with the SFD mounted on the faulted bearing does not change much comparing with that without the SFD. However, the amplitudes at fault characteristic frequencies decrease significantly due to the adoption of the SFD (from 1.04 × 10−6 m to 1.02 × 10−7 m at fi+f and from 5.3 × 10−7 m to 2.01 × 10−7 m at fi). Apart from that, the amplitudes at fault characteristic frequencies are even smaller than that of the fundamental frequency which may be a considerable influence on system response caused by the SFD mounted on the faulted bearing. It is found that the stability of the rotor system is enhanced due to the fact that fault characteristics are suppressed a lot by the SFD on the faults side compared to the normal side.


## 3.3.1. SFD Mounted on Bearing A

To study the effect of the SFD on the dynamic behavior of the rotor system with bearing fault, the SFD is mounted on one of the two bearings at a time. The spectral diagrams of the faulted system with the SFD mounted on the normal bearing at the speed of 100 rad/s, 150 rad/s are shown in Figures12 and 13.Figure 12
The spectral diagrams of the faulted system with SFD mounted on the normal bearing atω=100 rad/s and width = 1e-3 m.Figure 13
The spectral diagrams of the faulted system with SFD mounted on the normal bearing atω=150 rad/s and width = 1e-3 m.Comparing with the system without SFD (Figure8), it can be found that the amplitude of the fundamental frequency does not change, while the amplitudes at the fault characteristic frequencies and the sideband frequencies are decreased. For instance, the amplitude at 2fi+f decreased from 8.1 × 10−7 m to 6.7 × 10−7 m, that at 2fi decreased from 5.3 × 10−7 m to 4.8 × 10−7 m, and that at 2fi+2f decreased from 6.6 × 10−7 m to 5.3 × 10−7 m, while the maximum frequency still locates at 2fi+f the same as the system without SFD. It can be seen that the introduction of the SFD on the normal bearing does not have much effect on the frequency characteristics of the rotor system at low speed.


## 3.3.2. SFD Mounted on Bearing B

Figures13–15 are the spectral diagrams of the rotor system with the SFD mounted on the fault bearing at 100 rad/s and 150 rad/s, respectively.Figure 14
The spectral diagram of the faulted system with SFD mounted on the faulted bearing atω=100 rad/s.Figure 15
The spectral diagram of the faulted system with SFD mounted on the faulted bearing atω=150 rad/s.It can be seen that the amplitude at the fundamental frequency declines from 2.2 × 10−7 m (shown in Figure 7(b)) to 1.5 × 10−7 m (shown in Figure 13) when the rotating speed is 100 rad/s. The amplitudes at the fault characteristic frequencies also decrease a lot, such as from 8.1 × 10−7 m to 1.52 × 10−7 m at 2fi+f, from 5.3 × 10−7 m to 1.9 × 10−7 m at 2fi, and from 6.6 × 10−7 m to 6.5 × 10−8 m at 2fi+2f. In addition, dominant frequency shifts to 2fi.At 150 rad/s case, the amplitude of the fundamental frequency of the system with the SFD mounted on the faulted bearing does not change much comparing with that without the SFD. However, the amplitudes at fault characteristic frequencies decrease significantly due to the adoption of the SFD (from 1.04 × 10−6 m to 1.02 × 10−7 m at fi+f and from 5.3 × 10−7 m to 2.01 × 10−7 m at fi). Apart from that, the amplitudes at fault characteristic frequencies are even smaller than that of the fundamental frequency which may be a considerable influence on system response caused by the SFD mounted on the faulted bearing. It is found that the stability of the rotor system is enhanced due to the fact that fault characteristics are suppressed a lot by the SFD on the faults side compared to the normal side.


## 3.4. Time-Frequency Analysis

### 3.4.1. For Effect of Faults Characteristics

Time-frequency diagrams of the faulted rotor system at different rotating speeds are shown in Figure16. It is found that the energy of the signal concentrates around the 10x fundamental frequency, which is close to the peak at the 2fi+f frequency in the spectral diagram. The signal of the bearing fault performs as impacts in the high frequency range without any patterns, while the signal of the fundamental frequency is continuous in the same frequency range. Figure 16(b) indicates that the system energy centralizes at the fundamental frequency and between 5 and 10x fundamental frequency, and in Figure 16(c) the energy centralizes at 5x fundamental frequency. The amplitude of the max energy has a little change with the speed increasing.Figure 16
Time-frequency diagrams of the faulted system at (a) 100 rad/s, (b) 150 rad/s, and (c) 200 rad/s.
(a) (b) (c)The time-frequency diagrams of the faulted rotor system at different defect width were shown in Figures16 and 17. With the increasing of defects width, the distribution of signal energy changed a little, and the characteristic of the signal is changeless, but the amplitude of energy increased a lot with width increasing.Figure 17
Time-frequency diagrams of the faulted system at 100 rad/s with different defect width: (a)0.5e-3 m; (b) 1e-3 m; (c) 1.5e-3 m; (d) 2e-3 m; (e) 2.5e-3 m.
(a) (b) (c) (d) (e)


### 3.4.2. For Effect of SFD

Time-frequency diagrams of the faulted rotor system with the SFD mounted on bearing A (normal bearing) are shown in Figure18. It can be seen that the rotor signal in high frequency range is still remittent, while the distribution of the signal energy is much looser with some components moving to the higher frequency at 100 rad/s. With the speed increased to 200 rad/s, the signal energy moves to 5x fundamental frequency (fi+f in the spectral diagram), and the energy in high frequency is reduced. Thus the system signal energy does not show apparent changes when the SFD is mounted on the normal bearing side. Comparing to Figure 16, the energy of the signal has no obvious decreasing.Figure 18
The time-frequency diagrams of the system with the SFD mounted on the normal bearing at (a) 100 rad/s and (b) 150 rad/s and (c) 200 rad/s.
(a) (b) (c)Time-frequency diagrams of the rotor system with the SFD mounted on the bearing B (faulted bearing) are shown in Figure19. At low rotating speed, the energy of the signal decreases sharply especially at the frequency higher than 10x fundamental frequency (2fi+f in the spectral diagram). As the speed increased to 200 rad/s, the energy of signal shifts to the fundamental frequency and that of the fault signal at high frequency reduces a lot.Figure 19
The time-frequency diagram of the system with the SFD mounted on the faulted bearing at (a) 100 rad/s and (c) 150 rad/s and (c) 200 rad/s.
(a) (b) (c)


## 3.4.1. For Effect of Faults Characteristics

Time-frequency diagrams of the faulted rotor system at different rotating speeds are shown in Figure16. It is found that the energy of the signal concentrates around the 10x fundamental frequency, which is close to the peak at the 2fi+f frequency in the spectral diagram. The signal of the bearing fault performs as impacts in the high frequency range without any patterns, while the signal of the fundamental frequency is continuous in the same frequency range. Figure 16(b) indicates that the system energy centralizes at the fundamental frequency and between 5 and 10x fundamental frequency, and in Figure 16(c) the energy centralizes at 5x fundamental frequency. The amplitude of the max energy has a little change with the speed increasing.Figure 16
Time-frequency diagrams of the faulted system at (a) 100 rad/s, (b) 150 rad/s, and (c) 200 rad/s.
(a) (b) (c)The time-frequency diagrams of the faulted rotor system at different defect width were shown in Figures16 and 17. With the increasing of defects width, the distribution of signal energy changed a little, and the characteristic of the signal is changeless, but the amplitude of energy increased a lot with width increasing.Figure 17
Time-frequency diagrams of the faulted system at 100 rad/s with different defect width: (a)0.5e-3 m; (b) 1e-3 m; (c) 1.5e-3 m; (d) 2e-3 m; (e) 2.5e-3 m.
(a) (b) (c) (d) (e)


## 3.4.2. For Effect of SFD

Time-frequency diagrams of the faulted rotor system with the SFD mounted on bearing A (normal bearing) are shown in Figure18. It can be seen that the rotor signal in high frequency range is still remittent, while the distribution of the signal energy is much looser with some components moving to the higher frequency at 100 rad/s. With the speed increased to 200 rad/s, the signal energy moves to 5x fundamental frequency (fi+f in the spectral diagram), and the energy in high frequency is reduced. Thus the system signal energy does not show apparent changes when the SFD is mounted on the normal bearing side. Comparing to Figure 16, the energy of the signal has no obvious decreasing.Figure 18
The time-frequency diagrams of the system with the SFD mounted on the normal bearing at (a) 100 rad/s and (b) 150 rad/s and (c) 200 rad/s.
(a) (b) (c)Time-frequency diagrams of the rotor system with the SFD mounted on the bearing B (faulted bearing) are shown in Figure19. At low rotating speed, the energy of the signal decreases sharply especially at the frequency higher than 10x fundamental frequency (2fi+f in the spectral diagram). As the speed increased to 200 rad/s, the energy of signal shifts to the fundamental frequency and that of the fault signal at high frequency reduces a lot.Figure 19
The time-frequency diagram of the system with the SFD mounted on the faulted bearing at (a) 100 rad/s and (c) 150 rad/s and (c) 200 rad/s.
(a) (b) (c)


## 4. Conclusions

(1)
The characteristics of inner race defects on bearing can be concluded that there are all kinds of frequencies on the spectral diagram from low frequency to high frequency. Most of these frequencies are related to the characteristic multiple frequency of the inner race defect and the fundamental frequency. The signal of the bearing fault performs as impacts in the high frequency range without any patterns. (2)
The rotating speed and defect width had an obvious influence on the characteristics of inner race bearing. With the increasing of speed, the distribution of frequencies moves to the low frequency, and the amplitude of low frequency has great increasing, while the amplitude of the signal increases a little. With the increasing of the defect width, the amplitude of frequency and signal has great increasing while the distribution changes a little. (3)
The beneficial effect of the SFD strongly depends on the position of the SFD mounted. The vibration magnitudes of the rotor system show a slight decrease when the SFD is mounted on the unfaulted bearing while it is reduced significantly with the SFD mounted on the faulted bearing.Comparing with the existing methods of analysis on bearing defects, this analysis of this paper is focused on studying the characteristics of frequency distribution and explored the laws where the frequencies changed with the length of bearing defects and speed. This paper also explored the effects of SFD to the bearing inner race defect frequencies.



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*Source: 2489376-2017-01-04.xml*