# Raceway Defect Frequency Deviation of Full-Ceramic Ball Bearing Induced by Fit Clearance in Wide Temperature Ranges

Authors: Xiaotian Bai; Hao Zheng; Zinan Wang; Zhong Wang
Journal: Shock and Vibration
 (2021)

---

## Abstract

Full-ceramic ball bearings are widely applied in wide temperature ranges due to their excellent thermal shock resistance, and the condition monitoring and fault diagnosis are mainly conducted through the spectrum analysis based on the defect frequencies. However, the outer ring has a spinning motion in the temperature-related fit clearance, which leads to the deviation of raceway defect frequencies, and is not conducive to the fault diagnosis. In this paper, the temperature-related fit clearance is considered in the dynamic model, and defects are added on the inner raceway and outer raceway. The motions of the rings are calculated and analyzed in the frequency domain, and the trends of peak frequencies with temperature are investigated. Simulation and experimental results show that the spinning speed of the outer ring increases with temperature, and the defect frequencies exhibit obvious deviation in wide temperature ranges. In a temperature range of 500 K, the defect frequencies exhibit deviations of over 3%, which is obvious in the defect frequency identification. The results provide insights on the full-ceramic ball bearing dynamics and help with the fault diagnosis and status monitoring of the relevant devices.


---

## Body

## 1. Introduction

Rolling bearings are fundamental elements in rotary machines, whose operation stability is significant for the whole mechanical system. In many advanced areas such as aerospace and nuclear industries, rolling bearing systems often work in wide temperature ranges, thus providing higher requirements for the thermal shock resistance of rolling bearings. Full-ceramic ball bearings exhibit more stability in wide temperature ranges due to the small thermal deformation and are, therefore, more preferred over steel bearings in working conditions with wide temperature ranges [1, 2]. The balls and rings of full-ceramic ball bearings are made of engineering ceramic materials such as silicon nitride and zirconia, thus providing high stiffness and wear resistance to the bearing system. In the continuous operation of ceramic bearing systems, defects such as cracks and spallings often appear on the inner and outer raceways under impacts. The ceramic bearing components have poor resistance to surface cracking and spalling as a result of the unavailability of plastic binding phases, which results in the sharp degrading in performance when defects occur. Therefore, the condition monitoring and fault diagnosis for ceramic bearings are extremely important for the stable operation of related devices. Recently, the fault diagnosis is mainly conducted in the frequency domain, and the peak frequencies of the bearing system are compared with the preobtained defect frequencies for fault identification. For full-ceramic bearing systems, the outer ring becomes loose due to the temperature-related fit clearance. The motion of the outer ring leads to nonlinear interactions between bearing components, and the defect frequencies also vary obviously, which need to be studied in detail.Setting up dynamic models is one of the main approaches to the interactions between bearing components, and many studies have been carried out in the recent years [3–7]. Xi et al. [8] presented a dynamic model of the spindle bearing system with ball bearing and spindle housing and carried out investigations on the bearing performance in different working conditions. Bizarre et al. [9] formulated a ball bearing model with five degrees of freedom accounting for the effects of the elastohydrodynamic effect and suggested that the contact between bearing elements was affected by axial and moment loads and lubrication conditions. Li et al. [10] established a mechanical model of ball bearing with a localized defect on the outer raceway and analyzed the change of contact angles and load distributions with varying rotation speeds. Yan et al. [11] considered the skidding effect between balls and rings in the fault dynamic model and applied the impulse envelope signal phase in the fault signature enhancement to eliminate the order deviation and vagueness phenomenon. Molano et al. [12] built a bearing dynamic model with localized faults on the inner ring and outer ring for the Independent Cart Conveyor System and considered the modulation of load on the bearings and random contributions to get the fault features in the frequency domain. Yu et al. [13] took the time-varying effect caused by the distributed and localized faults into consideration and found that the faults not only influenced the system on the dynamic behaviors but also on the geometric topographies. It is generally accepted that the surface defects affect the interactions between bearing components by changing the contact forces, and the defect frequencies are obtained through the relative motions between rings and balls [14–16]. However, for full-ceramic ball bearings, the fit clearance between the outer ring and the pedestal changes greatly in wide temperature ranges. The relative motions between the rings need to be considered to ensure the defect frequencies.The fit clearance exists widely in bearing systems and causes nonlinear dynamic behaviors inevitably [17]. Chen and Qu [18] studied the effect of fit clearance on the vibration response and pointed out that the fit clearance led to the periodic contact between the outer ring and the pedestal. Cao et al. [19] considered the effect of fit clearance in the bearing-rotor-pedestal coupled model and found that multiple harmonics appeared in the response due to the nonlinear self-oscillation caused by the fit clearance. Mao et al. [20] proposed a model with the ring flexibility and pedestal contact force, and the results showed that the fit clearance made an impact on the system vibration by changing the load distribution. It has been proved in previous studies [21, 22] that the fit clearance for full-ceramic bearing had a nonlinear growth with the working temperature, and the outer ring made an orbit-spinning compound motion in the pedestal with the effect of friction torque. As a result, it can be predicted that changes will occur to the relative velocities between the balls and rings. The change of relative velocities directly leads to the deviation of defect frequencies and brings difficulties to the fault detection of full-ceramic bearing systems at different temperatures.Therefore, a precise dynamic model with faults and the effect of temperature-related fit clearance is quite essential. In this study, the interactions with the effect of fit clearance are analyzed, and the relative motions between balls and rings are obtained for the defect frequency deviation. The model is presented in Section2. The impacts of fit clearance on the outer ring motions are analyzed in Section 3. In Section 4, the calculation values are compared with the experimental results for verification. The trends of the motion with working conditions are discussed in Section 5. Conclusions are presented in Section 6.


## 2. Fault Dynamic Model with Fit Clearance

The full-ceramic bearings are usually installed in steel pedestals. For the convenience of assembling, the fits between the outer ring and the pedestal are usually set as clearances, leaving a positive clearance to make it easier to adjust the axial position of the bearing. However, when the working temperature changes, the difference in thermal deformation between the outer ring and the pedestal leads to the variance in fit clearance. Assuming that the thermal deformation of the components is uniform in all directions, the deformation anistropy of the ceramic material and the boundary effect that occurred at the edges are ignored. The bore diameter of the outer ring and the outline diameter grow synchronously, and the deformation is shown in Figure1.Figure 1
Thermal deformation. (a) The pedestal. (b) The outer ring.
(a)(b)In Figure1, the dashed lines show the outlines at the initial temperature T0, while the solid lines show the outlines after deformation. Here, D1 shows the initial diameter of the pedestal, and L is the initial pedestal width. D0 is the initial bore diameter of the pedestal, and d0 is the initial diameter of the outer ring. The initial fit clearance δ0 can be expressed as(1)δ0=D0−d0.And, the fit clearance after thermal deformation can be expressed as(2)δ0′=D121+λpΔT3−D12−D021+3λpΔT1+λpΔT−d0⋅1+λo⋅ΔT,where δ0′ is the fit clearance after deformation, λp is the thermal deformation coefficient of the pedestal, λo is the thermal deformation coefficient of the outer ring, and ΔT is the temperature range and can be expressed as(3)ΔT=T−T0,where T is the working temperature. Then, the contact model with clearance is shown in Figure 2.Figure 2
Contact model with clearance in various planes. (a) YOZ plane. (b) XOZ plane.
(a)(b)In Figure2, the coordinate system {O; X, Y, Z} is set as the reference system, in which O is the center of the pedestal bore, OX is the axial direction, and OY and OZ are radial directions. The coordinate system {Oi; Xi, Yi, Zi} shows the position of the inner ring, and {Oo; Xo, Yo, Zo} shows the position of the outer ring. Oi is set at the center of the inner ring, and Oo is set at the center of the outer ring. The balls are driven by the inner ring and slide along the outer ring raceway. di is the diameter of the inner raceway, dm is the pitch diameter, and do is the diameter of the outer raceway. Here, the cage is not shown for clarity. As the thermal deformation of the ceramic component is relatively small, the thermal deformation of the components is regarded as free expansion, which is the same as the outer ring. Only the radial motions of the bearing system are discussed in this paper, so the movements around OY and OZ are ignored. When the balls are running on the regions without defects, the motion of the inner ring can be described as(4)Fecosϕi−∑j=1NQijcosψj+FRijsinψj=miy¨i,(5)Fesinϕi−∑j=1NQijsinψj−FRijcosψj−mig=miz¨i,where Fe shows the centrifugal force, ϕi is the eccentric angle in {O; Y, Z}, Qij is the contact force acting on the inner ring, FRij is the friction force between the inner ring and the jth ball in the YOZ plane, ψj is the azimuth angle of the jth ball in {Oi; Yi, Zi}, mi is the mass of the inner ring, N is the total number of balls, and xi and yi are displacements of the inner ring, respectively. Here, the mass and force of the rotor are transferred and lumped on the inner ring, and the motion of the outer ring is shown as(6)∑j=1NQojcosϕj−FRojsinϕj−Qpcosϕo+Fpsinϕo=moy¨o,(7)∑j=1NQojsinϕj+FRojcosϕj−Qpsinϕo−Fpcosϕo−mog=moz¨o,(8)∑j=1NFRoj⋅dm+db1+λo⋅ΔT2−Fp⋅d01+λo⋅ΔT2=Jo⋅ω˙o,where Qoj is the contact force acting on the outer ring, FRoj is the friction force in the YOZ plane, mo is the mass of the outer ring, Jo is the rotary inertia, yo and zo are displacements of the outer ring along OY and OZ directions, ωo is the rotation speed around Oo, db is the ball diameter, Qp is the contact force from the pedestal, Fp is the friction force on the outer ring, ϕj is the azimuth angle in {Oo; Yo, Zo}, and ϕo is the eccentric angle of the outer ring in {O; Y, Z}. For the azimuth and eccentric angles, OY is set as the reference direction, as shown in Figure 3.Figure 3
Contact between the outer ring and the pedestal.In Figure3, eo is the distance between O and Oo and can be expressed as(9)eo=yo2+zo2,where ei is the distance between O and Oi and can be expressed as(10)ei=yi2+zi2.The azimuth ofψj can be considered to change uniformly as the balls are driven by the inner ring, and ϕj is related with ψj as(11)ϕj=ψj+Δϕ,whenarctanzo−ziyo−yi<ψj<π+arctanzo−ziyo−yi,ϕj=ψj−Δϕ,whenψj<arctanzo−ziyo−yi,orπ+arctanzo−ziyo−yi<ψj<2π+arctanzo−ziyo−yi,where Δϕ can be obtained through the cosine theorem as(12)cosΔϕ=di+dm2+do−dm21+λo⋅ΔT2−4yo−yi2−4zo−zi22di+dmdo−dm1+λo⋅ΔT2.When defects occur on the raceways, the balls pass through the defect area in sequence and bring about impacts due to the interrupted contacts, as shown in Figure4.Figure 4
Contact between the ball and the ring when the defect occurs on different raceways. (a) On the inner raceway. (b) On the outer raceway.
(a)(b)In Figure4, ψsi shows the azimuth of the defect on the inner raceway, ϕso shows the azimuth of the defect on the outer raceway, and the sizes of the defects are denoted by θsi and θso. For cases with a defect on the inner raceway, when the jth ball enters the defect area, it loses contact with the inner raceway at first and then has an impact with the raceway at ψj = ψsi. Fij is the contact force acting on the inner ring when ψj = ψsi and only counts when ψj = ψsi. θi shows the angle between Fij and ψsi, which can be obtained through(13)sinθi=didb⋅sinθsi2.Assuming that the motions of the balls on the raceways are pure rolling, the orbit speed of the balls can be expressed as(14)ωm=ωidi+ωodo2dm1+αo⋅ΔT.And,Fij can be expressed as(15)Fij=miωi−ωm2dicosθsi/22cosθi1+αo⋅ΔT.Therefore, when an impact occurs on the inner raceway, equations (4) and (5) should be revised as(16)Fecosϕi−∑j=1NQijcosψj+FRijsinψj+Fijcosψsi−θi=miy¨i,(17)Fesinϕi−∑j=1NQijsinψj−FRijcosψj+Fijsinψsi−θi−mig=miz¨i.For cases with a defect on the outer raceway,Foj is the impact force when ϕj = ϕso and only counts when ϕj = ϕso. θo is the angle between Foj and ϕso, which can be obtained through(18)sinθo=dodb⋅sinθso2.And,Foj can be expressed as(19)Foj=mjωm−ωo2docosθso/2−dbcosθo2cosθo1+λo⋅ΔT.Then, when an impact occurs on the outer raceway, equations (6)–(8) should be revised as(20)∑j=1NQojcosφj−FRojsinφj+Fojcosφso−θo−Qpcosφo+Fpsinφo=moy¨,o(21)∑j=1NQojsinφj+FRojcosφj+Fojsinφso−θo−Qpsinφo−Fpcosφo−mog=moz¨,o(22)∑j=1NFRoj⋅do1+λo⋅ΔT2−mjωm−ωo2docosθso/2−dbcosθo⋅tanθodo1+λo⋅ΔT24−Fpdo1+λo⋅ΔT2=Jo⋅ω˙o.Fp and Qp are taken into consideration only when the outer ring is in contact with the pedestal, that is,(23)eo≥δ0′2.Assuming that the contact force submits Hooke’s law within the small deformation, that is,(24)Qp=kp⋅eo−δ0′2.The friction forceFp is related with Qp when the outer ring is in contact with the pedestal, and the direction of Fp depends on the rotation of the outer ring, that is,(25)Fp=fp⋅Qp⋅z˙ocosϕo−y˙osinϕo+ωo⋅d0⋅1+λo⋅ΔT/2z˙ocosϕo−y˙osinϕo+ωo⋅d0⋅1+λo⋅ΔT/2,where fp is the friction coefficient between the outer ring and the pedestal. When eo<δ0′/2, it can be considered that the outer ring separates from the pedestal, and Qp = Fp = 0. Then, it is obvious that, as the working temperature increases, the fit clearance becomes larger, and the outer ring spins in the pedestal bore with the effect of Fp. It can be inferred that ωo has a complex variance in a wide temperature range due to the fit clearance. When ωo changes, the impact frequency also changes, as shown in equations (14)–(21), and the effect of the fit clearance can be revealed in the dynamic response of the inner ring and outer ring.


## 3. Numerical Simulation

To test the performance of the full-ceramic bearing system, numerical simulations at different temperatures are essential. The thermal deformation of the bearing and pedestal are expressed in equations (1)–(3), and the clearance between the outer ring and pedestal can be used as the geometry boundary conditions; then, the motions of the bearing can be obtained through equations (16) and (17) and (20)–(22). Assuming that, at time t = 0, the ball of j = 1 is located at ψj = 0°, the outer ring is in contact with the pedestal at ϕo = 0°, and the radial load is set as 100 N. Then, the dynamic response of the inner ring and outer ring can be obtained through iterative calculation, and the flowchart is shown in Figure 5.Figure 5
Calculation flowchart.Here, the initial temperature is set asT0 = 100 K, and the initial fit clearance δ0 is 0.003 mm. The structural parameters of the full-ceramic ball bearing and the pedestal are listed in Table 1.Table 1
Parameters of ceramic ball bearings and the pedestal.
ItemValueBearing pitch diameter (inch)1.318Contact angle (degree)0Bearing width (mm)15Ball diameter (inch)0.3125Number of balls8Pedestal diameter (mm)120Pedestal width (mm)10The pedestal is made of steel, and the balls and rings of the full-ceramic ball bearing are made of silicon nitride. The physical parameters of the bearing components are included in Table2.Table 2
Physical parameters of the bearing components.
ItemValueBearing density (kg/m3)3100Pedestal density (kg/m3)7850Bearing elastic modulus (Pa)3.3 × 1011Pedestal elastic modulus (Pa)2.16 × 1011Thermal deformation coefficient of the pedestal (1/K)1.25 × 10−5Thermal deformation coefficient of the bearing (1/K)2.8 × 10−6Contact stiffness of the pedestal (N/m)2.5 × 108


### 3.1. Dynamic Response with the Inner Raceway Defect in Different Temperatures

The maximum temperature range ΔT is set as 500 K in this study, and the working temperature T, which can be expressed as T = T0+ΔT, ranges from 100 K to 600 K. The rotation speed of the inner ring is set as 2400 r/min in the anticlockwise direction, and the radial load is set as 100 N. It is assumed that the shapes of the raceways are ideal arcs, and the balls are ideal spheres with identical diameters, the effects of surface waviness and shape errors are ignored. The friction coefficient between the bearing components is 0.1, and the friction coefficient between the outer ring and the pedestal is 0.2. The external shock is not taken into consideration. At the initial time t = 0, the defect is located at ψsi = 0°. The size of the defect is θsi = 2°, and the balls are not in touch with the bottom of the defect. The temperature step is 100 K, and the vertical speed z˙i is selected as the indicator. The time step for the calculation is 0.0004 s, and the dynamic responses in different temperatures are shown in Figure 6.Figure 6
Vertical velocities of the inner ring in various temperatures. (a)T = 100 K. (b) T = 200 K. (c) T = 300 K. (d) T = 400 K. (e) T = 500 K. (f) T = 600 K.
(a)(b)(c)(d)(e)(f)It can be seen from Figure6 that the velocity of the inner ring is approximate to sinusoidal variation, and the amplitude gradually increases with working temperature. The peak values appear periodically in the time domain, which is caused by the impacts on the defect area. To get the frequency components in the velocities, the results are processed by FFT with the analysis frequency of 2500 Hz, as shown in Figure 7.Figure 7
Frequency domain results for inner ring velocities.As indicated in Figure7, the frequency components contained in the inner ring velocities are related with the rotating frequency and defect frequency. Obviously, the peaks at 40 Hz and 80 Hz correspond to the rotating frequency and doubling rotating frequency, and the peak near 200 Hz is the defect frequency. The defect frequency and amplitudes at each peak in different temperatures are given in Table 3.Table 3
Peak frequencies and amplitudes of inner ring vibration in different temperatures.
Temperature (K)100200300400500600Amplitude at the rotating frequency (mm/s)1.701.751.861.992.142.35Amplitude at the doubling rotating frequency (mm/s)0.300.420.500.580.680.80Defect frequency (Hz)197.40198.65199.90201.15203.65204.90Amplitude at the defect frequency (mm/s)0.721.081.211.241.391.87The peak corresponding to the rotating frequency is the highest, which implies that the centrifugal force caused by the bearing clearance and fit clearance makes the most contribution to the inner ring vibration. The doubling rotating frequency can also be seen in the result, but the amplitude is much smaller. As the fit clearance increases with the working temperature, the maximum eccentricity grows, and the centrifugal force from the rotor and the inner ring increases to bring about the amplitude growth of the rotating frequency-related components. With the growth of the centrifugal force, the impact of balls crossing the defect area also becomes severer, and the amplitude of the defect frequency also increases with the temperature, as shown in Figure7.According to equation (22), the outer ring begins to rotate as the outer ring becomes loose with the increasing fit clearance. The outer ring has an opposite rotation direction to the inner ring, and the orbit speed of the balls decreases according to the pure rolling hypothesis. Then, the relative velocities between the inner ring and the balls increase, and the characteristic frequency for the defect on the inner raceway turns larger, which has the consistent trend as Table 3.


### 3.2. Dynamic Response with the Outer Raceway Defect in Different Temperatures

The working condition parameters are set the same as in the previous section, and the temperature also ranges from 100 K to 600 K. At the initial timet = 0, the defect is set at ϕso = 270°, and the defect has an opening angle of θso = 2°. The balls never touch the bottom of the defect during rotation. The time step is 0.0004 s; then, the vertical speed of the outer ring z˙o is calculated, as shown in Figure 8.Figure 8
Vertical velocities of the outer ring in various temperatures. (a)T = 100 K. (b) T = 200 K. (c) T = 300 K. (d) T = 400 K. (e) T = 500 K. (f) T = 600 K.
(a)(b)(c)(d)(e)(f)Similar with the inner-ring vertical dynamic response displayed in Figure6, the vertical vibration of the outer ring exhibits obvious periodicity, and the overall amplitude increases with temperature. The frequency domain result is shown in Figure 9.Figure 9
Frequency domain results for outer ring velocities.Compared with the inner ring velocities, the frequency domain results for outer ring velocities contain more peak frequencies. The peak frequencies at 40 Hz, 80 Hz, and 160 Hz are 1, 2, and 4 times rotating frequencies, respectively. The 3 times rotating frequency is close to the defect frequency, and the amplitudes at 3 times the rotating frequency are, therefore, hard to distinguish. The defect frequency and amplitudes at each peak in different temperatures are given in Table4.Table 4
Peak frequencies and amplitudes of the outer ring vibration in different temperatures.
Temperature (K)100200300400500600Amplitude at the 1 × rotating frequency (mm/s)1.892.022.192.402.612.83Amplitude at the 2 × rotating frequency (mm/s)1.041.141.221.351.481.58Amplitude at the 3 × rotating frequency (mm/s)0.340.620.420.620.810.92Amplitude at the 4 × rotating frequency (mm/s)0.180.210.250.340.390.42Defect frequency (Hz)122.44122.44123.69123.69124.94126.19Amplitude at the defect frequency (mm/s)1.491.952.111.672.362.89As shown in Figure9 and Table 5, the amplitudes at the characteristic frequencies related with the rotating speed generally increase with temperature, indicating that the centrifugal force, which increases with the enlarged fit clearance, is the main factor affecting the responses at these frequencies. When the outer ring makes a spinning motion reverse to the inner ring in the fit clearance, the orbit speed of the balls decreases, and the impact frequency caused by the outer ring defect increases according to equation (19). Therefore, the defect frequency grows with temperature, and the amplitude of the defect frequency also increases with the centrifugal force basically.Table 5
Comparison between the simulation and experiment results.
Temperature (K)Simulation result of inner raceway defect frequency (Hz)Experimental result of inner raceway defect frequency (Hz)Simulation result of outer raceway defect frequency (Hz)Experimental result of outer raceway defect frequency (Hz)100197.40197.80122.43122.80200198.65199.00122.43123.20300199.90200.40123.69124.00


## 3.1. Dynamic Response with the Inner Raceway Defect in Different Temperatures

The maximum temperature range ΔT is set as 500 K in this study, and the working temperature T, which can be expressed as T = T0+ΔT, ranges from 100 K to 600 K. The rotation speed of the inner ring is set as 2400 r/min in the anticlockwise direction, and the radial load is set as 100 N. It is assumed that the shapes of the raceways are ideal arcs, and the balls are ideal spheres with identical diameters, the effects of surface waviness and shape errors are ignored. The friction coefficient between the bearing components is 0.1, and the friction coefficient between the outer ring and the pedestal is 0.2. The external shock is not taken into consideration. At the initial time t = 0, the defect is located at ψsi = 0°. The size of the defect is θsi = 2°, and the balls are not in touch with the bottom of the defect. The temperature step is 100 K, and the vertical speed z˙i is selected as the indicator. The time step for the calculation is 0.0004 s, and the dynamic responses in different temperatures are shown in Figure 6.Figure 6
Vertical velocities of the inner ring in various temperatures. (a)T = 100 K. (b) T = 200 K. (c) T = 300 K. (d) T = 400 K. (e) T = 500 K. (f) T = 600 K.
(a)(b)(c)(d)(e)(f)It can be seen from Figure6 that the velocity of the inner ring is approximate to sinusoidal variation, and the amplitude gradually increases with working temperature. The peak values appear periodically in the time domain, which is caused by the impacts on the defect area. To get the frequency components in the velocities, the results are processed by FFT with the analysis frequency of 2500 Hz, as shown in Figure 7.Figure 7
Frequency domain results for inner ring velocities.As indicated in Figure7, the frequency components contained in the inner ring velocities are related with the rotating frequency and defect frequency. Obviously, the peaks at 40 Hz and 80 Hz correspond to the rotating frequency and doubling rotating frequency, and the peak near 200 Hz is the defect frequency. The defect frequency and amplitudes at each peak in different temperatures are given in Table 3.Table 3
Peak frequencies and amplitudes of inner ring vibration in different temperatures.
Temperature (K)100200300400500600Amplitude at the rotating frequency (mm/s)1.701.751.861.992.142.35Amplitude at the doubling rotating frequency (mm/s)0.300.420.500.580.680.80Defect frequency (Hz)197.40198.65199.90201.15203.65204.90Amplitude at the defect frequency (mm/s)0.721.081.211.241.391.87The peak corresponding to the rotating frequency is the highest, which implies that the centrifugal force caused by the bearing clearance and fit clearance makes the most contribution to the inner ring vibration. The doubling rotating frequency can also be seen in the result, but the amplitude is much smaller. As the fit clearance increases with the working temperature, the maximum eccentricity grows, and the centrifugal force from the rotor and the inner ring increases to bring about the amplitude growth of the rotating frequency-related components. With the growth of the centrifugal force, the impact of balls crossing the defect area also becomes severer, and the amplitude of the defect frequency also increases with the temperature, as shown in Figure7.According to equation (22), the outer ring begins to rotate as the outer ring becomes loose with the increasing fit clearance. The outer ring has an opposite rotation direction to the inner ring, and the orbit speed of the balls decreases according to the pure rolling hypothesis. Then, the relative velocities between the inner ring and the balls increase, and the characteristic frequency for the defect on the inner raceway turns larger, which has the consistent trend as Table 3.


## 3.2. Dynamic Response with the Outer Raceway Defect in Different Temperatures

The working condition parameters are set the same as in the previous section, and the temperature also ranges from 100 K to 600 K. At the initial timet = 0, the defect is set at ϕso = 270°, and the defect has an opening angle of θso = 2°. The balls never touch the bottom of the defect during rotation. The time step is 0.0004 s; then, the vertical speed of the outer ring z˙o is calculated, as shown in Figure 8.Figure 8
Vertical velocities of the outer ring in various temperatures. (a)T = 100 K. (b) T = 200 K. (c) T = 300 K. (d) T = 400 K. (e) T = 500 K. (f) T = 600 K.
(a)(b)(c)(d)(e)(f)Similar with the inner-ring vertical dynamic response displayed in Figure6, the vertical vibration of the outer ring exhibits obvious periodicity, and the overall amplitude increases with temperature. The frequency domain result is shown in Figure 9.Figure 9
Frequency domain results for outer ring velocities.Compared with the inner ring velocities, the frequency domain results for outer ring velocities contain more peak frequencies. The peak frequencies at 40 Hz, 80 Hz, and 160 Hz are 1, 2, and 4 times rotating frequencies, respectively. The 3 times rotating frequency is close to the defect frequency, and the amplitudes at 3 times the rotating frequency are, therefore, hard to distinguish. The defect frequency and amplitudes at each peak in different temperatures are given in Table4.Table 4
Peak frequencies and amplitudes of the outer ring vibration in different temperatures.
Temperature (K)100200300400500600Amplitude at the 1 × rotating frequency (mm/s)1.892.022.192.402.612.83Amplitude at the 2 × rotating frequency (mm/s)1.041.141.221.351.481.58Amplitude at the 3 × rotating frequency (mm/s)0.340.620.420.620.810.92Amplitude at the 4 × rotating frequency (mm/s)0.180.210.250.340.390.42Defect frequency (Hz)122.44122.44123.69123.69124.94126.19Amplitude at the defect frequency (mm/s)1.491.952.111.672.362.89As shown in Figure9 and Table 5, the amplitudes at the characteristic frequencies related with the rotating speed generally increase with temperature, indicating that the centrifugal force, which increases with the enlarged fit clearance, is the main factor affecting the responses at these frequencies. When the outer ring makes a spinning motion reverse to the inner ring in the fit clearance, the orbit speed of the balls decreases, and the impact frequency caused by the outer ring defect increases according to equation (19). Therefore, the defect frequency grows with temperature, and the amplitude of the defect frequency also increases with the centrifugal force basically.Table 5
Comparison between the simulation and experiment results.
Temperature (K)Simulation result of inner raceway defect frequency (Hz)Experimental result of inner raceway defect frequency (Hz)Simulation result of outer raceway defect frequency (Hz)Experimental result of outer raceway defect frequency (Hz)100197.40197.80122.43122.80200198.65199.00122.43123.20300199.90200.40123.69124.00


## 4. Experimental Investigation

To test the accuracy of the dynamic model with raceway defects at different temperatures, an experimental investigation is conducted on a rotor-bearing test rig to compare with the calculation results. The test rig is displayed in Figure10.Figure 10
Test rig used in the experiment.In Figure10, the ceramic bearings are installed in the steel pedestals, and a pedestal is placed in an incubator. The structural parameters of the bearing in the experiment are the same as in Table 1. The shaft is driven by the motor through the coupling, and the inner rings of the bearings run synchronously with the shaft. The rotation speed of the shaft can be adjusted manually through the motor frequency, and the plates on the shaft act as the radial load. The incubator is set for heat preservation and covers a bearing and a pedestal. The remote sensing thermometer is used to obtain the temperature, and the accelerometer is set to get the dynamic response. At the beginning, the liquid nitrogen is poured into the incubator to lower the working temperature. The rotation speed of the motor is set stable as 2400 r min−1; the weight of the rotor and plates is 100 N. Then, the working temperature gradually increases due to the heat generation and exchange, and the wide temperature ranges are simulated in the incubator. Enough liquid nitrogen can keep the initial temperature below 100 K, and the laboratory temperature is 298 K. After reaching the laboratory temperature, the working temperature continues to rise due to the heat production of the bearing and finally stops at about 340 K. To get the performance of the bearing with temperature ranges, the signals at T = 100 K, 200 K, and 300 K are collected. The bearing near the coupling is a healthy bearing, while the bearing in the incubator is with a raceway defect. The defects on the inner and outer raceways are initially located at the bottom positions, with ψsi = ϕso = 270°. The sampling frequency is set as 16384 Hz. Two cases with the inner ring defect and outer ring defect are investigated in the experiment, and the vertical accelerations at the pedestal are collected, as shown in Figure 11.Figure 11
Vertical accelerations at different temperatures with different defects. (a) Inner raceway defect. (b) Outer raceway defect.
(a)(b)Two cases with the inner raceway defect and outer raceway defect are shown in Figure11. The black solid lines show the performance at T = 100 K, while the red dashed lines and blue dot-dashed lines show the performances at T = 200 K and 300 K, respectively. In Figure 11(a), the peaks at 1 × and 2 × rotating frequencies are in consistence in different temperatures. However, for the defect frequency, which occurs at about the 5 × rotating frequency, the peak deviates with temperature. It can be seen that the peak frequency for the inner raceway defect increases with working temperature, and the amplitude also grows. Similar situations can also be seen in Figure 11(b), the peaks at 1 ×, 2 ×, 3 ×, and 4 × rotating frequencies remain unchanged in the wide temperature range, but the characteristic frequency for the outer raceway defect moves towards right. The amplitudes at the 1 × rotating frequency and defect frequency increase significantly and vary less at higher orders of rotating frequencies, and the trends are in consistent with the simulation results. The comparison between the defect frequencies in simulation and experimental results is shown in Table 5.As shown in Table5, the defect frequencies of the experimental results match well with the simulation results and have the same trends with working temperature. Therefore, it can be inferred that the dynamic model with temperature-related fit clearance is suitable for the analysis of the full-ceramic bearing system, and the defect frequencies deviate with the working temperature.


## 5. Discussion

As shown in the simulation and experimental results, when the working temperature of the full-ceramic bearing system increases, the fit clearance between the outer ring and the pedestal increases. Due to the friction forces from the balls and pedestal, the outer ring spins in the reverse direction to the inner ring, leading to the increase of inner raceway defect and outer raceway defect characteristic frequencies. On the contrary, the motion of the outer ring in the fit clearance makes the dynamic response of the bearing system more complex and brings about more frequency components related with the rotation speed. The outer ring is more affected by the friction and impact forces from the pedestal; therefore, compared with the dynamic response with the inner raceway defect, the dynamic response with the outer raceway defect contains more frequency components related with the rotation speed. When the working temperature is fixed, the rotation speed of the outer ring can be regarded as a constant value, and the rotation speed affects the defect frequency proportionally. As a result, the characteristic frequency of the outer raceway defect has less deviation with temperature compared with that of the inner raceway defect. In Table4, the defect frequency has no change from T = 100 K to 200 K and 300 K to 400 K. This is because the sampling frequency is not high enough, and the frequency deviation is smaller than the frequency step. According to the simulation results, the inner raceway defect frequency has a deviation of 3.1% and the outer raceway defect frequency has a deviation of 3.8% in the wide temperature range from 100 K to 600 K. Therefore, it can be inferred that the deviation should be considered to make it more accurate for the status monitoring of the full-ceramic bearing system.


## 6. Conclusions

The dynamic model of full-ceramic ball bearing systems with the effect of temperature-related fit clearance is established in this paper, and the motions of the bearing rings are calculated at different temperatures. The defect frequencies are investigated, and the simulation results are verified through experiments. Results show that when the temperature increases, the fit clearance grows due to the thermal deformation difference. The peak frequencies related with the rotation speed stay still, and the amplitudes grow due to the centrifugal force. The outer ring spins in the reverse direction to the inner ring, thus changing the relative motion between balls and rings and leading to the deviation of defect frequencies. The spinning speed of the outer ring increases with the temperature, and the characteristic frequencies for the inner raceway and outer raceway grow monotonously. The defect frequencies change in proportion, and the deviation of inner raceway defect frequency is smaller than that of outer raceway defect frequency. The defect frequencies at different temperatures can be revised through the results in this study, and the research findings provide theoretical foundations for fault diagnosis in wide temperature ranges. The results in this study also proved that the defect frequency deviation needs to be considered in the condition monitoring of full-ceramic bearing systems, and the defect frequencies need to be revised.



---

*Source: 6650798-2021-04-20.xml*