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

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.


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][14][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.

Model of the Rotor-Bearing-SFD System
2.1.SFD Model.The schematic of the SFD used in this paper is shown in Figure 1.The instantaneous pressure distribution   can be calculated using the Reynolds equations [18]: In order to solve the Reynolds equation, the Sommerfeld transformation was introduced [19]; after integrating twice   can be written as where ℎ s =  s (1 +  s cos ),   is the oil viscosity, Ω is the precession angular velocity of the journal,   is the radial where   is radius of SFD,  1 ,  2 , and  3 are classic Sommerfeld integral equations; they can be written as  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.
According to Harris' work [22], the contact stiffness of the ball bearing can be calculated as follows: where   is the load of ball,   is the contact deformation coefficient of the ball, and  , is the centrifugal force of the ball.The subscript "" refers to the ball number, and the superscripts "i" and "" 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: where  =   /2    is a nondimensional velocity,  =   is a nondimensional material parameter,  =   /   2  is a nondimensional load parameter,   is the ellipticity of the rolling element,   is the dynamic viscosity,  is the linear velocity,   = /(1−] 2 ) is the equivalent Young's modulus,  is the Poisson's ratio, and  is the pressure-viscosity coefficient.
Then the oil film stiffness between the rolling element and the inner race can be calculated as The oil film stiffness between the rolling element and the outer race,    , can be calculated in the same way.Thus, the total oil film stiffness of the bearing can be obtained: The global stiffness of the rolling element can be obtained: Accordingly, the radial contact load of a rolling element can be calculated as Then the supporting forces of the ball bearing in the and -directions are given as  where  inner is the additional displacement of the rolling element due to the inner race defect which is given as

Bearing Model with
where   is the radius of the rolling element and   is the width of the defect on inner race.The contact angle,   , of the th rolling element and the defect angle, Φ  , are given as where  is the number of the rolling element,  cage is the angular velocity of the cage, and  inner is the radius of the defect.
where  and  are the radiuses of the outer and the inner races, respectively.Substituting Eq. ( 12)-( 15), the additional displacement of the rolling element,  inner , per cycle can be obtained: 2.3.Rotor System Model.The rotor-bearing-SFD system is developed based on the Jeffcott rotor (shown in Figure 4) in which the ball bearing  (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  in Figure 4, and in this paper, the SFD mounted both  and  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.
The equations of motion of the dynamic system are derived as where   ,   , and   are the lumped masses of the rotor at the disk and the bearing positions respectively,  and   are viscous damping factors of the disk and the bearing, respectively,  is the stiffness of the shaft,  (,) ,  (,) ,  (,) , and  (,) are the supporting forces of the bearings in the and -directions,   is the lumped mass of the bearing outer race,   is the supporting stiffness of the SFD, and  , and  , are the supporting forces of the SFD in the and -directions.[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.

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].
Figure 8: The spectral diagrams of the fault system with bearing inner race defect at  = 100 rad/s and width 1 − 3 m.
By using this method, the model established in this paper is validated.

Effects of Rotating Speed.
For a ball bearing, the characteristic frequency of the inner race defect   can be calculated as where   is the diameter of the rolling element,   is the pitch diameter of the bearing,  is the number of the rolling elements, and  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 Figure 7.When the speed is at 50 rad/s, the maximum frequency is 4  +  (4.5−7 m), the maximum frequency became   (1.01−6 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.
From Figure 8, 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 2  , 2  + and 2  +2 are much higher than that at the fundamental frequency.In addition, the amplitudes at 3  −  and 2  −  are also greater than that at the fundamental frequency.Among the frequency components, the maximum amplitude at 2  +  (8.1 × 10 −7 m) is nearly four times that at the fundamental frequency.Besides, the sideband frequencies centering on   and its multiple frequencies with an interval  (e.g., 2  − , 2  + ) can be found in the faulted rotor system.
At 150 rad/s case, in Figure 9 for the faulted system, three dominant frequency components can be found which are 2  − 2,   + 2, and 2  − .The maximum amplitude is nearly 7.9 − 7 m at 2  − 2 which is far higher than Amplitude (m) Figure 10: Waterfall plot of inner race defect bearing system with width of defect at 100 rad/s.
the fundamental frequency.Similar to the 100 rad/s case, sideband frequencies from   to 2  can be found.Comparing to the faulted system at 100 rad/s, the amplitudes at the sideband frequencies increased while that centering at 2  decreased.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.

Effects of Defects Width.
From Figure 10, it is found that the maximum frequency is 2  +  no matter how width changed, with the increasing of width in range of (0.5 − 3 m, 2−3 m), the amplitude of 2  + is increased from 3.87−7 m to 1.381 − 6 m, with a little decrease when width at 2.5 − 3 m.The sideband besides   has great increasing while it, besides 3  , has no obvious changing.From Figure 11, the same phenomenon is found, only the maximum frequency is 2  − 2, and it increased from 3.67 − 7 m (at 0.5 − 3 m) to 1.569 − 6 m (at 2.5 − 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.

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 Shock and Vibration   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 Figures 12 and 13.Comparing with the system without SFD (Figure 8), 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 2  +  decreased from 8.1 × 10 −7 m to 6.7 × 10 −7 m, that at 2  decreased from 5.3 × 10 −7 m to 4.8 × 10 −7 m, and that at 2  + 2 decreased from 6.6 × 10 −7 m to 5.3 × 10 −7 m, while the maximum frequency still locates at 2  + 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.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.

SFD Mounted on Bearing B.
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 2  +, from 5.3 × 10 −7 m to 1.9 × 10 −7 m at 2  , and from 6.6 × 10 −7 m to 6.5 × 10 −8 m at 2  + 2.In addition, dominant frequency shifts to 2  .
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   +  and from 5.3 × 10 −7 m to 2.01 × 10 −7 m at   ).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.

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 Figure 16.It is found that the energy of the signal concentrates around the 10x fundamental frequency, which is close to the peak at the 2  +  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.
The time-frequency diagrams of the faulted rotor system at different defect width were shown in Figures 16 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.

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 Figure 18.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 (  +  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.Time-frequency diagrams of the rotor system with the SFD mounted on the bearing B (faulted bearing) are shown in Figure 19.At low rotating speed, the energy of the signal decreases sharply especially at the frequency higher than 10x fundamental frequency (2  +  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.

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.

Figure 2 :
Figure 2: Schematic diagram of the ball bearing.

Figure 3 :
Figure 3: The Inner race defect of the ball.
Inner Race Defect.When defect occurred on inner race of bearing, Figure2was transferred into Figure3in detail.The displacement of the th rolling element can be calculated as[6]

Figure 4 :
Figure 4: The model of the rotor-bearing-SFD system.

Figure 7 :
Figure 7: Waterfall plot of inner race defect bearing system with speed at width 1 − 3 m.

Figure 9 :
Figure 9: The spectral diagrams of the fault system with bearing inner race defect at  = 150 rad/s and width 1 − 3 m.

Figure 11 :
Figure 11: Waterfall plot of inner race defect bearing system with width of defect at 150 rad/s.

Figure 12 :
Figure 12: The spectral diagrams of the faulted system with SFD mounted on the normal bearing at  = 100 rad/s and width = 1 − 3 m.

Figure 13 :
Figure 13: The spectral diagrams of the faulted system with SFD mounted on the normal bearing at  = 150 rad/s and width = 1 − 3 m.

Figure 14 :
Figure 14: The spectral diagram of the faulted system with SFD mounted on the faulted bearing at  = 100 rad/s.

Figure 15 :
Figure 15: The spectral diagram of the faulted system with SFD mounted on the faulted bearing at  = 150 rad/s.

Figure 18 :Figure 19 :
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.
[19]he SFD, and   is the eccentric ratio of the bearing and given as   = √ 2 +  2 /  ,  ,1 and  ,2 are the SFD boundary pressures, and   is the bearing width.According to the Reynolds boundary condition,  ,1 and  ,2 are assumed equal to 0; the supporting force of the oil film can be obtained as[19] clearance