*3.1. Experimental System*

For verifying the validity of the established model, this paper designs and builds an inter-shaft bearing fault simulation test rig. The specific structure of the test rig is shown in Figure 9. The fault simulation experiment system consists of a motor drive, support system, rotor system, and data collection system.

**Figure 9.** Fault simulation system of inter-shaft bearing.

A NU202EM model from NSK Company (Shenyang, China) was used as an experimental bearing in this paper. Rectangular defects are artificially implanted on the rings and rolling surface of the inter-shaft bearing by the wire-cutting method. The width and depth of the defects on the surface of the IR and OR are 0.5 mm. The width and depth of the defects on the surface of the rollers are 0.1 mm, and the same defects run through the entire cylindrical roller longitudinally. A defective inter-shaft bearing is shown in Figure 10.

**Figure 10.** Inter-shaft bearing with local defects. (**a**) Normal statet; (**b**) outer-ring fault; (**c**) inner-ring fault; (**d**) roller fault.

## *3.2. Bearing Parameters*

During the experiment, the IR and OR rotate in the same direction and opposite direction, and the dynamic models of IR fault, OR fault and rollers fault are analyzed respectively. The inter-shaft bearing used is the NU202EM roller bearing. The specific parameters are shown in Table 1.

**Table 1.** NU202EM bearing parameters.


*3.3. Verification and Analysis of Dynamic Model of Inter-Shaft Bearing Fault* Verification of the Simulation Results of the Dynamic Model

#### (1) Normal State of Inter-Shaft Bearing

In the experiment, the speed of the OR was set at 1200 r/min and the IR speed was 300 r/min. The radial load was 1000 N and was applied to the inner surface of the IR. The fault characteristic frequency (FCF) of the bearing was calculated according to the empirical formula [40], as shown in Table 2.

**Table 2.** The FCF of normal state bearing.


When the inter-shaft bearing is in the normal state, the IR and OR rotate in reverse. Because the vibration signal of the inter-shaft bearing is nonlinear and nonstationary, the traditional Fourier transform can not accurately extract the impact characteristics of faults. Therefore, this paper uses Hilbert envelope analysis to research the vibration signal in the horizontal direction of the inter-shaft bearing [40]. Figure 11 is a time-domain signal and envelope spectrum of the numerical simulation signal of the dynamic model when the inter-shaft bearing has no faults. As can be observed in the time-domain signal, the bearing has no significant impact characteristics. It can be seen from the envelope spectrum that there is only the variable stiffness vibration frequency *fzc* and its double frequency 2 *fzc*, and the *fzc* in the spectrum is 110.2 Hz, which is close to the theoretical calculation value of 110 Hz. Figure 12 shows the time-domain signal and envelope spectrum of the experimental signal of the normal bearing. The characteristic frequency in the envelope spectrum is 114.7 Hz, which is different from the numerical simulation results. Due to the complex structure of the experimental system, the vibration amplitude of the inter-shaft bearing is small under normal conditions, and its signals are easily drowned out by the vibration signals caused by other system components. The bearing is far away from the measuring point, and the energy of the fault vibration signal will be attenuated, which will cause the vibration signal to be submerged by these noises.

**Figure 11.** Time–domain signal and envelope spectrum of the normal state (numerical simulation).

**Figure 12.** Time–domain signal and envelope spectrum of the normal state (experimental).

(2) Simulation of the Inter-Shaft Bearing with OR Fault

Set the OR rotating speed of the inter-shaft bearing at 300 r/min and the IR speed at 1500 r/min. According to the empirical formula of FCF, the FCF of the OR fault is obtained as shown in Table 3.

**Table 3.** The FCF of the OR fault.


Figure 13 shows the time-domain signal and envelope spectrum of bearing vibration when the OR of the inter-shaft bearing has local defects and the IR and OR rotate in the opposite direction. It can be seen from the time-domain signal that the bearing has obvious impact characteristics. The FCF of OR fault *f*0 and its double frequency can be clearly extracted from the envelope spectrum. The rotation frequency of OR *Fi* an also be extracted. There are also modulation frequencies such as *f*0 ± *F*0, *f*0 ± 2*F*0, and 2 *f*0 ± *F*0. *f*0 is 132.1 Hz, which is only 0.1 Hz different from the theoretical calculation value of 132 Hz. Therefore, it can be proved that the dynamic model established in this paper is accurate and effective for the simulation of reverse rotation.

**Figure 13.** Time–domain signal and envelope spectrum of OR fault (numerical simulation).

Figure 14 shows the time-domain signal and envelope spectrum of the experimental signal. There are also obvious phenomena of shock and amplitude modulation in the collected time-domain signals. The FCF of the OR fault can be clearly extracted as 130.9 Hz in the envelope spectrum, which is 1.2 Hz different from the simulation calculation result but within the allowable error range. This is due to the possibility of rolling element slippage during actual bearing operation. The experimental results also prove that the envelope spectrum contains the FCF of the OR fault *f*0 and its multiples component. There are also modulation frequencies such as *f*0 ± *F*0, *f*0 ± 2*F*0, and 2 *f*0 ± *F*0.

**Figure 14.** Time–domain signal and envelope spectrum of the OR fault (experimental).

(3) Simulation of the Inter-Shaft Bearing with the IR Fault

The IR rotation speed of the inter-shaft bearing is set at 300 r/min, and the OR rotation speed is 1200 r/min. According to the theoretical calculation formula, the FCF of the IR fault was calculated as shown in Table 4.

**Table 4.** The FCF of the IR fault.


Figure 15 shows the numerical simulation results of the dynamics model when the IR and OR of the inter-shaft bearing rotate in reverse. The inter-shaft bearing has obvious impact on the time signal. The FCF of the IR fault *fi* and its modulation sideband component can be observed in the envelope spectrum. The FCF is calculated as 165.1 Hz by the numerical simulation, which is basically the same as the theoretical value in Table 4. The rotation frequency of IR *Fi* and 2 *fi* can also be extracted. In the envelope spectrum, it can also be observed that all of the fault frequencies have sideband frequencies with *Fi* and its multiples as the interval.

Figure 16 is the experimental result of the IR fault, when the IR and OR rotate in reverse. The shock phenomenon of the signal can be clearly seen from the acquired timedomain signal, which is similar to the numerical simulation results. The FCF of the IR fault *fi* extracted from experimental signal is 164.8 Hz. Although there is an error between this value and the numerical simulation calculation value of the bearing fault, the error is only 0.3 Hz. The law of signal modulation frequency is consistent with the numerical simulation results. It also proves that the dynamic model of the inter-shaft bearing fault established in this paper is effective and accurate in simulating IR faults.

**Figure 15.** Time–domain signal and envelope spectrum of IR fault (numerical simulation).

**Figure 16.** Time–domain signal and envelope spectrum of IR fault (experimental).

(4) Simulation of the Inter-Shaft Bearing with Roller Fault

The OR rotation speed of the inter-shaft bearing is set at 300 r/min, and the IR rotation speed is 1500 r/min. The FCF of the roller fault are listed in Table 5, when the IR and OR rotate in the opposit direction.

**Table 5.** The FCF of the roller fault.


The roller not only rotates around its own axis but also revolves around the bearing center with the cage. When the roller generates a local defect, each rotation of the roller produces two impacts on the IR and OR, and the cage rotation frequency has an amplitude modulation effect on the impact signal. The numerical simulation results of the inter-shaft bearing dynamics model with roller faults are shown in Figure 17. From the envelope spectrum, some data that can be extracted include the FCF of the roller fault FCF *fb* and its double frequency, the rotation frequency *Fc* and its double frequency of the cage, and the sideband frequency with *fb* as the center frequency and *Fc* and its double frequency as the interval. The envelope spectrum shows that the *fb* and *Fc* in the numerical simulation signal of the dynamic model are 144 and 7.02 Hz, respectively. These numerical results is completely consistent with the theoretical calculation results. Figure 18 shows the experimental signal of the roller fault. The obvious shock signal can be seen from the time domain diagram, but the signal components are more complicated. The fault frequency can be extracted from the envelope spectrum, but other frequency components are not obvious.

**Figure 17.** Time–domain signal and envelope spectrum of roller fault (numerical simulation).

**Figure 18.** Time–domain signal and envelope spectrum of roller fault (experimental).

Compared with the above three cases of the bearing, the experimental signal envelope spectrum frequency component of the roller fault is different from the simulation signal. The fault signal is seriously interfered with by the noise signal, which makes it difficult to extract fault information. This is also one reason why the research results of such faults are published less frequently.

#### **4. Dynamic Response Analysis of the Inter-Shaft Bearing with Local Defects**

*4.1. Characteristics of Micro-Local Defects in Inter-Shaft Bearings*

Based on the model established above, the time–frequency characteristics of micro faults are analyzed. In the actual working process, it is often necessary to find and locate faults in time when the width of bearing defects is very small. Therefore, based on the established model, this paper studies the time–frequency characteristics of bearing micro faults. When the double rotors rotate in reverse, the vibration response of the OR fault is more obvious than that of the IR fault and roller fault. Therefore, this paper simulates the micro fault in the OR. Set the OR speed at 1000 r/min, the IR speed at 600 r/min, and the simulation results are shown in Figures 19–21.

Figures 19–21 show the simulation results of the time–frequency characteristics when the OR defect size is 0.1, 0.2 and 0.3 mm. When the defect size is 0.1 mm, the time domain signal can also see the impact characteristics, and its vibration amplitude is 2 m/s<sup>2</sup> at most. When the defect size is 0.5 mm in Figure 13, the vibration amplitude has exceeded 40 m/s2. When the defect sizes are 0.1, 0.2 and 0.3 mm, the time-domain amplitudes of them can be compared. As can be seen from the figure, the vibration amplitude of the bearing rises gradually with the increase in the defect size. Fault is easier to diagnose. When the defect size is 0.1 mm, the FCF of *f*0 is very obvious. The sideband frequencies have not appeared. Furthermore, there are a large number of interference frequencies. The above analysis shows that the effect and reliability of the fault feature extraction method based on time–frequency analysis are low for the early micro-fault diagnosis of the inter-shaft bearing.

**Figure 19.** Time–domain waveform and spectrum of the OR fault signal (0.1 mm defect).

**Figure 20.** Time–domain waveform and spectrum of the OR fault signal (0.2 mm defect).

**Figure 21.** Time–domain waveform and spectrum of the OR fault signal (0.3 mm defect).

#### *4.2. Simulation Analysis of Fault Characteristic Parameters (CP)*

Based on a dynamics model with a local defect, the key affecting factors of the fault CP, such as defect width, external load and working speed, are studied. The variation laws of the fault CP are obtained to provide a certain theoretical basis for bearing fault diagnosis and status inspection.

For research on bearing fault diagnosis, the CP, which are susceptible to bearing fault changes, are usually selected to reflect the characteristics of fault signals. CP include both dimensioned CP and dimensionless CP. According to the experience, the dimensioned CP that are sensitive to faults mainly include: maximum value (MV) *X*max, absolute mean value (AMV) *Xa*, effective value (EV) *Xrms* and amplitude of square root (AST) *Xr*. Dimensionless CP include kurtosis factor (KF) *Kv*, impulse factor (IF) *If* , peak factor (PF) *Cf* and shape factor (SF) *Sf* . Set the discrete signal sequence as A, then the calculation formula of fault CP are as follows [41]:

$$X\_{\text{max}} = \max(x(k)), k = 1 \dots N \tag{26}$$

$$X\_d = \frac{1}{N} \sum\_{k=1}^{N} |\mathbf{x}(k)|\tag{27}$$

$$X\_{rms} = \sqrt{\frac{1}{N} \sum\_{k=1}^{N} x^2(k)}\tag{28}$$

$$\chi\_{\mathbf{r}} = \left(\frac{1}{N} \sum\_{k=1}^{N} \sqrt{|\mathbf{x}(k)|}\right)^{2} \tag{29}$$

$$K\_v = \frac{\beta}{\sigma^4} \tag{30}$$

$$I\_f = \frac{X\_{\text{max}}}{|\overline{X}|} \tag{31}$$

$$C\_f = \frac{X\_{\text{max}}}{X\_{\text{rms}}} \tag{32}$$

$$S\_f = \frac{X\_{rms}}{|\overline{X}|}\tag{33}$$

where *β* is signal kurtosis, *β* = 1 *N N* ∑ *K*=1 (*x*(*k*) − *μ*)4; *σ* is the signal standard deviation; *μ* is the signal mean value.

#### 4.2.1. Effect of Defect Widths on CP

Based on the fault dynamic model of inter-shaft bearing with local defects on the OR, this paper studies the variation law of fault CP with defect widths. Set the radial load at 200 N to remain unchanged, and set the defect widths to change within 0~2 mm. The increment is 0.1 mm. The IR and OR of the inter-shaft bearing rotate in reverse, and the working speed of the IR is 6000 r/min, while that of the OR is 10,000 r/min.

Figure 22 shows the variation of the MV, EV, AST and AMV with the size of defect width. With the increase of defect width, the EV, AST and AMV are increasing gradually. But the increase is relatively smooth and approximately linear, the MV of the signal tends to increase gradually. With the increase of defect width, the fault of the bearing is aggravated, leading to the continuous increase of the vibration amplitude of the bearing. That is why numerical fluctuations in the increasing process and the dimensioned CP vibration parameters tend to increase.

**Figure 22.** Influence of defect width on dimensional CP.

Figure 23 shows the law of KF, IF, PF and SF of the fault signal with the changing of defect width. With the increase of defect width, KF, IF and PF increased first and then decreased. The SF is not changing significantly with the changing of defect width. It is known from the calculation formula of dimensionless CP that the above-mentioned variation law is caused by different relative growth rates of dimensionless CP on the dimensional CP of the numerator and denominator, with the increase of defect width in different periods of bearing faults.

**Figure 23.** Influence of defect width on dimensionless CP.

4.2.2. Effect of External Loads on CP

The external load variation range is 0–1000 N; the external load increment is 50 N, and the defect width is 2 mm. The IR and OR rotate in the opposite direction. The IR working speed is 6000 r/min, and the OR working speed is 10,000 r/min.

Figure 24 shows the variation laws of the dimensioned CP with the changing of radial loads. With the increases of radial load, EV, AST and AMV tends to increase. This suggests that the statistical parameters of vibration signal can represent the fault characteristics under a radial load. The increase in radial load leads to the decrease of bearing clearance, which makes the roller more easily make contact with the fault. The contact force between the roller and the raceway increases, and the energy of the vibration signal increases accordingly. When the radial load increases, MV firstly increases, then decreases and finally tends to be stable. This indicates that the increase in radial load causes the energy of the bearing vibration to increase; however, the peak of the vibration signal tends to be stable.

**Figure 24.** Influence of the radial load on dimensional CP.

Figure 25 shows the variation laws of dimensionless CP with the changing of radial loads. It can be seen that KF and IF is more sensitive to changes in the radial loads. The change trend of the above two parameters is to increase first and then decrease. PF and SF have no obvious change with the radial load, which is not suitable as a CP for fault diagnosis. With the increase in radial loads, the MV increases because the impact energy increases as the roller passes through the defect. However, with the continuous increase in the AMV of the signal, the ratio of them firstly increases, then decreases or tends to be stable.

**Figure 25.** Influence of the radial load on dimensionless CP.

4.2.3. Effect of Rotating Speeds on CP

Set the radial load at 200 N to remain unchanged. The defect width was 2 mm. The IR and OR rotated in reverse. The IR speed was 500 r/min, and the OR speed varied from 500 to 6000 r/min. The increment of speed was 500 r/min.

Figure 26 shows the variation laws of EV, AST, AMV and MV with the changing of speed. Furthermore, all the above dimensional CP tend to increase with the increase of speed. This indicates that dimensional CP can represent the fault characteristics.

**Figure 26.** Influence of working speed on dimensional CP.

Figure 27 shows the variation laws of KF, IF, PF and SF with a change in rotating speed. With the increase in rotating speed, SF does not change, but the KF, IF and PF firstly decrease and then increase. This is due to the difference in the relative growth rate of MV and AMV with the increase in rotating speeds.

**Figure 27.** Influence of working speed on dimensionless CP.
