Dynamic Modeling and Simulation Analysis of Inter-Shaft Bearings with Local Defects Considering Elasto-Hydrodynamic Lubrication

Abstract

As an important component of large engines, inter-shaft bearing is easily damaged due to its poor working conditions. By analyzing the time–frequency distribution rules of fault signals and the evolution law of micro-faults, the bearing failure mechanism can be revealed, and the bearing failure can be monitored in real time and prevented in advance. For the purpose of studying the mechanism of inter-shaft bearing faults, a 4-DOF (degree of freedom) dynamic model of inter-shaft bearing with local defects considering elasto-hydrodynamic lubrication (EHL) is proposed. Based on the established dynamic model, the impact characteristics and distribution rules of the fault signals of the bearing are accurately simulated, and the evolution law of the micro-faults is also analyzed. The effects of different speeds, loads and defect widths on maximum value (MV), absolute mean value (AMV), effective value (EV), amplitude of square root (AST), kurtosis factor (KF), impulse factor (IF), peak factor (PF) and shape factor (SF) are obtained. The findings show that the vibration amplitude of the bearing increases with the increase in defect size, and the faults are easier to diagnose accordingly. At the same time, PF, KF and IF are very sensitive to the initial failure of bearings. With the development of faults, the overall trend of AMV, AST and EV are relatively stable. The PF is sensitive to the change of rotating speeds and defect widths. The SF is insensitive to the change of rotating speeds, loads and defect widths. This lays a foundation for the research of monitoring and diagnosis methods of aeroengine inter-shaft bearing fault.

1. Introduction

Inter-shaft bearings are widely used in the supporting drive systems of twin-rotor aero engines. They are one of the necessary parts of the areoengine-supporting drive system. Due to the harsh working conditions of aeroengine, faults often occur in inter-shaft bearings.

At present, the dynamic modeling method of rolling bearing with local defects is widely studied. Rolling elements are equivalent to nonlinear springs, and a 2-DOF model is mainly for the dynamic analysis of a transient state during the running of the rolling bearing [1]. At the same time, Gupta [2,3,4,5,6,7,8,9] published many models to describe motion states and force states of each part and to consider the speed changes of each part and the corresponding effect of inertial force. However, the model does not consider the influence of damping, and it only simplifies the collision contact according to the elastic contact. Based on the McFadden model [10], Su [11,12] established the dynamic model of bearing with defects and distribution defects under variable loads and used this model to reveal the frequency characteristics of the two defects. Hu [13] proposed a 5-DOF dynamic model of deep-groove ball bearings. The model theoretically formulated the elastic deformation and nonlinear contact forces of bearings coupling dual rotors. Patel [14] established a 6-DOF dynamic model for deep-groove ball bearings and calculated the vibration response of single-point and multi-point faults under constant load. Based on the Hertz contact theory, Xi [15], Ma [16], Patel [17], Liu [18] and Cao [19] established bearing dynamic models with different degrees of freedom and analyzed the nonlinear response of rolling bearing with local defects.

Patil [20] considered the rolling element as a nonlinear contact spring and proposed a dynamic model to research the effects of the defect size on the vibration characteristics of the bearing. Based on the Patil model, Kulkarni [21] used cubic Hermite spline interpolation difference functions to simulate the pulse generated by bearing faults. Cui [22] established a nonlinear dynamic model, which used rectangular displacement excitation to simulate local defects. Khanam [23] proposed a dynamic model using a semi-sinusoidal curve to describe the displacement excitation. Wang [24] came up with a multi-body dynamic model to study the vibration response of cylindrical roller bearings with local defects and analyzed the influence of time-varying contact on bearing defects. The above research shows that the dynamic model of rolling bearing based on the Hertz contact theory, combined with the displacement excitation function, can accurately simulate the vibration signals generated by bearing surface faults.

Sassi [25] established a 3-DOF bearing vibration equation considering the influence of an oil film. The interaction between the rollers and the fault area were analyzed. Wang [26] developed a coupling model of the rotor bearings system, considering the oil film. The variation laws of oil films were analyzed based on this model. Yan [27] and Zhang [28] also considered the influence of the lubricating oil film of the bearing in the established dynamic model and proved that it can more accurately describe the real-time state of bearing operation when considering the influence of EHL on the bearing.

In addition to establishing models for rolling bearing with local defects, the researchers also proposed a variety of coupling modeling methods for fault bearing and rotor systems [29,30,31,32]. Niu [33] established a dynamic model considering the change in contact force direction of roller sliding and entering defects. Cao [34] proposed a dynamic model method of bearing, considering a bearing pedestal and rotor system. To sum up, research on the dynamic model of rolling bearing fault carried out by experts and scholars mostly focuses on the faults of conventional rolling bearing. There is not much research on the modeling of inter-shaft bearing fault dynamics. Rolling bearing and inter-shaft bearing have both similarities and differences. The modeling method of rolling bearing can be used as the basis of inter-shaft bearing modeling, but it can not completely describe the motion state of inter-shaft bearing. In this paper, the inter-shaft bearing is taken as the main research object. Based on the nonlinear Hertz contact theory, a local defect dynamic model of the inter-shaft bearing, considering the influence of time-varying displacement excitation (TVDE) and elasto-hydrodynamic lubrication (EHL), is proposed. The model is used to simulate and analyze the time–frequency distribution rules of fault signals and the evolution law of micro-faults.

Other parts of this paper are arranged as follow. The fault simulation experiment of co-rotating and counter rotating on the birotor bearing test rig is shown in Part 3. The fault characteristics of inter-shaft bearing with local defects are studied in Part 4, and the time–frequency characteristics of micro faults are also studied in this section. Finally, part 5 summarizes the conclusion of this paper.

2. Establishment of Fault Dynamic Model

2.1. Subsection Simplification and Assumption of Inter-Shaft Bearings

Inter-shaft bearing is an important component of aeroengine rotor systems and works between an HP (high-pressure) rotor and an LP (low-pressure) rotor. Unlike ordinary bearings, the inner ring (IR) and outer ring (OR) of bearings rotate at the same time. Depending on the rotation direction, the working speed of the bearing is the speed difference or sum of the two rotors speed. The supporting form of typical dual-rotor aeroengine inter-shaft bearing is shown in Figure 1.

The IR and OR of the inter-shaft bearing rotate at the same time, so it is considered to generate vibration frequently during operation. Considering that both the IR and OR have vertical and horizontal vibrations, a 4-DOF dynamic model is established in this paper. The model assumes that the rollers do not slip. Referring to the Patil model [20], the contact between rollers and rings is simplified as a nonlinear spring-mass system. The model is shown in Figure 2.

2.2. Nonlinear Hertz Contact Forces

On the basis of the nonlinear Hertz contact theory [35], Harris [36] derived the relationship between the nonlinear load and the displacement of the rolling bearing as shown in Equation (1).

$$F=K{\delta }^n$$

(1)

where K is the load deformation coefficient. $\delta$ is the radial displacement. Cylindrical roller bearing n = 10/9.

Since the rollers in the inter-shaft bearing has contact with both the IR and OR, the total contact stiffness is as follows.

$$K={\left[\frac{1}{{(1/K_i)}^{1/n}+{(1/K_0)}^{1/n}}\right]}^n$$

(2)

where $K_i$ is the contact stiffness of the IR. $K_0$ is the contact stiffness of the OR. For cylindrical roller bearing, $K=8.06\times {10}^4\times l^{8/9}$, where $l$ is the roller length.

${\delta }_j$, the radial deformation of the $j$ roller is

$${\delta }_j=(x_0-x_i)\cos {\theta }_j+(y_0-y_i)\sin {\theta }_j-C_r-H_d$$

(3)

where $x_0$ and $y_0$ are the displacement of the center of the OR along the x-axis and y-axis. $x_i$ and $y_i$ is the displacement of the center of the IR along the x-axis and the y-axis. ${\theta }_j$ is the angle between the center of the j_th roller and the x-axis. $C_r$ is the initial clearance between the roller and the raceway. $H_d$ is the TVDE when the roller passes through the defect.

$${\theta }_j={\omega }_{c}t+\frac{2\pi (j-1)}{Z}+{\theta }_{t0}$$

(4)

$${\omega }_c=\frac{1}{2}\left[{\omega }_i(1-\frac{d}{D_m}\cos \alpha )+{\omega }_o(1+\frac{d}{D_m}\cos \alpha )\right]$$

(5)

where ${\omega }_i$ is the angular velocity of IR of inter-shaft bearing, ${\omega }_{\mathrm{o}}$ is the angular velocity of OR of inter-shaft bearing, ${\omega }_c$ is the angular velocity of inter-shaft bearing retainer, ${\theta }_{t0}$ is the initial Angle of the first roller relative to the axis, $Z$ is the number of roller of inter-shaft bearing, $d$ is the roller diameter, $D_m$ is the pitch diameter of inter-shaft bearing, and $\alpha$ is the bearing pressure angle.

Substitute Equations (2) and (3) into Equation (2) to obtain the nonlinear Hertz contact force of the $j$ roller:

$$F_j=K{\left[(x_0-x_i)\cos {\theta }_j+(y_0-y_i)\sin {\theta }_j-C_r-H_d\right]}^{10/9}$$

(6)

The total nonlinear Hertz contact force received by the inter-shaft bearing is the sum of the nonlinear Hertz contact forces of all the rollers. The components of the total contact force on the x and y axes are:

$$\left\{\begin{array}{c}{F}_X=K{\displaystyle \sum _{j=1}^Z{\left[(x_0-x_i)\cos {\theta }_j+(y_0-y_i)\sin {\theta }_j-C_r-H_d\right]}^{10/9}\cos {\theta }_j}\\ F_Y=K{\displaystyle \sum _{j=1}^Z{\left[(x_0-x_i)\cos {\theta }_j+(y_0-y_i)\sin {\theta }_j-C_r-H_d\right]}^{10/9}\sin {\theta }_j}\end{array}\right.$$

(7)

2.3. Time-Varying Displacement Excitation

This paper assumes that the defect of the inter-shaft bearing runs through the entire raceway. Therefore, the defect length is larger than the roller, but the width of the fault is far less than the diameter of the roller. When the roller passes through the defect of the raceway, the roller will not contact the bottom of the defect, and the falling displacement of the roller center will be less than the defect depth.

The relationship between and bearing movement is shown in Figure 3. In the figure, $H_d$ is the time-varying displacement when the roller passes through the defect, $H$ is the defect depth, $B$ is width of the defect, and $H_e$ is the maximum deviation of the roller center when the roller passes through the defect. It can be seen from Figure 4 that the roller can be divided into three stages when it passes through the defect. The first stage is that the roller starts to enter the defect and completely enters the defect. At this time, the time-varying displacement $H_d$ increases gradually as the roller enters the defect; in the meantime, the roller always contacts the left side of the defect. In stage 2, the roller completely enters the defect. At this time, the roller contacts the left and right sides of the defect, and $H_d$ reaches the peak value of $H_e$. In stage 3, the roller enters the defect completely and leaves the defect just after. At this time, the roller only contacts the right side of the defect, and $H_d$ gradually decreases. To sum up, while the roller passes through the defect, $H_d$ increases first and then decreases with motion process. Therefore, the semi-sinusoidal function is used to describe the TVDE of the local defect to the roller. The maximum displacement excitation $H_e$ of the roller is:

$$H_e=d/2-\sqrt{(d/2{)}^2-(B/2{)}^2}$$

(8)

2.3.1. The TVDE of Defects in the OR

Figure 5 reveals the relationship between the rollers and the defect angle position when the IR and OR of the bearing rotation in the opposite direction. The angular velocity of the IR $w_i$ is clockwise, and the angular velocity of the OR $w_0$ is counterclockwise. When $w_i$ is smaller than $w_0$, the angle velocity of the cage $w_c$ is counterclockwise at this time, and ${\phi }_f$ is the defect angle. The counterclockwise direction is specified in the figure as the positive direction. ${\theta }_b{}_0$ is the angle of the first roller relative to the X-axis at the initial moment. At moment $t$, ${\theta }_b{}_i$ is the rotational angle of the i_th roller. ${\theta }_f{}_0$ is the angle relative to the X-axis at the initial moment of the defect. ${\theta }_f$ is the rotational angle of the defect. The TVDE $H_d$ generated when the roller passes through the defect is:

$$H_d=\left\{\begin{array}{l}{H}_e\sin \left(\frac{\pi }{{\phi }_f}(\mathrm{mod}({\theta }_f,2\pi )-\mathrm{mod}({\theta }_b{}_i,2\pi ))\right)\\ \hspace{1em}\hspace{1em}0\le \mathrm{mod}({\theta }_f,2\pi )-\mathrm{mod}({\theta }_b{}_i,2\pi )\le {\phi }_f\\ \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}else\end{array}\right.$$

(9)

where mod is the remainders. The expressions for ${\theta }_{bi}$ and ${\theta }_f$ are:

$${\theta }_b{}_i=\frac{2\pi }{Z}(i-1)+w_{c}t+{\theta }_b{}_0(i=1\dots Z)$$

(10)

$$\theta f=w_{o}t+{\theta }_f{}_0$$

(11)

2.3.2. The TVDE with Defects in the IR

Figure 6 shows the relationship of rollers and defects when the IR and OR rotate in the opposite direction. When $w_i$ is smaller than $w_0$, both $w_c$ and $w_0$ are in the same direction. Counterclockwise is the positive direction. The expression of TVDE $H_d$ is:

$$H_d=\left\{\begin{array}{l}{H}_e\sin \left(\frac{\pi }{{\phi }_f}\left(\mathrm{mod}({\theta }_{bi},2\pi )-\mathrm{mod}\left({\theta }_{f0}+2\pi -\mathrm{mod}(w_{i}t,2\pi ),2\pi \right)\right)\right)\\ \hspace{1em}\hspace{1em}0\le \mathrm{mod}({\theta }_{bi},2\pi )-\mathrm{mod}\left({\theta }_{f0}+2\pi -\mathrm{mod}(w_{i}t,2\pi ),2\pi \right)\le {\phi }_f\\ \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}else\end{array}\right.$$

(12)

$${\theta }_b{}_i=\frac{2\pi }{Z}(i-1)+w_{c}t+{\theta }_b{}_0(i=1\dots Z)$$

(13)

2.3.3. TVDE of Rolling Element Fault

Figure 7 shows the relationship between rollers and the defect angle position when the IR and OR rotate in reverse. Set $w_i$ to be clockwise rotation and $w_0$ to be counterclockwise rotation. When $w_0$ is greater than $w_i$, $w_c$ is counterclockwise, and the angle velocity of the roller $w_b$ is counterclockwise. The counterclockwise direction is the positive direction. Assume the roller 1 has a local defect fault. The defect angle is ${\phi }_b{}_f$. rack the motion of the first roller and establish another moving coordinate system on axis $om$. The counterclockwise direction is taken as the positive direction, and ${\theta }_b{}_{f0}$ is the initial angle of right defect relative to the $om$-axis. At moment t, the angle of the relative moving $om$-axis is ${\theta }_b{}_f$, and the TVDE $H_d$ of the bearing is:

$$H_d=\left\{\begin{array}{c}{H}_e\sin \left(\frac{\pi }{{\phi }_{bf}}\mathrm{mod}(2\pi +w_{b}t-{\theta }_b{}_{f0},2\pi )\right)\\ 0\le \mathrm{mod}(2\pi +w_{b}t-{\theta }_b{}_{f0},2\pi )\le {\phi }_{bf}\\ H_e\sin \left(\frac{\pi }{{\phi }_{bf}}(\mathrm{mod}(2\pi +w_{b}t-{\theta }_b{}_{f0},2\pi )-\pi )\right)\\ \pi \le \mathrm{mod}(2\pi +w_{b}t-{\theta }_b{}_{f0},2\pi )\le \pi +{\phi }_{bf}\\ \\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}else\end{array}\right.$$

(14)

2.4. Dynamic Model for Inter-Shaft Bearings with Local Defects

Based on the hypothesis theory of rigid rings and considering the eccentric load on the bearing, the inter-shaft bearing is established as a 4-DOF dynamic model. The contact stiffness, damping, nonlinear Hertz contact force, eccentric load, time-varying displacement and constant radial load are substituted into the dynamic equation to establish the dynamic model of an inter-shaft bearing fault.

$$\left\{\begin{array}{c}{M}_0{\ddot{x}}_0+c{\dot{x}}_0+K{\displaystyle \sum _{j=1}^Z\lambda {\delta }^{10/9}\cos {\theta }_j=W+F_{l0}\cos ({\omega }_{0}t)}\\ M_0{\ddot{y}}_0+c{\dot{y}}_0+K{\displaystyle \sum _{j=1}^Z\lambda {\delta }^{10/9}\sin {\theta }_j=F_{l0}\sin ({\omega }_{0}t)}\\ M_i{\ddot{x}}_i+c{\dot{x}}_i-K{\displaystyle \sum _{j=1}^Z\lambda {\delta }^{10/9}\cos {\theta }_j=W+F_{li}\cos ({\omega }_{i}t)}\\ M_i{\ddot{y}}_i+c{\dot{y}}_i-K{\displaystyle \sum _{i=1}^Z\lambda {\delta }^{10/9}\sin {\theta }_j=F_{li}\sin ({\omega }_{i}t)}\end{array}\right.$$

(15)

where $M_i$ and $M_0$ is the quality of IR and OR; $F_{li}$ is the eccentric load applied to the IR; c is the damping coefficient; $F_{l0}$ is the eccentric load applied to the OR. $W$ is the radial load perpendicular to inner surface of IR; $\lambda$ is the switch signal to indicate if the roller and the raceway are in contact, expressed as:

$$\lambda =\left\{\begin{array}{ll}1& \delta >0\\ 0& \delta \le 0\end{array}\right.$$

(16)

The dynamic differential equation is solved by the Newmark-$\beta$ method [37]. Taking $\gamma =1/2$ and $\beta =1/4$, this method has second-order accuracy and is an unconditionally stable method, which can accurately solve the fault dynamic model of inter-shaft bearing.

2.5. Fault Dynamic Model Considering the Influence of EHL

The inter-shaft bearing is running at a high speed between high-pressure rotors and low-pressure rotors. Therefore, the inter-shaft bearing needs real-time lubrication. There is an oil film between the rollers and the raceway, and it is always in the state of EHL under the dynamic pressure. The distribution of oil-film pressure in the inter-shaft bearing is shown in Figure 8. The oil-film thickness under the EHL state affects the tribological and dynamic characteristics of the friction pair. The oil pressure generated by the high-speed rotation of the bearing makes the oil film produce a “stiffening effect”. The EHL pressure curve is similar to the Hertz pressure curve in distribution. Therefore, in the dynamic model considering the influence of EHL established in this paper, the stiffness of bearing is considered as the series stiffness of oil-film stiffness and contact stiffness [37,38].

The paper assumed that the lubricating oil is at a constant temperature and that there is no end leakage, and the contact points of rollers and the rings do not slip. The Dowson–Higginson line-contact film-thickness formula is used to calculate the oil-film thickness between the roller and the IR and OR [39] as follows,

$$h_{\min }=\frac{2.56{\alpha }^{0.54}{({\eta }_{0}U)}^{0.7}{R}^{0.43}{L}^{0.13}}{E^{\prime }{}^{0.03}{W}^{0.13}}$$

(17)

where $\alpha$ is the viscosity coefficient of lubricating oil; ${\eta }_0$ is the dynamic viscosity of lubricating oil; $U$ is the average linear velocity of roller; $R$ is the equivalent radius of curvature; $L$ is the line contact length of roller; $W$ is the radial load; $E^{\prime }$ is the composite modulus of elasticity, $E^{\prime }=\frac{E}{1-{\nu }^2}$.

The inter-shaft bearing is characterized by the same rotation direction of the IR and OR. Assume the rotation speed of the IR is $n_i$, the radius is $R_i$, the rotation speed of the OR is $n_o$, and the roller radius is $r$. By defining the constant $\gamma =\frac{d}{D_m}\cos \alpha$, the average linear velocity $U$ between the IR and OR of the roller can be obtained according to Equation (5):

$$U=\frac{\pi }{120}{D}_m\left[n_i(1-\gamma )+n_0(1+\gamma )\right]$$

(18)

Since the inter-shaft bearing pressure angle $\alpha =0$, then

$$\gamma =\frac{d}{D_m}\cos \alpha =\frac{r}{R_i+r}$$

(19)

The equivalent radius is:

$$R={(\frac{1}{r}\pm \frac{1}{R_i+r\mp r})}^{-1}$$

(20)

where “−” is the contact equivalent radius of the IR and the roller; “+” is the contact equivalent radius of the OR and the roller.

Substitute Equation (19) into Equation (20), then:

$$R=r(1 \mp \gamma )$$

(21)

By substituting each parameter into Equation (17), the minimum oil-film thickness $h_i$ and $h_0$ of the rollers and the IR and OR can be obtained.

$$\left\{\begin{array}{c}{h}_i=0.21{\alpha }^{0.54}{({\eta }_0{D}_m)}^{0.7}{\left[n_i(1-\gamma )+n_0(1+\gamma )\right]}^{0.7}\\ \ast r^{0.43}{(1-\gamma )}^{0.43}{L}^{0.13}{E}^{\prime }{}^{-0.03}{W}^{-0.13}\hspace{1em}\hspace{1em}\\ h_o=0.21{\alpha }^{0.54}{({\eta }_0{D}_m)}^{0.7}{\left[n_i(1-\gamma )+n_0(1+\gamma )\right]}^{0.7}\\ \ast r^{0.43}{(1+\gamma )}^{0.43}{L}^{0.13}{E}^{\prime }{}^{-0.03}{W}^{-0.13}\hspace{1em}\hspace{1em}\end{array}\right.$$

(22)

The total thickness of the oil film is:

$$h=h_i+h_0=C_i{W}^{-0.13}+C_0{W}^{-0.13}$$

(23)

where $C_i$ is the coefficient in front of $W^{-0.13}$ in $h_i$; $C_0$ is the coefficient in front of $W^{-0.13}$ in $h_0$.

The oil-film stiffness of the rollers and IR and OR of the inter-shaft bearing can be obtained by Equation (23):

$$K_H=\frac{\mathrm{d}W}{\mathrm{d}h}={(0.13(C_i+C_0)W^{-1.13})}^{-1}$$

(24)

According to the calculation formula of series stiffness, the total stiffness of the rolling bearing considering the influence of EHL is:

$$K^{\prime }=\frac{1}{1/K+1/K_H}=\frac{K{K}_H}{K+K_H}$$

(25)

3. Experimental Validation of the Numerical Model

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.

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.

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. NU202EM bearing parameters.

Parameters	Value
Inner diameter (mm)	15
Outer diameter (mm)	35
Pitch diameter (mm)	25
Roller diameter (mm)	5
Number of rollers	11
Contact angle (°)	0
Radial clearance (μm)	12
Mass (kg)	0.07
Damping coefficient (Ns/m)	300
Bearing width (mm)	11

.

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. The FCF of normal state bearing.

Fault Form	Working State	$1 \times {\mathit{f}}_{\mathit{z}\mathit{c}}$	$2 \times {\mathit{f}}_{\mathit{z}\mathit{c}}$
Normal state	Counter-rotation	110 Hz	220 Hz

.

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 $f_{zc}$ and its double frequency 2$f_{zc}$, and the $f_{zc}$ 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.

(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. The FCF of the OR fault.

Fault Form	Working State	$1 \times {\mathit{f}}_0$	$2 \times {\mathit{f}}_0$
Outer-ring fault	Counter-rotation	132 Hz	264 Hz

.

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 $F_i$ an also be extracted. There are also modulation frequencies such as $f_0\pm F_0$, $f_0\pm 2F_0$, and $2f_0\pm F_0$. $f_0$ 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 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 $f_0\pm F_0$, $f_0\pm 2F_0$, and $2f_0\pm F_0$.

(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. The FCF of the IR fault.

Fault Form	Working State	$1 \times {\mathit{f}}_{\mathit{i}}$	$2 \times {\mathit{f}}_{\mathit{i}}$
Inner-ring fault	Counter-rotation	165 Hz	330 Hz

.

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 $f_i$ 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 $F_i$ and 2$f_i$ can also be extracted. In the envelope spectrum, it can also be observed that all of the fault frequencies have sideband frequencies with $F_i$ 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 time-domain signal, which is similar to the numerical simulation results. The FCF of the IR fault $f_i$ 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.

(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. The FCF of the roller fault.

Fault Form	Working State	$1 \times {\mathit{f}}_{\mathit{b}}$	$2 \times {\mathit{f}}_{\mathit{b}}$	Cage Frequency
Roller fault	Counter- rotation	144 Hz	288 Hz	7 Hz

, when the IR and OR rotate in the opposit direction.

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 $f_b$ and its double frequency, the rotation frequency $F_c$ and its double frequency of the cage, and the sideband frequency with $f_b$ as the center frequency and $F_c$ and its double frequency as the interval. The envelope spectrum shows that the $f_b$ and $F_c$ 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.

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 Figure 19, Figure 20 and Figure 21.

Figure 19, Figure 20 and Figure 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/s² 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.

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) $X_a$, effective value (EV) $X_{rms}$ and amplitude of square root (AST) $X_r$. Dimensionless CP include kurtosis factor (KF) $K_v$, impulse factor (IF) $I_f$, peak factor (PF) $C_f$ and shape factor (SF) $S_f$. Set the discrete signal sequence as A, then the calculation formula of fault CP are as follows [41]:

$$X_{\max }=\max (x(k)),k=1\dots N$$

(26)

$$X_a=\frac{1}{N}{\displaystyle \sum _{k=1}^N\left|x(k)\right|}$$

(27)

$$X_{rms}=\sqrt{\frac{1}{N}{\displaystyle \sum _{k=1}^N{x}^2(k)}}$$

(28)

$${\mathrm{X}}_{\mathrm{r}}={\left(\frac{1}{N}{\displaystyle \sum _{k=1}^N\sqrt{|x(k)|}})\right)}^2$$

(29)

$$K_v=\frac{\beta }{{\sigma }^4}$$

(30)

$$I_f=\frac{X_{\max }}{\left|\overline{X}\right|}$$

(31)

$$C_f=\frac{X_{\max }}{X_{rms}}$$

(32)

$$S_f=\frac{X_{rms}}{\left|\overline{X}\right|}$$

(33)

where $\beta$ is signal kurtosis, $\beta =\frac{1}{N}{\displaystyle \sum _{K=1}^N(x}(k)-\mu {)}^4$; $\sigma$ is the signal standard deviation; $\mu$ 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 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.

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 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.

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 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.

5. Conclusions

In order to clarify the fault mechanism of inter-shaft bearings, a 4-DOF local defect inter-shaft bearing of a dynamic model is established in this paper, which considers EHL and TVDE. A piecewise function is used in this model to describe the TVDE of the local defect on the surface of the inter-shaft bearing. At the same time, the influence of the oil film is considered to improve the accuracy of the fault dynamic simulation. In this paper, the frequency distribution of the fault evolution process is simulated and analyzed by the established model. The relationship between defect width, external load and rotating speed and fault CP is studied. The main conclusions of this paper are summarized as follows.

(1)

The fault dynamic model established in this paper can simulate the impact characteristics and distribution law of the inter-shaft bearing fault signals accurately, and the fault frequency calculation error is less than 1%.

(2)

With the increase of defect size, the vibration amplitude of the inter-shaft bearing also increases. When the width of the defect is less than 0.1 mm, the FCF and a large amount of interference frequently occurs in the envelope spectrum. This can interfere with the early micro-fault diagnosis of the inter-shaft bearing and reduce the effect of the fault diagnosis method based on time–frequency analysis.

(3)

The magnitude of the signal peak can reflect the impact force caused by the bearing fault. Normally, the MV increases with the increase in fault impact force since the change of MV is very sensitive to the fault in the early stage of fault; it is also very effective in monitoring the pitting corrosion fault on the bearing surface. AMV, AST and EV can reflect the magnitudes of signal energy. With the development of faults, the overall trend is relatively stable.

(4)

The KF and IF indicate whether there is impact component in signals, which is very sensitive to early bearing faults. The PF is sensitive to the change in rotating speed and defect width. The SF is the ratio of the EV to AMV. It has a certain indication effect on the fault, but it is not sensitive to fault change. Therefore, SF is not suitable as the CP for fault diagnosis.

In summary, an effective dynamic model of the inter-shaft bearing is established in this paper to accurately simulate the dynamic characteristics of the inter-shaft bearing with local defects. This study provides useful insights for the use of dynamic models to inspect and monitor the health of inter-shaft bearings of aeroengines.