#### Abstract

The high-speed train speed has a higher requirement for stability and operating safety with its operating speed increasing. The main focus of this paper is on the fault characteristics analysis of a high-speed train transmission system. The governing equation and boundary conditions of the transmission system are derived using the finite element method, in which a Timoshenko beam element is introduced to represent the wheelset shaft, and a rigid mass element is utilized to represent the gears, bearings, and wheels. To investigate the vibration response mechanism of a high-speed train transmission system, the critical speed of the system and its modal response are given. According to the types of high-speed train component fault features, the local fault features of gear, bearing, and wheelset are given. Healthy and faulty systems with localized faults in gear, bearing, and wheelset are studied. The characteristics of amplitude and frequency are verified numerically and the transmission characteristics of signal and relation are explained when there is local fault defect in the system. The results show that under different fault types, the time domain response of the system has obvious periodic impulse response faults, and the fault frequency band characteristic distribution is different in the frequency domain response. The simulation results are consistent with the experimental results.

#### 1. Introduction

The gear bearing wheel is a key component of high-speed train transmission systems. When the vehicles are travelling in a high speed, due to the fault of key components, a high level of vibration and noise may take place. Dynamic analysis of high-speed train transmission systems is the key to researching high-speed train running conditions and fatigue strength under complex excitation. Therefore, understanding how to accurately reveal the dynamic characteristics of the nonlinear high-speed train transmission dynamic model is very much in demand.

Previous investigations on transmission systems and shaft-disk characteristics mainly consisted of two basic methods, namely, the global and local approaches. In the shaft system aspect, Rzadkowski and Maurin [1] introduced a dynamic model of a flexible disk and studied natural frequencies of a rotating shaft-disk system. She et al. [2] investigated the influence of the disk’s position and flexibility on the critical speeds and natural frequencies of a coupling shaft-disk-blade unit. Al-Solihat and Behdinan [3] studied the nonlinear dynamic and force transmissibility characteristics of a rotor system supported by a suspension system with nonlinear stiffness and damping. Chiu et al. [4] explored blade-bending, disk-transverse, and shaft-torsion coupling vibration of a flexible shaft-disk system. However, a dynamic model of a rotating system with a beam hypothesis considering gear meshing has been used to explore the systems characterized by gear shaft [5, 6]. Sawalhi and Randall [7] established a modified model for a gearbox by taking into account the gear bearing effect and analyzing the stability of the system. Jianxiong et al. [8] studied the vibration characteristics of gear systems based on a multinode dynamic mode. Hanmant and Pratiher [9] researched the rotor-casing contact phenomena of a large deflection rotor-bearing system under a mass unbalance situation.

The above research mainly focuses on the vibration characteristics of rotors in order to study localized faults in gears. The flexible shaft model can investigate the vibration characteristics of key components. Kahraman et al. [10]used Timoshenko beam theory to propose a gear flexible-shaft dynamic model with consideration of the stiffness of the gear mesh and analyze the dynamics. Later, Kang et al. [11] extended the model to helical gears with the effects of gear eccentricity. Chen et al. [12] studied the critical speed of a gear transmission system using the finite element method. Based on a dynamic model of a wind turbine, Cho et al. [13] analyzed the dynamic behaviors of the gearbox and the natural frequencies of the gearbox. Ma and Xu [14] investigated the accurate calculation of gear stiffness for a nonlinear rotor model.

The literature of bearing dynamics mainly analyzed the dynamic behaviors of the fault bearing. Based on a fault model of the bearing, Harsha et al. [15] analyzed the bearing vibration characteristics caused by the raceway defects. Yang et al. [16] studied the misalignment and contact characteristics of the double-row tapered roller bearing. Furthermore, Yan et al. [17] established a model of a double-row tapered roller bearing that considered elastohydrodynamic lubrication. Recently, Wang et al. [18, 19] investigated the vibration characteristics of axle box bearings under intense wheel-rail excitation. Based on the vehicle-track coupled dynamics model, Huang et al. [20] investigated the fault frequency characteristics of axle box bearings.

As highlighted above, research has addressed the dynamics characteristics of key components of transmission systems. The main objective of this paper is to use the newly developed flexible transmission system dynamics model to investigate the fault characteristics of the key components of the system in a high-speed train. The test experiment is carried out to accurately represent the model by comparing the results from experimental tests and simulations. Based on different bearing, wheel, and gear fault types, the fault characteristics of key components of transmission systems are analyzed theoretically.

#### 2. Dynamic Model

As shown in Figure 1, a high-speed train transmission system is established that takes into account nonlinear factors such as the elastic support of wheel-rails, gear bearings, and flexible shaft elements.

##### 2.1. Rotor-Shaft Element

In finite element analysis, a Timoshenko beam with two nodes is often applied to model a shaft element. A typical element of the shaft is illustrated in Figure 2. Element coordinates of each node is: . Considering the lateral bending and torsion deformation, the kinetic energy of the shaft beam element can be written as

In the high-speed train transmission system model, the strain potential energy of shaft element is presented as

Here, and are mass density and cross section area, respectively. is the shear factor. is Young’s modulus and is the shear modulus of rotor material. and are pole moment of inertia and the moment of inertia.

The deformation of the node are used to depict the displacements of *Z* and *Y* direction, are used to represent the rotational displacements about *Z* and *Y* direction, is the torsion angle, and then the node function can be written as

Here, is displacement vector of element node. The spatial constraint matrices and can be written aswhere and are the shape functions of the shaft element [12].

Substituting the above equation (1)–(5) into the Lagrange equation,where is the external force, . The dynamics equation of shaft beam element is represented aswhere is the spin speed of the shaft.

##### 2.2. Meshing Gear Element Model

The model of meshing gear is shown in Figure 3. The node coordinates of meshing gear are . Kinetic energy of meshing gear is described as

Considering the effect of backlash, the potential energy of meshing gear element is given as

The influence of gear eccentric force on meshing gear element can be expressed as

Here, is the time-varying meshing stiffness of gears. is the comprehensive damping coefficient of gear meshing [12]. and , respectively, denote the radial of base circles of the driving gear and the driven gear. denotes the moment of inertia of the meshing gear. denotes the mass of meshing gear. and are the elastic stiffness and elastic damping, respectively. () is the torque of input and output.

Using the Lagrange approach to (6), the motion equation of meshing gear element is given aswhere , , , and are the mass matrix of meshing gear element, the damping matrix, the gyroscopic matrix, and the time-varying stiffness matrix, respectively.

##### 2.3. Bearing Element Model

Ignoring the axial force and the lateral motion and considering the normal contact force, the supporting bearing element model of tapered roller bearings is shown in Figure 4. Bearing element is assumed as a nonlinear contact. The contact force and moments between the roller and inner raceway can be described according to the Hertzian line of contactwhere is the Hertz contact normal force. is the unloaded contact angle. is the radius of bearing. denotes the race contact angle of the j-the rolling element.

Bearing contact element coordinates are . Applying the Lagrange method, the motion equation of bearing contact element can be obtained aswhere , , , and are mass matrix of bearing elements, damping matrix, stiffness matrix, and contact force matrix, respectively.

##### 2.4. Wheel Element Model

In Figure 5, the coordinates of the wheel element are . The kinetic energy and the potential energy of wheel can be expressed as

By using the Lagrange method, the motion equation of wheel-rail element can be written aswhere , , , and denote the mass matrix, the damping matrix, the gyroscopic matrix contact force matrix, and the support stiffness matrix, respectively, of the wheel-rail element [18].

##### 2.5. Equation of the High-Speed Train Transmission System

Considering the above factors, substituting equations (11)–(17) into (7), the equation of high-speed train transmission system can be expressed aswhere , *n* denotes the number of nodes. , , and are the mass matrix, the damping matrix, the gyroscopic matrix, and the stiffness matrix, respectively, of the high-speed train transmission system.

#### 3. Natural Frequency and Modal Functions

When the high-speed train transmission system is subjected to bending and torsion vibration, the homogeneous equation of rotor system can be presented as

In order to find the critical speed of the equation, the equation can be transformed into a linear equationwhere

Calculating the response characteristics of the system under different rotating speeds, the train transmission systems can be obtained from the Campbell diagram. When the speed of rotation is equal to eigenvalue , the critical speed of the transmission system is obtained, as shown in Figure 6. From the calculated Campbell diagram results, it can be seen that part of the natural frequencies change obviously with the increase of the speed, while the other part of the natural frequencies is basically not affected by the speed.

When the rotor speed changes, the value of the first five-order natural frequencies of the rotor system is obtained in Table 1.

The first-to-third-order mode of the high-speed train system is shown in Figure 7. The vibration amplitude of the mode shapes of the driving shaft changes little, and the vibration amplitude of the driven shaft changes obviously in Figures 7(a) and 7(b). The vibration of the driving shaft is large in Figure 7(c). Under the low-order mode of the system, the vibration amplitude of the driven shaft is more important. In the high-order mode of the system, the vibration amplitude of the driving shaft is especially important.

#### 4. Fault Feature of Gear Bearing Wheel

##### 4.1. Fault Feature of Gear with Tooth Root Crack

In order to study the fault characteristics of key components of the transmission system, the stress characteristics of key components must be considered. In the gear meshing element, the change in gear contact stiffness can effectively reflect the vibration characteristics of the gear system. The energy method [19] is used in this paper. The total potential energy was assumed to include four components: hertz energy , bending energy , shear energy , and axial compressive energy . The total potential energy can be expressed as follows:where is the meshing force. and are the radial and tangential forces, respectively. and represent Young’s modulus and shear modulus, respectively. is the width of tooth. When gears are meshing, the time-varying stiffness can be obtained bywhere , , , and represent the Hertzian, bending, shear, and axial compressive mesh stiffness, respectively [20].

When the gear root crack occurs, it causes a change in the bending deformation stiffness and shear deformation stiffness of the gear tooth. The crack length *q* = 2 mm. The calculating formula for the moment of inertia is shown in Figure 8, and the section area is as follows:where is the length of contact. is the distance between the meshing point of the gear tooth profile and the middle line of the gear tooth. is the height of the gear crack to the midline.

##### 4.2. Fault Feature of Gear with Surface Wear

As shown in Figure 9, when the gear surface wears, the formula of moment of inertia and section area of the gear is described aswhere and are shown in Figure 9. Other parameters are detailed in reference [21].

##### 4.3. Bearing Outer Race Defect

As shown in Figure 10, a local fault is considered as a small rectangular spall in the outer race. The position angle of defect is . The central angle of the defect is . The defect width is . The radius of the cylinder roller is . When the rolling body passes through the fault region, the normal displacement is . The displacement can be expressed as

When the tapered roller is in the defect region, the contact deformation of tapered roller can be formulated aswhere . are the number of roller and bearing clearance. The angular velocity of the cage is [5].

##### 4.4. Bearing Inner Race Fault

As shown in Figure 11, a local fault is considered as a small rectangular spall in the inner race. and are initial position angle and the central angle of the defect, respectively. The defect width is . When the rolling body passes through inner fault region, the contact displacement is . The displacement can be expressed as

When the tapered roller is in the fault region, the contact deformation of cylinder roller can be formulated aswhere . The angular velocity of rotor is .

##### 4.5. Polygonal of Wheels Fault

The extreme running conditions of high-speed trains often cause wheel deformation. As shown in Figure 12, the fault feature of the wheel surface deformation is considered a wheel polygonal with harmonic curve. The expression of surface deformation is as follows:where the angle is the initial phase. Wheel Angle is . The ripple amplitude of the wheel is .The radius of the wheel is .

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When the wheel experiences polygonal wear, the expression of the displacement relation between the wheel node and surface deformation is as follows:where the coordinates of the wheel is , wheel-rail contact angle , transfer matrix is as follows:

#### 5. Dynamic Simulation Analysis

In this paper, high-speed train transmission system is taken as the research object. Given system parameters, the length of the supporting shaft is . The outside diameter of shaft section is . The quality of the wheels is . The mass of concentrated inertia is . The numbers of teeth are , respectively. The module of gear is . The width of teeth is . The driven shaft bearing is NJ2232. The number of rollers is *N* = 17, are the inner and outer race radius of the bearing. The pitch diameter of the bearing is and the ball diameter . The drive shaft speeding is . Table 2 shows the parameters of the system. Table 3 shows the bearing gear fault parameters.

##### 5.1. Normal System Response

Figure 13 shows the time domain waveform and frequency spectrum of vertical response of the driven gear nodes. Under the influence of time-varying meshing stiffness, time history response shows that the amplitude of acceleration of driven gear has obvious gear meshing periodic *T*_{m}, as shown in Figure 13(a). Spectral response shows that FFT power density of acceleration of driven gear has clear gear meshing frequency *f*_{m}, 2*f*_{m}, and frequency doubling, as shown in Figure 13(b).

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##### 5.2. System Response under Gear Root Crack

During the driving gear occurs root crack, the time domain waveform and frequency spectrum of driven gear nodes are shown in Figure 14. The time domain response has an obvious period *T*_{s}, as shown in Figure 14(a), which is the rotating period of the crack gear. In the spectrum, some new frequency components appear around the natural and meshing frequencies. The gear fault frequency *f*_{s} is mainly concentrated in the gear meshing frequency band, as shown in Figure 14(b).

##### 5.3. System Response under Gear Wear

During the surface of the driving gear is slightly worn, the time domain waveform and frequency spectrum of vertical response of the driven gear nodes are shown in Figure 15. The system has obvious periodic impact response of gear in time domain response. *T*_{s} is the rotation period of worn gear shaft. The vibration amplitude of vertical response of the driven gear nodes increased in Figure 15(a). From the spectrum shown in Figure 15(b), the meshing frequency *f*_{m} and the corresponding components of the doubling frequency 3*f*_{m} can be clearly found. Moreover, the gear fault frequency is mainly concentrated in the gear meshing frequency band.

##### 5.4. Bearing Outer Race Response

During the outer race of the driven shaft bearing has local fault, the time domain waveform and frequency spectrum of vertical response of the fault bearing nodes are shown in Figure 16. The time domain response of the system has periodic impulse response, as shown in Figure 16(a). *T*_{o} is period of fault bearing outer ring. From the spectrum shown in Figure 16(b), there are bearing fault frequencies *f*_{o}. Moreover, the bearing fault frequency is mainly concentrated in the high frequency of the system.

##### 5.5. Bearing Inner Race Response

During the inner race of the driven shaft bearing has local fault, the time domain waveform and frequency spectrum of the fault bearing nodes are shown in Figure 17. *T*_{i} is the inner race rotation period of the fault bearing, as shown in Figure 17(a). From the spectrum shown in Figure 17(b), there are obvious frequencies of bearing fault *f*_{i}. The bearing fault frequency also is mainly concentrated in the high frequency of the system.

##### 5.6. Polygonal Wear of Wheels Response

When the left wheel of driven has *N *= 2 polygonal wear, the time domain waveform and frequency spectrum of the wheel node are shown in Figure 18. According to the time domain waveform shown in Figure 18(a), under the influence of wheel polygonal wear, time domain response has a period of 0.5*T*_{2} of strong impact. From the spectrum shown in Figure 18(b), there are 2*f*_{2} of driven shaft. When the wheel has local fault, the fault feature frequency is mainly affected by the different orders of the polygon and the response feature frequency is *N* × *f*_{2}. The fault feature is mainly distributed in the low-frequency band.

##### 5.7. Couple Fault Response

During the outer ring of the left axle box bearing, the driving gear and the wheel have local fault, the vertical vibration response of the fault bearing node are shown in Figure 19. According to the time domain waveform shown in Figure 19(a), there is an obvious impulse response *T*_{2} caused by a wheel fault. Compared with wheel fault, gear and bearing fault response is not obvious. Figure 19(b) shows the spectrum diagram. Local figure is shown in Figure 19(b) (part enlarge A), it has obvious wheel fault frequency *f*_{2} in the whole frequency band. Local figure is shown in Figure 19(b) (part enlarge B), the mesh frequency *f*_{m} and the gear fault frequency *f*_{s} is apparent. The local figure is shown in Figure 19(b) (part enlarge C), and the bearing fault frequency *f*_{o} is apparent and distributed in the high frequency band.

##### 5.8. Comparison of Simulation Experiments

The long-term tracking experiments of Fuxing Hao are presented in Figure 20. When the train is running at speed *n*_{1} = 3200 r/min, the vibration response of the bearing axle box is presented in Figure 21. Under the influence of the running environment, there are no periodic response characteristics in the time-frequency response in Figure 21(a). The acceleration response of the axle box is very small because the wheel-rail interference excitation is small in the early stages. In addition, there are obvious gear meshing frequencies and multiple frequency characteristics in the frequency response in Figure 21(b). Specifically, the simulation signals are basically the same as the vibration response characteristics.

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The vertical vibration responses of the bearing node without fault are shown in Figure 22. Compared to the experimental data, the acceleration response of the bearing is basically consistent, as shown in Figure 22(a). In addition, there are obvious gear meshing frequencies and multiple frequency characteristics in the frequency response, as shown in Figure 22(a).

#### 6. Conclusion

In this paper, the kinematic model of a high-speed train transmission system is established, and the fault characteristics of the bearing, wheel, and gear elements are studied.

The dynamic model of the system is established, which can better analyze the transmission relationship and the influence between bearings, wheels, and gears. The change in system response caused by faults in bearings, wheels, and gears is studied. When there is a gear fault, the vibration energy of the system of rotors is distributed in the low resonance frequency band. At the same time, the gear meshing frequency band has an obvious modulation phenomenon. When there is a bearing fault, the vibration energy of the rotor system is distributed in the high resonance frequency band. When the wheel is subject to polygonal wear, the fault feature frequency is mainly affected by the different orders of the polygon, and the response feature frequency is *N* × *f*_{2}. The fault feature is mainly distributed in the low-frequency band. When the system is couple fault, the impact caused by wheel fault is obvious. However, the frequency band position of each component of the system has not changed.

In the low-order mode of the system, the vibration amplitude of the driven shaft is more important. In the high-order mode of the system, the vibration amplitude of the driving shaft is especially important.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study was financially supported by the National Natural Science Foundation of China (grant nos. 12002142 and 51976087) and the National Natural Science Foundation of Shandong Province (grant nos. ZR202102180973 and ZR202102230860).