Effect of train traction on the wheel polygonal wear of high-speed trains

2.1. Finite element model

China's high-speed trains operate at high velocities, and the adhesion coefficient at the wheel-rail interface decreases continuously as the speed increases. Therefore, the creep force at the wheel-rail interface readily achieves a saturated state during driving and acceleration of high-speed trains. The saturated creep force is inclined to excite FIV in the WRS [27], thereby leading to the creation of cyclic uneven wear at the wheel-rail contact region [27, 17]. A finite element model of the WRS under rolling-sliding condition was established. Modal analysis was utilized to study the frequency-domain attributes of FIV in the WRS. The motion equations of the WRS were solved by implicit time integration method. The interplay between FIV of the WRS and the formation of PW on wheels was investigated to reveal the mechanism of PW on the wheels of high-speed trains in China. Fig. 1 illustrates the finite element model utilized for frequency domain analysis of PW on wheels. Complex eigenvalue analysis was conducted based on this model to analyze the vibration modes of FIV under saturated tangential force at the wheel-rail interface. To reduce computational cost, the model only included a single side of the wheel and rail, and the impact of axle was ignored. The nominal diameter of the wheel was 920 mm. The vertical suspension force, denoted as $F_v$, was substituted for the sprung mass in the model. The rail length measured 36,252 mm, about 56 times of the fastener spacing. The rail cant was 1/40. The rail was discreetly supported by ground springs and damping units replacing the fastener system. The spacing between adjacent fasteners was 650 mm, and the support length of each fastener was 130 mm [28]. The normal contact interaction at the wheel-rail interface was described using a face-to-face contact algorithm employing the penalty method. Coulomb law of friction was applied to characterize tangential contact behavior. The normal contact behavior at the wheel-rail interface was represented using the hard contact algorithm. To simulate the rolling process on a straight track, translation and rotational velocities were imparted to the wheel, ensuring that the wheelset's lateral displacement and yaw angle remained nearly negligible. Consequently, the lateral and rotational creepages at the wheel-rail interface can be regarded as zero. Consequently, only the longitudinal creepage at the wheel-rail interface was considered in the model. The displacement boundary conditions in the model were: the translational degrees of freedom in three directions at both ends of the rail was restricted. Meanwhile, all degrees of freedom of the nodes on the axle surface were coupled to a reference point on the wheel axis, with only the vertical degree of freedom of this reference point left unconstrained. The wheel and rail were discretized using incompatible hexahedral elements. Dense mesh was applied at the wheel-rail contact interface, while coarse mesh was used away from the contact interface. The model consists of a total of 111,040 solid elements and 135,570 nodes.

The dynamic response of the WRS was obtained through an explicit dynamic analysis and explore the time-domain traits of FIV in the WRS. Fig. 2 shows the established finite element model for time-domain analysis of PW on wheels. The dynamic response of WRS was determined using the implicit time integration method. In the dynamic model, the wheel rolled on the rail, and refined mesh was applied to the rail sections in contact with the wheel. The displacement boundary conditions for the wheel were: only the vertical, longitudinal translational degrees of freedom and rotational degree of freedom were retained. Other mechanical and displacement boundary conditions were consistent with the complex eigenvalue analysis model. The material parameters are listed in Table 1, and other relevant parameters are given in Table 2, the data for the coefficient of friction and stiffness, damping, and other important parameters were sourced from commonly used fastening systems in high-speed railways [29]. Typically, the wheel-rail coefficient of friction decreases with increasing train speed. At speeds around 300 km/h, the coefficient of friction in normal adhesion conditions is approximately 0.23. And we set the longitudinal creepage at 0.7 %, which can lead to a saturation of creep forces at the wheel-rail interface [22].

The transient dynamic response of the WRS is computed using the implicit time integration method through the following steps: (1) Apply the vertical suspension force for static stress analysis; (2) Apply translation and rotation velocities to the wheel to simulate the rolling-sliding motion of the wheel.

Fig. 1Finite element model of the frequency domain analysis of the wheel PW

Fig. 2Finite element model of the time domain analysis of the wheel PW

2.2. Complex eigenvalue analysis method

The equations of motion for the WRS can be established:

where $\mathbf{M}$, $\mathbf{C}$ and $\mathbf{K}$ represent the mass, damping and stiffness matrices of the WRS, respectively. $\ddot{x}$, $\dot{x}$ and $x$ denote the acceleration, velocity and displacement vectors of the nodes, respectively. The equation of motion Eq. (1) has the following characteristic equation:

where, $\mu$ signifies the eigenvalue, and $\varphi$ represents the eigenvector,

Due to the saturated creep force at the wheel-rail interface, the damping and the stiffness matrices of the system are asymmetric matrices. Eq. (2) has complex eigenvalues, which can be resolved using the subspace projection method to obtain the solution of Eq. (2), and further the solution of Eq. (1):

where $({\alpha }_k+i{\omega }_k)$ is the k-order eigenvalue of the system. When the real part $\alpha$ of the eigenvalue is positive, the system’s vibration will undergo exponential growth over time, resulting in instability. The equivalent damping ratio (hereinafter referred to as EDR) $\zeta =-\alpha /\pi \left|\omega \right|$ is generally used to evaluate the trend of instability. The system will become unstable and FIV will occur in the WRS when the EDR is negative.

2.3. Analysis of PW on wheel

The friction-wear model developed by Brockley [30] can qualitatively explain the correlation between FIV of the WRS and the formation of PW on wheels:

where $w$ denotes the volumetric wear rate, $K$ signifies the material wear constant, $H$ denotes the friction power between wheel and rail ($H=F\times v$, where $F$ is the tangential force at the wheel-rail interface, and v represents the sliding velocity at the wheel-rail interface). $C$ is the endurance frictional power, which is a constant. Under rolling-sliding condition, the tangential force at the wheel-rail interface $F=N\times f$, where $N$ is the normal contact force, and $f$ is the friction coefficient. Therefore, the wear rate $w$ exhibits a proportional relationship with the normal contact force $N$. If FIV of the WRS leads to oscillations of the normal contact force $N$, the wear rate $w$ will also fluctuate accordingly, resulting in periodic non-uniform wear of the wheel tread, eventually leads to the polygonization of the wheel.

3.1. Frequency domain

Fig. 3 depicts the frequency distribution of FIV in the WRS under high-speed train rolling-sliding condition. It is observed that the WRS has two unstable vibrations with negative EDRs, indicating the occurrence of FIV in the WRS. Instability is observed in these two vibrations at frequencies of 162.8 Hz and 622.7 Hz, respectively, with EDRs of –0.000275 and –0.000305 respectively. The mode shapes of these two vibrations with instability are depicted in Fig. 4. It can be observed that both unstable vibrations primarily manifest on the wheel. The 162.8 Hz unstable vibration is the yawing mode of the wheel, and the 622.7 Hz unstable vibration is the radial scaling mode of the wheel. The periodic yawing or radial scaling of the wheel can both lead to periodic oscillations of the friction power between wheel and rail, leading in periodic non-uniform wear of the wheel tread, resulting in the manifestation of PW. In accordance with the relationship between the order of PW and the unstable vibration frequency ($n=\pi f_{R}D/v$, where $n$ denotes the PW order, $f_R$ stands for the frequency of unstable vibration, D represents the nominal wheel diameter, and $v$ signifies the speed), the 162.8 Hz unstable vibration will lead to 5th-6th order PW, and the 622.7 Hz unstable vibration will lead to 21st-22nd order PW. The two orders of PW both exist in reality on the wheels of high-speed trains in China [1], among which the 21st-22nd order PW is high-order PW with the most severe detriment, but its cause remains controversial. The complex eigenvalue analysis results indicate that the 21st-22nd order PW has a close correlation with the periodic radial scaling of the wheel induced by saturated creep force under driving condition.

Fig. 3Distribution of frequencies of the unstable vibrations

Fig. 4Vibration shapes of the unstable vibrations

a)$f_R=$ 162.8 Hz

b)$f_R=$622.7 Hz

3.2. Time domain

Fig. 5 illustrates the alteration in the contact force at the wheel-rail interface under rolling-sliding condition. Fig. 5(a) presents the time-domain representation of the wheel-rail contact force, and Fig. 5(b) shows the frequency distribution of the wheel-rail contact force, which was obtained employing fast Fourier transform (FFT) to the signal in the time domain. The transient dynamic response of the WRS, obtained through employing implicit time integration, is extremely computationally expensive. Therefore, only 0.15 s of the dynamic response process was simulated, which is adequate to capture the time-domain traits of FIV in the WRS. Meanwhile, since the simulation time is very short, the change of wheel velocity is negligible, so the wheel velocity can be assumed to be constant during this period. As shown in Fig. 5(a), under rolling-sliding condition, the dynamic contact force at the wheel-rail interface oscillates significantly between 55-95 kN, and the average peak dynamic contact force reaching approximately 95 kN, about 1.27 times of the static value of 75 kN. Since there is no external irregularity in the model, the oscillation of the contact force can be attributed to the FIV of the WRS. As shown in Fig. 5(b), there are several peak values in the vibration amplitude spectrum between 560-800 Hz, among which the peak at 608.7 Hz is the most significant, indicating this frequency is the dominant vibration frequency. This is near the 622.7 Hz estimated by the complex eigenvalue analysis. However, there are also some discrepancies between the complex eigenvalue analysis and the transient dynamic analysis. For example, no peak appears around 162.8 Hz in Fig. 5(b), indicating the unstable vibration at 162.8 Hz estimated by complex eigenvalue analysis is unlikely to occur.

Fig. 6 illustrates the variation in friction force at the wheel-rail interface. Under rolling-sliding conditions, substantial oscillations can be observed in the friction force at the wheel-rail interface as well, with an oscillation range spanning 12.5-22.5 kN. The average maximum value of the dynamic friction force is about 1.30 times of the static friction force of 17.25 kN. As shown in the frequency distribution of friction force in Fig. 6(b), a significant peak emerges at 598.9 Hz in the amplitude spectrum of the friction force vibration. This frequency is near 608.7 Hz, where a significant peak in the wheel-rail contact force is observed, and is also proximate to 622.7 Hz as estimated by the complex eigenvalue analysis. Fig. 7 shows the fluctuation of vibration acceleration on the wheel axle. To eliminate the impact of wheel rotational acceleration, the analysis of vibration acceleration is conducted at the center of wheel rotation. It is evident that the wheel axle acceleration fluctuates violently with an average amplitude of approximately 50 m/s². Similarly, a significant peak also appears at 598.9 Hz in the wheel axle acceleration spectrum, which has a close frequency to peaks in the friction force and contact force spectrums.

The above analysis reveals that substantial vibrations arise in the contact force, friction force, and wheel axle acceleration during wheel rolling-sliding. These three vibrations have similar characteristics. The frequencies corresponding to the significant peaks in their vibration amplitude spectrums are consistent with the frequency of wheel radial scaling predicted by complex eigenvalue analysis. This indicates intense FIV occurs in the WRS under saturated creep force during rolling-sliding condition. The FIVs lead to fluctuations in the friction force and contact force at the wheel-rail interface, bringing about PW on wheels. The periodic radial scaling of the wheel induced by saturated creep force is a potential cause for the 21st-22nd order PW on Chinese high-speed train wheels.

Fig. 5Normal contact force at the wheel-rail interface

a) Time domain

b) Frequency domain

Fig. 6Friction force at the wheel-rail interface

a) Time domain

b) Frequency domain

Fig. 7Acceleration at the wheel axle

a) Time domain

b) Frequency domain