Study on the impact of wheel roundness defects on axle fatigue damage

2.1. Wheel flat model

Wheel flats can be categorized into new flats and old flats, and the models for these two types of wheel flats are different. This study adopts a wheel radius variation approach to describe the flat model. During one complete wheel rotation, the change in the wheel’s circumferential radius is defined as follows [23]:

where: $∆r\left(x\right)$ is the variation in the wheel’s circumferential radius; $h$ is the depth of the flat; $R$ is the nominal rolling radius of the wheel. In this study, $R=$ 0.43 m; $L$ is the length of the flat (in meters).

The new flat model is relatively idealized and does not frequently occur in practice. As the train operates at high speeds, new flats are typically worn down quickly, transforming into old flat models. Existing old flat models often use a cosine function to reflect the gradual change in flat depth, as follows [24]:

where: $L_f$ is the length of the flat (in meters).

Based on this, the wheel flat is transformed into a variation in wheel radius:

where: $a=L_f/\left(2R\right)$; $\beta$ is the wheel rotation angle (in radians).

2.2. Wheel polygonization model

Wheel polygonization is generally composed of multiple harmonics with different frequencies, amplitudes, and phase angles. To describe wheel polygonization, it is often simplified as a combination of simple harmonic waves. Under ideal conditions, wheel polygonization can be expressed as:

where: $∆r$ is the variation in wheel radius; $A$ is the amplitude of the order polygonization; $n$ is the order of wheel polygonization (number of sides); $\theta$ is the wheel rotation angle (in radians); ${\theta }_0$ is the initial phase angle of the order polygonization; $r$ is the actual radius of the wheel; $R$ is the nominal rolling radius of the wheel; $\omega$ is the angular velocity of the wheel.

The wheel roughness level $L_k^2$ (in units of dBre1μm) is defined by the following formula:

where:$r_k^2$ is the mean square value of the wheel out-of-roundness roughness $r\left(x\right)$ quantized within the $k$th 1/3 octave band; $r_{ref}^2$ is the reference value for wheel roughness.

Under international standard units, the center wavelength ${\lambda }_k$ of the frequency band is given by:

Within each 1/3 octave band, the narrowband spectral amplitudes are squared, summed, and divided by the number of calculation points to obtain $r_k^2$ [25].

3.1. Impact of wheel flats on axle dynamic stress

To investigate the influence of flat length and vehicle speed on axle dynamic stress, the dynamic response was calculated for a single flat positioned identically on both wheels. The dynamic stress was extracted at a cross-section located 35 mm inward from the gearbox side of the axle seat, corresponding in phase with the wheel flat. Simulations were conducted for flat lengths of 10 mm, 20 mm, 30 mm, 40 mm, and 50 mm under various operating speeds, and the results are presented in Fig. 2. Under the same flat length condition, the axle dynamic stress was observed to increase initially and then decrease with increasing speed, reaching a maximum value of 101 MPa at approximately 50 km/h.

Fig. 2Effect of flat spot length on stress amplitude at different speeds

Fig. 3Effect of flat spot length on stress amplitude at the same speed

Fig. 4Axle dynamic stress spectrum at 50 km/h with a 50 mm flat spot length

As shown in Fig. 3, which illustrates the relationship between axle stress and flat length at a speed of 100 km/h, the axle dynamic stress increases linearly with the length of the flat. Longer flats with greater depths result in higher axle dynamic stress. For a 50 mm flat, the dynamic stress data at 50 km/h is subjected to a Fast Fourier Transform (FFT), as shown in the frequency spectrum in Fig. 4. At 50 km/h, the wheel flat excites frequencies within the range of 350-400 Hz, which is close to the first-order bending frequency of the wheelset on one side.

3.2. Impact of wheel polygonization on axle dynamic stress

The influence of wheel polygonization on axle dynamic stress can be analyzed in terms of three key factors: polygon order, wave depth, and vehicle speed. Accordingly, simulations were performed to extract axle dynamic stress under polygonization conditions, wave depths, and speeds representative of typical high-speed train operations. As shown in Fig. 5 for a given polygon order, the maximum axle stress initially increases with speed and then decreases, reaching a peak within the simulated speed range. Fig. 6 illustrates the variation of maximum axle stress with different polygon orders and speeds at a fixed wave depth of 0.3 mm. It can be observed that higher polygon orders correspond to lower speeds at which the stress peaks occur.

The excitation frequency associated with polygonal wear can be calculated using the following formula:

where: $f$ is the excitation frequency of wheel polygonization (in Hz); $v$ is the vehicle speed (in km/h); $n$ is the polygon order; $d$ is the wheel diameter (0.92 m).

Using Eq. (9), the excitation frequencies corresponding to the maximum axle stress peaks for polygon orders of 18, 20, 22, and 25 are calculated as 475.73 Hz, 480.54 Hz, 475.73 Hz, and 480.54 Hz, respectively. These frequencies are close to the simulated fourth-order bending frequency of the wheelset at 490.11 Hz. Therefore, it is concluded that the increase in maximum axle stress may be due to the excitation of the fourth-order bending mode of the wheelset by wheel polygonization during operation. This excitation amplifies the dynamic forces on the wheelset, leading to a significant increase in axle stress.

Fig. 5Relationship between maximum stress and speed at different wave depths under 18th-order polygonal wear

Fig. 6Relationship between maximum stress and speed at different orders of polygonal wear with a 0.3 mm wave depth

By comparing the maximum axle stress under the same speed and polygon order, as shown in Fig. 7, the relationship between maximum axle stress and polygon wave depth for an 18th-order polygon at speeds of 200 km/h and 250 km/h can be observed. It is evident that the maximum axle stress increases with wave depth, exhibiting an approximately linear growth trend.

Fig. 7Relationship between maximum axle stress and wave depth

Fig. 8Relationship between maximum axle stress and polygonal order

When comparing maximum axle stress under the same speed but with different polygon orders at a fixed wave depth, as shown in Fig. 8, the stress response for polygon orders ranging from 18 to 25 at speeds between 250 and 350 km/h and a wave depth of 0.1 mm was analyzed. The results indicate that the maximum axle stress does not follow a clear trend with increasing polygon order.

Overall, axle dynamic stress is strongly influenced by both polygon wave depth and vehicle speed. As illustrated in Fig. 9, which shows the relationship between speed, wave depth, and maximum axle stress for a 20th-order polygon, the effect of wave depth on the axle is direct and pronounced, with stress increasing as wave depth grows. In contrast, the influence of speed is closely related to the modal characteristics of the wheelset. Consequently, the speed at which maximum axle stress occurs corresponds to the excitation frequency of the wheelset’s modal response. Notably, higher polygon orders require lower speeds to excite the corresponding wheelset modal frequencies.

Fig. 9Relationship between maximum axle stress, speed, and wave depth

4.1. Fatigue damage assessment of axle under wheel flat conditions

As noted previously, wheel flats have the greatest impact on axle dynamic stress at speeds around 50 km/h. However, in actual vehicle operation, the proportion of time spent traveling near this speed is relatively small. To better reflect real-world conditions, the fatigue strength assessment of the axle under wheel flat conditions is therefore conducted at a more representative speed of 300 km/h, which corresponds to the predominant operating range of high-speed trains.

The dynamic model was used to simulate the aforementioned conditions, and the corresponding stress-time history data were extracted. The simulated stress data were then processed using the rainflow counting method to obtain the stress spectrum. The number of cycles from the simulation was extended to cover a full wheel re-profiling period, and the axle fatigue damage was calculated based on standard fatigue strength assessment methods. The cumulative fatigue damage and equivalent stress amplitude for different flat lengths are summarized in Table 1.

As shown in Table 1, within a single re-profiling cycle (i.e., 300,000 km), fatigue damage to the axle caused by wheel flats increases with flat length. However, the overall damage remains very small and is insufficient to cause fatigue failure. Considering that high-speed trains operate with an average annual mileage of 600,000 km and have a design service life of 20 years, the total design mileage is approximately 12 million km. If a 50 mm flat were to persist throughout this period, the cumulative fatigue damage to the axle would reach only 0.0564, still well below 1. According to the Operation and Maintenance Regulations for Railway High-Speed Trains, which set a maximum allowable flat length of 20mm, the fatigue damage caused by wheel flats under normal operating conditions is therefore negligible.

4.2. Fatigue damage assessment of axles under polygonal wheel conditions

Since the characteristics of wheel polygonization change continuously in practice, it is challenging to accurately calculate the fatigue damage caused by polygonization over an entire re-profiling cycle. Therefore, equivalent simplifications are commonly applied in such calculations [15, 26]. Although the detailed development patterns of polygon orders remain poorly understood [19, 27], the dominant polygon orders are relatively well established: at a speed of 300 km/h, the dominant orders are 18th to 20th; at 250 km/h, the dominant orders are 23rd and 24th [28]. As illustrated in Fig. 10, which shows the evolution of wave depth for a 3rd-order polygon, previous research [29] indicates that wave depth growth exhibits abrupt changes, with the point of sudden change in wear rate referred to as the “transition mileage.” The smaller the initial polygon order and wave depth, the longer the transition mileage.

Based on statistical analyses of out-of-roundness measurements conducted on tens of thousands of wheels across multiple high-speed rail lines and train models, it was found that only 0.14 % of wheels had radial runout exceeding 0.3 mm, while the vast majority exhibited radial runout between 0 and 0.2 mm [1].

Therefore, in calculating the axle fatigue strength, two representative operating speeds are considered: 250 km/h and 300 km/h, corresponding to polygon orders of 24th and 20th, respectively. Due to the uncertainty in polygon order development, changes in polygon order are not considered, and the initial orders are assumed constant throughout the cycle. The initial wave depth is set to 0.02 mm, the maximum wave depth to 0.17 mm, and the transition mileage to 200,000 km. The specific calculation procedure is illustrated in Fig. 11.

Fig. 10Evolution trend of polygonal wave depth

Fig. 11Variation of wave depth with operating mileage

The evolution of wave depth over a single re-profiling cycle is calculated based on Fig. 11. Corresponding dynamic stress data for various wave depths are obtained through simulation, and the fatigue damage is evaluated following the same procedure described in the previous section. The calculated fatigue damage results for the axle under 24th-order polygon conditions within one re-profiling cycle are presented in Table 2, while the results for the 20th-order polygon conditions are summarized in Table 3.

The calculation results indicate that within a single re-profiling cycle, the cumulative fatigue damage to the axle under 24th-order polygon conditions is 9.78×10^-3, while under 20th-order polygon conditions, the cumulative fatigue damage is slightly higher at 1.17×10^-2. Overall, the damage caused by wheel polygonization remains relatively minor, and the corresponding equivalent stress amplitudes are well below the allowable strength limits. Therefore, it can be concluded that wheel polygonization within one re-profiling cycle induces only limited fatigue damage, insufficient to cause axle fatigue failure. However, compared with the condition of an ideal wheel, both 24th- and 20th-order polygons result in significantly greater fatigue damage to the axle.

Based on the wave depth evolution pattern illustrated in Fig. 11, if each re-profiling cycle follows the same development process, the cumulative fatigue damage to the axle after 40 re-profiling cycles would reach approximately 0.39 for 24th-order polygons and 0.468 for 20th-order polygons. Both values remain below the critical fatigue threshold of 1, indicating that the axle will not experience fatigue failure throughout its service life.

Furthermore, analysis of the fatigue damage per unit mileage reveals that, for both 24th- and 20th-order polygons, the damage per unit distance increases as the polygon wave depth increases. Examination of the damage values across different stages of the 20th-order polygon shows that although the mileage corresponding to larger wave depths is shorter, the total fatigue damage accumulated in these stages is considerably higher.

In summary, greater polygon wave depths lead to increased fatigue damage in the axle. Therefore, when significant polygonal wear is detected, timely wheel re-profiling should be performed to mitigate further damage and ensure operational safety.