Research on the Influence ofWheel Polygonization on Axle Stress

To study the influence of wheel polygonization on the dynamic stress of the wheel axle, a vehicle-track rigid-flexible coupling dynamic model was established. In the model, the wheelset, axle box, and track systemwere modelled as flexible bodies to consider the influence of elastic vibration. At the same time, the dynamic stress on key positions of the axle under the wheel polygonization excitation was measured on the high-frequency vibration test rig. +e accuracy of the model was verified by comparison with the test results. +e wheel axle stress under the excitation of the wheel polygonization with different orders, wave depths, and running speed was calculated.+e results show that the wheel polygonization can increase the amplitude of the axle dynamic stress, and the larger the wave depth of the wheel polygonization, the larger the stress amplitude. When the wheel polygonization frequency is close to the frequency of the wheelset elastic vibration mode, the wheelset first-order bending and second-order bending modes have a great influence on the axle stress.+e resonance vibration of the wheelset elastic modal can cause the dynamic stress on key positions of the axle increase sharply.


Introduction
With the continuous development of railway technology, high-speed railway has been developed rapidly all over the world, especially in China. So far, 30,000 kilometres of highspeed railway lines have already been in operation, and more high-speed lines are under design and planning. e running speed of the latest train "Fu-xing" reaches 350 km/h. China's high-speed railway has the characteristics of long running distance and fast driving speed. During the years of operation in China, some problems that need to be solved have gradually emerged. One of the most important problems is the phenomenon of wheel polygonization [1,2]. e wheel polygonization means harmonic wear in the longitudinal direction of the wheel tread [3,4]. e wheel polygonization can increase the wheel-rail force and exacerbate the vibration of the vehicle system, which can result in significant failures of the vehicle component. e wheel polygonization is a worldwide problem, existing in railway vehicles. e research on the wheel polygonization mainly focuses on the formation mechanism and its impact. rough experimental measurements and simulation calculations, Kalousek and Johnson [5] proposed to solve the wheel polygonization problem by reprofiling the rail and the wheel. Brommundt [6] studied the polygonal phenomenon by numerical calculation and the dynamics model. e interaction between the manufacturing error and the wheel's moment of the inertia leads to the wheel polygonization. By building the dynamic model of the ICE-1 high-speed train, Morys et al. [7,8] studied the production mechanism and evolution law of wheel polygonization. It is concluded that the wheel polygonization can cause a large change in the wheel/rail vertical force and stimulate the bending mode of the wheelset, leading to the vibration resonance and eventually aggravate the development of wheel polygonization. Meywerk et al. [9] studied the formation and evolution of wheel polygonization by simulation. It is concluded that the first and second-order vertical bending modes of the wheelset play an important role in the development of the out-of-roundness of wheel. Johansson and Dirk [10,11] also studied the wheel polygonization phenomenon by different models and proposed different polygonization formation mechanisms, respectively. Combining the vehicle/track system dynamics model and the wheel wear prediction method, Luo et al. [12] established a coupling model to study the polygonization wear on wheels and its impact on vehicle dynamics performance. Chen and Jin et al. [13,14] applied the finite element software to establish the dynamic model of the wheel/rail friction system and studied the influence of the wheel/rail contact parameters on the wheel polygonization wear. Li [15] analysed the generation mechanism of wheel polygonization for metro vehicles. e combination of experiment and simulation shows that the first-order bending mode is the main factor that forms the phenomenon of wheel polygonization. Cai [16] presents a detailed investigation of the mechanism of metro wheel polygonal wear using on-site experiments and numerical simulation, and they suggest that the P2 resonance is the main contributor to the high amplitude of wheel/rail contact forces in the 50∼70 Hz frequency range and the reason for subsequent polygonal wear. Li [17] considered that track vibration is an important factor in the generation of wheel polygonization based on a large number of measured data and simulation result. At the same time, many scholars have conducted research on the bad effects of wheel polygons. As early as in 1974, Jenkins and Stephenson [18] studied the impact loads of different types of out-of-round wheels and proposed an improved design of the wheel design. By building a vehicletrack dynamics model with flexible wheelset model, Wu [19,20] studied the influence of wheel polygonization on wheel-rail force and axle damage and also pointed out that wheel polygonization can stimulate the bending modes of wheelsets, thus causing adverse effects. Wang [21] studied the influence of wheel polygonization on vehicle system dynamics and proposed corresponding safety limits based on wheel-rail vertical forces.
rough the review of previous scholars, the research on wheel polygonization mainly focuses on two aspects, one is the formation mechanism of polygonal wheel and the other is the influence of wheel polygonization on the vehicle system. Previous studies on the influence of wheel polygonization mostly focused on the influence of vehicle dynamics parameters, such as the wheel-rail vertical force and the ride comfort. In fact, the polygonization has a great threat to the service life of the wheel and axle. erefore, in Section 1, a vehicle-track rigid-flexible coupling dynamics model is first established. en, a dynamic stress test on the rig is conducted, and the model is verified in Section 2. At last in Section 3, the influence of some important parameters of the wheel polygon on axle dynamic stress is analysed, and some conclusions are drawn in Section 4.

Rail
Model. e rail is modelled as a finite length Timoshenko beam, which is discretely supported on the track plate by fastener. e length of the rail model is 100 meters, which is long enough for simulation [22]. Both ends of the rail are considered to be fixed. e force applied on the rail is shown in Figure 1. e vertical vibration differential equations can be represented by While, the lateral vibration differential equations can be formulated as e torsion equation is where F wryj represents the vertical wheel-rail contact force of the j th wheel (j � 1, 2, 3, 4), F wryj represents the lateral wheelrail contact force, and the vertical and lateral forces of the discrete supports are marked as F szj and F syj (i � 1, 2, . . . , N s ), where N s is the number of supports. M Gj (j � 1, 2, 3, 4) is the moment induced by the wheel-rail contact forces, while M si (i � 1, 2, . . . , N s ) is the moment generated by the wheel-rail support forces, x wj (j � 1, 2, 3, 4) is the position of each wheel along the rail, x si is the position of each discrete support along the rail, ψ y represents the deflections of the rail around the z-axis, ψ z represents the deflections of the rail about the y-axis, δ indicates the torsion angle of the rail, G is the shear modulus of the rail material, K x is the torsional stiffness coefficient, A indicates the rail cross-sectional area, ρ is the mass density of the rail material, I z is the second moment of area of the rail cross-section about the z-axis, I y is the second moment of area of the rail cross-section about the y-axis, I 0 is the polar moment of inertia of the rail cross-section, and κ z and κ y are the torsional stiffness coefficients along the z direction and y direction. e above partial differential equation describing the deflections can be converted into a series of ordinary differential equations in terms of generalized coordinates using the modal superposition method: where q zk (t)q yk (t)q Tk (t),ω yk (t), and ω zk (t) are the generalized modal coordinates; Z k (x), Y k (x), Φ k , ψ yk , and ψ zk are the mode shape functions; and N indicates the number of modes.
For the differential equations from equations (1)-(5), the normalized mode function of the rail is applied, and the partial differential equation of the track is converted into a set of ordinary differential equations: Combining equations (7) and (6), the vibration response of the rail can be solved. Applying the obtained rail response to the vehicle dynamics model, the wheel-rail contact force in vertical direction can be solved using the Hertz contact model and the force in tangential direction can be calculated by the FASTSIM method provided in the SIMPACK platform.
x w4 x w2 e typical slab track structure is shown in Figure 2. It mainly includes two 60 kg/ m rails on the left and right, a fastener system that mitigates the impact of the wheel and rail, a high-stability track plate, the cement asphalt (CA) mortar layer, and the roadbed. In the simulation model, the fastener and mortar layer are simulated by a spring-damper unit. e rail is considered to be a Timoshenko beam to simulate the vibration behaviour of the rail, which has already been described. e elastic vibration of the slab track is simulated by the finite element method. e impact force caused by the wheel defects or the irregularity of the rail is transmitted to the slab through the rails and fasteners, causing vibration of the slab track. In turn, the deformation of the slab track affects the displacement of the rail, thus affecting the wheel-rail contact force, which cannot be ignored. Considering the vertical force F szi coupled with the fastener and the vertical force F pi provided by the mortar layer, the finite element model of the slab track is established. e vertical displacement response z s of the slab can be expressed by the following equation of motion: where Z i is the generalized modal coordinates vector corresponding to the i th mode, ϕ i is the i th mode shape function of the track slab, and N m is the total number of vibration modes considered in the slab model. In the model, a 100-meter long track section is considered, and each slab track is 6.5 meters long, 2.5 meters wide, and 0.3 meters thick. e finite element model is discrete using Solid 185 elements, each of which is discretely 45,000 cells. e calculation is performed on the ANSYS platform and set to a free boundary condition. In order to analyse the displacement response of the slab track, the first 30-order vibration modes of the slab track are considered, and the natural frequency is up to 594.74 Hz. e partial modal shape of the slab track is shown in Figure 3.

Vehicle-Track Rigid-Flexible Coupling Dynamics Model.
In this section, a multibody dynamics model of a high-speed railway vehicle is established, in which the frame, the wheelset, and the axle box are modelled as the elastic body. e modelling process of the vehicle system is not described in detail here, which has already been built in literature [3]. e rigid-flexible coupling dynamics model of the highspeed railway vehicle is shown in Figure 4. e wheel polygonization has a significant impact on the track and the vehicle system. erefore, it is not enough to consider the flexibility of the wheelset. e influence of the dynamic response of the track must also be considered. e key to the coupling of the vehicle system and the track is the interaction between the flexible wheelset model and the track model. erefore, the cosimulation calculation method is applied to the study of the wheel polygonization, as shown in Figure 5. e dynamic wheel-rail force calculated in the dynamics model is used as the input factors to evaluate the dynamic performance of the track. And the track displacement response is then integrated into the vehicle model with the SIMAT (SIMPACK-MATLAB) cosimulation interface.

Introduction of the Test Rig.
A dynamic stress test was conducted on the high-frequency vibration test rig at the State Key Laboratory of Traction Power, Southwest Jiaotong University. e schematic diagram of the test rig is shown in Figure 6. It is a high-speed rotary test rig driven by a 200 kW motor.
e maximum speed of the roller wheel is up to 4200 rpm with the diameter of 600 mm, and the hydraulic cylinder is loaded with counterweight above the bogie. e two hydraulic cylinders can apply a maximum load of 100 tons. A gearbox transmission is used between the motor and the roller wheel. By manually machining harmonic waves on the roller wheel tread, the wheel polygonization or track irregularities can be simulated. In the test, one wheelset of the bogie is driven by the motor, while the other is fixed. e roller wheel is processed into a 13 th order polygon with a depth of 0.05 mm. e wheel diameter of the test bogie is 920 mm. e wave length on the roller wheel is the same as the wave length of the 20 th order polygon on the wheel of the test bogie.
In order to study the influence of the wheel polygonization on the axle stress, the axle dynamic load data were collected and tested on the test rig. And the wireless telemetry data acquisition system is applied. Figure 7 shows the wireless data acquisition system. It consists of an acceleration sensor (or strain gauge), strain conditioning module, ICP conditioning module, thermocouple signal conditioning module, power supply, and signal transmitter. e working principle is that the acceleration sensor or the strain gauge is attached to the device under test, and the sensor is connected to the encoder through a corresponding conditioning module; then, the encoder wirelessly transmits the collected signal to the user through the signal transmitter. e equipment on the axle is powered by the electromagnetic induction.

Data Processing.
e test speed ranges from 0 km/h to 320 km/h, and the time frequency analyses the result of the axle at the transitional arc of the wheelset as shown in Figure 8. From the overall time-frequency result, it can be seen that the stress is mainly composed of low frequency and high frequency. e low-frequency component is the rotating frequency of the axle, and the high-frequency part is the wheel polygonization passing frequency. Moreover, it can be seen that the amplitude of the low-frequency portion is heavier than that of the high-frequency portion, which indicates that the vibration energy of the low-frequency component is dominating.
e high-frequency and low-frequency components are analysed separately. As shown in Figure 8, the low-frequency band is only displayed along 0∼50 Hz. e main component of this frequency is the rotating frequency of the roller wheel, which varies with the running speed. As the high-frequency band in the figure, it can be found that the vibration energy     stands out at the frequency around 170 Hz, 400 Hz, and 600∼650 Hz. rough the modal test on the test rig, we can determine that 170 Hz is caused by the excitation of the modal of the test rig, which leads to resonance, but this is not the focus of this study and will not be discussed. Compared with the modal calculation result of the wheelset in Figure 9,   the resonance band around 400 Hz is caused by the wheelset second-order bending mode, which is calculated to be 393 Hz. e resonance band about 600∼650 Hz is caused by the 624 Hz resonance of the third-order bending mode of the wheelset. e experimental results show that under the polygonal excitation, the elastic vibration mode of the wheelset can be excited, which will cause the increase of the axle stress. In order to ensure the operation safety of the axle, it is very urgent to study the effect of the wheel polygonization on the axle stress.

Model Validation.
Under the condition of the wheel polygonization, the dynamic model is used to calculate the dynamic stress of the axle. And the dynamic stress obtained by the rig test is compared to verify the accuracy of the dynamic model. e dynamics model and the test result are compared on a same place of the axle with the same running speed.
e test axle stress result with the running speed of 250 km/h is selected, and the position is the transition arc of the wheel seat, where the stress is the largest. e test result is shown in Figures 10(a) and 10(c). e simulation condition is the same with the 20 th order polygon wave depth of 0.05 mm, and the stress result at the wheel seat is extracted, as shown in Figures 10(b) and 10(d).
As shown in Figures 10(a)  rough the partial enlargement of the high-frequency band, it can be found that there are two frequencies belonging to the polygon excitation, and the amplitudes are extremely close to 2 and 3. By comparing the simulation and experimental results, it is shown that the calculation results of the dynamic model have relatively high accuracy and credibility, which is suitable for the analysis conducted.

Influence of Wheel Polygon on Axle Stress
As described in Section 2, it can be determined that the frequency of the wheel polygonization can excite the axle resonance. is section conducts research on this issue. First, the axle stress under the wheel polygonization condition and the ideal wheel condition is compared, as shown in Figure 11.
By comparing with polygonal wheels and ideal wheels, it can be explained that without polygonization, the bending stress of the axle is mainly caused by the static load of the vehicle carbody and the frame, and the main frequency is determined by the wheel rotating speed. e stress under the wheel polygonization has more high-frequency components. e static load part is the basis, its amplitude is generally steady, and the amplitude of the high-frequency part determines the magnitude of the final stress. e amplitude of the high-frequency partial stress is related to many factors.
is section studies the changes of axle stress under the wave depth and the polygon order. Shock and Vibration 7

Wave Depth.
e maximum stress of the axle as a function of the wave depth of different wheel polygonization is shown in Figure 12. e wheel has a 15 th order polygonization and the running speed is 280 km/h. It can be seen that as the wave depth increases, the maximum stress increases significantly, and the wave depth of the wheel polygonization has a great influence on the stress of the axle.
With the 15 th and 20 th order polygonization on the wheel, the maximum stress of the axle varies with the running speed and the wave depth is studied, as shown in Figure 13. e wave depth is listed as 0.02 mm, 0.05 mm, 0.07 mm, 0.1 mm, 0.15 mm, 0.2 mm, and 0.3 mm. By comparing the relationship between the stress and different wave depths, it can be confirmed that the amplitude of the axle stress is greatly affected by the depth of the polygon wave. And the greater the wave depth, the greater the stress. e maximum stress has two peaks at low speed and high speed, respectively. e two stress peaks of the 15 th order polygonization appear at 60 km/h and 280 km/h, while the 20 th order polygonization appears at 40 km/h and 210 km/h. As the polygonization order increases, the stress peaks appearing speed decreases. e axle stress is 52 MPa in the  ideal wheel condition. When the 15 th order wheel polygon has a wave depth of 0.15; the axle stress is 121 MPa, which is 2.3 times that of the ideal wheel condition. erefore, the wheel polygon has a significant influence on the axle stress.

Polygon Order.
To further discuss the influence of the polygon order on the stress peak appearing speed, more simulations are carried out. e influence of the polygon and the running speed on the axle stress is analysed, as shown in Figure 14. With the increase of the polygon order, the first peak appearing speed varies a little. However, the second peak appearing speed increases significantly. e polygonization excitation frequency can be calculated as follows: where v is the running speed, n is the polygon order, and d is the wheel diameter. According to Figure 14 and equation (10), the frequencies of the two peaks at different speeds are calculated, respectively, and the result is given in Table 1. Although the polygon order and speed are different, it can be found that the frequencies corresponding to the two peaks of each order polygon are relatively concentrated. e first peak frequency is concentrated at 72∼86 Hz, and the second peak frequency is concentrated at 402∼415 Hz. Considering the modal calculation results of the wheelset, it is reasonable to believe that 82 Hz, 393 Hz, and 432 Hz is the first-order bending  Shock and Vibration 9 mode frequency, the second-order bending mode frequency, and the torsional mode frequency, respectively. e first peak frequency is likely to be the first-order bending mode of the wheelset, and the second peak frequency is the secondorder bending mode or the torsional mode. e fast Fourier transform method (FFT) is performed to analyse the second peak stress data of each polygon order (under 0.3 mm wave depth condition). e result is shown in Figure 15(a). e amplitude around 390 Hz and 430 Hz is obvious. Considering the modal analysis, the second peak is caused by the resonance of the second-order bending mode (393 Hz) and the torsional mode (432 Hz) under the excitation of the wheel polygonization. It can be seen that the amplitude of the resonant stress of the polygon near 400 Hz has exceeded the amplitude of the wheel rotational frequency, which indicates that vibration induced by the polygonization is the main component when the polygonization with a magnitude of 0.3 mm occurs.   Similarly, the FFT method is performed to analyse the first peak stress data, shown in Figure 15(b). It can be found that the stress composition of each order is divided into two parts, one is the lower frequency, which is the wheel rotational frequency. e other part is concentrated on a range near 80 Hz, which is caused by the first-order bending resonance.

Conclusion
To study the influence of the wheel polygonization on axle stress, the cosimulation calculation method is applied to establish a vehicle-track rigid-flexible coupling dynamics model of a high-speed railway vehicle. And a dynamic stress of the axle on the vibration test rig with the wheel polygonization was described and used to verify the accuracy of the model. At last, the influence of the wheel polygon on axle stress was analysed by the dynamic model. e main conclusions drawn from the study results are as follows.
(1) In the case of ideal wheel, the axle dynamic stress has only one frequency component. When the wheel polygonization appears, it will have a significant impact on the axle dynamic stress, that is, a highfrequency component induced by the wheel polygonization is superimposed. e larger the polygon wave depth, the larger the wheel axle stress amplitude. (2) When the frequency of the wheel polygonization is close to the first-order bending mode and the second-order bending mode of the wheelset, resonance will occur and the dynamic stress in the key position of the axle can increase sharply. (3) e appearance of the wheel polygonization can have an obvious influence on the axle strength, and in the case of resonance, the axle stress can increase sharply. erefore, the occurrence of the wheel polygonization should be prevented as soon as possible, especially to avoid the polygonization with the frequency consistent with the structure natural frequency. For instance, the vehicle running at 350 km/h is sensitive to the 12 th order polygon, and the vehicle with 300 km/h is sensitive to the 15 th order polygon. Once the wheel polygonization is tested, some measures need to be implemented immediately, such as the wheel reprofile.

Data Availability
e data used to support the findings of this study have not been made available because the data that have been used are confidential and owned by CRRC Co., Ltd.

Conflicts of Interest
e authors declare that they have no conflicts of interest.