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Abstract

Wheel polygon amplitude can greatly affect wheel-rail vibration and sound radiation. Based on multi-body dynamics theory, a vehicle-track rigid-flexible coupling dynamics model was established. According to the actual running wear condition of the wheel, the wheel-rail vibration response was calculated and analyzed (the order of wheel polygons is 20, and the polygon amplitude is 0.01/0.02/0.03/0.04 mm, respectively). Together with the finite element/boundary model of the wheel, the calculated wheel-rail force was used as an external incentive to analyze the effects of polygon amplitude on the time-frequency domain of wheel noise. The research results show that: when the polygon order is 20, with the increase of polygon amplitude, the wheel-rail vertical force and the acceleration of wheel, rail and track slab increase gradually. It’s also found that the rail acceleration is obviously more sensitive to the amplitude than the track slab acceleration, while the vertical displacement of the rail and track slab is less sensitive to the polygon amplitude. At the same amplitude, the closer to the wheel rolling line, the more obvious the sound pressure decreases with the increase of height. At different amplitudes, the sound pressure at different positions will increase with the rise of the polygon amplitude. The root mean square value of sound power increases gradually with the addition of amplitude: When the amplitude changes from 0.01 mm to 0.04 mm, the calculated sound power increases by 4.1 dB.

Keywords

Wheel polygon wheel-rail vertical force vibration acceleration sound radiation sound power

Introduction

With the increasing speed and mileage of trains, the polygon phenomenon of wheels is becoming more and more serious. High-order polygon will cause high-frequency vibration of the wheel-rail system, and generate strong wheel-rail noise,¹ which will seriously affect the normal life of residents near the operating line and the safety of surrounding buildings, and even adversely affect the planned and proposed railway lines.²

In recent years, scholars at home and abroad have conducted a lot of researches on wheel/rail high-frequency vibration and noise. J. Nielsen³ investigated different types of wheel out-of-roundness, and simulated the influence of wheel/rail contact force and vehicle/track response under the condition of wheel out-of-roundness. Wu Yue and Han Jian⁴ analyzed the relationship between wheel polygon wear and wheel-rail force, and took the time domain wheel-rail force as external incentive to analyze the impact of wheel polygon wear on the vibration response of bogie axle box and frame. Song Zhikun^5,6 used ANSYS and SIMPACK software to establish a dynamics model of vehicle rigid-flexible coupling system, and studied the influencing factors in the development process of high-speed train wheel polygon, and proposed the limit value of wheel polygon. However, the above researches only studied the influence of wheel out-of-roundness on the abnormal vibration of the vehicle and track, while the impact on noise hasn’t been analyzed in detail.

The researches on wheel/rail noise in foreign countries were conducted a little earlier, and most of them were based on the Remington^7,8 and Thompson^9,10 models. The European Railway Institute developed the TWINS noise prediction model.^11,12 So far, TWINS is still the most accurate model for the calculation of wheel-rail rolling noise; however, the nonlinearity factor of wheel-rail contact was not considered in TWINS model, which will greatly reduce the accuracy of noise prediction under high-order wheel polygon wear. Xiao Huijuan² used the method of combining finite element and boundary element, and applied the wheel-rail force in the time domain to solve the finite element vibration response. She used it as a boundary condition to analyze the time-frequency domain characteristics of track noise. However, the imposed wheel-rail force did not take into account the wheel non-circularity. Meanwhile, in order to deeply analyze the transmission characteristics of wheel noise, inspirations from the acoustic characteristics of shell structure,^13,14 cylinder structure,¹⁵ and plate structure¹⁶ can be used for references.

In order to explore the effect of wheel polygon on wheel/rail vibration and noise, a rigid-flexible coupling dynamic model of a certain type of train was established based on multi-body dynamics theory. The wheel-rail vibration response was calculated and analyzed under four types of actual wheel polygon wear. Together with the finite element/boundary element model of the wheel, the calculated wheel-rail force was used as an external incentive to analyze the influence of polygon amplitude on wheel noise in this paper.

Rigid-flexible coupling dynamics model

Wheelset flexibility

The HYPERMESH software is used to establish a finite element model of the wheelset (Figure 1(a)). The wheel diameter is $920 m m$ , the tread profile is S1002CN, the elastic modulus is $2.1 \times 10^{5}$ MPa, and the density is $7.85 \times 10^{3} k g / m^{3}$ , Poisson’s ratio is 0.3. In the modeling process, the wheel and the axle are regarded as integral parts without considering the interference fit relationship between the wheel and the axle. According to the selection rules of sub-degrees of freedom,¹⁷ the main degree of freedom node of the wheel set are selected, and sub-structure analysis in ANSYS are performed (Figure 1(b)), and a flexible body conversion file is generated. Then the conversion file generated by ANSYS Software is imported into SIMPACK Sofrware to generate wheelset flexible body (Fbi file) (Figure 1(c)).

Figure 1.

Wheelset model. (a) Finite element, (b) Principal degree, (c) Flexible bodymodel of freedom model model.

The high-order bending mode of wheelset is selected to verify the accuracy of the flexible body model in SIMPACK. Table 1 shows the comparison between the modal intrinsic frequencies calculated in ANSYS and those in SIMPACK. It can be seen that the modal natural frequency error is less than 3%, and the mode shape is the same, which verifies that the master degree of freedom nodes selected above are reasonable, which can meet the accuracy of the model and can effectively shorten the calculation time. Therefore, the flexible body generated in SIMPACK can meet the accuracy requirements.

Table 1.

Analysis of intrinsic frequency pairs of high-order bending modes of wheelsets.

Order	Mode frequency calculated from ANSYS	Mode frequency calculated from SIMPACK	Difference rate (%)	Wheelset vibration shape
25	581.770	585.737	0.667	Wheelset third-order bending
26	581.770	586.002	0.722	Wheelset third-order bending
27	648.110	654.740	1.013	Wheelset fourth-order bending
28	648.110	655.167	1.077	Wheelset fourth-order bending
30	831.503	848.736	2.030	Wheelset fifth-order bending

The main vibration shapes of the wheelsets are shown in Figure 2(a)–(c):

Figure 2.

High-order bending mode of wheelsets. (a) Third-order (b) Fouth-order (c) Fifth-order.

Track flexibility

The flexible rail Fbi file generating process is the same as that of the wheelset. The elastic modulus, density and Poisson’s ratio are determined the same way as the wheelset. The stretch length is 63 m. The nodes of the main degree of freedom at the rail head and rail bottom are selected, respectively, as the wheel-rail contact and fastener connection reference point in SIMPACK. The finite element model and the main degree of freedom analysis model are shown in Figure 3(a) and (b). Different from the wheelset import, the configuration file (FTR file) of the flexible track needs to be written. The FLEXTRACK module in SIMPACK software is used to read the configuration file to realize the import of the flexible track. The track end is fixed with large stiffness and damping.

Figure 3.

Rail model. (a) Finite element model (b) Principal degree of freedom model.

In consideration of the complexity of the model and the calculating time, the under-rail system only takes account of the track slab, and does not consider the CA mortar layer and the base plate structure. First a three-dimensional drawing of the track slab is made in CAD, of which the size is 5.6 × 2.5 × 0.2 m, the elastic modulus is $3.6 \times 10^{4}$ MPa, the density is $2.5 \times 10^{3}$ kg/m^3, and the Poisson’s ratio is set at 0.2. The hexahedral meshing is made in HYPERMESH. The main degrees of freedom nodes on the upper and lower surfaces of the track slab are selected as the reference points for connecting the fastener and the ground in SIMPACK. Then, the modal reduction analysis is carried out in ANSYS, and a flexible body is generated in SIMPACK. The finite element model and the main degree of freedom analysis model are shown in Figure 4(a) and (b).

Figure 4.

Track slab model. (a) Finite element model (b) Principal degree of freedom model.

Establishment and verification of vehicle/track coupling dynamics model

Based on the theory of multi-body dynamics, a multi-rigid vehicle-track coupling dynamic model of the EMU trailer is established in SIMPACK (Figure 5(a)). The vehicle includes one car body, two frames, four wheelsets, and eight axle boxes. The wheelset, frame, and car body each has 6 degrees of freedom, including longitudinal (X-direction), lateral (Y-direction), vertical (Z-direction), side roll, nodding, and shaking. Only the nodding degree of freedom of the axle box is considered, and the whole vehicle includes 50 degrees of freedom. Then the wheelset and track flexible body established above are imported to replace the rigid body and a rigid-flexible coupling dynamics model is established (Figure 5(b)), in which the fastener distance is set to 0.625 m. The calculation of wheel-rail normal force in SIMPACK adopts Hertzian nonlinear elastic contact theory.

Figure 5.

Vehicle/track coupling dynamics model. (a) Multi-rigid body model (b) Rigid-flexible coupling model.

The vehicle rail rigid-flexible coupling dynamic model is further verified by using the wheel rail vertical force data¹⁸ measured on the ground. The wheel rail vertical force of the EMU passing through the ground measuring point at a running speed of 300 km/h was tested, and the vertical force of the left and right wheel rails of the 1-position axle of No. Six carriage obtained from the test is shown in Figure 6.

Figure 6.

Measured wheel rail vertical force.

It can be seen from Figure 6 that the maximum peak value of wheel rail vertical force is 116 kN. Figure 7 shows the time history curve of wheel rail vertical force obtained through simulation by using vehicle rail rigid flexible coupling model. It can be seen that the maximum value of wheel rail vertical force obtained by simulation is 118.62 kN. Compared with the measured data, the wheel rail vertical force error is 2.2%, which verifies the reliability of the model and proves that the model can be used for simulation calculation under wheel rail coupling.

Figure 7.

Wheel rail vertical force simulation results.

Analysis of the effect of wheel polygon wear on the vibration characteristics of the wheel-rail system

Numerical model of wheel polygon wear

During the operation of high-speed trains, the wheel tread will have flat scar phenomenon including local abrasion or peeling. After a period of time, the scratched edges and corners of the wheel will be quickly rounded. Wheel polygon wear is a special form of wheel irregularities, which is a periodic irregularity on the wheel tread.¹⁸ In order to study the wheel-rail vibration caused by the wear of wheel polygon, the out-of-round and radial deviation of the wheel tread are converted into uneven displacement of the wheel-rail interface, which is subsequently input into the coupled dynamics model. One form of adding polygon wear to SIMPACK is to input the polygon wear of the wheel into the coupling dynamic model by using the radius deviation method, which is generally used to study the vibration response of the measured wheel wear (different wavelengths and amplitudes), as shown in Figure 8(a); The other form is to input the ideal harmonic function (fixed wavelength and amplitude) into the coupling dynamic model, which is generally used to study the vibration response at a single wavelength or amplitude, as shown in Figure 8(b).

Figure 8.

Polygon mathematical model of the wheel. (a) Measured wheel polygon mathematical model, (b) Ideal wheel polygon mathematical model.

The radial deviation of the wheel is expressed as $∆ r (x) = R (x) - R_{0}$ (1)In formula (1), $x$ is the circumference of the wheel, R(x) is the actual radius of the wheel, and $R_{0}$ is the nominal radius of the wheel’s rolling circle.

The following is the form of polar coordinates $∆ r (θ) = R (θ) - R_{0} (0 \leq θ \leq 2 π)$ (2)

In formula (2), θ is the corresponding angle on the circumference of the wheel.

The wheel radial deviation ∆r(θ) can be expanded into a Fourier series with multiple harmonics, namely $∆ r (θ) = \sum_{i = 1}^{N} A_{i} \sin (i θ + φ_{i})$ (3)

In formula (3), $A_{i}$ is the amplitude of the i-th harmonic; $φ_{i}$ is the corresponding phase.

The wheel radial deviation ∆r(θ) is periodically expanded and transformed into a time history of wheel-rail irregularities $∆ r (t) = \sum_{i = 1}^{N} A_{i} \sin (i (\frac{V}{R_{0}}) t + φ_{i})$ (4)

Assuming the running speed of the train is v(m/s), the nominal wheel radius is R, and the wheel has N-order polygon wear, then the excitation frequency of the vehicle is $f = \frac{N \times v}{2 π R} (H z)$ (5)

Excitation input of wheel polygon

By tracking detection of the wheel wear of a certain type of EMU, the measured data are obtained and converted through Fourier series to get the order amplitude diagram as shown in Figure 9. It can be seen that the harmonic wear of six sets of wheels are all dominated by the 20^th order. Figure 10 shows the wear amplitude of each group of wheels along the circumferential direction. The results show that the polygon amplitude of each wheel is between 0∼ 0.05 mm. Table 2 shows the detailed wear of each wheel. The two methods of polygon wear input to the dynamic model have already been introduced in the previous contents. In order to explore the influence of polygon amplitude on wheel-rail vibration and noise, the wheel polygon is input into the model with an ideal single-amplitude harmonic function to calculate the vibration response.

Figure 9.

Wheel polygon order diagram.

Figure 10.

Wheel polygon amplitude diagram.

Table 2.

Amplitude of the polygon wear order of each wheel.

wheel	Dominant order	Amplitude\mm
3rd car, 1st axle, right wheel	20	0–0.045
5rd car, 2st axle, right wheel	20	0–0.02
7rd car, 1st axle, right wheel	20	0–0.02
7rd car, 2st axle, right wheel	20	0–0.02
7rd car, 4st axle, right wheel	20	0–0.02
8rd car, 1st axle, right wheel	20	0–0.045

The line is supposed to be a straight line without track irregularities, and the vehicle speed is set to be the same as the actual moving speed, 300 km/h. It can be obtained from the above table that the wear amplitudes of several groups of wheels are between 0∼0.05 mm, so the wheel polygon is set to be 20-order, and the amplitudes are set with 0.01 mm, 0.02 mm, 0.03 mm, 0.04 mm, respectively. The vibration responses under four working conditions are calculated and analyzed.

Analysis of wheel-rail force under different working conditions

The polygonal wear of the high-speed EMU will have a greater impact on the wheel-rail force, and shorten the life of the vehicle and track components. Figure 11(a) shows the time domain values of wheel-rail force under four working conditions. The values in the time range of 0.1 s ∼ 0.2 s are analyzed. It can be seen that under the excitation of the 20^th-order wheel polygon, the wheel-rail force changes periodically with time, and the bigger the amplitude of the polygon, the bigger the wheel-rail force will be.

Figure 11.

Wheel-rail vertical force. (a) Time domain value. (b) Maximum value.

In order to more intuitively analyze the effects of the amplitude on the wheel-rail force, the maximum value of the wheel-rail force under different amplitudes is extracted. The statistics are shown in Figure 11(b). When the amplitude is 0.01 mm, 0.02 mm, 0.03 mm, and 0.04 mm, respectively, the corresponding maximum wheel-rail force will be 94 kN, 110 kN, 126 kN, and 148 KN, respectively. It can be obtained that with the linear increase of polygon amplitude, the wheel-rail force increases basically linearly, which is consistent with the conclusion drawn by the reference.¹⁸

A spectrogram is obtained by Fourier transforming the wheel-rail vertical force, as shown in Figure 12. It can be seen that the main frequencies of wheel-rail force at different amplitudes are all 576 Hz, which is principally caused by the 20th-order wheel polygon, and can be calculated by formula (5). Through the analysis in reference,¹⁹ this frequency is close to the third-order bending modal frequency of the axle of the wheelset, which causes the resonance of the wheelset and makes the wheel-rail force increase greatly. It can also be seen in the frequency domain that the wheel-rail force will become bigger with the increase of polygon amplitude.

Figure 12.

Wheel-rail vertical force spectrogram.

Wheel-rail vibration analysis under different working conditions

The wheel high-order harmonic wear will not only affect the wheel-rail vertical force, but also have a great impact on the vibration of the wheelset, rail and track slab. During the tracking test of the service vehicle, wheel-rail vertical force is difficult to measure, and there are large errors in the measured results. Therefore, the vibration response of the wheel and the rail can be used as an important target to evaluate the effect of the wheel polygon.

The position of the center of mass of the wheelset is selected as the observation point to analyze the vertical vibration acceleration of the wheelset, as shown in Figure 13(a). It can be seen that the vibration acceleration of the wheelset changes periodically during operation. As the amplitude of the polygon increases, the vertical vibration acceleration of the wheelset gradually increases. Under the four working conditions, the maximum vertical vibration acceleration of the wheelset is 43.2959 m/s², 63.551 m/s², 89.4095 m/s², and 104.2200 m/s², respectively. The spectrogram of vibration acceleration is obtained by Fourier transforming method, as shown in Figure 13(b). It can be seen that the main frequencies of the wheelset acceleration under different amplitudes are all 576 Hz. As discussed above, this is obviously caused by the 20th-order wheel polygon.

Figure 13.

Wheel vertical vibration acceleration. (a) Time domain value. (b) Spectrogram.

Figure14(a) and (b) are the curves of rail acceleration and track slab acceleration with time, respectively. It can be seen from the figure that the waveforms of time history curve of the vibration acceleration of the rail are very similar to those of the track slab, and both can reflect the process of wheel load. When the wheel load approaches, the vibration acceleration first changes from small to large, and reaches the maximum when the wheel passes through the observation point. Then the wheel moves away, the acceleration gradually drops to zero.

Figure 14.

Time domain of vertical vibration acceleration. (a) Vertical vibration acceleration of rail. (b) Vertical vibration acceleration of track slab.

From the acceleration time chart, it can be seen that as the amplitude of the polygon increases, the acceleration of the rail and the track slab also shows an increasing trend. In order to more directly analyze the effect of the amplitude on the acceleration, the maximum accelerations of the rails and the track slab under different amplitudes are extracted and listed in Table 3. When the amplitude increases from 0.01 mm to 0.04 mm, the maximum rail acceleration increases from 198 m/s² to 665 m/s², there is an increase of 467 m/s². When the maximum track slab acceleration increases from 11 m/s² up to 37 m/s², there is an increase of 26 m/s². It can be seen that the rail acceleration is an order of magnitude larger than the track slab acceleration. This is due to the attenuation of the energy transferred from the rail to the track slab via fasteners.

Table 3.

Maximum acceleration under different amplitudes.

Polygon amplitude: mm	Maximum rail acceleration: $m / s^2$	Maximum track slab acceleration: $m / s^2$
0.01	198.3180	11.1979
0.02	342.5870	19.6394
0.03	459.6820	27.1685
0.04	665.9850	37.7778

The amplitude of the polygon also has a certain effect on the displacement of the rail and track slab. Figure 15(a) and (b) show the vertical displacement of the rail and the track slab, respectively. It can be seen that the displacement of the wheel load also changes from small to large, reaches the peak value, and then gradually changes to 0. The maximum displacement statistics under each amplitude are shown in Table 4. When the polygon amplitude increases from 0.01 m to 0.04 m, the maximum vertical displacement of the rail increases from 0.494 mm to 0.523 mm, and the maximum vertical displacement of the track slab increases from 0.126 mm to 0.128 mm. Therefore, it’s very obvious that the polygon amplitude has little effect on the vertical displacement of the rail and track slab.

Figure 15.

Time domain of vertical displacement. (a) Vertical displacement of rail. (b) Vertical displacement of track slab.

Table 4.

Comparison of rail and track slab displacement under different amplitudes.

Polygon amplitude: mm	0.01	0.02	0.03	0.04
Maximum rail displacement: mm	0.494	0.504	0.513	0.523
Maximum track slab displacement: mm	0.126	0.127	0.127	0.128

Analysis of the effect of wheel polygon on the wheel sound radiation

When the vibration wave inside the wheel propagates to the surface of the wheel structure, it will cause the wheel surface to vibrate and generate radiated noise. According to the noise generation principle of the wheel structure, the sound radiation results of the wheel are obtained by using the wheel-rail force excitation mentioned above, combined with the finite element method and the boundary element method.

The establishment of wheel finite element/boundary element model

The multi-body dynamics model can only derive the vibration response of a single node. It is unable to derive the vibration of the entire wheel surface for acoustic radiation analysis. Therefore, this paper establishes a unilateral wheel finite element model, without considering the axle, and imposes full constraints on the inner boundary of the wheel hub, as shown in Figure 16. The material parameters of the straight web wheel with an outer diameter of 920D are consistent with the wheelset parameters established above.

Figure 16.

Wheel finite element model.

In order to verify the correctness of the finite element model, a unit radial excitation is applied at the nominal contact point to calculate the displacement admittance of the finite element model, as shown in Figure 17, which is basically consistent with the displacement admittance result of reference.²⁰ It can be seen from Figure 17 that there are multiple peaks in the displacement admittance at the excitation point of the wheel, most of which are concentrated in the range of 1000 Hz ∼ 5000 Hz. These displacement admittance peaks are caused by the self-vibration of the wheel, which indicates that the self-vibration frequency of the wheel is mainly in the calculated frequency range, that is, many vibration shapes with radial displacement components are excited in the radial direction.

Figure 17.

Wheel displacement admittance.

The acoustic radiation of the wheel is analyzed by the boundary element method, and the acoustic boundary element model of the wheel is established. In order to prevent the sound leakage caused by the wheel hub hole during the calculation process, this paper uses an additional unit to plug the wheel hub hole, as shown in Figure 18. When establishing the boundary element model, the element length must meet the conditions of formula (6) $L \leq \frac{c}{6 f_{\max}}$ (6)

Figure 18.

Boundary element model of wheel.

In formula (6), $L$ is the unit length, $c$ is the speed of sound in the air, the value is 340 m/s, and $f_{\max}$ is the highest frequency calculated by the model. The maximum frequency calculated for the wheel in this paper is less than 5000 Hz. Therefore, when the unit size of the wheel is 11 mm, it can meet the calculation requirements.

Based on the results of the admittance response calculated above, the sound radiation power of the wheel is calculated to verify the correctness of the boundary element model. Figure 19 shows the calculated sound power of the wheel, which is basically consistent with the calculation results in reference.²¹ The sound power calculation result in this paper is slightly larger because the wheel diameter in this paper is 920 mm, while the wheel diameter in the reference is 915 mm.

Figure 19.

Sound power of wheel under unit load.

The effect of polygon wear on sound radiation

The time-history wheel-rail force under the 20^th wheel polygon wear and different amplitudes has been obtained. The wheel-rail force is used as an external excitation and applied to the nominal contact point of the wheel to analyze the transient vibration of the wheel. Then, the vibration response result of the wheel surface is used as the boundary condition to analyze the acoustic radiation in Virtual.lab software. The position of the plane sound field is 7.5 m, 15 m, 22.5 m, and 30 m away from the center of the wheel, and the size is 60 m × 60 m, as shown in Figure 20, which are represented by green, red, blue, and purple color bars, respectively. When the wheel is vibrating, the real-time characteristics of the sound field can be obtained.

Figure 20.

Planar sound field model.

In order to study the vertical distribution characteristics of the sound field, the sound radiation under a polygon amplitude of 0.03 mm is taken as an example, and the sound pressure time history results of the measuring points at different heights in the sound field are extracted for analysis. The observation field points are 9 m, 12 m, 15 m, and 18 m, respectively, above the center of the wheel.

Figure 21(a)–(d)are the time history curves of radiated sound pressure at the following heights: 7.5 m, 15 m, 22.5 m, and 30 m from the wheel rolling line, respectively. The propagation process of sound pressure can be seen from the figure above. From Figure 21(a), it can be seen that the sound pressure arrives at the position that is 9 m above the wheel center at 0.03 5s, 12 m above the wheel center at about 0.04 s, 15 m above the wheel center at about 0.04 s, and 18 m above the center of the wheel at about 0.06 s. At the same time, it is easy to see that the sound pressure waveforms at different positions and heights are basically the same.

Figure 21.

Time course of wheel sound radiation with amplitude of 0.03 mm. (a) 7.5 m away from the wheel rolling line. (b) 15 m away from the wheel rolling line. (c) 22.5 m away from the wheel rolling line. (d) 30 m away from the wheel rolling line.

The root mean square value of sound pressure at different positions and different heights are shown in Figure 22. It can be seen that the sound pressure decreases linearly with the height at the position 7.5 m away from the wheel rolling line, and when the height exceeds 15 m, the sound pressure starts to be lower than that at the position 15 m away from the wheel rolling line, where the trend of sound pressure decreasing with height is slightly eased; the sound pressure basically does not change with the height at both the position that is 22.5 m and 30 m from the wheel rolling line; it can be concluded that the closer it is to the wheel rolling line, the more obvious the sound pressure decreases with the increase of height.

Figure 22.

Root mean square value of sound pressure at different positions.

In order to study the effect of polygon amplitude on sound radiation, the field points of different positions with the same height of 9 m from the center of the wheel are taken as observation points. Figure 23(a)–(d) are the time-history curves of radiated sound pressure under different polygon amplitudes at positions that are 7.5 m, 15 m, 22.5 m, and 30 m away from the wheel rolling line, respectively. It can be seen that the sound pressure waveform is basically the same, and the process of sound pressure propagating is obvious. The sound pressure reaches the position that is 7.5 m away from the wheel rolling line at 0.03 s, 15 m away from the wheel rolling line at about 0.05 s, 22.5 m away from the wheel rolling line at about 0.07 s, and 30 m away from the wheel rolling line at about 0.095 s.

Figure 23.

Time history of wheel sound radiation at a height of 9 m from the center of the wheel. (a) 7.5 m away from the wheel rolling line. (b) 15 m away from the wheel rolling line. (c) 22.5 m away from the wheel rolling line. (d) 30 m away from the wheel rolling line.

The statistics changes of root mean square value of sound pressure with amplitude at different positions are shown in Figure 24. It can be seen that the sound pressure at different positions from the wheel rolling line rises with the increase of the polygon amplitude, and the growth trends are basically the same, showing a linear increase.

Figure 24.

Root mean square value of sound pressure at different positions.

By taking the polygon amplitude of 0.01 mm as an example, the sound pressure value at 7.5 m away from the wheel rolling line is 61.336 dB, the sound pressure value at 15m is 60.6597 dB, and the sound pressure value at 22.5 m is 59.5171 dB, and the sound pressure value at 30 m is 58.0528 dB. The sound pressure decreases by 0.6763 db, 1.1426 db, and 1.4643 db at every 7.5 m, respectively. It can be concluded that at the same height, the farther away it is from the sound source, the faster the sound pressure declines.

In order to explore the characteristics of sound radiation in frequency domain, the vibration response results obtained in the previous section are converted with DSP power spectrum to analyze the effect of polygon amplitude on sound power. Figure 25 shows the sound power of the wheels under different amplitudes. It can be seen that the waveforms of the sound power under different amplitudes are basically the same. They all increase with the rise of frequency within 0–200 Hz, and increase instantaneously at 576 Hz. This is due to the 20th order wheel polygon. The wheel vibration intensifies at this frequency, resulting in aggravated noise.

Figure 25.

Sound power at different amplitudes.

The root mean square values of sound power at different amplitudes are shown in Figure 26. It can be seen that as the amplitude increases, the root mean square value of sound power gradually increases. When the amplitude is 0.01 mm, the root mean square value of sound power is 44.7 dB, and when the amplitude is 0.04 mm, the root mean square value of sound power is 48.8 dB, with an increase by 4.1 dB.

Figure 26.

Sound power RMS at different amplitudes.

Conclusions

Based on the theory of multi-body dynamics, a rigid-flexible coupling dynamic model is established, which includes flexible wheelsets, flexible rails and track slab. Four actual polygon amplitudes are selected to study and analyze their effects on wheel-rail vibration and sound radiation.

(1) When the vehicle speed is 300 km/h and the polygon order is 20, with the increase of the polygon amplitude, the wheel-rail vertical force, and the acceleration of the wheel, steel rail and track slab gradually increase too. The acceleration of the rail is an order of magnitude larger than the acceleration of the track slab. The sensitivity of rail acceleration to amplitude is significantly higher than that of track slab acceleration, and the sensitivity of vertical displacement of rail and track slab to polygon amplitude is small and basically unchanged.

(2) The frequency spectrum analysis of the wheel-rail vertical force and the wheel acceleration show that both peak values are generated near 576 Hz. This is because the excitation frequency is 576 Hz when the vehicle speed is 300 km/h and the polygon order is 20, which is close to the third-order bending mode of the wheelset, and the resonance is caused accordingly.

(3) In the time-domain boundary element analysis, the propagation process of sound pressure can be seen. When the polygon amplitude is constant, the closer it is to the wheel rolling line, the bigger the field point sound pressure will be. At the same time, the closer it is to the wheel rolling line, the more obviously the sound pressure decreases with the increasing of height. When the distance away from the center of the wheel is constant, the farther away it is from the sound source, the faster the sound pressure attenuates.

(4) The waveforms of the sound power under different amplitudes are basically the same. They all increase with the rise of frequency within 0–200 Hz, and increase instantly at 576 Hz. This is due to the reason of 20th-order wheel polygon. As the amplitude increases, the root mean square value of the sound power gradually increases: When the amplitude changes from 0.01 mm to 0.04 mm, the calculated sound power increases by 4.1 dB.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This research was funded by the National Natural Science Foundation of China (No.52075032,No.52027807),China National Railway Group Science and Technology Program grant (N2022J009) and China Academy of Railway Sciences Corporation Limited Science and Technology Program grant (No. 2021YJ269,No.2022YJ280).

ORCID iD

Zhi-kun Song

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Influences of wheel polygon amplitude on wheel-rail vibration and sound radiation

Abstract

Keywords

Introduction

Rigid-flexible coupling dynamics model

Wheelset flexibility

Track flexibility

Establishment and verification of vehicle/track coupling dynamics model

Analysis of the effect of wheel polygon wear on the vibration characteristics of the wheel-rail system

Numerical model of wheel polygon wear

Excitation input of wheel polygon

Analysis of wheel-rail force under different working conditions

Wheel-rail vibration analysis under different working conditions

Analysis of the effect of wheel polygon on the wheel sound radiation

The establishment of wheel finite element/boundary element model

The effect of polygon wear on sound radiation

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References