Hybrid Numerical Analysis Models and Experiment Research for Wheel–Rail Noise of Urban Rail Vehicle

Abstract

For urban rail vehicles operating at speeds ranging from 60 to 250 km/h, the dominant source of radiated noise is the wheel–rail interaction. Finite element modal analysis was conducted on the wheelset, rails, and track slab. A multibody dynamics model under straight-line condition was established. It was a rigid–flexible coupling dynamics model, including the rigid vehicle body, flexible wheelsets, flexible rails, and flexible track slabs. Dynamic simulation calculations were carried out in this model to obtain the wheel–rail forces. The finite element and boundary element models of wheels and rails were established using simulation software to obtain the results of wheel–rail noise. The sound pressure levels on the surfaces of wheels and rails were calculated under the operating conditions of 120 km/h, 140 km/h, 160 km/h, and 200 km/h in the straight-line condition. The variation law of the frequency distribution of wheel–rail noise with the change in speed was obtained. The variation fitting function of wheel–rail noise SPL with speeds was obtained. Within the speed of 200 km/h, as the speed increased, the total value of wheel–rail SPL basically shows a linear growth. The simulation analysis results were compared with the experiment results. It indicated that the simulation results were reasonable. The simulation models are of great significance for the noise prediction in train design and manufacturing.

1. Introduction

In densely populated metropolises, convenient urban rail transit systems have gained increasing popularity. During train operations on tracks, the vibrations of the vehicle, track, and slab structures interact and mutually influence each other, forming a complex train–track–slab coupled dynamic system [1,2,3].

For rail vehicles operating at speeds ranging from 60 to 250 km/h, the dominant source of radiated noise is the wheel–rail interaction [4]. According to the generation mechanisms, wheel–rail noise can be categorized into three types: rolling noise, impact noise, and curve squeal [5]. Additionally, based on its radiation sources, wheel–rail noise can be further divided into contributions from the wheel and the rail [6]. Among these, wheel–rail rolling noise arises from the excitation of short-wave irregularities on the rail surface, which induces wheel–rail vibrations that propagate through the air as sound energy [2,7].

The train operation’s force can cause the track’s vibration, while the track simultaneously exerts dynamic forces on the vehicle [8]. This reciprocal interaction between the train and the track constitutes a dynamic coupling relationship, termed train–track dynamic coupling [9]. With the advancement of simulation technologies, computational modeling has provided effective solutions for addressing such coupling problems. By investigating the vibration characteristics of wheels and track structures during train operation, a three-dimensional coupled vibration simulation model of an urban rail train-track–slab system has been established based on multibody system dynamics and the Finite Element Method [10,11]. These models enable systematic analysis of wheel and track vibrations during train operations, thereby providing a rational basis for vibration and noise reduction design in urban rail transit systems [12,13].

The wheel–rail surface roughness is introduced into the wheel–rail contact system as irregularities. At the contact patch, the roughness undergoes the contact patch filtering effect, generating dynamic excitation for the system. It is termed as wheel–rail combined roughness excitation [14]. This combined roughness subsequently generates wheel–rail interaction forces through the mobility of the wheel, contact interface, and rail, referred to as wheel–rail interaction [15]. Under the excitation of these interaction forces, the wheel and rail produce dynamic responses manifested as wheel vibrations and rail vibrations [16,17]. The rail vibrations further transmit a portion of their energy to the foundation structure through the under-rail slabs, inducing vibrations in the foundation structure. These structural vibration energies propagate outward via the air medium, resulting in acoustic radiation from the wheel, rail, and foundation structure [18,19]. These radiation components form the acoustic field response of wheel–rail rolling noise in the external environment. It is necessary to control wheel–rail vibration acoustic radiation [20].

For train–track dynamic analysis, rigid–flexible coupling models are predominantly employed. However, due to the large number of degrees of freedom (DOFs) in flexible bodies, such models involve extensive nonlinear computations, consuming significant computational resources and time [21]. To address the complexity of rigid–flexible coupling simulations, the substructure analysis method is applied [22]. This approach constructs a superelement matrix for repetitive units within the flexible body, which can be reused iteratively during simulations, reducing computational time and memory usage. The analytical workflow includes model establishment, material property assignment, mesh generation, constraint coupling setup, modal analysis, determination of main DOFs, and superelement matrix generation [23].

Noise analysis methodologies primarily include three approaches: the Finite Element Method (FEM), the Boundary Element Method (BEM), and the Statistical Energy Analysis (SEA). For solving noise radiation from wheel–rail systems, the BEM offers the highest accuracy [24]. The BEM transforms the boundary value problems of partial differential equations in elastomechanics into boundary integral equations, integrating the mesh discretization techniques of FEM. Consequently, the BEM exhibits distinct advantages in solving mid-to-low frequency noise radiation problems. As a numerical computation method, BEM is categorized into direct and indirect formulations. Both approaches are based on the Helmholtz wave equation but differ in their solution strategies. The direct formulation is applicable only to internal or external acoustic fields, whereas the indirect formulation simultaneously resolves both internal and external fields [25]. The acoustic problem addressed in this study is confined to the external acoustic field of the wheel–rail system, thus, the direct formulation is adopted for noise simulation.

The aim of this work was to carry out a numerical simulation for wheel–rail noise and to verify the digital model obtained by comparing it with experimental results. Based on the multibody dynamic modeling, the rigid–flexible coupling dynamics analysis of vehicles, rails and track slabs was carried out. The lateral and vertical wheel–rail forces under the operating conditions of 120 km/h, 140 km/h, 160 km/h, and 200 km/h were obtained in Section 3. The FEM and BEM models of wheels and rails were established to obtain the wheel–rail vibration and noise in Section 4. The simulation analysis results were compared with the experiment results from wheel–rail noise experiments conducted at 200 km/h in Section 5.

This study has the following contributions. Firstly, these hybrid numerical analysis models, which include rigid–flexible coupling dynamic model and wheel–rail noise model, can be applied in the prediction and research on wheel–rail dynamics and noise. It provides the theoretical basis and method for studying the influence on wheel–rail noise. The proposed methodology in this paper was based on the system dynamics of vehicle, wheel, rail and fundament. The influence of wheel–rail modal vibration on wheel–rail noise was considered in the entire system. The analysis of wheel–rail noise was not limited to the reaction between wheel and rail. It included the influencing factors of vehicles. Secondly, simulation results can be used as references for noise design and control of urban rail vehicles. The findings are of great significance for the noise control in train operation.

2. Theoretical Analysis

The wheel–rail interaction was considered in the system of train and rail. The method of rigid–flexible coupling dynamics was introduced and used to establish the train–rail vibration model. Then, the BEM for solving wheel and rail’s acoustic radiation was introduced. At last, the flowchart of simulation method process in this paper was given.

2.1. Rigid–Flexible Coupling Dynamics

When analyzing substructure responses, the total displacement or stress component at a specific point within the substructure is given by

$$u=u_0+\left[L_u^R\right]\Delta u^R$$

(1)

$$\sigma ={\sigma }_0+\left[L_{\sigma }^R\right]\Delta u^R$$

(2)

In Equations (1) and (2), $\left[L_u^R\right]$, and $\left[L_{\sigma }^R\right]$ represent the transformation matrices between the retained DOFs of the substructure and its displacement (stress) components.

The internal force vector of the substructure can be expressed as [6]

$${\bar{I}}^R=[\bar{M}]{\ddot{u}}^R+[\bar{D}]{\dot{u}}^R+[\bar{K}]\Delta u^R$$

(3)

where, ${\ddot{u}}^R$, ${\dot{u}}^R$, and $\Delta u^R$ denote the acceleration, velocity, and displacement variables of the retained DOFs, respectively; $[\bar{M}]$, $[\bar{D}]$, and $[\bar{K}]$ are the reduced mass, damping, and stiffness matrices of the substructure.

Since the substructure response is linear, the virtual work of the substructure is as follows [26]

$$\delta W=\left[\begin{array}{cc}\delta u^R& \delta u^E\end{array}\right]\left(\left[\begin{array}{c}\Delta P^R\\ \Delta P^E\end{array}\right]-\left[\begin{array}{cc}{K}^{\mathrm{RR}}& K^{\mathrm{RE}}\\ K^{\mathrm{ER}}& K^{\mathrm{EE}}\end{array}\right]\left[\begin{array}{c}\Delta u^R\\ \Delta u^E\end{array}\right]\right)$$

(4)

where $\Delta P^R$ and $\Delta P^E$ represent the nodal forces acting on the substructure during structural loading, excluding self-equilibrated preloads from the substructure.

$$K=\left[\begin{array}{cc}{K}^{\mathrm{RR}}& K^{\mathrm{RE}}\\ K^{\mathrm{ER}}& K^{\mathrm{EE}}\end{array}\right]$$

(5)

where $K$ is the reduced stiffness matrix.

The internal DOFs $u^E$ exist only within the substructure. Thus, the conjugate equations for $\delta u^E$ in the substructure are fully balanced:

$$\Delta P^E-K^{\mathrm{ER}}\Delta u^R-K^{\mathrm{EE}}\Delta u^E=0$$

(6)

The displacement variables of the internal DOFs $\Delta u^E$ can be expressed in terms of the retained DOF displacements $\Delta u^R$ as follows:

$$\Delta u^E={\left(K^{\mathrm{EE}}\right)}^{-1}\left(\Delta P^E-K^{\mathrm{ER}}\Delta u^R\right)$$

(7)

For static analysis of the substructure, the reduced stiffness matrix is

$$[\bar{K}]=K^{RR}-K^{RE}{\left(K^{EE}\right)}^{-1}\Delta P^E$$

(8)

The load cases acting on the substructure are represented by the following load vector:

$${\bar{P}}^R=\Delta P^R-\left(K^{RE}\right){\left(K^{EE}\right)}^{-1}\Delta P^E$$

(9)

From Equation (7), all substructure displacements and stresses can be derived from the computed boundary nodal displacements or substructure nodal displacements. Therefore, the substructure method is applicable to dynamic analysis of large-scale structures, as the order of the boundary stiffness matrix $\left[K^{EE}\right]$ is significantly lower than that of the original substructure stiffness matrix, drastically reducing computational complexity.

Based on the mass matrix $M$, stiffness matrix $K$, damping matrix $D$, and shape functions $U$ of the elastic structure’s finite element model, the modal mass matrix $\overline{M}$, modal stiffness matrix $\overline{K}$, damping matrix $\overline{D}$, and generalized force vector $\bar{p}$ can be derived as follows [9]:

$$\begin{gathered} \left\{\begin{array}{c}\begin{array}{c}\bar{M}=U^{T}MU\\ \bar{K}=U^{T}KU\\ \overline{D}=U^{T}DU\end{array}\\ \bar{p}=U^{T}p\end{array}\right. \\ U=\left(\begin{array}{cc}{u}_1& u_1\cdots u_l\cdots u_n\end{array}\right) \end{gathered}$$

(10)

where, $u_{\mathrm{l}}(l=\mathrm{1,2},\dots ,n)$ denotes the $l$ mode shape of the structure; $n$ is the number of modal DOFs; and $p$ represents the interaction forces between the wheel and rail.

For orthogonal shape functions, the natural mode shapes are orthogonalized static shapes. The equations of motion for the elastic structure can be simplified into $n$ second-order differential equations [27]:

$$\overline{m_l}{\ddot{z}}_l+\overline{d_l}{\dot{z}}_l+\overline{k_l}{z}_l={\bar{p}}_l$$

(11)

where, $\overline{m_l}$, $\overline{d_l}$, $\overline{k_l}$, and $\overline{p_l}$ $\left(l=\mathrm{1,2},\dots ,n\right)$ denote the modal mass, modal damping, modal stiffness, and generalized force corresponding to each mode shape $u_l$, respectively.

2.2. BEM for Acoustic Radiation Characteristics

The BEM for solving acoustic radiation employs the surface acoustic pressure and normal vibration velocity as boundary conditions. Assuming the distance from an arbitrary point Y in the acoustic field to a point X on the structural surface is r, the following equations were as follows [7]:

$${\nabla }^{2}p(r)+k^{2}p(r)=0$$

(12)

$${\displaystyle \frac{\partial p}{\partial n}}=-i\omega \rho v_n$$

(13)

$$\underset{r\to \infty }{lim}\left[r\left({\displaystyle \frac{\partial p}{\partial r}}-ikp\right)\right]=0$$

(14)

Here, Equation (13) represents the fluid–structure interface boundary condition, while Equation (14) corresponds to the Sommerfeld radiation condition for far-field acoustic waves.

By applying Green Function, the Helmholtz wave equation is transformed into the Rayleigh equation [28]:

$$p\left(Y\right)=i\omega \rho {\int }_S{G}_p\left(XY\right)v_{n}dS$$

(15)

where $G_p$ denotes the Green’s function for the half-infinite space, and $G_p={\displaystyle \frac{e_{ikr}}{4\pi r}}$.

Discretizing and solving the above equation yield the following acoustic field solution:

$$\left\{P\right\}=\left[D{]}_p\right[V{]}_n$$

(16)

where $\left\{P\right\}$ is the acoustic pressure vector at field point Y, $[D{]}_p$ is the coefficient matrix, and $[V{]}_n$ represents the normal vibration velocity on the surface.

2.3. Modeling and Analyzing Processes

The overall simulation process adopted in this paper is shown in Figure 1. It indicated an overall description of the simulation method process. The simulation modeling and calculation were carried out in accordance with this process in the following content.

Compared with traditional analytical methods, firstly, the boundary conditions were different. Secondly, the established model included the parameters of vehicle body, bogie, and suspension system, so different parameters can be adjusted based on this model for analysis in the future. Thirdly, the frequency range of previous analysis on vehicle dynamics generally did not exceed 2000 Hz. However, in this paper, to meet the high-frequency characteristics of noise analysis, the frequency range of dynamics was raised to 5000 Hz.

3. Dynamic Modeling of Rigid–Flexible Coupling

The analysis of wheel–rail vibration noise induced by urban rail transit lines involved a complex system comprising the train, rails, track slabs, and foundation. This study established a rigid–flexible coupling dynamic model of the urban train–track–slab system through multibody dynamics methods and techniques.

3.1. Flexible Body Modeling and Modal Analysis

The wheelsets, rails, and track slabs were modeled as flexible bodies. FEM models of the wheelsets, rails, and track slabs were developed in the software Abaqus for modal analysis and substructure analysis, generating data files containing geometric, mass, stiffness, damping, modal, and nodal coordinate information of each structure. These files were then imported into the multibody dynamics model of the urban train in the simulation software Simpack. Thereby the modeling of rigid–flexible coupling dynamic system can be completed.

The finite element models of the wheelset, rail, and track slab are illustrated in Figure 2. The wheelset was discretized using hexahedral solid elements with a mesh size of 0.03 m, comprising 16,960 elements, and its modal analysis frequency range was set to 50–5000 Hz. The rail, with a length of 18.6 m, was also modeled using hexahedral solid elements at a mesh size of 0.05 m, resulting in 39,432 elements, and its modal analysis frequency range was similarly defined as 50–5000 Hz. The track slab was simplified to a plate in the dynamic model, and the material could be considered as uniform concrete. Other complicated structures and heterogeneous materials, such as fastenings and under-rail vibration pads, were simplified into stiffness and damping parameters in the simulation model. The track slab was discretized with shell elements at a mesh size of 0.06 m, containing 4692 elements, and its modal analysis frequency range was specified as 50–5000 Hz.

Considering the influence of mesh independence, different numbers of elements were applied for modal analysis. The modal frequency values of each order were compared. According to the analytical results, it was found that the number of elements for the wheelset was required to exceed 10,000; the number of elements for the rails was required to exceed 20,000; and the number of elements for the track slabs was required to exceed 3000. Due to computing power and computing time, the final mesh number of wheelset was 16,960; the final mesh number of rail was 39,432; and the final mesh number of track slab was 4692.

The modal analysis results of flexible wheel rail are shown in Figure 3 and

Table 1. Modal analysis results of flexible wheelsets.

Modal Order	Modal Frequency/Hz	Modal Shape
2, 3	87.59	First-order bending of wheelset
4, 5	147.26	Second-order bending of wheelset
6	250.29	Reverse umbrella type
7, 8	296.63	Third-order bending of wheelset
9–12	366.79	Wheel bending deformation
13	370.54	Umbrella-shaped in the same direction
14, 15	585.51	Fourth-order bending of wheelset
16, 17	843.10	Wheelset fifth-order bending

.

The first-order vibration of the steel rail is shown in Figure 4. The pinned–pinned vibration of the rails would be introduced into the dynamic analysis.

3.2. Establishment of Rigid–Flexible Coupling Model

The modeling procedure of the used FE software and Simpack software is described in Figure 5.

According to the chart flow, the modal analysis of flexible body was carried out via Abaqus software. The flexible binary input files, generated by the flexible body modal analysis and substructure analysis, were imported into Simpack. These files were used when modeling in the Simpack software. The rail and track slab were connected via force elements to simulate the fastenings and under-rail pads in the Simpack software, while the lower parts of the track slabs were connected to the ground through force elements. The flexible wheelsets were replaced and connected with the rigid bogie via force elements.

The specific parameters of force elements used in the simulation model are listed in

Table 2. Parameters of dynamic simulation model.

Variable	Value	Variable	Value
Longitudinal stiffness of primary suspension	1300 kN/m	Longitudinal stiffness of secondary suspension	120 kN/m
Lateral stiffness of primary suspension	1300 kN/m	Lateral stiffness of secondary suspension	120 kN/m
Vertical stiffness of primary suspension	1280 kN/m	Vertical stiffness of secondary suspension	195 kN/m
Damping coefficient of primary suspension	5 kN·s/m	Damping coefficient of secondary suspension	20 kN·s/m

.

The rigid–flexible coupling dynamic model of the urban train, track, and slab was collectively formed, as illustrated in Figure 6.

The wheel–rail irregularities (roughness) in the dynamic simulation model were calculated using the US V-Class track irregularity spectrum. To meet the simulation requirements for wheel–rail noise in the frequency range of 50–5000 Hz, the upper and lower frequency bounds of the roughness spectrum were adjusted based on the train speed specified in the simulation. The Power Spectral Density (PSD) for vertical, longitudinal, and lateral rail irregularities (i.e., geometry irregularity, alignment irregularity, and gauge irregularity) are shown in Figure 7, respectively.

The time-domain vertical and lateral wheel–rail irregularities along the longitudinal distance of the track are depicted in Figure 8, respectively.

3.3. Results of Wheel–Rail Force

Based on the above model, dynamic simulation calculations were carried out. Due to the analysis frequency range of the wheel–rail noise being 5000 Hz, the time step of the simulation calculation was set to 0.0001 s. After the analysis of time independence, when the calculation time length was 0.8 s, the calculation result of the wheel–rail force reached stability. Considering the time cost of simulation calculation, the calculation time duration was selected as 1 s.

By using the US V-Class spectrum track irregularity standard, the frequency spectra of the vertical wheel–rail force and lateral wheel–rail force under the straight-line operating condition at the speed of 120 km/h were obtained. The corresponding frequency spectra of these forces are illustrated in Figure 9.

The above results indicated that for the straight-line operating condition, the vertical wheel–rail force was much greater than the lateral force. Vertical wheel–rail force was mainly concentrated in the frequency range below 1000 Hz. There were also certain amplitudes in the high-frequency range.

As mentioned above, the frequency spectra of the vertical wheel–rail force and lateral wheel–rail force at the speed of 140 km/h were obtained. These are illustrated in Figure 10.

The frequency spectra of the vertical wheel–rail force and lateral wheel–rail force at the speed of 160 km/h were obtained. These are illustrated in Figure 11.

The frequency spectra of the vertical wheel–rail force and lateral wheel–rail force at the speed of 200 km/h were obtained. These are illustrated in Figure 12.

As the speed increased from 120 km/h to 200 km/h, both the vertical wheel–rail force and the lateral wheel–rail force increased. The maximum amplitude of the vertical wheel–rail force increased approximately from 3 kN to 8 kN. The maximum amplitude of the lateral wheel–rail force increased approximately from 250 N to 470 N. The obtained wheel–rail force was used as excitation for noise analysis in the following steps.

4. Analysis of Wheel–Rail Noise

The wheel–rail force obtained in the previous section was applied as the excitation in this section. The noise analysis models of wheel and rail were established by using the BEM. The calculation results of wheel–rail noise were obtained.

4.1. Prediction Model for Wheel–Rail Noise

Based on the three-dimensional geometric models of the wheel and rail, FEM and BEM models were established. The geometric models of the wheel and rail were imported into the software and appropriately partitioned before mesh generation. To prevent acoustic leakage at the wheel hub during calculations, the wheel hub was filled with supplementary elements. The maximum computation frequency for the wheel and rail was 5000 Hz, with six elements per wavelength. Consequently, the BEM mesh size must not exceed 11 mm. To mitigate high-frequency vibration interference caused by end-reflected waves, the rail model length must exceed eight fastener segments. Therefore, a rail segment spanning ten fastener segments (i.e., 6 m) was selected for vibration simulation analysis, while the central four fastener segments (i.e., 2.4 m) were used for boundary element analysis of rail radiated noise. An infinite rigid acoustic boundary condition was applied 0.05 m below the rail head to simulate the reflection effect of the track slab. The boundary element radiation model of the wheel is shown in Figure 13a, and that of the rail is presented in Figure 13b.

4.2. Result Analysis

During the experiment measurement of exterior noise from rail vehicles, the measurement location depended on the train speed. For the operating speed below 200 km/h, the measurement position was generally set at 7.5 m from the track centerline and 1.2 m above rail top. This measurement position is called the standard point. In order to analyze the simulation results and compare them with the experiment results, the same position was selected in the simulation model. The simulation results of the sound pressure level (SPL) were selected for analysis at this position (7.5 m from the track centerline, 1.2 m above rail top).

The results of wheel–rail noise analysis under the rated speed of 160 km/h are in Figure 14 and Figure 15.

Under the 160 km/h straight-line operating condition, the wheel–rail interaction force generated surface SPL contour maps for the wheel and rail, as shown in Figure 16.

The calculated noise radiation contour maps for the wheel–rail system under the 160 km/h straight-line condition are also illustrated in Figure 17.

4.3. Comparison of SPL at Different Speed

Comparative analysis was conducted under various speed conditions. Under the excitation of wheel–rail dynamic forces, the 1/3-octave band spectra of wheel–rail-radiated SPL were calculated at speeds of 120 km/h, 140 km/h, 160 km/h, and 200 km/h. These spectra were obtained by superimposing the radiated SPL at the standard points (7.5 m from the track centerline, 1.2 m above rail level), as illustrated in Figure 18.

Comparison of wheel–rail-radiated SPL spectra at different speeds is illustrated in Figure 19. It indicated that as the speed increased, the values of SPL at each frequency increased accordingly. The SPL value was the largest within the frequency range of 800 to 1000 Hz. Within this frequency range, as the speed increases from 120 to 200 km/h, the SPL increases by approximately 6 to 7 dB(A). The control of wheel–rail noise needs to be targeted at this frequency range.

The variation trend of the total SPL with speed is illustrated in Figure 20.

When speed increased from 120 to 200 km/h, the total SPL increased about 8 dB(A). The variation relationship of wheel–rail noise SPL with speed was basically linear. The linear variation function was obtained through linear function fitting. The linear variation function was given as follows:

$$y=76.9+0.104x$$

(17)

The analysis of the above simulation results has been carried out. It indicated that the spectra of wheel–rail noise at different speeds were obtained. The variation law of the frequency distribution of wheel–rail noise with the change in speed was obtained. Then, the variation law of the total value of wheel–rail SPL of wheel–rail noise with speed was obtained by using a fitting function. Within the speed of 200 km/h, as the speed increased, the total value of wheel–rail SPL basically shows a linear growth.

5. Comparative Analysis of Test Results

Wheel–rail noise can be tested by installing microphone sensors at the standard points (7.5 m from the track centerline, 1.2 m above rail level). Experimental equipment is also illustrated in Figure 21.

The data acquisition device was Hottinger B & K acquisition module. The type of acquisition module was 3053, and the frequency range was 0–25.6 kHz. The type of microphone sensors used in experiments was 4189. The dynamic range of microphone sensors was 16.5–134 dB, and frequency range was 20–20,000 Hz. Based on the frequency ranges of the data acquisition device and microphone sensors, the sampling frequency of the experiment was set as 25.6 kHz. So, the frequency of data analysis reached 12.8 kHz, which could sufficiently meet the requirements of high-frequency components.

The experiment results of wheel–rail noise in 200 km/h operation were obtained. The simulation analysis results were compared with the experiment results from wheel–rail noise experiments conducted at 200 km/h. The results are illustrated in Figure 22.

It can be found that the calculated value is smaller than the measured result. This is because in the actual testing process, the measured noise results not only include wheel–rail noise but also aerodynamic noise, which cannot be excluded in the test due to the installation method of the sensor. The error may also be caused by the roughness of the wheel and track. Because the roughness of the wheel and track cannot be exactly the same as that used in the simulation calculation. The measured results also included the ground reflection effect. The frequency range of the maximum value of the calculated value and the measured value is basically consistent.

The organization and implementation conditions of train operating experiments are extremely complex and difficult. The cost of conducting the operating experiments is extremely high. The validation is limited to experimental data at a single operating speed of 200 km/h, even though simulations were performed for multiple speeds. However, based on the theoretical analysis and experimental results derived from a large number of previous studies, there was a relationship between the total SPL of wheel–rail noise and train’s speed. This relationship can be used to obtain the validation results at other speeds through linear interpolation and linear fitting [29,30]. The SPL of wheel–rail noise at other speeds, including 120 km/h, 140 km/h, and 160 km/h, was obtained. They were compared and verified with the simulation results. The verification results are shown in the following

Table 3. Total SPL of simulation result and validation result at different speeds.

Operating Speed	Simulation Result	Validation Result
120 km/h	89.62 dB(A)	91.5 dB(A)
140 km/h	91.46 dB(A)	93.4 dB(A)
160 km/h	93.25 dB(A)	95.2 dB(A)
200 km/h	97.93 dB(A)	99.8 dB(A)

.

The total SPL results of the simulation were verified. It can be seen that the error at each speed did not exceed 2 dB(A). The error was very small, and the variation tendency was consistent. The error’s reasons were discussed in the previous spectral analysis. Therefore, the validation to include experimental results of multiple speed operating conditions could considerably strengthen the reliability and generality of the conclusions.

6. Conclusions

Numerical analysis of wheel–rail noise for urban rail vehicles was carried out using hybrid numerical analysis models.

(1) Finite element modal analysis was conducted on the wheelset, rails, and track slab. As the flexible body structure in the multibody dynamics model, a multibody dynamics model under linear conditions was established, including the rigid–flexible coupling dynamics model of multi-rigid body vehicles, flexible rails, and flexible track slabs.

(2) Dynamic simulation calculations were carried out in the model to obtain the lateral and vertical wheel–rail forces under the operating conditions of 120 km/h, 140 km/h, 160 km/h, and 200 km/h in the straight line. In this paper, we only discussed the straight line’s condition. There are differences between the straight line’s model and the curve’s model for dynamic calculation. In the future, it is necessary to establish a curve’s model for dynamic calculation considering the wheel–rail noise through curves. The finite element model of curve rail needs to be introduced during establishing the dynamic model. In terms of the calculation process steps, the simulation model and calculation method of wheel–rail noise passing through the curve will be analyzed in the future.

(3) The finite element and boundary element models of wheels and rails were established using simulation software to obtain the prediction model of wheel–rail vibration and noise.

(4) The sound pressure levels on the surfaces of wheels and rails, as well as the noise radiation results of wheels and rails, were calculated under the operating conditions of 120 km/h, 140 km/h, 160 km/h, and 200 km/h. The SPL value was the largest within the frequency range of 800 to 1000 Hz. Within this frequency range, as the speed increases from 120 to 200 km/h, the sound pressure level increases by approximately 6 to 7 dB(A). When speed increased from 120 to 200 km/h, the total SPL increased about 8 dB(A). The variation function of wheel–rail noise SPL and speed was basically linear.

(5) The simulation analysis results were compared with the experiment results from wheel–rail noise experiments conducted at 200 km/h. It indicated that the simulation results were reasonable. The error was small, and the simulation model was reliable. In the future, other experiments can be conducted under different speeds to further verify the model and calculation results.