1. Introduction

To ensure a comfortable, safe, and efficient transportation, it was critical to scientifically assess the dynamic performance of vehicle-track systems under varying loads; consequently, a vehicle-track interaction system was formulated. The vehicle-track interactions are essentially nonlinear, while for lowering the modelling complexity, increasing the computational efficiency, and broadening the model application, it was rather popular to simplify the vehicle-track systems as a coupled and linearized system.

In the last decades, the models aiming at describing the vehicle-track interactions have experienced a rapid development from the earliest two-dimensionally vertical models [1–8] to more advanced three-dimensionally spatial ones. Concerning the system nonlinearity, Sun et al. [9] presented a 3D wagon-track system dynamics model, where a wagon system with 37 degrees of freedom (DOFs), a four-layer track, and the wheel-rail interfacial contacts were comprehensively considered; Zhai et al. [10] presented a theoretical framework and methodologies for characterizing the 3D nonlinear vehicle-track interactions; recently, Xu and Zhai [11, 12] made pioneering and the most advanced work in developing temporal-spatial stochastic models for train-track (bridge) interactions, in which the finite element method (FEM) and random theory were introduced to construct the track systems and to achieve the random vibration analysis, respectively.

With the development of nonlinear models, the methodologies are constantly developed to mathematically investigate the vehicle-track (bridge) interactions in the field of linearization modelling. See for instance, Zeng and Guo [13] made pioneering work on building a train-bridge time-dependent model by an application of the principle of a stationary value of total potential energy of dynamic system [14]; Yang and Wu [15] proposed a versatile element that is capable of considering various vehicle-bridge interaction (VBI) effects; however, the condition of wheel-rail separation is neglected in both of the these two models. Some other related researches can be consulted in [16, 17]. Moreover, Zhang et al. [18] regarded the vehicle as a spring-mass-damper system and the track as an infinitely long substructural chain consisting three layers of rail, sleeper, and ballast, and the vehicle and the tracks were coupled by linear springs; based on this developed model, the pseudoexcitation method (PEM) was introduced to achieve the high-efficient frequency characteristic analysis for system indices. Nguyen et al. [19] presented simulations for vehicle-substructure dynamic interactions and wheel movements, where the two compatibilities, i.e., force equilibrium and geometrical relation, are satisfied in the wheel-rail interfaces.

Additionally, the train-track dynamic simulations under transient conditions also show remarkable significance to railway maintenance, assessment, rolling fatigue, safety, etc. See for instance, Handoko and Dhanasekar [20] pointed out that the traction and braking forces are seldom considered in the practice of wagon dynamics simulation, although these forces will greatly modify the wheel-rail contact parameters and then the wheelset dynamics; Liu et al. [21] also studied the mechanism of wheelset longitudinal vibration by analysing the process of wheel/rail rolling contact; generally, the track defects can cause profound effects to the dynamics of the railway wagon; Zhang and Dhanasekar [22] noticed this problem and published a model for the dynamics of wagons subject to braking/traction torques on a perfect track by explicitly considering the pitch degree of freedom for wheelsets [23, 24] and extended this model for cases of lateral and vertical track geometry defects and worn railhead and wheel profiles; Grossoni et al. [25] examined the dynamic behaviour at a rail joint using a two-dimensional vehicle-track coupling model, where the influence of the number of layers and the number of elements between two sleepers and the beam model are investigated. Laterally, Zong and Dhanasekar [26] considered the gap between rail joints to account for thermal movement and to maintain electrical insulation for the control of signals and/or broken rail detection circuits; besides, railhead can provide high stresses due to the passage of heavily loaded wheels through a very small contact patch [27], and a multibody dynamic model was developed in [28] to accurately model and analyse the track dynamic behaviour in vicinity of rail discontinuities; recently Zong and Dhanasekar [29] provided a idea of simplifying the design of the IRJs consisting of only two pieces of insulated rails embedded into a concrete sleeper; Ling et al. [30] presented a formulation for a passive road-rail crossing involving stiffened edges of the raised road pavement to minimise the risk of failure of wheel-rail contact using a nonlinear three-dimensional multibody dynamics model; additionally, a series of work on impact derailment due to lateral collisions between heavy road vehicles and passenger trains at level crossings and the associated derailments had also been conducted in [31–34].

In the above presentations, though the nonlinear and linear vehicle-track interaction, models have been established by many researchers, while the applicability of the linear model and its comparison to the nonlinear model have rarely been reported. In this paper, a contribution, which is not beneficial to train-track dynamic to severe transient impact/rolling issues, was made to develop a highly practical and efficient model, and fully deriving the equation of motion for the linearized vehicle-track interactions. The coupling mechanism between a railway vehicle and the tracks in relation to wheel-rail interactions and system responses that are of great interest to railway engineers will be revealed.

2. WRI Element

In the whole vehicle-track model presented in Figure 1, the wheel-rail interactions play a key role in determining the dynamic trajectories of vehicle/track behaviours. In this part, a wheel-rail interaction (WRI) element will be developed to depict the wheel-rail interaction mechanism from aspect of linearization.

2.1. Displacement Vector

For a wheel-rail contact pair, as shown in Figure 2, the displacement vector, ${\tilde{X}}_{\text{wr,}k}$, with order of ${\tilde{T}}_{\text{wr}}\times 1$, can be expressed by

$$\begin{array}{}{\tilde{X}}_{\text{wr,}k}={\left[\begin{array}{cccccccccccccccc}{x}_{\text{w},k}& y_{\text{w},k}& z_{\text{w},k}& {\psi }_{\text{w},k}& L_{v\text{,}k}& R_{v\text{,}k}& L_{z\text{,}k}& R_{z\text{,}k}& L_{h\text{,}k}& R_{h\text{,}k}& L_{y\text{,}k}& R_{y\text{,}k}& L_{n\text{,}k}& R_{n\text{,}k}& L_{x\text{,}k}& R_{x\text{,}k}\end{array}\right]}^{\text{T}},\end{array}$$ ()

where the superscript “T” denotes the transposition of vectors or matrices; the subscript “k” denotes the kth wheel-rail contact element; x_w, y_w, z_w, and ψ_w are the longitudinal, lateral, vertical, and yaw motions of the wheel; by assuming the track irregularity as virtual displacement of the rail at the wheel-rail contact positions, L_v and R_v are the vertical irregularities of the rail at the left and right side, respectively, L_h and R_h are the lateral irregularities of the rail at the left and right side, respectively; moreover, L_z and R_z, respectively, denote the vertical displacement of the left- and right-side rail at the wheel-rail contact points; in the same manner, L_y and R_y denote the lateral displacement of the left- and right-side rail at the wheel-rail contact points, respectively; L_n and R_n denote the angle of torsion of the left- and right-side rail around the Y axis, respectively; and L_x and R_x denote the longitudinal displacement of the rails with respect to the left and right wheel-rail contact points, respectively.

In normal wheel-rail contacts, it is assumed that the vertical motion of the wheel is nonindependent and being totally determined by the rail vertical displacement, namely,

$$\begin{array}{}{z}_{\text{w},k}=\frac{1}{2}\left(L_{v\text{,}k}+R_{v\text{,}k}+L_{z\text{,}k}+R_{z\text{,}k}\right),\end{array}$$ ()

with

$$\begin{array}{c}{L}_{z\text{,}k}={\mathbf{D}}_{R,z\text{l,}k}{\mathbf{N}}_{R,z\text{,}k}^{\text{T}},R_{z\text{,}k}={\mathbf{D}}_{R,\text{zr,}k}{\mathbf{N}}_{R,z\text{,}k}^{\text{T}},\\ {\mathbf{D}}_{R,\text{zl,}k}=\left[\begin{array}{cccc}{W}_{R1,\text{l}}& {\theta }_{R1,\text{ly}}& W_{R2,\text{l}}& {\theta }_{R2,\text{ly}}\end{array}\right],\\ {\mathbf{D}}_{R,\text{zr,}k}=\left[\begin{array}{cccc}{W}_{R1,\text{r}}& {\theta }_{R1,\text{ry}}& W_{R2,\text{r}}& {\theta }_{R2,\text{ry}}\end{array}\right],\\ N_{\text{r},1}=132-{\left({\zeta }_{\text{w},k}/l_{\text{wr}}\right)}^2+{\left({\zeta }_{\text{w},k}/l_{\text{wr}}\right)}^3,\\ N_{\text{r},2}={\zeta }_{\text{w},k}\left[12-\left({\zeta }_{\text{w},k}/l_{\text{wr}}\right)+{\left({\zeta }_{\text{w},k}/l_{\text{wr}}\right)}^2\right],\\ N_{\text{r},3}=32{\left({\zeta }_{\text{w},k}/l_{\text{wr}}\right)}^2-{\left({\zeta }_{\text{w},k}/l_{\text{wr}}\right)}^3,\\ N_{\text{r},4}={\zeta }_{\text{w},k}\left[{\left({\zeta }_{\text{w},k}/l_{\text{wr}}\right)}^2-\left({\zeta }_{\text{w},k}/l_{\text{wr}}\right)\right],\\ {\mathbf{N}}_{R,z\text{,}k}=\left[\begin{array}{cccc}{N}_{\text{r},1}& N_{\text{r},2}& N_{\text{r},3}& N_{\text{r},4}\end{array}\right],\end{array}$$ ()

where D_R,zl and D_R,zr are the nodal displacement vector used to calculate the vertical displacement of the rail beam at the left and right side along the X axis; W and θ_y are the nodal vertical displacement and angular displacement of the node around the Y axis, in which the subscripts “l” and “r” denote the left- and right-side rails, respectively and the subscripts “R1” and “R2” denote the left and right nodes with respect to the rail beam element, respectively; ζ_w denotes the distance between the left node of the rail element and the wheel-rail contact position; and N_R,z is the Hermite polynomial interpolation used for Euler–Bernoulli beam.

When the wheels lose contact with the rails, the vertical motion of the wheel is independent, by assuming z_w in Equation (1) as an independent degree of freedom (DOF). In the numerical simulation, the inequalities below are applied to judge whether or not the wheelsets are contacting to the rails [35] (set the left wheel-rail contact side as an example):

$$\begin{array}{c}\frac{\left|\left(L_v+L_z\right)-z_{\text{w}}\right|}{\min \left(\left|L_v+L_z\right|,\left|z_{\text{w}}\right|\right)}<{\xi }_1,11-{\xi }_2\le \frac{L_v+L_z}{z_{\text{w}}}\le ,\mathrm{when}\left\{\begin{array}{c}{z}_{\text{w}}>0,\\ L_v+L_z>0,\end{array}\right.\\ 11-{\xi }_3\le \frac{\left|z_{\text{w}}\right|}{\left|L_v+L_z\right|}\le ,\mathrm{when}\left\{\begin{array}{c}{z}_{\text{w}}<0,\\ L_v+L_z<0,\end{array}\right.\\ \max \left(\left|L_v+L_z\right|,\left|z_{\text{w}}\right|\right)\le {\xi }_4,\mathrm{when}\left\{\begin{array}{c}{z}_{\text{w}}<0,\\ L_v+L_z>0,\end{array}\right.\end{array}$$ ()

where ξ_i, i = 1,2,3,4, is a small positive number depending on the analytical precision.

3. Vehicle-Track Interaction Model

As illustrated in Figure 1, a ballastless track system was formulated with a moving vehicle subjected to constant velocity. The railway vehicle was modelled as a four-axle mass-spring-damper system, which consisted of a car body, two bogie frames, four wheelsets, and two stage suspensions. The car body and bogie frames have 6 DOFs each including motions of longitudinal X, transverse Y, bounce Z, roll Φ, yaw ψ, and pitch β with respect to their mass centroids, respectively, while for the wheelsets, only the motions of longitudinal, transverse, vertical, and yaw were considered. The total number of DOFs of vehicle system was therefore 34. The vehicle proceeded with velocity v in the longitudinal direction.

In the track system, the rails were modelled as spatial Bernoulli–Euler beam supported by discrete viscoelastic supported with finite length, and the rail pads were assumed as linear spring-damper elements. Moreover, slab tracks were assumed to be thin plate elements connected to subgrade through cement asphalt mortar (CA mortar) layer, which was regarded as continuous plane springs and dashpots.

4. Numerical Examples

In the numerical examples, it is assumed that a railway vehicle runs with a constant velocity of 350 km/h on the ballastless slab-tracks. For constructing the vehicle-track model, a track segment element is taken between two adjacent rail pads [40]; moreover, a cyclic calculation model proposed by Xu et al. [41] is applied to increase the computational efficiency lowered by the high DOFs of the long track finite element model, and the boundary effect of the tracks can be reduced fairly well.

The physical parameters of the vehicle and the tracks can be referred in Tables 1 and 2. The track irregularity, as the excitation of the wheel-rail interactions, is obtained by a track irregularity probabilistic model [42] where the measured track irregularities from a high-speed line of China are used as the data sources.

4.1. Comparison between the Linear Model and the Nonlinear Model

To investigate the efficiency and feasibility of this presented model, a comparative analysis is conducted against the nonlinear model developed by Xu and Zhai [11]. In the nonlinear model, the three-dimensional wheel-rail nonlinear contacts are considered comprehensively.

Figures 3 and 4 show the comparisons on the lateral and vertical accelerations of the car body. As seen from these two figures, though there exists unavoidable discrepancy between the models on response amplitudes, the tendency of the time-domain responses of the linear model coincides well with that of the nonlinear model; moreover, the dominant frequencies of the car body accelerations drawn from these two models are almost the same. For the vertical acceleration of the car body, the absolute maximum values are 1.6 m/s² and 1.41 m/s²; with respect to the nonlinear model and linear model, the dominant frequencies are 1.17 Hz and 6.25 Hz. While for the lateral acceleration, the absolute maximum values of the nonlinear and linear model are 0.145 m/s² and 0.144 m/s², respectively; besides, the domain frequencies for these two models are 1.72 Hz, 5.46 Hz, and 10.94 Hz and are similar, but there exists a slight difference in spectral densities inevitably.

Instead of calculating the wheel-rail force using the nonlinear Hertzian contact theory [43]:

$$\begin{array}{}{F}_{\text{v}}=K{\delta }^{32\text{/}},\end{array}$$ ()

where K is the Hertz contact coefficient and δ is the wheel-rail relative elastic compression, the wheel-rail vertical force F_v in this paper is obtained by introducing D’Alembert’s principle:

$$\begin{array}{}\sum _i\left({\mathbf{F}}_i-m_i{\mathbf{a}}_i\right)\Delta {\mathbf{r}}_i=0,\end{array}$$ ()

where i is an integer indicating a variable; F_i is the total applied force on the i-th particle; m_i is the mass of the i-th particle; a_i is the acceleration, and Δr_i is the virtual displacement of the i-th particle; F_vconsists of the rolling force F_roll, vertical inertial force F_iner, and gravitational force F_grav, namely, F_v = F_roll + F_iner + F_grav.

As an example, the forces acting on the left side of the wheelsets can be written, respectively, as

$$\begin{array}{c}{\mathbf{F}}_{\text{roll},i}=\left(\frac{m_c{\ddot{y}}_c}{4}-\frac{I_{\text{cz}}{\ddot{\psi }}_c}{4l_c}\right)\frac{H_{\text{cb}}+H_{\text{bt}}+H_{\text{tw}}+R_w}{2b_0}-\frac{I_{\text{cx}}{\varphi }_c}{8b_o}+\frac{m_{\text{Gq}}{\ddot{y}}_{\text{Gq}}}{4b_0}\left(H_{\text{tw}}+R_w\right)-\frac{I_{\text{Gqx}}{\varphi }_{\text{Gq}}}{4b_0}-{\left(-1\right)}^{i+1}\frac{I_{\text{Gqz}}{\ddot{\psi }}_{\text{Gq}}}{2l_t}\frac{H_{\text{tw}}+R_w}{2b_0},{\mathbf{F}}_{\text{roll},i}=\left(\frac{m_c{\ddot{z}}_c}{4}-\frac{I_{\text{cy}}{\ddot{\beta }}_c}{4l_c}\right)+\frac{m_{\text{Gq}}{\ddot{z}}_{\text{Gq}}}{2}-{\left(-1\right)}^{i+1}\frac{I_{\text{Gq},y}{\ddot{\beta }}_{\text{Gq}}}{2l_t}+m_{w,i}{\ddot{z}}_{w,i},\\ {\mathbf{F}}_{\text{grav},i}=\frac{m_{w}g}{2},\end{array}$$ ()

where the subscript “i”, i = 1,2,3,4, denotes the ith wheelset; the subscript “Gq” represents the front bogie frame, which is only applicable for i = 1 and 2; when i = 3 or 4, the parameters “m_Gq,” “${\ddot{y}}_{\text{Gq}}$,” “${\ddot{z}}_{Gq}$,” “I_Gq,x,” “${\ddot{\varphi }}_{\text{Gq}}$,” “I_Gq,z,” “${\ddot{\psi }}_{\text{Gq}}$,” “I_Gq,y,” and “${\ddot{\beta }}_{\text{Gq}}$” should be substituted by “m_Gh,” “${\ddot{y}}_{\text{Gh}}$,” “${\ddot{z}}_{\text{Gh}}$,” “I_Gh,x,” “${\ddot{\varphi }}_{\text{Gh}}$,” “I_Gh,z,” “${\ddot{\psi }}_{\text{Gh}}$,” “I_Gh,y,” and “${\ddot{\beta }}_{\text{Gh}}$,” respectively, in which the subscript “Gh” represents the rear bogie frame; b₀ denotes the half of the lateral distance between the two wheel-rail contact point at the initial condition; R_wis the nominal rolling radius of the wheel; and H_tw is the vertical distance between the centroid of bogie frame and the central line of the wheelset.

Figure 5 illustrates the discrepancy of the wheel-rail vertical force between the nonlinear model and the linear model due to the implementation of different wheel-rail coupling models. As observed from Figure 4, there is a significant correlation between the results of these two models; see for instance, the sections possessing violent wheel-rail interactions are almost the same, but the forces derived by the nonlinear model are locally larger than those of the linear model, especially at high frequency ranges shown in Figure 5(b), because the time interval of integration of the nonlinear model should be greatly smaller than the linear model for guaranteeing the stability of time integration. Moreover, Figure 6 shows the dynamic responses of the rail lateral displacement at both time and frequency domains, from which one can observe that the results of these two models coincide well with each other. The linear model has accurately characterized the dominant frequencies of rail lateral displacement in the whole frequency range of 1–200 Hz though there are differences in spectral values.

In the numerical calculation, if the railway vehicle moves on the track for one second, the computation time for the linear model is 15 sec, while for the nonlinear model, it is 156 sec in MATLAB^®, and thus the computational efficiency has been significantly improved.

4.2. Random Vibrations concerning Randomness of Creep Coefficients

Creep coefficient possesses key influence on wheel-rail tangential interactions. Through numerical investigations, it is noticed that the lateral creep coefficient C_Y in the linearized model will majorly determine the vehicle-track lateral dynamics under irregular excitations. Undoubtedly, the creep coefficients are time-dependent and show randomicity. For obtaining more complete results, there is a necessity to treat the model as an ergodic dynamical system where the most influential parameters, e.g., creep coefficient, should be considered with an ergodicity.

In this study, in order to ensure the reliability of dynamic assessment of a linear model, further work on random vibration analysis will be probed into, where the detailed procedures can be illustrated as follows:

• (1)

  To reveal the full amplitude and probability characteristics of the creep coefficients, the track irregularity probabilistic model [42] and the nonlinear vehicle-track coupled model [11] are applied to characterize the probability density function (PDF) of C_Y.

As an example, Figure 7 plots the PDF fitting results of lateral creep coefficient, from which it is found out that the τ location-scale distribution can be properly used to reflect the probability property of the creep coefficients, where τ location-scale distribution can be expressed by

$$\begin{array}{}\mathbf{f}\left(x\right)=\frac{\mathbf{\Gamma }\left(\upsilon +12/\right)}{\sqrt{\upsilon \pi }\mathbf{\Gamma }\left(\upsilon /2\right)}1+{\left({\left(x-\mu /\sigma \right)}^2/\upsilon \right)}^{-\left(\upsilon +1\right)/2},\end{array}$$ ()

where Γ(⋅) is the gamma function and μ, σ, and υ denote the location, scale, and shape parameter, respectively.

• (2)

  Since the PDF of C_Y denoted by p_Cy is obtained in the above step, the methods for random parameters selection presented in Ref. [5] can be implemented to get the represented random field of C_Y, that is, Ω_Cy.

• (3)

  Set H(·) as the operator of vehicle-track interaction model and loading Ω_Cy and track irregularity vector T_Irre into H(·), the random vibration of vehicle-track systems R_indi can be calculated out consequently, that is,

$$\begin{array}{}{\mathbf{R}}_{\text{indi}}={\mathbf{H}}_{{\mathbf{\Omega }}_{\text{Cy}}}\left({\mathbf{T}}_{\text{irre}}\right),\end{array}$$ ()

where the subscript “indi” denotes the dynamic indices of vehicle-track systems.

Figure 8 plots the up and down boundary of power spectral density (PSD), denoted by P_u and P_d, respectively, of rail lateral displacement and wheel-rail lateral force, respectively, within which the confidence probability is 99%, that is,

$$\begin{array}{}\mathbf{P}{\mathbf{r}}_{u-d}\left(\omega \right)={\int }_{P_u\left(x\right)}^{P_d\left(x\right)}{f}_{P_{i\text{ndi}}}\left(x;\omega \right)dx=0.99,\end{array}$$ ()

where Pr(·) denotes the probability function; ω is the frequency; $f_{P_{\text{indi}}}\left(x;\omega \right)$ is the PDF of spectral densities against specific frequency ω; and x is the spectral density. The red lines are the PSD results of the nonlinear model.

It can be observed from Figure 8 that the boundary PSDs derived by the linear model can properly envelop the results of the nonlinear model, based on the mathematical relationship between spectral density and time-domain amplitude, that is,

$$\begin{array}{}A=\sqrt{\frac{1}{\pi }{\int }_{{\omega }_{i-1}}^{{\omega }_i}P\left(\omega \right)d\omega }.\end{array}$$ ()

It is obvious that time-domain responses of the linear model will inevitably envelop those of the nonlinear model. Therefore, with an implementation of the random vibration analysis, the results derived by the linear model can be provided as a reference to guarantee the safety margin of the vehicle-track dynamic behaviours.

However, it should be noted that though the linear model is capable of characterizing the characteristic frequencies of dynamic indices, the chosen of the creep coefficient will exert significant influence on the response amplitude; moreover, the responses of wheel-rail lateral force at low frequency of 10 Hz below will increase violently, which should be treated as one of the deficiencies of the linear model with an emphasis. It is therefore a rather important work to choose the creep coefficient optimally.