# Theoretical Studies on the Longitudinal Inhomogeneity of Track Stiffness and a Track Status Estimation Method

Authors: Wen Bai; Lei Xu
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

Due to material multiplicity, rigid-flexible hybrid and environmental variations, and so forth, the properties of the tracks will be inevitably changed due to the cyclic loads temporally and spatially. In this paper, a theoretical study is conducted to clarify the influence of the longitudinal inhomogeneity of track stiffness on system responses and then a general state estimation method is proposed to inversely predict the parametric distribution of the tracks. To achieve these purposes, a 3D nonlinear vehicle-track coupled model is firstly constructed, where track systems are established by the finite element method with random parameters. Based on this dynamic model, the distribution characteristics of system responses due to random parametric excitations are properly revealed. The numerical studies show that the longitudinal inhomogeneity of track supporting stiffness has significant effects on system responses; besides, the response distributions are varied against different parametric distributions. Finally, a representative example is demonstrated to show the effectiveness of the proposed track state estimation method.


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## Body

## 1. Introduction

Railway tracks serve as an essential element in supporting the running vehicles and guiding their dynamical motions. Compared to vehicles, the railway tracks are characterized by the complexity in material multiplicity, rigid-flex hybrid, longitudinal large scales, environmental changes, and so forth. Apart from the construction deviations, the track status, generally regarded as the track geometric such as track irregularities, and parametric status, such as the stiffness softening, are constantly affected by the cyclic vehicle loads and are nonuniformly and randomly evolved against the railway operations.In most of the vehicle-track dynamic simulations [1–8], track irregularities at time or frequency domain are treated as the main system excitations, and the vehicle and the tracks are considered as steady and constant systems. The longitudinal inhomogeneity and randomness of the tracks and their dynamic influence are neglected. In the field of vehicle-track dynamics nowadays, irrespective of it being in local-micro or in global-macro, it seems to be a growing trend to meticulously take into account the parametric excitation in a way approaching the essences. Generally, the parametric excitation is originated from the inhomogeneity of system properties, especially with uncertainties.In previous researches, the cognitions of the track parameter unevenness are mainly characterized from frequency domain. Wu and Thompson [9] investigated the wheel/rail interactions and system responses considering the spatially varying stiffness of a ballasted track using a quasi-static method. In their method, the time-domain model is equivalently transformed from the space-varying receptance based on the linearity of the tracks and wheel-rail interactions. Nélain and Vincent [10] carried out a deep research into the influence of sleeper periodicity, randomness and variance of the sleeper stiffness, and the foundation stiffness along the track on the parametric unevenness. Additionally, Verachtert et al. [11] investigated the relation between the unevenness of slab or track bed and the perceived unevenness derived by a train in frequency domain.Specific to systematic dynamics, Mao et al. [12] conducted an interesting work in random dynamic analysis of a train-bridge system, where the random system parameters are considered, while the longitudinal inhomogeneity of the system is not involved; similar work can also be found in [13, 14]. Xu and Zhai [15] made an extensive research on vehicle-track stochastic dynamics, in which the randomicity of the vehicle-track system in excitations and dynamic parameters is wholly considered. Besides, the full view of track irregularities is depicted in matrix forms to construct a highly efficient probabilistic model. From the perspective of microanalysis, Corrêa et al. [16] regarded the granular medium as a stochastic heterogeneous continuum medium and calibrated a heterogeneous continuum model to model the railway ballast samples.However, a universal method where the nonlinear wheel-rail contacts and the random track system with parametric unevenness along the track subject to a vehicle loading are not fully considered for studies in three-dimensional (3D) space. Besides revealing the dynamic effects of track inhomogeneity in vehicle-track dynamics, another challenge that derives from the influenc of track inhomogeneity on system response is how to estimate the track parametric status from the responses conversely. Although it might be possible to experimentally detect the track parameters or support stiffness within a small track section, it is difficult to measure the parametric state at large scales.Au et al. [17] described a method for the vehicle parameters subjected to multispan continuous bridges using genetic algorithm, where the vehicle is modelled as a 2-degree-of-freedom system. Similar work can be consulted in [18, 19]. These approaches are practical rather than simplified systems only. Recently, Onat et al. [20] proposed a method to estimate the maximum friction coefficient in wheel-rail interfaces based on a sigma point Kalman-filter-based scheme. Up to now, there have been limited reports in literature which provide some theoretical guidance in the estimation of track parametric status, especially the parameters randomly distributed along a large length of the tracks.In this paper, we pay special attention to clarifying the influence of track parametric inhomogeneity in a spatially longitudinal direction on system responses. Based on the clarification of the dynamic effects of track longitudinal inhomogeneity, a further research on inverse approach of estimating the track parametric status will be proposed. The random theories have indicated that only statistical information is required to construct a random model; thus, the estimation of the track parameters is concentrated on estimating the parametric distributions.Hence, instead of the experimental researches and validations, the work presented is mainly conducted from a theoretical view using numerical methods.


## 2. Track Longitudinal Inhomogeneity and Its Coupled Dynamics with the Vehicle

Figure1 shows two commonly used track types: ballasted track and ballastless track.Figure 1
Typical track systems.In this paper, the railway tracks are modelled as a ballastless system with rails, track slabs, and a winkle foundation. The rails and the track slabs are connected by discrete rail pads, and the cement asphalt (CA) mortar is applied to continuously provide elastic supports. Both the rail pads and CA mortar are treated as linear spring-dashpot elements.In the track slab system, the concrete track slabs including the rails are essentially continuum medium in longitudinal direction. Therefore, we will meticulously construct their dynamic matrices using finite element method (FEM).


### 2.1. Matrix Construction for the Tracks with Inhomogeneity

Here a “Lateral Finite Strip and Slab Segment” method [21] is applied to describe the shape function of the track slab. For arbitrary point o within the track slab element (TSE), as shown in Figure 2, its vertical displacement can be expressed as(1)Wo=NVδij,with(2)NVo=1−3syol2+2syol3−syol−2syol2+syol3l3syol2−2syol3syol3−syol2l,δij=Wi,θXi,Wj,θYjT,in which Wi and θXi denote vertical displacement and angular displacement around X-axis for point i and Wj and θXj denote vertical displacement and angular displacement for point j; syo is the local Y-coordinate of point o; l is the length of TSE along Y-axis.Figure 2
Top view of the TSE.Meanwhile, the vertical displacement and angular displacement inδij can be further expressed as(3)Wi=NViWa,θYa,Wb,θYbT,θXi=NTiθXa,θXbT,Wj=NVjWc,θYc,Wd,θYdT,θXj=NTjθXc,θXdT,with(4)NVi=1−3sxiw2+2sxiw3−sxiw−2sxiw2+sxiw3w3sxiw2−2sxiw3sxiw3−sxiw2w,NTi=1−sxiwsxiw,NVj=1−3sxjw2+2sxjw3−sxjw−2sxjw2+sxjw3w3sxjw2−2sxjw3sxjw3−sxjw2w,NTj=1−sxjwsxjw,in which sxi and sxj are, respectively, the local X-coordinate for points i and j; w is the width of TSE along X-coordinate. θY is the angular displacement around Y-axis.Accordingly, by combining equations (1) and (3), the interpolation function for vertical displacement of o can be written by (5)Nv,osxo,syo=NV1NV,NV2NT,NV3NV,NV4NTsxi=sxj=sxo.It is assumed that the dynamical parameter, for example, vertical stiffness of the cement asphalt mortar, follows a functionυ=fυx,y. Obviously, the stiffness matrix induced by the vertically elastic interactions between the track slabs and the subgrade can be calculated:(6)Kυe=fυsxo,syo∫−w/2w/2∫−l/2l/2Nv,oTNv,odsyodsxo.Certainly, the matrix induced by the elastic deformation energy of the thin-plate track slab element can be obtained by(7)Kse=∫−l/2l/2∫−w/2w/2BTDBdsxodsyo,with(8)B=−∂2Nv,o∂syo2∂2Nv,o∂sxo2∂2Nv,o∂syo∂sxoT,D=Essxo,syoHssxo,syo3121−μsxo,syo21μsxo,syo0μsxo,syo10001−μsxo,syo2,in which Es denotes Young’s modulus of the track slab; μ denotes Poisson’s ratio; Hs denotes the height of the track slab.To other matrices of the track components (e.g., the rails) with continuum materials, the same manner as above can be applied, in which one needs to replace the specific parameters with spatially varied function (discrete).The detailed expressions have been listed in [6]; no more illustrations are therefore given for brevity.Two other issues for the construction and computation of the random track system in large length will be encountered:(i)
Computational consumption for large matrices(ii)
Boundary condition in long-length calculationFor solving the above matters, a cyclic calculation method presented in [22, 23] will be applied in the computations.


### 2.2. Construction of Coupled Dynamics with the Vehicle

Following the work presented in [1], the track vibration accompanied by its coupling effects to a moving vehicle or train can be depicted mechanically and physically, as shown in Figure 3. In this work, the vehicle is considered as a multi-rigid-body system, including one body, two bogie frames, and four wheelsets. Each body has five degrees of freedom, that is, bounce ZV, transverse YV, roll θV, yaw ψV, and pitch βV.Figure 3
Vehicle-track 3D model. (a) Side view; (b) end view [22].
(a)(b)Without loss of generality, the dynamic equations of motion for the vehicle-track interactions can be expressed by(9)MVV00MTTX¨VX¨T+CVV00CTTX˙VX˙T+KVV00KTTXVXT=FVFT,in which the subscripts “V” and “T” denote the subsystems of the vehicle and the track, respectively; M, C, and K denote the mass, damping, and stiffness matrix, respectively; F denotes the loading vector.In equation (9), the details of dynamic matrices in the equations of motion have been presented in [6].Besides, the interactions between the vehicle and the tracks are mainly transmitted through the wheel-rail interfaces, where the wheel-rail contacts/creepages execute complex motions.In this work, the wheel-rail nonlinear contacts at three-dimensional (3D) space are depicted. A wheel-rail coupling model [1] is introduced, as shown in Figure 4. In this model, the relative motions between the full degrees of a wheelset and the vertical, lateral, and angular motions of the rail are accurately constructed from geometries and algebraic equations with nonlinearity.Figure 4
Wheel-rail solid contact model [6].The dynamic loads derived by this wheel-rail coupling model have been listed in [15].


### 2.3. Simple Comparisons with FASTSIM

Wheel-rail interaction forces have been known as the key technique of vehicle-track dynamics, which is also an aspect that is notoriously difficult to quantify theoretically. In this part, a comparison between this dynamic model and FASTSIM by Kalker [24] is presented with respect to wheel-rail forces, where track irregularities are used as system excitations.Figure5 shows the comparisons of wheel-rail longitudinal and lateral forces, respectively. It can be clearly observed from Figure 5 that the wheel-rail forces derived by this model agree well with the results of FASTSIM. Accordingly, the time-varying process of wheel-rail interactions including amplitude distribution and frequency components can be wholly demonstrated. Further validations about this vehicle-track interaction model have been presented in [6].Figure 5
Comparison between FastSim and this dynamic model. (a) Wheel-rail longitudinal force; (b) wheel-rail lateral force.
(a)(b)


## 2.1. Matrix Construction for the Tracks with Inhomogeneity

Here a “Lateral Finite Strip and Slab Segment” method [21] is applied to describe the shape function of the track slab. For arbitrary point o within the track slab element (TSE), as shown in Figure 2, its vertical displacement can be expressed as(1)Wo=NVδij,with(2)NVo=1−3syol2+2syol3−syol−2syol2+syol3l3syol2−2syol3syol3−syol2l,δij=Wi,θXi,Wj,θYjT,in which Wi and θXi denote vertical displacement and angular displacement around X-axis for point i and Wj and θXj denote vertical displacement and angular displacement for point j; syo is the local Y-coordinate of point o; l is the length of TSE along Y-axis.Figure 2
Top view of the TSE.Meanwhile, the vertical displacement and angular displacement inδij can be further expressed as(3)Wi=NViWa,θYa,Wb,θYbT,θXi=NTiθXa,θXbT,Wj=NVjWc,θYc,Wd,θYdT,θXj=NTjθXc,θXdT,with(4)NVi=1−3sxiw2+2sxiw3−sxiw−2sxiw2+sxiw3w3sxiw2−2sxiw3sxiw3−sxiw2w,NTi=1−sxiwsxiw,NVj=1−3sxjw2+2sxjw3−sxjw−2sxjw2+sxjw3w3sxjw2−2sxjw3sxjw3−sxjw2w,NTj=1−sxjwsxjw,in which sxi and sxj are, respectively, the local X-coordinate for points i and j; w is the width of TSE along X-coordinate. θY is the angular displacement around Y-axis.Accordingly, by combining equations (1) and (3), the interpolation function for vertical displacement of o can be written by (5)Nv,osxo,syo=NV1NV,NV2NT,NV3NV,NV4NTsxi=sxj=sxo.It is assumed that the dynamical parameter, for example, vertical stiffness of the cement asphalt mortar, follows a functionυ=fυx,y. Obviously, the stiffness matrix induced by the vertically elastic interactions between the track slabs and the subgrade can be calculated:(6)Kυe=fυsxo,syo∫−w/2w/2∫−l/2l/2Nv,oTNv,odsyodsxo.Certainly, the matrix induced by the elastic deformation energy of the thin-plate track slab element can be obtained by(7)Kse=∫−l/2l/2∫−w/2w/2BTDBdsxodsyo,with(8)B=−∂2Nv,o∂syo2∂2Nv,o∂sxo2∂2Nv,o∂syo∂sxoT,D=Essxo,syoHssxo,syo3121−μsxo,syo21μsxo,syo0μsxo,syo10001−μsxo,syo2,in which Es denotes Young’s modulus of the track slab; μ denotes Poisson’s ratio; Hs denotes the height of the track slab.To other matrices of the track components (e.g., the rails) with continuum materials, the same manner as above can be applied, in which one needs to replace the specific parameters with spatially varied function (discrete).The detailed expressions have been listed in [6]; no more illustrations are therefore given for brevity.Two other issues for the construction and computation of the random track system in large length will be encountered:(i)
Computational consumption for large matrices(ii)
Boundary condition in long-length calculationFor solving the above matters, a cyclic calculation method presented in [22, 23] will be applied in the computations.


## 2.2. Construction of Coupled Dynamics with the Vehicle

Following the work presented in [1], the track vibration accompanied by its coupling effects to a moving vehicle or train can be depicted mechanically and physically, as shown in Figure 3. In this work, the vehicle is considered as a multi-rigid-body system, including one body, two bogie frames, and four wheelsets. Each body has five degrees of freedom, that is, bounce ZV, transverse YV, roll θV, yaw ψV, and pitch βV.Figure 3
Vehicle-track 3D model. (a) Side view; (b) end view [22].
(a)(b)Without loss of generality, the dynamic equations of motion for the vehicle-track interactions can be expressed by(9)MVV00MTTX¨VX¨T+CVV00CTTX˙VX˙T+KVV00KTTXVXT=FVFT,in which the subscripts “V” and “T” denote the subsystems of the vehicle and the track, respectively; M, C, and K denote the mass, damping, and stiffness matrix, respectively; F denotes the loading vector.In equation (9), the details of dynamic matrices in the equations of motion have been presented in [6].Besides, the interactions between the vehicle and the tracks are mainly transmitted through the wheel-rail interfaces, where the wheel-rail contacts/creepages execute complex motions.In this work, the wheel-rail nonlinear contacts at three-dimensional (3D) space are depicted. A wheel-rail coupling model [1] is introduced, as shown in Figure 4. In this model, the relative motions between the full degrees of a wheelset and the vertical, lateral, and angular motions of the rail are accurately constructed from geometries and algebraic equations with nonlinearity.Figure 4
Wheel-rail solid contact model [6].The dynamic loads derived by this wheel-rail coupling model have been listed in [15].


## 2.3. Simple Comparisons with FASTSIM

Wheel-rail interaction forces have been known as the key technique of vehicle-track dynamics, which is also an aspect that is notoriously difficult to quantify theoretically. In this part, a comparison between this dynamic model and FASTSIM by Kalker [24] is presented with respect to wheel-rail forces, where track irregularities are used as system excitations.Figure5 shows the comparisons of wheel-rail longitudinal and lateral forces, respectively. It can be clearly observed from Figure 5 that the wheel-rail forces derived by this model agree well with the results of FASTSIM. Accordingly, the time-varying process of wheel-rail interactions including amplitude distribution and frequency components can be wholly demonstrated. Further validations about this vehicle-track interaction model have been presented in [6].Figure 5
Comparison between FastSim and this dynamic model. (a) Wheel-rail longitudinal force; (b) wheel-rail lateral force.
(a)(b)


## 3. A General Method to Estimate the Status of the Tracks

The reality of track parameters can be expressed as(10)Xk≔xk1,xk2,…,xkN,with xk=xk,1,xk,2,…,xk,m,…, where the subscript “k” denotes the time point, “m” denotes the mth track parameter, and there are N spatial values for each track parameter vector xk,j, j=1,2,…,m.The state-space equations of the vehicle-track system can be further written by(11)Zkt∣X=HXk,uk,wk,t=0,ttol.In the above equation,t denotes the time-varying process of the system responses; H⋅ denotes the measurement function, which will be substituted by the vehicle-track model here. Thus, Zk will denote the system responses. uk denotes other system factors; wk represents noise signals.Obviously, it is a difficult work to measure the real status of tracks, especially with large scales. One is therefore expected to inversely and approximately estimate track status (the distributions of track parameters) from the conventional dynamic responses and estimated status, namely,Zk and Xk−1.The system responses are mainly induced by two factors: wheel/rail irregularities and inconstant mechanical-physical properties longitudinally and evolved. Therefore, the noise in equation (11) contains both the measurement error and the effects of irregularities.In this present study, an efficient vehicle-track interaction model, denoted by℘, had been fully constructed, where the physical-mechanical behaviors and interactions of the vehicle-track systems can be accurately depicted. The system responses derived by the theoretical model can therefore be calculated by(12)Z˜k=℘Xk,Ω,where Ω denotes other system parameters including wheel/rail irregularities.Now a problem will be firstly encountered: we have no idea what the longitudinal distribution of the track parameter looks like. By applying a multi-correlated-variable Monte Carlo method [15], we can randomly generate a pseudoparameter matrix:(13)Mk|X=x1,1sx1,2s…x1,Nsx2,1sx2,2s…x2,Ns⋮⋮⋱⋮xV,1sxV,2s…xV,NsV×N×m,with ℓxp,gs,xp,hs=R1m−g−hR1m−R0m/N, in which the superscript “s” denotes the simulated value; V denotes the total number of samples; R0m and R1m denote the minimum and maximum correlation coefficients for the mth parameter, respectively; ℓxp,gs,xp,hs denotes the correlation coefficient between the gth and the hth parameter columns.Loading theseV parameter samples into the vehicle-track interaction model, one can obtain the system response matrix consequently; that is,(14)Z˜kt∣MX=℘Mk|X,Ω.In reality, the real track parameters follow a specific probability distribution, namely,xk∼fx,kx , and fx is the probability function of the parameter k. Thus, the key work lies in building up the relationship between fx,kx, Z˜kt|MX, and Zkt|X and estimating fx,kx.In stochastic analysis, we more likely intend to clarify the distribution of a variable in probability rather than attempting to estimate the deterministic values. Following the basic principle, the closer the correlation of system responses between simulation and measurement is, the more approachable to reality system parameters used in simulation are.The steps below are therefore designed to achieve the estimation offx,kx: (1)
To increase the precision of estimation, the comparison between responses of calculation and measurement, that is,Z˜kt|MX and Zkt|X, are conducted within multiple time segments, Δta=ttol⋅a−1/A,a/A, a=1,2,…,A.(2)
The uncertainty of the responses will be estimated by the correlation between the calculation and the measurement at time portionΔta.(15)Wτka=covZ˜kΔta|MX,ZkΔta|XσZ˜k,ΔtaσZk,Δta,where the subscript “τ” denotes the type of the responses, calculation, or measurement: τ=Z˜korZk, respectively; za denotes the system responses, za∈Z˜kt|MXorZkt|X; cov⋅ denotes the estimator of covariance and σdenotes the standard deviation. Obviously, WZ˜kka is a matrix with order of V×A, while WZkka is a matrix with order of 1×A.(3)
The weight coefficients evaluating the contribution of the parameters on system responses can be obtained by(16)γl,akx=1∑l=1Vwl∑l=1Vwlδx−l,withwl,a=pzl,a|xl,a∝1/2πQ1/2e−WZ˜kkl,a−WZkka2/2Q, l=1,2,…,V, where Q is the variance of the measurement noise, Q=0.05∼0.4, and pz|x is the posterior distribution of the response under certain track conditions.(4)
Based on equation (16), the estimated distribution of the parameters will be assessed by(17)f˜x,kx=∑a∑lγl,akxfl,a|Mk|Xx,where fl,a|Mk|Xx indicates the probability distribution of the parameters in the parameter matrix.Finally, the estimation offx,kx can be approximated by assuming fx,kx≅f˜x,kx.Sincefx,kx has been estimated approximately, an acceptance-rejection algorithm, as illustrated in Algorithm 1, can be introduced to randomly simulate the spatial series of track parameters Xk.Algorithm 1:Acceptance-rejection algorithm for simulating spatial parameters with longitudinal inhomogeneity.
x1=minx, x2=maxxp1=minfx,kx, p2=maxfx,kxforn=1:Nx˜=Urx1,x2, p˜=Urp1,p2p˜′=interp1x,fx,kx,x˜whilep˜>p˜′x˜=Urx1,x2, p˜=Urp1,p2p˜′=interp1x,fx,kx,x˜endaddx˜ to XkendreturnXkNote:Ur represent the function generating uniformly random numbers


## 4. Numerical Examples

In the numerical examples, it is assumed that the vehicle runs with a constant velocity of 300 km per hour on the tracks. The main parameters of the vehicle and the tracks are listed in Tables1 and 2, respectively [1].Table 1
Main parameters of the railway vehicles used in the simulation.
NotationParameterValueMcCar body mass (kg)46280MtBogie mass (kg)3300MwWheelset mass (kg)1780IcxMass moment of the inertia of the car body about theX-axis (kg m2)149970IcyMass moment of the inertia of the car body about theY-axis (kg m2)2267765IczMass moment of the inertia of the car body about theZ-axis (kg m2)2139900ItxMass moment of the inertia of the bogie about theX-axis (kg m2)2673ItyMass moment of the inertia of the bogie about theY-axis (kg m2)1807ItzMass moment of the inertia of the bogie about theZ-axis (kg m2)3300IwxMass moment of the inertia of the wheelset about theX-axis (kg m2)949IwyMass moment of the inertia of the wheelset about theY-axis (kg m2)118IwzMass moment of the inertia of the wheelset about theZ-axis (kg m2)967KpxStiffness coefficient of the primary suspension along theX-axis (MN/m)14.68KpyStiffness coefficient of the primary suspension along theY-axis (MN/m)6.47KpzStiffness coefficient of the primary suspension along theZ-axis (MN/m)1.176.KsxStiffness coefficient of the secondary suspension along theX-axis (MN/m)0.167KsyStiffness coefficient of the secondary suspension along theY-axis (MN/m)0.167KszStiffness coefficient of the secondary suspension along theZ-axis (MN/m)0.323CpzDamping coefficient of the primary suspension along theZ-axis (kN s/m)9.8CsyDamping coefficient of the secondary suspension along theY-axis (kN s/m)39.2CszDamping coefficient of the secondary suspension along theZ-axis (kN s/m)9.8LcSemilongitudinal distance between bogies (m)8.75LtSemilongitudinal distance between wheelsets in a bogie (m)1.25R0Wheel radius (m)0.43Table 2
Main parameters of the railway tracks used in the simulation.
NotationParameterValueEElastic modulus of the rail (N/m2)2.059 × 1011m¯rMass of the rail per unit length (kg/m)60.64I0Torsional inertia of the rail (m4)3.741 × 10−5IyRail second moment of the area about theY-axis (m4)3.217 × 10−5IzRail second moment of the area about theZ-axis (m4)5.24 × 10–6kzrpFastener stiffness in the vertical direction (N/m)4.0 × 107kyrpFastener stiffness in the lateral direction (N/m)1.5 × 107czrpFastener damping in the vertical direction (N s/m)5.0 × 104cyrpFastener damping in the lateral direction (N s/m)5.0 × 104kycamEquivalent lateral stiffness coefficient of the CAM (N/m)6.5 × 107kzcamEquivalent vertical stiffness coefficient of the CAM (N/m)2.4 × 108cycamEquivalent lateral damping coefficient of the CAM (N s/m)3.1 × 104czcamEquivalent vertical damping coefficient of the CAM (N s/m)5.4 × 104


### 4.1. Influence of the Longitudinal Inhomogeneity of Track Stiffness on System Responses

To clearly reveal the dynamic influence of the longitudinal inhomogeneity of track stiffness on system responses, track profile irregularities will not be taken into account in the vehicle-track interactions in this subsection. It is assumed that the values of a track parameter satisfying specific probability characteristics are randomly distributed along the railway line instead of a constant.Three types of probability distributions are considered: uniform distributionUX, normal distribution NX, and log-normal distribution log−NX. To guarantee the comparability for different conditions of property inhomogeneity, the random parameter values should be within almost the same ranges. The equations below are therefore satisfied:(18)FUx−10.01=FNx−10.01=Flog−Nx−10.01,FUx−10.99=FNx−10.99=Flog−Nx−10.99.where F⋅ denotes the cumulative probability function and the superscript “−1” denotes the inverse function.Take the response distribution of car body accelerations, wheel-rail forces, displacements, and accelerations of the rail and the track slab as examples. For normal distributions, the coefficient of variation (CV) is denoted by(19)CV=σμ,where σ is the standard deviation and μ is the mean. CV=0.2 in this example.Figure6 illustrates the statistical mean PDF of wheel-rail vertical force, rail vertical displacement, and track-slab vertical displacement with 200 samples simulated by different forms of distributions of rail pad vertical stiffness. From Figure 6, one can observe that the longitudinal variability of rail pad vertical stiffness shows a significant influence on the probabilistic distribution of system vertical vibrations. Besides the variation of divergence and response amplitude of a specific index accompanied by the changes of the parametric distribution, the longitudinal uniform distribution exerts more effects on the variance of system responses compared to the other two, normal distribution and log-normal distribution.Figure 6
The PDF of dynamic indices specific to different longitudinal distribution of rail pad vertical stiffness. (a) Wheel-rail vertical force; (b) rail vertical displacement; (c) track-slab vertical displacement.
(a)(b)(c)For quantitatively characterizing the uncertainty of a random signal, the concept of entropy can be introduced by(20)Hz=E−lnpz,where H denotes the entropic value, E⋅ denotes the mathematical expectation, ln⋅ denotes the logarithm with bottom e, and pz denotes the probability function.As an example, Figure7 shows the entropic distribution of the wheel-rail vertical force, from which one can observe the following: first, the uncertainty of the wheel-rail vertical force excited by the uniformly distributed rail pad vertical stiffness is significantly higher than those excited by normal and log-normal distributions; second, it can be observed from Figure 7(b) that the statistical uncertainty of system responses fluctuates slightly, which means that once the parametric distribution is confirmed, we can approximately estimate its influence on the distribution of system responses.Figure 7
The entropy of PDF of wheel-rail vertical force due to variation of rail pad vertical stiffness. (a) Entropic value for specific samples; (b) the statistical mean value of entropy under different total samples.
(a)(b)Figure8 further plots the PDF of wheel-rail lateral force and rail lateral displacement affected by the deviations of rail pad lateral stiffness. In this example, for showing the influence of the magnitude of rail pad lateral stiffness on system lateral vibrations, the stiffness coefficients following log-normal distribution are given higher values, as shown in Figure 9.Figure 8
PDF of dynamic indices due to longitudinal inhomogeneity of rail pad lateral stiffness. (a) Rail lateral displacement; (b) wheel-rail lateral displacement.
(a)(b)Figure 9
The lateral stiffness of 200 samples for a specific rail pad.It can be observed from Figure8 that the longitudinal inhomogeneity of rail pad lateral stiffness also shows distinctive effects on the distribution characteristics of the wheel-rail lateral force and the rail pad lateral displacement. Obviously, the larger the lateral stiffness of the rail pad, the smaller the wheel-rail lateral force, and the larger the rail lateral displacement. It is similar to the results derived above where the uniform distribution shows higher influence on the uncertainty of system response compared to the other two distributions.In Figure10, the PDF of rail vertical displacement is exposed, from which one can observe that the longitudinal distribution of the rail pad lateral stiffness has little influence on the system’s vertical vibrations.Figure 10
The influence of longitudinal distribution of rail pad lateral stiffness on the distribution of the rail vertical displacement.The results for the dynamic response induced by the longitudinal inhomogeneity of vertical stiffness of the cement asphalt mortar (CAM) (subgrade support layer) are shown in Figure11. The vertical stiffness of the CAM shows great influence on the responses of track components, for example, the rails and the track slabs. However, its influence on the wheel-rail interaction forces is relatively small.Figure 11
The PDF of dynamic responses due to the nonuniformity of vertical stiffness of CAM. (a) Rail vertical displacement (mm); (b) track-slab vertical displacement (mm); (c) wheel-rail vertical force (kN).
(a)(b)(c)


### 4.2. Estimation of the Distribution of Track Parameters

From the numerical analysis above, one can recognize that the rail pad stiffness, especially the vertical stiffness, shows significant influence on wheel-rail forces and rail vertical displacement. Hence, we therefore put an emphasis on estimation of the distribution of rail pad vertical stiffness.The track irregularities used in vehicle-slab track interactions from a high-speed line are shown in Figure12. Besides, the longitudinally nonuniform rail pad vertical stiffness is assumed to follow normal distribution with CV of 0.05, 0.10, and 0.20.Figure 12
Track irregularities used in the dynamic model. (a) Vertical irregularity; (b) lateral irregularity.
(a)(b)In the estimations, the following computational conditions are considered:(1)
Only track irregularities are loaded as the system excitations, where the system responses are denoted byZt|Ω(2)
Track irregularities and randomly distributed rail pad vertical stiffness (which is assumed as the real distribution of “measurement” in the theoretical analysis) are loaded as the system excitations, where the responses are denoted byZt|X(3)
The responses excited by a group of correlated variables of rail pad stiffness and the specific track irregularities are denoted byZt|MXFigure13 plots the time-domain responses of rail vertical displacement with CV = 0.2, from which one can observe that the rail vertical displacement changes significantly against time once with the longitudinal inhomogeneity of vertical stiffness of rail pads, that is, Zt|Ω versus Zt|X. The response assemblage, that is, Zt|MX, is the result of a set of excitations (parametric and irregularity), which has properly covered the responses Zt|Ω and Zt|X.Figure 13
The responses of rail vertical displacement due to different excitations with CV of rail pad vertical stiffness = 0.20.According to the method presented inSection 3, we can estimate the probability density function (PDF) of the vertical stiffness of the rail pad, as shown in Figure 14. As seen from Figure 14, the estimated distribution of the rail pad vertical stiffness coincides well with that of assumption in various variations: CV = 0.05, 0.10, and 0.20. Through probability entropy calculations, the entropy difference between estimation and assumption is, respectively, 0.002, 0.01, and 0.008 for conditions CV = 0.05, CV = 0.10, and CV = 0.20. From the analysis in Section 4.1, one has recognized that once the longitudinal distribution of a track parameter is determined, the system responses it caused can be approximately estimated with a slight deviation.Figure 14
Comparisons between estimation and measurement estimated by the rail vertical displacement. (a) CV = 0.05; (b) CV = 0.10; (c) CV = 0.20.
(a)(b)(c)Undoubtedly, one can also estimate the distribution of the rail pad vertical stiffness through the wheel-rail vertical forces. The corresponding results are shown in Figure15, from which it can be clearly observed that estimation of the stiffness distribution also approaches those of assumption.Figure 15
Comparisons between estimation and measurement estimated by the wheel-rail vertical forces. (a) CV = 0.05; (b) CV = 0.10; (c) CV = 0.20.
(a)(b)(c)From the numerical examples, it can be seen that the present method is highly effective. However, limitations have also been notified in its implementation to engineering practices; that is, the CV of track parameters is a prerequisite in the simulation. An abundant measurement data is therefore required to build the mapping relationship between system responses and track parametric distributions.


## 4.1. Influence of the Longitudinal Inhomogeneity of Track Stiffness on System Responses

To clearly reveal the dynamic influence of the longitudinal inhomogeneity of track stiffness on system responses, track profile irregularities will not be taken into account in the vehicle-track interactions in this subsection. It is assumed that the values of a track parameter satisfying specific probability characteristics are randomly distributed along the railway line instead of a constant.Three types of probability distributions are considered: uniform distributionUX, normal distribution NX, and log-normal distribution log−NX. To guarantee the comparability for different conditions of property inhomogeneity, the random parameter values should be within almost the same ranges. The equations below are therefore satisfied:(18)FUx−10.01=FNx−10.01=Flog−Nx−10.01,FUx−10.99=FNx−10.99=Flog−Nx−10.99.where F⋅ denotes the cumulative probability function and the superscript “−1” denotes the inverse function.Take the response distribution of car body accelerations, wheel-rail forces, displacements, and accelerations of the rail and the track slab as examples. For normal distributions, the coefficient of variation (CV) is denoted by(19)CV=σμ,where σ is the standard deviation and μ is the mean. CV=0.2 in this example.Figure6 illustrates the statistical mean PDF of wheel-rail vertical force, rail vertical displacement, and track-slab vertical displacement with 200 samples simulated by different forms of distributions of rail pad vertical stiffness. From Figure 6, one can observe that the longitudinal variability of rail pad vertical stiffness shows a significant influence on the probabilistic distribution of system vertical vibrations. Besides the variation of divergence and response amplitude of a specific index accompanied by the changes of the parametric distribution, the longitudinal uniform distribution exerts more effects on the variance of system responses compared to the other two, normal distribution and log-normal distribution.Figure 6
The PDF of dynamic indices specific to different longitudinal distribution of rail pad vertical stiffness. (a) Wheel-rail vertical force; (b) rail vertical displacement; (c) track-slab vertical displacement.
(a)(b)(c)For quantitatively characterizing the uncertainty of a random signal, the concept of entropy can be introduced by(20)Hz=E−lnpz,where H denotes the entropic value, E⋅ denotes the mathematical expectation, ln⋅ denotes the logarithm with bottom e, and pz denotes the probability function.As an example, Figure7 shows the entropic distribution of the wheel-rail vertical force, from which one can observe the following: first, the uncertainty of the wheel-rail vertical force excited by the uniformly distributed rail pad vertical stiffness is significantly higher than those excited by normal and log-normal distributions; second, it can be observed from Figure 7(b) that the statistical uncertainty of system responses fluctuates slightly, which means that once the parametric distribution is confirmed, we can approximately estimate its influence on the distribution of system responses.Figure 7
The entropy of PDF of wheel-rail vertical force due to variation of rail pad vertical stiffness. (a) Entropic value for specific samples; (b) the statistical mean value of entropy under different total samples.
(a)(b)Figure8 further plots the PDF of wheel-rail lateral force and rail lateral displacement affected by the deviations of rail pad lateral stiffness. In this example, for showing the influence of the magnitude of rail pad lateral stiffness on system lateral vibrations, the stiffness coefficients following log-normal distribution are given higher values, as shown in Figure 9.Figure 8
PDF of dynamic indices due to longitudinal inhomogeneity of rail pad lateral stiffness. (a) Rail lateral displacement; (b) wheel-rail lateral displacement.
(a)(b)Figure 9
The lateral stiffness of 200 samples for a specific rail pad.It can be observed from Figure8 that the longitudinal inhomogeneity of rail pad lateral stiffness also shows distinctive effects on the distribution characteristics of the wheel-rail lateral force and the rail pad lateral displacement. Obviously, the larger the lateral stiffness of the rail pad, the smaller the wheel-rail lateral force, and the larger the rail lateral displacement. It is similar to the results derived above where the uniform distribution shows higher influence on the uncertainty of system response compared to the other two distributions.In Figure10, the PDF of rail vertical displacement is exposed, from which one can observe that the longitudinal distribution of the rail pad lateral stiffness has little influence on the system’s vertical vibrations.Figure 10
The influence of longitudinal distribution of rail pad lateral stiffness on the distribution of the rail vertical displacement.The results for the dynamic response induced by the longitudinal inhomogeneity of vertical stiffness of the cement asphalt mortar (CAM) (subgrade support layer) are shown in Figure11. The vertical stiffness of the CAM shows great influence on the responses of track components, for example, the rails and the track slabs. However, its influence on the wheel-rail interaction forces is relatively small.Figure 11
The PDF of dynamic responses due to the nonuniformity of vertical stiffness of CAM. (a) Rail vertical displacement (mm); (b) track-slab vertical displacement (mm); (c) wheel-rail vertical force (kN).
(a)(b)(c)


## 4.2. Estimation of the Distribution of Track Parameters

From the numerical analysis above, one can recognize that the rail pad stiffness, especially the vertical stiffness, shows significant influence on wheel-rail forces and rail vertical displacement. Hence, we therefore put an emphasis on estimation of the distribution of rail pad vertical stiffness.The track irregularities used in vehicle-slab track interactions from a high-speed line are shown in Figure12. Besides, the longitudinally nonuniform rail pad vertical stiffness is assumed to follow normal distribution with CV of 0.05, 0.10, and 0.20.Figure 12
Track irregularities used in the dynamic model. (a) Vertical irregularity; (b) lateral irregularity.
(a)(b)In the estimations, the following computational conditions are considered:(1)
Only track irregularities are loaded as the system excitations, where the system responses are denoted byZt|Ω(2)
Track irregularities and randomly distributed rail pad vertical stiffness (which is assumed as the real distribution of “measurement” in the theoretical analysis) are loaded as the system excitations, where the responses are denoted byZt|X(3)
The responses excited by a group of correlated variables of rail pad stiffness and the specific track irregularities are denoted byZt|MXFigure13 plots the time-domain responses of rail vertical displacement with CV = 0.2, from which one can observe that the rail vertical displacement changes significantly against time once with the longitudinal inhomogeneity of vertical stiffness of rail pads, that is, Zt|Ω versus Zt|X. The response assemblage, that is, Zt|MX, is the result of a set of excitations (parametric and irregularity), which has properly covered the responses Zt|Ω and Zt|X.Figure 13
The responses of rail vertical displacement due to different excitations with CV of rail pad vertical stiffness = 0.20.According to the method presented inSection 3, we can estimate the probability density function (PDF) of the vertical stiffness of the rail pad, as shown in Figure 14. As seen from Figure 14, the estimated distribution of the rail pad vertical stiffness coincides well with that of assumption in various variations: CV = 0.05, 0.10, and 0.20. Through probability entropy calculations, the entropy difference between estimation and assumption is, respectively, 0.002, 0.01, and 0.008 for conditions CV = 0.05, CV = 0.10, and CV = 0.20. From the analysis in Section 4.1, one has recognized that once the longitudinal distribution of a track parameter is determined, the system responses it caused can be approximately estimated with a slight deviation.Figure 14
Comparisons between estimation and measurement estimated by the rail vertical displacement. (a) CV = 0.05; (b) CV = 0.10; (c) CV = 0.20.
(a)(b)(c)Undoubtedly, one can also estimate the distribution of the rail pad vertical stiffness through the wheel-rail vertical forces. The corresponding results are shown in Figure15, from which it can be clearly observed that estimation of the stiffness distribution also approaches those of assumption.Figure 15
Comparisons between estimation and measurement estimated by the wheel-rail vertical forces. (a) CV = 0.05; (b) CV = 0.10; (c) CV = 0.20.
(a)(b)(c)From the numerical examples, it can be seen that the present method is highly effective. However, limitations have also been notified in its implementation to engineering practices; that is, the CV of track parameters is a prerequisite in the simulation. An abundant measurement data is therefore required to build the mapping relationship between system responses and track parametric distributions.


## 5. Conclusions

In this paper, the dynamic influences of the longitudinally distributed track parameters on system responses are systematically studied using stochastic and statistical approach, and a new approach for estimating the parametric distribution of the tracks was developed theoretically.From the numerical studies, several conclusions can be drawn accordingly:(1)
The longitudinal inhomogeneity of track stiffness shows significant influence on vehicle-track interactions. Besides, the distribution characteristics of the system responses are varied against the difference of distribution of track stiffness.(2)
The mean entropy value of wheel-rail vertical force is increased by about 8% when it is assumed that the rail vertical stiffness follows uniform distribution instead of normal and log-normal distribution. Besides, it is observed that the response dispersion degree tends to converge when the random samples are larger than 50.(3)
The nonuniformity of track vertical stiffness shows larger influence on wheel-rail forces and track structural deformations in comparison with the lateral stiffness.(4)
It could be a practical way to assess the effects of random distribution of track parameters on the uncertainty of system responses.(5)
The state estimation method newly developed in this work is fairly effective in estimating the track parametric distributions.In the future, three aspects are urgently needed to study: the first is a more physically approachable model to simulate the vehicle-track dynamic performance with respect to spatial variability and nonlinearity of track properties, the second is the development of a highly accurate and efficient measurement method to detect the vehicle-track system responses, and the third is an exploration of practical methods to estimate the track parametric distribution through the vehicular responses in the condition of low signal-to-noise ratio.



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*Source: 8847633-2021-04-21.xml*