Modeling and Characterization of Complex Dynamical Properties of Railway Ballast

Abstract

The nonlinear mechanical properties of ballasted tracks along railways result in complex dynamics of the vehicle–track systems. Employing localized characterization of ballast and a simplified model might underestimate the vehicle–track system’s dynamical responses and safety. This paper presents a new dynamical stiffness model of railway ballast by incorporating the ballast’s time-varying, nonlinear viscoelastic, and elastoplastic properties. The new nonlinear stiffness model is a versatile tool that comprehensively characterizes the ballast characteristics of displacement-dependent stiffness, frequency-dependent stiffness, hysteresis, and time/space-varying features. These features, widely reported in previous experimental research, can now be effectively understood. Conventionally, to characterize time/space-varying characteristics of ballast along the track, ground-penetrating radar (GPR) has been used as the most efficient approach to survey railway networks quickly and to infer track dynamical properties. Aiming to improve the present technique for characterizing time/space-varying properties of ballast stiffness by using a GPR signal, the adaptive optimal kernel time–frequency representation (AOKTFR) method is used to process a typical GPR signal from a railway ballast scanning. It is found that the results of AOKTFR exhibit a clear time-varying pattern and precise frequency modulation. In contrast, the conventional time–frequency methods failed to give a clear time-varying pattern. The results illustrate that AOKTFR is a practical approach for processing the time-varying nonlinear signal of GPR and correlating it with the time-varying nonlinear stiffness of ballast.

1. Introduction

Classical research on railway vehicle dynamics has focused on the detailed multi-body modeling of vehicle systems by only incorporating simplified dynamic behaviors of the track system that supports the vehicle [1,2]. Analyzing the whole vehicle and track system systematically is necessary for improved design of the vehicle and track systems [3,4,5]. Much research has been done on vehicle–track models to explore the complex dynamics and interactions between vehicles and railway tracks [6,7]. Most vehicle–track system models consist of a vehicle sub-model, including car body, bogies, and wheels, and a track sub-model, including rails, fastening structure/sleeper, ballast, and subgrades. With the help of this kind of model, greater understanding has been developed in different aspects.

On the other hand, railroad track structures’ structural adequacy and maintenance challenges have received considerable attention. The vertical stiffness of ballasted railway tracks, a crucial index in defect detection and condition monitoring, has been the subject of extensive research [8,9,10,11]. Most of this research has focused on measuring methods and modeling track vertical stiffness. However, most studies on track modeling and experimental characterizations have used conventional linear approaches [12,13,14,15,16,17,18,19,20]. There is a consensus that the complex dynamics problem of vehicle track systems must be regarded as nonlinear and that nonlinear models must be well formulated. This consensus underscores the importance of developing nonlinear models to understand the complex dynamics problem of vehicle track systems. Furthermore, as the properties of the tracks could be inevitably changed due to the cyclic loads temporally and spatially, the track stiffness has space-varying and time-varying properties.

Much research has been conducted to model and measure the complex stiffness of tracks for design, monitoring, and maintenance, which have been widely discussed in recent research. Uyulan and Gokasan [21] discussed the nonlinear dynamic characteristics of a railway. Qi et al. [22] analyzed the nonlinear dynamical properties of granular ballast using a five-parameter model. The well-known dynamical features of the nonlinear stiffness of ballast dependence on deformation and frequencies have been established [23,24,25]. Alabbasi and Hussein [26] discussed the elastoplastic properties of ballast and nonlinear load-deformation behavior under cyclic loading.

Punetha et al. [27,28] proposed a simple geotechnical rheological model for simulating the viscoelastic-plastic response of a ballasted railway substructure. Kaewunruen and Tao [29] proposed a simplified nonlinear dynamic model of ballast to quantify the frequency-dependent features in flood situations. Bai and Xu [30] studied the space-varying track stiffness associated with the longitudinal inhomogeneity of the track using a track status estimation method. Olamide and Lam [31,32,33] studied the detection of railway ballast damage using various nonlinear ballast stiffness and deformation models. Tahiri et al. [34] conducted a nonlinear analysis of the ballast influence on the train railway bridge using a Duffing oscillator model. Liu et al. [35] measured the ballast bed’s frequency response function (FRF) curves under different temperature and humidity conditions. They analyzed the ballast bed’s vibration characteristics and evolution laws under different conditions.

Fatin et al. [36] investigated the nonlinear stiffness of a railway ballast with slight strain. Shi et al. [37] reviewed the stiffness irregularity of railway ballasted tracks. Robles et al. [38] studied a track dynamic model with the evolution of rail corrugation in sharp curves. Farooq et al. [39] used a viscoelastic rheological track model to evaluate the track lateral response. Angie et al. [40] investigated railway track acceptance, which is the inverse of stiffness. They characterized the effect of the ballast stiffness at the rail and found it impacts the rail response at low frequencies, whereas it does not affect the structure at high-frequency ranges. Jain et al. [41] characterized space-varying stiffness and distributions on the dynamic behavior of railway transition zones. Powrie [42] reviewed recent research and future directions of railway track substructures. He indicated that the key performance indicators for railway tracks are the track support stiffness, the vertical alignment or level, and the rate at which the level deteriorates due to plastic settlement associated with plastic strain.

In addition, Dafert et al. [43] estimated complex ballast stiffness based on measurements during the dynamic track stabilization. Yunyun et al. [11] comprehensively reviewed vertical stiffness measurement methods and values as well as effective ballasted track parameters and their contribution to railway track condition monitoring, indicating that vertical stiffness is one of the most influential factors for recognizing track vertical behavior. Germonpré et al. [44] studied the effects of longitudinal track unevenness and space-varying track stiffness on railway-induced vibration. Nasrollahi et al. [45] studied a benchmark of calibrated track models for the simulation of differential settlement in a transition zone using field measurement data. Despite research covering most aspects of the complex dynamical properties of ballast, the complex nonlinearity of ballast has not been comprehensively modeled. Ballast is a homogeneously graded material that comes from hard rock. Describing the dynamic stiffness and damping of the time-varying nonlinear granular material is crucial to modeling the whole vehicle–track system.

Analytical modeling of ballasted tracks has been developed for static and dynamic analysis. In static analysis, the ballast layer is mainly modeled as a simple linear elastic spring with a constant stiffness. In dynamic analysis, other structural elements, such as viscous dampers, are used to model the ballast layer analytically. Analytical models of ballasted tracks have been widely used to study ballasted track dynamic behavior due to their low requirement of computational time.

The previous analytical modeling approaches have several drawbacks in quantifying the actual mechanical behavior of the ballast layer, such as the amplitude and frequency dependence of stiffness and hysteresis. It is noted that ballast exhibits complex non-linear, time-dependent load-deformation behavior under cyclic loading as elastoplastic material with a rheology mechanism. The stiffness and damping properties of the ballast layer exhibit space-varying and time-varying features due to ballast densification and deterioration under complex operation conditions, such as cyclic loading, temperature change, wetness change, ballast particle size/shape changes, and fouling due to long term service/operations. These have been widely reported in previous experimental research, such as [21,22,23,24,25,26,27].

From a material perspective, the conventional analytical approach of modeling the ballast layer does not represent the real complex behavior of railroad ballast under cyclic loading, in which ballast behaves as nonlinear elastic-plastic-viscous material with nonlinear load-deformation behavior. However, in previous modeling, the track system, particularly the nonlinearity of ballast, has not been comprehensively included. Multi-body simulations of vehicle–track systems have been routinely carried out using commercial software such as Vampire 5.0, NUCARS 2024.2.0, and VI-Rail 16.0 [46,47,48,49,50].

These programs simulate the dynamic response of a vehicle on a track by simply assuming a simple uniform ballast and track support stiffness. As part of the track structure, ballast is made of a homogeneously graded material from hard rock, with a diameter ranging from 3 to 6 cm. It is crucial to describe the complex dynamic properties of this granular material to model the whole track and vehicle–track systems. However, the research on the theoretical modeling of ballast has not comprehensively characterized the nonlinear stiffness properties. In principle, ballast stiffness relies on interactions of particles/granules and frictional interlocking, which renders complex nonlinear properties in ballast under dynamic loading. The simple idealization of ballast stiffness has been used for multi-body simulations and to develop a non-destructive field-testing technique for monitoring railway components. The existing modeling of ballast stiffness is only considered as part of the nonlinear properties of ballasts [22,24,25,26,29,43].

Numerical modeling of railway ballast has been investigated to simulate the complex dynamical behavior of ballasts observed in experiments. Most finite element models (FEM) of ballasted tracks in the literature model the ballast layer as a continuum of homogenous material by discretizing the ballast layer into very small elements [6,51]. Although the continuum assumption works in most cases, the insight visualization of stresses and displacement cannot be correctly evaluated using FEM because the discontinuous ballast layer is modeled as a continuum material. The challenge of FEM is to choose an appropriate material model that simulates the complex discontinuous ballast material behavior.

The discrete element method (DEM) is a numerical method used to solve mathematical problems associated with discrete characteristic materials like granular material [52,53,54,55,56]. DEM has been used to model ballast where discontinuous property is considered. DEM can visualize the microscopic mechanical behavior of ballast under loading. The initial settlement, volumetric change, particle breakage, force chain distribution, particle displacement, particle shape, particle size distribution, and ballast fouling can be investigated using DEM. The research based on DEM correlates well with the ballast rheology mechanisms and the effect factors from the experiments. DEM is a powerful tool that can analyze the mechanical behavior of ballast microscopically and macroscopically. The main limitation of DEM is the massive requirement of computational time. In the literature, DEM is used mainly to validate experimental tests of railway ballast on a small scale.

However, previous analytical modeling of railway ballast for multi-body simulations and condition monitoring/evaluation needs to comprehensively cover the complex mechanical behaviors of ballast from experiments and numerical analysis. To fill this gap, this paper proposes a new model to comprehensively quantify the nonlinear, rheological, elastic-viscous-plastic properties of ballast and cover the experimentally observed phenomena of time/space-varying, displacement-dependent, and frequency-dependent stiffness and hysteresis.

On the other hand, the conventional ballast models used in multi-body simulations have been mainly idealized based on conventional experiments. The standstill track stiffness measurement can be performed with the traditional jack-loading method, impact hammer method, falling weight deflectometer method, and track-loading vehicle method. The continuous methods to measure track stiffness include the unbalanced loading laser displacement method, deflection basin deformation rate method, eccentricity excitation method, and the GPR method, which has been widely applied recently.

The dynamic stiffness of railway ballast is considered a time-varying quantity, considering the effect of contaminations, fouling, degradations, and wet content. The ballast stiffness also has a feature of high spatial variability due to section-by-section ballast deterioration and maintenance along the railway network. Railway ballast experiences high deterioration under loading and environmental conditions. By nature, some of the most notable track deteriorations are located on the ballast, which has high nonlinearities.

Stiffness is one of the basic performance parameters for railway track load-carrying capacity. Efficient and accurate stiffness measurement has been considered the foundation for further development of railway engineering and, therefore, has excellent theoretical and practical significance. Conventional methods for dynamic property characterization have been used for specific locations at the level of specific track sections. They are hard to use for quick and continuous measurements at the level of the railway network. GPR has been widely used for railway ballast inspection at high speeds of standard trains, by which the ballast dynamics properties can be inferred.

Since the ballast layer is the most dominating component of the track stiffness, the use of GPR makes the indirect estimation and monitoring of track and ballast stiffness possible, and the application of GPR to assess the ballast condition and stiffness has been widely developed [7,10,57,58,59,60,61]. Currently, for surveying and analyzing long sections of transport infrastructure, the mainstream detection method is GPR detection [57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79]. Even though GPR detection/inspection for ballast layers has matured, some challenges still need to be address, such as combining GPR results with track stiffness.

Previous studies have revealed that the overall electrical properties of the base material can be used to infer its strength characteristics, assuming that information on the layered properties of the track substructure can be used to indirectly infer the modulus and stiffness of the track and ballast [58,59]. These correlations can be used to predict the deformation properties of ballast and its dynamic behavior under heavy axle load in different conditions by using a GPR survey.

Based on this technique and similar tests on different types of ballast under different conditions, a criterion can be developed to detect the railroad high-risk areas and predict the engineering properties of ballast through GPR surveys. The time-varying property of ballast GPR signals was investigated in [60]. The study investigated the frequency spectrum of GPR in the time domain using the short-time Fourier transform (STFT) method, in which the authors demonstrated the variation trends of the frequency energy when the ballast was subect to different fouling conditions. Ref. [61] introduced an algorithm to extract the magnitude spectra at salient frequencies and then to classify railway ballast conditions with the support of vector machines. Ref. [65] introduced a theoretical framework to bridge the working principles of electromagnetic and mechanical systems for GPR application on ballast. Ref. [66] investigated a method to assess railway track conditions according to GPR data in the frequency domain.

The above methodologies successfully showed the reliability and effectiveness of GPR in assessing ballast stiffness and time-varying dynamic properties. Al-Qadi et al. [67] used time–frequency and Wavelet techniques to process ballast GPR data to evaluate ballast degradation due to fouling detection. Alzarrad et al. [77] used STFT and Wavelet techniques to process ballast GPR data for efficient railroad inspection. Conventional time–frequency methods for ballast GPR signal, such as STFT or spectrogram, which is the squared magnitude of STFT, have the inherent weaknesses of cross-term errors and inaccuracy for nonlinear signals. On the other hand, Wavelet transform has worse accuracy for spectrum values compared with STFT performance due to its disadvantages of shift sensitivity, poor directionality, and lack of phase information.

To further illustrate the time/space-varying property of the developed stiffness model of ballast, a detected ballast GPR signal is characterized using adaptive optimal kernel time–frequency representation (AOKTFR) [80,81,82]. In principle, the AOKTFR method has higher time–frequency resolution than conventional time–frequency methods and is more suitable for nonlinear time-variant systems. Compared with conventional time–frequency methods, AOKTFR has been proven to have a higher resolution and better performance in suppressing cross-terms [83,84,85].

The present analysis demonstrated that the AOKTFR of the ballast GPR signal exhibits specific time/space-varying properties. In contrast, the conventional time–frequency analysis of the same ballast GPR signal fails to clearly exhibit the time/space-varying properties. The results of the AOKTFR of the ballast GPR signal exhibit strong time-varying patterns and frequency modulation, which characterize the time-varying properties of the ballast. Based on the analysis, the time-varying nonlinearity of the ballast system is illustrated. This research also sheds light on the non-destructive testing of the ballast system and health monitoring of the track system by using advanced time–frequency analysis of the GPR signal.

2. Modeling of Complex Dynamical Properties of the Railway Track and Ballast

Understanding the interaction between vehicles and the track and the track system stiffness is vital to the safety and comfort of trains [2,3,4,5,6,46,47,48,49]. The stiffness of the ballasted railway track is primarily provided by resilient components such as the rail pads and ballast. The stiffness of track components changes as their condition deteriorates due to ballast damage and contamination. On the other hand, deviations in track component stiffness can also accelerate the track degradation process and lead to new track defects. Hence, track stiffness should be appropriately monitored during the whole life cycle of railway tracks. In addition, most of the dynamic train–track interaction models require the stiffness values of ballast as inputs. Therefore, accurate modeling and precise identification of the stiffness of track components from field measurement are very much desired.

Most vehicle dynamics commercial programs have limited track components, such as ballasts. Assumptions must be made to simulate the vehicle and track performances when using these programs. The stiffness of vertical tracks or track components (k) can be defined in the simplest form as the ratio between load (F) and deflection (z) as a function of time (t) [2,7,24,25]:

$$k\left(t\right)={\displaystyle \frac{F\left(t\right)}{z\left(t\right)}}$$

(1)

Commonly, the stiffness of the different components of the track structure is nonlinear. Moreover, at significant load intensities, increases in nonlinear stiffness are a consequence of ballast compaction. As the load value increases, the ballast deformations increase these contact surfaces, thus increasing the ballast stiffness. In addition to the load value, ballast stiffness depends on excitation frequency (f).

The time/space-varying ballast stiffness is one of the basic causes of differential track settlement, which has a primary influence on track geometry deterioration. The basic cause of the emergence of spatially varying track stiffness is the change of variable ballast.

Traditionally, track behavior analysis under vertical load is based on the beam on an elastic foundation model and Winkler’s hypothesis. This approach assumes that the behavior of the superstructure and substructure elements is linear and that there is proportionality between the load and the deflection, with the track and component stiffness having constant values. The improved approach to the stiffness definition includes both inelastic and nonlinear behavior of the superstructure and substructure elements, the difference between the stiffness under static and under dynamic load, and the variations of the stiffness along the track. A typical ballasted railway track system can be modeled as a layered structure that comprises rails, fasteners/pads/sleepers, and ballast, as well as foundations. Track stiffness can be defined as the combined effect of different track components.

Figure 1 shows the schematic model of the Quart vehicle–track system. It has various system components in a vertical direction, in which subsystems describing the Quart vehicle and the track are spatially coupled by the wheel–rail interface.

In the vehicle sub-model, two double-axle bogies support both ends of the body, and the bogies are connected to the wheels at one end and the body at the other end through the suspension. The track sub-model in Figure 1 represents the typical ballasted track structure, consisting of rails, pads/sleepers, ballast, and the subgrade/foundation. The rail can be regarded as a continuous Bernoulli–Euler beam, which is discretely supported at the rail–sleeper junctions by three layers of springs and dampers, representing the elasticity and damping of the pad/sleeper, the ballast, and the subgrade/foundation. The rail support modulus Kr can be expressed by the layers of springs with different stiffnesses [2,3,7,8,18,25,26,27,44]:

$$K_r={\displaystyle \frac{1}{1/k_p+1/k_t+1/k_b+1/k_s}}$$

(2)

where k_p represents the rail pad stiffness, k_t represents the sleeper stiffness, k_b represents the ballast stiffness, and k_s is the foundation soil stiffness. The following equation can express the motion of the vehicle subsystem:

$$M_v{\ddot{x}}_v+C_v{\dot{x}}_v+K_v{x}_v=F_v\left({\dot{x}}_v,x_v,{\dot{x}}_t,x_t\right)+F_{ext}$$

(3)

where ${\ddot{x}}_v$,${\dot{x}}_v$, and $x_v$ represent the accelerations, velocities, and displacements of the vehicle subsystem; $M_v$ represents the mass of the vehicle; $C_v, K_v$ are the damping and the stiffness originating from the vehicle subsystem, to describe nonlinearities of the suspension; ${\dot{x}}_t,x_t$ are the velocities and displacements of the track subsystem; and $F_v\left({\dot{x}}_v,x_v,{\dot{x}}_t,x_t\right)$ represents the system load, which represents the nonlinear contact forces between the wheel and rail. The parameter ${\dot{x}}_v,x_v$ of the vehicle and ${\dot{x}}_t,x_t$ of the track decide the model. $F_{ext}$ represents the external forces, including gravitational forces and forces resulting from the centripetal acceleration when the vehicle is running through a curve.

The equations of the vertical motion of rails have been expressed according to fourth-order partial differential equations:

$$E{I}_y{\displaystyle \frac{{\partial }^4{Z}_r\left(x,t\right)}{\partial x^4}}+m_r{\displaystyle \frac{{\partial }^2{Z}_r\left(x,t\right)}{\partial t^2}}=-{\displaystyle \sum }_{i=1}^N{F}_{svi}\left(t\right)\delta \left(x-x_i\right)+{\displaystyle \sum }_{j=1}^4{P}_j\left(t\right)\delta \left(x-x_{wj}\right)$$

(4)

In Equation (4), Z_r(x, t) represents the vertical displacements of the rail; m_r represents the rail mass per unit length; ρ represents the rail density; EI_y represents the rail bending stiffness to the Y-axle; F_svi(t) represents the vertical dynamic forces at the ith rail-supporting point; P_j(t) represents the jth wheel–rail vertical forces; and δ(x) represents the Dirac delta function. Therefore, the final equation of the track subsystem is an equation including sleepers and ballast blocks, the model of which can be represented by individual rigid masses, which can be expressed as

$$M_t{\ddot{x}}_t+C_t{\dot{x}}_t+K_t{x}_t=F_t\left({\dot{x}}_v,x_v,{\dot{x}}_t,x_t\right)$$

(5)

where $M_t$ represents the mass of the track structure; $C_t$ and $K_t$ are the damping and the stiffness of the track subsystem; ${\ddot{x}}_t$ represents the accelerations of the system; and $F_t\left({\dot{x}}_v,x_v,{\dot{x}}_t,x_t\right)$ represents the load of the track subsystem, to represent the nonlinear wheel–rail forces obtained by the wheel–rail coupling model.

Many existing research studies have investigated various nonlinearities in the track system and classic nonlinearities in the train system. Ballast is vital to track alignment, stability, and sustainability. The mechanical behavior of railroad ballast has been interpreted by modeling based on large-scale experiments. However, the existing basic ballast models have not considered amplitude-dependent stiffness, frequency-dependent stiffness, and complex damping properties, although experimental evidence has proven these properties to be critical. To comprehensively quantify the nonlinear properties of ballast systems in the framework of rheology theory, this paper deploys and extends the general multiple parameters model for characterizing time/space-varying amplitude-dependent stiffness, frequency-dependent stiffness, and hysteresis in ballast systems.

Rheological theory has been used to quantify the nonlinear elastoplastic-viscoelastic properties of materials with multiple nonlinear spring elements, viscous damping, and frictional elements [86,87,88,89,90,91]. In numerical simulations of ballast systems, a generalized Maxwell model has been used to establish constitutive relations on a material level regarding stresses and strains. The proposed model in this paper extends the understanding from the material level to the structural component level. The straightforward extension to a structural level is given in this research in terms of forces and displacements. Moreover, since the experimental data are widely established as forces and displacements, the proposed parameter model can be used for system identification and facilitating efficient, accurate inspection.

To quantify the complex ballast stiffness, the proposed model is based on a generalized Maxwell model with multiple parameters [90,91], which is a combination of nonlinear elastic, friction, and viscous portions. The model is based on a superposition of elastic, friction, and viscous forces, which reflects the time/space-varying amplitude-dependent stiffness, frequency-dependent stiffness, and rate-dependent hysteresis. The detailed model of ballast supporting force characterizing the frequency-dependent, amplitude-dependent, and history-dependent stiffness is given as follows. Figure 2 shows the schematic model of the generalized model for the rheological ballast system. It is a generalized Maxwell model with spring/damper/sliding friction elements, comprehensively characterizing the ballast system’s nonlinear and time/space-varying viscous-elastic-plastic properties.

$$F\left(x,\dot{x}\right)=F_e+F_f+F_v$$

(6)

where $F_e$ is the non-linear elastic force, $F_f$ is the ballast internal interaction or friction force, $F_v$ is viscous force. For steady harmonic excitation $x=x_{0}sin\omega t,$ the amplitude of each force can be expressed as

$$\left\{\begin{array}{c}{F}_e=k_t{\displaystyle \frac{2d_t}{\pi }}\tan {\displaystyle \frac{\pi x}{2d_t}}+a\left(y\right)\hfill \\ F_f={\displaystyle \frac{F_{fmax}}{2x_2}}\left(\sqrt{{x_2}^2+{x_0}^2+6x_{2}x}-x_2-x\right)\hfill \\ F_{vRc}=\left({\displaystyle \frac{{\left(\omega c_1/k_1\right)}^2}{1+{\left(\omega c_1/k_1\right)}^2}}{k}_1+{\displaystyle \frac{{\left(\omega c_2/k_2\right)}^2}{1+{\left(\omega c_2/k_2\right)}^2}}{k}_2\right)x\hfill \\ F_{vIm}=\left({\displaystyle \frac{c_1}{1+{\left(\omega c_1/k_1\right)}^2}}+{\displaystyle \frac{c_2}{1+{\left(\omega c_2/k_2\right)}^2}}\right)\omega x\hfill \end{array}\right.$$

(7)

where $F_{vRe}$ and $F_{vIm}$ are the real and imaginary force components of complex viscoelastic force, in which $x_0$, $\omega ,$ and t are displacement amplitude, circular excitation frequency, and time. $F_{fmax}$ is the maximum friction force, whereas the displacement $x_2$ is the displacement required to gain a friction force as large as $F_f=$ $F_{fmax}/2$. a(y) is the time/space-varying term in the elastic force, dependent on spatially variable y(t) along rail locations due to the variation of ballast properties. $k_t$ is the stiffness coefficient, $d_t$ is the characteristic thickness representing the location of the elastic curve asymptote. $c_1$, $c_2$ are the viscous damping coefficients, and $k_1$, $k_2$ are the stiffness coefficients. A typical dynamical stiffness versus amplitude and frequency based on the model is plotted in Figure 3, with parameters $k_1=$ 35 MN/m, $k_2$= 250 MN/m, $c_1$ = 300 kNs/m, $c_2$ = 40 kNs/m, $F_{fmax}$ = 140 kN, and $x_2$ = 2 mm. Figure 4 shows a typical hysteresis loop of the force versus harmonic displacement based on the model, illustrating the complex stiffness nonlinearity.

The new nonlinear dynamic stiffness model enables comprehensively characterizing the dynamic properties of displacement, frequency dependence, and hysteresis, which have been widely recorded in previous experimental research [3,8,24,25,38,44,57,58,59]. In contrast, previous models in the literature only quantify some of these properties.

Moreover, the above stiffness model consists of a time/space-varying element. Previous research suggests that the ballast stiffness has time/space variability, significantly contributing to track deterioration rates. Thus, it must be considered when designing and maintaining railway tracks. The nonlinear model in Equation (7) characterizes the ballast’s time/space-varying nonlinearity at local or specific locations by variable component a(y). In contrast, ballast has nonuniformity, and y(t) is a variable along the track. Furthermore, even at the exact location, the stiffness and damping properties of the ballast layer are not constant with time due to ballast densifications and deteriorations under cyclic loading and after long-term service.

The above multiple parameter stiffness model of railway ballast is in the framework of rheology theory. The model comprehensively describes railway ballast’s time/space-varying, nonlinear, viscoelastic, elastoplastic, and hysteresis properties. Multiple-parameter models can be used effectively for multi-body vehicle–track system simulation and efficiently for quick testing of assets and diagnosis. The parametric model provides the benefit of computational simulation being cost-friendly and experimental stiffness estimation being cost-effective.

3. Detecting Time-Varying Dynamic Properties of Ballast Using GPR

Since railway ballast properties are highly varied along the rail network, vehicle–track system modeling, evaluation, and validations are required to be implemented and verified at the network level instead of locally at the section level. Efficient measures to characterize track and ballast time/space-varying dynamic properties using GPR along railways have been widely applied in the community [57,58,59,60,61,62,63,64,65,66,67,69].

As a very efficient method to obtain physical and geometric information for the railway ballast, the GPR technique is based on the transmission/reception of electromagnetic waves into the ground in a given frequency band. The main advantage of this approach is that drive-by high-speed sampling and minimum or non-destructive sampling are required to calibrate the system. The inhomogeneity of materials with different electromagnetic properties and the interface between different structured layers can lead to changes in the position and amplitude of the signal peaks, from which information on layer thickness and material properties can be inferred.

Figure 5 illustrates a cross-sectional schematic of a railway substructure and the associated trajectories (A-scans) from a single GPR measurement, which include a signal generator (transmitter), transmitting and receiving antennas, and a recording device (receiver). The transmitter generates a pulse to detect objects and transmits it to the transmitting antenna. When an electromagnetic wave strikes an object whose electrical properties are different from other materials, part of the wave’s energy is reflected to the receiving antenna. This energy is then sent to the receiver for storage and display.

Because the frequency spectrum of the GPR return reveals the characteristics of the materials on the electromagnetic wave path, the frequency features are used to categorize ballast conditions automatically. Generally, the inference, identification, and classification of the GPR system comprise three steps: pre-processing, extracting features, and classification, as illustrated in Figure 6.

The conventional approach to GPR detection and analysis is based on a linear framework in which linearity similarity is used to correlate mechanical dynamics with electromagnetic system/electric circuit systems characterized by discrete, specific natural frequencies. STFT has been used widely in GPR signal processing for frequency analysis. The method obtains data in the time and frequency domains by tracking changes in the frequency spectrum over time (or depth) and ballast properties (e.g., the correlation between the ballast’s electromagnetic and mechanical properties).

Conventionally, in the test analysis using GPR, the ballast system is assumed to be linear and time-invariant. The output of GPR is assumed to be a convolution of the incident signal or using the superposition principle. However, the ballast system is an inhomogeneous and non-isotropic nonlinear system, and its property is a time-variant nonlinear system in which the convolution or superposition principles cannot be used for high accuracy. As such, the existing approaches to extract features by using deconvolution, such as STFT, have limitations.

4. Characterization of Dynamic Properties of Ballast GPR Signal Using AOKTFR

In the following, a typical ballast GPR signal is analyzed using advanced time–frequency analysis in the context of a time-varying nonlinear framework. In principle, the sub-band time–frequency analysis-adaptive optimal kernel time–frequency representation (AOKTFR) method has been proven to have the advantages of higher time–frequency resolution than conventional time–frequency methods and is more suitable for nonlinear time-variant systems [80,81,82,83,84,85].

Compared with other methods, AOKTFR has been demonstrated to have a higher resolution and a better performance to suppress cross terms, which can provide new ideas for studying the properties of time-varying nonlinear signals. In the field of time-varying signal analysis, the advantages of AOKTFR are comprehensively demonstrated through comparison with various time–frequency methods such as Wigner distribution, Choi-Williams distribution, spectrograms, page distribution, and Rihaczek distribution [92,93,94]. The AOKTFR of the analyzed signal x(t) can be expressed as

$$\begin{array}{c}AOK\left(t,f\right)={{\displaystyle \int }}_{-\infty }^{+\infty }{{\displaystyle \int }}_{-\infty }^{+\infty }A\left(\tau ,v\right)\varphi \left(\tau ,v\right)exp\left[-2\pi \left(vt+f\tau \right)\right]dvd\tau \\ A\left(\tau ,v\right)={{\displaystyle \int }}_{-\infty }^{+\infty }x\left(t+{\displaystyle \frac{\tau }{2}}\right)x^{\ast }\left(t-{\displaystyle \frac{\tau }{2}}\right)exp\left(-j2\pi v\tau \right)d\tau \end{array}$$

(8)

$\Phi \left(\tau ,v\right)$ is a 2D radial Gaussian $\Phi \left(\tau ,v\right)=e^{-r^2/2{\sigma }^2\left(\psi \right)}$

$$A\left(t,\tau ,v\right)=\int s^{\ast }\left(u-{\displaystyle \frac{\tau }{2}}\right)\cdot {\omega }^{\ast }\left(u-t-{\displaystyle \frac{\tau }{2}}\right)\cdot s\left(u+{\displaystyle \frac{\tau }{2}}\right)\cdot \omega \left(u-t-{\displaystyle \frac{\tau }{2}}\right)\cdot e^{jvu}du$$

(9)

The AOKTFR can effectively suppress the cross-term while keeping a high time–frequency concentration kernel function. The optimal kernel function can be obtained by minimizing auto-term distortion by passing as much auto-term energy as possible for a kernel to suppress cross-terms. AOKTFR time–frequency representation is an efficient method to track the instantaneous frequency and energy variation change with time.

Figure 7 shows a trace of the GPR signal from clean ballast. The sensing frequency of GPR is 1 GHz. Similar types of GPR signals have been widely characterized in the literature. Figure 8 and Figure 9, respectively, show the FFT and power spectrum of the GPR signal. Figure 10 shows the spectrogram of the GPR signal, and Figure 11 shows the AOKTFR time–frequency representation of the GPR signal.

The results show that the output of conventional spectrogram time–frequency analysis differs from that of AOKTFR analysis. The salient difference can be seen by comparing Figure 10 and Figure 11. The AOKTFR time–frequency representation strongly exhibits time-varying and precise frequency modulation, whereas the spectrogram does not. The obtained spectrum features from the spectrogram and AOKTFR of the GPR signal are obviously different in different frequency bands.

Previous research only used conventional time–frequency analysis to process the ballast GPR signal in the time–frequency domain. In the literature, the advanced methods for classifying and identifying the GPR signal of a ballast system are all based on conventional time–frequency analysis, using the extraction of magnitude and energy spectra at specific frequencies and bands [66,67,68,69,70,71,72,73,74,75,76,77].

Since conventional time–frequency analysis, such as STFS or spectrograms, has the inherent weakness of cross-term errors and is not suitable for complex nonlinear signals, AOKTFR can, by nature, improve the existing results for GPR signal analysis, classification, and identification.

5. Conclusions

This paper develops a new dynamics model of the railway ballast system. As the most sensitive and dominant component in track stiffness, the dynamic stiffness of the ballast layer is comprehensively modeled with multiple parameters in terms of time/space-varying displacement-dependent stiffness, frequency-dependent stiffness, and hysteresis. Based on published experimental results of real ballast on tracks and published numerical analysis of rheological properties of ballast at the material level, the proposed parametric rheological model extends to the ballast structure level for the first time, which can be used for multiple-body system simulation as well as system identification and diagnosis.

Furthermore, the aim is to characterize the time-varying dynamic properties of ballast using a GPR signal; a typical ballast GPR signal is investigated using AOKTFR and a conventional time–frequency method spectrogram. AOKTFR results exhibit time-varying and frequency modulation patterns, whereas conventional time–frequency analysis results do not show the time-varying properties due to the inherent errors of cross terms. This research helps to correlate the time-varying nonlinear properties of the ballast system model with the GPR measurements. It helps to understand/infer the time-varying nonlinear properties of ballast from GPR measurements. The research also sheds light on the non-destructive testing and health monitoring of ballast systems through advanced time–frequency analysis of GPR signals for higher accuracy as compared to conventional time–frequency analysis.

This paper contributes two novelties. The first is the presentation of a new approach that is well-suited to comprehensively model complex nonlinear ballast stiffness, and the second is a study identifying a better approach for characterizing the time-varying frequency of the ballast GPR signal than conventional time–frequency approaches used for correlation with stiffness.