# Modeling and Characterization of Complex Dynamical Properties of Railway 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 subjectof 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. [PERSON] and [PERSON] [21] discussed the nonlinear dynamic characteristics of a railway. [PERSON] 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]. [PERSON] and [PERSON] [26] discussed the elastoplastic properties of ballast and nonlinear load-deformation behavior under cyclic loading. [PERSON] et al. [27; 28] proposed a simple geotechnical rheological model for simulating the viscoelastic-plastic response of a ballasted railway substructure. [PERSON] and [PERSON] [29] proposed a simplified nonlinear dynamic model of ballast to quantify the frequency-dependent features in flood situations. [PERSON] and [PERSON] [30] studied the space-varying track stiffness associated with the longitudinal inhomogeneity of the track using a track status estimation method. [PERSON] and [PERSON] [31; 32; 33] studied the detection of railway ballast damage using various nonlinear ballast stiffness and deformation models. [PERSON] et al. [34] conducted a nonlinear analysis of the ballast influence on the train railway bridge using a Duffing oscillator model. [PERSON] 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. [PERSON] et al. [36] investigated the nonlinear stiffness of a railway ballast with slight strain. [PERSON] et al. [37] reviewed the stiffness irregularity of railway ballasted tracks. [PERSON] et al. [38] studied a track dynamic model with the evolution of rail corrugation in sharp curves. [PERSON] et al. [39] used a viscoelastic rheological track model to evaluate the track lateral response. [PERSON] et al. [40] investigated railway track acceptance, which is the inverse of stiffness.