# Dynamic Analysis of CRTS II Ballastless Track-Vehicle System Based on Moving Unit Method
## 1 Introduction
[14] provided a finite element equation for an infinite Euler beam on a linear viscoelastic Kelvin foundation under simple harmonic moving loads; using this method, they obtained the equations for the immobile point loads of orbital beams under kinematic loads. Inspired by these ideas, [PERSON] et al. [15] used moving coordinates to solve moving vehicle-track vibration problems and named the method the moving element method (MEM). Since then, many researchers have applied MEM to solving vehicle-track problems. [PERSON] et al. [16] modeled the ballasted track model in three layers: a continuous rail layer, a discrete steeper layer, and a track bed; the dynamic response of the train-track system was calculated using the moving unit method. [PERSON] et al. [17] established a ballastless track unit with a three-layer Euler beam model as the basis, derived the corresponding stiffness, mass, and damping matrices, and investigated the dynamic characteristics of the train and ballastless track system, but they neglected the effect of discrete supports and the non-linear relationship between wheel and rail contact. It has been shown through existing domestic and international research that the use of MEM simulations to calculate the dynamic response of the track can significantly reduce the size of the track calculation model, providing more efficient, accurate, and effective calculations. Track unevenness has a significant effect on the coupled vibration of the vehicle-track system and is very harmful to the high-speed rail track system. In this regard, a large amount of research has been produced by academics at home and abroad. In the time domain dynamic analysis, the most commonly used methods to describe track unevenness are the deterministic function method and the stochastic process method. The deterministic function method assumes that the track irregularity has a sine or cosine curve shape; for instance, [PERSON] et al. [18] proposed a two-dimensional Timoshenko beam vehicle-track coupling numerical model to investigate the influence of simple harmonic function track unevenness on the track mat stiffness. In the stochastic process method, a time-domain sample of the stochastic unevenness is extracted from the power spectral density (PSD) of the track; for instance, [PERSON] et al. [6] treated the track vertical uneveness as a Gauss smooth stochastic process and investigated the dynamic response of the vehicle and track system. [PERSON] et al. [7] used the Fourier inversion method based on the German track uneveness spectrum to obtain the track spectrum at wavelengths greater than 1 m for the effect of random process irregularities on the dynamic response of the ballastless track. [PERSON] et al. [19] selected a high-speed rail track and bridge system for nonlinear dynamic analysis, proposed the distribution law of the seismic residual shift of high-speed rail track and a bridge system under different seismic intensities, and introduced the evaluation method of the seismic power spectrum density (PSD) of track uneveness. [PERSON] et al. [20] proposed an effective method to predict the ground vibrations caused by metro trains, taking into account random track uneveness. In order to study the impact of ballastless track uneveness on China's high-speed railways, the China Academy of Railway Sciences proposed a ballastless track uneveness spectrum for China's high-speed railways in 2014, which better reflects the current overall design and construction level of ballastless tracks in China [21]. In this article, on the basis of the moving unit method, a vibration analysis model of the coupled high-speed rail vehicle-ballastless track system is established and a MATLAB calculation and analysis program is developed, taking into account the discontinuity of the prefabricated track plate and the discrete support of the trackpad in CRTS II ballast-less track.