## 1 Introduction  
To better understand the influences of various factors such as train speed on the performance of railway structures, the dynamic analysis of railway formations under a train load has become increasingly in-depth. Some existing works [1, 2, 3] reported that under higher train speed conditions, the railway track and underlying ground show significant vibrations, with a substantial vertical displacement or dynamic stress. Therefore, it is essential to study the dynamic response of the track formation on soft soil under moving train loads. The vibration of the railway structure was commonly affected by the train speed, axle load, embankment height, and stiffness. Many scholars [4, 5, 6, 7, 8, 9] discussed the effect of these factors according to experimental, analytical, or semi-analytical methods. For example, [PERSON] et al. [10] studied the vibration velocities of a piled raft supported embankment and foundations in soft soil, with different underground water levels, using an experimental method. [PERSON] et al. [11] investigated the ground vibration of a semi-infinite space superposition beam model of layer soils under static and dynamic harmonic loads. [PERSON] and [PERSON] [12] proposed a time-domain semi-analytical solution, to analyze ground responses under moving loads, while considering various vehicle velocities. [PERSON] et al. [3] studied the dynamic response of slab track in soft soil under a moving train load using a semi-analytical method. The research indicated that the vibration of soft soil under the conditions of a high-speed train is more obvious than under the condition of a low-speed train. On the other hand, the vibration behavior of the track formation has also been investigated based on numerical simulations [13; 14; 15; 16; 17; 18; 19]. For example, the dynamic response of subsoil under the static and dynamic loads of moving trains was studied by [20; 21]. In addition, a numerical model was proposed by [PERSON] and [PERSON] [22], to analyze changes in track vibration and train speed. The results showed that a high-speed train causes a sizeable dynamic response. [PERSON] [23] pointed out that train speed and soil stiffness jointly affect the vibration of subgrade soil. [PERSON] et al. [24] established a 2.5 D FEM, to simulate the layered soil vibration caused by high-speed trains. The results showed that track irregularities with a smaller wavelength caused greater vibration than those with a longer wavelength. More importantly, the stress-time history of a beam bridge under a dynamic moving load was calculated using the transient dynamic method (TDM) by [PERSON] et al. [25]. The above research demonstrated that train speed makes a remarkable contribution to the vibration of railway structures. However, it is difficult to deal with nonlinear and geometrically complex problems in engineering practice using the analytical method. The data obtained by field measurements and laboratory model tests under complex practical conditions have a high reference value, but the cost is high and the process is time-consuming. With the development of high-performance computers, various numerical methods have been widely used to solve the dynamic response problem of variation for geotechnical problems. On the basis of existing studies and combined with the principle of elastic dynamics, this paper established a coupled dynamics model of a train-ballastless track-layered subgrade-foundation. The transient dynamic method (TDM) was used to analyze the dynamic response of the railway formation under a high-speed train load. First, by comparing the results of the SSM recommended in the code design [26], the reliability of the proposed method in analyzing the vibration response of a railway line was verified. Then, the effects of train speed and embankment stiffness on the dynamic responses of tracks on soft soils, induced by moving train loads, were evaluated.