## 1 Introduction  
The theory of dynamic pile-soil-cap interaction is an important work in earthquake engineering, transportation engineering and machinery foundation design [1]. Over the past several decades, the dynamic interaction between pile and soil has been widely investigated, especially in the research of the low-strain reflection theory of a single pile and its application in the field of pile integrity detection. Large amounts of work have been conducted on the technology of detecting defective piles, large-diameter piles and pipe piles [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. However, the dynamic interaction between a pile with cap and the soil has not been sufficiently studied due to the complexity of the pile-soil-cap interaction. There are three methods for the studies on the dynamic responses of piles with caps including theoretical analysis [14, 15, 16, 17, 18, 19, 20], numerical simulations [21, 22] and in situ or indoor model tests [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. The analytical method, despite its difficulty and challenges, is still widely used by researchers as it helps to understand the inherent mechanisms in the dynamic interaction of the pile-soil-cap system. The vertical and lateral vibrations are extensively investigated to give insight into the dynamic responses of pile foundations [35, 36, 37, 38, 39, 40, 41, 42, 43]. The theoretical investigations of the vertically dynamic response of a pile foundation involve two keys: soil simulation and pile-soil interface description. In the early stage, a single-phase elastic medium was used to simulate the soil. In recent years, the theory of saturated or unsaturated porous medium was gradually developed to simulate the dynamic interaction between the pile and soil [44; 45; 46; 47; 48; 49; 50; 51; 52; 53]. However, with the development of engineering in high-latitude and cold regions, the application of pile foundation has been highly popularized in permafrost or seasonal permafrost areas. Frozen soil is a special kind of soil that is a complex multiphase structure with solid particles, ice, water and gas. A prominent feature of frozen ground engineering is that there is a freezing/thawing phenomenon with the change in temperature that will have a significant impact on the dynamic responses of the pile foundation [54; 55; 56]. Based on the theories of frozen porous media established by [PERSON] [57] and [PERSON] [58], [PERSON] et al. [59] and [PERSON] et al. [60] obtained the analytical solution of the vertical vibration of a single pile in frozen porous media. Immediately after, the analytical solution for pile-soil system vertical vibrations in a double-layered stratum under the freezing and thawing processes of a saturated porous medium was derived by [PERSON] et al. [61]. However, the effects of the slippage between the pile and soil and the mass of pile cap were not considered, and it is worth noting the lack of validation of the relevant model test data. In reality, the slippage between the pile and soil is a very common phenomenon when the pile is subjected to large external excitations. [PERSON] and [PERSON] [62] investigated the dynamic interaction of pile-soil structure in liquefied sand by FLAC3D. They proposed an interface model to simulate the dynamic response between the pile and soil, which in effect was that the non-linearity of the soil was considered by using hysteretic damping. [PERSON] et al. [63] considered the slip contact between the pile and soil and established a numerical FEM model of the vertically dynamic responses of a pile foundation. Numerical results also revealed that the resonant frequencies were overestimated, and their corresponding amplitudes were underestimated when the perfect bonded model was adopted. Although the numerical method has developed rapidly in recent years, there are some difficulties in parameter acquisition and constitutive relationship determination.