## 6 Conclusions  
In this study, a new pile-soil-cap model was developed to investigate its vertical vibration in a double-layered stratum under the freezing and thawing processes of a saturated porous medium, taking into account the slippage effect between the pile and soil. Based on the theoretical model, the degradation of the four conditions described in the paper was verified, and the comparison between the existing solutions illustrated the need to take the pile-soil slippage effect. Then, a dynamic model test of the pile-soil-cap in a self-made ground-freezing system was performed. The reasonableness of the model was further validated by comparing the proposed solution with the experimental results. Parametric analyses were also carried out to discuss the effects of the cap mass, freezing and thawing depth, soil temperature, dynamic shear modulus of unfrozen soil, pile bottom-support stiffness and support parameters at the pile cap bottom on the amplitude-frequency curve of pile-soil-cap. The main findings can be summarized below, as follows:  
* Based on the theory of composite porous media, a simplified axisymmetric model of the vertical vibration of a single-pile foundation in a frozen/thawed stratum is established, and the analytical solution of the dynamic response of the pile cap is obtained. The present solution taking into account the slippage between the pile and soil is more reasonable than the perfect contact model, which overestimates the resonant frequency and underestimates the amplitude of the single-pile foundation;
* Both the comparisons of the present solution with the existing experimental results of a vertical vibration model test for a pile cap foundation in unfrozen soil, and the present experiment in an artificially freezing/thawing ground show that the slippage model for pile foundation can agree well with the experimental results;
* The mass of the pile cap has an obvious influence on the vertical vibration of the pile foundation, which should not be ignored. The thickness of the active layer, the freezing temperature, the shear modulus of the unfrozen soil and the contact parameters of the pile and soil also have a significant influence on the vibration of vertical the pile foundation;
* The slippage parameters in the proposed method can be calibrated according to different pile and soil types, which needs to be further studied through a large number of experiments.  
Figure 12: Effects of the support coefficients of the pile cap on the vertical amplitude–frequency curves. (**a**) \(k_{ms}\) and (**b**) \(c_{ms}\).  
**Author Contributions:** Q.L.: methodology, supervision, validation, resources, writing--review and editing and funding acquisition. X.L.: methodology, formal analysis, investigation, data curation, software and writing--original draft. M.W.: methodology, supervision, resources and funding acquisition. L.H.: conceptualization and resources. W.D.: writing--review. J.L.: data curation. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by the National Natural Science Foundation of China [grant numbers 52108347]. The authors are grateful to the editorial board and reviewers of this paper.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.