## 2 Mathematical Model  
In this paper, a pile embedded in frozen soil was studied. The bottom of the pile was assumed to be an elastic half-space, and the top of the pile was subjected to an arbitrary vertical excitation load. The overlying frozen soil layer is described as a homogeneous and isotropic frozen saturated porous medium composed of soil particles, ice and water. By ignoring the interaction between the solid and the ice phases, the governing equation is [30; 36; 37; 38; 39; 40]:  
\[\overline{R}\
abla\
abla\cdot\overset{\rightarrow}{u}-\overline{\mu}\
abla \times\
abla\times\overset{\rightarrow}{u}=\overline{\rho}\frac{\partial^{2} \overset{\rightarrow}{u}}{\partial t^{2}}+\overline{A}\frac{\partial \overset{\rightarrow}{u}}{\partial t} \tag{1}\]  
where \(\overset{\rightarrow}{u}\) is the displacement field vector, and \(\overset{\rightarrow}{u}=\left\{\overset{\rightarrow}{u}\overset{(s)}{u}, \overset{\rightarrow}{u}\overset{(f)}{u},\overset{\rightarrow}{u}^{(i)}\right\}^ {\text{T}}\) represents the solid, liquid and ice displacements of the saturated frozen porous medium, respectively. \(\overline{R}\), \(\overline{\mu}\), \(\overline{\rho}\) and \(\overline{A}\) are coefficients of the frozen saturated porous medium, which can be expressed as:  
\[\overline{R}= \begin{bmatrix}R_{11}&R_{12}&0\\ R_{21}&R_{22}&R_{23}\\ 0&R_{32}&R_{33}\end{bmatrix}  \tag{2}\]  
\[\overline{\mu}= \begin{bmatrix}\mu_{11}&0&0\\ 0&0&0\\ 0&0&\mu_{33}\end{bmatrix}  \tag{3}\]\[\overline{\rho}= \begin{bmatrix}\rho_{11}&\rho_{12}&0\\ \rho_{21}&\rho_{22}&\rho_{23}\\ 0&\rho_{32}&\rho_{33}\end{bmatrix}  \tag{4}\]  
\[\overline{A}= \begin{bmatrix}b_{11}&-b_{11}&0\\ -b_{11}&b_{11}+b_{33}&-b_{33}\\ 0&-b_{33}&b_{33}\end{bmatrix}  \tag{5}\]  
where \(R_{11}=K_{1}+\frac{4}{3}\mu_{11}\), \(R_{12}=R_{21}=C_{12}\), \(R_{22}=K_{2}\), \(R_{23}=R_{32}=C_{23}\), \(R_{33}=K_{3}+\frac{4}{3}\mu_{33}\). \(K_{1}\), \(K_{2}\) and \(K_{3}\) represent the bulk moduli of solid, pore fluid and ice, respectively. \(C_{12}\) and \(C_{23}\) denote the elastic coupling coefficients of pore fluid and solid and ice, respectively; \(\rho_{11}\) is the density of the soil particles, \(\rho_{22}\) is the density of the fluid, \(\rho_{33}\) is the density of ice, \(\rho_{12}=\rho_{21}\) is the density of solid-liquid interaction, \(\rho_{23}=\rho_{32}\) is the interaction density of ice and liquid phases; \(b_{11}\) and \(b_{33}\) are the viscosity coupling coefficients of soil skeleton and pore fluid, and ice skeleton and pore fluid, respectively; \(\mu_{11}\) and \(\mu_{33}\) are the shear moduli of soil skeleton and ice, respectively. According to the previous research results of pile dynamics [2; 3; 10; 27], for most concrete solid piles with large slenderness (i.e., the slenderness of the pile is greater than 10), the pile can be assumed to be an elastic and circular homogeneous one-dimensional rod. \(H\), \(a\), \(\rho_{\text{b}}\) and \(E_{\text{b}}\) are the length, diameter, body density and elastic modulus of the pile, respectively. The governing equation of the vertical vibration of the pile can be written as [30]:  
\[E_{\text{b}}\tau a^{2}\frac{\text{d}^{2}w_{\text{b}}}{\text{d}z^{2}}-\rho_{b} \tau a^{2}\frac{\text{d}^{2}w_{\text{b}}}{\text{d}t^{2}}=f(z) \tag{6}\]  
where \(w_{\text{b}}\) denotes the displacement of the pile, \(f(z)=-2\pi a[\tau_{zz}^{(s)}(a,z)+\tau_{zx}^{(i)}(a,z)]\) is the friction stress around the pile, and \(\tau_{zx}^{(s)}\) and \(\tau_{zx}^{(i)}\) denote the traction of the pile shaft produced by the soil skeleton and the ice skeleton, respectively. In this paper, the interaction of the pile-soil system was simplified by considering it as the combination of the upper homogeneous isotropic saturated frozen porous medium and the underlying elastic half-space, as shown in Figure 1a.