Abstract: The piles with irregularly shaped cross-sections are increasingly prevalent in engineering applications, yet the theoretical researches in this area remain relatively scarce. In response to the current state of researches, the Hamilton's principle and variational calculus are employed to derive the governing equations for the model of irregularly shaped pile-viscoelastic soil interaction in the Cartesian coordinates. Utilizing the COMSOL, a two-dimensional model for soil with irregular boundaries is established to solve the governing equations for the soil, overcoming the challenge of solving the soil displacement function caused by irregular boundaries. MATLAB is employed to solve the control equations for the piles using the methods for calculating the boundary values, and subsequently, an iterative program is developed in the MATLAB to perform the coupled calculations of the aforementioned equations. A theoretical model for analyzing the vertical dynamic responses of irregularly shaped piles is established. The semi-analytical solution of the theoretical model is compared with the existing analytical solutions to validate the reliability of the proposed method. Finally, the influences of the cross-sectional parameters of irregularly shaped piles, the pile-to-soil modulus ratio and the slenderness ratio of the piles on their head impedance are discussed. The results indicate that as the external load frequency increases, the influences of the cross-sectional shape of the piles on their pile head impedance gradually increase, with the irregular effects of an H-shaped pile being more pronounced compared to those of the X-shaped and rectangular piles.