## 8 Conclusions  
A new3D axisymmetric model was presented to describe the longitudinal vibration of a large-diameter floating pile in viscoelastic surrounding soil with radial inhomogeneity, taking the effects of three-dimensional wave propagation and lateral inertia of the pile shaft into account. The corresponding analytical solution was deduced and validated via comparisons with existing solutions. Parametric analyses were also conducted to examine the influences of Poisson's ratio of the pile shaft, pile length and viscous damping coefficient, as well as the degree and radius of disturbed surrounding soil, on the longitudinal impedance of pile shaft. The relating results demonstrate that:  
1. Both the resonance frequency and amplitude of longitudinal impedance decrease with the increasing Poisson's ratio of the pile shaft in the high-frequency range, while the Poisson's ratio of the pile shaft has a negligible effect on the longitudinal impedance of the pile shaft in the low-frequency range. 2. The oscillation amplitude and resonance frequency of the longitudinal impedance increase with the decrease of pile length in the high-frequency range and this tendency becomes more significant as the frequency increases. In addition, the shorter the pile length, the greater the difference in the longitudinal impedance. 3. The viscous damping coefficient of the surrounding soil has an obvious influence on the longitudinal impedance of the pile shaft within the high-frequency range. With the increase of  
Figure 11: The effect of the radius of strengthened soil due to a construction disturbance on the longitudinal impedance of a pile shaft: (**a**) true stiffness; (**b**) equivalent damping. Figure 10: The effect of the degree soil is strengthened due to a construction disturbance on the longitudinal impedance of pile shaft: (**a**) true stiffness; (**b**) equivalent damping. the viscous damping coefficient of the soil, both the resonance frequency and amplitude of the longitudinal impedance decrease. 4. The degree of disturbed surrounding soil has a significant effect on the oscillation amplitude and frequency of longitudinal impedance. In contrast, the change of the radius of disturbed soil has an influence only on the oscillation amplitude, while the influence on the resonance frequency is negligible. 5. The presented analytical model and solution are suitable for the longitudinal vibration problem of a large-diameter pile in viscoelastic surrounding soil with radial heterogeneity, especially when the pile slenderness is low. Furthermore, the proposed solution can be easily degenerated to describe the longitudinal vibration problem relating to a large-diameter floating pile in radially homogenous soil or a pile with fixed-end supports. The pile vibration problem investigated in this manuscript is mainly focused on wave propagation within a pile-soil system where the excitation force intensity is low. The theoretical analysis will overestimate both the stiffness and damping of piles due to the assumption of perfect bonding between the pile and the soil when the excitation force intensity is high. Conceptualization, Methodology, Software, and Writing--Original draft preparation, K.M. ; Supervision, Writing--Review and Editing, Resources, Formal analysis, Project administration, and Funding acquisition, C.C. ; Data curation and Investigation, Z.L. ; Supervision and Visualization, H.L. ; Software and Validation, H.P. All authors have read and agreed to the published version of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 51878109, 51778107 and 51578100), the Fundamental Research Funds for the Central Universities (Grant No. 3132019601), China's Scholarship Council (CSC No.201806570004), and the Cultivation project of Innovation talent for doctorate students (CXXM2019 BS008)  
The corresponding author would like to acknowledge the support from the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology.