## 7 Results and Discussions
### Parametric Analyses  
Due to the consideration of the lateral inertia effect of the pile shaft, the present solution from the Rayleigh-Love rod model can easily be reduced to the one from the Euler-Bernoulli rod model by setting \(v\mathbb{P}\to 0\). The effect of Poisson's ratio on the longitudinal impedance at the pile head is shown in Figure 5. It is clear that both the resonance frequency and amplitude of the pile head's longitudinal impedance decline with the increasing Poisson's ratio in the high-frequency range, while the effect of Poisson's ratio on the longitudinal impedance can be neglected in the low-frequency range. Furthermore, it illustrates the limitation of the Euler-Bernoulli rod model (\(v\mathbb{P}\to 0\)) to describe the longitudinal vibration of a large-diameter pile, compared with the Rayleigh-Love model. With the aim to further illustrate the difference between the Rayleigh-Love rod and Euler-Bernoulli models, two cases for Poisson's ratio with \(v\mathbb{P}=0\) and \(v\mathbb{P}=0.3\) are used in the following analyses, besides other parameters. Figure 6 depicts the effect of the pile length on the pile head's longitudinal impedance. It is clear that both the resonance frequency and amplitude of the longitudinal impedance rise with the decrease of pile length in the high-frequency range and this tendency becomes more significant with the rising frequency. In addition, the shorter the pile length, the greater the difference in longitudinal impedance between the two cases with \(v\mathbb{P}=0\) and \(v\mathbb{P}=0.3\). This result demonstrates that  
Figure 4: Comparison between the present solution (\(c_{j}^{\text{s}}\to 0\) and \(v\mathbb{P}\to 0\)) and the results of [PERSON] et al. [22]: (**a**) true stiffness; (**b**) equivalent damping. Figure 5: The effect of Poisson’s ratio of pile shafts on the longitudinal impedance of pile shafts: (**a**) true stiffness; (**b**) equivalent damping. the present solution derived from the Rayleigh-Love model is suitable for the longitudinal vibration of a large-diameter pile, especially when the pile slenderness is low. Furthermore, it can be seen from Figure 7 that the viscous coefficient of soil has an obvious influence on the longitudinal impedance of the pile shaft within the high-frequency range. With the increase of the soil's viscous damping coefficient, both the oscillation amplitude and frequency of the longitudinal impedance decrease. Figures 8 and 9 depict the effect of the degree and radius soil is weakened due to a construction disturbance on the longitudinal impedance at the head of the pile shaft, respectively. It is clear that the oscillation amplitude and resonance frequency rise with the increase of the degree to which the soil is weakened, and the tendency becomes significant in the high-frequency range. In contrast, only the oscillation amplitude of longitudinal impedance is augmented with the enlargement of the radius of the weakened soil due to a construction disturbance, and the influence of the weakened soil's radius on the resonance frequency is negligible. Figure 6: The effect of the pile length on the longitudinal impedance of the pile shaft: (**a**) true stiffness; (**b**) equivalent damping. Figure 7: The effect of the viscous damping coefficient on the longitudinal impedance of the pile shaft: (**a**) true stiffness; (**b**) equivalent damping. The influences of the degree and radius of strengthened soil on the longitudinal impedance of the pile shaft are illustrated in Figures 10 and 11, respectively. It is observed that the oscillation amplitude and frequency both decrease with the increase of the degree to which the soil is strengthened, which is significant in the high-frequency range. Differently, only the oscillation amplitude of longitudinal impedance becomes smaller with the increase of the strengthened soil's radius due to a construction disturbance, and the influence of the strengthened soil's radius on the resonance frequency can be practically ignored.